Eighth Symposium NAVAL HYDRODYNAMICS ~HYDRODYNAMICS IN THE OCEAN ENVIRONMENT ARC-179 Office of Naval Research Department of the Navy _ ec Nua ne riled |, How T bbT2hOO TOEO O WO 0 1OHM/18W Ae 7 ar 7 Wea ei Ve 4 Bah ae i Eighth Symposium NAVAL HYDRODYNAMICS HYDRODYNAMICS IN THE OCEAN ENVIRONMENT sponsored by the OFFICE OF NAVAL RESEARCH the NAVAL UNDERSEA RESEARCH AND DEVELOPMENT CENTER and the CALIFORNIA INSTITUTE OF TECHNOLOGY August 24-28, 1970 Rome, Italy MILTON S. PLESSET T. YAO-TSU WU STANLEY W. DOROFF MARINE Editors ,noGr, BIOLOGICAL ae bn LABORATORY 4 Pe ADE TL LE ELITE LLYN 3 é LIBRARY AC ay $ eins VBOBS HOLE, MASS. OFFICE OF NAVAL RESEARCH— DEPARTMENT OF THE} NAWYH. © I. Arlington, Va. PREVIOUS BOOKS IN THE NAVAL HYDRODYNAMICS SERIES “First Symposium on Naval Hydrodynamics,’’ National Academy of Sciences—National Research Council, Publication 515, 1957, Washington, D.C.; PB133732, paper copy $6.00, 35-mm microfilm 95¢. “Second Symposium on Naval Hydrodynamics: Hydrodynamic Noise and Cavity Flow,” Office of Naval Research, Department of the Navy, ACR-38, 1958; PB157668, paper copy $10.00, 35-mm microfilm 95¢. “Third Symposium on Naval Hydrodynamics: High-Performance Ships,”’ Office of Naval Research, Department of the Navy, ACR-65, 1960; AD430729, paper copy $6.00, 35-mm microfilm 95¢. “Fourth Symposium on Naval Hydrodynamics: Propulsion and Hydroelasticity,’’ Office of Naval Research, Department of the Navy, ACR-92, 1962; AD447732, paper copy $9.00, 35-mm microfilm 95¢. “The Collected Papers of Sir Thomas Havelock on Hydrodynamics,” Office of Naval Research, Department of the Navy, ACR-103, 1963; AD623589, paper copy $6.00, microfiche 95¢. “Fifth Symposium on Naval Hydrodynamics: Ship Motions and Drag Reduction,’ Office of Naval Research, Department of the Navy, ACR-112, 1964; AD640539, paper copy $15.00, microfiche 95¢. “Sixth Symposium on Naval Hydrodynamics: Physics of Fluids, Maneuverability and Ocean Platforms, Ocean Waves, and Ship-Generated Waves and Wave Resistance,” Office of Naval Research, Department of the Navy, ACR-136, 1966; AD676079, paper copy $6.00, microfiche 95¢. “Seventh Symposium on Naval Hydrodynamics: Unsteady Propeller Forces, Funda- mental Hydrodynamics, Unconventional Propulsion,”’ Office of Naval Research, Depart- ment of the Navy, DR-148, 1968; AD721180; Available from Superintendent of Docu- ments, U.S. Government Printing Office, Washington, D.C. 20402, Clothbound, 1690 pages, illustrated (Catalog No. D 210.15:DR-148; Stock No. 0851-0049); $13.00. NOTE: The above books, except for the last, are available from the National Technical Information Service, U.S. Department of Commerce, Springfield, Virginia 22151. The catalog number and the price for paper copy and for microform copy are shown for each book. Statements and opinions contained herein are those of the authors and are not to be construed as official or reflecting the views of the Navy Department or of the naval service at large. For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $10 Stock Number 0851-0056 li PREFACE Continuing in an uninterrupted manner since 1956, the biennial symposia on naval hydrodynamics convened for its Eighth Symposium, August 24-28, 1970 at Pasadena, California. This conference was jointly sponsored by the Office of Naval Research, the Naval Undersea Research and Development Center, and the California Institute of Technology. The technical program in this series is traditionally structured about a limited num- ber of topics of current interest in naval hydrodynamics. In the case of the Eighth Symposium, “‘Hydrodynamics in the Ocean Environment’’ was selected as the focal theme not only because of the present widespread research interest and activity in this subject but also in recognition of 1970 as the inaugural year of the ‘International Decade of Ocean Exploration.” This motif for the Eighth Symposium was also aptly reflected in the banquet address to the participants by Rear Admiral O.D. Waters, USN, then Ocean- ographer of the Navy. The organization and management of a meeting of this magnitude requires the atten- tion and energy of a large number of people over a long period of time. To Dr. Harold Brown, President of the California Institute of Technology, to Captain Charles Bishop, Commander, Naval Undersea Research and Development Center, and to all the various members of their organizations who contributed in many different ways to the success of the Eighth Symposium, the Office of Naval Research is deeply indebted, and to them we extend our heartfelt gratitude and appreciation for a job well done. It is particularly appropriate, however, to acknowledge the specific roles of Professor Milton S. Plesset and Professor T.Y. Wu of the California Institute of Technology and Dr. J. Hoyt of the Naval Undersea Research and Development Center who as a group carried the lion’s share of the responsibility for the detailed planning and day-to-day management of the Eighth Symposium. We take special pleasure in acknowledging the invaluable assistance of Mrs. Barbara Hawk, secretary to Professor Plesset, who in a most gracious and efficient man- ner carried out a multitude of important tasks in support of the Symposium. In addition, Mrs. Hawk, together with Mrs. Alrae Tingley, were responsible for the preparation of the typescript which was used in the publication of these proceedings. Mr. Stanley Doroff of the Office of Naval Research played his usual critical role, participating actively in every aspect of the planning and execution of the arrangements for the Eighth Symposium. feb bg u~ RALPH D. COOPER Director, Fluid Dynamics Program Office of Naval Research ill ba : ie vi" ti, Jaki 7 ‘ r tid ; cig i? Ce, ] ety my , JOAN 5 *1'* ict iyi nH wy, j } ; 7 iy Cole ; % cl wat | pi A aut idee rT re Ses am pad yey fiat Pond nys titans, Wiis ok HCN Geet ase SY OVE SES EN wpe! cegineenlge sii aa a hacseve7, wg yoga i] nual ie mC ori! Ge. 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Se 7 er ) oa on i - win Pe Ga Ara cel Ti hier se ' Se Ypeh ated 1 Oe, es ie. . - ¢ gofees ol a erengau 769 Reb 9 TRA eee ws oy ie * re, 2 - Fad Pt 7 tt anobosteial ach-h ao sneltagitesoal vt eed, tes » 4 Nisagor |i i es 5 ape a“ yen, ai ; Tae es fue S45 og, Lin-ERO ates Siege ia Pe wees ae 4 7 ann ise bas ' yiitindy E54, A ae rete erty ey tat thy alne 5 f ; : aonen’t .olitboxetd TAMA), Chet 6M pots coe ostiaa Se aan BM fo caiane thy Sao ¥ Pits ; hy . anya Sitaw Ratcy 5 ultidey ‘f Sie pro iiyTawh) s «bil dio ad) mara ¥ easily . sicllislleat aa, odieol eRe CCTM VA re: Grated ane wer Aracrs' cud ye haem ey ay. “a Filth $87 a : ue j . i an) i r | oT - 7 : TSUNAMIS G. F. Carrier Harvard Untverstty Cambrtdge, Massachusetts I. INTRODUCTION An understanding of the coastal inundation caused by Tsunamis requires the piecing together of several studies. Among the poten- tially important characterizing features of the phenomenon are: the temporal and spatial distribution of the ground motion which initiates the Tsunami, the distance from the source to the target area in ques- tion, the bottom topography of the intervening ocean, and the topag- raphy of the coastal area itself. These are all discussed in some detail in [1], a manuscript which was prepared in conjunction with a longer series of lectures than this one. In order to avoid excessive duplication of publication, we content ourselves here with a brief summary of that material. As will be evident, many details remain to be explored; unfortunately, there is no evidence to suggest that even a more comprehensive understanding of the phenomena will sug- gest procedures for alleviating the intensity of Tsunami inundation. Il. INITIATION AND DEEP WATER PROPAGATION The wave generated by a submarine earthquake is large enough in lateral extent and small enough in amplitude so that a linear theory is completely adequate for an analysis of the propaga- tion over deep water. However, the propagation path is so long that dispersion and its attendant changes in wave shape cannot be ignored. Accordingly one can adopt either the classical linear theory of gravity waves or the Boussinesque formalism to study the early stages of the wave propagation. When either is done, for a basin of constant depth, H, it is convenient to present the results in terms of a particular family of initial ground motions. We discuss here the waves which result when the ground motion is given by F, (x,t) = ier. [ - tal 5(t). When the half width, L, of the distrubed region is "small," the wave which arrives at a distance x, will have been greatly affected by dispersion; when the width of the generating ground motion is Carrter longer, smaller distortions of the wave will be apparent at xp». Figures ita, 1b, and ic illustrate the quantitative aspects of the foregoing statement. In the notation of those figures, l= 4en When the depth of the water is 3 miles and x,= 3000 miles, the three cases shown represent ground motions whose half widths are 0, 19 and 33 miles. Figure id indicates the wave which ensues when the ground displacement is given by F= Fy (x,t) - F) (x +520 5t) with a = 10. That is, the ground motion has a dipole character rather than a general subsidence or elevation. The persistent lore that the second or third crest of the Tsunami penetrates more than the first makes it interesting to speculate (in view of Figs. 1) that many initiating ground motions may be of dipole form. III RUN-UP ON A PLANE BEACH When the wave encounters a sloping shelf along which the water depth generally goes to zero, the wave steepens and becomes greater in amplitude. Accordingly one no longer can rely on a linear theory. However, the shelves of real interest are such that the distance along the wave trajectory above sucha shelf is short enough so that dispersion in this region is not of any real importance. There is a non-iinear, non-dispersive shallow water theory which leads to tractable problems when the depth of the basin is linear in one horizontal coordinate and when the entire phenomenon is independent of the other. Thus, we can regard the results re- ferred to in Section 2 as the input information for a study in which we ask how such waves climb up a sloping shelf. The analysis which accompanies such a study involves only the solution of a linear equation whose interpretation in the non-linear context is explicit and accurate. The result of interest is the ratio of the run-up, No, (the vertical distance above sea level to which water encroaches) to the wave height, 1,, at the edge of the shelf. One interesting result is this: Hor @ = 10 - -176 qe we hot cay, Tsunamts . qt *31a OStt ——OblL wait Oell Ol 0011 0601 0801 0201 0901 0S0i Ov0l 0¢0l 0201 0101 0001 066 086 OOl= VHdIv 09 ey "Sha x Ostt Obl l O¢ll raul OL OO 0601 0801 0201 0901 0S0L OvOl O¢0l 0201 O10! 0001 066 086 O'O= VHdIV og Carrter Oflt x zi oll DOL 0601 0801 0201 x ool 0601 0801 0201 0901 0901 0s01 0S01 pol PT °Sta Ob0l O¢0l 0201 OT °StaT O¢0l 0201 0101 0101 000! 0001 066 066 086 016 096 O'Ol = VHd 1V 086 06 096 O'O€ = VHd1V Tsunamts where A = 4.2 if the ground motion is upwardor A= 5.6 if it is downward. The corresponding results for other values of L can easily be found (the calculation requires only the use of the method of stationary phase). The dependence on the shelf slope, 0, is that which would be found for monochromatic waves whereas the depen- dence on x_ is a consequence of the dispersion during the deep water propagation. IV. DEEP OCEAN TOPOGRAPHY If there were systematic variations in the water depth between, say, the Aleutians and the equatorial Pacific, one might expect that the relative intensities of the Tsunamis (with Aleutian source) which were incident on different Pacific islands might differ because of mid-ocean refractive effects. Exhaustive studies of this effect have certainly not been completed but the indications are that this is not a major reason for the different response at (for example) Wake and Hawaii. One might also anticipate that the irregular deep ocean topographical variations could seriously modify the wave which propagates across the ocean. This possibllity has been analyzed treating each event as a member of an ensemble of phenomena each of which take place over a topography which is itself a member of a stochastically described collection of random topographies. This is motivated loosely by the fact that the one-dimensional topography between any given source and any given target differ from that associ- ated with any other source-target pair, and the fact that the topog- raphies are so poorly known that little else can be done. The result of this study indicates that the ratio of intensity at x, of the wave over the irregular bottom to that over constant depth is characterized by 2 el ee iar ana where € is the ratio of the average irregularity height to the average depth and L = 2nN where N is the number of wave lengths of the monochromatic wave whose scattering is being studied. For wave lengths in the spectral region of major interest, the effect of this facet of the wave propagation seems to be of relatively small im- portance too. V. ISLAND TOPOGRAPHY When the wave encounters an island, the lateral scale of that island has the same order of magnitude as much of the important part of the wave length spectrum. Thus, the pretense that the wave climbs a plane shelf must be corrected. The refractive effects so Carrier implied cannot be estimated readily py geometric optics methods at such wave lengths and one must resort to numerical procedures. The results of such studies are depicted in Figs. 2 and 3, taken from Lauterbacher [2]. Figure 2 indicates the variations of intensity with position on a given island and Fig. 3 indicates the extent of this effect for different ratios of wave length to island size. = WAVE INCIDENT AT 0° 9 2 < q O 8S iL —= ar 7 S Y 6a = < 5 <¢ Ww 7 Ww > 4s < < = [33 = 2s = = = 15 x< = aq 0 0 xX 8 & 8 2 8 4 8 COAST POSITION (MEASURED IN RADIANS AROUND ISLAND CENTRE) Fig. 2. Maximum wave amplification at coast (OAHU) Tsunamts L/R = 1.67 Patel BEACH MAXIMUM "op! iP) BEACH MAXIMUM WAVELENGTH )/L Fig. 3. Ratio of two-dimensional to one-dimensional maximum wave amplitude on beach. L, island diameter at ocean floor; R, island diameter at beach. ACKNOW LEDGMENT This work was supported in part by the Office of Naval Research under contract N00014-67-0298-0002 and in part by the Division of Engineering and Applied Physics, Harvard University. REFERENCES 1. Carrier, G. F., "The Dynamics of Tsunamis," to appear in the Proceedings of the Summer Symposium on Mathematic Prob- lems in Geophysics, 1970. 2. Lauterbacher, C. C., "Gravity Wave Refraction by Islands," J. Fluid Mechs., Vol. 41, Part 3, pp. 655-672, April 1970. 7 : ia a on! hint Wo ER Ay? Nas. s ii ALA ae ee ae | 4 rip i TiDe? aie oy 4h aa o) queed 1, Oh OO ATS oy aree j a K a i TT; 14 ; iw # PATL MY ates Mae &:1 oe ine fig ’ mys ef if i , 4 Oh Aw otek é 5 (te a se vies May cee Alia ah meer lsh aul aoe ony adh, Sgn ied Lone a sedis sy ai Se) ringers (ies ue ety fate vil ia ee) Ch 4 rina pes fi oO eer 4 ee see i heen ‘ in “ R i a as. “ =f wt 4, 08) i. a if ui see Any the a) in ant. pe ae ym | Al hi # h G: : "a 4 5 “ a Th ae ve sf he ; fi, Pr d oe } 7 7 i # 4 nf % (a ® Sou,” +k Sot hae gy ot ™\, Wi ia 4 hb wl Fa) fa t a 1 1a? at : ie: a ings’ ee wolf i ee) aie erage a es on © 1s Caer 6. me wy eee } a at ¥, eel ae wp i — aoe i ante just ne sy: AiDaeseormces ac > = i =. 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INTRODUCTION Since the comprehensive review on wind wave generation by Ursell [1956], there have been renewed, intensive studies, theoreti- cal as well as experimental, on the subject. Although significant contributions have been made by many investigators, the final goal of achieving a basic understanding of the fundamental mechanism of energy transfer between a turbulent air stream and the ocean has not been realized. A unified, comprehensive theory of wind wave gener- ation must provide adequate explanation of the energy transfer between the two media at all stages of wave growth from capillary waves to sea swell. In the absence of sucha unified theory, a convenient classification of various flow regimes in wind-wave generation may be made by use of the ratio of water wave celerity C and the air shear velocity u* at the interface. When C®# ats the dominant mechanism of energy transfer between air and water is the "viscous mechanism," characterized by the critical layer being within the laminar sublayer and treated by Miles [ 196 at. When C¥% 10 u’, the critical layer is outside the laminar sublayer and the dominant mechanism of energy transfer is the "inviscid mechanism" (Miles [1957, 1967] and Benjamin [ 1959]) with transfer arising from the normal pressure acting on the interface and the neces- sary phase angle between the pressure distribution and the progres- sive wave. As pointed out by Longuet-Higgins [ 1969], neither of the above theories accounts for two well-established features of wave generation: (1) the existence of some wave energy in a frequency range corresponding to waves traveling faster than the mean free- stream velocity and (2) the damping of a swell by an adverse wind. The experimental investigations of Sutherland [1967], Hires [1968], and Chang [1968] and many others are limited to the viscous range. Because of the high Reynolds number in a typical wind blowing over the ocean surface, the viscous mechanism can be safely neg- lected as irrelevant to full-scale wave energy transfer. Hence, Miles' inviscid model has received most attention and been widely employed in comparisons with experimental data obtained in full scale (ocean) and laboratory simulations. 11 Hsu and Yu The dearth of systematic measurements taken under controlled conditions closely comparable to those of Miles' model was a moti- vation for our research program at Stanford University. In order to examine the applicability of Miles! inviscid theory, experiments were designed for measuring the wave induced perturbation pressure or inviscid Reynolds stress under steady-state and unsteady-state conditions. Other experiments were also devised for measuring the growth of mechanically generated waves subjected to wind action. From these measured wave growths, the growth factor of Miles was calculated. The objectives of this paper are to present a summary of our experimental data in the inviscid range, to compare our data and other existing data to the theory and the ocean observations, and to suggest specific and fruitful avenues for further study. Il. A BRIEF REVIEW OF THE THEORY To facilitate presentation and discussion of the experimental data, a brief outline of the assumptions, key equations and results of Miles' inviscid, shear-flow theory are presented below. The deep-water, wave profile is assumed to be a progressive, sinusoidal wave, expressed as n ='avexp! hik(x:- Gt)], kay<< 1 (1) where a is the amplitude, k= 2n/L is the wave number, lL is the wave length, and C is the wave celerity. The assumptions of irro- tational, incompressible water motion lead to the existence of velocity potential. By substituting the velocity potential in the linearized Bernoulli equation and evaluating the result at the free surface, one obtains the equation of motion governing the propaga- tion of a small amplitude, surface wave 2 Py OMe © Pele ey, age Pa (2) where g is the acceleration caused by gravity, p, is the mass density of water, and p, is the aerodynamic pressure caused by the wind stream. Miles [1957] assumed the aerodynamic pressure p, has the form p, = (2 + iB)p,U, kn (3) where pg is the mass density of the air, U, is a reference speed for the air, and @ and f are, respectively, the in-phase and 12 Laboratory Investtgattons on Atr-Sea Interactions out-of-phase non-dimensional-pressure-coefficients. The phase angle @ is g = tan’! (B) (4) The constants @ and 8 were determined by solving an inviscid Orr- Sommerfeld equation which represents the perturbations (caused by the water wave at the interface) to a wind shear-flow described by an assumed logarithmic, mean velocity distribution Uly) = U, In = (5) fe) where y is the vertical distance from the mean water surface and Y, is the roughness height. The effect of the impressed aerodynamic pressure P, on the surface wave can be evaluated by solving Eq. (2). It follows that the complex wave celerity 2 1 p fU a : pesslipe gd © 1 Bonet C ei ae (2 +i6)(ct) J (6) where C, = (g/k)’?. Substituting Eq. (6) into Eq. (1) yields 4 p thins a= a, exp ls kC, co (<4) Bt] (7) where a, is the amplitude at t=0. It is convenient to measure the growth of wave amplitude as a function of fetch x in a wind-wave channel. The dynamic equi- valence, valid for x >>L, is given by Phillips [1958] as x= Sot, me where C,/2 is the group velocity of a deep-water wave. Conse- quently, the fetch-dependent amplitude growth a is 2 Ke are a= a, exp [£2 © utpal o w where a, is now the wave amplitude that would exist without wind 13 Hsu and Yu action ior-at. x = 01. The total energy per unit of surface area E of a small- amplitude, sinusoidal, progressive wave is If the energy corresponding to a, is E,, Eq. (8) may be rewritten as 2. 2 E = E. exp [££ x’u’ px] (9) ° EP, | The out-of-phase pressure component BP is responsible for the energy transfer from the air stream to the wave. Experimental results are presented in the non-dimensional form log, tie = AF (10) where F is a non-dimensional fetch k?2 HPS ete and A= (2 £2) log ee toa x10. Py fe) : for pes 1 gm/cm°* and Sp 0.00118 gm/cm’*. III. LABORATORY INVESTIGATIONS 3.1. Techniques of Simulation 3.41.1. Moving wavy-boundary (steady-state) In attempts to verify Jeffreys' sheltering hypothesis, Stanton, et al. [1932], Motzfeld [1937], Thijsse [1951], and Larras and Claris [1960] measured pressure distributions over stationary, solid, and two-dimensional sinusoidal boundaries in either a wind tunnel or a water channel. In the light of the critical-layer mecha- nism proposed by Miles, these stationary wavy-boundary experiments 14 Laboratory Investtgattons on Atr-Sea Interacttons cannot be regarded as an adequate, steady-state simulation of the wind-generated wave problem, because the critical layer in the experiments is of zero thickness and, hence, the critical level lies on the stationary boundary. All of the above experiments, with the exception of Thijsse's indicated a smaller sheltering coefficient than that anticipated by Jeffreys, who expected the pressure distribution to be out-of-phase with the wave (in accordance with Miles' inviscid theory). The resulting small sheltering coefficient may be attributed to either viscous or finite wave-amplitude effects. For amore realistic steady-state simulation of wind-generated waves and demonstration of the importance of the critical-layer mechanism of energy transfer, the wavy boundary must be moving with a speed equal to the wave celerity and opposite to the direction of mean free-stream. An important advantage in this simulation is that the flow field is steady. Consequently, measuring techniques are greatly simplified. The first successful moving, wavy boundary experiment and its resultant presentation of the normal pressure distribution on the boundary were reported by Zagustin, et al. [1966, 1968]. Subse- quently, Ott, et al. [1968] extended the Zagustin investigation and used refined experimental procedures to achieve better experimental accuracy. Small amplitude waves with a length of 3 ft and amplitude of 0.65 in. were used. Because of the limited capability of the experimental facility, G/U. was limited to approximately 0.75 (U, is the air velocity at the edge of the boundary layer). 3.1.2. Flexible wall with progressive waves (unsteady- state) Kendall [1970] described a series of experiments on wind- wave simulation in a low turbulence wind tunnel. The wavy wall was the floor of the constant pressure test section of the tunnel. The surface of the wavy wall was composed of neoprene rubber sheet which was constrained to form a series of sinusoidal waves (length = 4 in. and height = 0.25 in.). The rubber sheet was supported from beneath by a series of ribs which were connected to individual. circu- lar eccentric cams. Each cam was positioned with proper phase difference on a common cam shaft expending the length of test section. Rotation of the cam shaft caused each rib to execute a reciprocating vertical motion and thus a progressive wave form was produced. Reversing the direction of rotation of the cam shaft produced waves traveling in the opposite direction, giving - 0.5 < C/U, <0.5. The boundary conditions for the two methods of wind- generated wave simulation described above deviate slightly from those of a true air-water interface. If the fluid particle velocity in a wave motion is small compared to the wave celerity (true for small amplitude waves), the moving wavy boundary simulation approximately satisfied the boundary conditions. In the flexible wall experiment the surface 15 Hsu and Yu particle motion resulting from the flexure of the rubber sheet was a backward-rotating 3:1 ellipse as compared with the forward- rotating circle of deep water waves. Again, the boundary condition was approximately satisfied for small amplitude waves. 3.1.3. Wind-wave research channel (unsteady-state, true air- water interface) The physical features of the Stanford facility were reported by Hsu [1965]. The channel is approximately 6 ft high and 3 ft wide and has a usable test section length of 75 ft. At the downwind end, there are a beach to absorb wave energy and a centrifugal fan to produce the wind in the channel. At the other end, the air is drawn vertically through a system of filters and then carried hori- zontally on to the water surface at the beginning of the test section by a converging elbow. A hydraulically-driven, horizontal-displace- ment, wave-generating plate is located 17 ft upstream of the test section. This distance is sufficient to allow generated waves to become fully established prior to being subjected to wind action. Sinusoidal waves, ranging in frequency from 0.2 to 4 cps, can be generated. The maximum wind speed is approximately 70 fps with a nominal water depth of 3 ft inthe channel. A limitation of the present facility is that the wind and the propagating waves move in the same direction. 3.2. Measurements of Wave-Induced Perturbation Pressures Because the flow field was steady in the moving wavy-boundary experiment (Sec. 3.1.1), two conventional, but small (1/32 in. O.D), pitot-static probes were used for all the velocity and pressure measure- ments. The reference probe was located in the free-stream while the other probe could be moved to any distance from the moving boundary by atraversing mechanism. Realizing that the traversing probe must be aligned with the local flow direction, we mounted this probe in a special rotating device. These probes were connected to a Pace P-90 differential pressure transducer through a manifold system which provided selective readings of dynamic pressure or pressure differential between the two probes. The pressure measurements in the flexible wall experiments (Sec. 3.1.2) were made through static holes in the flexible surface. Essentially, a length of metal tubing in the form of a loop was used to connect the static hole and the pressure transducer. The loop served to cancel the unwanted pressure gradient generated by the motion of the tubing. Because the thickness of critical layer was small in the wind- wave channel experiment (Sec. 3.1.3), the measurements of pertur- bation pressure were obtained by use of a specially-designed wave following system. Again, the perturbation pressure is the difference between the pressure at the air-water interface and that in the free 16 Laboratory Investtgattons on Atr-Sea Interacttons stream and was monitored by two identical pressure sensors, one at the interface and the other in the free stream, through a Pace P-90 differential pressure transducer. The whole system was allowed to follow the wave motion so that the lower pressure sensor was kept a fixed distance from the instantaneous air-water interface and inside the critical layer. The unwanted pressure signal caused by the motion of the system was determined by calibration tests and removed in the final data reduction. 3.3. Measurement of Wave Growth A series of experiments was run to measure wave growth rate in the Stanford wind-wave channel. Small-amplitude, deep- water waves with frequencies varying from 0.9 to 1.4 cps were used with the maximum wind speed ranging from 12 to 44 fps (fan speed of 100-300 rpm). Time records of wave profiles were obtained with capacitance-wire sensors at seven locations spaced at 10 ft intervals along the centerline of the test section. Air velocity distributions were taken at six intermediate locations with a conventional pitot- static probe. Although the mechanically-generated waves were initially of small amplitude and closely sinusoidal, they become steep and some- what non-sinusoidal with increasing fetch in response to the wind action. The true wave profile could be viewed as a superposition of a mean wave and a spectrum of ripples. Therefore, a phase averag- ing procedure was adopted to determine the mean wave profile at each fetch and fan speed. The mean wave profile at each phase angle was the result of averaging 35 waves in the time series. The stream function fitting technique introduced by Dean [1965] and outlined for this application by Bole and Hsu [1967] was used for evaluating the kinetic and potential energy of each mean wave profile. Finally, the total wave energy at each location of the test section was adjusted for wave energy dissipation due to viscous action. The dissipation was determined experimentally for conditions without wind. Along with the mean wave profile, the ripple variance of the water surface about the mean wave profile at each phase angle of the wave and the mean ripple variance and standard deviation for all the phase angles were calculated. The ripple variance is, of course, proportional to the potential energy contained in the ripple. IV. RESULTS AND DISCUSSION 4.1. Water Surface Roughness (Unsteady-State, True Air- Water Interface) When mechanically generated waves were subjected to wind action, ripples were always present and were superposed on the waves. Thus, the water surface can no longer be regarded as smooth 17 Hsu and Yu and its roughness can be described by the ripple standard deviation o of the water surface elevation about the mean-wave profile. It was observed that o increased with wind speed at the same fetch. In general, o increasedas y,u /v increased. The values of o are listed in Table 1 and vary from about 0.001 to 0.039 ft. From a least-square fit of the velocity profile Eq. (5) to the measured data, values of U, and y, can be obtained and hence the values of y,, ky,, and B (see Sec. 2). The values of yg are com- piled in Table 2 and vary from 0.004 to 0.011 ft. The values in Table 1 and Table 2 show that the ripple standard deviation is larger then the critical layer thickness in all cases. It seems that the surface roughness or ripples should destroy the organized actions of vorticity which Lighthill [1962] presented as the physical expla- nation of Miles' instability mechanism. Thus, Miles' interpretation of the energy transfer mechanism -(adopted from Lin [1955]) as the perturbation Reynolds stress working against the mean velocity pro- file at the critical layer is severely strained by the existence ofa ripple layer large enough to obliterate the critical layer. The potential energies of wind+generated ripples with and without mechanically generated waves are presented in Table 3. The presence of the generated waves decreases ripple energy sig- nificantly. Although there are many irregularities, ripple energy generally decreases as wave frequency increases. Exceptions occur at 300 rpm and 60 ft fetch where the 1.2 and 1.4 cps waves are breaking and ripple energy is sharply decreased. Sample power spectra of the ripples superposed on a 1.1 cps wave were obtained by subtracting the mean 1.1 cps wave profile from the original water surface elevation time series, The remaining time series, which contains only ripple variation, was then spectral analyzed. The resulting power spectra of wind- generated ripple with and without mechanically generated waves is exhibited in Fig. 1. Spectral peaks for the two cases appear at about the same frequency, but spectral density is drastically reduced when waves are present. The two possible reasons for ripple attenuation in the presence of waves are | a. sheltering effects retard ripple generation by the wind, b. non-linear wave-wave interactions cause ripple energy to be dissipated and to be transferred to the waves as suggested by Longuet- Higgins [1969] (see later discussion on wave energy). 4,2. Mean Wave Profiles Mean wave profiles were determined by phase-averaging over records of 35 waves. A sample of corresponding pairs of mean wave profiles and their corresponding original recordings for 18 Laboratory Investigations on Atr-Sea Interactions TABLE 1. RIPPLE SPECTRUM STANDARD DEVIATION o x 10° (ft) Fetch Mechanical Wave Frequency ( ee eee ee NououwnrF Noo pre 3 3 6 5 6 8 8 9 8 0 8 2 * Waves breaking. 19 Hsu and Yu CRITICAL LAYER THICKNESS TABLE 2 e y, xX 10° (ft) cps) “~ > 13) i=) o S iow oO u fy 7) > wo = ec @ 13) cial S wo G io) Oo = 20 Laboratory Investigattons on Atr-Sea Interacttons TABLE 3. RIPPLE SPECTRUM POTENTIAL ENERGY Values x 10° (£t-1b/£t7) Fetch Mechanical Wave Frequency (cps) NFrFO FrFRrRFOOO * Waves breaking. Zi Hsu and Yu 200 RPM ——— I. cps wave No wave Frequency (cps) Fig. 1. Influence of 1.1-cps wave on 200-rpm ripple spectra 1 cps and 1.3 cps waves at 300 rpm is given in Figs. 2 to 5. A close examination of these figures (wave motion is toward the left and the usual Sanborn attenuation scales are marked) reveals that the posi- tions assumed by the ripples influence the mean wave profile. For example, ripple superposition in the 1.0 cps wave caused the mean wave crest to become flattened. However, ripple superposition on the 1.3 cps wave did not cause mean wave profile distortion. Many of the records show ripples superposed in such a way that the crest was sharply peaked. The existence of such conditions may enhance separation of the air flow over the wave surface. A source of error arises from the fact that mean wave pro- files were distorted, and yet their energy was compared with that of Miles' pure sinusoids. An expression consisting of a cosine and sine plus their two higher harmonics was least-squre fitted to each of the mean wave profiles (see below). Results indicated that the total 22 ° Laboratory Investigattons on Atr-Sea Interacttons (99S 7°Q = UOTSTATP yejuoztz1oy J) wida QoE te s8ujpiooer uroques sdo-Q*t "€ *S1q (ft) Surface Elevation Water uidi go¢g 7e SoTtjord oaem ueourt sdo-g*y “7 *3Iq (Bop) ejbuy esoyg 0 06 os! O12 ce) 06 osi O12 ose Wd OOF (49) 49004 23 Hsu and Yu (298 Z°O = UOTS]A]Tp Teqwuoztoazy JT {dur -yeoiq soAaeMm = *q*m) urdi QOE ye sSujpIooer uroques sdo-¢*], “Gc *3Tq yoyey _ 81D9S (ft) Surface Elevation Water fo} °o widi go¢€ ve soTyyord ovaem uvout sdo-¢*y, *F “817 (Sep) ejBuy esoud te) 06 os! ol2 fo) 06 os! ol2 o9¢ Wd OO€ sdo ¢-|=43 (43) Ydbed 24 Laboratory Investigattons on Air-Sea Interactions error introduced by representing the mean wave by these five terms was not greater than 5 per cent in any case. 4.3. Wave Energy The total wave energy (potential plus kinetic) was calculated by the method developed by Dean [1965] through least-square- fitting an analytic stream function to the mean measured wave profiles. Details of the procedure were presented by Bole and Hsu [1967]. In order to compare the measured wave energy with Miles' prediction, wave dissipation in the channel, determined experimentally under the condition of no wind, as a function of fetch was added to the measured wave energy. Figure 6 shows the energy ratio E(x)/E(0), as a function of downstream distance for a 1.4 cps wave subjected to various wind speeds, while the results in Fig. 7 are for a constant wind speed (300 rpm) acting on waves of various frequencies. The data was then reduced to the non-dimensional fetch F defined in Eq. (10). The final results, compared with Miles' inviscid theory, 40 50 Oo 10 20 30 x--Fetch (ft) Fig. 6. 1.4-cps wave growth vs. x AD Hsu and Yu 300 RPM (e} 10 20 30 40 50 x-- Fetch (ft) Fig. 7. 300-rpm_wave growth vs. »¥ are exhibited in Figs. 8 and 9. In nearly every case, experimental values fall well above the theoretical line. By assuming the growth to be totally dependent on F inthe form of Eq. (9), we calculated a ratio of the experimental and theoretical B values and present the results in Table 4. The f-ratios vary from about 1 to 10. The mean ratio for all frequencies and rpm's is about 3. The total spectral energy of the ripples in most of the experi- mental cases was no more than about 20 per cent of the mechanically- generated wave energy. Phillips [1966] argued that non-linear inter- actions between waves should be weak. Hence, our procedure of measuring the growth of a single wave within a spectrum should be a valid means of evaluating the parameters necessary for comparisons with Miles' theory. 26 Laboratory Investtgattons on Atr-Sea Interacttons 6.0 5.0 40 E(F) 30 E(0) 2.0 0.04 0.03 fe) 100 200 300 400 500 600 Fig. 8. 1.4-cps wave growthvs. F 0.06 E(F) r(x 0.04 300 RPM (wb)- Wave Breaking 0.02 te) 200 400 600 800 1000 1200 : Fig. 9. 300-rpm wave growthvs. F 24 Hsu and Yu TABLE 4, RATIO OF EXPERIMENTAL TO THEORETICAL B Mechanical Wave Frequency (cps) (ey [oopaofia [ie [aa | a) — | on, WNIOROC TAMHREUYN AUARNOS YAWWON AAPA N c= OorwWwWn sb -—_ 1 ~~ ° OrFwWrN Oo RNY W OF * * ° AIRO’ OROWN OMDMDUID WHWWNEH TONWO CHOMOhHO NOWNNY NONNOS WEEN O PRE Ne PP PND WHWNR YYNWEH PNW 2 9 9 -8 3 2 3 7 8 oak .8 4 5 5 6 2 al 1 1 8 FPrRFoONNDND FPRNYNDY UWWPRQY FPR RH Ww Ww OFrPrFr FrFrRFOrF NNWNHBR OF KF O ee 2. Ie 3 3. us Ue @s 4, 6 3 3. 2. 3. 2. Us 4, 2. 2. is NWR wD WwW PRR PN FPR OFR BWNYHO WHWHWWH DOANE > * Waves breaking. On the other hand, our experimental evidence indicated that the wind- generated ripples riding on the mechanically- generated waves had a tendency to break on the crests rather than in the troughs. Longuet-Higgins [1969] showed that, in breaking, the ripples may impart a significant portion of their momentum to the longer waves in a strong non-linear interaction. Mollo-Christensen's [1970] field observations showed that there were relatively high peaks of energy in a high frequency band located near the crest of the main wave. It is difficult to conclude that Mollo-Christensen's data, taken in a confused sea, whether these high frequency peaks are produced as a result of the breaking waves, caused by wave 28 Laboratory Investigations on Atr-Sea Interactions groups of different frequency overtaking one another, or partly by the generation of high frequency waves on the wave crest. To fully investigate the non-linear, wave-wave interactions and to establish the role of ripples in the transfer process, measurements similar to those of Mollo- Christensen and additional detail measurements of the velocity field below the air-water interface should be carried out under the controlled conditions of a laboratory simulation. The incompatibilities of the Miles’ mathematical model with the natural wave-growth environment, as discussed in the previous section, were anticipated by Miles. He stated in this 1957 paper that "our model cannot be expected to have more than qualitative significance for rough flow." It would appear that a more realistic model and an improved theory of energy transfer cannot be formu- lated until detailed studies of the structure of air flow near the air- water interface are carried out. 4.4. Non-dimensional Pressure Coefficients The non-dimensional pressure coefficients @ and B ob- tained from the various techniques of laboratory simulation are exhibited in Figs. 10 and 11 as functions of ky,. Comparison between the measured values of the in-phase pressure coefficient @ (steady- state, moving-wavy-boundary; unsteady-state, wind-wave channel) and the Miles' theory is shown in Fig. 10. The experimental values of the out-of-phase pressure coefficient B, evaluated from wave growth measurements, are shown in Fig. 11. Although there is considerable scatter in the experimental data, the deviation from the inviscid theory is clearly evident and is consistent with the results of the wave growth measurements. Because of the limited capability of the experimental facility in the steady-state, moving- wavy-boundary experiment, experimental values were limited to ky, = 0.1. The experimentally determined phase angle g obtained from the three different methods of laboratory simulation -- moving-wavy- boundary, flexible boundary with progressive waves, and wind-wave channel -- is shown in Fig. 12 as a function of C/u’. In view of the uncertainties among investigators in determining u values, the experimental phase angle as a function of C/Ug@ and their cor- responding theoretical values are shown in Fig. 13. The JPL-data includes negative values of C. Because the measured velocity pro- files varied to some extent with Ug and C as discussed by Kendall [1970] , theoretical values of g for the case in which Um = 5.5 in./sec and C=0 were calculated. In an attempt to detect flow separation in the region near the air-water interface in the wind-wave channel experiments, pressure measurements over waves of various amplitudes with constant fre- quency were made. The measured phase angles for two wave frequencies, 0.6 and 0.78 cps, are shown in Fig. 13. The scatter 29 Hsu and Yu Stanford Data: |. Pressure Measurement f=0.4cps C=8.64fps a: & 2.5" f=O.5cps C=8.25fps a: wIl.15" f=O0.6cps C=7.55 fps a: o1.15" 01.5" 62.10" $3.10 f=O0.78cps C=6.40fps a: 01.25" 92.12" 063.15" Il.Steady State Moving Belt Simulation asO;75'% -X#3,0" \[Casi92 tps = x a Inviscid Theory > io > 2 4” 26" 8 1072 2 4 6 810"! 2 4.) BR ky, Fig. 10. Comparison between measured and theoretical values of a vs. ky, 30 Laboratory Investtgattons on Air-Sea Interacttons °° SA g jO Son[ea [ed]},ie1090y} pue perznseou usemjeq uosjzedurog “Ty *8qq %y 18 9 »& 2 pUlg= 9. 8 2 2.018 9 » 2 ¢0ls 9 » 2 el 91 e: jUaWaINSDAW ABsauzy 2ADM ‘III = $d} 26°€=9 oe =X 1o2:0= oz uol,OjnWIS 419g Bulaow a404S Apoays ,she® ,220@ ,GZ1%:0 sdjyOv'9=9 sd9g2'0=3 ,Olee ,0OV'2@ ,GI® ,Stl® :0 sdj GGz=9 sdd9Q9'O=43 ,Gli@ :0 sdj Gz°e=9 sd9Go=43 b2 1g cg :0 sd} ~9'8=9 sd9 pO =3 sUaWaINSDEW a4nssald °| :040Q psoyudss 82 | Hsu and Yu xD *sA & JO SONTeA Ted]}Je1O9Y4} pue pernseos UseMjoq uosjaeduioy °7y *8tq ty Ol- oo! sd) G¢G‘2 =) sd99°O=3 8d} GZ°g=9 sd9G‘O=3 as Ov! sdsp9'@=9 Sdd p'O=3 09! Ksoay, ~~~ ' 7 os! x = sdj 26'¢€=9 O'S=X 1G2°0=0 uolyOINWwIS 412g Bulaow a4yn0is Apoays ‘ji sale ® ,el2oO {Sele :0 sd} Ob'9=9 sd981°0=5 ,Oree® ,O1z2e Sie ,SIi® :0 sdy GG2=d sd9 9°0 =5 ,stl@ ;0 sd$ G27e=9 sdd9 G°O =3 829 :0 $d$ 9'B=D $49 H'O =3 JURZWAINSDAW AINSS9Jg “| ;04DGQ P4sojudyS 2998/4, 9OIY 228%, G50 -04100 df ¢ (8920168q) 3Z Laboratory Investigations on Atir-Sea Interactions ©n/D “sad jo sanyea [eojje10ey} pue pernseew usemjoq uosyieduiog oy 9/1 v4 Aa | ot 8°0 9°0 vO 20 O sd) ¢G'2=9 9°02} sd} G2'8=9 S‘O2}3 8d} o9'8=9 9°05} Asoay, ~7~7-'-— Xe $d) 26'€=9 0'e =X ,GZ'O=20 UOJOINWIS 419g BUIAOW a4D4S Apodea4s wGlE@ ,2tZ© ,G21@ :0 sdyOop'9=9 sdogz‘oO =3 sole ® ,Ol2e@ ,SI® Srl %:0 sdigguzz=9 sdo 9°09 =} ,cll @:0 sdyg2e=5 sdog'o =3 1S 29:0 SdypQBe=D 8d2 HCO =3 {URW aINSOOW aINsSsald "9084, O'OIV ‘9984, 2°10 "99Sfy DOIVY “DASAWGGO ‘99S; HHA “99S/, ZED :?N w w w 2'0- "ey 81a v'0- 06 ole) Oll 02! O¢ | d Ov! OSI (s90u60q) O09] OZ! Osi ;D}0Q psoyudis nO v=X ,S2lO20 :0410Q dt 33 Hsu and Yu of the experimental data precludes any definite conclusion about possible flow separation. Although a unified theory is needed to describe the relationship between the phase angle » and + C/U F the experimentally determined phase angles in the inviscid range do indicate a correct trend compared with Miles' theory. V. CONCLUSIONS The accumulated laboratory experimental evidence obtained at Stanford and elsewhere indicates general support for Miles' inviscid theory of energy transfer between air stream and progressive waves through the phase shift of the aerodynamic pressure at the interface. However, the experimental growth rate is considerably in excess of Miles' prediction, being approximately three times larger. The most fruitful avenue for further study would appear to be to reexamine the necessary simplifying assumptions in the Miles' inviscid model. The incompatibilities near the air-water interface suggest that detailed experimental investigations of this region are essential before an understanding of the energy transfer mechanisms and the conditions under which they occur can be fully established. The effects of turbulence, possible flow separation, ripple super- position and boundary layer development are complex, but could be modelled and fruitfully studied in laboratory simulations. REFERENCES Benjamin, T. B., "Shearing Flow over a Wavy Boundary,’ J. Fluid Mech., 6, 161-205, 1959. Bole, J. B. and Hsu, E. Y., "Response of Gravity Water Waves to Wind Excitation," Stanford Univ. Dept. of Civil Engineering Téch. Rep.“Nos 794, (1967 « Chang, P. C., "Laboratory Measurements of Air Flow over Wind Waves Following the Moving Water Surface," CERv8- 69PcC1i8, Colorado State Univ. , 1968. Dean, R. G., "Stream Function Representation of Non-Linear Ocean Waves," J. Geophys. Res., 70, (18), 4651-72, 1965. Hires, R. I., "An Experimental Study of Wind Wave Interactions," Tech. Rep. No. 37, Chesapeake Bay Inst., Johns Hopkins Univ. , 1968. Hsu, E. Y., "A Wind, Water-Wave Research Facility," Stanford Univ., Dept. of Civil Engineering Tech. Rep. No. 57, 1965. Kendall, J. M. Jr., "The Turbulent Boundary Layer Over a Wall With Progressing Surface Waves," J. Fluid Mech., 41, Pt. 2, 13 April 1970, pp. 259-282. 34 Laboratory Investtgattions on Atr-Sea Interacttons Larras, H. and Claria, W., "Recherches en Souffleries sur L'Action Relative de la Houle et du Vent," La Houille Blanche, 6, 647-677, 1960. = Lighthill, M. J., "Physical Interpretation of the Mathematical Theory of Wave Generation by Wind," J. Fluid Mech., 14, 385-398, 1962. ~~ Lin, C. C., The Theory of Hydrodynamic Stability, Cambridge Univ. Press, London, 1955. Longuet- Higgins, M. S., "A Non-Linear Mechanism for Generation of Sea Waves," Proc. Roy. Soc. A, 1969. Miles, J. W., "On the Generation of Surface Waves by Shear Flow," J. Fluid Mech., 3; 185-204, 1957. Miles, J. W., "On the Generation of Surface Waves by Shear Flow, Part 4," J. Fluid Mech., 13, 433-477, 1962. Miles, J. W., "On the Generation of Surface Waves by Shear Flow, Part 5," J. Fluid Mech., 30, 163-175, 1967. Mollo-Christensen, E., "Observations and Speculations on Mechanisms of Wave Generation by Wind," Dept. of Meteorology, MIT, 1970. Motzfeld, H., "Die Turbulent Stromung an Welligen Wanden," £4. Angew. Math. Mech, ; 17, 193-212, 1937. Ott, R., Hsu, E. Y. and Street, R. L., "A Steady-State Simulation of Small Amplitude Wind-Generated Waves," Stanford Univ. Dept. of Civil Engineering Tech. Rep. No. 94, 1968. Phillips, O. M., "Wave Generation by Turbulent Wind Over a Finite Fetch," Proc. 3rd Natl. Congr. Appl. Mech., pp. 785-789, 1958. Phillips, O. M., The Dynamics of the Upper Ocean, Cambridge Univ. Press, New York, . Stanton, T. E., Marshall, D., and Houghton, R., "The Growth of Waves on Water Due to the Action of Wind," Proc. Roy. Soc., Ser. A., MSs Pp. 283-293, 1932. Sutherland, A. S., "Spectral Measurements and Growth Rates of Wind-Generated Water Waves," Stanford Univ. Dept. of Civil Engineering Tech. Rep. No. 84, 1967. 35 Hsu and Yu Thijsse, J. T., "Growth of Wind-Generated Waves and Energy Transfer," National Bureau of Standards, Washington, D.C., Circular No. 512, 281-287, 1951. Ursell, F., "Wave Generation by Wind," Survey in Mechanics, Cambridge Univ. Press, 1956. Zagustin, K., Hsu, E. Y., Street, R. L., "Turbulent Flow Over Moving Boundary," J. of the Waterways and Harbor Div., Proc. ASCE, 397-414, 1968, Zagustin, K., Hsu, E. Y., Street, R. L., and Perry, B.; "Flow over a Moving Boundary in Relation to Wind-Generated Waves," Stanford Univ. Dept. of Civil Engineering Tech. Rep. No. 60, 1966. 36 AIR-SEA INTERACTIONS: RESEARCH PROGRAM AND FACILITIES AT IMST M. Coantic and A. Favre IMST Marseille, France ABSTRACT This research concerns the small-scale physical pro- cesses responsible for mass, momentum and energy exchanges between the atmospheric surface layer and the oceans. Their theoretical study has been undertaken. It outlines the importance of turbulence and the influence of recip- rocal interactions between the various transfer pro- cesses. It has led to the design of an experiment where the natural phenomena shall be partially simulated, ina large laboratory facility. This one combines a micrometeorological wind tunnel with a 40 meters long wave tank, under controlled tem- perature and humidity conditions. It has been extensively tested with a one-fifth scale model. It is presently under construction, and wiil be operative by 1971. Instrumental studies have also been undertaken, and results obtained in the measurement of turbulence in water flows. I. INTRODUCTION The knowledge of energy exchange processes between atmos- phere and oceans appears of major interest for oceanography as well as for meteorology. These two media have indeed to be considered as elements of a single system, for the dynamical and thermodynam- ical evolution of each of them largely depends on interactions through their common boundary. 37 Favre and Coanttie One of the essential steps in the solution of the air-sea inter- action problem lies in the understanding of small-scale processes in the air and water layers adjacent to the interface, where the various forms of energy are either transferred or converted, while going from one medium to the other. The experimental study of these phenomena involves a detailed and delicate exploration ofa region whose thickness is of the order of the wave height. Now, experiments performed at sea are subjected to such environmental constraints that the accuracy and repeatability of measurements seems necessarily limited. It has therefore appeared useful to complement field studies by laboratory experiments, where an ex- tensive investigation is feasible under exactly repeatable conditions and with the possibility to control independently each of the govern- ing parameters. This is the program which has been undertaken atI.M.S.T., and which is described in the present paper. The preliminary steps of this program have included: collection of information about cur- rent research; attempt of a critical survey of existing knowledge, in order to find out definite research objectives; and a first theoreti- cal study of the physical mechanisms of air-sea interactions, and of their governing parameters. These studies have led to the con- clusion that it would be feasible to obtain, in the laboratory, a partial simulation of the atmospheric-oceanic energy exchange processes, provided that a sufficiently large facility could be realized. The following steps of the program have then comprised: the preliminary design of this facility, combining a micrometeorolo- gical wind tunnel with a 40 meters long wave tank; the realization of a one-fifth scale model, and its use for various preliminary tests and experiments; the detailed design and the building of the large wind-wave facility; and, last but not least, the development of various theoretical and instrumental researches. The purpose of the present paper is to introduce the various objectives and results of our research program, and to describe the facilities which have been, or are being, realized. Due to space limitation, that presentation will be limited to a rather short account referring to previous publications for more details, when possible. The plan adopted is logical rather than chronological: - Theoretical studies; - Setting up the characteristics and design of the large air- sea facility; - Model tests; - Building of the air-sea facility; - Studies of measuring instruments and methods. At last, we shall try to draw some preliminary conclusions about this program, the prospects it opens, and its possible appli- cations. We shall also have the pleasure to express our thanks to the many individuals and organizations who have contributed to its realization. 38 Atr-Sea Interactions; Program at IMST Il. THEORETICAL STUDIES 1. The Physical Mechanisms of the Ocean-Atmosphere Interaction As it is well known, the small-scale transfer of energy be- tween atmosphere and oceans occurs following four various mecha- nisms, sketched by Fig. 1: a) Radiation, including: i) short-wave radiation from the sun on the sea surface, which is partially reflected and absorbed over a more or less large depth under the interface; ii) long-wave radi- ation coming from the atmosphere and from the sea, and involv- ing a radiative transfer process between the interface itself and the atmospheric layers (see II.3). b) Evaporation (or condensation), and turbulent convection of water vapor, which, due to the very high latent heat of vaporization of water, leads to a turbulent latent enthalpy transfer from the sea surface to the atmosphere. c) Turbulent convection of sensible enthalpy, resulting from tem- perature differences between adjoining points of the system. RADIATION TURBULENT ENTHALPY KINETIC ENERGY TRANSFER TRANSFER Incident Reflected Z WY] j Yj j ay YY YY YY YY Yyy / Yy YY YY : : ; ' Gaseous phase 4 4. L S Interface Q $ Liquid phase Zz __ RESULTING ENERGY TRANSFER = S+L+0:S'+Q' Short wave Long wave Received § Emitted Sensible Fig. 1. Schematic display of energy transfers in the vicinity of the ocean-atmosphere interface. 39 Favre and Coantie d) Transfer of kinetic energy across the turbulent boundary layers on both sides of the interface, of which the most obvious effect is the generation of waves. Information on these mechanisms can be gathered in many books, ranging from meteorology (e.g. Brunt [ 1939], Haltiner and Martin [1967] , Roll [1965]) to oceanography (e.g. Lacombe [ 1965], Phillips [1966], Sverdrup [1957]), or devoted to atmospheric tur- bulence (e.g. Lumley and Panofsky [1964], Monin and Yaglom [1966], Priestley [1959]). One of the first steps of our program has been to attempt to review the physical laws and equations governing the ensemble of these phenomena (see Coantic [ 1968]). The main conclusion that can be reached is that, although the above types of transfer have been analyzed and listed separately, they are not independent, and the key of the problem lies in their reciprocal interactions. For instance, processes a) and b) set in action very large amounts of energy, whereas c) and chiefly d) are responsible for much smaller exchanges. However, the kinetic energy transfer, which enters as the smallest term in the energy balance, strongly influences the turbulent evaporation and convection processes. Infact, except for certain radiation effects, air-sea interactions are essentially governed by turbulence. This is only one aspect of the aforementioned reciprocal inter- actions. Other ones will appear, for instance, when considering the boundary conditions for the various variables at the interface, or when expressing the conservation of the different energy fluxes, as schematized on the lower part of Fig. 1. Furthermore, two most important peculiarities are displayed when comparing the present case to the more classical problem of simultaneous heat, mass and momentum exchange between a fluid flow and a more or less rough surface. In the latter case, the turbulent convective processes, if not completely understood, are sufficiently well known to allow a good estimate of the various transfer rates. However, the methods of computation therein developed are not applicable here for two main reasons: - On one hand, because the boundary is no longer static, and pos- sesses "dynamic rugosities" capable of yielding and absorbing momentum with large variations of the ratio of the tangential shear stresses to the normal pressure forces. This fact will have conse- quences difficult to ascertain, not only upon the dynamical exchange mechanism but also upon the degree of "Reynolds analogy" be- tween this process and those concerning exchange of scalar vari- ables. On the other hand, because, due to the well known stratification effects in the atmosphere, heat and humidity can no longer be con- sidered as "passive scalar containments." This means that the turbulent structure of the boundary layer, and the transfer rates themselves, are strongly modified by the direction and intensity of the vertical heat and humidity gradients. 40 Atr-Sea Interacttons; Program at IMST As discussed in our previous publications (Coantic [ 1968], Coantic et al. [1969]), and in Part III of the present paper, the preceeding considerations have been the basis for the settling of our research program and the design of our simulation facility. Some aspects of the problem are already being the subject of theoretical investigations, which we shall now mention shortly. 2. Wave and Current Generation by Wind The transfer of mechanical energy from air to sea has two main consequences: the development of currents and turbulence in the upper ocean, and the generation and amplification of waves. This latter process can be broadly described as follows: the turbulent atmospheric boundary layer exerts on the water surface normal and tangential stresses, with steady, periodic and random components. As a consequence of these stresses and of the gravity and capillarity restoring forces, motions of a wavy character are generated at the interface. As soon as their amplitude becomes appreciable, non- linear effects are developed, which result in a modification of the airflow structure and, hence, of the applied stresses, the existence of a continuous wave spectrum and the production of turbulent energy in the sea. The wave amplitude is then limited by the dissipative action of turbulence and viscosity. As mentioned earlier, the understanding of this complex mechanism is essential to elucidate, not only the dynamical, but also the thermodynamical aspects of air-sea interactions. A careful study has, accordingly, been undertaken (Ramamonjiarisoa [ 1969, 1970]), first of existing theories (based on models proposed by Miles and Phillips) and later on of more recent developments in the researches of Stewart, Mollo-Christensen, Longuet-Higgins, Hasselmann and Reynolds, among others. This helped us in identify- ing some points that have to be subjected to experimental study, namely: the existence of separation after the wave crests, the phase shift between surface pressure and elevation, the spatial and temporal variations of Reynolds stresses, and of the turbulent structure of the flow in general. Our future measurement program has been estab- lished in consequence, taking advantage of the possible use of the space-time correlation technique, and of numerical data processing methods to separate the "mean," the "phase average" and the "tur- bulent" parts of each variable. 3. Interaction of Turbulent and Radiative Transfers Another typical example of reciprocal interactions between the various modes of energy transfer near the air-sea interface is the simultaneous transport of sensible enthalpy by turbulent con- vection, and by infrared radiation. The turbulent heat flux is usually assumed constant with height in the atmospheric surface layer. However, the validity of this hypothesis is known to be questionable, due to a possible vertical variation of the infrared 41 Favre and Coantte radiative flux (see e.g. Munn [ 1967]). This problem has been approached theoretically, using semi- empirical expressions fitted to the emissivity curves, and assuming logarithmic temperature and humidity profiles. A first approxima- tion of the radiative heat flux divergence is thus obtained analytically, as a function of the surface layer parameters (Coantic and Seguin [1970]). Numerical values of the infrared flux gradient, dq,/dz, in the first ten meters of the marine atmosphere are shown in Fig. 2, for two different wind velocities (A: Uj) =3 m/s; B: Ujo = 9 m/s); two sea surface temperatures (cases 1,2: 8, = +5 CG; cases 3,4: 8, = + 20 °C); and two temperature differences (cases 1,3: 0,9 - 89 = - 5 °C; cases 2,4: O19 - 0g = +1 °C). The resulting vertical vari- ations of the turbulent heat flux, shown by Fig. 3, are seen to attain unexpectedly large values, of the order of 30 to 40%, when wind velocity is low and humidity is high. These results are considered as preliminary. If confirmed, they could lead to a reinterpretation of some experimental data, and should appeal to an extension of turbulent transfer theories to the case of a variable heat flux. -08 -04 (a) : +04 a mw/em¥m Fig. 2. Computed vertical variations of radiative flux divergence for various atmospheric situations 42 Atr-Sea Interacttons; Program at IMST d~) WV Za | V SS, OQ 6./ Sy) O Qi Q2 Q3 04 Fig, 3. *Relative vertical variation of the turbulent heat flux, for various atmospheric situations 4. Water Vapor Turbulence and Its Measurement The turbulent transfer of humidity in the lowest atmospheric layers is, as mentioned earlier, one of the principal mechanisms for the exchange of energy between air and sea. In addition, this process governs the mean distribution and turbulent structure of specific humidity in the lower levels, and thus exerts an essential influence on electromagnetic wave propagation. Therefore, the contemplated studies require the measurement of humidity fluctu- ations, whose levels and scales have to be estimated to delineate suitable measuring devices. This estimation has been obtained by: a) studying the equa- tions governing the mean distributions, turbulent fluxes and levels of fluctuations of humidity b) examining the known experimental data; and c) predicting the form of the spectrum from the Kolmogorov- Obukhov theory (Coantic and Leducq [1969]). Figure 4 compares the predicted spectral behavior of turbulent humidity fluctuations (after some shift towards lower frequencies), and recent measurements by Miyake and McBean [1970]. Considering the experimental under- estimation of the high frequency part of the spectrum, the overall agreement is not too bad. Once the estimate is made, it is then possible to define specifications for devices measuring humidity turbulence, for use either in the field or in the laboratory (see VI. 3). 43 Favre and Coantte LOG(FX SPECT) © DEW POINT @ LYMAN ALPHA ry ©°, LOG(FZ/U) -$0 ~20! -90-x0,0 4 4014 [20 Fig. 4. Comparison between predicted spectral behavior of turbu- lent humidity fluctuations, and measurements by Miyake and Mc Bean [ 1970] 5. Two-Phase Processes in the Vicinity of Air-Sea Interface The equations governing the mean properties of air-sea inter- actions are usually written in an earth fixed Eulerian frame of refer- ence, and different sets of equations have to be used in the gaseous and in the liquid phase. Due to the unsteady random character of the interface, this means that an appreciable part of the system has, strictly speaking, to be treated as a two-phase flow. If one wants to take into account the obviously important effects of sea spray in the lower atmosphere, and of air bubbles in the upper ocean, the necessity of considering two-phase effects is still more clear. Prompted by chemical and nuclear engineering problems, notable progress has been gained these last years in the analytical and empirical description of such processes. We plan to apply the methods therein developed to the study of the two-phase portion of the ocean-atmosphere system. Ill. STUDY OF CHARACTERISTICS AND DESIGN OF THE SIMU- LATING FACILITY 14. Conditions for Modelling Small-Scale Air-Sea Interactions In consequence of the physical mechanisms of air-sea energy exchanges, the planned laboratory experiments will concern the structure of turbulent velocity, temperature and humidity boundary 44 Air-Sea Interacttons; Program at IMST layers obtained at the interface between an airflow and a water mass. The three basic processes of momentum, heat and mass transfer will be effectively realized by controlling air velocity, temperature and humidity, and water velocity and temperature. Furthermore, appropriate heating or cooling will provide an approximate repre- sentation of the most important radiation effects. However, such experiments will be really useful in modelling the atmospheric-oceanic phenomenon, only if the three aforementioned specific features: turbulent atmospheric structure, stratification effects, and interface motion, are at least partially reproduced. This seems feasible, provided that a sufficiently large facility can be realized. 2. Simulation of the Atmospheric Dynamical Structure It is well known that the atmospheric surface layer motions can be simulated in the laboratory, in so-called "micrometeorologi- cal wind-tunnels" (see e.g. Pocock [ 1960], Cermak et al. [ 1966], McVehil et al. [1967], Mery [1968]). In short, these motions are characterized, on one hand by extremely high values of Reynolds number, and on the other hand, by stratification effects corresponding to appreciable values of Richardson number. For a good modelling, these dimensionless numbers have to keep significant values in the laboratory flow. For the latter, this implies rather large tempera- ture differences, and low wind velocities. In consequence, to pre- serve sufficiently high Reynolds numbers while observing cumulative stratification effects, it is necessary to build large facilities. Simi- lar conclusions are reached if one considers the problem of main- taining the ratio between the roughness height at the surface and the boundary layer thickness or the Monin-Obukhov length, or if one requires the reproduction of an appreciable Kolmogorov inertial range. For these reasons, the test section length of micrometeoro- logical wind tunnels reaches several tens of meters and the velocity range is of the order of a few meters per second, while provision is made for creating temperature differences of several tens of degrees centigrade. The main characteristics of our project are as follows: - Length of the water surface forming the interface in the test section: 40 meters. - Air velocity range: 0.5 to 14 meters per second. - Maximum temperature and specific humidity differences: 30 °C, and 25.10°° Kg water by Kg air. The estimated performance of the facility is sketched by Fig. 5, which shows the rather wide range of dimensionless parameters that should be covered. The general scheme of the tunnel is given in Fig. 6. It is a closed-circuit wind-tunnel, with several rather 45 Favre and Coantie unusual dispositions, dictated by specific requirements. For instance, to obtain stable functioning at the lowest velocities, the return circuit's area has been purposely reduced; the total head loss has been increased by tightly finned heat exchangers acting as flow equalizers just upstream the settling chamber; and the diffusors have been fitted with vortex generators and stabilizing vanes. The test section's area is 3.20 by 1.45 meter, and the overall size of the facility 61 by 7.50 meters. The wind velocity can be continuously varied from 0.5 to 14 meters per second, with a relative accuracy of 2.10°°, using a helicoidal fan driven by a variable speed D.C. motor with electronic regulation. ENE Ae ‘el a FAS SRE 0 10 20 30 40 Xm oy 0 500 i000 1500 ~~ 2000 =—=——{) = "ims SESSA U = 4mis (a) ( b ) a =: 20 30 40 Xm Cd) Fig. 5. Estimated performance of the facility: a) Reynolds number, and boundary layer thickness; b) Sensible and latent en- thalpy fluxes; c) Wind-Waves' age and significant héight; d) Richardson number. 46 Air-Sea Interactions; Program at IMST D, SECTION A FPP ERTTOOOT TTT TOTS TTT Parmer eae SP se eC Lr, SE ie ee Nn gl, eee) oO—L-O SECTION B Fig. 6. General scheme of the wind-water tunnel 3. Reproduction of Heat and Mass Transfer Processes Supposing a convenient flow structure has been obtained, the existence of nonzero temperature and partial water vapor pressure differences between the water surface and the incoming air flow will be sufficient to cause turbulent convective processes of mass and sensible and latent enthalpy similar to those encountered in the atmospheric boundary layer. The equations governing these transfers being linear with respect to temperature and humidity, these last variables can be fixed on grounds of experimental convenience, as long as stratifi- cation effects do not arise. The estimated values of flux Richardson number, computed at one-quarter boundary layer thickness and for a temperture difference amounting to 25 °C, are displayed in Fig. 2(d) as a function of longitudinal abscissa and velocity. At the highest velocities, Richardson number is clearly negligible, and temperature and humidity differences will be chosen, to improve experimental accuracy, at the highest levels authorized by the equipment's capabilities (see Fig. 2(b), and below). On the other hand, at the lowest velocities, temperature and humidity can no longer be considered as scalar passive contaminants, and their differences will be chosen in order to obtain a given Richardson number, i.e. a given effect on the dynamical structure. 47 Favre and Coantte A first approximation representation of radiative beat ex- changes seems also to be feasible in the laboratory. At the small scale we are interested in, the main effect of short wave solar radi- ation is a global elevation of the oceanic temperature, that can be reproduced by heating the water mass. Due to the radiative trans- fer process mentioned in II.3, the reproduction of infrared heat exchange is more delicate, but its primary effect yet remains the cooling (or occasionally heating) of the interface itself. This localized heat sink shall be simulated, either by increasing the cooling produced at the same place by evaporation, or by lowering the temperature of the ceiling of the test section, and thus con- trolling the radiative heat exchange between this wall and the water surface. The designed facility will allow independent control of air and water temperatures in the 5 °C - 35 °C range, with an accuracy of the order of 0.1 °C. The relative humidity of air entering the test section will be varied from 60% to 100%. Fig. 7 schematizes the main components of temperature and humidity control system: cooling and drying (by condensing) coils, heating coils and vapor injectors in the air circuit; cooling and heating heat exchangers in the water circuit; heat generator, frigorific unit with cooling tower, steam boiler, regulating system. The working principle is represented using the temperature-mixing ratio diagram. =e HEATER 160000 cu HEATER 90000 cal /b at if Fig. 7. Schematic diagram of temperature and humidity control systems 48 Atr-Sea Interactions; Program at IMST 4. Reproduction of Interfacial Motions The problem of obtaining laboratory waves statistically similar to those encountered over the oceans has been thoroughly studied these last years, with the view, either to perform more realistic structural tests, or to experimentally investigate the mechanism of wave generation by wind. The works of Veras [| 1963], Hidy and Plate [ 1965], Hsu [1965], Gupta [1966], and of the Waterloopkundig Laboratorium [ 1966a,b] can be cited among many others. The unsymmetrical, randomly varying and three-dimen- sional waves existing in nature can be simulated only at the cost of building large laboratory facilities. The so-called "wind-wave tunnels" reach one hundred meters in length and several meters in width, with smooth and parallel side walls and an efficient absorbing beach at the end. The main characteristics of waves naturally generated by wind along our 40 meters long tank have been forecast from the preceeding references and are shown by Fig. 2(c). It is clearly possible to generate gravity waves of appreciable amplitude, and thus to cover a nonnegligible range of Froude numbers. However, the "wave age," i.e. the ratio of the celerity of propagation, C, of dominant waves to the wind velocity, U, remains low, especially at the highest velocities. The same is true of the ratio C/U~ (where U+ is the friction velocity in the boundary layer), which is known as an important parameter in the wave generation process. Asa matter of fact, these two ratios control the relative magnitude of normal and tangential stresses exerted by wind on water, with im- portant consequences upon the various energy exchange mechanisms (see II.1). It is therefore necessary to have the possibility to act on these parameters, by controlling the wave height and celerity independently of wind velocity. This will be done by means ofa wavemaker set at the beginning of the water channel and conveniently randomly actuated. It is known that the combined action of sucha device and of wind blowing will result, after some distance, ina satisfying wave pattern. The details of that part of the equipment are sketched by Fig. 8. A new type of wavemaker, comprising a fully submerged wave plate connected to the tank by means of bellows, has been imagined. This arrangement allows to realize a fairly smooth joining of air and water flows, even in the presence of waves. The end of the channel will be equipped with an absorbing beach made from parallel tubes with a 7° slope. A slight water movement (between 0.1 and 0.01 m/s) necessary for cleaning and temperature controlling purposes, will be insured by«a recirculating 35 HP heli- coidal pump. * By A. Ramamonjiarisoa. 49 Favre and Coantte SMOOTH AIR JOINING CONTRACTION KOO OL eT eke) — 2: eC ar NOES Va a PAE TNy, gt SO PAU ACS xr $3,0¢ Waistes =f IE 2 a ASS 8 WATER aie Se ogo BS s Fig. 8. Details of junction between air and water flows, and arrange- ment of the submerged wavemaker. 5. Further Details and Conclusions At each step in the design of the facility, we endeavored to improve the simulation of the natural phenomenon and some of the arrangements taken to this end have just been described. A peculiar problem was set by the parasitic boundary layers which unavoidably originate along the side walls and ceiling of an elongated working section, and which are known to result in cumbersome secondary motions. The dispositions adopted to reduce these effects, thereby improving the representation of the unlimited atmospheric-oceanic system, are represented by Fig. 9. The cross sectional shape of the working section (see Fig. 9(a)) has been designed with a height/ width ratio of 1 to 2.2, and furthermore fitted, like in the Water- loopkundig Laboratorium [1966a,b] design, with vertical plates restricting the span of the useful water surface to 2.62 meters. The lateral quays thus realized will limit the parasitic dynamical as well as thermodynamical effects in the central part of the working section, where the measurements will be performed. To further improve the two-dimensionality of the flow, and to prevent the inter- action of the studied boundary layer with that one developing on the working section's ceiling, boundary layer control devices will be used. As shown by Fig. 9 (b), they combine boundary layer suction (by means of slots or porous walls) and blowing (through slots), taking advantage of the possibilities of tangential blowers. At last, it will be necessary, specially at the lowest wind velocities, to artificially trigger transition, and eventually to in- crease the boundary layer thickness, by means of devices similar to those studied by Counihan [1969] or Campbell and Staden [ 1969]. 50 Atr-Sea Interacttons; Program at IMST Suction through porous wall Suction and blowing by slots (b) Fig. 9. Details of test section: a) Section view, showing the lateral quays; b) Boundary layer control devices. All the foregoing will make clear that we have tried to insure an acceptable simulation of the main aspects of air-sea interactions. Entire modelling, with full similarity, cannot, of course, be attained. We believe, however, that experiments where the various physical mechanisms are effectively put in action, and where basic parameters possess significant values, will realize a partial simulation of natural exchanges, thus affording the possibility of interesting investigations. 51 Favre and Coantte IV. PRELIMINARY ONE-FIFTH SCALE MODEL TESTS A one-fifth scale model of the large air-sea interaction facility has been built. The primary object was to check and im- prove various design characteristics; altogether it was also planned to perform instrumental studies, and to execute preliminary small- scale scientific experiments. This scale model is a detailed reproduction of all parts of the large facility, including not only the aerodynamic and hydraulic elements, but also the equipment controlling heat and humidity ex- changes. A view of the wind-water tunnel, and of the control console, is given in Fig. 10. Fig. 10. General picture of the model 52 Atr-Sea Interactions; Program at IMST 1. Overall Aerodynamic Tests The first use of the model has been to test the global aero- dynamic performance of the facility. The initial design, represented by the upper part of Fig. 11, suffered from several imperfections resulting in a low power factor and in an inadequate working stability. Detailed flow explorations have led to successive amendments in the geometry of the model (see Pouchain [1970] and Coantic et al. [1969]). The final design, shown in the lower part of Fig. 11, offers satis- factory performance, and has therefore been adopted in the later building of the large facility, the aerodynamic characteristics of which have been predicted from the model tests. SECTION A SECTION B,C INITIAL SECTION D ee — hea SECTION E,F 0 In FINAL Fig. 11. Improvements in aerodynamic design of the model 2. Flow Exploration and Improvement in the Working Section A deeper flow study of the working section has then been per- formed, of which typical results, obtained for a wind velocity of 8.3 m/s, are displayed by Fig. 12. It can be seen that the situation is good in the entrance section, with a very flat velocity profile, and a turbulence intensity below 0.002 (the effects that can be dis- cerned near the water surface are the consequence of artificial boundary layer thickening in the final part of the contraction). The further growth of the various boundary layers, and the fact that they 53 Favre and Coantte ©1=280m ° 4 e 121 Um/s X= 160 mm X=2800mm X=5600 mm ae | ein ee e—e— is e—e—2— o— 0 —0 —-0 —0— 0— O— 9 __ | | lyr Oss Lam dL ~L—h—A—A DD hb — a—a— b—D—A—A OO 0 OO Ome a OOO OO 0 0 OO Om XN = - 300 - 200 - 100 0 100 200 300 Fig. 12. Flow characteristics in the test section: a) Vertical velocity profiles; b) Horizontal velocity profiles; c) Vertical distributions of turbulence intensity. 54 Atr-Sea Interacttons; Program at IMST begin to join together towards the end of the working section, leading to a fully developed channel flow, are also apparent. As already mentioned (see III.5), it is therefore necessary to take steps to restrict the development of the lateral and upper bound- ary layers. Tests performed under different conditions have proven that the contemplated control method was efficient in this respect (Pouchain [1970]). Fig. 13 illustrates typical results obtained for various blowing rates (i.e. ratios of jet velocity to mean flow veloc- ity), while sucking through a porous wall and blowing across a 15 millimeters height slot. The improvement in velocity distribution is clear. As shown by Fig. 14, the turbulent intensity distribution is also ameliorated. Some problems related to pressure perturbations still have to be solved, but, on the whole, the method appears as promising. No blowing ese With blowing Uj /U, = 105 ------ - Fig. 13. Improvement in test section's flow by means of boundary layer control: a) Flow configurations for three rates of blowing; b) Variation of boundary layers! thickness. 55: Favre and Coanttie X=5750 X=157 =0 e= 15mm Fig. 14. Effects of blowing on turbulence intensity distributions. he Hydraulic Tests The hydraulic performance of the facility has also been sub- jected to various tests. The functioning of the water recirculating circuit has been controlled. The working of the new submerged wavemaker, and of the absorbing beach, has also been found satis- factory. Observations of waves generated by wind in the model tank, such as those shown by Fig. 15, suggests they qualitatively possess the three-dimensional random structure typical of oceanic wind waves. Measurements of wave spectrum at different fetches along the working section have just been done, and the results displayed in Fig. 16 compare favorably with those of previous studies (see the references in III.4). The spectral shape and evolution strongly suggests the existence of nonlinear effects transferring energy from higher to smaller frequencies, as recently postulated by Longuet- Higgins and Mollo-Christensen. 56 Aitr-Sea Interactions; Program at IMST Fig. 15. Sample view of wind-waves obtained for a 4 m/s velocity and a 4 m fetch. Fetch e F=840 mm Vv F=2340 mm ° F=3840 mm °o F=5340 mm 5 F=6840 mm 10° 10° Fig. 16. Evolution of wind-waves' spectra as a function of fetch along the model's test section. Mm ~] Favre and Coantie 4. Tests of Temperature and Humidity Control Systems Various working tests of that part of the equipment have been executed. The validity of the previously chosen control methods has been checked, the obtainable temperature and humidity range has been controlled, and the stability of regulating loops has been tested. After some improvements, the overall thermodynamic per- formance of the model has been correct. A further study of temperature repartitions upstream and inside the working section has then been undertaken, for different flow and thermal conditions. Typical results are shown by Fig. 17, where an initially isothermal airflow, and the development of thermal boundary layers can be observed. The temperature distribution in the entrance section is usually good; except in extreme cases of large heating and velocities of the order of one meter per second, where parasitic stratification effects appear. U.=260ms 8,=52°C — Twnt27 °C ee papers eras ues PLIPIIG We Att / f yyy / , mae ++ | 300 200 X=0 100 seer = A 30 20 10 Cc 30 20 10 0 “ eee) 30 20 10 0 8-8, C ee) 30 20 10 0 Fig. 17. Temperature Distribution in the Model's Test Section V. CONSTRUCTION OF THE LARGE AIR-SEA INTERACTION FACILITY In view of the rather considerable size of the designed wind- wave tunnel, its erection was not possible inside I.M.S.T. 's main building. It was therefore decided to build a new laboratory, includ- ing the large air-sea facility, its auxiliary equipments and a group of offices, workshops and laboratories, and located in the new Marseille-Luminy Campus. Its floor plan is shown by Fig. 18. 58 Air-Sea Interacttons; Program at IMST Echelle (Ale Eee Metres Seufflerie SE 16 a a a | Fig. 18. Air~Sea Interactions Laboratory Floor Plan The preliminary design of the facility has been determined by I.M.S.T., and its detail drawings set up with the aid of architect and engineering offices. The construction works have been planned in three stages: a) Erection of buildings, concrete structural parts of the tunnel, electric equipments, temperature and humidity con- trol systems; b) Fitting up of the main elements of air and water circuits, including static parts (tunnel walls etc.) as well as pump and fan; c) Completion and equipment of the facility, placing control apparatus and such parts as wave-maker and boundary layer control devices in position. Works have been started in January 1969, and step a) is fully completed from several months (see Fig. 19). Step b) is now nearly achieved, and the first run of the wind-water tunnel is planned for the end of the present year.” Our program forecasts about one more year for the execution of step c), and the beginning of strictly scientific experiments by the end of 1971. These experiments will concern: first the dynamic exchange process alone; then, the heat and mass transfer processes; and later on the effects of stratifica- tion upon these three mechanisms. The execution of this program will clearly take several years. VI. RESEARCHES RELATED TO INSTRUMENTATION PROBLEMS Some of the anticipated experiments obviously necessitate, either the development of new measuring instruments and methods, or the adaptation of existing ones. Corresponding researches have been undertaken since the early stages of our program. ¥ Note added in proof: this has been achieved by November 1970. 59 Favre and Coantie a o Bees ae a'e's's| ae ae ee@dexs: “te 8 ‘ ’ SKRSSSSSREOHTS oR, 1 > ae : e ‘ ~ > Fig. 19. Constructions! progress by May 1970: view of refrigerating coils, and of concrete contraction and test section. 1. Turbulence Measurements in Water Flows A first work has been devoted to the measurement of velocity and temperature mean and fluctuating values in water flows, and of the associated turbulent momentum and heat fluxes. The adaptation to this problem of the well known hot-wire technique has been studied experimentally. An apparatus including a tubular water channel was constructed to that end, and hot-wire sensors were manufactured. The theoretical and experimental study of the performance of various types of wires and films has resolved satisfactory methods of cali- bration and measurement, particularly for commercially manufactured conical hot films for which dimensionless cooling laws have been pro- posed, and for slanting wedge-shaped films. The intensity and the spectrum of turbulence, and the Reynolds stresses themselves, have been determined inside a circular conduit, with a comparable degree of accuracy to that attainable in air flows (see Fig. 20). Later on, the effects of water temperature variations upon the hot-film response 60 Atr-Sea Interacttons; Program at IMST | io-8- COANTIC AIR © D=- 765mm fe) D= 257,8mm 10°’- RESCH WATER @ D- 44 mm Fig. 20. Turbulence measurements in water flows: a) Comparison of turbulence spectra measured in dynamically similar air and water flows. 61 Favre and Coantte e LAUFER © MARTIN @ BALOWIN © MICKELSEN COA NTIC AIR © SANOBORN Temperature cste © WEISSBERG ° ei ay" => mm © SANDBORN Intensite cste © ASHKENAS » D= 76mm Gg D=258 @ NEWMANN et LEARY @ GAVIGLIO RESCH WA TER e o Re 39000 « Re 83000 — Computed Fig. 20. Turbulence measurements in water flows: b) Comparison of turbulence intensities measured in dynamically similar air and water flows; c) accuracy checking of measured shear stresses. 62 Atr-Sea Interacttons; Program at IMST have been thoroughly studied (see Fig. 21), and measurements of the intensity of temperature turbulence have been executed. These results are given in a number of publications: Resch [ 1968, 1970], Resch and Coantic [1969], Ezraty and Coantic [1970], Ezraty [1970]. They show conclusively that, subject to some pre- cautions, turbulence measurements can be quite accurately per- formed in water flows, using hot-film anemothemometers. 2. Measurements of Turbulent Fluctuations of Humidity After the theoretical study reported in II.4, the development of a water vapor turbulence measuring technique has been undertaken. Various methods have been considered: psychrometry, dew-point measurement, use of hot-wire, absorption of Lyman alpha or infrared No eG os, Sg a A ©, -17 34 NuzNu-=> -B (a Tape” ae Ke ! — git 4 gf 059) toc R where ‘oma is a coefficient which, for the sake of simplicity, will be written as ‘oq in the following. The wave amplitude is then given by 84 Hxploston-Generated Water Waves 2 / ment) les (4s) J,(kR) cos (kr - tVk tanhk), — (22) ag the two "cavity parameters" n, and R being embodied within our previous expression for the envelope amplitude, A. It is through empirical determination of these two parameters that we hope to correlate theory and experiment. 4.2 Experimental Correlation While y. and R cannot be experimentally measured, they can be determined indirectly from Rees and eats which are characteristic of the source disturbance, and also measurable. Hence, we seek to relate k,,, and n,,,r to the characteristics of the explosion by experiment, and n, and R to ky, and Nmar by theory. The expression giving kg, in terms of nN, and R is GA ie a 0 (23) since this expression defines the maxima of the wave envelope; the least non-zero value of k for which the above expression holds is K ax? For k__ >3 (relatively deep water) vi = /2 = const max be > TdV/dk ! and V2 mg J,(kR) . (24) mheretore, Kk can be determined from the first turning value of max the Bessel function J3(kR); viz for Kay = 4.20. (25) Our other measurable, mgr, may now be related to 1) and R by evaluating Ang, (OF Tmax) at k = Kkmqe When this is done and the resulting expression is simplified, we have Wott = 10ST ae (26) All that remains now is to relate "mgr and Kg, to the characteristics of the explosion; these are W, explosive yield in pounds of TNT, Z, detonation depth in feet, and D, the water depth 85 Le Mehauté in feet. A large volume of experimental data (with small chemical changes in relatively deep water) has been obtained at the Waterways Experimental Station, Vicksburg, Mississippi, from which the following empirical relations were deduced: * at 0.54 i limagt oO. Wi WeCeds Pie O> Fos =~ 0-25 (27a) K riage) BeOS ae Ow ieee =10 w?4 L. Ges -0.25> oo32-7.5 (27b) -0.3 * ft ee Ss soo Ww at Z Insufficient Data wos <7 gees, (2-7) The products Teer given above were determined from the empiri- cal data of Fig. 5. Corresponding data for shallow water explosions and other aspects of explosion-generated waves in shallow water are beyond the scope of this presentation, and the reader is referred to LeMéhauté [ 19714]. YIELD O W =0.50 Ibs TNT fe) = 2.00 lbs TNT A 10.00 Ibs TNT + 125.00 Ibs TNT e 385.00 Ibs TNT MONO 1966 © HYDRA IL -A a =9,250.00 Ibs TNT = 14,500.00 Ibs TNT 0 ay 2. = -4 -5 a6 =7 -8 z/w2:3 Fig. 5 An empirical scaling fit relating the maximum wave height “max With distance from explosion r*, yield and depth of explosion (data provided by Waterways Experimental Station) 86 Exploston-Generated Water Waves Figure 6 presents examples of the matching between theoreti- cal wave envelope and wave records due to a 9, 620 lb TNT explosion. The slight irregularities in the symmetry of the recorded wave trains are attributed to partial shoreline reflection interferring with the radiating wave trains (Hwang et al. [1969]. But, in general, the computed wave envelopes agree fairly closely with the observed amplitudes. 4.3 Limitiations of the Model Due to Scale Effects An examination of Fig. 5 reveals that the bulk of data upon which predictions are based are restricted to yields from one-half to a few hundred pounds of TNT. One wonders then just how reliable extrapolation to very large yield (say, 10'9 pounds of TNT) would be. The limited data available from nuclear explosions is insufficient to resolve this problem. Comparison between crater data in soft materials for both nuclear and TNT explosions suggest that the laws of similitude may be applied to contained explosions but may not apply over a large yield range for venting detonations. In particular, the shock wave from a nuclear explosion travels much faster in air than in water, which is not the case for a TNT explosion. We may infer several things, however, just from the nature of the scaling parameters given by Eq. (27). Consider, for example, the groups n*,,.r*/W°54 and Z/W®3. In each, the exponent of W was chosen to best compress the data of Fig. 5 into a single curve, since W represents an energy, dimensional analysis suggests that Nmaxt/(W/pg)'/2 and Z/(W/pg)!/4 are appropriate scaling parameters, although similar conditions also require that other parameters, such as atmospheric pressure and sonic velocity in water, should also be scaled with yield. These conditions are never satisfied experimental- ly, and it is therefore not surprising that exponential scaling alone is not satisfactory. Moreover, the fact that the parametric coefficients vary with Z means that the phenomena are not simply scalable (Pace et al. | 1969]). Lastly, the lack of evidence for an u.c.d. at large yields suggests that the generation process is fundamentally different. For small yields (and subsequent small depth at burst) hydro- static pressure is small compared to atmospheric pressure; for large yields the reverse is true. In the former extreme, dimensional analysis suggests 1/3 power scaling; in the latter, 1/4 power scaling. In an analogous review of earth crater scaling, (Chabai [1965]) has proposed an "overburden scaling law" in which the scaling exponent varies between these two extremes, but without convincing improve- ment in agreement to the experimental data. 87 Distance Between Gage and SZ 3,600 ft. lll Het 568 neem ee ll sare ST a 7 (feet) iil i mci : Bi t (seconds = Fig. 6 Compari of OSI 1966 Mono Lake experiments with theory 88 Exploston-Generated Water Waves 4.4 Energy Coupling The deficiencies of simple exponential scaling are more appar- ent when considering the efficiency of energy coupling into water waves. The analytic source models discussed above are linear, and thus the total wave energy is equal to ns potential energy of the source model; i.e., proportional to 16 2@R2, But, in view, of Eq. (26), the empirical relations given in Eq. (27) imply that 16 2R2 ~ 8 which obviously cannot be true for all yields, since it ‘states that wave energy increases faster than explosion energy under geometri- cally similar conditions. It is also pertinent to recall that the calcu- lating of energy based on the theoretical source model may lead toa significant error; since only the first wave train has been watched with experiments, it may happen that the following wave train contains less energy than the theoretical model, as due the dissipative mechanism which influences the high frequency waves. Keeping in mind these reservations, it is found that the energy in the wave train is 2 Ey = 126(n 2) ft=Lb, Then, inserting the value of n,,,r in terms of yield and water depth, it is found that at lower critical depth, the efficiency e is e = 0.0074 WO-%E 1% (W is in pounds). At upper critical depth, the increase of efficiency with yield within the range of available experiments is even more pronounced, For example, e which is 1% in the case of 0.5 lbs of TNT has been found to be 6% in the case of an explosion of 375 pounds, which implies that n° = w-6! at upper critical depth. Such results cannot, of course, be extrapolated to atomic yield. Since the fraction of yield energy appearing as waves is only a few per cent for the largest tests so far conducted, we are faced with the problem of trying to distinguish very small energy differ- ences in normalizing analytic models to actual experiments. While the present models provide adequate predictions for the largest waves over an impressive range of yields (0.5 - 64,000,000 lbs TNT equivalent), it is recognized that important phenomenological factors, such as atmospheric pressure, shock interaction, and cavity stability have been neglected, each of which can reasonably be expected to influence wave formation to some extent. What is really surprising is that such simple models work as well as they do, considering the great complexity of the process of explosive wave generation. 89 Le Méhauteé ACKNOWLEDGMENT The writer has had the opportunity of collaborating with a number of researchers who have deeply contributed to establishing the present state of the art. The original contributions of Dr. Li-San Hwang, Manager of the Hydrodynamics Group at Tetra Tech, Inc. and Mr. David Divoky have been of invaluable assistance in assembling and editing this material and verifying formulation and notation. Dr. William Van Dorn of Scripps Institution of Oceanography and Mr. Robert Whalin of the Waterways Experiment Station have also significantly contributed to the contents. Dr. Van Dorn revised this manuscript and made many most pertinent suggestions. Mr. John Strange provided the writer with the set of experimental data on wave generation obtained by the Waterways Experiment Station. This study was sponsored by the Office of Naval Research, Contract No. N00014-68-C-0227, under the technical management of Mr. Jacob L. Warner. REFERENCES Chabai, A. J., "On scaling dimensions of craters produced by buried explosives," J. Geophys. Res., vol. 70, no. 20, pp. 5075- 5098, 1965. Chan, R. K. C., Street, R. L. and Strelkoff, T., "Computer studies of finite amplitude water waves," Tech. Report No. 104, Stanford University, ONR Contract No. Non 255(71)NR-62-320, June, 1969. Hwang, L.-S., Fersht, S. and Le Méhauté, B., "Transformation and run-up of tsunami type wave trains on a sloping beach," Proc. 13th Congress I.A.H.R., vol. 3, pp. 131-140, 1969. Kajiura, K., "The leading waves of tsunami," Bull. of Earthquake Res. Inst., vol. 41, pp. 535-571, 1963. Kranzer, H. C. and Keller, J. B., "Water waves produced by explosions," J. App. Physics, vol. 30, no. 3, 1959. Kreibel, A. R., “Cavities and waves from explosions in shallow water," URS Research Co., Report No. URS-679-5, DASA Contract No. N0014-67-C-045, 1968. Mader, C. L.., "Fortran BKW: a code for computing the detonation properties of explosives," Los Alamos Scientific Laboratory of the University of California Report No. LA-3704 under Atomic Energy Commission Contract No. W-7405-ENG. 36, 1967. 90 Exploston-Generated Water Waves Le Méhauté, B., "Explosion-generated water waves," Advances in Hydrosciences, Academic Press, New York (publication pending), 1971. Pace, C. E., Whalin, R. W., Sakurai, A. and Strange, J. N., "Surface waves resulting from explosions in deep water," Report No. 4, Waterways Experiment Station, Vicksburg, Mississippi, 1969. Snay, H. G., "Hydrodynamic concepts selected topics for underwater nuclear explosions," NOL TR 65-52, DASA-1240-1(2) U.S. Naval Ordnance Laboratory, September 15 (AD-803-113), 1966. Stoker, J. J., "Water waves," Interscience Publishers, Inc., New York, New York, 1957. Whalin, R. W., "Contributions to the Mono Lake Experiments," NESCO Report S 256-2, ONR Contract No. Nonr-5006(00), 1965. Whalin, R. W., "Research on the generation and propagation of water waves produced by underwater explosions (U)," National Marine Consultants Report NMC-ONR-64, Part II: A Prediction method (CONFIDENTIAL), 1965. 91 J Bo ee ter a bate ren gun wotee Ty oe iaoard) Ara os eaots Simeshask. | aeowes: 280 TYE j ; ri 2 r j ehee d wri s3 we . ‘ Set ee nsl Be : ‘ _ - en ih | a 4 “ det, 4b Aa Ca ay bh ay SN Seated) Oa © Ky? : se Rercie be etree cnqiiete bike. 5. ste oN els cst LenstWl We Ds 5 . 50 RY: gydag eeocpercey ak hr) a Let TmeR LST, BANE MA ensiaue “a guid eMoar |, OtdE APO PEERS Pe ks PS! Pie sees i a? 4) -olggsidaa 201M, , " + bhi Veo pay BER] ty} ' TY ve 41 3 (hie, BG). ADA CoA phar oligo fee9, at thei gy Pad Ware? Joie cok ts Sok Oe a proach) syarmaterGD 4 at ; | . 7 r yal “yiet 2ha Hush pono, egy att .2ovaem Toi "eel a a | t i SA) Hxok wok ~ehto twat ; ral 1 : : 7 f nel hf , $ t ; 7 o bey >, , Sts, iv ™ | lag sh g MIE rt) TOs H oat orerk Sait i rpoohwd isto en W ht | Mey gs ‘ 14 | tn, wee vo Tuts - - 3 a . ) ‘ (GQ9G0UE -t hows sat S9R1 INOW. ANE OS ch groan QO023M4 ait 2a! Yeo,ew 215 THe oe getg bre AOL ate NSH 22 mw to7_,e 2a FH LW OF ct : ” ead Aas 1 eS 0 he Wa tee ee ae aR AT - or ROYS foitoibe | L ty tthh tmet y Apo Nr its ut aj a! i - 4 ; We ‘ 4 i i j ‘ " H fh i P ; eo i j 7 i ig? Cy a Ae f ‘ | ie aa } . Vewlbe 1 q i ; pot pee *, " ' r F ; in ri : 7 4. : yon eae 4 — aa bi i iy ele ! £ Lo.) i 5 t ieut : te cS \ 5 ee . * j s an i L4 i Ee Sas he rT f « f ih 1S i my hae Ui Bh ged in} ATi) j r H j f & r ee ; a f A x his a j ; fh r pe HYDRODYNAMICS IN THE OCEAN ENVIRONMENT Monday, August 24, 1970 Afternoon Session Chairman: J. Wehausen University of California, Berkeley Page Resonant Response of Harbors (The Harbor Paradox Revisited) 95 Jie We Miles, University of California, San Diego Unsteady, Free Surface Flows; Solutions Employing the Lagrangian Description of the Motion 117 C. Brennen, A. K. Whitney, California Institute of Technology Two Methods for the Computation of the Motion of Long Water Waves -- A Review and Applications 147 R. L. Street, R. K. C. Chan, Stanford University, and J. E. Fromm, IBM Corporation An Unsteady Cavity Flow 189 D. P. Wang, The Catholic University of America 93 RESONANT RESPONSE OF HARBORS (THE HARBOR PARADOX REVISITED) John W. Miles Untversity of Californta San Dtego, California I. INTRODUCTION We consider the surface-wave response of a harbor to a prescribed, incident wave in an exterior half-space on the hypo- thesis of linearized, shallow-water theory, an ideal fluid, anda narrow mouth, invoking the equivalent-circuit techniques that have proved so efficient in attacking analogous problems in acoustics and electromagnetic theory. These techniques offer significant advantages in practice: (i) the sub-problems of external radiation, channel coupling, and internal resonance may be attacked separately; (ii) the equivalent-circuit parameters may be expressed as homo- geneous, quadratic forms that may be simply approximated without solving the complete boundary-value problem; (iii) observed values (including those from model experiments) of dominant parameters, such as resonant frequencies, may be incorporated in preference to, or in place of, theoretical values; (iv) empirically determined dissipation parameters (resistances) may be incorporated; (v) ana- log computation, both conceptual and electrical, may be invoked to expedite understanding of the resonant response. Referring to Fig. 1, we consider a harbor H that opens to the sea through a narrow mouth M ina straight coastline, x =0. Let Ci(x;y) =5V, exp {- jk (x cos 0, + y sin 6; )} (1.4) br =¢; (- x,y) (1.2) be the complex amplitudes of the incident and specularly reflected A more detailed version of this work has been published elsewhere [ Miles, 1971]. 95 Miles Fig. 1. Schematic diagram of harbor opening on straight coast line; ¢€;, €; and O, are, respectively, the incident, specularly reflected, and scattered waves (from x = 0) waves on the hypothesis of the monochromatic time dependence exp (jwt), where € denotes free-surface displacement (we omit the modifier complex amplitude of throughout the subse- quent development), k is the wave number, and Vj = 2¢;(0,0) is a measure of the excitation of the harbor through M. By narrow, we imply a/R << 1 and ka<<1i, (2. 32,6) where a is the width of M, and R is a characteristic dimension of H. These restrictions imply that the motion within H is small, and that the energy of the motion induced by V; (or, more precisely, by the pressure pgV,) is dominantly kinetic and concentrated near M (the narrowness of which implies locally high velocities), except in the spectral neighborhoods of the resonant frequencies of the harbor. An appropriate measure of this dominant motion is the flow through M, say I, which, by hypothesis (linearized theory), must be simply proportional to V;. We regard Vj; and I as the voltage and current at the input terminals of an equivalent circuit and seek a description of the resonant response of the harbor in terms of the voltages induced in this equivalent circuit. The input impedance, Z; = V,/I, for the configuration of Fig. 1 may be resolved (see Fig. 2a) into a series combination of a radiation impedance, Zu= Rat jXy- and a harbor impedance, Zy= jXy» where 1 i XI tl? /o, and X,|I|*/w are respectively proportional to the power radiated from i through M (in the form 96 Resonant Response of Harbors (The Harbor Paradox Revistted) of a scattered wave, (,), the non-radiated energy stored in the ex- terior half-space, and the energy stored in the harbor (we also could incorporate an empirical, resistive component in Z,, say Ry, to account for an energy dissipation proportional to Ry tls). We infer from the solution of the corresponding acoustical radiation problem [ Miles 1948; §3 below] that both Ry and Xy are bounded, positive- definite functions of w, by virtue of which we may regard them as single resistive and inductive elements, respectively (although neither Ry nor Xy has the same frequency dependence as its elementary, electrical counterpart). We infer from the analogy with the corre- sponding acoustical resonator [ Morse 1948, §23] that Zy comprises an infinite sequence of parallel combinations of inductance L, and capacitance C,, which bear a one-to-one correspondence to the natural modes of the clos, harbor and resonate at the corresponding frequencies, w, = (L,C,) , together with a single capacitor Cg, which corresponds to the degenerate mode of uniform displacement, for which wo= 0. The solution within H may be expanded in this infinite set of modes, with the root-mean-square displacement and the kinetic and potential energies in the n'th mode being proportional to the voltage across C, and the energies stored in Ly and Cp, respectively. The arguments of the preceding paragraph suggest that the individual modal impedances are important only in the neighborhoods of their respective resonant frequencies, and hence that Z,may be approximated in the neighborhood of w=wW, bya lumped inductance, say Ly, in series with either Cg or the single, parallel combination of L, and C,, such that the energy in all modes but the n'th is proportional to L [I “. The corresponding equivalent circuit is shown in Fig. 2b (we give a quantitative derivation of this equivalent circuit in §§2 and 3). Fig. 2. Equivalent circuit for harbor opening directly at coastline: (a) implied by (3.2); (b) implied by (3.2) and (4.6). 97 Miles The voltage-amplification ratio, « |v,/V,|, provides a measure of the resonant response in the cee ere of w = w,. The zero'th mode, in which the harbor acts like a Helmholtz reso- nator, is unique in that the equivalent circuit reduces to a series combination of Ray Lyt ly, and C, and exhibits a simple, series- resonant behavior path a resonant frequency, say Wp, that is deter- waar by a balance between the potential energy stored in H, 2 ce es the kinetic energy stored in the vicinity of M, = f° + ‘( The results for the rectangular harbor Lites and Munk 19 peaoce that the sharpness of the Helmholtz resonance is ec me by = {log (R/a)} (1.4) and that oe o(s!/%y, ae ="O(1/- 6): and Qo= O(1/5) (1.5a,b,¢)} as a/R > 0, where Gq. is the peak value of @ ae and Q, is the ratio of the resonant Goceeney, to the half- -~power bandwidth of the resonance curve for the n'th mode. The resonant response of the harbor in the higher modes is strikingly different than that of a simple, series-resonant circuit in consequence of the proximity of the parallel-resonant frequency, Wp, at which Z; = oo, and the series-resonant frequency, w,, at which |Z, | has a minimum and @, = >> 1. We show in §4 that @,= , + O(8), G,= O(1/8), and Q,=0(1/8) (n# 0) (1.6a,b,¢) It follows from (1.5) and (1.6) that narrowing the harbor mouth does not affect the mean-square response to a random excitation in the spectral neighborhood of w= w, (which response is proportional to @, to afar if the bandwidth of the random input is large compared with ” Bt_) "except in the akg We mode, but that the response in that mode increases inversely as 6! Miles and Munk [1961] overlooked the proximity of parallel and series resonance in the higher modes and arrived at the erroneous conclusion that narrowing the harbor mouth would increase wan /O,, for all modes, rather than only the Helmholtz mode, and designated the phenomenon as "the harbor paradox." In fact, as pointed out by Garrett [1970], this qualitative conclusion is inconsistent with their quantitative results, which actually imply (1.6) for the higher meds in a narrow rectangular harbor. Garrett also showed that Ga 2/6) is similarly invariant for excitation of a circular harbor through an open bottom and correctly conjectured that the result holds generally for the higher modes in any harbor. In brief, the harbor paradox originally stated by Miles and Munk 98 Resonant Response of Harbors (The Harbor Paradox Revisited) holds only for the Helmholtz mode and otherwise must be replaced by the weaker paradox that narrowing the harbor mouth has no effect on the mean-square response of the higher modes to a random input in the absence of friction (narrowing the mouth increases friction, thereby decreasing the response, in a real harbor). It follows that the higher modes are not likely to be strongly excited, but that the Helmholtz mode may dominate the response of a harbor to an exterior disturbance that has significant energy in the spectral neighborhood Of Wo. Carrier, Shaw and Miyata [1970] consider a harbor that communicates with the coast through a narrow canal and find that both @, and Qo are significantly increased (as might be inferred from the analogy with the classical Helmholtz resonator; cf. Rayleigh [1945], §307). We show in §5 that such a canal is analogous to an electrical transmission line and may be replaced by a symmetrical, four-terminal network for the calculation of V, (see Fig. 3). The analogy with the transmission line rests on the hypothesis that only plane waves are excited in the canal. An examination of the effects of higher modes shows that the elements of the four-terminal network may be appropriately generalized, but that the plane-wave approxi- mation is likely to be adequate if the breadth of the channel is less than a half-wavelength. Fig. 3. Canal and equivalent circuit for the plane-wave approximation. The impedances Z,, = Z,, and Z,, are given by (5.4) The precise determination of Z, and Z,, requires the solu- tion of an integral equation for the normal velocity in M (or, in the case of an intervening canal, a pair of integral equations for the normal velocities across the terminal sections of the canal). The formulation of §§2 and 3 yields variational approximations to Zy and Z, that are invariant under a scale transformation (i.e. a 99 Miles change in the mean value) of the velocity in M and stationary with respect to first-order variations of this velocity about the true solu- tion to the integral equation (cf. Miles and Munk [1961] and Miles [ 1946, 1948, 1967]; we omit the explicit formulation of the integral equation and further discussion of the variational principle in the present development). The resulting representation of Zy is rela- tively insensitive to the geometry of H and yields a simple, ex- plicit approximation that depends essentially only on ka. The cor- responding representation of Z, requires Green's function (subject to a Neumann boundary condition) for the closed harbor, the explicit, analytical construction of which is possible only for those boundaries (rectangular, circular or circular-sector, and elliptic or elliptic- hyperbolic sector) that permit separation of variables; however, we may infer the matrix representation of this Green's function for a polygonal approximation to an arbitrarily shaped harbor from Lee's [1971] collocation solution of the general problem. We give explicit results for a circular harbor in §6 with special emphasis on the Helmholtz mode. It appears from these results that a large harbor with a short entrance or a small harbor with an entry canal of length comparable with R may resonate in the Helmholtz mode under tsunami excitation. II HARBOR IMPEDANCE Let x and y be the Cartesian coordinates in the free sur- face, t the time, w the angular frequency,’ h the depth, 1/2 c = (gh) and k =" (2.1a,b) the wave speed and wave number, ¢€ the free-surface displacement, a the x-component of the particle velocity, 6 and u the corre- sponding complex amplitudes, such that {£(x,y,t),i(x,y.t)} = @[{0(x,y),ulx,y)}e”’], (2.2) where ® implies the real part of and j=y-1, r= udS (dS = h dy) (2.3) M the flow through M, Vee a u* av ) ae dy (2.4) * a weighted measure of the displacement in M, where u_ is the 100 Resonant Response of Harbors (The Harbor Paradox Revistted) complex conjugate of u, Zy= V/L= n| i‘ u ay|* i tu* dy (225) the harbor impedance, and * * P= ZR (ogn J tu dy} =z pgR(VI ) (2. 6) the rate at which energy flows through M. We may regard eV, 6I, (a/B)Z,,, and aPR(VI") as the voltage, current, impedance, and power in an equivalent electrical circuit, where the constants of proportionality, @ and £6, may be chosen to obtain convenient electrical units. The choice a=86=1 is implicit in the discussion in §1, but not in what follows except as noted. Solving the shallow-water equations (Lamb [1932], §189) for an assumed velocity in M, subject to the boundary condition that the normal derivative of €, n° V6, vanish on B, the lateral boundary of the free surface in H, we obtain &(x,y) = (jw/g) \ G(x,y30,n)u(0,n) dy (2:7) where G(x,y3,7) = » (ke - Ky! Walxs yal >7)> (2.8) is the point-source Green's function for H, the w, are the nor- malized eigenfunctions for the closed harbor, and the summation is over the complete set of these functions. The wy, are real and satisfy (v2 + 1) Up 0) (x,y in Hi); (2. 9a) (n> V)b, = 0 on B, (2. 9b) and i. Yn, dA = 6, (2. 9c) where k, are the eigenvalues (resonant wave numbers), and 6mn is the Kronecker delta. We designate the degenerate (but non-trivial) Miles solution corresponding to w= const. by n= 0: me eO) oe Ae (2. 10) where A is the area of H. We also note that more explicit results may require the use of two indices to count off the individual modes, The exact determination of the assumed velocity u(0,y) requires u and ¢ to be matched across M to the corresponding solution of the exterior boundary-value problem (see §3 below). This matching condition yields an integral equation for u(0,y), the exact solution of which in finite terms does not appear to be possible; however, simple approximations to u(0,y) are capable of yielding excellent approximations to Zy and Z, by virtue of the associated variational principle (cf. Miles [1946, 1948, 1967] and Miles and Munk [1961]). We proceed directly to such approximations by intro- ducing the normalized trial function f(y), such that HO ,Alasell /AY Ely), (sy) dgtayits (2.14a,b) M In the subsequent development, we neglect the dependence of f(y) on k and assume that it depends only on the geometry of M. The validity of this approximation, which also implies that f(y) is real, depends essentially on the antecedent approximation ka << 1. Substituting (2.11) into (2.4) and (2.7), combining the results in (2.5), and invoking (2.8), we obtain v=! tf * ay (2.12) M and Zi » Lins (2.13) n where . 2 . p- Ww a w n a (geallf, wel = (gaa). ee is the modal impedance, and pp is a dimensionless measure of the excitation of the n'th mode through M (note that p,o=1 and Z, = 1/jwA). The Z, in the equivalent circuit appear in series, Z, as a capacitor, and each of the remaining Z, asa parallel combina- 102 Resonant Response of Harbors (The Harbor Paradox Revisited) tion of an inductor and capacitor, L,= tin/(weA) and C,= A/\in. The dominant terms in Zy as w—~0O are Zo and the sum of the inductive reactances obtained by neglecting w* relative to wa in the remaining Lane III. RADIATION IMPEDANCE The solution of the shallow-water equations in the exterior half-space (x <0) for a prescribed incident wave, say Ci(x,y), and the assumed velocity u(0,y) in the harbor mouth is given by [ Miles and Munk 1961] C(x,y) = O(x,y) + O)(-x,y) + O.(x.y), (3. 1a) where t.(x,y) = - 3 (w/g) | He [klx? + fy-nl?)”?Juio,n) an (x = 0), (3. 1b) He is a Hankel function, the first two terms on the right-hand side of (3.1a) give the solution for total reflection from the plane x = 0 (as would occur if M were closed), and ¢, is the scattered wave. Substituting u into (3.1) from (2.11), setting x = 0, and then sub- stituting the result into (2.12), we obtain Vee Vi 2/23 (3.2) where Vi = 2{ t.f* dy (3. 3a) M = 26.(0,0) (ka << 4) (3. 3b)! is the equivalent exciting voltage of the incident wave, and Za $ (w/e) | Hy (kly-n|)£ (y(n) dn dy (3.4) M“M The definition of Vi implicit in (1.1) corresponds to the approxi- mation (3.3b). 103 Miles is the radiation impedance of the harbor mouth. The equivalent circuit corresponding to (3.2) is sketched in Fig. 2a. The velocity distribution in M for ka<< i corresponds to that for potential flow. Normalizing this distribution according to (2.11b), we obtain fly) = 0 [ay -yV* (ly| Oe, (4.2) n n where Gul) = pa [Va/Vi | = wae |Zn/(Zyt Z| (4.3) is the amplification factor for the n'th mode, and Kk = k°A = w(A/gh) (4.4) 104 Resonant Response of Harbors (The Harbor Paradox Revisited) a a dimensionless measure of (the square of) the frequency (similarly, BA). 2 Invoking (3.3b) on the hypothesis a PK << 1, we obtain of = mraty | for the (temporal) mean-square elevation of 2Ci, by vintue of le (4.2) reduces to =o?) Qk). (4.5) 2 The hypotheses (1.3a,b) imply |Z,| << + Z| for each of the modal impedances in the summation of (2.13) except in the neighborhood of kK = K,, where the sum may be approximated by ZyF (jo /c*) [A +p,(x, - «71, (4. 6a) where A, = y Barlen? (4. 6b) m=0 being excluded from the summation. Invoking (2.14), (3.6), and (4.6) in (4.3), we obtain Gol) = fh u? + [ado(a) - 1]*5°V? (4. 7a) and Gn(ic) = wl? $2 (ie = wal +L - tg - wel f?, (4. 7b) where A(x) = Ay + Aj(ka), (4. 8a) A, = A,+Aj(k.a) (n #0). (4. 8b) The peak values of Gy, are given by mel -I/2 Go = 2Ko and G,= 2n, A, (n # 0), (4. 9ayvb) where K = he is the series-resonant point determined by ~ ~ ~ -| KyA(K,) = 1 and Ky, = K, + pA, (n # 1). (4.10a,b) ~ The amplification factor drops off sharply on both sides of K = K, 105 Miles andis O(1/A,) for |K - k,| >> 41/A,. The point kK =k, corresponds to parallel resonance (Zp =o), for which the total flow through M vanishes (I = 0) whilst o* remains of the same order as o;. We define the Q of the resonant response near K = Kp as the ratio of the resonant frequency to the half-power bandwidth, such that [ the frequencies at the half-power points are proportional to fra (143 Q:')] G[xK,(1 + Gr )] = GL (4.11) Substituting (4.7) into (4.11) and invoking (4.10), we obtain the first approximations ~-| ~ Qo = 2k =G, (4. 12a) and il 2 4~-,2 Q, = 2h, KA, = 2 KAG,- (4. 12b) Now suppose that the incident wave is random with the power spectral density Sj(f), such that © oe - S| (£) df (w = 2rf), (4.13) 0 where f is the frequency. Generalizing (4.5), we obtain 2 c=) f S,(£) |G,(«) | af (4.14) fe] n for the power spectral density in the harbor. Substituting (4.7) into (4.14), invoking w= ck/VJA, and calculating the contribution of the resonant peaks at ™ = w, on the hypothesis that their bandwidths are small compared with those of S,(f), we obtain o° = (gh/a)'” », PS) (Eq) (4.15) n where Be ee a Ghd 2 ye aol P, = (4m) Ky an [it (O,/Ka) (kh - Re dd (4. 16a) 0) - 5 Pol Gr (Qn/ ha oO) (4. 16b) 106 Resonant Response of Harbors (The Harbor Paradox Revisited) is the power-spectrum-amplification factor for the n'th mode. Sub- stituting (4.9), (4.10) and (4.12) into (4.16b), we obtain Pp =4Ki”, (4.17) from which we infer that narrowing the harbor mouth does not affect the mean response to a random input except in the Helmholtz mode, but that it does increase significantly the response in that mode [ this conclusion ignores the increase in viscous dissipation that would be associated with narrowing the mouth]. V. EQUIVALENT CIRCUIT FOR CANAL We now interpose a cana of breadth b and length £ between the harbor and the coast, as shown in Fig. 3, and obtain the equivalent circuit on the assumption that only plane waves need be considered in the canal. This approximation is strictly valid only for kb << 1, but a more complete analysis shows that the effects of the cross- waves (y-dependent modes) are not likely to be significant for kb 3 to the right of the dashed line. Resonant Response of Harbors (The Harbor Paradox Revisited) Fig. 7. Power-spectrum-amplification factor for Helmholtz mode in circular harbor. Kol > $ to the right of the dashed line. Miles 1000 (m,s) = (0,1) 300 PF 100 Qms 30 Fig. 8. Q,. for the first five modes ina circular harbor. The dashed portions of the curves correspond to ka> 1 114 Resonant Response of Harbors (The Harbor Paradox Revisited) The period for the Helmholtz mode is given by T, =r,/¢ = an(A/gh)? KY? (6.7) Choosing R=1000' and h = 20', we obtain Ty = 2\,/rR minutes, which approximates typical tsunami periods (20 - 40 minutes) for \,/2mR in the range of 5-10 (see Fig. 5). We infer that a large harbor with a short entrance (£/R << 1) or asmall harbor witha canal (£/R ~ 0.3-3) may act as a Helmholtz resonator under tsunami excitation. REFERENCES Bartholomeusz, E. F., "The reflexion of long waves at a step," Proc. Camb. Phil. Soc., 54, 106-18, 1958. Carrier, G. F., Shaw, R. P. and Miyata, M., "The response of narrow mouthed harbors ina straight coastline to periodic incident waves," J. Appl. Mech. (in press), 1970. Garrett, C. J. R., "Bottomless harbors," J. Fluid Mech., 43, 443-49, 1970. Lamb, H., Hydrodynamics, Cambridge University Press, 1932. Lee, J. J., “Wave induced oscillations in harbors of arbitrary shape," J. Fluid Mech., Boy (2-93, 1975. Miles, J. W., "The analysis of plane discontinuities in cylindrical tubes," Parts Tand II. J. Acoust. Soc. Am., 17, 259-71, 272-84, 1946. Miles, J. W., "The coupling of a cylindrical tube to a half-infinite space," J. Acoust. Soc. Am., 20, 652-64, 1948. Miles, J. W., "Surface-wave scattering matrix for a shelf," J. Fluid Mech.., 23, 755-67, 1967. Miles, J. W., "Resonant response of harbors: an equivalent-circuit analysis," J. Fluid Mech., 46, 241-65, 1971. Miles, J. W. and Munk, W. H., "Harbor paradox," J. Waterways Harb. Div., Am. Soc. Civ. Engrs, 87, 111-30, 1961. Morse, P. M., Vibration and Sound, New York: McGraw-Hill, 1948. Rayleigh, Lord, Theory of Sound, New York: Dover, 1945. pa ‘ dy Dilacas its ail on ia eS a rikoonw UNSTEADY, FREE SURFACE FLOWS; SOLUTIONS EMPLOYING THE LAGRANGIAN DESCRIPTION OF THE MOTION Christopher Brennen, Arthur K. Whitney Caltfornta Institute of Technology Pasadena, Caltfornia ABSTRACT Numerical techniques for the solution of unsteady free surface flows are briefly reviewed and consideration is given to the feasibility of methods involving param- etric planes where the position and shape of the free surface are known in advance. A method for inviscid flows which uses the Lagrangian description of the motion is developed. This exploits the flexibility in the choice of Lagrangian reference coordinates and is readily adapted to include terms due to inhomogeneity of the fluid. Numerical results are compared in two cases of irrotational flow of a homogeneous fluid for which Lagrangian linearized solutions can be con- structed. Some examples of wave run-up on a beach and a shelf are then computed. I. INTRODUCTION There are many instances of unsteady flows in which analytic solutions , even approximate ones, are not available. This is par- ticularly true of free surface flows when, for example, non-linear waves or even slightly complicated boundaries are involved. Though analytical methods are progressing, especially through the use of variational principles (Whitham [1965]) and, in some cases, the non-linear shallow water wave equations yield important results (Carrier and Greenspan [ 1958]) there is still a need for numerical methods. Indeed, numerical "experiments" can be used to comple- ment actual experiments. Until very recently numerical solutions in two dimensions £17 Brennen and Whitney invariably seemed to employ the Eulerian description of the motion though the Lagrangian concept has been used for some time in the much simpler one-dimensional case (e.g. , Heitner [ 1969], Brode [ 1969] ) and to make small time expansions (Pohle [1952]). Perhaps the best known of these Eulerian methods is the Marker-and-Cell technique (MAC) begun by Fromm and Harlow [ 1963] and further refined by Welch, et al. [1966], Hirt [1968] , Amsden and Harlow [1970] , Chan, Street and Strelkoff [1969] and others. The most difficult problem arises in attempting to reconcile the initially un- known shape and position of a free surface with a finite difference scheme and the necessity of determining derivatives at that surface. In the same way, few solutions exist with curved or irregular solid boundaries. In steady flows, mapping techniques have been em- ployed to transform the free surface to a known position (e.g. , Brennen [ 1969]). It would therefore seem useful to examine the use of parametric planes for unsteady flows. The Lagrangian description in its most general form (Lamb [ 1932]) involves such a plane and by suitable choice of the reference coordinates, the free surface can be reduced to a known and fixed straight line. However a discussion of other parametric planes and mapping techniques is included in Section 3. The major part of this paper is devoted to the development of a numerical method for the solution of the Lagrangian equations of motion in which full use is made of the flexibility allowed in the choice of reference coordinates. For the moment, we have restricted ourselves to cases of inviscid flow. Very recently, Hirt, Cook and Butler [1970] published details of a method which employs a Lagrangian tagging space but is otherwise similar to the MAC tech- nique. This is further discussed in Section 4B. Il. LAGRANGIAN EQUATIONS OF MOTION The general inviscid dynamical equations of motion in Lagrangian form are (Lamb [ 1932]): < Ya Ze P, 1 (Xy-F) | Xbp +(%y-G) P Yep (2-H) Zoe +S 7 Pep =o (1) pal Y, Ze P, where X,Y,Z are the Cartesian coordinates of a fluid particle at time t, F, G, H are the components of extraneous force acting upon it, P is the pressure, p the density and a, b, c are any three quantities which serve to identify the particle and which vary con- tinuously from one particle to the next. For ease of reference (X,Y,Z) are termed Eulerian coordinates, (a,b,c) Lagrangian co- ordinates. Suffices a,b,c,t denote differentiation. Lagrangtan Solutions of Unsteady Free Surface Flows If Xo, Y,, Z, is the position of a particle at some reference time ty (when the density is p,) then the equation of continuity is simply Q(X,Y,Z) _ , (Xp, Yo, Zo) (2) Ae eter, Frequently it is convenient to define a, b, c as identical to Xp», Yo, Zo; thus reducing the R.H.S. of (2) to p 9; however it will be seen in the following sections that flexibility in the definition of a, b, c is of considerable value when designing numerical methods of solution. If the extraneous forces, F, G, H, have a potential 922 and p, if not uniform, is a function only of P then, eliminating 2 + P/p from (1): 0 ee ar (UpX, - U.X, + V,Y, - V-¥, + W,Z, - W,Z,) =—b= 0 ) oT: Br (UcXq- UgXe +t VeY, - Va¥_ + WyZ, - U,Z,) = ae = 0 (3) ) oT 5-H (UgX, - UpX, a Vows = Vii%e Le Wo2p - W, 2Z,) i “5 = 0 where, for convenience, the velocities Xy> Y,> Z, are denoted by U, V, W. The quantities [,, [,, [3 are related to the Eulerian vorticity components, 6,, 5, 63 by PD, = 6,(¥,2, - ¥,2,) + (2.x, - ZX) + O,(X,¥, - X,¥,) Ty = 6)(%Zq - YgZ_) + SA(Z,Xq - ZX.) + 05(X, ¥, - X,Y.) (4) r; 7 CY oZp ~ YpZ) C(2,X, - Z,X,) " C3(X,Y, e X,Y) (Thus, of course, vorticity changes with time are due solely to changes in the coefficients of the L.H.S. of (4) which, in turn, represents stretching and twisting of the vortex line.) Given the vorticity distribution €(X,Y,Z) at some initial time, t,, T(a,b,c) (which is independent of time) may be obtained through Eqs. (4) and used in the final form of the dynamical equations of motion, namely Eqs. (3) integrated with respect to time. ts Brennen and Whitney For incompressible, planar flow the equations reduce to Continuity: XqYp- YgXp = F(a,b) (5) (or differentiated w.r.t. t): U,Y,- U,Yy ete VaX, = Vix. = 0 (6) Motion: UX, - U,X, + UA Ss - Vee =) = ia.) (7) By introducing the vectors Z=X+t+iY and W= U - iV, (6) and (7) conveniently combine to: ZW», = Zp Wa i I(a,b). (8) Other types of flow have also been investigated. For example, in the case of a heterogeneous, or non-dispersive stratified liquid in which p is a function of (a,b), Eq. (8) becomes: t ! | ZaWy~ ZpWq =~ (T(asiTaey, ~ 5) (Xp - FMP, ~ PG) te] + (4, - G)(p,¥, - p,¥,) odtee M9) The integral term therefore manufactures vorticity. The methods developed for a homogeneous fluid in Sections 4A to D are modified in Section 4E to include such effects. III. OTHER PARAMETRIC PLANES It may be of interest to digress at this point to consider other parametric planes (a,b), which are not necessarily Lagrangian. That is to say the restrictions X,(a,b,t)=U, Y,(a,b,t) = V are abandoned so that U,V are no longer either Eulerian or Lagrangian velocities. Provided J = 8(X,Y)/8(a,b) # 0, or ow, the equation for incompressible and irrotational planar flow remains ZW, - Z,W, = 0- (10) To incorporate one of the advantages of the Lagrangian system, it is required that the free surface be fixed and known, say on a line of constant b. Then the kinematic and dynamic free surface conditions are respectively 120 Lagrangian Solutions of Unsteady Free Surface Flows (U =. X%,)¥,-.(V - YK 50 (11) (U, + F)Xq + (Vz; + G)¥, + (U - X,)U,g - (V - ¥4)Vq = 0. (12) Now a useful choice concerning the (a,b) plane would be to require the mapping from (X,Y) to be conformal. Then, of course, (10) simply reduces to the Cauchy-Riemann conditions Ug, = - Vp, U,p= V, sothat W =U - iV is an analytic function of c=a+tib or of Z. In this way, John [1953] has constructed some special, exact analytic solutions. The kinematic condition, (11), has the particular solution W(a,t) = Z,fa,t) on.the free surface, which implies W(c,t) = Z,(c,t) by analytic contingation. If, in addition, + (F +iG) = iZ,K(c,t) (13) where K is real on the free surface, then the dynamic condition thereon is also satisfied. John discusses several examples for various choices of the function K. The potential of such methods may not have been fully realized either analytically or numerically. In the latter case, however, the conformality of the (X,Y) to (a,b) mapping is not necessarily a great advantage, whereas a fixed and known free surface position most certainly is. The digression ends here and the following sections develop a Lagrangian numerical method from the equations of Section 2. IV. A NUMERICAL METHOD EMPLOYING LAGRANGIAN COORDINATES A method for the numerical solution of incompressible, planar flows is now described. It attempts to take full advantage of the flexibility in the choice of Lagrangian coordinates. A. Time Variant Part The method uses an impiicit scheme with central differencing overtime, t. Thus Z Pia, b) is determined at a series of stations ab riee: distinguished by the integer, p- Knowledge of velocity values, EY , at a midway station p +2 enables y Ated (a,b) to be found from Z’ through the numerical approximation p+l pri Zs, Zt TZy (error order TZeee) (14) 121 Brennen and Whitney where 7 is the time interval. Acceleration values, Zi , needed in the free surface condition (Section 4C) are approximated by (Zo - Zz CAWG (error order 7T4Z 4444). Thus the main part of the solution involves finding P, ‘. knowing Z? Z; and their previous values. The first time step (from p=0 to p=t1) requires a little special attention. Clearly Z °(a,b) is chosen to fit the required initial conditions. But further information is required on a free surface which will enable the accelerations in that condition to be found (see Section 4C). B. Spatial Solution A method of the present type is restricted to a finite body of fluid, S. However, S, could be part of a larger or infinite mass of fluid if an "outer" approximate solution of sufficient accuracy was available to provide the necessary matching boundary conditions at the interface. The region, S, need not be fixed intime. It would indeed be desirable, for example, to "follow" a bore. In a great number of cases of widely different physical ge- ometry including all the examples of Section 6, it is convenient to choose S to be rectangular inthe (a,b) plane. This rectangle (ABCD, Fig. 1) is then divided into a set of elemental rectangles. The motion of each of these cells of fluid is to be followed by deter- mining the Z values at all the nodes. NODE NUMBERING IN FREE SURFACE CONDITION: D a Fig. 1. The Rectangular Lagrangian Space, S, Showing the Numbering Conventions Used 122 Lagrangian Soluttons of Unsteady Free Surface Flows Making the assumption of straight sides the actual area ofa cell in the physical plane is A= 3[(X, - X)(¥Y, - ¥,) - (X,- X,)(¥, - Y3)] (15) Number suffices refer to the four vertices, numbered anticlockwise; other node numbering conventions are shown in Fig. 1. If this area is to remain unaltered after proceeding in time from station p to pti through Eq. (14) then Imag {(Zo - Za)’(W, - We) - (Z, - Zg) (Wp - Wa) } pti/2 Pp. ae + r{(U, - U,V, - V4) - (Up - U,V, - Vz} + 2A AD =0=R, (16) where the terms on the L.H.S., second line are numerical cor- rections required to preserve continuity more exactly and prevent accumulation of error over a large number of time steps. The nu- merical value of the L.H.S. at some point in the iterative solution is termed the continuity residual, Rg. Assuming linear variation in velocity along each side of the cell, evaluating the circulation around 1234 and setting this equal to the known, initial circulation, I, yields (in the case of a homo- geneous fluid): Real {(Z, - Z,)(W, - Ws) - (Z, - Z3)(Wp - W,)} - 20, =0= Ry (17) Slight hesitation is required here since, for validity, the Z and W values in this equation should relate to the same station in time. But by choosing to apply it at the midway stations and substituting zPrve Ze + (r/2)ze" the T terms are found to cancel and (17) per- sists when the values referred to are Z and Ww?*!# R, is the circu- lation residual. The modification of (17) in the case of a hetero- geneous fluid is delayed until section 4E. Combining (16) and (17) produces the cell equation: (Z, - Z,)(W, - W,) - (Z,, - Z,)(W, - W,) Main Part Zar - Aa i Continuity T t+ ir {((U,- U,)(V, - Vay - (U,- U,)(V, - V,)} + Corrections - 21, Permanent Cell Circulation Term 123 Brennen and Whitney 1 + Te {(Wig Wier a)(Z, - Z,) Higher = (Wis ots Wio = W3 - Wz) (Z4 - Z3) Order + (W, + Wi. - We - 3)(Z, - Zs) Correction - (W, + W,, - W, - W,)(Z, - 2,)} if required =0=R,+iR, =R, the cell residual. (18) The higher order correction, included for completeness, allows the shape of the cell sides and the variations in velocity along them to be of cubic form. Without it the neglected terms are of order Z Wppp» ZabWap, etc., with it they are of order Za Wbppbppp etc. Values referred to are Z? and Wt [Ptl2 vPtv2, Though this derivation of the cell equation is instructive, it can be obtained more directly (except for the continuity correction) by integration of (8) over the area of the cell in the (a,b) plane (using Taylor expansions about the center of the cell). p The cell equations must now be solved for weve (W'F="PV); Z being known, in order to proceed in time. In a recently published paper, Hirt, Cook and Butler [ 1970] take a rather different approach in which the (a,b) plane is employed merely as a tagging space. The equations are written in essentially Eulerian terms, no derivatives with respect to a,b appearing. The numerical method (LINC) is similar to that of the MAC technique (Fromm and Harlow [1963], Welch, et al. [1966] , Chan, Street and Strelkoff [1969], etc.) and involves solving for the pressure at the center of a cell as well as for the vertex velocities. Advantages of the method described in the present paper are: the pressure has been eliminated (though this may be disadvantageous in compressible flows); no special treatment is required for cells adjacent to bound- aries; inhomogeneous density terms are relatively easily included. However, since the LINC system is based on the Eulerian equations of motion, the inclusion of viscous terms is more easily accomplished than in the present method where such an attempt leads to horrendous difficulties. C. Boundary Conditions To complete the specifications, a condition upon waits is required at each of the boundary nodes. ot Bis usually takes the form of an expression connecting U and V °. For example, solid boundaries, whether fixed or moving in time, may be prescribed by a function, F(X,Y,t) =0. Then the required relation is 124 Lagrangtan Solutions of Unsteady Free Surface Flows p+v2 F(x? + tu y+ ey 4) = 0 (19) Dynamic free surface conditions are simply constructed from Eqs. (1). If, for example, the only extraneous force is that due to gravity, g, inthe negative Y direction, the condition on a free surface suchas AB, Fig. 1, is XyXq t (Yes + g)Y, = a a 7 (20) a” Nea 3/2 (Xq + Ya) where T is the surface tension if this is required. Unlike the field Eqs. (8) or (18) these boundary conditions may not be homogeneous in all the variables. In a given problem only the boundary conditions are altered by different choices of typical length, h (perhaps an initial water depth), and typical time, say yh/g inthe above example. Then, using the same letters for the dimensionless variables, g and T/p in Eq. (20) would be re- placed by 1 and S = T/pgh’. The numerical form of that condition used at a free surface node suchas 0 (Fig. 1) is: ¥ 2 +1/ = a x)"(Ue i Us ip FAY ay (Ve ies Vo on 7) = 18(R - 5) (24) where F is assessed at each node as _ LOK, - XY, + ¥, - 2¥,) - (, - ¥,MX, + X, - 2X0] ae ee ee ee e E / Cae R e : [oxesany ty wey and the accelerations have been replaced by the expressions given in Section 4A. Again, Eq. (21) relates Up, to Shae since all other quantities are known. If the liquid starts from rest at t =0 (as in the examples of Section 6) then difficulties at the singular point t = 0 can be avoided by choosing to apply the condition at t= 7/4 rather than t =0. Using Zy, = 22)"/r and Z=Z°+ (7/4)zV2 at that station the special boundary condition becomes 1/2 \/2 Us {(X, - Xs)" +F(U, - Uy} + (v, - va)” (vy? + 7g /2) = 0 (22) 125 Brennen and Whitney in the case of zero surface tension. D. Method of Solution It remains to discuss how the equations may be solved to find at every node. Due to the non-linear terms in (18) and some boundary conditions as well as to the fact that a good estimate of w?’*'2 can be made from values at previous time stations, a simple iterative or relaxation scheme was employed. Such a method in- volves visiting each cell in turn and adjusting the W values at its vertices in such a way that repetition of the process reduces the cell residuals, R, to negligible proportions. But, on arrival ata particular cell, there are an infinite number of ways in which its four vertex values can be altered in order to dissipate the single cell residual. However, experience demonstrated that a procedure based on the following changes (AW, 9 3anq4) was superior in convergence and stability to any of the others tested: wti72 AW, = - AW; = wiR(Z, - Z3)/8A (23) AW, = - AW, = wiR(Z, - Z,)/8A Here w is an overrelaxation factor and A is the area of the cell, which is unchanged with time and given by the expression (15). These incremental changes have a simple and meaningful physical inter- pretation. As can be seen from Fig. 2, they are a combination of two changes, one representing pure stretching and the other pure rotation, which dissipate respectively the continuity and circulation components of the residual. Having visited each and every cell, the boundary conditions were then imposed. Where these were Sen in the form A.U?*'/#+ B.V?*!2+ C=0=R,, A,B,C being constants and Rgthe residual, the following ehanges" were made, the choice being based upon experi- ence: p+i/2 AU A| Re se op te (24) Ayetv2 al (A® + B*) The whole process was then repeated to convergence, E,. Inhomogeneous Fluid In a non-dispersive, inhomogeneous fluid, p(a,b), which is independent of time, will be prescribed through the initial choice of Z (a,b). Indeed in many cases it will be convenient to choose Z° in 126 Lagrangian Soluttons of Unsteady Free Surface Flows 3 4 Fig. 2(a). The Cell in the Reference Plane (a,b) ee MQ2=- gq (22-24) 4Q niet (Zaza) 2°! Ga '427 44 R. —_——— AQ, = - gh (Z,-Z3) INCREMENTAL VELOCITY CHANGES INCREMENTAL VELOCITY CHANGES WHICH DISPERSE THE CONTINUITY WHICH DISPERSE THE CIRCULATION RESIDUAL, R_ (PURE STRETCHING) RESIDUAL, R,; (PURE ROTATION) Fig. 2(b). The Cell in the Physical Plane (X,Y) such a way that p is some simple analytic function of (a,b). This is particularly desirable because by substituting for p, pg, py in Eq. (9), this can then be integrated over a cell area (as in Section 4B) to produce a convenient additionalterm, 9 on the L.H.S. of the cell Eq. (18). Since the expression for gre will depend upon that choice of p(a,b) an example will illustrate this. If p is to be constant along the free surface, AB, Fig. 1, and along the bed, CD, it may be possible to choose Z° such that p is a linear function of b, say p= Ppcop(1 + yb) where Y= Pap/Pep - 1 and b=1 on AB. Then, 127 Brennen and Whitney p+i2 p-V/2 Qio34 a G i234 +1/2 a ve! p = -31n (1 - p) [{(y, FULUSSU) ° = 40,40, +0, Fu) p x {X,- X,-X,+X,} + -\/2 Pp +{V, +Vp tV, +¥,)°"'7-(V, +Vp V5 4Vg) ? + amg} {¥, -¥,-¥, +¥%} ] Lt p pri/2 p 2 t+ 2-5 in (t-w} [(x,-x)°(u, 40,0" - (u, tu)? 7} p p+i/2 p-i/2 - (X,- X3) {(U, + U,) a (U;+U,) } + =F + (¥, -¥,) {(v, tv.) = (vy, - Vv, + 27g} +1/ p-i/2 ~ (Xg- Ya) {(Vp + Va) = (Vg +V_) + 278} where p = yAb/(1 + yb3,), b3, being the b value on side 34 of the cell and Ab the difference across each and every cell. The first term is of order p, the second order p*. The boundary conditions are usually identical to the homogeneous case. V. ACCURACY, STABILITY, CONVERGENCE AND SINGULARITIES A. Accuracy If the cell equation, (18), is used without the higher order spatial correction, an indication of the errors due to neglected higher order spatial derivatives can be obtained by assessing the value of that correction and inferring its effect upon the final values of W. Unfortunately, the mesh distribution and mesh size required for a solution of given accuracy will not be known a priori and can only be arrived at either by trial and error or by using some technique of rezoning. The latter method in which cells are subdivided where and when the violence of the motion demands it, can be difficult to pro- gram satisfactorily and has not been attempted thus far. Errors due to higher order temporal derivatives are most fa regulated by ensuring that, for each cell, both T| W, "Wel /IZ, - Z| and T|W,- W,|/|Z,- Z, are comfortably less than unity. A workable rule of thumb can be devised in which a suitable Tt for a particular time step is determined from the W and Z values of the preceding step. 128 Lagrangian Soluttons of Unsteady Free Surface Flows By Stability of Cell Relaxation Suppose the central member, cell A, of the group of cells shown in Fig. 3 contained a residual Ry which was then dissipated according to the relations (23). Transfer functions, Dag, Dg, etc., will describe the residual changes, AR,, etc., in the surrounding cells where AR, = wD,,R,, etc. (24) Fig. 3. Z-Plane For example Dag = {(Z, - Zis\(Zp- Z4) - (Zig - Z4(Z, - Zs) }i/8A, where A, is the area of cell A. For convergence of the relaxation method it is clearly necessary that the w for each cell be chosen so that all w|D| are significantly less than unity. It is instructive to inspect the case in which all the cells are roughly geometrically similar inthe Z plane. Then ld? - a3 Dial = [Daal = Dae = iD ~ ors vam =i 129 Brennen and Whitney dd 4 [Dacl = [Dye] = |Dacl = |Dasl * oa = ao ye where d,, dp are the lengths of the cell diagonals. For square cells, Y, =Y,=0 and the situation is stable. However difficulties may arise when the cells are very skewed or elongated and it is in such situations, in general, that care has to be taken with the relaxation technique. C. Observation on the Cell Equation One feature of the basic cell equation, (18), itself demands attention. Note that without the higher order spatial correction, the residuals, R in all of the cells (of Fig. 3) remain unaltered when the W or Z values at alternating points (say the odd numbered points of Fig. 3) are changed by the same amount. Such alternating "errors" must be suppressed. Some damping is provided by the higher order spatial correction since it is not insensitive to these changes. But experience showed this to be insufficient unless all the boundary conditions also inhibited such alternating "errors." Solid boundaries usually provide adquate damping. For instance, in Fig. 4(a) fluctuations in U on BC, DA andin V on BC are obviously barred. But the free surface provides little or no such suppression and as will be seen in the next section this can lead to difficulties. It is of interest to note that some of the solutions of Hirt, Cook and Butler [1970] exhibit the same kind of alternating errors. In the MAC technique, neglected higher order derivatives of the diffusion type and with negative coefficients (a "numerical" viscosity) can lead to a numerical instability if not counteracted by the introduction of sufficient real viscosity. In the present method, as with that of Hirt, Cook and Butler [1970], the convection terms which cause that problem are not present. The higher order spatial correction does contain terms of diffusion order, but it cannot be directly correlated with a viscosity since viscous terms are ofa 2 different form (i.e., like f vVxy I dt). Also, the higher order spatial correction has a beneficial rather than a destabilizing effect. D. The Free Surface By including previously neglected derivatives, the numerical free surface condition (without surface tension) is found to correspond more precisely to: 2 A {XqXt+ * YolYr+ t 1)} + ier { XeaaX t+ + YoaalYt+ t 1)} 2 4 be 7 {XX t YaYererd = 0 (26) 130 Lagrangtan Solutions of Unsteady Free Surface Flows where Aa isthe a difference across a cell and the second and third terms constitute truncation errors. Inspect this in the light of a linearized standing wave solution (see Section 6A), i.e., X=a-Mcocos Jkt sin ka’e*” iT} Y = bikM cos vik t cos ka ef? where the variables are non-dimensionalized as in Section 4C and k is the non-dimensional wave number in the a,b plane. Then, the second and third terms of Eq. (26) will be insignificant provided 2 2 k (Aa) respectively. Or, in terms of a wavelength, \ = 21/k: R. >> 2 and T<< 45 AX (27) since Aa® AX, the X difference between points on the free surface. The first condition states the inevitable; namely, that the solution will be hopelessly inaccurate for (a,b) plane wavelengths comparable with the mesh-length Aa. Given that the first condition holds then the second says that 7 << 84AX. For a travelling wave system the same condition states that T should be less than the time taken for a wave to travel one mesh length. This constitutes a restriction on T which is usually more stringent than that of Section 5A. If, for example, the depth of the fluid is divided into N intervals andthe X difference across each cell is of the same order as the Y difference then T should be less than 8/N. A more difficult problem arises when the first condition is considered alongside the fact, ascertained in the previous section, that the field equation provides little or no resistance to disturbances whose wavelength is equal to Aa. The only resort would seem to be to some artificial damping technique which would eliminate or sup- press these small wavelengths. The technique used in the examples to follow was to relax the W values on the free surface such that w = pw'?° + (1 - B)W* where W'S° was the value indicated by the free surface condition, W* the value which would make the numeri- cal equivalent of Woagg, be zero at that point and B was slightly less than one half, 1314 Brennen and Whitney E. Singularities Successful numerical treatments of singularities depend upon the availability of analytic solutions to the flow in the neighborhood of that point. For example, at a corner between solid walls the velocity varies as the (1 - $)/B power of distance from that junction where f is the included angle. If this is /2 (as at points C or D, Fig. 4(a)) the variation is linear and thus the numerical estimate of the circulation around the cell (see Section 4) in such a corner is a good one. Where the angle is not 1/2 (D, Fig. 4(c)) errors will occur due to the non-linear variation of velocity, but corrective pro- cedures are easily devised. A great deal less is known about the singularities at a junction of a free surface and a solid boundary. If the wall is static and verti- cal (A, Fig. 4(a)) so that X,, = X,= Xp;= 0, etc., it follows from the equation of motion that if Y, =0 at t=0 then it is always zero for irrotational flow; the tangent to the free surface at the wall is always horizontal. Thus the free surface condition without surface tension is automatically satisfied at such a junction and only weak singular behavior is expected. But a similar analysis of the case when the wall begins to move at t=0 (remaining vertical) indicates that Yt must be infinite at the junction (B, Fig. 4(a)) at t = 0, the singularity being logarithmic in space. An extension to t#0 has not so far been obtained. One approach might be a Fourier analysis of the step in X;; 80 that the steadily oscillating solutions of Fontanet [1961] could be used. These suggest that Y,, becomes finite for t>0. SLOPING BEACH X Mactth Xl amexe FIG. 4(b) FIG. 4(d) 132 Lagrangtan Solutions of Unsteady Free Surface Flows In the examples to follow (see Figs. 4(a) to (d)) satisfactory numerical solutions could be obtained by ignoring all but one of the singularities. The exception was the shoreline, point A, Fig. 4(c). If B is the angle between the tangent to the free surface at A and the horizontal then correlating the two boundary conditions yields: (Zia = - el? /(cot B cos a + sin a) (28) Thus the sign of B determines the direction of the acceleration up or down the beach, If the fluid starts from rest at t=0, B=0, then (Z;+;+)t:0 = 0 and successive differentiation of the basic equation (8) and the boundary conditions yields (for irrotational motion): TT. - ee i Zrists Zate? Spire 2 tate. =O OF ».unless o@ = 3 (29) Z Z Z. =0oro, unless a= "or = ttittt *“atttt > “brttt ? 4 6 These relations suggest a behavior which is logarithmically singular in time at t=0 unless @=n/2n, n integer. Roseau [1958] found similar logarithmic singularities in periodic solutions for the general case which excluded @=1/2n and another set of particular angles (see also Lewy [1946]). But a systematic analysis of the singular behavior (especially for t# 0) has not as yet been completed. Rather, since the relations (29) no longer necessarily hold if the condition of irrotationality near that point is relaxed, the problem was circum- vented numerically by replacing the circulation condition on the single cell in that corner by the condition (28) at the point A and the time = 0 was avoided by applying (28) at t = 7/4 just as was done with the general free surface condition (Section 4C). Note that strong singularities could be introduced by unsuitable mapping to the (a,b) plane. VI. SOME RESULTS INCLUDING COMPARISONS WITH LINEAR SOLUTIONS A. Lagrangian Linearized Solutions Linearized solutions to the Lagrangian equations are obtained by substituting K=at6&, Y=B +n into the equations of continuity and motion and neglecting all multiples of derivatives of § and n. For incompressible and irrotational planar flow the Cauchy-Riemann conditions 6 = - nb, &=Nq result so that € tin, and therefore Z-c (where c=atib) is an analytic function of c. In the absence Brennen and Whttney of surface tension the free surface condition reduces to Ey ten, =0 (g = 1 in the dimensionless variables) (30) only when the additional assumption that 1+ << gis made. In this way harmonic solutions can be obtained for some simple problems. In passing, it may be of interest to compare Lagrangian linearization with the more common Eulerian type, at least in some simple cases. For travelling waves on an infinite ocean the first order Lagrangian terms are precisely those of Gerstner's waves. The Eulerian solution must be taken to the third order to achieve this waveform. On the other hand, while the Eulerian solution is always irrotational the Lagrangian only approaches it. Thus the comparitive accuracy of the two methods depends upon what particular feature of the flow is under scrutiny. A comparison of the works of Zen'kovich [1947] and Penney and Price [1952] for standing waves on an infinite ocean demonstrates the same features. B. Example One, Figs. 4(a),, 551627 9:85 9% and 10 In the example of Fig. 4(a), the liquid is initially at rest in the rectangular vessel BCDA; between t=0 and t=T the side BC moves inward according to 2 X,dt) = M sin nt /2T for Ot Tt eh _ | kr kr)“ Meek ee tank | 134 Eigse 6s Lagrangtan Soluttons of Unsteady Free Surface Flows t/Y¥ = 0.0 8.0 12.0 16.0 20.0 2U.0 26.0 + X o ¢ SYMBOL © © 4 90.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 X Linearized solution to example 1: M=0,53, T= 327, T=0.53, showing free surface position at a selection of times, t. %/¥ = 0.0 8.0 12.0 16.0 20.0 24.0 26.0 + xX o SYMBOL © O 4 MESH 65X9 POINTS 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 Xx Numerical solution to example 1: M=0.53, T=327T, 7 =0.53, showing free surface position at a selection of times, t £35 VAS) 1.4 Fig. 7. 1.0 o2 0.0 Bigs: 8. Brennen and Whitney t/Y¥ = 0.0 4.0 8.0 12.0 16.0 20.0 23.0 26.0 30.0 SYMBOL © © 4 + X © # X& Z —~——--- Se ee ee ee 1.0 2.0 3.0 4.0 5.0 6.0 a0 8.0 9.0 10.0: X Linearized solution to.example 1:.M = 1.16, T = 167, = C.60; showing free surface position at a selection of times, t t/Y = 0.0 4.0 8.0 12.0 16.0 20.0 23.0 26.0 30.0 SYMBOL © © 4 + X © # XX Z MESH 65 X 9 POINTS 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Numerical solution to example 1: M= 1.16, T = 167, T=0.60, showing free surface position at a selection of times, t 136 Lagrangtan Solutions of Unsteady Free Surface Flows = N t/Y¥ = 0.0 4.0 8.0 12.0 16.0 20.0 22.0 © A + X © 4 SYMBOL >| ks oOo N [0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 X Fig. 9. Linearized solution to example i: M = 2.00, T= 167, T=0.48, showing free surface position at a selection of times, t pease aa | a | iE t/y = 0.0 4.0 8.0 12.0 16.0 20.0 22.0 SYMBOL 0 10) a ate x © Gr MESH 65X9 POINTS 2.4 2.0 20.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 xX Fig. 10. Numerical solution to example 1: M=2.00, T=167,7 =0.48, showing free surface position at a selection of times, t ve Wg Brennen and Whttney and £ is the a difference between the walls AD and BC. In Figs. 5 and 6, 7 and 8, 9 and 10 the numerical and linearized free surface shapes are compared for three cases of increasing wave amplitude. As the amplitude increases the similarity between the two diverges; both the wave velocity and the build up on the wall become progressively greater in the numerical solution. Note also that, especially in Fig. 10, the peak of the wave is much sharper than in the linearized solution. For amplitudes less than that of Figs. 5 and 6 the results were almost identical. GC. Example Two, Figs. 4(b), 11, and 12 The second example, Fig. 4(b), introduces moving and curved solid boundaries; the liquid is disturbed from rest by a bed uplift of the form: 2 2 = For Xi 2 For. Xi Xi wo uke’, Yop = 0 ali at Within certain extreme limits on M and T this causes a surface wave immediately above the bed disturbance which then spreads out to each side and is followed by a depression wave over the bed uplift. The linearized solution is 00 Vee eS e) +) R _ [itann (75) awe coat ete k=l + Bt) ain (SEE | (32) where R, = M {sin (5X2 : in (GX) 1/20? ce (Bis e(x2 = X1))"} Vy Ee tanh Ens 2sin a for 0 <<" T., =(Z2 for. t > T = my rT B,(t) =o, cos vtti-(1+o,) cost for O x x Zz 90.0 1.0 2.0 3.0 4.0 5.0 X Fig. 11. Linearized solution to example 2: M=0.344, Xi = 0.75, X2=2.12, T = 67, T=0.35, free surface positions at selection of times, t t/Y¥ = 0.0/3.0 5.0 7.0 10.0 13.0 16.0 16.0 20.0 SYMBOL © © 4 + X © # X& Z MESH 40 X9 POINTS 50.0 1.0 2.0 3.0 4.0 5.0 X Fig. 12, Numerical solution to example 2: M=0.344, X1 = 0.75, X2= 2.12, T =67, T=0.35, free surface positions at selection of times, t 13g. Brennen and Whitney r, = sech® (3£)/[E% - 4] and £ is the a difference between the vertical walls. For T of the order of 2 or 3 and for values of M upto0.3, at least, there was virtually no difference between the numerical and linearized solutions. Figures 10 and 11 in which M = 0.344 demonstrate this. D. Example Three, Figs. 4(c), 13, 14, 15. A Sloping Beach By altering the condition on the boundary AB of example one and employing the shoreline treatment of Section 5E, the interaction of the waves with a sloping beach could be studied. In Fig. 13 a small wave appraches a 27° beach. As the horizontal inclination of the tangent to the free surface at the shoreline (B) decreases, the shoreline (A) accelerates up the beach until B becomes positive. The acceleration then reverses (as in Eq. (28)) and the wave reaches maximum run up. The backwash is extremely rapid and positions t/r = 21, 22 suggest that this causes the small wave which is follow- ing the main one to break. By this time the cells have become very distorted and the mesh points excessively widely spaced to allow further progress. A similar succession of events takes place with the larger wave and smaller beach angle (18°) of Fig. 14. Note in this case,the large run-up to wave-height ratio. In neither of these cases does there appear to be any tendency for the main wave to break on its approach run. Indeed the reaction with the beach is similar to the behavior predicted by Carrier and Greenspan [ 1958] in their non-linear shallow water wave analysis. The wave amplitude was further increased and the beach slope decreased to 9° in an attempt to produce breaking on the approach run. A preliminary result is shown in Fig. 15. Variations in the application of the free surface condition and in the shoreline treatment have, as yet, failed to remove the irregularities in that solution. A stronger shoreline singularity coupled with an insufficiently rigorous treatment of it may be to blame. An optimistic viewer might detect a breaking tendency. E. Example Four, Figs. 4(d), 16. A Shelf One final example is shown in Figs. 4(d) and 16 where the wave travels up a shelf, created by changing the boundary condition on CD, Fig. 1. Excessive vertical elongation of the cells on top of the shelf caused this computation to be stopped at the last time shown. (At this point the wave height/water depth ratio on the shelf is of the order of 2.) However, one can detect a splitting of the wave into two waves as might be expected from the theory of Lax [1968]. 140 Lagrangian Soluttons of Unsteady Free Surface Flows t/r = 8.0 11.0 14.0 17.0 19.0 21.0 22.0 SYMBOL oO Oo & + x Oo eS MESH 30X9 POINTS 20.0 1.0 2.0 3.0 U.0 5.0 6.0 7.0 X Fig. 13. Example 3 with M=0.30 and T=67T, T=0,.571. The beach slope is 27°. Free surface positions at selection of times, t ce | t/y = 8.0 12.0 14.0 17.0 19.0 21.0 23.0 25.0 EK SYMBOL © © 4 + x © # ®& i} MESH 30X9 POINTS © we 20.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 X Fig. 14. Example 3 with M=0.60 and T=87, 7T=0.571. The beach slope is 189, Free surface positions at selection of times ,t 141 Brennen and Whitney =e Je ee N th = 12.0 16.0 20.0 21.0 22.0 22.8 23.7 2U.5 | SYMBOL oO © -A + x © a: x | MESH 51 X9 POINTS : oO 20.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 X Fig. 15. Example 3 with M=2.00, T=167, T=0,481. The beach slope is 9°. Free surface positions at selection of times, t 2.0 2.4 t/Y = 8.0 10.0 11.4 12.2 13.1 13.9 14.3 SYMBOL CROP ay et a xX nO a 4 MESH 80X9 POINTS 2.0 50.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 X Fig. 16. Example 4 with M =2.00, T=167T, T=0.481. Shelf defined by X1=4.41, X2=5.16, HR=0.3. Free surface positions at selection of times, t 142 Lagrangtan Soluttons of Unsteady Free Surface Flows VII. CONCLUDING REMARKS Rather severe examples were taken in order to test the limiting characteristics of the method developed. Provided the various interval limitations were adhered to only two problems arose which could prematurely conclude a computation. First, excessive elongation of the cells in regions of the most violent motion could cause the mesh points to be excessively widely spaced; rezoning could, however, make it possible to continue. Secondly, it would appear that a more detailed knowledge and treatment of some singu- larities is required. Work on this, and especially on the shoreline singularity of example three, is in progress at the moment. Other types of examples which have been only briefly investi- gated thus far are: the matching with a semi-infinite region in which some analytic solution is used; the inclusion of surface tension; the extension of the method to three dimensions; examples in which the fluid is inhomogeneous. It is hoped to present such results in the near future. The authors are deeply appreciative of the kind and considerate help given by Professor T. Y. Wu. This work was partially sponsored by the National Science Foundation under grant GK 2370 and by the Office of Naval Research under contract N00014-67-A-0094-0012. REFERENCES Amsden, A. A. and Harlow, F. H., The S.MAC method: A numerical technique for calculating incompressible fluid flows, Los Alamos Scientific Laboratory Report LA-4370, 1970. Biesel, F., "Study of wave propagation in water of gradually varying depth," in Gravity Waves, U.S. National Bureau of Standards NBS Circular 521, 1952. Brennen, C., "A numerical solution of axisymmetric cavity flows, au J. Fluid Mech., 37, 4, 1969. Brode, H. L., "Gas dynamic motion with radiation: a general numeri- cal method," Astronautica Acta, 14, 1969. Carrier, G. F. and Greenspan, H. P., "Water waves of finite amplitude on a sloping beach," J. Fluid Mech., 4, 1958. 143 Brennen and Whitney Chan, R. K-C., Street, R. L. and Strelkoff, T., Computer studies of finite amplitude water waves, Stanford University Civil Engineering Technical Report No. 104, 1969. Fontanet, P., Théorie de la génération de la houle cylindrique par un batteau plan, Thesis, University of Grenoble, 1961. Fromm, J. E. and Harlow, F. H., "Numerical solution of the prob- lem of vortex street development," Physics of Fluids, 6, 1963. Heitner, K. L., A mathematical model for the calculation of the run-up of tsunamis, Thesis, California Institute of Technology, 1969. Hirt, C. W., The numerical simulation of viscous incompressible fluid flows, Proceedings of the 7th ONR Symposium, Rome, 1968. Hirt, C. W., Cook, J. L. and Butler, T. D., "A Lagrangian method for calculating the dynamics of an incompressible fluid with free surface," J. of Computational Physics, 5, 1970. John, F., "Two-dimensional potential flows with free boundaries ," Communs. Pure and Appl. Math, 6, 1953. Lamb, H., Hydrodynamics (6th Ed.), Cambridge University Press, 1932. Lax, P. D., "Integrals of non-linear equations of evolution and solitary waves," Communs. Pure and Appl. Math., 21, 1968. Lewy, H., "Water waves on sloping beaches," Bull. Amer. Math. Soc. , 52, 1946. Penney, W. G. and Price, A. T., "Finite periodic stationary gravity waves ina perfect liquid," Phil. Trans. Roy. Soc., A, 244, 1952. Pohle, F., "Motions of water due to breaking of a dam, and related problems," in Gravity Waves, U.S. National Bureau of Standards Circular 521, 1952. Roseau, M., "Short waves parallel to the shore over a sloping beach," Communs. Pure and Appl. Math. , ii, 1958. Sekerz-Zen'kovich, Ya. I., "On the theory of standing waves of finite amplitude on the surface of a heavy fluid," (R) Dokl. Akad. Nauk. SSSR, (N.S.), 58, 1947. 144 Lagrangtan Solutions of Unsteady Free Surface Flows Wehausen, J. V. and Laitone, E. V., "Surface Waves," Handbuch der Physik, Vol. IX, Fluid Dynamics III, 1960. Welch, J. E., Harlow, F. H., Shannon, J. P. and Daly Be Js The MAC method, Los Alamos Scientific Laboratory Report No. LA-3425, 1966. Whitham, G. B., "A general approach to linear and non-linear dispersive waves using a Lagrangian," J. Fluid Mech., 22, 2, 1965. _ 145 ‘ ; - = x | ~ . | its) , \ [ . ¥ TWO METHODS FOR THE COMPUTATION OF THE MOTION OF LONG WATER WAVES — A REVIEW AND APPLICATIONS Robert L. Street Robert K. C. Chan Stanford University Stanford, California and Jacob E. Fromm IBM Corporatton San Jose, Caltfornia I. INTRODUCTION The continuing evolution in speed and capacity of digital com- puters has encouraged the development of many computationally oriented methods for analysis of the movement of waves over the surface of the ocean and onto the shore. Carrier [1966] gave analy- tical techniques requiring numerical evaluation for the propagation of tsunamis over the deep ocean and for the run-up on a sloping beach of periodic waves that do not break. He noted that linear theory is valid in the deep ocean and over much of the sloping shelf; thus, non- linear theory is needed only in specific regions where the nonlinear contributions to the dynamics are important. However, his nonlinear, approximate theory was developed only for the plane flows. An ex- tension and application of Carrier was made by Hwang, et al. [1969]. They studied the transformation of non-periodic wave trains ona uniformly sloping beach using the nonlinear shallow water wave equation and the Carrier-Greenspan transform. This transform fixes the moving, instantaneous shoreline of the physical plane to a single point in the transformed plane. Although the analysis deals only with plane flows and does not handle breaking waves, it does predict wave run-up and reveals a significant beat phenomenon. To study nonlinear effects and/or to account more completely for the waves' reaction to arbitrary ocean topography and boundaries, it is natural to turn to numerical methods and their accompanying computer codes. The simplest of the numerical methods are repre- sented by the refraction techniques of Keulegan and Harrison [ 1970] 147 Street, Chan and Fromm and Mogel, et al. [1970]. These methods, based on linear, geo- metric-optics theory, are applicable to arbitrary bottom topography but can predict neither breaking nor run-up. They also neglect reflection and diffraction effects. The computer code is very simple, requiring step-by-step solution of Snell's law and a wave intensity equation over a grid of bottom depths. Vastano and Reid [1967] described a procedure employing a numerical integration of the linearized long wave equation to study tsunami response at islands. They concluded, by comparison with analytic solutions for special cases, that their numerical model gave an adequate representation of the solution to the linearized equations. Their paper indicated that the work was a lead-in to a more general treatment of arbitrary bottom topography. At the island they useda vertical cylinder that penetrates the surface and thereby restricts the movement of the instantaneous shoreline. No run-up can be cal- culated in the usual sense. In another approach, Lautenbacher [1970] used linear, shallow water theory to study run-up and refraction of oscillating waves of tsunami-like character on islands. His method allows for the moving, instantaneous shoreline and, of course, for superposi- tion of individual results from monochromatic waves. Working from an integral equation formulation and employing a Hankel function representation of the far-field radiation condition similar to that used by Vastano and Reid [1967] , Lautenbacher used a grid of dis- crete points to numerically integrate the integral equation of his model. Combining his work with Carrier [1966], Lautenbacher was able to estimate total tsunami run-up from a distant source. He also emphasized the importance of refractive focusing effects. The numerical methods aimed specifically at modelling non- linear effects take three forms: a. Approximate, plane-flow models for arbitrary or sloping beaches and based on approximate equations. b. Exact plane-flow models. c. Quasi-three-dimensional models, The approximate, plane-flow models are represented by the work of Freeman and LeMéhauté [1964], Peregrine [1967], Heitner [1969], Street, et al. [1969], Camfield and Street [1969] , and Madsen and Mei [1969a, 1969b]. With the exception of Heitner [1969], the authors used Eulerian coordinates. Freeman and LeMéhaute [ 1964] applied the method of characteristics to the nonlinear, shallow water wave equations for plane flow. They described a method for com- puting the shoaling of a limit-height solitary wave on a plane beach, predicting the point of breaking inception by the crossing of character- istic lines and computing the subsequent bore development and run-up at the shoreline. A term was added to the equations to correct the 148 Computatton of the Motion of Long Water Waves assumed hydrostatic pressure distribution beneath the wave, the assumption not being valid for finite amplitude waves. Camfield and Street [1969] used a refined correction term, but the results were not entirely satisfactory in either case. Peregrine [1967] and Madsen and Mei[1969a] derived approxi- mate, nonlinear, governing equations for the propagation of long waves over slowly varying bottom topography. In these equations the verti- cal component of motion is integrated out of the computation so a single-space-dimension problem is all that remains. Madsen and Mei [1969a] showed that, while they and Peregrine used different approaches and solution methods, the equations are the same when presented in the same variables. Furthermore, Madsen and Mei [1969a] explained that these nonlinear equations, obtained under assumptions similar to those leading to cnoidal waves in the case of horizontal bottoms, give a uniformly valid description of long wave problems as long as breaking does not occur. In particular, their equations were derived under the condition that the Ursell parameter 2 U, = ye = O(1) (1) ie) where 1 is a measure of wave amplitude, L, is a characteristic wave length and do is the water depth. Thus, the nonlinear, govern- ing equations are of the Korteweg-deVries type (KdV) that have per- manent solutions, e.g., cnoidal waves, for the case of horizontal bottoms. Madsen and Mei[1969a] demonstrated that, although the equations pertinent to each of the three groups of long waves (Airy where U,>>1, KdV where U, = O(1) and Linear where UU, << 1) have different mathematical solutions, the features characterizing each group are all contained in the equations derived under the assumption of waves of the KdV type. The method of characteristics was applied by Madsen and Mei [1969a, 196 9b] to solve their equations for initial, boundary value problems involving solitary waves and periodic waves on plane slopes and a shelf; their equations, like those of Peregrine [1967], are applicable to general, uneven bottoms. Street, et al. [1969] pre- sented a numerical model APPSIM and results based on the Peregrine [1967] method, but employing initial and boundary conditions similar to those of Madsen and Mei[1969b]. These methods reproduced the nonlinear breakdown on a shelf of a solitary wave, the breakdown having previously been observed only in experiments [Street, et al., 1968]. Furthermore, comparison shows quantitative agreement _ amongst these methods and relevant experiments. Run-up cannot be calculated with these models which employ the vertical beach (or island) face that was used by Vastano and Reid [1967]. However, Peregrine's [1967] derivation included the two horizontal space dimensions, while Madsen and Mei [1969a] did not. Accordingly, as an extension of APPSIM, a quasi-three-dimensional model 149 Street, Chan and Fromm APPSIM2 is based on Peregrine's equations and is described in detail in Section III below. Heitner [ 1969] presented a nonlinear method based on Lagrangian coordinates and a finite element representation of the fluids. His theory retains terms representing the kinetic energy of the vertical motion; thus, like the methods described just above, Heitner's approximate method permits permanent waves to propa- gate. Unlike those methods, Heitner's formulation gives a repre- sentation of wave breaking inception, bore formation and run-up on a linear beach for plane flows. The exact, plane-flow models are represented by the work of Brennen [1970], Hirt, et al. [1970] , and Chan and Street [1970]. All are based on the exact equations of motion; however, Chan and Street [1970] work in Eulerian coordinates, Brennen [1970] uses Lagrangian coordinates and equations, and Hirt, et al. [1970] em- ploy Lagrangian coordinates but retain the Eulerian form of the overning equations. Based on the Marker-and-Cell (MAC) method Welch, et al. , 1966], Chan and Street's model for water waves is called SUMMAC and is discussed in Section IVofthis paper. Brennen has applied his technique to tsunami generation by ocean floor move- ments and to run-up on abeach. Hirt, et al., made applications to wave sloshing in a tank by way of verification of their method, called LINC, which has wide application in ocean wave analyses as well. All the exact, plane-flow methods mentioned employ some form of finite- difference or cell representation of a discrete grid of points. Finally, the quasi-three-dimensional methods are represented by the work of Pritchett [1970] and Leendertse [1967] and by the extension to two horizontal dimensions of the APPSIM program of Street, et al. [1969] discussed above. Pritchett presents a code for solving incompressible, two-dimensional, axisymmetric, time- dependent, viscous fluid flow problems involving up to two free surfaces. The basic equations are exact; heuristic models for tur- bulence simulation are used. Scalar quantities such as heat and solute concentration can be traced, and the fluid may be slightly non- homogeneous. Most of the variables, their placement in the compu- tational mesh, and the free surface treatment are those used in MAC [ Welch, et al., 1966]. Leendertse [1967] developed a computational model for the calculation of long-period water waves in which the effects of bottom topography, bottom roughness and the earth's rotation were included. The equations of motion are vertically integrated so only the two horizontal space-dimensions remain (much in the manner of Peregrine, [1967]), but while Peregrine [1967] , Madsen and Mei[1969a, 1969b] and Street et al. [1969] retain terms to account for the vertical accelerations of the fluid, Leendertse [1967] does not. His equations become the usual nonlinear shallow water wave equations with added terms to account for the bottom roughness and earth's rotation. He 150 Computatton of the Motion of Long Water Waves focuses his attention on the modelling of long waves such as tsunamis in areas with irregular bottom topography and complicated ocean boundaries. His computation uses a space-staggered scheme (where velocities, water levels, and depth are described at different grid points) and a double time step operation in which the time integral is considered over two successive operations in a manner designed to make effective use of the space-staggered scheme. Among the papers mentioned in this Introduction, only Leendertse [1967], Welch, et al. [1966] and Chan and Street used computer graphic display for output of results. The value of graphic display is illus- trated in the results presented in the remainder of this work. II. THE PRESENT WORK Street, et al. [1969] gave a progress report on the develop- ment of computer programs fortwo numerical, finite-difference models for the study of long water waves. These models and their accompanying programs were, as noted above, given the acronyms APPSIM and SUMMAC. Both were based on representation of the motion of inviscid, incompressible fluids in terms of the Euler equations of motion in Eulerian coordinates. Flow boundary con- ditions were derived from physical requirements and the governing equations at the boundaries. The mathematical models thus obtained were then transformed to numerical, finite difference models for the purposes of computation. In 1969 the study had been confined to plane flows, but the numerical results had been verified by compari- son with experiments and the work of others. The models were to provide detailed flow field data in the portion of the wave shoaling process where nonlinear effects are significant, but breaking has not occurred. Our approximate simulation (APPSIM) is based on the method of Peregrine [1967] and supplemented by the work of Madsen and Mei [1969a, 1969b]. APPSIM was implemented for quasi-two-dimensional, plane flows (vertical motion integrated out). For the purpose of implementing, testing, and verifying the program and method, we simulated the propagation of solitary waves on a stepped slope which represents the configuration of the continental slope and shelf, i.e., we examined long waves in moderately shallow water. The key criteria to be satisfied were a. Solitary waves propagate stably on a horizontal bottom. b. Solitary waves decompose into undular bores when the waves propagate onto a stepped slope [ Street, et al., 1968]. — c. Wave heights must be in good quantitative agreement with available experimental data. As reported by Street, et al. [1969] APPSIM met these criteria. Street, Chan and Fromm The successful application of APPSIM to examples of plane motion of long waves indicated that the method could be applied to a quasi-three dimensional simulation (two horizontal space dimensions with vertical effects represented in once integrated equations) of the motion of waves over arbitrary bottom topography. We have ex- tended APPSIM to handle general bottom topography and both solitary and oscillatory wave inputs. The new method is called APPSIM2 and presented in SectionIII below. An objective of our exact simulation was to provide detailed information about wave processes near the shore and at the ocean- structures interface. The Stanford- University- Modified Marker-and- Cell (SUMMAC) method computes time-dependent, inviscid, incom- pressible fluid flows with a free surface; the method is suitable for analyzing two-dimensional flows. Initially, we simulated the propa- gation of solitary waves in a horizontal channel filled with fluid to unit depth and with vertical end walls, The solitary wave propagation problem possessed several key features: a. The theories for the wave motion against the wall were not in agreement with experiments. b. The solitary wave should propagate stably (without change of form) in zones not near the channel walls. c. Perfect reflection from the walls should occur. We undertook a significant modification of the MAC method to create a numerical scheme suitable for water wave simulation. As reported in 1969, the resulting SUMMAC simulation met the criteria of stable propagation and perfect reflection of solitary waves and resolved the disagreement between theory and experiment for motion against a vertical wall. The successful application of SUMMAC to the initial example indicated the possibility of employing a modification of the same technique to attack a variety of other problems. We have subsequently studied the generation of water waves by a periodic pressure pulse and the shoaling and run-up of solitary waves on a stepped slope and on plane beaches. A summary of the presently implemented SUMMACG method and results for the periodic pressure pulse problem are pre- sented in Section IV of this paper. An evaluation of the numerical qualities of SUMMAC and a report of the shoaling and run-up studies are given in Chan, et al. [1970] and Chan and Street [1970b]. 152 Computatton of the Motion of Long Water Waves Ill. APPSIM2 3.14. The Governing Equations and Auxiliary Conditions Dimensionless variables are defined below and in Fig. 1 where the physical domains considered are also illustrated. The variables are * * -! (x sy en d*)d, (Xess od) /2 t—1 (e/a) (u*, v*) (gd) /? (u,v) where those on the left-hand sides are dimensionless and d, is the depth in the deepest part of the simulated wave tank. Here, x and y are fixed, Cartesian coordinates in the horizontal plane; u(x,y,t) and v(x,y,t) are the corresponding mean (vertically-averaged) horizontal fluid velocities; n(x,y,t) is the free surface shape, measured from the still or undisturbed water line in the tank, and d(x,y) is the depth of the still water. de (ER oma Aeromax (ae), b. Submerged Seamount (Plan View) Fig. 1. Definition of symbols and simulated wave tanks for APPSIM2 153 Street, Chan and Fromm For waves of the KdV type that we wish to study, U, = O(1). If € = No/do (2) and o = dg/L. <1 (3) for long waves, then from Eq. (1), Uy=e€/o*=1 and €=0% This is the relation on which Peregrine [1967] based his expansion-in-a- parameter analysis. He pointed out also that d(x,y) = O(1) and the derivatives of d equal O(c) are necessary restrictions; otherwise, the variations in the depth of water are shorter than the incident waves and tend to generate shorter waves, thus upsetting the scheme of the approximations. Under the above conditions Peregrine [1967] obtained the momentum equations 1 1 .2 u, tuu, +vu, tn Fs d[ (du),, + (dv) yy] =i bug Vaylt (4) OSS yg Oy (dull, -5alV- (du)l -Faa[V-U] +z aalV- Ul (14) where u = (u,v) and V = (8/8x, 8/8y). Ifthe bottom is flat in the neighborhood of Wall 1, d,=d,y=0 and Eq. (11) becomes (6 teas i (12) But, if n(0,y,t) = n(t), then until some reflection from a shoal or beach in the tank returns to x =0, uy(0,y,t) = 0 (13) and, hence, v,(0,y >t) = 0 (14) 155 Street, Chan and Fromm Given (0,y,t) = n(t) and Eqs. (13) and (14) and using Eqs. (4-6) at Wall 1, one can uniquely determine u, v and 7 there for the case of a prescribed incoming wave. The conditions at the remaining walls are, of course, not changed. The appropriate initial conditions for Eqs. (4-6) are the initial values of the dependent variables, viz., n(x,y,0) = ni(x,y); ulx,y,0) = uj(x,y)s = v(x,y,0) = vj(x,y) (15) 3.2. The Difference Representation and Computation Scheme For numerical computation the region 0Sx=L,,0Sy=1, is covered by a grid of discrete mesh points with a spacing Ax= Ay = 4 and calculations are carried forth with time steps At. To allow for proper representation of the boundary conditions the grid indices (i,j) run over the intervals (1,M) and (1,N) respectively and points (1,j), (Mj), (i,1) and (i,N) lie outside the tank walls, e.g., x=0 is equivalent to (2,j), etc. Consequently, L, = (M-2)A and Le=(N-2)A (see Fig. 1). In the finite difference representation of the differential equations and auxiliary conditions, central space-differences are always used; both forward and time-splitting schemes are used for time-differences. The differential equations of motion, eqs. (4-6), lead to a highly nonlinear and coupled set of difference equations. These are solved iteratively, using a predictor-corrector method. If u, v and n are known at all the grid points at t!.e n' time level, the following scheme leads to calculation of u, v and 7 at the nti!? level. First, u, v and 1 are predicted at the nti level by use of the nonlinear, shallow water wave equations (Eqs. (4-6) with their right-hand sides set to zero). With the superscript P indicating the predicted value, the difference equations are, after rearrange- ment for computation, P At i) aaa er 2 (aj jng My) F Vly — yj) F Tiny > m1) (16) py Ale aCAt oe joo" \+ E “a At Mig = Mig ~ Ty Cig (Giely + Niet - Gi-ny - Tieng) F diy F TiMCGiedy ~ Bi-1j PN inc Milas i Moiel ce dtiel es Silas nij-0)) (18) 156 Computatton of the Motton of Long Water Waves t where variables without superscript are known at the nf time level. Second, the x-momentum, Eq. (4) is used to obtain a difference equation for u values at the nti™ Jevel. Central space-differences and time-splitting are used about the point (i,j,n+43); this leads to a difference equation that is implicit. Now we gather the u™! terms from the j'? row on the left-hand side of the equation and all other terms on the right-hand side. The heart of our procedure is to use uP, vP and 7? values in lieu of the u™*!, vy"! and n*! terms which appear on the right-hand side of the equation, these terms being mostly inthe j-1 and jt1 rows. The result, when rearranged for computation is, for each j, n+l n+l nel _ Au + BM + Cull =H, 15 3,4,060,M-2 (19) where 2 AE 1 di = - pelle ie age A 4A bea af 2A* ( 3 dijdj.1,) adj, B= Ll + aSu, 3A At 4 dy aa Ms | en eS 2 dij diey ) : At E = ig’ BA ij iat Uj _4;) At P AGA Vip lijet = Mifare Bie At P P — ZA Miatj - Nien t Niet) - Ti-g) dij + eae (- Fig Piany * 24 j4iy - Fi) 4 P P P FZ (distjet Vietiet ~ FietjetVi-tjet ~— Fiety r%i6 jet p midis ip ¥ - dj a faljel i+ljel Vieljel i-ljal Vi-tjet IOS tei peljat daar qae) 157 Street, Chan and Fromm ‘ Visi) T 20;j - Yi-tj Pe 64 1, P P P P FAV atjeh 7 Vic ljeht Viatjat PoVi-ajat 5 Vixtfel S:Viedjon tT Viste bY intjt )) Equation (19) must be solved for all i simultaneously for a given j. The matrix of coefficients of the unknowns in Eq. (19) is tridiagonal and is quickly and easily solved by a tridigonal-matrix equation solver employing Gaussian Elimination. The process is repeated for each j until the ee are known. Appropriate boundary conditions are introduced at the ends of the j'® row in each computation. Third, the y-momentum Eq. (5) is used to obtain a difference equation for the v values at the nti™ Jevel. The result is entirely equivalent to that for the u values, viz., the third-order terms on the right-hand side create a naturally implicit system so time-splitting is used. Now, however, we use vP and nP values for all i-1 and iti points along a given column of implicit equations (i = constant, 3 <= j= N-2) and we use the u™! values just computed. The result, when rearranged for computation, is ~ nel ~ nel n+l . AVii + Bj, + Cv iin =£ , j= 3,4,.-.,N-2 (20) where ~ 2dij B=1 + Sy 34 At 1 / dj an Vi ae ae dys dij) At e P E=Vijy - ZA ay (Via een a Vii io) ) ot rail aia) Pega, Vise Vise, & Vij-1 At Ee esl ic PE Aa Bijeii Myel = ja) t 158 Computatton of the Motton of Long Water Waves dij 1 n+l n+l nel 2ae \ 4 (diggjat Ciatjel ~ Fienjet Bi-tjet — Tietj-1 Uistj-1 n+l a Gi ype Pie aj ~ Gigtjetietjel + Gj-tjet Uj - Ijel Be Gi gtjet ia tj-t zs dij. 4i-tj-1) a Fijat Vijel + 2dijvij - dij-1Vij-t) 2 dj; 1 ,_nel nel n+] + n+| = an (5 (Wis ijel ~ Upetjet ~ Vie tye 7 Bprj-r ~ Uistjel " Uj ijel t Gietj-1 Uj-1j-1) - vijr + 2vij - vij-t) Now, Eq. (20) must be solved for all j simultaneously for a given i. Again we use the tridiagonal-matrix equation solver. The process is repeated for each i until the vjj are known. Fourth, the continuity Eq. (6) is used to obtain 3 difference equation for the 7 values at the nti™ jevel. The u™ and v"* values are used in an equation that, as suggested by Peregrine [1967], uses an average for u and v values, but forward differences for "1 values. The result, rearranged for computation, is the explicit equation nel At ul t+ uij = 1 i} Bip. Ty xa ( 2 (di.iy + Misty - Fi-tj - Ning) 4 n+l nel FS (dij t mig hlinng + Yisty - inj — Mn n-l ne TVileh oc Vilas “iclre wat vit + vij 2. (dijet i Nijel ~ dij et ie ni} (21) This equation is used to compute nit! for 2512 M-1., 2-2 j and then the first iteration, the predictor iteration, at the n time level is complete. = N-1, th If now the second through fourth steps above are repeated with 59 Street, Chan and Fromm P values replaced by the nt 11” values just calculated, the accuracy of the solution is increased; this is the corrector iteration. Numeri- cal tests showed that the computations remained stable if at least two iterations were used (one predictor, one corrector). The u™, v +l and n+! values obtained in the second and the third iteration agreed to at least four significant figures after several hundred At steps in simulation of solitary wave motion onto a shelf (Fig. fia). Boundary conditions in difference form were derived from Eqs. (7-10) in the case of solitary wave simulation. The wave was started well inside the tank walls which were held rigid. For ex- ample, for Wall 1 in Fig. 1 we have from Eqs. (7-10) at any time level 1), = Ngjr Vij = V3j> 4ej= 0, and Wyj = - Ug} Other walls have similar conditions. For input of an oscillatory wave propagating in the positive x-direction at x = 0 we prescribe 2nto.- At « (nti) (22) fo) where ‘to is the amplitude (usually small) and Lo is the wave length. The celerity Cy is taken to be unity in nondimensional terms. From Eq. (14) we have v,, = v3). Because n3* and 74; are computed explicitly in the tank region, ni; is obtained by polynomial interpolation according to the second-order formula n+l n+l n+l n+l Me ig ee Ee, (23) Finally, the continuity difference Eq. (21) is used for points (2)) where it has not been previously employed to relate uf! to the values inthe interlor, j= 2. With u i known as a function of | Ugj» Uzi, etc., Eq. (19) can be used in 2S i= M-2 and the ui found; vi values are replaced by v{, values in the first iteration. Figure 2 is a flow chart for the APPSIM2 computations. These were performed on an IBM 360/91 system. For a typical computation with A =0.25, At=0.22, M= 154, N = 54 and 126 time steps the program required about 360K bytes (90K words) of core storage and 4 minutes of CPU time (about 1/30 minute per time step). The stability of the method is discussed in Sec. 3.4 after presentation of computational results. 3.3. Results and Discussion To illustrate the focusing effect of wave refraction and the reaction of waves to a shelf geometry, solitary and oscillatory waves were shoaled over the bottom topography shown in Fig. la. The water depth in the tank was 1.0 while the depth of the shelf was 0.4. 160 Computation of the Motton of Long Water Waves REMARKS BOTTOM, dj; SOLITARY OR OSCILLATORY WAVE IN +X- DIRECTION WAVE IN CALC u,v,7 (shallow water eqns) CALC PREDICTED VALUES uP? y? pP EXPLICIT KOUNT = 0 use uP PnP CALC u®*!, KOUNT = KOUNT +1 IMPEICLT. IN X- DIRECTION use unt! yP nP IMPLICIT IN Y-DIRECTION EXPLICIT (uses advanced velocities, however) SET uP sult! pps noth yPaynrl KOUNT 2 2 WRITE/TAPE DUMP T= TOP;STOP Fig. 2. APPSIM2 flow chart The deep and shallow portions are connected smoothly by a cosine curve. For the solitary wave simulation, the pertinent parameters were L, = 38, Lg= 13, x,= 14.5, X%,= 19.5, yy= 7.75, Yg= 2-75, A=0.25, At=0.2165, and no= 0.1. The wave was started with its crest lying along x,= 8.0 and propagated in the positive x-direction toward the shelf. Chan and Street [1970a] showed that the effective half-length of a solitary wave of amplitude No = 0.1 is about 11 so it is necessary to correct the initial Boussinesq [ Wiegel, 1964] wave profile for the influence of the wall at x = 0; however, it was unnecessary to correct the leading portion of the wave for the bottom influence. The initial u-velocity distribution was calculated from Eq. (6) under the assumptions that v=0, n = n(x,t) and the wave is moving at a constant speed Co=1+0.5 7 [Wiegel, 1964] with constant form. In addition At was selected in accordance with the Courant-type condition A Ats=— Co Results of the solitary wave computation are shown in Figs. 3-5. 164 Chan and Fromm Street, HH Hr Ht , wry 4H arr ae StH HHH xt ao a oH oa HHH tHE HH +H SR, oper poccths sposcpeseccpe. HHH HEHEHE Yt Se HE HEHEHE HEHE EERE EEE HE EEE OHHEE T=213.0 T=19.5 HOHE tt i Et Ap : E : E E EEE EEE EEE HE HE EEEEEEHEE EE EEE EEE EEE HEEEEEEEHEEHE EEE HEE EEE EEEEHEEEEEE HELE EEEEEEHEEEEEEHEEEEEE HEEL HEHEHE HE —— oo EEEEEEEEHEEEEEEE HEHEHE SHEEEEEEE HH Het — tHE —H4-H AHH HE OHH HHEHE bhatenee 26.0 162 T= Free surface maps for a solitary wave on the shelf 6) Fig. Computatton of the Motion of Long Water Waves r= 19-5 Fig. 4. Velocity map (v) for t= 19.5 0.2 0.1 7 0.0 18 20 22 24 26 28 30 32 34 36 “O52 x LEGEND: - SCALE EXAGGERATED FOR ILLUSTRATION ---APPSIM (2-DIMENSIONAL SIMULATION) ——APPSIM2(3-DIMENSIONAL SIMULATION ) 0.1 hea x ” = 0.0 2 4 6 8 10 12 14 #16 18 20 22 24 26 28 30 32 34 36 x Fig. 5. Wave profiles for two- and three-dimensional propagation 163 Street, Chan and Fromm The evolution of the free surface is illustrated by free sur- face maps in Fig, 3, while Fig. 4 shows a v-velocity map for = 19.5. The maps are printed during program execution by a sub- routine that scales the variable values on a range running from a minimum of zero to a maximum of ten. Only odd numbers are printed at their corresponding node points. These maps are ex- tremely useful for initial interpretation of the data, Later, quanti- tative studies of results can be made because the u, v, n fields are stored on tape after every five time steps and maps are made after every 20 to 40 steps. Thus Fig. 5 illustrates a quantitative comparison between the two-dimensional results and the three- dimensional simulation at t= 19.5 for y=0 and y=L»2. Both the effect of wave refraction and the nonlinear response of the flow are evident. As another example, Fig. 6 contains pictures of the develop- ment of the yn, u, and v fields for oscillatory waves shoaling ona shelf. The pertinent parameters were L, = 66, L,= 41, x;= 25.5, X5= 45.5, yr = 25.5, ye = 15.5, 4=0.5, At=0.5, and 2) = Hy = 0.05. The input wave at Wall 1 (Fig. 1) had eerie length Ly '=(20 and speed C,=1.0 sothe period Ty = 20. The actual computed length was essentially. 20 also. In this case we sought to simulate a large region so A was large; even so 458K bytes of core storage were required for the program. In spite of the rather coarse grid the computed properties of the waves were smooth and well behaved. T= 21.5 T= 44.5 T=68.5 | i 7-CONTOURS U-CONTOURS V-CONTOURS Fig. 6. Shoaling on a shelf (oscillatory waves) 164 Computation of the Motton of Long Water Waves The contour lines in Fig. 6 were computed by a plotting pro- gram developed by Schreiber [1968]. The facility used was an IBM 2250 graphic display unit in which the computed contour lines are projected on a TV screen. The contour plots were recorded by photographing the surface of the screen. Several motion pictures have also been made with this apparatus. Two 2250 units are used when films are made. First, as noted above the computed field values are stored on tape during the APPSIM2 computer run. Later, a special program calls up the tapes and transforms the field data to contour lines. These are transmitted to the 2250 units. One is used as a control console to monitor picture quality and to set the movie camera speed. The second unit has a 16 mm movie camera mounted on it and focused on the screen. The camera operation is synchronized with the suc- cession of contour plots flashed on the TV screen. Titles are also constructed on the screen and filmed. Judicious editing transforms the 16 mm film into a useful and interesting movie. As the sequence of Fig. 6 shows, the evolution of the flow fields is particularly instructive. Because all the pertinent parameters are usually shown simultaneously on the screen with the contour plots, quantitative interpretations of the contour information can be made directly from the graphic display. For example, at t = 68.5 the maximum wave height along the line y =0 is about 0.06, while along y=Lap, the height is 0.05 (i.e. , the wave is unaffected by the shelf at this time). In Fig. 6, the n-plot increment for t = 68.5 is 0.01, the maximum value is n = 0.064 and the minimum is n = - 0.026. For oscillatory waves it is necessary to have some detailed printout in addition to the graphic display for quantitative analyses because the contours are not marked with their contour level values. Finally, we simulated long wave amplification by a circular submarine seamount and compared our results with the experimental values of Williams and Kartha [1966]. The following pertinent param- eters exactly match one of their experimental runs for a half-seamount (Fig. 1b) with non-dimensional parameter X = 2mb/L,= 3.0: xc= 56.7 is the distance from the wave generator to the peak of the island; b = 7.73 is the radius of the base of the seamount, T = 16.6 is the wave period, No= 0.0082, d=1.0 beyond the seamount base, € = 0.116 is the submergence of the seamount at its peak and L,= 23.2. The tank length L, = 116 was selected to prevent re- flections from reaching the seamount. The amplification ratio H;/H, = Ay was calculated where H,= 2n, and Hj = the trough-to- crest distance on waves at the island peak where d=e€e. The experimental Af = 2.42, while A; = 2.46 according to the refraction theory of Mogel, et al. [1970] and As = 2.70 accord- ing to APPSIM2 for a seamount whose shape was given by d(x,y) = (41.0 - aS) +€ 165 Street, Chan and Fromm with eres re x)" eon? and q=1.0 (the shape factor). This was a linear seamount with a sharp peak where the first derivative of d(x,y) is discontinuous and the higher derivatives are undefined (cf., Sec. 3.1). Unfortunately, the difficulty of resolution of the island features near the peak was compounded by the fact that to simulate the experimental conditions within our nominal core allotment of 500K bytes we had to use A = 0.5 where 4 =0.25 would have been preferable. As a consequence, we believe, of the coarse grid the short waves generated by the island wave response are not properly resolved, being of the order of one or two grid divisions. The solution, therefore, while not unbounded, appears unstable. On the other hand, Williams and Kartha [1966] did not report on the sea-state near the islands and our results might be physically reasonable. Atest using q = 2.0 which gives a dome-like island produced an A; = 4.65 for X= 2mb/L, = 3.0 which is slightly beyond the range of the experiment. However, this A¢ value lies, as does our result for q = 1.0, within the uncertainty band of re- sults presented by Williams and Kartha. 3.4. Stability Analys is Initial calculations with APPSIM2 and no iteration, viz., operating with only a single predictor/corrector step produced some- what ragged results after several tens of time steps. Accordingly, a linear stability analysis was made to examine the amplitude pro- perties of the computational scheme. For the stability analysis we set dj; = 1 and defined the constant parameters B= At+ v;/4 (24) y=At+n/4 R= At/A The equations defining the computational scheme were linearized by considering only difference quantities as variables and treating the remaining terms as the constants of Eq. (24). Thus, the prediction Eqs. (16-18) become 166 Computation of the Motion of Long Water Waves p 1 { Uy = 4a - FU - Bien) - Play - Bij.) - $ Renietj - Nj-1)) (25) =) 4 | Mp a) Pipe Mijn) ini nij-0 (26) 4 1 nig = Nii - > eM - Ni-tj) - > PCnijed - Nij-1) - S(vARMuieyy - vig) - GOYFR Mv qa - Ye) (27) The remaining Eqs. (19-21) of the corrector step were treated ina similar manner; where P values appeared, the values from Eqs. (25-27) were introduced. To test the resulting linearized form of Eqs. (19-21) we introduced the Fourier component solution (or error) —ex ilo, x + oy) iwt e W-=w where we seek to determine if w values, either real or complex, exist such that W is a solution of the difference equation and where * u —> te * = Vv = constant nN Ole 2m/ry for representative wave lengths \, and XX» in the x- and y-directions respectively. Now, let p = e!¥4! so —> — KK n i(o,x+o,y) (= W=Wu (28) at any point in time and space. Thus, we insert Eq. (28) in the linearized u, v, 1 equations and obtain 167 Street, Chan and Fromm Aiy AQ yg] U x * [A] W = |ag, agp - ail v =) (29) * ay 7930. 331)" where the aij = ajj(%,Bsy,R,As0, s05,p). Because Woe OF Sin general, a Fourier component solution can exist and Eq. (29) can be satisfied only if |A| = 0, viz., only if a set of eigenvalues exists for the matrix A. The condition |A| = 0 leads to a determination of bs linear stability depends on the amplitude of p. If |p| 1 the solu- tion by Eqs. (16-21) would be termed linearly stable. Analysis of the coefficient matrix A in Eq. (29) is complex and is not reproduced here, but the key results are as follows. First, if 6, = 20A/)d, and 85 = 2nA/r», then for finite At, \, and Nos lim |p. | = il A—=0 Similarly, for finite A, \, and \3, lien, |e = At—0 The solution scheme is stable in these limit cases. Second, the case 6,=0, namely, \,= 00, was investigated. Then, ln | = f(a, y,9,) for assumed A and R. This approach leads to a complex, cubic equation with a possible root Ju,| = 1 + Oe) (30) and a possible root pair with maximum modulus equal to or slightly exceeding unity, i.e., lu2,3lmax = 1 (34) Specifically, in a solitary wave computation with A= 0. 5“and Mt-=\0525, we have for n= 0.1, y =0.05, a= 0.05, p=0, R=Oe 168 Computatton of the Motton of Long Water Waves and -7 Ips] e 1+7xX10 <1 +o0(a) lo3lmox © 1.005 <1 + O(a) for \,;> 2A. For i, = 24, |p| =1 exactly. Forsythe and Wasow [ 1960] suggest that the errors may be controllable and lead to a stable computation when [els 2 O(At) (32) In the present case a@< At because a= uAt/A and u/A <1; accordingly, the condition (32) can be written as |p. | = 1+ O(a). Our computational experience with APPSIM which has 0,=0 always and does not use iteration also suggests that the computation is stable, at least for several hundred time steps. APPSIMZ2, however, because of its coupled (u,v, 7) equations and the propagation of error from the u-field where |ujj | is relatively large compared to the v-field and n-field, does require at least one iteration (which tends to make the computation more like an implicit scheme) to retain a smoothly varying solution on a smoothly varying bottom topography. 3.5% Prognosis The computational results indicate that APPSIM2 is a useful means of studying the evolution of flow fields in wave shoaling over smoothly varying bottom topography. However, the method requires considerable computer storage and moderate execution time. Thus, APPSIM2Z, which models nonlinear processes in nonbreaking waves, should be used only when nonlinear effects are expected to be sig- nificant, other methods (cf., SectionI) being appropriate otherwise. As Madsen and Mei [1969a] indicate, the equations of KdV type used in APPSIM2 should make the method applicable to a wide range of long wave problems. Two futher steps should be made in the development of the method. First, the linear stability property could be improved by introduction of a second-order central difference method for the convective terms in Eqs. (18) and (21). This central difference [ Fromm, 1968] leads to a modification of Eq. (31) such that l2,3 lear =1 for most components of interest. Alternatively, Eq. (21) can be made an entirely implicit equation for ni; [cf., Eq. (19)]; this will eliminate the growing contribution represented by Eq. (31) and 169 Street, Chan and Fromm caused by the explicit nature of the ate equation. Second, a series of simulations of specific hydraulic models should be made to deter- mine the grid size A required to resolve the smallest significant feature of a problem and to determine the sensitivity of the simula- tion to discontinuous bottom topography. IV. SUMMAC Chan and Street [1970a] proposed the SUMMAC computing technique as a tool for analyzing two-dimensional finite-amplitude water waves under transient conditions. The method is, as noted above, a modified version of MAC which was developed by Welch, et al. [1966]. The essence of the initial modifications consisted of a rigorous application of the pressure boundary condition at the free surface and extrapolation of velocity components from the fluid interior so that inaccuracy in shifting the surface boundary is kept at a minimum. The objective of this section is to provide a summary of the SUMMAC method, of its application to water wave problems and of a number of new improvements added to SUMMAC since Chan and Street [1970a] was written. 4.1. Summary of the Method The fluid is regarded as incompressible and the effect of viscosity on the macroscopic behavior of flow is considered to be negligible. The entire flow field is covered with a rectangular mesh of cells, eachof dimensions 6x and dy. The center of each cell is numbered by the indices i and j, with i counting the columns in the x-direction and j counting the rows in the y-direction ofa fixed Cartesian coordinate system (Fig. 7). The field-variable values describing the flow are directly associated with these cells [ Welch, et al. 1966]. The fluid velocity components u and v and the pressure p are the dependent variables while the independent variables are x, y and the time variable t. In addition to the cell system which represents the flow field by a finite number of data points, there is a line of marker particles whose sole purpose is to indicate where the free surface is located. These hypothetical particles may or may not represent the actual fluid particles at the free surface, depending on whether one chooses the Lagrangian or the Eulerian point of view to calculate the motion of free surface, The marker-and-cell system provides an instantaneous repre- sentation of the flow field for any particular time. When an initial set of conditions is given, the entire fluid configuration can be ad- vanced through a small but finite increment of time 6t. First, the pressure for each cell is obtained by solving a finite-difference 170 Computatton of the Motion of Long Water Waves j= JMAX- ~-|~ -- Fig. 7. Cell setup and position of variables Poisson's equation, whose source term is a function of the velocities. This equation was derived subject to the requirement that the resulting finite- difference momentum equations should produce a new velocity field that satisfies the continuity equation (conservation of mass). The finite- difference equations of motion are then used to compute the new velocities throughout the mesh. Finally, the marker particles are moved to their new positions, their velocities being interpolated from the nearby cells. The new flow configuration now serves as the initial condition for the next time step and the foregoing procedure is repeated as many times as necessary for the investigation. With proper choice of 6x, dy and 6t, the SUMMAC algorithm is capable of yielding so- lutions that are computationally stable and also reasonably faithful in simulating the physical phenomena. Dimensionless variables are used throughout (cf., Sec. 3.1). The governing equations for an incompressible, inviscid fluid are Pa Tb t+vyz—=-y t+ By, (33) 171 Street, Chan and Fromm dv dv dv _ dp and du BV _ 9 (35) ox By Here, p is the pressure; gy, and gy are the x and y components of the gravity acceleration whose absolute value is g and t is the time variable. Also, if the direction of gravity is the same as the -y direction, then g,=0 and gy=- 1. Boundary conditions are easily derived for the fluid motion at the solid walls of the tank (cf. , Chan and Street [1970a]). For incom- pressible fluids with very low viscosity, such as water, it is suffi- ciently accurate to use at the free surface the single condition p= Pg (x; t) (36) where pg is the externally applied pressure at the free surface. Under usual circumstances p,= 0, but it can also be prescribed as a function of x and t for some problems. As shown in Fig. 7 the computation region is divided into a number of rectangular cells. The fluid pressure p is evaluated at the cell centers, while u is defined at the mid-point of the right-hand and left-hand sides of the cell and v is defined at the midepoint of the upper and lower sides. Then, for the cell (i,j) the following set of equations are derived from Eqs. (33) and (34): rei 7 ied t dt gy + (Pij - Piety) (37a) Tuite 7 oar t-5t gy ot ot (Pi.y, - Pij) (37b) “el = Viieg + OB, +e (Pij - Pijst) (37¢) Viel = Vijek + 6t gy + s (Pij-1 - Pij) (37d) In the above equations, variables with the superscript nt1 are related tothe nti‘ time step. Variables lacking a superscript are evaluated at the n't step. Thus, Eqs. (37) are suitable for updating the values of u and v about the cell (i,j). The "eonvective contributions" u* and v* are Computatton of the Motion of Long Water Waves * du du Wish) = Yiedj SE iis By itd (38) * av ov Vijed = Vijes + St ¢ - Ret ES 5 V By Dlist (39) * * n+l n+ where Vid) and Vij contribute to uj. j and vjjsk » respectively, through the convection process and ( ) represents Me as yet un- specified finite difference approximation of the enclosed terms. Before Eqs. (37) can be employed to compute the new velocities, the p field must be obtained. Consider the finite-difference conti- nuity equation [see Eq. (35)] we Fe adele ied n+l Ey ie j Bites j ij+ ~ Yij- _ Dij = a Pe ee (40) Substituting Eqs. (37) into Eq. (40) and requiring Dj, = 0 leads to the pressure equation a 21: $DO: 1; as + pi —— Pi+tj Pi-lj + Pij+l Pij-! + - Pij =( 5x2 Sy 2 Rij) (41) Here Z= 2(<5 + 52) (42) sj 1 ee aE 4 L ee ee ij iied - Vij-d a a ptt ot : ee M2) Near the free surface "irregular stars" (Fig. 8) must be used to de- rive an appropriate pressure equation so that, in the discrete sense, the free surface condition p =p, is applied to the exact location. Let ), No, Na, Ng be the lengths of the four legs of the irregular star (Fig. 8) and p,, Py», P3: Pg be the value of p at the ends of these legs. Then, it can be shown the irregular-star pressure equation is = — 179730 Pi 7 TPs + "4P2 + Pa yp. 44 Pi ~ 2mang t 17 Fes ns nn, wea peg Equation (44) reduces to Eq. (41) when Eq. (44) is applied to an interior cell. Street, Chan and Fromm Fig. 8. Irregular star for p calculations The hypothetical particles that mark the free surface are moved to their new locations according to their locally interpolated values of u and v. Fora given particle k we find the velocity component u, for the particle by making a Taylor series expansion about the nearest data point of the u field. Similarly, a series expansion about the nearest data point of the v field gives v,, the y-component of the particle velocity. With u, and v, available, each free surface marker particle is advanced by the following formulas: ri —s WW =] + _ = [o4] ct (45) n n+l yy + ve dt —- Hl where x and Yk refer to the position of the yh particle at the nt" time step. Also, the particle velocities are evaluated at the advanced, i.e., nti , time step. The quantities u and v are not defined outside the fluid domain, but they are needed to carry out the computations using Eqs. (37) and (43) and the particle velocities near the free surface. We calculate these undefined u and v values by a simple linear extrapolation from the fluid interior. A complete set of initial data -- the u and v fields and the position of a line of particles depicting the free surface, are needed to start the computation. The initial pressure p needs to be known only approximately, such as a hydrostatic distribution, because the p field is solvable if u and v are given. A704 Computatton of the Motton of Long Water Waves The evolution of fluid dynamics is calculated in "cycles," or time steps. At the start of each cycle the source term Rjj for each cell is evaluated by Eq. (43). The pressure p is computed only for those cells whose centers fall in the fluid region; either Eq. (41) or Eq. (44) is used as appropriate. The successive-over- relaxation method is used to solve the p field. The iteration is terminated when (m) (m-1) i CP <€ (46) for every cell, where (m) means the m'" iteration and eP isa predetermined small positive number. The accuracy in solving pij at the n~ time step has a direct bearing on the accuracy of satisfy- ing the continuity equation Di; nl 0 [Eq. (41)] at the nti" step. Smaller values of Dij We result len smalles €p are used, However, there is little improvement in reducing Di; for €,< 10°” because the round-off level of the computer has been reached. Now Eqs. (37) yield the new velocities. Then each marker particle is advanced to its new position by Eqs. (45). Thus a cycle is completed and the next one can be started immediately. The convective contributions given by Eqs. (38) and (39) can be approximated by a wide variety of finite difference formulas. Chan and Street [1970c] show that, while the original MAC and early SUMMAC equations used a first-order explicit method, second- order explicit methods are better. Of the two second-order explicit schemes studied, the so-called "upstream" difference alone rather than in a "phase-averaged" procedure yields better results in prob- lems where free surface waves are present. In,this upstream dif- ference, if Whim represents either Vises jp OF wii) > then for the case when Um ™0 and Vom >0 Use Won Gi Wm. l +E ens Wy) 4 (8-1) co ; Da + Fen) (47) where W pry = Weim + (wg tm) + (a-1) (Wp om - 20 im of W pm) (48) and 175 Street, Chan and Fromm n n ite StU pm , 6 = 5tV em 6x dy We examined the finite difference convection equations by the extended von Neumann method in which nonlinear equations are first linearized. The resulting criteria were jaj=i1; |pi[ =1 (49) A second criterion dt 1 tx < re! (50) where C =the surface wave celerity, was derived by considering the propagation of the free surface waves. These were simple linear analyses and can only be used as guidelines in choosing the time increment 6t for given 6x and dy. Because numerical dispersion is quite severe for short wave components, care must be exercised to provide adequate resolution for all the important features in the flow. As a rule of thumb, the smallest significant flow feature must be represented by at least ten cells. In both the MAC and SUMMAC a line of particles was used to mark the free surface position. A pair of (xy, y,) values were associated with the k' particle at the nth time step. Then Eq. (45) was employed to calculate (xe x.) This procedure is really a Lagrangian method that tends to be unstable after a large number of time steps. The problem is not serious for simulating solitary waves [cf., Chan and Street, 1970a]. But, in calculating periodic waves a given particle is moved up and down as each wave passes. In the process a small number is systematically added to and then subtracted from x, and y, contributing to very large round-off errors. In addition, there is no restraint on the individual particle positions because each is calculated independently of the others. To overcome the difficulty with moving particles, an alterna- tive approach using the Eulerian point of view can be developed. The flow region is divided by a number of vertical lines with equal spacing 4 and n is now the height of the free surface measured from the reference level y = 0 at the channel bottom. The horizontal posi- tions of these vertical lines are fixed and we only compute the change in n along each vertical line as time passes. The kinematic condition at the free surface, from the Eulerian viewpoint is 176 Computatton of the Motion of Long Water Waves ee (51) Many difference schemes may be developed to approximate Eq. (51). Our tests show that the forward implicit method with the difference equation qu _ nr al al th 5 Ue Be Tek oo ke (Seed ; ) (52) is one of the best. A stability analysis shows that Eq. (52) leads to a linearly stable computation with slight dampling. Numerical tests were carried out in the context of a simple physical problem whose exact solution was known, viz., a solitary wave ina horizontal channel. Among five alternative combinations of surface and correction term treatments tested, that using Eq. (52) and Eqs. (47) and (48) was the best. Now consider 6t. The maximum fluid speed in the above tests was u,,,* 0.30 and 6x =0.5 while 5y = 0.1. According to Eq. (49) max 6x _ 0.50 _ 1.67 or 0.30 Umax The speed of the surface wave is C=1.18. The Courant condition [ Eq. (50)] would require ox 0,50 = But the Courant condition should also be observed in computing the free surface. Because we used the spacing 4 =0.05 at the free surface, the condition A > 0205. bt < & = pq = 0.0424 must be satisfied. Therefore, the most restrictive condition is &t < 0.0424. In all the test examples, 6t =0.05 was used. This is slightly larger than the estimated maximum allowable 6t, but no distortions or instabilities were noted. However, the result of seriously violating the Courant condition, i.e., using 6t = 0.10, was large non-physical distortions that suggest one has to be careful about the choice of 5t. Street, Chan and Fromm Experiments were also performed on the problem of generating periodic waves by pressure pulse (Sec. 4.2). Instability at the free surface became explosive after 600 time steps when the particle method was used. Using the forward implicit method, we were able to calculate up to more than 3000 steps and there were still no signs of instability. The numerical tests described above indicated that it is ad- vantageous to use the second-order upstream difference method to compute the convective contributions to a ; and vn - For the free surface calculations, the forward implicit scheme is best. However, the particle method of computing the free surface need not be dismissed altogether. The Eulerian method is restricted to waves in a channel whose two ends are vertical walls. If the water surface has an advancing front, such as a solitary wave climbing on a slope [ Chan and Street, 1970b] , the particle method is the only choice, When 6t is small enough and the particle velocities are evaluated at the ntit? time step, the particle method does provide a stable solution. However, the particle method should not be used in the simulation of periodic waves over long periods of time. 4.2. Results and Discussion As an example, periodic pressure pulses were used to generate a train of oscillatory waves in a channel of constant depth (Fig. 9). The fluid is entirely at rest at t= 0. Then, the pressure distribution on, athe Xq R, = F(X,T) Fig. 9. Setup of pressure pulse problem 178 Computatton of the Motton of Long Water Waves p, sin (=e) . [eee nea for, OS x= xq Ps = (53) 0 for x > 4 is applied to the free surface. Here p, is the amplitude of the pressure pulse, Tp, is its period and xg is the horizontal length of the surface subject to the prescribed pressure. Equation (53) was employed by Fangmeier [1967] in solving the same type of problems using time-dependent potential flow equations. In the first case, a channel of the length L, = 30.0 was used. The computation domain consists of 80 X 24 cells, each with 6x = 0.30 and Sy = 0.10. Weused pp=0.10, Tp=7.6 and xq=4.0 in Eq. (53) to generate the surface disturbances. The development of the u field is shown in Fig. 10. The plot increment is 0.025 per contour line with u = 0.0125 on the contours closest to the ends of the channel. At t = 10.0 the leading wave leaves the generating area and progresses to the right. At t = 43.493 the first wave runs up the right-hand wall and reflection begins to interfere with the on- coming waves. As a result, a standing wave pattern occurs when t= 72.986 to t= 84. 233. | T2#10.000 T= 78.734 T= 72.986 T= 84.233 Fig. 10. Periodic waves (u contours) 179 Street, Chan and Fromm } \\ /} 7} \ J | | / TV. y ZA GINAIAI\\ JIN Soa | WWIII T= 10.000 T= 78.734 | BIAS = 4 LO T=#30.000 T= 80.984 T= 43.493 T= 62.484 LOI = ROO “A T= 72.986 T= 84.233 Fig. 11. Periodic waves (v contours) In Fig. 11, the time history of the v-field is shown. The plot increment for the contours is also 0.025 per line. On the chan- nel floor, v=0.0. The first contour above the floor has v= +0.0125 if it is in front of the wave crest, and v = - 0.0125 if it is at the back. The sparse contours on the right-hand side of the channel at t = 80.984 and t = 84.233 indicate that when the standing waves reach their peaks the fluid velocity almost becomes zero temporarily. This phenomenon is caused by the interaction of the reflected and incident waves that tend to alternately enhance and cancel each other. In Fig. 12 we used along channel with L; = 60.0. Thus the "progressive" wave patterns can be analyzed before the reflection sets in. The wave train is composed of a group of dispersive waves. The amplitude increases from the leading wave to the third wave. It then decreases on the following waves. This observation suggests that the nonlinear response of the fluid system is somewhat out of phase with the forcing function at the surface. Therefore, it appears that pure nondispersive periodic waves cannot be generated by the disturbance described by Eq. (53) unless the amplitude p, is very small. Because of its symmetrical profile, we selected the fourth wave in Fig. 12 and compare it in Fig. 13 with Stokes' second-order and third-order theories [ Wiegel, 1964]. Good agreement with the 180 Y/do 0.75 1.25 1.00 0.50 0.25 Computation of the Motton of Long Water Waves 24.00 30.00 X/do 36.00 Fig. 12. A train of nearly periodic waves T= 7.60 Ls 6.70 H= 0.55 d= 0.9348 23 24 Fig. 13. 25 26 X/do 27 SUMMAC STOKES' SECOND ORDER THEORY STOKES' THIRD ORDER THEORY 28 29 30 Comparison of wave profiles 181 60.00 Street, Chan and Fromm third-order theory is found. To obtain a meaningful comparison with the profiles of the Stokes' waves, a Fourier analysis was performed on the profile computed by the SUMMAC method. The SUMMAC wave profile in Fig. 13 can be expanded in a Fourier series of the Stokes form [ Wiegel, 1964]. The coefficients can be evaluated by the standard procedures in calculus. The first ten coefficients have been com- puted and compared with those for the Stokes' theories. From the trend of each coefficient, it appeared that as the order of approxi- mation increases the Stokes' wave converges to our numerical solution. Also, in comparison of wave speeds we find good agree- ment with Stokes' third-order theory. The difference is within 0.4 per cent. In Fig. 14 the distribution of u under the wave crest and the wave trough is compared with Stokes’ theory. The SUMMAC method predicts a much lower u velocity under the crest than Stokes' solu- tions. This discrepancy is probably caused by the fact that the numerical simulation was made in a channel of finite length which is a closed system and the waves have not quite reached the steady state, while the Stokes’ waves hold for an infinitely long channel. Nevertheless, the slope of the u-distribution (i.e., 8u/8y) is very close to that of the third-order theory. (a) (b) UNDER WAVE UNDER WAVE TROUGH CREST Y/ do ° SUMMAC —— STOKES' SECOND ORDER THEORY ——— STOKES' THIRD ORDER THEORY Fig. 14. Distribution of u under wave crest and trough 182 Computatton of the Motton of Long Water Waves 1.50 DIRECTION OF MASS TRANSPORT 1.25 STILL WATER LEVEL 1.00 Y/do 0.75 B 3 % a 8 a et oe Ti.2s 11.80 11.75 12.00 te.e9 12.50 12.75 13.00 13.25 13,50 13.75 14,00 X/dy Fig. 15. Motion of fluid particles The paths of the fluid particles are plotted in Fig. 15. We selected three fluid particles which lie on the vertical plane x = 12.0 at t=0.0. Their initial vertical positions are y = 6.0, 0.5 and 1.0, respectively. The instantaneous particle positions are plotted at every 5 &t's (6t = 0.05). Each particle moves in an oscillatory pattern which completely differs in nature from the translation motion in a solitary wave. The surface particle travels in a quasi-elliptic orbit but never returns to its original position. Thus, there is a net mass transport in the direction of wave propagation near the free surface. At half water depth the scale of the orbits is smaller and the current (mass transport) is opposite to the wave direction. On the channel bottom the particle merely goes back and forth hori- zontally and the "backward current" is also larger there. Because the wave channel in our simulation is a closed system, the fluid carried along by the surface waves must return in the opposite direction in the lower fluid layers. Finally, a comparison was made with the numerical solutions of Fangmeier [1967]. The qualitative agreement was good, as was the agreement in the wave phase; however, the SUMMAC method gave a much better treatment of the free surface that markedly re- duced the height of the largest of the waves as compared to Fangmeier's simulation. Street, Chan and Fromm 4.3% Prognosis The successful application of the SUMMAC technique to several physical problems indicates its usefulness as an engineering research tool for analyzing the dynamics of water waves in two space dimensions. It is capable of providing accurate quantitative results as well as qualitative descriptions [see, e.g. , Chan and Street, 1970b]. In addition, rapid advance in the design of high-speed com- puting systems makes numerical modelling economically feasible. While it is possible to employ the SUMMAC technique to attack a wide variety of water wave problems, some limitations inherent in the method must be noted. First, as a result of achieving a high degree of accuracy in applying the free surface pressure con- dition by using irregular stars, waves after breaking cannot be simu- lated. When breaking occurs, the computation must be terminated. Second, only non-turbulent flows are considered in our model. Although laminar viscous damping has little effect on large scale wave motions, energy dissipation due to the turbulence can be sig- nificant. However, a recent study by Pritchett [1970] shows that it is feasible to implement a heuristic simulation of turbulence in the MAC framework. ACKNOW LEDGMENT This research was supported in part by the Fluid Dynamics Branch, Office of Naval Research, through Contract Nonr 225(71), NR 062-320. REFERENCES Brennen, C., "Some Numerical Solutions of Unsteady Free Surface Wave Problems Using Lagrangian Description of the Flow," 2nd International Conf. on Numer. Meth. in FluidDyn., Berkeley, Calif. , Springer-Verlag, Pub. , September, 1970. Camfield, F. E., and Street, R. L., "Shoaling of Solitary Waves on Small Slopes," J. Waterways and Harbors Division, ASCE, V. 95, No. WW1, Proc. Paper 6380, pp. 1-22, February, 1969. Carrier, G. F., "Gravity Waves on Water of Variable Depth," J. Fluid Mech., Vol. 24, Pt. 4, pp. 641-660, April 1966. Chan, R. K. C.,*‘ and Street, R. L., "A Computer Study of Finite- Amplitude Water Waves," J. Compt. Physics, Vol. 6, No. 1, August 1970 [1970a]. 184 Computatton of the Motton of Long Water Waves Chan, R. K. C., and Street, R. L., "Shoaling of Finite- Amplitude Waves on Plane Beaches." Proc, 12th Conf. on Coastal Enginer- ing, Washington, D.C., ASCE, September 1970 [1970b]. Chan, R. K. C., and Street, R. L., "SUMMAC -- A Numerical Model for Water Waves," Stanford C. E. Dept. T. R. No. 135, Stanford, Calif., August 1970 [1970c]. Chan, R. K. C., Street, R. L., and From, J. E., "The Digital Simulation of Water Waves -- An Evaluation of SUMMAC," 2nd International Conf. on Numer. Meth. in Fluid Dyn., Berkeley, Calif. , Springer-Verlag, Pub. , September, 1970. Fangmeier, D. D., "Steady and Unsteady Potential Flow with Free Surface and Gravity," Ph.D. Dissertation to U.C. (Davis), Davis, Calif. , 1967. Forsythe, G. E., and Wasow, W. R., Finite-Difference Methods for Partial Differential Equations, J. Wiley and Sons, Inc., 1960. Freeman, J. C. and LeMéhauté, B., "Wave Breakers on a Beach and Surges on a Dry Bed," J. Hydr. Div., ASCE, V. 90, No. HY2, pp. 187-216, March, 1964. Fromm, J. E., "Practical Investigation of Convective Difference Approximation of Reduced Dispersion," IBM Research Report, RJ531, IBM Res. Lab., San Jose, Calif., 1968. Heitner, K. L., "A Mathematical Model for Calculation of the Run- up of Tsunamis," Earthquake Engrg. Res. Lab. Report, Calif. Inst. Tech., Pasadena, Calif., May 1969. Hirt, C. W., Cook, J. L., and Butler, T. D., "A Lagrangian Method for Calculating the Dynamics of an Incompressible Fluid with Free Surface," J. Compt. Physics, V. 5, No. 1, April, 1970. Hwang, L. S., Fersht, S., and LeMehaute, B., "Transformation and Run-up of Tsunami Type Wave Trains on a Sloping Beach," 13th Cong. of IAHR, Kyoto, Japan, 31 Aug. to 5 Sept. 1969. Keulegan, G. H., and Harrison, J., "Tsunami Refraction Diagrams by Digital Computer," J. Waterways and Harbors Div., ASCE, V. 96, No. WW2, Proc. Pap. 7261, pp. 219-233, May 1970. Lautenbacher, C. C., "Gravity Wave Refraction by Islands," J. Fluid Mech., V. 41, Pt. 3, pp. 655-672, 29 April, 1970. 185 Street, Chan and Fromm Leendertse, J. J., "Aspects of a Computational Model for Long- Period Water- Wave Propagation," RAND Memo RM-5294-PR, Santa Monica, Calif., May 1967. Madsen, O. S., and Mei, C. C., "Dispersive Long Waves of Finite Amplitude over an Uneven Bottom, MIT Hydro. Lab. Rep. No. 117, Dept. of Civil Engrg. , Cambridge, Mass. , November 1969 [ 19692] : Madsen, O. J., and Mei, C. C., "The Transformation of a Solitary Wave over an Uneven Bottom," J. Fluid Mech., V. 39, Pt. 4, pp. 781-792, 15 December, 1969 [1969b]. Mogel, T. R., Street, R. L., and Perry, B., "Computation of Along- shore Energy and Transport," Proc. 12th Conf. on Coastal Engrg., Wash., D. C., ASCE, September 1970. Peregrine, D. H., "Long Waves ona Beach," J. Fluid Mech., V. 27; Pt. 4, Ppp. 815-827, 1967. Pritchett, J. W., "The MACYL6 Hydrodynamic Code," Info. Res. Assoc., IRA-TR-1-70, Berkeley, Calif., 15 May, 1970. Schreiber, D. E., "A Generalized Equipotential Plotting Routine for a Scalar Function of Two Variables," IBM Research Rep. RJ-499 (No. 10673), Computer Applications, New York, 24 May, 1968. Street, R. L., Burges, S. J., and Whitford, P. W., "The Behavior of Solitary Waves on a Stepped Slope,"Stanford C. E. Dept, T.R. No. 93, Stanford, Calif., August 1968. Street, R. L., Chan, R. K. C., and Fromm, J. E., "The Numerical Simulation of Long Water Waves -- Progress on Two Fronts," Proc, Intl. Symposium on Tsunamis and Tsunami Res., East/West Center, Honolulu, Hawaii, October 1969. Vastano, A. C., and Reid, R. O., "Tsunami Response for Islands; Verification of a Numerical Procedure," J. Marine Res., V. 25, Ne. 2; pp. 129-139, 1967. Welch, J. E., Harlow, F. H., Shannon, J. P., and Daly, B. J... "The MAC Method -- A Computing Technique for Solving Viscous, Incompressible, Transient Fluid- Flow Problems Involving Free Surfaces," Los Alamos Sci. Lab. Rep. LA-3425, 1966. Wiegel, R. L., Oceanographical Engineering, Prentice Hall, Englewood Cliffs , New Jersey, 1964. 186 Computation of the Motton of Long Water Waves Williams, J. A., and Kartha,'K. K., "Model Studies of Long Wave Amplification by Circular Islands and Submarine Seamounts, " Hawaii Inst. of Geophys., Final Report (HIG-66-19), November, 1966. 187 wee AN UNSTEADY CAVITY FLOW D. P. Wang The Catholte Untversity of America Washington, D.C. I. INTRODUCTION A perturbation theory for two-dimensional unsteady Cavity flows has been formulated by the present author and Wu [1965]. In that formulation we regard the unsteady part of the motion as a small perturbation of a steady cavity flow already established. This already established steady cavity flow will be called as the basic flow. Our perturbation expansion is carried out in terms of a set of intrinsic coordinates (s,n) of the basic flow. The coordinate s is the arc length measured along a streamline in the direction of the basic flow, and n the distance measured normal to a streamline. An illustration is given in Fig. 1, where the solid lines represent the basic flow configuration, AB represents the wetted side of the solid body, AI and BI, the two branches of the cavity wall which is a free surface. Also shown in Fig. 1 is the unsteady perturbed flow configuration represented by dotted lines. The unsteady ee h(s,t) - |= - = aa Fig. 1 Illustration of an unsteady perturbation flow * This paper will henceforth be referredtoas W. 189 Wang displacement of the free surface and the solid body from their cor- responding locations in the basic flow is denoted by n = h(s,t). If the wetted side of the solid body and the free surface of the basic flow are taken to be n=0,h(s,t) is assumed to be a very small quantity. In W we have found that the linearized kinematic and dynamic boundary conditions on the free surface for the unsteady perturbation potential $, are oe se tage on n=0 (1) and 0, (2) Bh vg Stee gn ons and that the boundary condition on the solid body is ee = oP + ee (a,b) on n=0., (3) In the above equations R and q, are respectively the radius of curva- ture and the constant speed on the free surface of the basic flow, and q, is the speed of the basic flow. The + (or -) sign onthe right- hand side of (2) holds for the upper (or lower) branch of the cavity wall; these signs are necessary to make R a positive quantity. We should mention here that in obtaining (2) we have assumed that the cavity pressure remains unchanged during the unsteady perturbation. If we regard q,/R as an equivalent gravitational acceleration and the s-coordinate rectilinear, then (1) and (2) are in the same form as the linearized free surface boundary conditions in water wave problems. Thus we expect that the centrifugal acceleration q?/R due to the curvature of the basic flow streamline should play the role of a restoring force in producing and propagating the surface waves along the curved cavity wall. The purpose of the present paper is to use this perturbation theory to study some unsteady behavior of the Kirchhoff flow when the solid plate is in small harmonic oscillations. II. THE BASIC FLOW In this paper we consider the basic flow to be a flat plate held normal to an incoming uniform stream of infinite breadth, with-a cavity formation of infinite length as shown in Fig. 2. This is the so-called Kirchhoff flow. Both the speed of the incoming stream and the length of the plate AB are taken to be unity. A set of Cartesian coordinates (x,y) with its origin at the stagnation point C is chosen as indicated in Fig. 2, where the point I denotes the point at infinity. 190 An Unsteady Cavity Flow z— plane Fig. 2 The basic flow and its conformal mapping planes The solution of this problem can be obtained by the Levi-Civita method in terms of a parametric variable ¢ (Gilbarg [ 1960]), and we simply give it below for subsequent use. The complex potential f,, velocity w, and the complex variable z=x+iy are f.-7(t+¢) (4) w= Sy > (5) and £94 Wang g 1 z=) Wo dc , (6) where K=33— (7) The flow region in various planes can also be found in Fig. 2. For points on the plate AB, we may deduce from (6) x= > (sin 20 - 4cos 0~ w~ 20), (8) where @= Arg, when © is onthe circular arc ACB, and -nr2Z=0s0. (9) III UNSTEADY PERTURBED FLOW It is shown in W that by eliminating h between (1) and (2) and transforming (s,n) and R tothe variables fy and Wo a single free surface boundary condition in complex variable form can be obtained, which is Re [L(f,)] = 0, (10) where f, = >, + ip, (11) is the complex perturbation potential, and the linear differential operator L is L= (s+ wr) -{elSG (Fe tor) - Se or (12) It is also shown in W that the boundary condition (3) on the solid body can be transformed into Im gel = - alte + oe (ach) (13) 192 An Unsteady Cavity Flow where h is regarded as a given function of s and t. Since along the solid body df, =q,ds, which is purely real, (13) may be written as * dh Im f =- f 37 As = qoh, (14) where we have set Imf, to zero at s = 0, the stagnation point of the basic flow. For the basic flow considered and from the definition adopted for the intrinsic coordinates (s,n), we note that along CB do = Wo: (s,n) = (x,y), (15) and along CA Go = - Wo: (s,n) =- (x,y). (16) If we denote the prescribed motion of AB as y= 1,(x,t) instead of n = h(s,t), with the aid of (15) and (16), (14) becomes x 8 im t= - J) sapl dx - won. (17) Let us assume that the prescribed motion of AB is given by n (x,t) = € cos ot, (18) where e€ is avery small constant quantity and w is the frequency of oscillation. For convenience¢ in the following analysis, let us intro- duce an imaginary unit j = a which is regarded as different and non-interacting with the imaginary unit { used in defining the com- plex variable z=xtiy. If we agreed that only the real part with respect to j of a quantity is meaningful to us, we may write (18) as mi, Goat) = ee). (19) To avoid any confusion in the notation, from now on when we mention the real (or imaginary) part of a function ¥, denoted by Re (or Im 3) as it has been used so far, we mean the real (or imaginary) part of 3 with respect to i, not with respect to j, evenif 3 con- tains j. Only when the final result is obtained shall we take the real part with respect to j as our solution. If we assume that the disturbance has already been applied for a long time so that the entire flow is in harmonic oscillation, we may write the complex velocity potential f(z, t) as 193 Wang f(z,t) = f(z)e!™’, (20) If we substitute (20) into (10) we may write the free surface boundary condition as Re H=0 (21) where 2 5 fe a 625, Abd ae Sea at a [o? + jo in (4 awe) ¢, (22) df df, df,/ df, df, Wo df which is an analytic function defined in the flow field. We may also write 2 i df 1 [ dw, ye H = ag et 2jo- In Wo aa Wo dz Wo ( She 1 dw - |w rare aa peattcel |e. (23 [u? +; = CoeerR ) ) Since the differential operator L is purely real on the plate AB, the boundary condition (17) may be expressed in terms of the analytic function H. By a straightforward application of L on (17) and by the use of (19) and (20), the boundary condition on AB may be written as Im H= y, (24) where 2 2 =€ [- ok + jo(w crs ova + [wrx jo(w, + me, dfy df ; 4 d z - jo(w, - 3 )] Ge tm wo + Jo atv + 2)I. (25) The boundary conditions expressed in the forms of (21) and (24) may be used to determine H for points in the interior of the flow field. However, to obtain a physically acceptable H, other boundary con- ditions have to be imposed on H. We shall assume that the free surface displacement due to the unsteady disturbance of the plate AB has to be bounded everywhere. This condition can be satisfied if the free surface displacement is bounded at the separation points A,B and at the point at infinity. 194 An Unsteady Cavity Flow It is shown in W that the curvature of the free surface of the basic flow is 1 . | Ge de : (26) R Wo df, where q,= 1 inour problem. From the local mapping behavior near z| = 00 between the w,-, f,- and z-planes shown in Fig. 2, we see that Wo ti” .agr (27) and o> = iz as lz| — ©, (28) where a, is a constant. In the following analysis we always use an to indicate some constant. The substitution of (27) and (28) into (26) gives us { -3/2 er Olz] ) (29) on the free surface as |z | — oo. Since we assume that the free sur- face displacement near the point at infinity has to be bounded, then, from (2) and (29), we have, near the point at infinity, spi B=. 00 For harmonic oscillations, (30) suggests that we may write = Ale )efO as |z| + &, (31) since along the free surface of the basic flow 0/8@s = 8/@f). The substitution of (31) into (2) gives j -f dA _jwlt-f) pea as |z| — oo. (32) is) In view of (29) and (28), (32) ipplies that along the free surface near ies the point at infinity A = O(z , at most, in order that h be bounded there. With h being bounded at infinity and having a form shown 195 Wang in (32), we can obtain, from (1), that on the free surface 99,/8n = o(z7') as |z|-—- oo. These results indicate that along the free sur- face near the point at infinity both f and df/dz should vanish. Since the unsteady disturbance is mainly a surface phenomenon, it is not unreasonable for us to assume that f and df/dz also vanish in the interior of the flow field near the point at infinity. Therefore, we assume that HO as |z| — oo. (33) This rules out the possibility that there is any induced circulation around the point at infinity. Based on the assumption that h is bounded at infinity and the result that on the free surface 909,/8n = o(z) as |z | —> oo, an integration of (1) will show that if oo Gel) **#5>0 (34) in the neighborhood of the points A, B with r the distance from these points, h will be bounded at A,B. Condition (34) is also necessary in order that the pressure be integrable over the plate AB. To facilitate the determination of H, let us introduce a transformation , (35) G=T- (7? - 1ylf2 where the cut in the LG hep cnaks is taken along the straight line between -1 and 1, and (72-1)'/* +7 as |t| +m, -aw 1 (40) - 27(1 - reife on |Re T | 0 as |t| > o, (43) since near | z | = 0 Ta aces (44) To satisfy (43), b, = 0 for i= Zs (45) Due to the symmetry of our problem, which implies Im f=.0 on Cl, (46) and due tothe fact that the differential operator L is purely real on CI, we require that be O. (47) This leaves only the constant b,; undetermined. After carrying out the integrations in (42), we may write H(t) = ooS95 papery | j(Ku)?M,(7) + (Kul?M, (7) 2 QnjKor + b\7%{(7?-1)'? ], (48) where al M, (7) = 07%(7°-4)(47 +9) +7771)? [ 20- 5 -4(1 tn)7"] 4 Array? in — ore aty os(a), (49) 198 An Unsteady Cavity Flow M(t) = 2ur(672-5) + w%{572-1) - 2(77-1)/* [4G + (546m) 77] + 7(4-572)(72-1)/2 In Zoe + (2-37%)(72-1)'® (7), (50) G = Catalan's constant = 0. 915965594, (51) and | = -| or) = WesBe err) 8 ~tw=cos ¢=0, (52) which cannot be expressed in terms of elementary functions. It is not difficult to see from (48) that as | 7 | — oo, the dominating term is the one containing b,, which is of the order eae : Since the remaining terms in (48) are obtained from the integral shown in (42), they are, therefore, of the order |7 ee This indi- cates that the b, term is the most important term for the flow field near the point at infinity. With H given by (48), (22) may be regarded as a linear, second order, ordinary differential equation for f. If we transform the independent variable from f, to G and make the following change of dependent variable 1/2 f2 F(t) (S72) acy (53) The differential equation (22) is readily reduced to a°*F ae where 2 1/2 df dw jwfo G(t) = ($7) (Ge) eH, (55) f, and wy, as functions of ¢ are given by (4) and (5), and H is given by (48) with 7 as a function of ¢ given by (35). To help us to understand the properties of the Eq. (54), let us make the follow- ing change of variables 199 Wang ge ioe (56) F imei” , which transforms (54) into ar y (1 - 8jK 2p) = = 4ice?* 5 ap? - 8j)Kw cos 2B)F =- 4iGe §. (57) This is Mathieu's differential equation. The flow region now occupies in the B-plane a semi-infinite strip shown in Fig. 4. In principle, a general solution of (57), or (54), can be obtained. If we denote the solutions of the homogeneous equation of (54) by F,(¢) and F,(), then a general solution of (54) is C C F(¢) = WOR) fF (¢) \ F,(A)G(A) dd - F,(6) \ F, (A) G(A) art, 3 4 (58) where dF W(F|,F,) = art F, - SF, (59) which is a constant. B-plane (7774 40) (37774 ,0) Fig. 4 0 | | €— plane 7/6 Fig. 5 A conformal mapping plane of the basic flow region bounded between ICAT 202 An Unsteady Cavity Flow as |é | — oo, and if the distance between the boundary lines of D does not tend to zero as |&]|— oo in any subdomain of D, then, a uniformly valid expansion of U in terms of Airy functions can be obtained. Olver [ 1957] later extends the result to the case when yp“ is a large complex parameter. For our problem, all the above requirements for a uniformly valid expansion in terms of Airy functions are satisfied except that the parameter pw” in our case is jKw, where j is an imaginary unit independent of the imaginary unit i used in the complex variable §. Since i and j do not interact and j may be regarded as a real quantity so far as the imaginary unit of i is concerned, we assume that Olver's result is applicable here and write the two solutions of (64) as = Ai[(jKu)” é][1 + 0(<)] (70) and Algae = Mi +o) ast o-oo, (71) where Ai(X) is the Airy function with argument X, which may be expressed either as the sum of two converging series or as an integral given in the following (Jeffreys & Jeffreys [ 1956]) 27i/3 a / xs- Ls? Ai(x) = a e * dg. (72) mi eee Substituting x =(jKa)'/* into (72) and manipulating the result, we can show that AY(JKu)'”¥] = 3{(1-17) ai[(aKey!E) + (1H) Ail(-iKa)”"E]}, (73) where Ge and (eile will be taken as ewe and aris respec- tively. Therefore, from (63) oN *) =(§ %) ie 1/3 1/3 ~2(F by {(1-ij) Ail (Ko) 6] + (1+ij)Ai[(-iKe) 6]}. (74) Similarly, 203 Wang dé -1/2 F, = (e) Us, dé -1/2 ori/3 ‘ ~2 (Se) (1-4) ailGK aye 8] + (1445) Aili a)'3 0? 7}, (75) In (74) and (75) dé -1/2 £ 1/4 CaM Ga ae sh where € is given by (62) and x given by (60). With F, and F, given in (74) and (75), we may find W(F, ,F,), which is 1/3 jw/6 -iw/3 W(F, , F,) kw s(Ka) e Z (77) or WR Fy ays liners Geren, (78) 6) Let us now express the solution F interms of the 7 variable. From (4), (5) and (35), we have fo= KT; (79) 2 (SP) = 4K?7?(7? = PE tS 4)'72}2 ; (80) and away". 3 [per Sa = ty (81) (az ) ~ V2 ~ ~ pts When we change the variable of integration in (58) from © to T, we need the quantity d€/d7, which may be obtained from (35) and is Se - [7 - (r? - 1)? cr? - ty. (82) F 2 If we denote G(t)d$/dt by g(t)e!“*, then with the aid of (55), (48), (79), (80), (81) and (82) 204 An Unsteady Cavity Flow te fitr ty7r&-'4) 34] (r2_ 1)'72 [j(Kw)°M, (7) +(Ke)*M,(7) g(T) = - 2njKwt + b,7%(7? - 1)7]. (83) In (83) we note that although b, is purely real with respect to i, it may be complex with respect to j; we also note that the term 2mjKwT is extremely small as compared to the term j(Kew)® M (T) as w—+ oo, and since both of them are dependent on j, so we may neglect 2mjKwT from (83) and write vee | [atest ety tT] [j(K&)°M, (r) Ag SORE +(Ka)'M,(7) + b,7%(7?- 1)'7] . (84) The asymptotic form of F can now be expressed as 1 io WE Fy File] i. F,[ t(o)e** - Fal oc | F,[t (o)] e(o)eX* ao}, (85) Gg where o is the integration variable in the T-plane and F,(¢), F4(¢), g(7) and W(F,,F,) are given in (74), (75), (84) and (77). Substituting all the necessary results obtained above into (20); noting the relations (1 £ij)> = 2(4 ij) (86) (1 +ij)(41 - ij) = 0 when we are taking the real part of (20) with respect to j, we obtain the complex velocity potential f,; as ee (eis eae (reqir®=1 ~ 4) = Pivolteucskae ; at mY Meneses) { Aif(iKs)'E] Ail(Ke)"e?"/%] . me 205 Wang Ai[iK w)!3 © 77%] Ai[(iK w)'72 ti} EAN g (01+ B,g,o)] de ' Ti _jw(t-Kr7+Ko 7) | +l ye e* { aif(-iKu)' 6] Ai[(-iKu)”4 @ M/E) oe ree -27ri/3 - Ail(-iKw) 6] aif(-ika)’? EEA)! [e,(0) B g3(c)] dc (87) where § and x as functions of T are given by t 2.972 | 3ri/4 do ee “a | (2 - 1)'4 and x= [( BS -r] (89) € and X indicate the functional values of & and y when the variable 7 is replaced by the integration variable o, g,(7) = Saar pie MoM (7) + Ka)? M7) (90) g(T) = earn al i(K.w)?M,(T) +(Kw)*M,(7)] , (91) SA ie Tift + yr? - 1) - 4), (92) B, is an arbitrary complex constant which is derived from b,, and B, is the complex conjugate of B,. The lower limits of integration shown in (87) are chosen for our convenience. Therefore, when necessary, homogeneous solutions of the form 206 An Unsteady Cavity Flow A(t) elt) ATK yy! ale n 2 ‘ Altre Ai[(iKw)'” a ; 95 A(T) eK ail(-iK al] ae and -iw(t-Kr2) -27i/3 A(T)e Ailcikay” e ay él, where 1/4 A(t) = —2) (94) T - V7? -1-i may be added to (87). We shall now study the behavior ofthe solution (87) for |r| > R, >> 1, 0 = Arg? = 1/2. 4A circular arc of radius Ry, 4s drawn in the o-plane as indicated in Fig. 6. Also shown in that figure is a hyperbola S representing the equation Re [ i(o# - 7?)] =0. Fig. 6 Paths of integration for the solution given in Eq. (87) when 7 is large 207 Wang If we are travelling along S inthe direction of increasing Imo and if we restrict ourself to the first, second and fourth quadrants of the o-plane only, then on the right-hand side of S Re [ i(o?-77)] <0, and on the left-hand side Re [i(c*-7?)] > 0. When either Im 7 or ReT is zero, S degenerates into the positive real and imagi- nary o-axes. From (88), we note that Scan: (95) and 1 T Arg § =zArgTts as |t| + c (96) Therefore, for 7 and w large, the arguments of the Airy functions appearing in (87) are large. Their asymptotic representations are (cf. Jeffreys & Jeffreys [1956]), AR cep 2 Ll WwW C—O aw (Kay 274 when ~-carge R, -1/4 -| \/4 i/8 -3/4 Bee ema ee Alo any. ee is (98) And it is easy to see, as some related explanation has been given in the paragraph after (52), that as |7| > R, eel 4. 7h, 9/4 Xx g,(T) t B g3(7)] a B, 2 (99) -|/4 om — _/4, 7i/8 9/4 x [ g,(7) ~ B g,(7)] = Sat oe T : Let us denote the a of the potential represented by the first inte- gral in (87) by f, (e Haleatize ss» lt fey Js eh Gao Cee er ae T mi +iw(t-Kr + ko*) -27i/3 xf eee { Ail(axia)”* €] aif (Ku)? 7”) oe : Ail (iKo)!> ee Ail(iKw)”° =] \ Creal g (oc) +B, ga(c)] do (100) For any 7 with |7| >R, and 0= Arg 7 = 1/2, the path of inte- gration in (100) = always be chosen to be Lj, which is a path coming from ooe * to 7, lying completely on the right-hand side of the hyperbola S and outside the circular arc lo | =R,. A typical L; is shown in Fig. 6. Since along L; |o| >R,, we may substitute the expansion given in (97), (98) and (99) into (100) and obtain B See Kt 2K) aS, mae COV8 oi ah @(E, é) do, (101) 1(2Kw) Ly where (6,2) = 2 eA fKag” - PY). (102) 209 Wang We note that along Li the hyperbolic sine function in (101) may be- come exponentially large, however, since L,; lies on the right of S, thefactor exp [iKw(c?-7?)] will be overwhelmingly small there. Let us now integrate (101) by parts once to get Tt 2 2 (e) i iw(t-Kt tka ) oy < Eee 4 fe Bsinh @(€,€) n(2Kw)/273/4 (2iKw) Z jw(t-Ke 9k o*) S Ua... ov = ri/4 © [2° sinh ®(€,&) 008 oss wi/4 + 04 Se cosh &(E 2) ac} (103) The integrated part in (103) ig identically zero; when we evaluate it at the upper limit, sinh ®(€,6) = 0, and at the lower limit, the factor exp iKwo’) is overwhelmingly small. If we integrate the integral (103) successively by parts and note the relation that wi/4 wW/2 ae -/2 2 Bets CK oe pee -* : (104) 37i/4 -yY2 where gue dé /do a a ee o is obtained from (88), we can show that (e) i iwt - fe (105) 41(Kw) Since: j= O(z'/2) as |z | — oo, from (104), we see that the contri- bution to the potential due to f/®) is of order |z[’? for |z| large. This type of potential is acceptable. Now, let us denote the Bart of the potential represented by the second integral in (87) by anaes | ALL De ZE (g yy ( FB iutt- Kr+Ko*) 2 eee (Ku)! (T= Vr2-1 =i) a x {ail (-1Ka) €] ai(-iK we ?”8} ~ Ail(-ika)/Fe°?7”E) aif(-iKw)'7E] | (E/2)'4[ g,(0)+B, g,(0)] ac. (106) 210 An Unsteady Cavity Flow To investigate the behavior of ee for |7|>R _, let us divide the region in the first quadrant of the o-plane outside the circular arc |o| = Rj into two parts, D; and D3. D,, as shown by the shaded area in Fig. 6, is bounded by the circular arc |o| = Rj, the imagi- nary o-axis ye the hyperbola S' representing the equation Re [ i(o?- R?e””“)] = 0. D, is bounded by the real c-axis, |o| = R, and S'. For T in Dg, we may choose the path of integration in (106) to be Ly, whichis similar to L, except now it lies on the left-hand side of S. Atypical Ly is shown in Fig. 6. Usinga process similar to that used to obtain (105) from (100), we may obtain, from (106), (Ww) eByi i -I SS a i 2 4 1(Kw) (107) When 7 is in Dj, the hyperbola S will be extremely close to both positive axes of the o-plane, S degenerates into the axes when T lies on the imaginary o-axis. For this case we have to deform the simple path Le into L3+ La, as shown in Fig. 6, in order that the factor exp [-iKw(o?-77)] will not become oo eee large along the path. Lz is a path coming from coe ”'’* to 0 on the lower-half o-plane, from there along the real o-axis towards o = 1, turning a small circle to the upper side of the real c-axis and along it to 0 on the upper-half o-plane, to circumscribe the cut in the o-plane, and then leaving 0 to coe3/4, Path La comes from ocoe?™'/4 to 7, lying completely on the left-hand side of the hyperbola S and outside the circular arc |o | = R,;. The integration along Lg is convergent; near both ends of the path the integral is exponentially small, near o=0 M,(c) and M,(c), which appear in g,(c), are of order o% and o respectively and g3(c) is of order G2: near o = 1 only Mj(c) contributes to ga(c) a square-root type of, singu- larity. The latter property of M,(c) is the reason that (Kw) M,(c) is being kept in the integrand together with the term (Kw)?Ma(c). We may remark here that if we did not neglect the term 2mjKw7 appearing in g.(c) from g(7) in (83), we would have aterm of the form 2miKwo. Since the presence of such a term would not affect the property of the integrand along L, near,o»=0 and 1, and since it is one order in w smaller than the (Kw) M,(c) term, its neglect is justified. Let us now denote the contribution to fm from the inte- gration along L, by Z(7), 2€ (B/x yi Fleiwtt-Ked ps Z(T) = —= Ai[(-iKw)°~ € (Ku)? posing a e { i[(-i ] wer m4 ; woe™/4 Saray Ai[(-iKu)'” patel (E/x)'4 | (co) +B, g.(o)] do - [ 3 244 Wang @omi/4 -27i/3 =f slaiptouna) ee aes Kee yesh rica) PET EAM 3 [e,(o) +B,g,(o)] ao} . (108) We note that when 7 is in D, the, Air TA Ail (=iKw) Vee exponentially large and Ail (5 iKw)!/? eat is exponentially small therefore, in order. that {\" tend to zero as |7| + oo in D,, we require that the coefficient Ai[(-iKw)'/3&] be zero. This deter- mines the constant B,, or B,, : 2 5, - Pi aaes pee on Aaa | (a g(c) av/ L3 ; 2 i Ps ateary Per Ey GQ) *g.(o) do. (109) L 3 With B, given by (109), Z(7) becomes exponentially small when tT isin D,. We shall not attempt to evaluate By, explicitly in this paper, however, in view of (91) and (92) we may conclude that B, = O[(Ku)*]. (110) Since Ly is outside the circular arc lo | = R,;, we may substitute the expansions given in (97), (98) and (99) into the integral along Ly to obtain a Tt x= -iw(t- Kr 4K o2) 41 X oM sinh [ 5 eel a Fa ie ae recy do. (111) £") Now, if we apply the method of integration by parts to the integral along Ly, we can show that for 7 in D,, with Z(T) being expo- netially small, Be i ea (112) AnibK wo) Summing up all the results obtained in (105), (107) and (112), we 202 An Unsteady Cavtty Flow have ei i f| ~ ——s (B,e 4n(Kw) T as [il 0. (113) From (113) we may derive the following results: (i) From (110) we may conclude that f,* eKw. (ii) For 7 lying on the real 7T-axis, which corresponds to the cavity wall of the basic flow, f, is purely imaginary; this indicates that near the point at infinity the perturbation velocity wy, = 0f,/8z is always perpendicular to the original cavity wall. (iii) If we recall that 7 = O(z'/#) as | 7 | — oo, we can see that the perturbation velocity is of order |z lees for large values of |z|; this, together with the result stated in (ii), implies that the unsteady free surface displacement tends to zero as |z|—~ oo. (iv) Along the imaginary T-axis, which corresponds to the line of symmetry of the flow, f, is purely real; this means that there is no velocity component normal to the line of symmetry which, of course, is what we should expect. It should be pointed out here that the order of magnitude and the direction of the perturbation velocity on the free surface near the point at infinity agree with the results obtained by Wang and Wu [1963] in the study of small-time behavior of unsteady cavity flows. Finally, we shall investigate the behavior of the solution (87) near the separation point 7= 1. From (4), (35) and (5), the pertur- bation velocity w, may be written as Of Of Ww, Halas st. (114) Equation (114) indicates that the singular behavior of w, near 7 = 1 can be studied from that of df /€87 near T= 1. Let us now differ- entiate f; given by (87) with respect to 7. The differentiation of f,; with respect to T may be viewed as consisting of four parts; the differentiation of the 7 appearing in the limits of integration, the differentiation of the factors exp (+iKwr*), the differentiation of the Airy functions with respect to 7, and the differentiation of the factor in front of the curely brackets in (87). Only the latter two parts produce terms of the form a_(T* - 13" @ near T = 1; all the other parts either give zero or a finite contribution to w,. Therefore, condition (34) and the condition that the pressure is integrable over the plate AB are satisfied. Since the solution given by (87) behaves properly at infinity and at the separation point, we conclude that it is the solution of the problem; no additional solution of the homogeneous equation, as shown in (93) needs to be added. 243 Wang ACKNOWLEDGMENTS I wish to express my appreciation to Professor T. Y. Wu for useful discussions during this research. I am also indebted to my wife Yvonne for typing this manuscript. REFERENCES Gilbarg, D., Jets and Cavities, Encyclopedia of Physics, Ix, Berlin: Springer-Verlag, pp. 369-71, 1960. Jeffreys, H., Asymptotic Approximations , Cambridge University Press, pp. 52-9, 1962. Jeffreys, H. & Jeffreys, B. S., Methods of Mathematical Physics, 3rd Ed., Cambridge University Press, pp. 508-11, 1956. Langer, R. E., "The solution of the Mathieu equation with a com- plex variable and at least one parameter large," Am. Math. Soc., Trans. 36, pp. 637-95, 1934. Muskhelishvili, N. I., Singular Integral Equations, Groningen, Holland: P. Noosdbett std .',). pp. TOociz: 1946. Olver, F. W. J., "The asymptotic solution of linear differential equations of the second order for large values of a parameter," Phil. Trans., Roy. Soc. London, 247A, pp. 307-68, 1954. Olver, F. W. J., "Uniform asymptotic expansions of solutions of linear second-order differential equations for large values of a parameter," Phil. Trans., Roy. Soc. London, 250A, pp. 479-517, 1958. aaa Wang, D. P. & Wu, T. Y., "Small-time behavior of unsteady cavity flows," Arch. Rat. Mech. & Analy., 14, pp. 127-52, 1963. Wane, D. Pr. & Wu, To iv. General formulation of a perturbation theory for unsteady cavity flows," J. Basic Eng., ASME, Trans. D; 87, pp. 1006-10, 1965. 214 HYDRODYNAMICS IN THE OCEAN ENVIRONMENT Tuesday, August 25, 1970 Morning Session Chairman: J. K. Lunde Skipsmodelltanken, Trondheim, Norway Page Deep-Sea Tides 217 W. He. Munk University of California, San Diego Stability of and Waves in Stratified Flows (ag he) C. Yih, University of Michigan On the Prediction of Impulsively Generated Waves 239 Propagating into Shallow Water P, van Mater, Jr., U.S. Naval Academy and E. Neal, Naval Ship Research and Development Center 215 of lb 4 i “" P| (i ipa tir WA) ae aj ‘ iat. - Ge oh Vis ! al ten to om i TMAMWONIvWa 4 AASOO SMT. it 2OWAA WYQOR c J *) "y ei cs woe bee it . ftv i ; : 2) bev Ber & si a é rere joy , ‘ hey? Peto 3 ttT* shA 4 i - : f ; ica 4 4 1% - ; 4 Feit) J, , t Pcie rea! wy : eit a Wet fit a ; 7 i Y : ¥ , i ye ‘ é o 4 ve ‘ . ¢ fy : hve 26 aie! it \ x 4 Weert .ittiodhags f “saddest itbatria «tae fi : Leis PA re 90 , BL st Le a waa fe fy me ' P : ‘ve . : Fi ee i ass 7 , 2h Ve a Ce | SU aiG be | siete LSS to (ile tila Lisa Rts, ah; a award piers 16 Sh pov B YT Ps ae he : » MBB MOL, E a, vile- 44) uy “ae ; nove W bata weno ylewlas geil 26 paves : Gee yh 761sW wollade sdctt saltwed baa sa apavk, Javan oe wihbes ts leht %, LS ee sek - wt * % preg * |, 24: siigobvedl ons 5459498. qide, Ley Vi «toon = = a" the 120d (a \ ref hiae i a Hy te 4tone hee 44 iow x My 8 | ‘ ht te i | ee Cy By Aras A 2ts 7 U iy 7] DEEP-SEA TIDES Walter H. Munk Untversity of Caltfornta san Diego, Caltfornta ABSTRACT The classical Laplace tidal theory, when applied in numerical form to the world's ocean basins, does not yield results in good accord with observations. In part, this may be due to density stratification and in- ternal tides (coupled to external tides); and in part to dissipation at the ocean boundaries. At a given port the spectrum of the observed tides shows a complicated line structure superimposed over acontinuum. The continuum rises at the frequencies where the lines are clustered, probably as a result of internal tides. Tide dissipation leads to an exchange of angular mo- mentum between the spin of the earth and the orbit of the moon. As a result of this spin-orbital coupling, the length of day and month are both increasing. Ob- servations of the moon since 1680, of Babylonian eclipses and of the structure of Devonian tropical coral (which give the number of Devonian days per year) confirm these calculations. To untangle these problems, it is probably necessary to make observations in the deep sea, relatively re- moved from the scattering and absorbing boundaries. Such observations have now been made for the last three years, and they yield relatively clear pictures of the deep-sea tidal pattern. The tides in the northeast Pacific can be roughly accounted for by superposition of a northward-traveling Kelvin wave (trapped by rotation to the boundary) and a southward-traveling non- trapped Poincaré wave. In order for the calculations to be realistic, they need take into account the tidal yielding of the sea floor. rae We | STABILITY OF AND WAVES IN STRATIFIED FLOWS Chia-Shun Yih Universtty of Michigan Ann Arbor, Michigan ABSTRACT A theorem giving sufficient conditions for stability of stratified flows, which is a natural generalization of Rayleigh's theorem for shear flows of a homogeneous fluid, is given. Sufficient conditions for the existence of singular neutral modes, and consequently of unstable modes, are also presented, and in the development the possibility of multi-valued wave number for neutral stability of the same flow is explained. Finally, neutral waves with a wave velocity outside of the range of the velocity of flow (non-singular modes) are studied, and results concerning the possibility of these waves are given. In addition, Miles' theorem [ 1961] on the stability of stratified flows for which the Richardson number is nowhere less than 1/4, and Howard's semi-circle theorem [ 1961] are extended to fluids with density discontinuities, I. INTRODUCTION The stability of stratified flows of an inviscid fluid has been studied in a general way, i.e. , without specifying the actual density and velocity distributions, by Synge [1933], Yih [1957], Drazin [1958], Miles [1961, 1963], Howard [1961], and others. Of these, Miles has made particularly substantial contributions to the subject. However, many questions still remain open. Among these are the following: (i) Miles [ 1961] showed that if the Richardson number is nowhere less than 1/4, the flow must be stable. This is a sufficient condition for stability. What can one say regarding the stability of the flow when the Richardson number is less than 1/4 in part or all of the fluid? Are there then some sufficient conditions for stability not 219 Yth covered by Miles' criterion? What, in fact, is the natural generalization of Rayleigh's theorem on the sufficient condition for stability of a homogeneous fluid in shear flow? (ii) Are there some sufficient conditions for instability? (iii) Miles [| 1963] has shown that the wave number at neutral stability can be multi-valued for the same flow, in an actual calculation for a special density distribution and a special velocity distribution. Is there an explanation for this, even if not completely general? (iv) Do internal waves with a wave velocity outside the range of the velocity of flow exist? How many modes are there? What is the character of each mode? In this paper the questions posed above will be answered in as general a way as possible. By "general" I mean "without numerical computation." Although special calculations for special flows, involving the use of computers, are important because they often give us insight into and understanding of the subject, and sometimes are of practical interest, results obtained in a general way are often more useful. The question naturally arises: Can general results be continually improved and sharpened, albeit with increasing cost in labor, but without the use of computers? The answer to this question necessarily reveals the attitude of the respondent more than anything else. My answer to it is in the affirmative, and the results contained in this paper, aside from whatever interest or merit they may have for those cultivating the subject, are given to substantiate my faith. In addition, some straightforward extensions of Miles' theorem mentioned in (i) above, and of Howard's semi-circle theorem [ 1961] , are made to make these theorems applicable to fluids with discon- tinuities in addition to continuous stratification in density. Il DIFFERENTIAL SYSTEM GOVERNING STABILITY If U and p denote the velocity (in the x-direction) and the density, respectively, of the primary flow in the absence of distur- bances, and u and v denote the components of the perturbation in velocity in the directions of increasing x and y, the linearized equations of motion are ip (uy F Ua +O) = - Bes, (1) p(v, + Uv,) = - py - BP» (2) in which subscripts indicate partial differentiation, t denotes time, 220 Stabiltty of and Waves tn Strattftied Flows p is the deviation of the pressure from the hydrostatic pressure in the primary flow, p is the density perturbation, g is the gravi- tational acceleration, and The equation of continuity permits the use of a stream function w, in terms of which the velocity components can be expressed: se v= a (3) The linearized form of the equation of incompressibility is p, + Up, tvp'=0, (4) in which plz a, dy If n is the vertical displacement of a line of constant density from its mean position, the kinematic relationship le ea (5) holds. All perturbation quantities will be assumed to be periodic in x and have the exponential factor exp ik(x- ct), so that from (5) and (3) we have wes(U=-c)y, u==([(U=c)n]t, v= ik(U-e)7. (6) Then (1) and (4) give =p(U-c)’n' and p=-P'n. (7) Writing n(x,y st) = Fly)eik- et) (8) 221 Yth and substituting (6) and (7) into (2), we have, with B denoting -p'/p, [p(U-c)*F']' + pl Bg-k'(U-c)*]F = 0, (9) which is the equation used by Miles [1961] and Howard [ 1961] to study the stability of stratified flows. Miles [1961] assumed U to be monotonic and U and ) to be analytic in his studies. Howard [1961] was able to prove Miles' theorem (on a sufficient condition for stability) and to obtain his own semi-circle theorem without these hypotheses. But both of them assumed ‘p to be continuous, and considered the upper boundary to be fixed as well as the lower one. We shall now show that the theorems of Miles and Howard can be generalized to allow density discontinuities. The mean velocity U (though not necessarily U') will be assumed continuous. Let there be n surfaces of density discontinuity, and let the free surface, if there is one, be the first of such surfaces. The densities above and below the i-th surface of density discontinuity will be denoted by (p,), and (pg); , respectively, and we shall define (Ap); by (Ap), = (py - p,); « (10) The interfacial condition can be obtained by integrating (9) in the Stieltjes sense in an arbitrarily small interval containing the discon- tinuity under consideration, and is, with the accent indicating differ- entiation with respect to y, [p(U-c)*F'], - [p(U-c)*F'], = - gApF, (11) to be applied at any surface of discontinuity. At a free surface Py vanishes, and (11) becomes (U-c)*F' = pF, (12) which is the free-surface condition, to be applied at y=d, d being the depth. If the upper surface is fixed instead of free, the condition there is F(d) = 0. (12a) The boundary condition at the bottom, where y=0, is F(0) = 0. (13) 222 Stability of and Waves tn Stratifted Flows Ill. EXTENSION OF MILES' THEOREM Following Howard [ 1961], we set eu w2r, where W=U- cc. Then (9) can be written as (pwG')' - [(pU')'/2 + pw + pw'(U'*/4 - gB]G= 0. (14) The boundary condition at the bottom is G(0) = 0. (15) The interfacial conditions (11) become Py (WG! - U'G/2), - p (WG! - U'G/2), = gApw'G, (16) to be applied at the surfaces of density discontinuity, and in particu- lar the upper-surface condition becomes WG'-U'G/2=gW'G, or G(d)=0, (17) depending on whether the upper surface is free or fixed. Multiplying (14) by G", where the asterisk indicates the com- plex conjugate, and integrating from the bottom to the first surface of density discontinuity and then from discontinuity to discontinuity throughout the fluid domain, and utilizing (15), (16), and (17), we have (Vewtia'l? +k |G/7] +( Gun |Gl*/2 + (51 u'?/4 - g6] w'|G/w/? -) gaaw*|c/w|?-) (GU, - @UIJIGP/2=0, U8) in which each of the integrals is over the entire fluid domain exclusive of the surfaces of density discontinuity (i.e., it is a summation of integrals over the layers of continuous density distributions), and the summation is over the discontinuities, including the free surface if there is one. If the flow is unstable, c, > 0, and the imaginary part of (18) is Valiot? +x%iGiy +f alae - u'7/4] |o/wl* + ) ga,pla/wl? =o, i (19) 223 Yth from which it is again evident that if gb = U'*/4 everywhere in the fluid exclusive of the interfaces and the free surface (if there is one), the flow must be stable. IV. EXTENSION OF HOWARD'S SEMI-CIRCLE THEOREM Equation (9) can be written as (pW2F')' + p(Bg - k*W*) F = 0. * Multiplying this equation by F , the complex conjugate of F, inte- grating throughout the fluid domain and using the boundary or inter- facial conditions (11), (12), and (13), we have (ewilr' P+ lel -(seplFP- Dealer =o, (20) in which the summation is over the surfaces of density discontinuity, and the integrals extend throughout the fluid exclusive of the surface of discontinuity in density, The real and imaginary parts of (20) are (at u- op)? - It lel? +e"1F 1 -loe6iFl - ) sae lFl =o, (21) ¥t Hy eed) Zee Bal, 2c,4 plu cote, +k FL) 20. (22) Writing G-nlley teri; we obtain from (22) (ue = ¢. $a, (23) then from this and from (21) we obtain (ut thea tel o + \ gop [F[? + Ded lF 2 (24) If a and b are respectively the minimum and the maximum of U, 224 Stability of and Waves tn Stratifted Flows so that a= U=b, we have o=((u- au-va={u%- (a +») vO +tab\ Q =[c, te, - (a +b)c, + ab] |0 + geBlFP +) evlFl, | after using (23). This means that [c, - (a +b)/2]’ +07 =[(b - a)/2l’, (25) that is, the complex wave velocity c for any unstable mode must lie inside the semi-circle in the upper half-plane, which has the range of U for diameter. Thus Howard's semi-circle theorem is recovered. -2 -2 From (19) and noting that |W| Sc, , we deduce that kc) = max (u'?/4 - gB) (26) remains valid even if there are surfaces of discontinuity in density. In (26) we exclude these surfaces in the evaluation of B. It is easy to see that (26) contains Miles' theorem. V. SUFFICIENT CONDITIONS FOR STABILITY Miles' theorem gives a sufficient condition for stability. But it certainly does not guarantee instability if the local Richardson number J(y) defined by Sy) = #6 (27) ww! is less than 1/4 in part of the fluid or even all of the fluid. We shall sharpen Miles' sufficient condition for stability by deriving two theorems which constitute, more than anything hitherto known, the natural generalization of Rayleigh's theorem for the stability of a homogeneous inviscid fluid. For the discussion in this section it is more convenient to use the stream function w= f(y) etk(x-ct) . (28) Comparison with (6) and (8) shows that Zio Vou f(y) = (c - U)F(y). (29) In terms of f(y), the governing equation (9) becomes ~ (OU jer det eee (pf')' + Sly Te t=30., (30) Equation (30) can be made dimensionless by the use of the new variables (31) a3 iv ipr esp pow c= p Bi” MA datz U Vv? c where p, is a reference density and V a reference velocity. Then (30) becomes, after the circumflexes are dropped, pf) 9 |i ic eae fo (32) ay in which everything is now dimensionless, the accents indicate differ- entiation with respect to the dimensionless y, @=kd (33) is the dimensionless wave number, and N = ga/v* (34) is actually the reciprocal of the square of a Froude number. The appearance of N does not necessarily signify the importance of sur- face waves, since it appears even if the upper boundary is fixed. The fact that it is associated by multiplication to p' indicates that the entire term represents the effect of gravity in a stratified fluid in shear flow. Henceforth in this paper we shall consider rigid boundaries only, for which the boundary conditions are £(0) = 0 and f(1) = 0, (35a,b) to be imposed on the function f in (32). 226 Stabitltty of and Waves in Stratified Flows It is then clear that the system consisting of (32) and (35a,b) gives, for a non-trivial solution, a relationship Fi (a7,N,c) = 0. (36) Since c is complex, (36) has a real part and an imaginary part. When c; is set to zero and c, eliminated from the two component equations, a relationship F,(@,N) = 0, (37) if one such exists, gives the neutral-stability curve. It is possible, however, that c is real for all values of @ and N, in which case c. = 0 inthe entire N-a@ plane, and then of course there is no neutral-stability curve because one component equation of (36) is Cir 0, and the other is simply (36) itself, with the c therein real, In this section, we shall assume rr and U to _be continuous, analytic, and monotonic. Furthermore, we assume p'< 0 throughout. We now recall the following known results: (i) If J(y) is not less than 1/4 for the entire fluid domain, then the flow is stable [ Miles 1961], (ii) If c; #0 then c, must be equalto U at some point in the flow, as a consequence of the semi-circle theorem of Howard [1961], and (iii) If an eigenfunction exists for (c,, @,, N,), then near that point c is a continuous function of @ and N, [ Miles 1963 and Lin 1945]. Under the assumptions we have made on p and U, and in view of the known results just cited, we conclude that the non-existence of any singular neutral mode, which is a mode with a real c equal to U at some point in the flow, implies the non-existence of unstable modes. The reason is as follows. In the N-@ plane there is always a region of stability. For we can imagine g andhence J(y) to in- crease indefinitely, until J(y) is everywhere greater than 1/4, which is attainable since B is nowhere zero. Thus there is a region of large N for which the flow is stable. If unstable modes exist there must then be a stability boundary dividing the region of stability from the region of instability, and hence a neutral-stability curve. As we approach that curve from the region of instability, c, being within the range of U so long as c; #0 and continuous in @ and N so long as c is an eigenvalue, according to (iii) above, in the limit, when c; = 0, c, must be within the range of U, i.e., the limiting mode must be a singular neutral mode. Hence the non-existence of a singular neutral mode implies the non-existence of unstable modes. 221 VER In fact even the existence of special singular neutral modes for which c equals the maximum or minimum of U does not imply the existence of contiguous unstable modes, as a consequence of the semi-circle theorem of Howard. Hence we need not be concerned with these special border cases. In demonstrating the non-existence of unstable modes it is sufficient to demonstrate the non-existence of singular neutral modes with a 1/4 everywhere. In his demonstration he actually showed that a singular neutral mode with a J(y,) > 1/4 at the place y=y, where U=c is impossible. Hence we need only consider the case Hy), = 1/4 in our search for the non- existence of singular neutral modes. For J(y,) = 1/4, one solution of (32).is £, = (y - y,)!/?w, (38) where w, = 1 +Aly - y,) +e. (39) with a= [un GBs yam] , (y=5) (40) pu' sp c provided U' does not vanish at y = y,.- [ We shall consider mono- tonic U only. Hence this restriction on U' does not affect our results in this paper.] The other solution is found by assuming it to be of the form f,h, substituting it into (32), and_solving for h. The result, after division by a constant (which is p, or p at Y)? is f= £, In (y-y,) - [2A + (In pi’ (y-y)/ 11 + Bly-y,) teeel, (41) where B is aconstant. Now the Reynolds stress defined by eS 1S. Die 9 (42) where the bar over uv means time or space average, can be ex* pressed interms of f as p,V 2ac.t T = S a(f'f"), e Te (43) 228° Stabtlity of and Waves in Stratified Flows in which the asterisk denotes the complex conjugate, and the t, now interms of d/V, is ee Oo ees as is f, Considering the singu- lar neutral case, for which c, = 0, it is easy to see from (40) and (41) that f'f* is real for y > Ye and equal to ~in: for ty 0, and if (pu)! and (In p)" are positive throughout, then singular neutral modes are impossible. 229 Yth Proof. We have shown that it is necessary only to consider the case Jy, )< 1/4. We may consider f, only, since the proof for f_ is the same, and since the solution ‘is either f_or f. Nowat y=y, we have f,=0. Near ¥, ve have Since p' is negative and U' and (pU')' are positive, U" is positive. Thus U-c is greater than Ujz for z>0O. On the other hand -p'/p is less than (-p TP). for < > y,» since (In p)" is posi- tive. We know that for small positive z Q is negative, as can be seen from (48). Hence for any z>0O the term Nee (U - c)” is less than pJ, /z* and Q is negative. Equation (48) exhibits the behavior of Q near Yor Let the bracket in (47) be denoted by - G. Then since Q is negative and U-c is positive for y>y,, and since p' is negative and (pU')' positive, G must be positive for y>y,- Multiplying (47) by w and integrating between y, and 1, we have | (p2>%ww'), - if 2°“ pw! + Gw’) dan O, (49) y c where the subscript i indicates that the parenthesis is evaluated at y = 1. Note that the integral in (49) is convergent in spite of the simple pole in two terms contained in G-- one of which in Q, as indicated by (48). Equation (49) clearly shows that w(1) cannot be zero. .Hence the theorem. Another theorem is Theorem 2. If e and _ U_ are continuous and analytic, with p! negative and U_ positive, and if U" and (In) are negative throughout, then singular neutral modes are impossible. The proof for this theorem is similar to that for Theorem 1. The only modification demanded for clarity is that instead of (44) we should write f(y) _ 2h) /2., (2) with z now definedas y, - ye The equation corresponding to (47) is now 230 Stability of and Waves in Strattifted Flows (50) in which, it must be emphasized, all accents indicate differentiation with respect to y, not z. The rest is strictly similar to the proof for Theorem 1, except the range of integration is between z=0 and z=y, (or between y=y, and y = 0), and we want to show w#0 at y = 0. Note also that U"<0O now guarantees (pU')'<0. Since the non-existence of singular neutral modes implies the non-existence of unstable modes, we have also Theorem 3. If p and U are continuous and analytic, with p' negative and (ay positive, and if either (pU")' and (In) " are both positive throughout, or U and (In P) are negative throughout, the flow is stable. This theorem is the natural generalization of Rayleigh's theorem for inviscid homogeneous fluids in shear flow. Previous attempts at this generalization [ Synge 1933, Yih 1957, Drazin 1958] have produced the result that there must be stability if (in dimensional terms) 2Bg(U - c,) Ju- cl? _ (su')? _ does not change sign. This criterion is not useful because it involves not only c, but also c;. VI. SUFFICIENT CONDITIONS FOR INSTABILITY Sufficient conditions for instability have seldom been given in studies of hydrodynamic stability. In giving some such conditions, we shall also be able to explain why the @ can be multi-valued for the same N, at neutral stability. We assume that p and U are analytic, that p' = 0, and that at a point where p'=0, U" is also zero. The value of U at that point will be denoted by U,, for we shall consider the possibility of having c equalto U at that point. We demand that at any other point where U=U,, p'=0= U" must be satisfied. If U is mono- tonic, of course there is only one point at which U = U,. = Under the assumptions made, p" must be zero at Y,* since p' is never positive, andnear y, 231 Yth p' = poly - y,)- If ps were not zero p' would be positive for y slightly larger than vo With this realization, it is immediately clear that the bracket in (32) has no singularity at y_. Let us denote the bracket in (32) by the sumbol B, which is a function of y, @, and N. Then if m is the minimum of B/p_ between two points y, and y,, with 2 O0Syb. All this is in contrast with waves propagating in a layer of homogeneous fluid with a free surface and in shear flow. In that case [ Yih 1970], if U is monotonically increasing with y, waves of all wave lengths can propagate with c greater than b, and only sufficiently long waves can propagate with c less than b. ACKNOWLEDGMENT This work has been supported by the National Science Foundation. REFERENCES Drazin, P. G., "On the Dynamics of a Fluid of Variable Density," Ph.D. Thesis, Cambridge University, 1958. Howard, L. N., "Note on a Paper of John W. Miles," J. Fluid Mech., Vol. 10, pp. 509-512, 1961. Lin, C. C., "On the Stability of Two-dimensional Parallel Flows, Part II," Quart. Appl. Math., pp. 218-234, 1945. Lin, C. C., The Theory of Hydrodynamic Stability, Cambridge University Press, 1955. Miles, J. W., "On the Stability of Heterogeneous Shear Flows," J. Fluid Mech., Vol. 10, pp. 496-508, 1961. Miles, J. W., "On the Stability of Heterogeneous Shear Flows, Part 2aK A Fluid Mech, ? Vol. 16, PPpe 209-227, 1963. L Synge, J. I., "The Stability of Heterogeneous Liquids," Trans. Roy. Soc.’ Can., Vol. 27, pp. 1-18, 1933. Yih, C.-S., "On Stratified Flows in a Gravitational Field," Tellus, Vol. 93 lpps 220-227. 1957, Yih, C.-S., "Surface Waves in Flowing Water," to be published in J. Fluid Mech. in 1971. 254 Stability of and Waves in Strattfied Flows DISCUSSION L. van Wijngaarden Twente Institute of Technology Enschede, The Netherlands The flow with a free surface of a fluid, homogeneous in density, but with inhomogeneous velocity distribution, is a special case of your class of stratified fluids. Burns [ 1953] considered this case and I guess his results are comprised in yours. When viscosity is allowed for, the problem becomes much more complicated. It may be of interest to note that Velthuizen and 1[1969a, 1969b] studied this prob- lem taking viscosity into account. We obtained results essentially different from Burn's results, which is due to viscous effects. At large Reynolds number the flow can be divided in an inviscid region and viscous regions at the critical layer and at the bottom. At the outer edge of the viscous layer at the wall the Reynolds stress cannot be put equal to zero a priori because a stress may build up in the wall layer. REFERENCES Burns, J. C., "Long Waves in Running Waters," Proc. Camb. Phil. Soc. 49, 695, 1953. Velthuizen, H.G.M. and L. v. Wijngaarden, J. Fluid Mech. 39, 4, 817, 1969a. Velthuizen, H.G.M. and L. v. Wijngaarden, IUTAM Symposium on Instability of Continuous Systems, Herrenalb, Sept. 1969. 235 Yth REPLY TO DISCUSSION Chia-Shun Yih Untverstty of Michigan Ann Arbor, Mtchtgan It is well known that Rayleigh's sufficient condition for stability of inviscid fluids flowing between rigid boundaries is satisfied by a parabolic velocity profile, whereas plane Poiseuille flow, which has this profile, has been found by Heisenberg and Lin to be unstable at sufficiently large Reynolds numbers, when viscous effects are taken into account. Since the present paper is a study of the stability of inviscid fluids,and, in particular, Rayleigh's criterion for stability is generalized in it, Professor van Wijngaarden's position that the consideration of viscosity may force us to modify some of the con- clusions in the paper is easily acceptable. In considering viscous effects, however, it is not entirely self-consistent to assume a horizontal mean flow with a free surface, as Velthuizen and Professor van Wijngaarden have done [ 1969a,b], since such a flow obviously cannot be maintained, and must in time attenuate to a state of rest. This is not to say that any conclusion of instability reached by them is without significance, for instability of a transcient nature may well occur, with the disturbances growing for a short duration of time. In this regard the results of Benjamin [1957] and Yih [1963] for surface waves ina fluid layer flowing down an inclined plane are relevant. They found that the speed c, of long surface waves, be they unstable, neutral, or stable, exceeds the maximum speed of flow. The absence of long waves propagating up- stream supports Professor van Wijngaarden's claim in connection with Burn's result, which is supported by a study [ Yih 1971] of waves in a flowing inviscid liquid. But the nonexistence of a critical layer renders rather less cogent the argument given in Professor van Wijngaarden's discussion. On the other hand, this nonexistence sub- stantiates the conclusion made in Yih [1971] (and similarly in this paper) regarding the nonexistence of singular neutral modes, since the velocity U in laminar flow of a viscous fluid down an inclined plane is parabolic, with a constant U". We also recall that Tollmien's sufficient condition for insta- bility [ 1935] of an inviscid fluid is not much affected by the considera- tion of viscosity, at least when the Reynolds number is large, and hope that the same is true with the sufficient conditions for instability presented in this paper. 236 Stability of and Waves in Strattfied Flows REFERENCES Benjamin, T. B., "Wave formation in laminar flow down an inclined plane," J. Fluid Mech., 2, pp. 554-574, 1957. Tollmien, W., "Ein allgemeines kriterium der instabilitat laminarer geschwindigkeitsverteilungen," Nachr. Ges. Wiss. Gottingen, Math. Phys. Kl., Fachgruppe 1, 1, pp. 79-114, 1935. Yih, C.-S., "Stability of liquid flow down an inclined plane," Phys. of Fluids, 6, pp. 321-334, 1963. (The other references are given either in the paper itself or in Professor van Wijngaarden's discussion.) 237 t ms yard Lord ser burs’ “ “® ,43 ae he ‘ l C ,simataod PetTBLg = ; »aientiol ue ‘Oat MT “as 7 rorr } rae 91 ‘eh ON THE PREDICTION OF IMPULSIVELY GENERATED WAVES PROPAGATING INTO SHALLOW WATER Paul R. Van Mater, Jr. United States Naval Academy Annapolis, Maryland and Eddie Neal Naval Shtp Research and Development Center Washington, D.C. ABSTRACT This report treats the problem of the propagation ofa dispersive wave system generated impulsively by a surface explosion in deep water into shoaling and shallow water regions. A topography consisting of an arbitrary bottom profile with parallel straight contour lines is assumed. The linear theory of impulsive wave genera- tion for water of uniform depth is used as a basis for evaluating the spectral energy of the wave system ata point in deep water distant from the explosion. Con- servation of energy is then invoked to extend the pre- diction to propagation over a bottom of variable depth. A cnoidal wave theory is introduced to describe the changes in form of the individual phase waves and the wave envelope as the system enters the shallow water regime. The effect of wave refraction at locations other than along an axis normal to the bottom contours is treated. Empirical criteria are incorporated to pre- dict the occurrence of wave breaking, the decay of wave height after breaking, and the attainment of stability in the reforming wave. Attenuation of wave height due to dissipation of energy at the fluid boundaries is also considered. All of the above elements have been in- corporated in a computer program. Details of the computational procedure are described in an appendix. Typical predictions made using this program are dis- played for small, moderate, and large source strengths. The agreement of the predictions with experimental observations is discussed qualitatively, but no experi- mental data are included. 239 Van Mater and Neal I. INTRODUCTION The nature of wave systems yenerated impulsively by explo- sions at or below the water surface is of natural interest to re- searchers in the naval community because of the ship behavior which results from such an environment. When water-wave systems enter shallow water they undergo changes in form which may have an adverse effect on motions depending on the size of the waves relative to the ship or small craft. While the motivation of this work from our point of view is ultimately the prediction of ship behavior ina shallow-water explosion-generated wave environment, this paper is confined to the prediction of the forcing function -- the wave system. In itself this case presents an interesting means of study- ing the shoaling behavior of a dispersive wave system, an area which has received surprisingly little attention. Previous efforts in the direction of predicting impulsively- generated wave systems entering water of variable depth stem from the work of Dr. William Van Dorn (cf. Van Dorn and Montgomery [ 1963]) and have been confined to the prediction of wave envelopes. This paper should be viewed as a second generation of the Van Dorn model. As a starting point we shall tabulate some of the rather com- plex effects which occur in shallow water. Not all of these will be considered in the present prediction scheme but the exclusions will be noted. (a) In deep water, phase and group velocities depend pri- marily on wave frequency giving rise to the well-known characteristic of the system known as frequency disper- sion. As the system moves into water whose depth is small compared to the lengths of the waves in the system this frequency dependence weakens and dependence on water depth and wave height strengthens. Waves which in deep water moved through the group at phase velocities up to twice the group velocity now become nearly frozen in their position in the group. (b) The nearly sinusoidal form of the waves in deep water changes to one of sharp crests separated by long flat troughs. An asymmetry about the horizontal plane develops in which the crest height above the still water line is greater than the trough depth. The maximum slope of the waves increases. This last feature is of particular importance in ship motion prediction. (c) As wave height becomes significant with respect to the water depth and the wave nears the breaking point the leading face of the wave steepens and a wave slope asym- metry develops. As the slope of the face of the wave near the crest approaches the vertical the wave becomes 240 Impulstively Generated Waves Propagating into Shallow Water (d) (e) (£) (g) (h) (i) irreversibly unstable and breaking follows. In the final stage before breaking there is an abrupt increase in wave height known as "wave peak-up." Breaking may fall into one of three broad categories: plunging, spilling, or surging, After breaking the wave continues as a spilling wave until it either runs up on the beach or reforms as a stable wave. Energy dissipation accompanying breaking reduces the wave height. In the case of dispersive systems entering shallow water a low-frequency oscillation is superposed on the wave train. This "wave set-up and set-down" is caused by the transport of mass with the system, particularly when breaking or near breaking occurs, and the resulting counterflow. When an element of a wave crest passes over a bottom contour line obliquely it is refracted so as to be more nearly aligned with the bottom contour line, If the bottom contours are not parallel straight lines a focusing of wave energy can occur at "caustic points" and consider- able wave height enhancement can result. The height of waves in shallow water is attenuated by energy losses due to bottom friction, bottom percolation, internal friction, and surface contamination. Incoming waves which encounter steep offshore bars or beaches may be reflected seaward. Under the right con- ditions standing wave systems of surprising severity may be produced. Non-linear instabilities in a shallow-water wave may cause it to decompose, or split into two or more com- ponent waves. The hazardous "double rollers" are often- times an example of this. A different type of decomposi- tion may occur when a wave passes over an offshore bar and nearly breaks but then recovers. One or more smaller waves known as "solitons" may be shed from the back of the larger wave. Little is known about these waves at present. All of these features can affect ship and small craft operations in shallow water and have been included to emphasize the complexity of the overall problem. Not all will be attempted in the prediction scheme presented in this paper. For our purposes we will consider a wave system resulting from an explosion in deep water of nearly uniform depth, which propagates into shallow water over a terrain represented by parallel straight line bottom contours. We will 241 Van Mater and Neal include change of wave form with wave slope symmetry retained, wave asymmetry about the horizontal plane, wave peak-up, wave breaking, wave height attenuation after breaking, stability of the reforming waves, wave refraction along paths other than normal to the bottom contours, bottom friction, and surface contamination, Excluded are: wave slope asymmetry and change of wave form close to breaking, wave set-up and set-down, the presence of caustics, bottom percolation and internal friction, wave reflection, non-linear decomposition, and solotonic shedding. The system will not be carried all the way into the beach. Specifically, a linear theory for impulsively-generated waves in water of uniform depth is invoked to describe the waves in deep water at a large distance from the source. From this point a different linear theory based on conservation of energy per unit frequency is employed to depict the system as it moves into a region of shoaling topography. The integral expressions in this theory are evaluated numerically using the conditions at each of a series of closely-spaced stations as input for evaluating conditions at the next station. As the system progresses into shallow water its frequency- dispersive nature gradually disappears and non-linear features dominate in the wave form and propagation velocities. A non-linear cnoidal wave theory is matched numerically to the previous solutions to carry the system inthis region. The cnoidal theory is used to describe the profiles of the individual waves in the system and the asymmetry of the system about the horizontal plane. To treat wave breaking existing experimental evidence has been reexamined and an improved criterion for wave breaking is incorporated. From the same experimental source empirical formulations are developed to account for wave height attenuation after breaking and the attainment of stability in the reforming wave. The Van Dorn formula for bottom friction and surface contamination is used to account for these effects. The system is computed not only along an axis normal to the bottom contours but also along a series of rays which emanate radially from the source and change direction continuously due to refraction as the waves move inshore over the shoaling water. All these features have been incorporated in a computer pro- gram. Results of this program for a specified bottom profile and for several source strengths are presented as figures. For the researcher working on similar type problems perhaps the most useful part of the paper will be the computational procedure which is dis- cussed in some detail in an appendix. II WAVE GENERATION Treatment of the subject of water waves produced by a local disturbance has a long history beginning with Cauchy [1815] and Poisson [ 1816] each of whom independently solved the classic two- dimensional wave problem which bears their names. In recent 242 Impulsively Generated Waves Propagating into Shallow Water years Kranzer and Keller [| 1959] , Kajiura [1963], Whalin[1965a], and Whalin [1965b] have made significant contributions on the sub- ject. The last cited is an extension of the Kajiura work and appears to be the most general treatment on the subject. The theory as presented by Whalin relates an initial distri- bution of impulse, surface velocities, and surface deformations to the waves produced at some distance from the source in water of finite but uniform depth. To a certain extent the choice of a source model is arbitrary in that several models may give an adequate fit to experimental data with each having its own particular advantages and disadvantages. No physical reality is assigned to the source model in terms ofthe dynamics of the explosion; fortunately, how- ever, the dimensions of the source model have been found scalable in terms of explosive yield so that useful predictions can be made. A source model that has been found to give good agreement with experiment is a paraboloidal cavity given by: nz) = 2d, fe) - 5| a (1) 0 eae The collapse of this cavity at time t=0O generates the wave system. In cases of large explosions where the dimensions of the cavity are not small compared to the water depth, this model yields a poor prediction and a different source model, perhaps utilizing an initial time dependency should be employed. According to Whalin the surface elevation, (r,t), for an axially symmetric surface deformation is: 00 nr .t). = ae (0) o cos (Qt) J,(or) do (2) 0 where mo) is the Hankel transform of the initial surface deforma- tion 7(r): oO Alo) =| n(z)Jo(or) tr dr. (3) All unprimed quantities in these equations have been non-dimensional- ized using the water depth, h. Primed variables indicate the cor- responding dimensional quantities. rod, = dimensionless radius and height of initial surface . deformation = ro'/h, d,'/h 243 Van Mater and Neal = dimensionless radius to field point = r'/h r n = dimensionless elevation of water surface = 7'/h h = water depth o = dimensionless wave number = Kh K = wave number = root of equation: we = gk tanh kh Q = dimensionless frequency = (w*h /g)!/2 w = frequency, radians /sec t = dimensionless time = t'Ve/h t' = time, sec. The integral in (2) is evaluated using the method of stationary phase (cf. Stoker [1957]). After doing this and making the substitu- tions the result is: n(r,t) = 7982. [- aoe - J,(or,) + cos (or - Mt) (4) Here, v = group velocity = 3[ (0/x) tanh ale [41+ (20/sinh 20)]. Eq. (4) is valid at distances from the source that are large in com- parison with the radius of the cavity and in water of uniform depth. A point is selected which satisfies these conditions and the spectral energies of the system evaluated as will be discussed in the next section. III WAVE PROPAGATION OVER A BOTTOM OF VARIABLE DEPTH The extension of the solution to regions of variable depth is based on a conservation of energy approach originally presented by Van Dorn and Montgomery [ 19631. The equation presented therein evaluated the spectral energy, that is the energy per unit frequency of the system, for the special case of propagation along a ray normal to parallel bottom contour lines. The derivation is presented here in a slightly different way to permit its extension to include refractive propagation. The topography considered is represented in Fig. 1 which also shows the coordinate system and a typical refracted wave ray. The following assumptions apply: (a) wave frequency remains constant throughout the region of wave travel and is unaffected by refraction (b) energy is transported at group velocity in a direction normal to the wave crest (c) energy per unit frequency is conserved between adjacent wave orthogonals. 244 Impulsively Generated Waves Propagating into Shallow Water WAVE FRONT eae — (DEEP WATER) ORIGIN STATION | 40 59 64 71 76 100 Fig. 1. Bottom topography and wave refraction The approach will be to consider the energy patch between two adjacent wave rays, S, and S,, and between two adjacent frequencies, w and w + dw, and to establish the dimensions of this patch as a function of path position. While the total energy of the patch remains constant the energy density changes with patch size and this, in turn, determines the local wave amplitude. Establish a rectilinear coordinate system, (x,y), with x-axis normal to the parallel bottom contours and the beach. Orient the curvilinear coordinate system (s,n) shown in Fig. 2 to the median between rays. Since Kk = K(w,h) only the magnitude of K,, Kz, and Kk will be the same but due to the curvature of the orthogona s the directions will be different, Let kx, have components (Z, »m, ) parallel to (s,n). Since the patch 66 6&, is small £, + fl, = ri m, = m,=m. It may be shown by applicatinn of Snell's Law that the head of the vector kK, will shift to the right along line AB in Fig. 3 main- taining a constant we secke on the beach as the patch moves inshore. Then, from Fig. 3 m, cos 8, = m cos 8 = mg cos @ . (5) i] and also, m, = Ky59%, (6) 245 Van Mater and Neal Fig. 2. Propagation of an energy- patch over a variable-depth bottom b STRAIGHT BOTTOM CONTOURS PARALLEL TO THE BEACH ARE ASSUMED Fig. 3. Wave number vector diagram for a shoaling bottom 246 Impulsively Generated Waves Propagating tnto Shallow Water so that, _ Cos Q ~ cos 6 60, - (7) Since dw is small the frequency dependent change in the trajectory of the phase wave orthogonals may be neglected. The change in the patch width due to the geometric spreading of the rays as the patch moves through a distance, ds, will be m 27 ds. The patch width will be sti i et ath = ds 0) Sf = 20, - £08 Sods : (8) cos Next, consider the patch length, 6§&. If 6t is the interval required for the patch to pass a given point, i, on s, then where v, is the group velocity at point i. The time interval, 5t, may also be represented as follows _ 6t dt = =— dw ata save! a8 = aw — (1/v) ds Ow o Vv 0 dw i i 1 av 1 Ov 8x =- dol — ~— ds = - dw — = ds. 0 v2 w 0 wt aK Ow Since v= 8w/8K, 1 90 it = - aw | niet dg bye OK 247 Van Mater and Neal i ) 1 &t = ao OK (+=) ds. (9) The patch length becomes, i 8& = v, aw J ee (=z) de (10) If E(w) is the energy per unit frequency of the source dis- turbance then the energy between orthogonals in a frequency band, dw, near the source will be 2505 E(w) dw a Since the energy of the patch has been assumed constant this will also be the value at the field point, i, and this in turn may be equated to the local wave energy: 2 1 E(w) dy “B80 Zips n OG65;, (14) where nj is the local wave amplitude. Substituting the expressions for 5% and 6€ from Eqs. (8) and (10) gives: i i 260, _ 1 zi cos 0 2 a7 1 E(w) dw aS Pg Ni 2880 cos ds | [v, aw | OK (=) ds | ° The final result is, Bled of 222% a6] [of & (ste) a]. ca a a ee al B, The first bracketed factor, @,, represents the effect of geometric spreading between rays while the second bracketed factor, B;; represents the effect of the spatial stretching of energy between adjacent frequencies, or frequency separation. Computationally, E(w) will be evaluated using the results of the previous section, Eq. (4), at a point sufficiently removed from the source and over bottom depths nearly enough uniform to satisfy the conditions on application of that equation. From that point on inshore new a's and B's will be computed numerically for each 248 Impulstvely Generated Waves Propagating into Shallow Water field point desired. Treating E(w) as a constant a new nj = n(w) will be computed. Details will be discussed in the appendix on com- putational procedure. The phase of the waves in water of variable depth requires attention. The phase function of Eq. (4), cos (or - Qt), for uniform depth may be rewritten as Ww 1 cos (K = +) r Vv since t'=r'/v'. Now, however, in water of variable depth wave number and group velocity will depend on location as well as fre- quency. The argument must now be represented in integral form with the integration performed along the path, s. The phase term now becomes cos | («-5) as'| (12) where K = k(h,w), v' = v'(h,w), and h = h(s'). The central assumptions involved throughout this development have been that the system is linear and conservative and that the energy per unit frequency remains constant. The assumptions are quite viable as long as the water is of deep to moderate depths, say one-half wave length or deeper. Inshore of this point the system becomes progressively more non-linear, non-conservative and the frequency assumption more vulnerable. An evaluation of the linear assumption will appear in the section. IV. NON-LINEAR FEATURES OF THE SHALLOW WATER SYSTEM The previous section carried the wave system from a region of uniform depth into a region of variable depth; however, the des- cription of the system retained its linear character. We have des- cribed earlier the change in form of shallow-water waves from one of sinusoidal form to one of sharp crests separated by long flat troughs with an associated horizontal plane asymmetry. In this section a particular non-linear theory, the cnoidal wave theory of Keulegan and Patterson [ 1940], will be incorporated to modify the form of the waves and the wave envelope. The first of the cnoidal family of wave theories was presented by Korteweg and de Vries [1895]. Because the wave elevation was given in terms of the Jacobian elliptic cn function they coined the work cnoidal to describe the resulting wave form. Subsequent contri- butions, in addition to the Keulegan and Patterson paper cited above, 249 Van Mater and Neal have come from Keller [1948] , Benjamin and Lighthill [ 1954] , Laitone [1960] , and Laitone [1962]. Masch[1964] has computed the shoaling characteristics of cnoidal waves assuming constant power. Iwagaki [1968] has simplified cnoidal wave equations to a form which he calls "hyperbolic waves." Masch and Wiegel [ 1961] have pro- vided extremely useful tables of cnoidal wave functions. A paper by Le Méhauté, Divoky, and Lin [1968] motivated, in part, the choice of the Keulegan and Patterson theory for incor- poration in this prediction scheme. These authors reported shallow- water wave experiments and compared the results with twelve differ- ent wave theories. Their finding was that none of the theories was uniformly satisfactory but that the cnoidal theory of Keulegan and Patterson was the most generally satisfactory. For the shortest waves linear theory was the best but failed rapidly as the wave length was increased. Stokes' second order and Laitone's second order were consistently worst. Stokes' third and fifth order were better but not as good as linear theory. In terms of the wave profile the Keulegan and Patterson cnoidal profile gave the overall best agree- ment and was accurately placed with respect to the still water line. This latter finding has also been confirmed by Adeymo [ 1968]. The central equations adapted from Keulegan and Patterson for use here are: n'(r',t!) = - ng! + Hon? _— (kr' - ot'), | (14) \f2 =e b-(2)+) (2) @-2g8)] > a my! . K(k) - BO), ag y ar (16) kk” K(k) “H 16 2 meek K(k)] (17) 2 OW 2 } a = ie : ea LG} : (2) + (=) O73. 3 BY) | (18) (primes indicate that the variable is_in the dimensional coordinate system.) The expression, L?H/h>, may be recognized as the Ursell parameter, although, in fact, Stokes was the first to identify it. The parameter gives a measure of the linear or non-linear nature of the system. Linear theory is generally applicable for values less than 1 while cnoidal theory is most appropriate for values greater than 10. The cn function displays a character that is particularly useful Impulsively Generated Waves Propagating into Shallow Water for this application. When the modulus, k, assumes its minimum value, zero, cn reduces to cosine. When it assumes its maximum value, unity, it reduces to the hyperbolic secant, sech. Since cos? reduces to cos by the double angle formula and since the form ofa solitary wave is given by a sech* function, the cn function by appropriate choices of k describe the complete transition from sinusoidal waves to solitary waves. As the value of the modulus increases from zero toward unity the wave crests become more sharply peaked and the troughs longer and flatter; the height of the crest above the still water line increases and the depth of the trough decreases. The wave form is symmetrical about a vertical through the wave crest so that no wave slope asymmetry is reflected. Thus the cnoidal feature may be used to improve the realism of the phase waves in several ways: (a) to give the non-breaking waves a more realistic profile (b) to introduce asymmetry of the waves about the still water line (c) to increase the velocity of the waves in very shallow water. This feature has not been utilized in this application. Computationally, the frequency, w, and the water depth, h, are defined in the given frequency and spatial array. The wave height, H, is obtained from the linear theory of the previous section. The elliptic modulus, k, is computed from an iterative solution of Eq. (18). K(k) and E(k) are computed from a series expansion in terms of k. Further details appear in the appendix on computational procedure. The application, then, involves the use of a theory developed by irrotational, periodic, non-dispersive waves of permanent form in water of uniform depth to represent a wave system which is dis- persive and not periodic passing over a bottom of variable depth and in which vorticity due to bottom friction is present at least to some extent. The assumptions implicit in this extension are that the phase waves assume a form appropriate to their local peoquency within the group and that this frequency content changes only slowly, and further that rotationality effects are small. The latter assumption appears to be the most vulnerable. V. WAVE BREAKING AND ENERGY DISSIPATION Despite abundant literature on the subject of wave breaking there does not exist today a fully-adequate mathematical description of the process, and, in fact, much of the experimental evidence is contradictory and subject to wide scatter bands. In the case of this analysis the need is for a criterion for wave breaking which relates the wave frequency, w, the linearly computed wave height, H, and 254 Van Mater and Neal the water depth, h. In addition expressions are needed which deter- mine the wave height decay after breaking and the point at which spilling waves regain their stability. In the absence of an adequate theoretical base a purely empirical approach will be used. Experimental evidence has been plentiful. Iverson [1952] and Morison and Crooke [ 1953] made classical contributions. Nakamura, Shiraishi, and Sasaki [1966] presented what is perhaps the broadest range of data on breaking and decay after breaking. A fairly complete bibliography of other works on the subject appears in Van Mater [1970]. No uniformly satisfactory criterion which predicts both the occurrence and the location of wave breaking has yet been developed. A commonly used but crude criterion is that a wave breaks when the wave height-to-water-depth ratio is equal to or greater than 0.78. The Nakamura et al. paper previously alluded to contains all this information covering a rather broad range of beach slopes and wave conditions, and on this basis it was selected to develop the criteria needed for this application. The paper is essentially a report of experimental data with no analytical comparisons or proposals. A comparison of the shoaling coefficients inferred from the Nakamura data shows lower values for the gentler bottom slopes than would be obtained by linear theory. This gives rise to suspicion of prominent frictional effects in the experiments. Although the wave height may have been attenuated by friction as the waves moved inshore it seems reasonable to assume that the characteristics, w, H, and h, at the breaking point would not be severely affected. More vulnerable, perhaps, is the rate of decay after breaking and the length of the surf zone reported inthe paper. Nevertheless, the scale of the experi- ments is approximately the size of those with which we are concerned. The following formulas are a result of reworking and fitting curves to the Nakamura data, (a) Wave breaking occurs if: 5 wh (150)"" bs oth sian ‘ H ~ 2 t+ logi, (wh/g 20.13, S20.01) zn \/4 p< (10 ae -1.1) +1og ee +0.10 (wh/g<0.13, S= 0.01) (19) where S = tangent of the angle of bottom slope. A plot of this criteria for several bottom slopes is given in Fig. 4. 252 Impulsively Generated Waves Propagating tnto Shallow Water 7s (10 es ai + logie( 92) +10, — <13 s 2.01 Fig. 4. Wave breaking criteria (b) Decay of wave height after breaking is given by, yes h H, = Hy (72 (20) where, H,, h,= wave height and water depth at a point after breaking H,_, h, = wave height, water depth at breaking b’ “b The expression is for two-dimensional waves. In the explosion- generated wave case to account for geometric spreading the expres- sion must be multiplied by the ratio a, /a, where @ is obtained from Eq. (12). (c) The equation for wave stability after breaking is: [8 = aa (0.85 - 0.40 _ (21) The wave heights in Eqs. (17) - (21) reflect experimentally- measured quantities; however the wave height information we have at hand is computed from the linear theory of Eq. (12). Linear theory is known to underpredict wave shoaling as the wave height becomes a substantial fraction of the water depth. In addition, in the final stages before breaking the wave front slows more rapidly than the back. Associated with this developing wave slope asymmetry 253 Van Mater and Neal is a rapid increase in wave height known as "wave peak-up" which occurs just before breaking. The effect has been observed by Le Méhauté, Snow, and Webb [1966]. On the basis of the experiments reported in that source, Van Dorn, Le Méhauté, and Hwang [ 1968] state that the increase in wave height due to peak-up is 40% of the linearly computed value. Computationally, the wave height computed on the basis of Eq. (12) is multiplied by a factor of 1.40 to account for peak up and tested against Eq. (19) for each point in the spatial and frequency array. If breaking occurs the increased wave height is retained as H, for use in Eqs. (20) and (21). If no breaking occurs the linear value of wave height is retained. The Van Dorn boundary dissipation equation for impermeable bottoms and modified for a wide basin is: /2 ang PUD conan aoa where, v = kinematic viscosity of water and points 1 and 2 are successive points in the direction or propagation. VI. RESULTS AND DISCUSSION In this section the computer program predictions for three source strengths, corresponding to small, moderate, and large explosions, will be described and discussed. No experimental data are included but comments will be made on the agreement between the predictions and experiments which have been conducted. The theory and the computational procedure outlined in the preceding sections are applicable to any arbitrary bottom profile with parallel straight contours and gentle slopes. However, to compare the theory with experiment a specific bottom profile, that of a test basin at the Waterways Experiment Station, Vicksburg, Miss., was introduced as an input to the computer program. Through the courtesy of that laboratory access was granted to data from a series of experiments conducted there. The data has not yet been formally published by WES at this time, so it cannot be reproduced here; however general comments on the agreement between predicted and experimental results will be made. The computer results are non-dimensionalized on the basis of the water depth at the explosion, hg Previous notation is used except that the water depth, h, is taken as the depth at the origin, ho. 254 Impulstvely Generated Waves Propagating into Shallow Water Radial distance along axis: r= 2 /ho Distance along path: s =s'/hg Wave elevation: y= 7) /B Offset of path from axis: y=y /o Source strength: W= yh. In the last definition Y is the explosive yield in pounds of TNT, equivalent. For dimensional consistency the exponent should be 1/4; however experiments have shown that exponents from 0.26 to 0.30 (depending on the submergence of the source) provide better scaling. The value 0,3 is taken here. Results of time histories along the axis for three different source strengths are presented in Figs. 5 - 7: Figure 5, small explosion, W = 0.139 Figure 6, moderate explosion, W = 0.166 Figure 7, large explosion, W = 0.224. time —=t=t'V gh, Fig. 5. Prediction of wave system on axis for small explosion Van Mater and Neal —— 7x107 —— time —= ttV9g7h5 Fig. 6. Prediction of wave system on axis for moderate explosion —— 7110 1210.67 STA.59 re '/hg= 9.83 20 30 40 50 60 70 80 90 time —= t= Vg/ho Fig. 7. Prediction of wave system on axis for large explosion 256 Impulstvely Generated Waves Propagating into Shallow Water Predictions are displayed for the five stations, 59, 64, 71, 76, 100 shown in Fig. 1. The first station, station 59, may be considered to be in the transition range from deep to shallow water for the larger waves in these explosions. The remaining stations are all in progres- sively shallower water. A frequency range was chosen which would permit the computation of the first four wave groups. The full four groups are shown at station 59, Fig. 6; however since only the first two groups are of practical interest these are the only ones shown in the remaining displays. Wave breaking occurred for the large explosion but not for the two smaller explosions. The individual phase waves which have either just started to break or are continuing to break are indicated by an asterisk in Fig. 11. The irregular shape of the envelope at stations 71, 76, and 100 is caused by the fact that breaking and wave (envelope) height decay after breaking have already occurred for these frequencies within the envelope. Typically, breaking will start within a small frequency band then spread to adjacent frequencies as the envelope moves inshore. The effect of refraction is shown in Fig. 8 for the moderate explosion case only. The wave trains correspond to those along the ray families 0 9* = 0°, 20°, and 40°. Mean offsets of the path from the axis are indicated. Three stations, 59, 76, and 100 are shown to give a representative effect. ¥#¥'/ho 78.49 " ns 10% ——_—_— 60 70 80 90 100 time —=t=t'V9/ho Fig. 8. Prediction of wave system along refracted rays for moderate explosion Z57 Van Mater and Neal Early runs on the computer used the cavity dimensions sug- gested in Van Dorn, Le Méhauté, and Hwang [1968] and the wave peak-up, wave breaking, decay, and stability criteria which have been outlined previously. To improve agreement between the theory and observation adjustments were made to some of these empirical coef- ficients. First, cavity dimensions were adjusted for the best fit with the results shown below. ; 24 Yo = ae — eel C,., Cy, small explosion 7.80 1,53 moderate explosion 7.82 1.33 large explosion 8.14 2.03 recommended by Van Dorn 9.60 2.80 For wave breaking the peak-up ratio was changed from 1.40 to 1.50, breaking and decay criteria were not changed. The criteria for stability after breaking was changed as follows 2 Ha = Hp (0.90 - 0.40 2he) hg hp 24 Stable where H, is now taken as the height before peak-up. The trajectory of an individual wave ray is dependent on the initial angle at the origin, 0), bottom profile, and frequency. Con-= sequently absolute convergence of a family of wave rays representing an array of frequencies is not possible. To give the best overall conformity over the spatial domain, 9 is adjusted with frequency, i.e. 05 = Oo(w). For the conditions assumed a simple linear varia- tion in @(w) was found to give a variation in path lengths which generally fell within a 1% band and a variation in offsets from the axis which fell within a 5% band, The control angle upon which 6,(w) is based is designated gene With the background now established the following remarks may be made regarding the agreement between the predicted, or to put it more accurately, the hindcast wave system and the wave system observed in experiments. (a) For the two smaller explosions, W = 0.139 and W =0.166, agreement of the envelopes at stations 59 and 64 is very good. The envelopes of the first group are underpredicted at stations 71 and 76 but overpredicted at station 100. This infers that the linear theory underpredicts shoaling wave height enhancement in very 258 Impulsively Generated Waves Propagating into Shallow Water shallow water, and also that the boundary dissipation formula used in the program does not provide sufficient attenuation. The assym- metry of the envelopes show particularly good agreement and is one of the strong features of the program. The observed trough levels at stations 71 and 76 is slightly lower than predicted. This is attributed to wave set-down due to the presence of a counterflow, or backwash current. (b) For these two smaller explosions the theory in general predicts the correct number of waves in the envelope. Agreement in phase is poor at station 59 but better at subsequent stations. There is also some grounds for suspicion of zero-set clock errors in the experi- mental data so that it is difficult to make definitive statements on this subject. The same suspicion makes it difficult to comment on the agreement of the phase velocity of the observed waves. (c) The agreement in regard to the form of the waves in these smaller explosions is especially impressive, The change from sinu- soidal form to cnoidal form quite accurately represents the observed (d) The observed waves were nearly fully attenuated by the middle of the second wave group. Stronger attenuation in the higher- frequency range is required in the boundary dissipation formula. (e) Time of arrival (based on linear group velocity) of the wave groups is in very good agreement indicating that transporting energy at linear group velocity, even in the presence of noticeable viscous effects, remains valid into quite shallow water. (f) The fit of the envelope for the large explosion, W = 0.224, at station 59 is only fair. The observed envelope of the first group reaches its maximum ata later time. It appears that for this source strength the explosion may no longer be considered to occur in deep water and that a different source model, perhaps one yielding a higher-order Bessel function, is indicated. In addition the observed troughs of the large waves were much lower than those predicted. Again the probable cause is the presence of an observed strong back- wash current which would have the effect of depressing the trough level. The presence of backwash currents is not reflected by the theory. (g) The agreement in phase is quite good at stations 59 and 64, but not perfect, After wave breaking sets in the phase agreement de- teriorates. One of the observed waves in the vicinity of the first node decomposed into two component waves both of which eventually broke. Otherwise, the theory predicted the correct number of waves. (h) For the large explosion breaking is predicted for the second wave at station 64. Actually, the third and fourth waves broke at this 259 Van Mater and Neal location and the second wave did not break until a subsequent station. A different source model giving a later peak to the first envelope would probably also correct this discrepancy. The theory predicts about the right number of waves breaking at subsequent stations, although there is disagreement in some cases on which individual waves break. The decay rate after breaking appears to be about right for the envelope heights in the surf zone match quite well. As before there is still distortion of the envelope due to backwash effects. The stability criteria seems to give a surf zone of about the right length, but the data are inadequate to say conclusively. Comparison witha much larger number of experiments is needed to fine tune these empirical breaking relationships; however impulsively-generated wave system furnish an ideal vehicle for such studies. (i) The form of the breaking waves and the near-breaking waves is very poorly predicted by this program. The wave slope asymmetries which develop near and at breaking are not reflected at all in the cnoidal wave form. For future generations of the program the incorporation of a theory developed by Biesel [1952] is under con- sideration. (j) Despite the fact that the "correction factor" approach in account- ing for wave peak-up is somewhat distasteful it appears useful at this evolutionary stage. At the minimum the peak-up correction factor should be refined to reflect the influence of w, H/h, and bottom slope. Clearly what is needed is a computationally useable non-linear theory which predicts this phenomena. (k) No comments can be made on the quality of predictions along the refracted rays. Such data were taken in the WES experiments but are not presently available for comparison. In summary it may be said that for small and moderate size explosions the theoretical and empirical program presented gives good predictions of envelope shapes and asymmetry, wave form, and times of arrival of wave groups in shallow water. Wave height enhancement in very shallow water and viscous attenuation are some- what under-predicted. The quality of the prediction of the phase of individual waves remains to be established. For large explosions the present source model gives only a fair prediction of envelope shape. Group times of arrival and form of non-breaking waves con- tinues to be well predicted. Form of near-breaking waves is poorly predicted. Counterflow currents have a prominent influence when waves are very large, but the presence or effect of such currents is not predicted. The location, size, and extent of the surf zone appears satisfactory based on a limited comparison. 260 Impulsively Generated Waves Propagating into Shallow Water ACKNOWLEDGMENTS The work described in this presentation was performed at the Naval Ship Research and Development Center, Washington, D. C. under the joint sponsorship of the Defense Atomic Support Agency and the Naval Ship Systems Command, Task Area SR 104 0301, Task 0583. The authors wish to express their gratitude to the following persons for assistance and support in this project: Dr. Ming-Shun Chang Naval Ship Research and Develop- ment Center Dr. Hun Chol Kim Korean Institute of Science and Technology Professor T. Francis Ogilvie University of Michigan Mr. John Strange U.S.A.E. Waterways Experiment Station Mr. Raymond Wermter Naval Ship Research and Develop- ment Center Miss Claire Wright Naval Ship Research and Develop- ment Center REFERENCES Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, U. S. Dept. of Commerce, Nat. Bureau of Stds., Appl. Math. Series 55, 1964. Adeyemo, M. D., "Effect of Beach Slope and Shoaling on Wave Asymmetry," Proc. 1ith Conf. Coastal Engineering, ASCE, 1968. Benjamin, T. B. and Lighthill, M. J., "On Cnoidal Waves and Bores," Proc, Royal Soc., A.; vs 224; 1954. Biesel, F., Gravity Waves, .U. S. Dept, .of Commerce, Nat. Bureau OL otdsee Circulars 21.. 1952. Cauchy, A. L. de, Mem. de1'Acad. Roy. des Sciences (Memoir dated 1815), 1827. Iverson, H. W., Gravity Waves, U. S. Dept. of Commerce, Nat. Bureau of Stds., Circular 521, 1952. 264 Van Mater and Neal Iwagaki, Y., "Hyperbolic Waves and their Shoaling," Proc. 11th Conf. Coastal Engineering, ASCE, 1968. Keller; JB), Comm. Appl..;Math:.,.v. 1.1948. Keulegan, G. H. and Patterson, G. W., Jour. Res., Dept. of Commerce, Nat. Bureau of Stds., v. 24, 1940. Kajuira, K., Bull. Earthquake Res. Inst., Japan, v. 41, 1963. Korteweg, D. J. and de Vries, G., "On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves," Philo. Magazine (Br.), V Series, V1 29; 1895. Kranzer, H. C. and Keller, J. B., "Water Waves Produced by Explosions," Jour. Appl. Physics, v.30, n. 3, 1959. Laitone, E. V., "The Second Approximation to Cnoidal and Solitary Waves," J. Fluid Mech., Vol. 9, 1960. Laitone, E. V., "Limiting Conditions for Cnoidal and Stokes Waves," J. Geophysical Res., v. 67, n. 4, 1962. Le Méhauté, B., Divoky, D. and Lin, A., "Shallow Water Waves: A Comparison of Theories and Experiments," Proc. 11th conf, Coastal Engineering, ASCE, 1968. Le Méhauté, B., Snow, G. F. and Webb, L. M., Nat. Engr. Science Co., Rpt. S245A, 1966. Masch, F. D., "Cnoidal Waves in Shallow Water," Proc. 9th Conf. Coastal Engineering, ASCE, 1964. Masch, F. D. and Wiegel, R. L., "Cnoidal Waves, Table of Functions," Council on Wave Research, The Engineering Foundation, Richmond, Calif., 1961. Morison, J. R. and Crooke, R. C., U.S. Army Corps of Engineers, Beach Erosion Board, Tech. Memo. 40, 1953. Nakamura, M., Shiraishi, H. and Sasaki, Y., "Wave Decaying Due to Breaking," Proc. 10th Conf. Coastal Engineering, ASCE, 1966. Poisson, S. D., Mem. de l'Acad. Roy. des Sciences, 1816. Stoker, J. J., Water Waves, Interscience, Chap. 6, 1957. Van Dorn, W. G., Le Méhauté, B. and Hwang, L., Tetra-Tech, Inc. Rpt..TC-130, 1968. 262 Impulsively Generated Waves Propagating tnto Shallow Water Van Dorn, W. G. and Montgomery, W. S., Scripps Inst. Ocean. Ref. 63-20, 1963. Van Mater, P. R., Nav. Ship Res. and Dev. Ctr. Rpt. 3354, 1970. Whalin, R. W., "Water Waves Produced by Underwater Explosions: Propagation Theory for Regions Near the Explosion, " Jour. Geophysical Res.; v. 70, ne. 22, 1965a. Whalin, R. W., Nat. Engr. Science Co. Rpt. S 256-2, 1965b. Wwieoel, R. L., Oceanographical Engineerin » Prentice Hall, 1964. APPENDIX 1 COMPUTATIONAL PROCEDURE This appendix discusses the details of implementing the theory outlined in the previous sections in a computer program which will predict the wave system in shallow water. Initially, a monotonically decreasing bottom profile is assumed with parallel straight bottom contours as shown in Fig. 4. Actually, the specific profile used in this program was chosen to conform to that of a test basin at the U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss. in order to permit comparison of the analytical predictions with experiments performed there. A polar coordinate system is established with the origin at the point of the explosion and the axis taken normal to the bottom contours. The axis is divided into a number of closely spaced stations, indexed i, with i=0O atthe origin. The frequency range of interest is also divided into a number of closely spaced frequencies, indexed j. A number of rays, or orthogonals, indexed k, are established emanating from the origin. The local angle of the orthogonal with the normal to the bottom contours varies with the frequency, w, and the water depth at a given location is identified as 6jj,. Because of the frequency dependence absolute congruence of the trajectories of the orthogonals is not possible. To give the best overall conformity with respect to both location and path length the initial angle of the orthogonal at the origin, Oj, is adjusted with frequency. Thus the index k identifies a family of orthogonals which have approximate but not precise spatial agreement, except, of course, on the axis. Throughout indexical notation is for array identification only and tensor convention is not implied. A starting station, i=I, is selected in deep water sufficiently distant from the explosion for Eq. (4) to be valid. That theory strictly 263 Van Mater and Neal is for water of uniform depth, but for practical purposes as long as the minimum depth (at i =I) is greater than half the length of the longest waves with significant energy results will be quite satis- factory. Accordingly the depth used in Eq. (4) is taken as the depth at i=I. Equation (4) may be rewritten in indexical notation as /2 1 qi Hee I Bie Se [paw | cc. allie) a ijk ij (ea) do ij Nijk = Bijn COS (Kj Sijk - &jtijx) (24) where, Bijx = envelope elevation function in dimensional system, (ft) Nijk = wave elevation in dimensional system, (ft) ro,dg = radius and depth of cavity in dimensional system, (ft) Stik = path length to i station along ira ray. Also indexed j since path varies with frequency oj, = Kjhi Vij = group velocity = /gh, - alah Cj I (1 LNs ) (25) Bo EN oo sinh 20); J3 = third-order Bessel function of the first kind =) . Vgh; . — vil)" ete 2qjj - 20); cosh2ojj toij __ 1 ) ty J sinh 2075; J (26) The wave number, Kij » obtained from the equation 2 and may be approximated in closed form by Ki. = #1 (coth SEL FoeN cs g 2 2 1/2 Vere: (28) 264 Impulsively Generated Waves Propagating into Shallow Water The Bessel functions J,(z) and J\(z) are computed from series expansions given in Aueaniowite and Stegun [1964] (eqs. 9.41-9.46). J,(z) and J3(z) are then computed from the recursion relation Jn (2) + Ing (z) = (2n/z)Jp(z). The time of arrival of each frequency at the starting station is computed from the relation. S;; ijk It is now possible to compute the wave spectral energy at the starting station by applying Eq. (12). That equation, rewritten for numerical integration, is: a. = ik 2k PTjk (30) where : 1 “Tk » Re. canoe aaa (31) I Bik = Vy » a (<2) : ee 3 Ae Sy pecs reco i=O \j (1 ciaece el As! (32) cos 0. -l, j,k The last factor in Eqs. (31) and (32) represents the incremental distance along the path where As' is the station spacing on the axis. The energy may now be carried forward from station to station as: Beene Uk or 265 Van Mater and Neal 12 ieee Tial, i kZiel, j,kPiolik = ijk jn Pijr Thus atk Bijk \/2 t -= ! t { Niet, isk az SP heres | S (33) Similarly; Bl. = Blix ijk Pijk qe (3.4) tote OUR La: ty kPisi, jk In the computer program it is the wave system envelope, B', that is carried forward to the next station and the new value at that station computed from Eq. (34). The phase term is then applied to obtain the wave elevation. The envelope function B'(w) is symmetrical about the SWL by definition. This corresponds well to the observed envelopes at the starting station in deep water, but as the system moves into shallow water the envelope and the phase waves develop asymmetries about the SWL which we seek to describe by the application of the cnoidal theory as previously discussed. The first problem is to calculate the elliptic modulus, k, for once this parameter is known, all other cnoidal properties may be computed directly. Two difficulties are immediately realized in the determination of the elliptic modulus. First, Eq. (18) does not admit an explicit solution in k. Secondly, the form of cnoidal waves becomes quite sensitive to k as k approaches unity. For example, there is a noticeable difference between the form of the wave determined by k* = 0.99990 and that determined by k?= 0.999990. Further, explosion parameters of interest require the determination of modulus values as large as k”"=1-10°. Thus the following procedure was employed to efficiently and accurately determine k from Eq. (18). Write Eq. (18) as 2 2 2 2 diwih gaSmiy pee ee as H 1 Aes | oar aie Ee Cad Ge) ta > eae We then seek the roots of the equation g(k) = 0 (35) 266 Impulstvely Generated Waves Propagating tnto Shallow Water Now, smaller roots of g(k), say 0< ages oe 10, are readily obtained by iteratively searching for zeros of g(k) in successively finer increments. Larger roots of g(k), say 1-10% Ki-1,j,k * S00 6. t witii, (38) ge i-l,j,k i=O where i 1 As! ' — e a eee ‘ike > vi] C08 Onin (39) i=0 In Eq. (39) tij, is the time of arrival of the jth frequency com- ponent at location (i,k) and is printed out for each frequency at each location. Introducing the cnoidal phase term of Eq. (14) the wave ele- vation becomes Nijk = - H2ijx + Hijxon” [ sun) (Hi jx)» ix | ° (40) The Jacobian elliptic functions can all be expressed in terms of theta functions, and can be computed from the resulting infinite series. However, in this program the elliptic function cn in Eq. (40) is evaluated to any specified degree of accuracy using Landen's transformations. Let m=k’, m, = € -m. Then for m sufficiently small such that m* and higher powers are negligible, we have the follow- ing approximations for the Jacobian elliptic functions sn(u,m) = sin u - 0.25 m(u - sin ucos u) cos u (41) cn(u,m) = cos u t+ 0.25 m(u - sin ucos u) sinu (42) dn(u,m) = 1-0.50 msin u (43) For m sufficiently close to unity such that mi and higher powers are negligible, we have the approximations tanh u + 0.25 m,(sinh u cosh u - u) sech” u (44) sn(u,m) cn(u,m) = sech u - 0.25 m, (sinh u cosh u - u) tanh u sech u (45) dn(u,m) + sechu + 0.25 m,(sinh u cosh u + u) tanh u sechu (46) 268 Impulsively Generated Waves Propagating into Shallow Water Using the following transformations, intermediate values of the parameter m are reduced or increased such that the above approximations are applicable. To increase the parameter, let Arno 1 - me : Pl tm ee p, = (7) vie emer 1 + P, = Then - \/2, sn(v,p) cn(v,p) sn(u,m) = (1 +p, ) ante op (47) dn(u,m) = (1 - (1 - p'”)) sn’(v,p) (48) : a P| dn(v,p) 2 sn‘ (v p) cn(u,m) = (1 - (1 - P, UC Rea (49) To decrease the parameter, let 15: Be u P= (+=) ’ va— EFT é oy ie m, ip Then an{u;m) = (1 +p") “peek (50) i +p “an (v>p) dn(u,m) =r(1 = 2 ___sn(v,p) (51) > PU TF pent(v ep) I CERN css BLN (52) i oh p* sn*(v,p) Note that in both the descending and ascending Landen trans- formations, sn and dn are required in order to compute cn. In the computer program values of m greater than 0.6 are computed using the ascending transformation. Values of m less than or equal to 269 Van Mater and Neal 0.6 are computed using the descending transformation. The transfor- mations are reapplied until higher powers of m or m, are deemed negligible. The currently used cutoff value is m*(m?) = 10 Both of the Landen transformations converge quite rapidly. Thus the cutoff parameter value is attained in three of fewer applications of the pertinent transformations. Some computational difficulty may be experienced in evalua- ting the hyperbolic functions used in the ascending Landen's transfor- mation, for large values of the argument u. This problem can be alieviated somewhat by reducing the cn argument, u, to its principal value - 4K(k) Su S 4K(k). Further difficulty may be resolved by using the descending transformation throughout the modulus range where applicable. When k =u, the cn* term in Eq. (40) reduces to cos? (Wijx)/ 2 and Hijj, = H2ijx = (Hijk)/ 2 = B'. Thus, for k=0, Eq. (40) reduces to Nijk = B' cos Wijk, which is the usual wave elevation equation. The matter of the frequency dependence of the trajectories of the wave orthogonals has been discussed briefly. In principle it would be possible to compute an initial angle 0 jx for each frequency and at each station which give a path length and a path offset from the axis that would fall within established error limits. Such an iterative procedure would increase the computation time enormously and was rejected on this basis. Several schemes were tried in attempting to find a simple rule for the choice of 99), which would give reasonable conformity in a given family of trajectories. The simplest rule turned out to be the best. A linear distribution of Qik was chosen according to the following relation: * Ojk = Sojkl 1 - 0.04(w - 0.2)] (54) where Ocik is a control angle for the family of trajectories. The choice of the above relation is quite an arbitrary one and a different and more complicated bottom topography could necessitate a different function or the iterative procedure discussed above. The refraction angle at each station along a given path is com- puted from Snell's Law: Bi ik = arcsin Koj_sin Bojk (55) J Ki 270 Impulstively Generated Waves Propagating into Shallow Water The path length and offset from the axis are given by ro As' =i\ = » cos 011 5 x (56) t 239 ! 1 = ° Yiik =) As' * tan 85 i,k ; (57) The system is tested for wave breaking by applying Eq. (19) to the envelope height, Hjj,, increased by a factor of 1.40 to account for non-linear peak up. If the test succeeds and breaking occurs then the increased value of envelope height is retained as H, for use in computing decay after breaking, Eq. (20), and wave stability, Eq. (21). Indexing the breaking point as i=b and any location in the surf zone beyond the breaking point as i= a, these equations become: ; haV'25-58a / apy Hoy = Hy (7) Gr (58) ; 2 ee = pik (0.85 - 0.40 Sita) , (59) o “Stable bjk 8 Once stability is found the envelope height, or rather the symmetrical envelope elevation B = H/2 is carried forward in the usual way. Since the test for breaking is applied to the envelope, strictly, wave breaking can be considered to occur only if a phase wave crest occurs within the breaking band of frequencies. As a practical matter the prediction of the exact arrival of the phase wave is the most difficult and least reliable part of the whole procedure, so that the surf zone should be considered to extend over any region where breaking is predicted for any frequency within the envelope. The boundary dissipation equation, Eq. (22), is applied to the envelope height between successive stations outside the breaking zone. In the present computation the array consists of 100 stations, 120 frequencies, and 3 families of wave rays CH = 0°, 20°, 40°). The computer program is listed in Van Mater [1970]. The program requires 23 minutes of running time on the IBM 7094 and provides about 14,000 lines of output. 271 5 ‘ tng ' iho)” : ; ass file rs sf 4 ich it t " a i re r t £) 1 eet fee ES * ) 40 tA “ “oa oa } AF | $ nt Ph) RS 0 ) Molla thawte any ' est 4. f Me 4 » qin'd Gel n j haga A hy ( { ad \é ¢i ht D R VR ita coe ete RY iti fs aod bole 21S Siren! o dre oe itRend f ey irk Reve eee Levene T is ets bide yee) tgilioe ct ae LP at ee vy Mugs war. i £4 Grieve 46hdelns cf sete “vara mM bt Heruea HYDRODYNAMICS IN THE OCEAN ENVIRONMENT Tuesday, August 25, 1970 Afternoon Session Chairman: J. Hoyt Naval Undersea Research and Development Center, Pasadena Page Three Dimensional Instabilities and Vo-tices Between Two Rotating Spheres 215 J. Zierep, O. Sawatzki, Universitat Karlsruhe On the Transition to Turbulent Convection 289 Ruby Krishnamurti, Florida State University Turbulent Diffusion of Temperature and Salinity: -- An Experimental Study S11 A. H. Schooley, U.S. Naval Research Laboratory Self-Convecting Flows 321 M. P. Tulin, Hydronautics, Inc. and J. Shwartz, Hydronautics-Israel, Ltd. and Israel Institute of Technology 273 1 wade Hh owt SB ay © ‘ T ttc | 9 ache se 7 4 4 voto gk eta f _ i P| 7s ~ : Moose ae a eg s | i ee t THREE DIMENSIONAL INSTABILITIES AND VORTICES BETWEEN TWO ROTATING SPHERES J. Zierep and O. Sawatzki Untversttdt Karlsruhe Karlsruhe, West Germany We study the motion of a viscous medium between two concen- tric rotating spheres. Investigating this type of flow is an extension of the well-known contribution of G. I. Taylor [1], who studied the motion between two rotating cylinders. Due to the action of the centrifugal force instabilities are possible. The main difference between the two flow fields is that for the spheres the centrifugal force is a function of the latitude. We have here an instability ina three-dimensional flow and it is possible that there exist different flow regimes -- stable and unstable ones -- side by side. This prob- lem is closely related to the cellular convection flow, especially to those existing over nonuniformly heated surfaces [2, 3, 4]. The thermal buoyancy corresponds to the centrifugal force, the nonuniform heating corresponds to the latitudinal dependence of the centrifugal force, Now some fundamental things about the used apparatus. * ihe experiments have been done primarily with the inner sphere (Aluminum) rotating and the outer one (Plexiglass) fixed. The gap was filled with silicon oil that contained aluminum powder as flow indicator. Measured has been mainly the frictional torque that keeps the angular velocity of the inner sphere constant. The temperature in the gap was con- trolled by thermocouples and photographs have been taken of the different flow configurations. The measurements have been done by Ritter and Wimmer [ 6] as part of their master thesis. Figures 1 and 2 show the results for two different gap widths. Plotted is the friction torque coefficient 6, over a Reynolds number that ranges from 10! to 10§ Covering this wide range has been accomplished by 1) using silicon oils with viscosities between 3 and 1000 c St and 2) by varying the angular velocity from 0 to 200 revo- lutions /sec. In principle we have here independent of the width of the gap three different domains of fluid motion. For small Reynolds numbers Sic aang Arm ar RG ga For detailed information see [5]. 275 Zterep and Sawatzkt R, = 79.95 mm s 10" Du: Motion in a spherical gap Sms & (Re Fig. 1 CY (Re) for the relative gap width s/R, = 0.0527 10 | Motion in a spherical gap | Coe 28.8 Re <1) R, = 67,80mm R, = 79,95 mm 10° s =12,15 mm 2 “0 Re = Ri i. Cu es | M -} 1 Ce = — | 9, 25. 2 a aay Ry -wy-S / Ss Ta ,, = =: aa R, = 455 | Pte 2] 10 1 wy 5 6 10' 107 10° 10 10 0 Big. 2 o,,(Re) for the relative gap width s/R, = 0.18 276 Instabilities and Vorttices Between Two Rotating Spheres the Navier Stokes equations give Cy ~ 1/Re, the law for creeping flow. Surprisingly this holds up to Re = 3.3 + 103 for the small gap and up to Re = 600 for the larger one. Next to this regime follows one of laminar boundary layer type with ¢,~ 1/VRe. Finally the turbulent flow regime with C,~ 1 Re is Peaches after passing some possible instable flow configurations in the transition region. In general we have this behavior in all cases but quantitatively there are important differences depending on the relative width of the gap. The reason for this behavior is the multitude of the possible flow configurations. First we study the case of the small relative width of the gap. For low Reynolds numbers (for instance Re = 10) the streamlines are concentric circles around the axis of rotation (Fig. 3). With Fig. 3 For small Reynolds number the streamlines are concentric circles around the axis of rotation. s = 5mm, Re= Rw, /v = 10 increasing Reynolds number the streamlines change to spirals (Fig. 4). Close to the rotating sphere the spirals are moving from the poles to the equator but close to the fixed sphere the spirals are moving from the equator to the poles. The inner and the outer spirals join and form closed curves. With passing the critical Taylor number Ta = 41.3 Taylor vortices begin to develop close to the equator. It is remarkable that the critical Taylor number here has the same value as for the concentric cylinders. The axes of these vortices have spiral form and end free in the flow field. (Fig. 5). From 277 Zterep and Sawatzkt Fig 4 For larger Reynolds number the streamlines are spirals, s = 5mm, Re = 2350 Fig 5 For large Reynolds number the vortex axes become spirals and end in the flow field. s = 1.05 mm, Re = 27,000, Ta = 41.6 278 Instabiltittes and Vortices Between Two Rotating Spheres the end of the vortices up to the poles the laminar flow remains stable. With increasing Reynolds number, the axes of the vortices become wavy (Fig. 6) and the flow turns turbulent after passing some intermediate states (Figs. 7, 8}. In this case photographs still show a remarkable distinct structure of the flow. Fig. 6 The vortex axes become wavy for very large Reynolds number. s = 2mm, Re = 16,800, Ta = 68.8 For the larger relative width of the gap this matter is much more complicated. In the transition from laminar to turbulent flow we found that altogether five basically different but reproducible main modes are possible. In the torque diagram all these modes are noticeable and they are remarkably stable as soon as they have become existent. For this reason we called them "stable instabilities." In the experiment these different modes can be established by apply- ing a suitable acceleration of the angular velocity. In analogy to the rotating cylinders [7] we have here a case of nonuniqueness. The mode of instability that is finally realized depends on the initial con- dition given by the experimentator. Now the main modes I - V shall be discussed briefly. I. In spite of having an overcritical state no vortices become visible. The transition to the turbulent flow occurs by passing through mode II, that is described below. Mode I is characterized by the fact 219 Zterep and Sawatzkt Fig 7 The’turbulent)motion. ‘s = 3.5 mm, Re = 53,500, Taii522 Fig. 8 The turbulent motion. s = 2mm, Re = 158,000, Ta = 648 280 Instabilities and Vortices Between Two Rotating Spheres that in the field between the two boundary layers of the rotating and the fixed sphere -- according to the large gap -- a flow is established that moves with a constant but smaller angular velocity than the inner rotating sphere. Obviously this type of motion prevents or at least delays the development of the Taylor vortices. A similar pattern is known to exist also in the gap between two discs [ 8] with one of them rotating. II. This regime is characterized by a flow with vortices that begin at the poles (Fig. 9). The axes of these vortices are inclined Fig. 9 Motion of Mode Il. s = 12.15 mm, Re = 8,300, Ta. =/630 slightly to the streamlines close to the fixed sphere. With increasing Reynolds number these vortices advance around the poles to the equator. The axes become more and more wavy and finally the flow turns turbulent. The physical explanation of these vortices is by no means evident. We have the conjecture that we have here a situation analogous to the occurrences close to a free rotating sphere [9] or disc [10]. Very often these vortices are called Stuart vortices. Contrarily to the familiar pattern of vortices, rotating with alternating direction the Stuart vortices rotate all in the same direction. III. Two Taylor vertices develop symmetrical to the equator. Outside the vortex zone we have mode I flow. Surprisingly at the equator -- where the centrifugal force has its maximum -- the flow 281 Zterep and Sawatzkt Fig 10 Sketch of Mode III Fig 1% Motion of Mode III. s = 8 mm, Re = 8,130, Ta = 318 282 Instabilities and Vortices Between Two Rotating Spheres is directed inward (Figs. 10, 11). This can be explained by a cellular motion in the field between the pole and the vortex that forces the vortex to rotate in the mentioned direction. The result is the sink flow at the equator. IV. Two pairs of Taylor vortices develop symmetrical to the equator but now with an outward motion at the equator (Figs. 12, 13). Mode III is a limit case of IV reached by increasing angular velocity. The cell close to the equator becomes smaller and smaller and in the limit the flow reverses at the equator. V. This is an unsteady version of mode III. Vortices, generated at the equator, leave the equator under a small angle of about 10° (Fig. 14) and move on spiral trajectories to the pole. It is interesting to see that the critical Taylor number increases with increasing gap width. Corresponding calculations for rotating cylinders with arbitrary gap width, done by KirchgaBner [141], agree very well with our experimental results for spheres having the same direction (Fig. 15). The explanation for this is that in our case the instability first begins at the equator and we have there locally a similar situation as in the case of the two cylinders. \S SSS Fig. 12 Sketch of Mode IV 283 Zterep and Sawatzkt Fig. 13 Motion of Mode IV. s = 12.15 mm, Re = 2,660, Ta = 201 Fig. 14 Motion of Mode V. s = 12.145a0m, Re =i ,210, La =95.6 284 Instabilities and Vortices Between Two Rotating Spheres 35 000 rods Motion in a cylindrical gap if 30 000 Re. Arwyrs y 25 000 2 Re kr 15 000 10 000 5 000 0 Eno &. 7. 8 IS 17 19 21 Fig.15 The critical Reynolds number for rotating cylinders [11] and the corresponding measurements for the spherical gap As far as theory is concerned we have treated three problems. Without going into details we give a short summary. a. In case of fully laminar flow and a small relative gap width the differential equations can be solved by using an approximation method like that of v. Karman - Polhausen. The results are simple expressions for the streamlines. Close to the walls these are loga- rithmic spirals that fit very well to the experimental results (Fig. 4). b. Mode 1 -- with larger relative gap -- can also be treated easily. For the region close to the fixed and the moving sphere esti- mations can be used for the boundary layer thicknesses. For a first approximation results for the boundary layer of a rotating disc [ 12] can be used. Between these two boundary layers we have an already 285 Zterep and Sawatzkt mentioned full laminar flow. The Navier Stokes equations in spherical coordinates give a very simple solution here with the velocity linear in r. The analytical expressions for the three flow regions can be combined and a short and simple calculation gives a torque coefficient that fits surprisingly well to our experimental results. c. The Stuart vortices -- already mentioned in connection with mode II -- all rotate in the same direction. Experiments done with a rotating free sphere [9] have confirmed this type of vortices. A simple cinematic consideration shows how one comes from the Taylor-Gortler vortices that have an alternating direction of rotation to the Stuart vortices. For this it is only necessary to superpose a suitable flow field to the Taylor-Gortler vortices with a flow direction cross to the vortex axes. With other words: to get Stuart vortices it is necessary that the Taylor vortices become embedded in a suitable flow field. We are dealing here with a real three-dimensional effect that changes our vortex model. One realizes easily that for a rotating free sphere or in the gap between two spheres of which one is rotating the just mentioned situation exists. REFERENCES [1] Taylor, G. J., "Stability of a Viscous Liquid Contained Between Two Rotating Cylinders," Phil, Trans. A 223, 289-293, 19236 [2] Zierep, J., "Thermokonvektiv Zellularstromungen bei inkonstanter Erwarmung der Grundflache," ZAMM 41, 114-125, 1961. [3] Koschmieder, E. L., "On Convection of a Nonuniformly Heated Plane," Beitr. z~ Phys. d. Atmos., 39, 208-216, 1966. [4] Muller, U., "Uber Zellularstromungen in Horizontalen Flussig- keitsschichten Mit UngleichmaBig Erwamter Bodenflache, " Beitr. z. Phys. d. Atmos., 39, 217-234, 1966. [5] Sawatzki, O., and Zierep, J., "Das Stromfeld Zwischen Zwei Konzentrischen Kugelflachen, von Denen die Innere Rotiert, " Acta Mechanica, 9, 13-35, 1970. [6] Wimmer, M., and Ritter, C. F., "Die Stroémung im Spalt Zweier Konzentrischer Kugeln," Diplomarbeiten, Univ. Karlsruhe, Lehrstuhl fur Stromungslehre 1968, 1969. [7] Coles, D., "Transition in Circular Couette Flow," J. Fluid Mech. 21, 385-425, 1965. 286 Instabilities and Vortices Between Two Rotating Spheres [8] Schultz-Grunow, F., "Der Reibungswidstand Rotierender Scheiben im Gehause," ZAMM 15, 191-204, 1935. [9] Sawatzki, O., "Das Stromungsfeld um eine Rotierende Kugel," Acta Mechanica, 9, 159-214, 1970. [10] Gregory, N., Stuart, J. T., and Walker, W. S., "On the Stability of Three-dimensional Boundary Layers with Application to the Flow Due to a Rotating Disc," Phil. Trans. Roy. Soc. A248, 155-199, 1955. [11] Kirchgafner, K., "Die Instabilitat der Stromung zwischen zwei rotierenden Zylindern gegeniiber Taylor-Wirbeln fur beliebige Spaltbreiten, ZAMP 12, 14-30, 1961. [12] Cochran, W. G., "The Flow Due to a Rotating Disc," Proc. Camb. Phil. Soc. A 140, 365-375, 1934. DISCUSSION L. van Wijngaarden Twente Institute of Technology Enschede, The Netherlands The description of mode I reminds me of the result that Batchelor [ 1956] derived for laminar flow with closed streamlines of fluids with small viscosity: Thin boundary layers on solid boundaries separated by a region of constant vorticity. This result was derived for two-dimensional flow. In your case the flow is three- dimensional, but it might be that the same conditions which are neces- sary for Batchelor's result hold in this case of mode I. REFERENCES Batchelor, G. K., "On steady laminar flow with closed streamlines at large Reynolds number," J. Fluid Mech., Vol. 1, p. 177, 1956, 287 Zterep and Sawatzkt REPLY TO DISCUSSION J. Zierep Untversttdt Karlsruhe Karlsruhe, West Germany The general condition for the existence of closed streamlines, given by Batchelor in the cited reference, can be applied to the present case of a flow with rotational symmetry. We obtained infor- mation about the velocity distribution that has been confirmed by our analytical analysis, following a different path. 288 ON THE TRANSITION TO TURBULENT CONVECTION Ruby Krishnamurti Flortda State Universtty Tallahassee, Florida EINE. RODUCTION The heat flow out of the sea floor has been observed in close to 2000 measurements; the mean value for all the oceans is found to be 1.4 X 10° cal/cm’sec. [Lee and Uyeda 1965] This is three orders of magnitude smaller than the solar heating at the sea surface and is surely negligible in any budget of the upper oceans. Yet, because this heat flux is imposed from below, it may be of some consequence in the dynamics of the abyssal circulation. If this heat were to be transferred purely by gonduction through the sea water, a temperature gradient o of 10.°C°/m would be required. The largest depth across which such a gradient can exist without con- vective overturning is determined by the critical value of the Rayleigh number R, which is defined as follows: where g is the acceleration of gravity, @ the thermal expansion coefficient, kK the thermal diffusivity, p the kinematic viscosity, and d is the depth of the layer in consideration. This largest depth that can transfer the imposed heat flux by conduction is only around 3 cm. If there are regions or time periods of the abyssal oceans in which horizontal advection of heat is not the dominant process, then this vertical convection, with its attendant vertical mixing of nutrients, can be an important process. Some understanding of convective processes can be gained from laboratory studies of a horizontal layer of fluid which is heated from below and cooled from above. The following is a review of such laboratory studies and also a report of some recent experiments in rotating and non-rotating systems. * This is contribution No. 33 of the Geophysical Fluid Dynamics Institute. 289 Krishnamurtt II. TRANSITION TO TURBULENT CONVECTION IN A NON- ROTATING LAYER OF FLUID Unlike the fast transition to turbulence in plane parallel shear flows, the horizontal convecting layer undergoes a number of discrete transitions, remaining in each régime for a finite range of Rayleigh number. The transition to turbulent convection appears to result in the following manner: at sufficiently low values of the Rayleigh number the fluid system is stable to all small disturbances. As the value of R is increased the system becomes unstable to one kind of disturbance. As R is increased still further the fluid becomes un- stable to more kinds of disturbances. At sufficiently large Rayleigh numbers the flow is unstable to so many kinds of disturbances, each occurring with uncontrolled phase, that the flow may be called tur- bulent. Before discussing the first three of these transitions, the experimental apparatus will be described. Apparatus One of the possible designs of experimental apparatus is shown in Fig. 1. The fluid layer occupies a region 51 by 49 cm, with a depth that can be chosen (usually between 1/2 and 5 cm). CONSTANT O76, © 2a. O10 -Ov@ 2 8 Se GE y j © y SoS ee YY [recoroer | g VOLTAGE. ) elbow ZLEEY O TRANSFORMER ai g @LO.O. Oo Ou oe = = (ey) ALUMINUM 6061 FLUID e METHYL STYROFOAM GU METHACRYLATE INSULATION Fig. 1. Apparatus 290 On the Transttton to Turbulent Convectton The plexiglass tank containing the fluid also contains four blocks of aluminum 6061 T 651. Two of the blocks are 4 in. thick, two are iin, thick, each is Z0 in. by 20 in. wide. The electrical heater, which is a fine mesh of resistance material embedded in silicon rubber, is attached to the bottom of the lowest block, which is 4 in. thick. The heat input is controlled by a variable transformer backed by a constant voltage transformer of the line voltage. Above this lowest aluminum block is a low-conductivity layer of methyl metha- crylate. A layer of liquid sufficiently thin that it never convects for the temperature gradients occurring in these experiments effects constant thermal contact between the layers. Above this low con- ductivity layer is a block of aluminum 1 in. thick; above this is the convecting fluid, whose depth is defined by plexiglass spacers. The arrangement of blocks above the convecting layer is symmetric to that below except that the cooling is accomplished by cooling fluid from a constant-temperature circulator flowing in channels in the uppermost aluminum block. The channels for incoming and outgoing flows are side by side in order to minimize horizontal temperature gradients. The channels were cut in a complicated pattern and spaced so that the separation of channels was not close to an integral multiple of the expected convection cell size. The maximum flow rate of the cooling fluid is 2+ 5 gal/min. This apparatus was used in the studies which will be described with air, water, and silicone oils. For convection in mercury, the aluminum blocks were replaced by copper blocks of which two are 2 in. thick, two are 1 in. thick and each is 20 in. by 20 in. wide. The thermal conductivity of the aluminum is about three orders of magnitude larger than that of the oils. The thermal conductivity of copper is 50 times as large as that of mercury. This is, of course, an attempt to approach the ideal condition of perfectly conducting boundaries. With poorly conducting boundaries a horizontal tempera- ture ripple corresponding to the cellular structure in the convecting fluid penetrates into the boundaries and may control transitions to different cellular structures. Also the metal acts as a diffuser of any horizontal temperature variations arising from the discrete nature of the cooling channels. The large mass of metal (approxi- mately 400 1b of aluminum or 700 lb of copper) acts as a large heat capacity so that temperatures in the blocks are very stable. The heat transported by the convecting liquid is measured by concentrating the temperature gradient across the poor conductor in the manner devised by Malkus [1954]. In the steady state the heat H transported by the fluid is the average of the heat conducted across the two poor conductors: ig ye Tos on Dee H=k,—7q es : where Kp and kp are the molecular conductivities of the low con- ductivity layers, dp is the depth of the layer, and T,, T,, Ts and 29d Krishnamurtt T, are the temperatures of the four aluminum blocks. The sub- scripts are ordered from bottom to top. The conductivities kp and k' are measured in terms of that of the liquid when it is known that the liquid layer is in a state of steady conduction. Then the following relations hold: Ty AVEL wi) ) Dpeeeignw pa Tail Ty k, —22—3 = k, 2 =k) —3 4 d Bi) a, pi Bidgig 1° where k, is the molecular conductivity of the fluid. Thus, once the conductivity and depth of the poor conductors is determined, a measurement of the temperatures in the four metal blocks allows the determination of the Rayleigh number and the heat flux. Fine aluminum flakes suspended in the liquid were used to visualize the flow. The aluminum flakes become aligned in a shear flow, and because they are flakes, reflect light more strongly in certain directions , depending upon the direction of the shear and of the illumination. In a uniform shear, the brightness is uniform; where there is a differential shear, there will be corresponding bright and dark regions. In the case of water, alumimm flakes would not stay in suspension sufficiently long, so another tracer called 'rheoscopic fluid AQW 010' was added to the water. This tracer displays differential shears, just as do the aluminum flakes, but remains in suspension about 10 times as long. Since the fluid layer is bounded above and below by opaque boundaries, the plan form of convection is obtained by viewing the flow from the side as shown in Fig. 2. The tracers were illuminated at mid-depth by narrow overlapping beams of collimated light from two 2 W zirconium arc lamps. The two beams directed at each other allow visualization of shear regions at both positive and negative angles to the line of sight. This line of sight is perpendicular to the beam. As the light beam is moved horizontally, illuminating differ- ent regions of the fluid, a camera is moved horizontally on a threaded Illumination Illumination at x4 at Xp Camera Camera position A position B Fig. 2. Geometry for photographing plan form of convection 292 On the Transttton to Turbulent Convectton rod in order to keep the illuminated region in focus. Simultaneously, the back of the camera rolls on an inclined plane since the camera is free to rotate about an axis through its lens. Thus, different regions of the fluid produce images on different parts of the film. In this way, one obtains a picture of the flow pattern as if one were viewing from above. For each steadily maintained external condition, the steadi- ness or non-steadiness of the resulting flow was to be determined. This was found to be too difficult by simply observing moving tracers through the fluid since there were gentle time dependencies with time scales of the order of several minutes to several hours. In order to have a record of the flow at an earlier time against which to compare the flow at a later time, the following photographic technique was devised. The apparatus used is shown schematically in Fig. 3. Two narrow overlapping beams of light illuminate aluminum flake tracers along a line inthe x-direction, say, through the fluid. The beam remained fixed in space throughout the obser- vation time. The camera was free to rotate about an axis through its lens. With the camera aperature open, a synchronous motor drew a wedge under the back of the camera at a rate determined by the time scale of the time dependence of the flow. Thus, the photo- graph displays an (x,t) representation of the flow, where t is the time coordinate. At t = 0, the camera recorded alternating bright and dark regions, corresponding to the cellular structure, as a narrow strip of image across the film. When the flow was steady, the cell boundaries remained fixed in time, thus producing straight lines parallel to the t-axis on the photograph. With the beam near the top (or bottom) of the convecting layer, the tracer particles have an x-component of velocity which is given by the slope of the trajectories inthe (x,t) representation. 1Convecting liquid ~N Light beam ane | p—F —nn Fig. 3. The apparatus for photographing the time evolution of flow 293 Krishnamurti The studies that will be described here were performed as externally steady, fixed heat flux experiments. Rayleigh number and heat flux were measured for fluids having Prandtl number from 10° to 104. The Rayleigh number ranged from 10° to 108, Except in the cases of air and mercury, the "pDlan form" of the convection was obtained by viewing from the side. The time dependence was determined by both the (x,t) photographs and thermocouples internal to the fluid. In the cases of air and mercury time dependence was determined only by the internal thermocouples. The First Transition In the order of increasing R, the first transition occurs at the well-known critical Rayleigh number R,. This is a transition from the conduction state to one of steady cellular convection. It occurs independently of the Prandtl number Pr where Pr = v/Kk. The nature of the flow and the change in slope of the heat flux curve have been predicted and experimentally verified. For the vertically symmetric problem the only stable finite amplitude solution of the infinite number of possible steady solutions is the two-dimensional roll [Schluter, Lortz and Busse 1965]. With a vertical asymmetry, such as that produced by changing mean temperature or by variation of material properties (v¥,xX,@) with temperature, the conduction state is subcritically unstable to finite amplitude disturbance, and the flow near the critical point is hexagonal [ Busse 1962; Segel and Stuart 1962; Krishnamurti 1968a,b| - Inthis discussion we restrict our attention to the case in which rolls are the realized flow just above Re. As the heat flux, and hence the Rayleigh number, are increased above Re, steady two-dimensional rolls continue to be the observed flow up to approximately izZ.R.; ter, 10< Pr < 10%. The size of the rolls becomes larger in this range, as shown in Fig. 4, where the wave-number 6 is plotted against Raleigh number. This increased size of the cell might be rationalized by an argument such as the follow- ing. By averaging over the entire fluid the non-dimensionalized tem- perature equation in the Boussinesq approximation one finds H = Rom + (w9) where H is the dimensionless heat flux, o, is the vertical tem- perature gradient averaged over the entire fluid, w is the vertical velocity, 6 is the departure of the temperature from a horizontal average, and brackets indicate averaging over the entire fluid. Thus Ro, is the heat flux due to conduction, (w®8) is the convective heat flux. As the externally imposed heat flux is increased such that R exceeds Rg, the fluid transfers this larger flux through the cor- relation (w®). Consider a fluid parcel near the lower boundary. Its temperature 9 is limited by the thermal diffusivity of the fluid material. As H is continually increased, the fluid is forced to 294 On the Transtitton to Turbulent Convectton The Second Transition The only theoretical study of stability of two-dimensional con- vection in the Rayleigh number range of the second transition is that of Busse [1968]. He shows that for infinite Prandtl number, two- dimensional rolls having wave-number ff within a finite band (see Fig. 4) are stable to a restricted class of infinitesimal disturbances provided that R< 22,600. If R > 22,600 rolls are unstable for all B. Busse shows further that the roll plan form is then unstable to a disturbance of rectangular form with one side along the original roll axis. It is not known from this theory whether the resulting flow above 22,600 is steady. It is also not known how the selection of B from this band of possible wave-numbers occurs. Laboratory studies [ Krishnamurti 1970a] show that two- dimensional rolls do indeed become unstable near this Rayleigh number, which will be labelled R,. The "plan forms" (obtained from the side) are shown in Fig. Ba where that on the left shows rolls below Ry, that on the right shows the flow pattern above Ry. The three-dimensional disturbance that forms on the rolls above R,, is consistent with Busse's instability to a rectangular distur- bance. Since the method of photography displays regions of strong shear, the hypotenuse of the rectangle should appear bright. Thus, the nature of the growing mode (which is found experimentally to attain a steady state) is in agreement with Busse's result. It may be noted that the rectangular disturbance of his theory is one with symmetry in the vertical. The point of transition is also in good agreement with that computed by Busse, for that wave-number 6 which occurs in the experiment. Figure 5b shows the same transi- tion when a circular boundary of plexiglass has been inserted into the rectangular region. Both Davis [1967] and Segel [1969] show that spatially modulated rolls will line up with their axes parallel to the short side of a rectangular container. In the almost square container, there appeared to be little preference of orientation of the rolls; rolls were seen along the line of sight as well as perpen- dicular to the line of sight in two different repetitions of the same experiment. The preference of rolls to line up with their axes parallel to the short side may be re-expressed as a preference of the rolls to meet the boundaries rather than lie along the boundaries. This effect is displayed in Fig. 5b. Presumably circular rolls did not develop because the plexiglass has thermal conductivity so close to that of the fluid that there was negligible distortion of the con- duction temperature field and no fringing of the isotherms since there was fluid outside of the ring. Associated with this change from steady two-dimensional to steady three-dimensional flow, there is observed a discrete change in slope of the heat flux curve (Fig. 5a). This corresponds to the second change of slope observed by Malkus [ 1954]. Ry, showed no 295 Krtshnamurtt move more rapidly to transport this increased heat flux. If the fluid must move faster, then the cells must be larger in order to allow the hot rising fluid to be in the vicinity of the cold upper boundary for a sufficiently long time to lose its heat before sinking and repeating the process. Although there are many ways in which the fluid could have transferred the increased heat flux, moving more rapidly with increased cell size is one ofthem. Of course, if the cells become very large, the viscous dissipation of energy near the horizontal boundaries would slow down the flow and defeat its own purpose. This will be discussed later. It is seen in Fig. 4 that, when the cell size is allowed to evolve freely (without being forced as in the ex- periments of Chen and Whitehead [ 1968]), approximately one-half of Busse's stability diagram is filled with observations, but the domain B > f, is conspicuously bare. 2x 10° 1x10 Rt 5x10? Unstable 3x10 2x10 d (cm) R increasing R&R decreasing Pr iGai 1-2 x ® Pr 57 2 Oo Pr 10? 2 + ® Pr 0:86 x 10° 3 ® @) Pret x 10* 5 A A Pr 0-85 x 104 2 * Fig. 4. The observed cell size plotted on Busse's stability diagram for two-dimensional rolls 296 On the Transtitton to Turbulent Convection The Second Transition The only theoretical study of stability of two-dimensional con- vection in this Rayleigh number range is that of Busse [1968]. He shows that for infinite Prandtl number, two-dimensional rolls having wave-number f within a finite band (see Fig. 4) are stable toa restricted class of infinitesimal disturbances provided that R < 22,600. If R> 22,600 rolls are unstable for all B. Busse shows further that the roll plan form is then unstable to a disturbance of rectangular form with one side along the original roll axis. It is not known from this theory whether the resulting flow above 22,600 is steady. It is also not known how the selection of B from this band of possible wave-numbers occurs. Laboratory studies [Krishnamurti 1970a] show that two- dimensional rolls do indeed become unstable near this Rayleigh number, which will be labelled R,. The "plan forms" (obtained from the side) are shown in Fig. 5a where that on the left shows rolls below Ry, that on the right shows the flow pattern above Ry. The three-dimensional disturbance that forms on the rolls above Ry, is consistent with Busse's instability to a rectangular disturbance. Since the method of photography displays regions of strong shear, the hypotenuse of the rectangle should appear bright. Thus, the nature of the growing mode (which is found experimentally to attain a steady state) is in agreement with Busse's result. It may be noted that the rectangular disturbance of his theory is one with sym- metry in the vertical. The point of transition is also in good agree- ment with that computed by Busse, for that wave-number £8 which occurs in the experiment, although the selection mechanism of that B is not understood. Figure 5b shows the same transition when a circular boundary of plexiglass has been inserted within the rectangu- lar region. Both Davis [1967] and Segel [1969] show that spatially modulated rolls will line up with their axes parallel to the short side of a rectangular container. In the almost square container, there appeared to be little preference of orientation of the rolls; rolls were seen along the line of sight as well as perpendicular to the line of sight in two different repetitions of the same experiment. The preference of rolls to line up with their axes parallel to the short side may be re-expressed as a preference of the rolls to meet the boundaries rather than lie along the boundaries. This effect is dis- played in Fig. 5b. Presumably circular rolls did not develop because the plexiglass has thermal conductivity so close to that of the fluid that there was negligible distortion of the conduction tem- perature field and no fringing of the isotherms since there was fluid outside of the ring. Associated with this change from steady two-dimensional to steady three-dimensional flow, there is observed a discrete change in slope of the heat flux curve (Fig. 5). This corresponds to the second change of slope observed by Malkus [ 1954]. Ry, showed no (Ae hts Krtshnamurtt HEAT FLUX vs RAYLEIGH NUMBER PRANOTL NUMBER = 860 SHOWING THE SECOND TRANSITION. a HEAT FLUX « 10 Fig. 5a. RAYLEIGH NUMBER x10“ Heat flux plotted against Rayleigh number showing the second transition. Photographs show the corresponding change in plan form. The Prandtl number is 860. 298 On the Transttion to Turbulent Convectton Fig. 5b. Photographs showing the plan form within circular side walls. The transition is the same as in Fig. 5a. The Prandtl number is 860. definite Prandtl number dependence in the range 10 < Pr < 107. There was a marked hysteresis both in the heat flux and plan form as the Rayleigh number was increased then decreased past Ry . This transition is shown by the curve labelled II in the régime dia- gram (Fig. 10). The Third Transition The third transition in order of increasing R occurs ata Rayleigh number which will be labelled R,,,;. It marks a change from steady three-dimensional to time-dependent flow, and has associated with it a discrete change in slope of the heat flux curve (Fig. 6) [Krishnamurti, 1970b]. The change in slope was gmeasured for each of the fluids shown in Fig. 10, with 10° “@ T, there must be a transition from the conduction state to a time de- pendent flow as the Rayleigh number is increased beyond the critical. The apparatus consisted of a fluid layer 1 or 2 cm in depth, 18 inches in diameter in the horizontal direction. The fluid was bounded below by a 2 in. thick aluminum block containing an electri- cal heater which is a fine mesh of resistance material. Above the fluid layer was bounded by a glass plate over which the cooling fluid circulated. Photographs taken from above by a camera rotating with the fluid are shown in Fig. 11. Figure 11a shows rolls, Fig. 11b shows the cross instability forming on the rolls, of the same kind found in 304 On the Transitton to Turbulent Convectton 10° or q TURBULENT FLOW : o TIME 2 DEPENDENT® 6 3-DIMENSIONAL +105 = —| g0 2 A pow e a e es { =a Sins =e 3 7 STEADY, 3- °DIMENSIONAG FLOW ° © Sie : 4 ° ae Se sy x mr y IER ° 8 Pr= © > ° ° ° Zi0*+ & : 3 ti ae i wa Fe STEADY 2-DIMENSIONAL ° FLOW | 5 10° 10? 10" 1 fe) 10? 10° 10* PRANDTL NO. Fig. 10. The régime diagram. The circles represent steady flows, the circular dots represent time-dependent flows. The stars represent transition points. The open squares are Rossby's observations of time-dependent flow, the squares with a dot in the centre are Willis and Deardorff's obser- vations [1967b] for turbulent flow. The triangle is Silveston's [1958] point of transition to time-dependent flow. the non-rotating case. Figure iic shows the break-down of rolls and waves forming on them. The disturbance forms an angle of 58° + 2° to the original roll axis, exactly as predicted by Kuppers and Lortz. Figure iic is atransient state; iid is the final steady state. In this state the over-all wavy pattern was not observed to change with time but the internal striations representing regions of strong shear were segn to change with a time scale of the order of one minute. (Here d‘/v = 40 sec, d*/k@1 hr). Figure 12 shows the régime diagram for the rotating convection. The observed criti- cal Taylor numbers compare only approximately with those computed by Kuppers [1970] for finite Prandtl number and rigid boundaries. The observed transitions occurred at T, =1.5X10° for Pr= 6.7, R@ R, andat T,=7%10* for Pr=10%, R= R,., The predicted values are T, =7X10° for Pr=1, Te=1.7X10° for Pr=5, and T, = 2103 for Pr— oo. 305 Krtshnamurtt Fig. 11. Rotating Benard Convection, showing cross and wave instabilities on rolls (ajlresetee Ry Ts 114 X40") Pe (Bb) -Reei9ii2 Reset 1 & 102, Pe instability (c)i, Ruste UR. § T= 207) X h0> ay Px developing of waves (d)) RS 21 Ro) T= 2. 7x40, Pz 6.7 rolls 6.7.07 8S 10°; showing the I 119% showing developed waves 306 On the Transition to Turbulent Conveetton x STEADY FLOW Po ss Wo fe INSTABILITY STEADY a = x = 6. FLOW 3 Pes 60 CROSS 8 INSTABILITY f 5 13 af 6 WAVE 7} INSTABILITY > UNSTEADY 12 a4 a fo) i Fs FLOW ee ee @ ® @ Wi 2 STEADY . 13 @ e e to) 107 TAYLOR NUMBER RAYLEIGH NUMBER 3 9 8 ° \ UNSTEADY : STEADY TWO- FLOW DIMENSIONAL a7 WAVE INSTABILITY FLOW 102 TAYLOR NUMBER 2 3 45678910 Fig. 12. The régime diagram for rotating Bénard convection IV. SUMMARY AND CONCLUSIONS Series of externally steady, fixed heat flux experiments were performed to measure Rayleigh number, heat flux and changes in flow of horizontal, non- rotating convection for 2.5X10°= Pre 0.85 X 10% and 103< Ra<10% The régime diagram summarizing these experiments is shown in Fig. 10. Each of the curves I, II, III and IV marks a transition with a change of slope in the heat flux curve. The first is the transition from the conduction state to one of steady two-dimensional convection in the form of rolls. There is a second transition characterized by the following properties: (i) There is a discrete change of slope of the heat flux curve at Rayleigh number R,, near 12 R,, showing no definite Prandtl number dependence in the range 10 < Pr< 104, 307 Krishnamurtt (ii) There is a change in the flow pattern from two-dimen- sional rolls to a three-dimensional flow which is periodic in space and steady intime. The change occurs ata Rayleigh number coinciding with R,, to within the error in determining Rj. (iii) There is hysteresis in the heat flux as well as in the flow patternas R is increased from below or decreased from above, indicating that the transition is caused by a finite amplitude instability. The third transition is indicated by curve III in Fig. 10. Above this curve, the flow is time dependent with a slow tilting of the cell in the vertical and a faster oscillation which has the nature of hot or cold spots advected with the mean flow. Transition to disorder is seen to result from an increased number and frequency of such oscillations. Higher transitions observed by Malkus [1954] and confirmed by Willis and Deardorff [1967a] have not been discussed. The small amplitude nonlinear theories have been quite suc- cessful in a small neighborhood of the critical point Rg. The obser- vation that transition to turbulence occurs near Re for small Prandtl number in non-rotating convection, and for T> T, for rotating convection, indicates the possibility of gaining further understanding of transition to turbulence through the nonlinear theories. The research reported here was supported by the Office of Naval Research Contract N-00014-68-A-0159 and by grant number GK-18136 from the National Science Foundation, REFERENCES Busse, F. H., Dissertation, University of Munich. (Translation from the German by S. H. Davis, the Rand Corporation, Santa Monica, California, 1966), 1962. Busse, F. H.,"On Stability of Two-Dimensional Convection in Layer Heated from Below,"J. Math. and Physics, 46, 140, 1968. Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Oxford, 1961. Chen, M. M. and Whitehead, J. A. ,"Evolution of Two-Dimensional Periodic Rayleigh Convection Cells of Arbitary Wave- Numbers," J. Fluid Mech., 31, 1, 1968. Davis, S. H., "Convection in a Box: Linear Theory," J. Fluid Mech. s 30, 465, 1967. 308 On the Transitton to Turbulent Convection Fultz, D. and Nakagawa, Y., "Experiments on Oven Stable Thermal Convection in Mercury," Proc. Roy. Soc. A. 231, 198, 1955, Krishnamurti, R., "Finite Amplitude Convection with Changing Mean Temperature, Part 1, Theory," J. Fluid Mech., 33, 445, 1968a. -_ Krishnamurti, R., "Finite Amplitude Convection with Changing Mean Temperature, Part 2, An Experimental Test of the Theory," J. Fluid Mech., 33, 457, 1968b. Krishnamurti, R., On the Transition to Turbulent Convection, Part I, The Transition From Two- to Three-Dimensional Flow," J. Fluid Mech., 42, 295, 1970a. Krishnamurti, R., "On the Transition to Turbulent Convection, Part 2, The Transition to Time Dependent Flow," J. Fluid Mech., 42, 309, 1970b. Kuppers, G. and Lortz, D., "Transition From Laminar Convection to Thermal Turbulence in a Rotating Fluid Layer," J. Fluid Mech., 35, 609, 1969. Kuppers, G., private communication, 1970. Lee, W. H. K. and Uyeda, S., Terrestrial heat flow, Washington, D.C., pp. 87-190. (American Geophysical Union, Geophysi- cal Monograph series no. 8), 1965. Malkus, W. V. R., "Discrete Transitions in Turbulent Convection," Proc. Roy. Soc. A225, 185, 1954. Rossby, H. T., Dissertation, M.I.T., 1966. Schluter, A., Lortz, A. and Busse, F., "On the Stability of Steady Finite Amplitude Convection," J. Fluid Mech., 23, 129, 1965. ae Segel, L. A., "Distant Side- Walls Cause Slow Amplitude Modulation of Cellular Convection," J. Fluid Mech., 38, 203, 1969. Segel, L. A. and Stuart, J. T., "On the Question of the Preferred Mode in Cellular Thermal Convection," J. Fluid Mech., 13, 289, 1962. Silveston, P. Leis Forch. Ing. Wes. 24, 29-32, 59-69, 1958. Veronis, G., "Celular Convection with Finite Amplitude in a Rotating Fluid," J. Fluid Mech., 5, 401, 1959. 309 Krishnamurtt Willis, G. E. and Deardorff, J. W., "Development of Short-Period Temperature Fluctuations in Thermal Convection," Phys. Fluids, 10, 931-937, 1967a. Willis, G. E, and Deardorff, J. W., "Confirmation and Renumber- ing of the Discrete Heat Flux Transitions of Malkus," Phys. Fluids, 10, 1861, 1967b. 310 TURBULENT DIFFUSION OF TEMPERATURE AND SALINITY: — AN EXPERIMENTAL STUDY Allen H. Schooley U.S. Naval Research Laboratory Washington, D.C. ABSTRACT Stratified temperature and salinity conditions in water have been established in a small laboratory tank. A method for making measurements and cal- culating the eddy diffusivities of temperature and salinity for different controlled levels of turbulence are described. The ratio of temperature and salinity molecular diffusivities is on the order of 100. The ratio of temperature and salinity eddy diffusivities, for the most turbulent conditions studied, is 14. The dissipation of turbulent power density (P) due to viscous friction was found to be on the order of 10’ larger than the power density (P') consumed in changing the thickness of the pycnocline. The experiments hint that P/P' may be relatively constant over a range of turbulence. If this is assumed to be true, there exists the possibility of estimating temperature (D) and salinity (D') eddy diffusivities by knowing the change of density, (Ap), and (Ap),, with time (At) for a given depth differ- ence (Ah). Plots of D and D!' in cm/sec vs. (P'/n)'4 =110(Ap/At)'/2(Ah) in sec”', are shown where (n) is the dynamic viscosity of water. 344 Schooley I. INTRODUCTION The oceans are dominated by several turbulent processes. For each turbulent situation there are "eddy" diffusivities of tem- perature and salinity that are much larger than the molecular diffusivities. In spite of the difficulties in measuring eddy diffusi- vities at sea, there is considerable, though incomplete, literature on the subject [ Neumann and Pierson, 1966]. Since turbulent ocean processes are inherently uncontrollable, several exploratory labora- tory experiments were conducted in late 1964 and early 1965. This paper is the first publication of the results of these preliminary experiments. Il APPARATUS. Figure 1 shows the test cell where the experiments were conducted. It has transparent plastic walls and is 30 cm long, 9 cm high, and 2.5 cmthick. The bottom 4 cm was filled with either room temperature distilled water or a 0.25% solution of sodium chloride, depending whether thermal or salinity diffusion was to be studied. The molecular thermal diffusivity of pure water at at- mospheric pressure and 20°C is 0.00143 cm*#/sec (4% less than sea water). The molecular diffusivity of an aqueous NaCl solution is 0.0000141 cm*/sec (9% less than sea water) according to Hill [ 1962]. Convenient distilled water and NaCl solutions were used instead of sea water because of these relatively small differences. Fig. 1. Experimental cell. Temperature difference sensor near center. Salinity difference sensors at right. L shaped wire at left produces turbulence on demand. Turbulent Diffuston of Temperature and Salinity When a thermal experiment was to be conducted, the top 4 cm of the cell was filled with water having a temperature several degrees above room temperature. A thin piece of balsa wood was floated on the top of the bottom layer and warm water introduced through a small nozzle directed perpendicularly to the top of the balsa wood. This procedure deflects the downward momentum of the warm water flow to the horizontal and filling was accomplished with a minimum of vertical mixing with the cooler more dense water below. Whena salinity diffusion experiment was to be conducted, a top layer of pure water was introduced the same way. In this case the bottom water contained the salt solution. A repeatable amount of turbulence was introduced by mechani- cally moving a stiff insulated wire back and forth at the interface between the two layers at a controlled rate and for a controlled length of time. The wire is shown in Fig. 1 at the 4 cm level where the interface was located when the cell was filled. (The cell is empty in Fig. 1.) The top of the "L" shaped wire is coupled to a mechanical system outside the picture. It is the bottom part of the "L" that was rotated back and forth laterally at the interface pro- ducing turbulence when desired. Table I gives the specifications for generating the amounts of turbulence that were used. Table I. Specifications of the Turbulence Generator Turbulence No. Mixer Dimensions me ee oe nes 0 - 0 4 1.8 mm diam, 7 cm long 10.8° 2.5 2 : : 5.4 When a thermal experiment was being conducted, the center sensor was used. It consists of a two element copper-constantan thermopile with junctions 2 cm apart. The upper junction was placed 1 cm above and the other i cm below the interface of the upper warm and the lower cooler water. The output of this sensor was 30 microvolts for each degree difference in temperature. It was connected to a small commercial micro-voltmeter and recorder which gave a time record of the temperature difference, 1 cm above and 1 cm below the interface. The record of vertical temperature difference decay, with time, gives data related to the effect of mole- cular diffusion when no turbulence js introduced. The vertical tem- perature difference decay, with controlled amounts of turbulence, was recorded by using a succession of carefully timed turbulent pulses interspaced with short intervals of quiescence. When a salinity diffusion experiment was being conducted the two sensors on the right in Fig. 1 were used. They are identical Schooley probes consisting of two closely spaced platinum wires set in epoxy. The conductivity of the solution at the points where each of the probes were located was read from a meter scale. The upper probe was placed one half cm above the interface and the other one half cm below. Standard salt solutions were used to calibrate the conductivity of the probes in order to measure salinity. This calibration was non- linear and temperature corrections were necessary. Figure 2 illustrates the technique that was devised to facilitate measuring eddy diffusivity by the use of a series of turbulent pulses of the same amplitude and time. The ordinate of this chart is a record of the temperature difference (0 to 1.67°C/cm) measured by the thermopile shown in Fig. 1 vs. time, which progresses from left to right. The record starts at the upper left corner where the temperature difference starts to decrease due to molecular diffusion. At horizontal chart position #5 the mixer shown in Fig. 1 was acti- vated for 15 seconds and then stopped for about 8 minutes. This 8 minute pause allows the pulse generated internal waves to damp out so that the temperature difference due to turbulent eddy diffusion can be measured and separated from the relatively slow molecular diffusion. Again at chart positions #6.4 and #7.8 similar 15 second pulsed of turbulence were introduced. The total time of turbulence was thus only 45 seconds, and three successive temperature differ- ences due to eddy diffusion were recorded at three 15 second intervals of turbulence. The effect of eddy diffusion of NaCl was measured by the same technique using the conductivity sensors. 5 a a a $f Fy 2& SS SSS SS = S22 = Ey SSS oe eee ——- = =>" = = ee eo ee ee ————— SSS ———— ( 4¢ Ses Duke GLI une coca ain NOs ye GUN OLY c: Gioia eM Goo Glad - he Aare Fig. 2. Temperature difference AT vertically vs. time. The decay of AT by molecular diffusion is interrupted three times by turbulent pulses lasting 15 sec each. Turbulent Diffusion of Temperature and Salinity III. THERMAL DIFFUSION The model assumed is shown schematically in Fig. 3. At zero time ty and depth Zo, a semi-infinite region of water at the temperature T>, is assumed to be brought together with a semi- infinite region of water at temperature T,. Further, it is assumed that z, and T, are essentially constant for t,, to, «+. . Eventually as t becomes large, the semi-infinite model breaks down in prac- tice because the effective value of T, decreases. Fig. 3. Schematic presentation of temperature diffusion as a function of water depth z andtime t. Semi-infinite depth is presented vertically with z, a reference. Attime to a sharp temperature discontinuity is assumed, and T, defined as (T, - T,)/2. At t, diffusion has started and To remains constant. For long times t,, t, the semi- infinite model breaks down because the effective value of T, does not remain constant in practice. The heat flow for this model is governed by the one-dimensional diffusion equation 2 OT ot OT bvains> Sse (1) where D is the diffusivity, usually expressed in cm?/sec. Due to vertical symmetry the problem can be formulated using the T, part of Fig. 3, where T, =(T, + T,)/2 is the reference temperature. Taking T, = 0, z,=0, t,=0 the initial and the boundary conditions for z>0O and t=0 is T(z;0) = Ty = (T, - T|)/2- For 2=0 and t>0; T(0,t) = T.. Schooley The analytical solution to this well-defined problem is T(z,T) = Tel 1 - erf(x/2/Dt)] + T, (2) where x/2,/dt erf(x/2VDt) = 2/Vr iy at ae OQ From (1) the gradient of T atthe boundary z, is oT| _ To oz Zo wDt or oT TT, 4 a 0 E |-3(@) (3) Zo and 2 D= (re/m| 4 /(32) (4) Zz 2 Since I5/2 is constant for any one experiment, a plot of 1/t vs. (AT/Az)* for the data points should give a straight line through the origin with slope Dr/T>. Figure 4 is such a plot for the experiment of Fig. 2. A mean square fit gives a slope of 0.66 with a correlation coefficient of 0.99, Since Tp = 2.45°C, in this case the eddy diffusivity is D = 2.45°(0.66)/m = 1.26 cm®*/sec. In practice all plots of the experimental data do not yield perfectly straight lines, particularly for larger values of t (smaller values of 1/t) than are shown in Fig. 4. Calibration and experimental errors are always present. In addition, as is illustrated in Fig. 3, the effective value of T, is not constant for extended lengths of time (t = ty > tq) because the experiments were necessarily conducted in a finite size container. However, Fig. 4 does represent con- sistency of the data with the simple analytical theory under the assumptions that have been made. 316 Turbulent Diffuston of Temperature and Salinity -20 | | IA Natvaz)? 26, c/em)* (sec") 1/t Fig. 4. Sample of experimental data showing (AT /dz)? is a linear function of 1/t. This is in accord with theory that diffusi- vity D = (T2/n)[(1/t)/(AT/Az)*] IV. SALINITY DIFFUSION The substitution of S for T in (4) is all that is necessary. Thus 2 = (ss/n) [4/38 (5) O° where D' is the diffusivity of NaCl dissolved in water in cm®/sec, S, the mass concentration of salt in gm/cm’, t, the diffusion time in seconds, z, depth in cm, and Sb, the initial salinity discontinuity in gm/cm”*, 317 Schooley 0/0! 100 63 40 25 15 10 THERMAL DIFFUSIVITY IN WATER, D (cm/sec) 01 Pg 01 00001 0001 001 O1 A 1 DIFFUSIVITY OF NaC! IN WATER, D! (cm?/sec) Fig. 5. Thermal diffusivity D vs. diffusivity of NaCl in water D' for zero turbulence at lower left, and increasing turbulence toward the upper right. V. DISCUSSION OF RESULTS Figure 5 shows thermal diffusivity D vertically, and salt diffusivity D' horizontally. The point marked (X) at the lower left represents the handbook values for the two molecular diffusivities. The nearby circular point was determined experimentally when the mixer of Fig. 1 was not used. The point nearest the center of Fig. 5 is an average of four experiments using turbulence #1 as listed in Table I. The maximum deviations from the mean are shown. The upper right point is for turbulent condition #2. The mean value was derived from seven experiments. Again maximum deviations from the mean are shown. In Fig. 5 a straight line has been drawn connecting the molecular diffusivity point with the point of maximum eddy diffusivity. The intermediate point is somewhat below this line but there is clearly not enough experimental data to determine the shape of the curve. At the top of Fig. 5 a scale shows how the ratio of D/D' decrease with increasing turbulence. Turbulence in a stratified fluid manifests itself in two ways. 318 Turbulent Diffuston of Temperature and Salinity There is heat energy liberated due to viscous friction. Also, a part of the turbulent energy is dissipated in changing the potential gravi- tational energy of the pycnocline by changing its thickness. The power density associated with a change in potential energy can be shown for water to be approximately 2 P' = aio peel ergs/cm?® sec (6) where g is acceleration of gravity in cm/sec”, (Ap) is either (Ap), or (Ap), symbolizing the change in density due to temperature or salinity differences across the pycnocline in gm/em®, (Ah) the change in thickness of the pycnocline in cm, and (At) =(t,- t,) in sec. For tur- bulence #1, P/=0.0074 ergs/cm?® sec. For turbulence #2, P} = 0.055. The power density due to viscous friction P was estimated by measuring the temperature rise in the water due to turbulence #1 and #2 being maintained for measured lengths of time. For turbulence #1 this was about P, = 0,014(107) ergs/cm* sec. For turbulence #2, P, = 0.082(107). The ratio of P,/Pj = 1.9(10’), and P,/P,=1.5(10’). Thus, it appears that the power density due to viscous friction is on the order of 10’ greater than the power density associated with a change in the pycnocline thickness. However, the ratios for the two conditions of turbulence are different only by about 25%. This is interesting, for if it should turn out that P/P' is relatively constant over a practical range of turbulence, temperature and salinity diffusivities could be estimated directly from the time rate of change of pycnocline thickness (after internal waves are filtered out). Possible application to the ocean is intriguing. For physical and dimensional reasons, let us divide P' by the dynamic viscosity of water n = 0.01 gm/cm sec, and take the square root. Equation (6) then becomes 1/2 (P'/n)"”? = 110(Ap/At)’*(Ah) 1 /sec (7) This equation contains variables that are relatively easy to measure and has the dimension of vorticity. It is plotted in Fig. 6 for the average values of the variables used in the small scale laboratory experiments. 319 Schooley (B)be 190 (44)F cam (even Fig. 6. Tentative extrapolation of experimentally determined diffusion coefficients D and D' vs. variables that are relatively easy to measure at sea. VI. ACKNOWLEDGMENT I am grateful to Prof. Hakon Mosby, Geofysisk Institutt, Bergen, Norway, for his interest and early participation in this exploratory project. I am indebted to Albert Brodzinsky, NRL, for applying his skill in mathematics and physics to the problems of data analysis. REFERENCES Neumann, Gerhard, and W. J. Pierson, Jr., Principles of physical oceanography, Prentice-Hall, Englewood Clits, N.J->» pp. 392-421, 1966. Hill, M. N. (Editor), The sea, Vol. 1, Wiley and Sons, N.Y., Dp. 27; 4962. 320 SELF-CONVECTING FLOWS Marshall P. Tulin Hydronauttes, Incorporated Laurel, Maryland and Josef Shwartz Hydronauttes-Israel, Ltd. and Israel Instttute of Technology ABSTRACT A theory for the motion of two-dimensional turbulent vortex pairs in homogeneous media has been developed based on separate velocity scaling of the internal and external flow fields involved in the motion and taking into account variations in volume, circulation, momen- tum, and energy. Based on the results obtained from this theory (I) a simplified theory (II) is derived to deal with the rising motion of turbulent vortex pairs in stratified media. The theoretical results are compared with systematic experimental observations. In theory (I) the ratio of internal to external velocity scales, \, is introduced as an important variable and the theory is specifically derived for the two limiting cases of weak (W ~ 1) and strong ( >> 1) circulation. The weak circulation theory leads to results similar to those obtained in the past using theory based on com- plete similarity and momentum conservation; i.e., z~t!/3, The strong circulation theory leads to results which depend very much on the way in which vorticity from the shear layer is ingested into the vortex pair. When ingested so as to cause annihilation (cancellation) of the ingested vorticity, the asymptotic trajectory is z~t'/2, Under the same conditions the velocity ratio, Ww, increases toward an asymptotic value, and the virtual momentum coefficient for the motion tends to zero. As a result, the asymptotic motion (assuming vorticity annihilation) corresponds to a motion with complete similarity and with energy conservation. 3214 Tultin and Shwartz A comparison of experimental observations of rise versus time and radius versus height with theory (I) lend strong support to the strong circulation theory and suggest that ingested vorticity may be largely annihilated. Based on these finding for homogeneous flows, a sim- plified theory (II) for stratified media was developed upon the assumptions: (i) the motion is determined by conservation of volume, mass, and energy (neg- lecting vorticity and momentum); (ii) complete simi- larity (dR/dz = 6, a constant). Good agreement was found between the predictions of this theory and the results of systematic experiments, and particularly for the maximum rise of height. NOTATION a Density gradient in surrounding fluid, a = (1/pe)(dpe/dz) A Initial buoyancy parameter (theory), Eq. (44) A, Vorticity mixing coefficient, Eq. (9) b Half distance between cores of vortex-pair B Stratification parameter (theory), Eq. (45) c Constant Cy Energy dissipation coefficient D Energy dissipation parameter, Eq. (46) E Total energy of convected mass G Experimental buoyancy parameter, Eq. (44) j Geometrical parameter, j= 0 for planar geometry, j = 1 for axial symmetry k Virtual potential energy coefficient, Eq. (38) Kin Virtual mass coefficient K Virtual kinetic energy coefficient, Eq. (37) M, Vertical component of total momentum n Parameter defined by Eq. (47) r Radial distance from center of rising mass, r* = Eg? + ne + co R Mean radius of rising mass R Non-dimensional mean radius of rising mass, R= R/R, S) Experimental density stratification parameter, Eq. (53) 322 Self-Convecting Flows t Time me Non-dimensional time, t = Wot/z, t ax Time at which the maximum height of rise is reached u,v,w Velocity components, see Fig. 14 W Vertical velocity of rising mass WwW Non-dimensional vertical velocity of rising mass, W = W/W, we Vertical velocity of ideal vortex-pair Z Height of rising mass center above its virtual origin Zz Non-dimensional height of rising mass center, Z = z/Zo Zmax Maximum height reached by rising mass B Modified entrainment coefficient Y Local vorticity, Eq. (7) I Total circulation about a single vortex €,n,»6 Coordinate system, see Fig. 14 p Local density inside convected mass Pe Density of surrounding fluid p, Average density of convected mass, Eq. (43) Ap Density difference, Ap = pi - Pe Wy Velocity ratio, W,/W, Le Non-dimensional vertical momentum, (M,/p)/W,,R° Subscripts ( ), Initial conditions ( ); 7odaternal { }, External I. INTRODUCTION Ideal Vortex-Pairs. Flow visualization studies carried out by Scorer [1957] , Woodward [1959] and Richards [1965], indicate that the shear layer which is formed between a moving isolated mass of fluid and the stationary surrounding medium tends to roll up and create a flow field which resembles (in two dimensions) the one associated with two line vortices of equal strength but opposite sign, separated by a distance 2b, so-called "vortex-pairs." The possi- bility of vortex-pair motions in an inviscid fluid was considered and analyzed over 100 years ago by Sir W. Thomson [1867]. His analysis Tultn and Shwartz applies only to an idealized vortex-pair in which each vortex has a highly concentrated core which is set into motion only by the influence of the other vortex. Such an ideal vortex-pair moves through the surrounding fluid in a direction perpendicular to the plane joining the vortex cores and with a velocity W determined only by the pair separation, 2b, and the circulation about a single vortex, [, according to the relation ee A unique feature of this idealized vortex-pair motion is the existence of a closed streamline and a finite captured mass, as indicated by the oval in Fig. 2a. Thomson [1867] calculated the semi-axes of the oval-shaped captured mass to be 2.09b and 1.73b so that the cross-sectional area is approximately 3.62 7b“ and the ratio of width to thickness is 1.21. Under certain circumstances it is entirely possible that carefully balanced vortex-pairs, approximating Thomson's ideali- zation, can be formed. The motion around the vortex centers must be affected by viscosity in a real fluid, but as long as the viscous cores do not extend close to the bounding closed streamline, the flow within and without this streamline may be so closely matched that no large shearing motions or accompanying drag are associ- ated with the motion of the captured mass. In fact, nearly ideal vortex-pairs are sometimes found in the wakes of lifting surfaces, Fig. ta, and are known as "contrails," see Scorer [ 1958] and Spreiter and Sachs [1951]. Of course, the concentrated vorticity in the vortex cores tends to diffuse, and does so rapidly when the flow in the core is turbulent. Turbulent Vortex-Pairs. The probable short lifetime of ideal vortex-pairs under turbulent conditions gives special impor- tance to vortex-pairs whose behavior is governed by turbulent entrainment; indeed, it is these kinds of motions which are most commonly observed in nature, as in the case of a mass of fluid forced out rapidly through an aperture, Fig. ib, or in the convection of isolated masses in nature, Fig. ic, or in the bent-over and rising chimney plume. Turbulent vortex-pairs are characterized by the fact that the interior motion does not match the outer flow at the boundary of the captured mass, so that a region of high shear exists there, accom- panied by the production of vorticity and by turbulent entrainment. In other words, these vgrtex-pairs move with a velocity W not equal to the velocity W derived from Thomson's model, Eq. (1). We may, in principle, generalize Thomson's model to con- sider those cases where the velocity of translation, W, has a more 324 Self-Convecting Flows SIVWYSHL ONIVYNDDO ATIVSNLYN GNNOYV GNV NI MO74 (2) sijed-xoj10A jo sojdurexy *] “317 JNLYIdV NV HONOYHL GiIDYOd GINTS ‘NOILOW JAISINdWI (9) -- L33HS X3LYOA ONITIVYL (2) 325 Tultn and Shwartz (S905 94} YIM BupyAouUL ToATESqoO) sajed-xo9jJIOA pozyTetouen °7 *31q (4M >M) (aM< M) (2M=M) UIWd XJLYOA GIdOTSAIAYIAO (2) YlWd XILYOA GIdOTSAIGYIGNN (4) wIVd XALYOA TV3IaGl (°) 3:26 Self-Convecting Flows general relation to the velocity w* which characterizes the internal, rotational motions of the vortex-pair. Two distinct cases suggest themselves, in theory. In one case, W> W , and the convected mass loses volume to the surrounding fluid and continually shrinks in size; we denote this vortex-pair as "underdeveloped," see Fig. 2b. In the other case, W < W", and the "overdeveloped" vortex- pair gains mass through the entrainment of exterior fluid, Fig. 2c. Of these it is the latter motion which is most commonly observed in nature and forms the main subject of this work. Within an overdeveloped motion, the velocity at the boundary inside the vortex-pair, as seen by an observer moving with it, will be larger than the velocity of the surrounding fluid just outside the boundary of the pair. Accompanying the velocity gradients thus created across this boundary, shear stresses are exerted by the vortex-pair on the surrounding fluid, resulting in the entrainment of outer fluid and a general increase in the volume of the convected mass, see Fig. 3. Within the high shear zones at the boundary on either side vorticity of sign opposite to that within the respective SHEAR AND ENTRAINMENT me Tian INGESTED VORTICITY MIXES HERE Fig. 3. Entraining vortex pair (W, > W,) 327 Tulitn and Shwartz interior is created, and is pulled down around the bottom of the rising mass toward the plane of symmetry. To the extent that the ingested vorticity remains on its own side of this plane, the vorticity within the interior will be steadily reduced; of course, ingested vorticity of opposite sign does have a chance to mix and thus to annul itself, depending on the efficiency of mixing. Should effective annihilation of ingested vorticity occur, then the initial total vor- ticity within one side of the pair would be conserved in time. As for the kinetic energy implicit in the motion of the vortex pair, it must be continuously reduced with time due to turbulent dissipation. Self-Similarity in Vortex-Pair Motions. It is a striking characteristic of free turbulent flows in homogeneous media at sufficiently high Reynolds numbers that, under similar circumstances, the flows at different points in space or time can usually be reduced from one to another upon normalization by an appropriate length and velocity scale (self-similarity). This is true, for example, of the flow at different downstream sections of turbulent jets and wakes. It is therefore natural to expect that a turbulent vortex-pair exhibits complete self-similarity during its life time, and this assumption has been made in all theoretical treatments of the subject, starting with Morton, Taylor, and Turner [1956]. Two important consequences of this complete similarity are: (1) cons eryation of the ratio of internal and external velocity scales, w/w , during the motion; (2) linearity of the length scale of the convected mass with the dis- tance traveled from a virtual origin. This latter result, predicting that the traces of the side boundaries of the convected mass form a wedge, is independent of the dynamics of the motion and serves to provide a check on self- similarity. In fact, a number of previous experiments on self- convecting masses claim to confirm this behavior to a reasonable approximation, see, e.g., Scorer [1958] , Woodward [1959] and Richards [1965]. It is, obvious to ask whether a "natural" value of the velocity ratio W/W", or the same thing, of the constant B = dR/dz is ob- served, independent of the original circumstances giving rise to the convected mass. The answer seems to be no. In the present experi- ments, two distinctly different ranges of value of dR/dz differing by a factor 2, have been repeatedly measured; these correspond to two different stroke lengths in the apparatus used to originate the vortex motions, Furthermore, although the present data may be claimed to correspond "in a reasonable approximation" to a constant value of dR/dz, yet quite consistent deviations from linearity exist between the traces of pair radius and distance traveled, see Fig. 4. These deviations are such that dR/dz seems actually to increase throughout the observed motions. 328 RISE HEIGHT, (z - z,)/R, Fig. 4. Self-Convecting Flows K aK Eq. [240] Ge = 1/2, = 0.38 RADIUS, R/R, The variation of vortex-pair radius with height in a homo- geneous medium, experiment 329 Tultn and Shwartz In view of these facts, and for other reasons, it seems desirable to attempt a more general theory of the motion of turbulent vortex pairs, based on the assumption of separate velocity scaling of the internal and external motions; i.e, allowing W/W” to vary continuously. Afterwards, a simplified theory pertaining to motions in stratified media will be developed and the results compared to experimental observations, Il THEORY (HOMOGENEOUS FLOWS) Separate Similarity of Internal and External Flows. We visualize the vortex-pair motion to be divided into internal and ex- ternal flow fields, separated by a thin region of high shear, which also forms the boundary of the captured mass, see Fig. 3. We assume that each flow field is itself self-similar. Internal. — (x Wy, (zy): = W, (t) + w, (5:4) (2) External. Welxsyst) = Welt) - we(% 5%) (3) and similarly for the other velocity components. We let Y= W,/W,, where yw is, in general, not constant in time as it is in the case of complete similarity. We choose Wj, as the circumferental velocity averaged over the inner boundary of one-half of the vortex pair and W, the same except averaged over the outer boundary. (The inner and outer boundaries are separated by a thin shear layer.) Volume Changes. The volume of fluid comprising the vortex pair increases continuously with time due to entrainment into it. Because of the similarity assumed, the rate of entrainment of volume must (in two dimensions) be proportional to a characteristic velocity and a characteristic length. We take for the former, the velocity difference W, - W, a(cR°) dt = 2nR(W; - We) > @(¥) (4) or, 330 Self-Convecting Flows S = Te (y - 1) + aly) = Qh - tay) (5) Note that the dependence of the proportionality constant @' on the velocity ratio ~ has been left unspecified. Circulation Changes. The circulation I about one half of the vortex pair is, on account of the inner similarity, proportional to the length scale and inner velocity scale. On account of the way in which W, was defined, T ewi[ee“t] a) Finally, the change in circulation may be related to the flux of vorticity: aE . Laws of Motion. Limiting Cases; Weak Circulation (W — 1). In this case the inner and outer flows are almost matched and the deviation from the ideal vortex motion is small. The laws of motion in their appropriate form become, Volume. & = (p- 1) * a@"(4) (5a) Vorticity. anes =- Aja'(1) +» (pb - 1° > W, (10a) Momentum. K (1) ° W,R° CONS te (14a) Energy. d(WeR) We sse? K(1) S784 = - (58) WR Colt) (17a) Combining (5a) and (10a) leads to the result, | W, < cep A, #0 (18) Whereas, (iia) requires Ww, * = or z=tl/3 (19) so that the presumed motion can take place only if A, =0 or A, = (W - 1)-'. Combining (17a) and (10a) leads to the requirement that the velocity ratio be constant and have the value 3533 Tultn and Shwartz v=tt[oenytttha | (20) ° @ At the same time, dR/dz is required to be constant and to have the value, = dR _/ We Col) iin PAE Secs age (sp) ; KU) te We leave till later a discussion of comparison with experiment, but we may note now that the prediction (19) is similar to that of the previous theory based on complete similarity. Strong Circulation (J >>1). In this case the interior circu- lation is very strong relative to the ideal value and the deviation from the ideal vortex motion is large. The appropriate laws of motion are, Volume. GR Sepa ut 7am a (W) (5b) Vorticity. AMAR) = aa'(y) + Wi > (10b) Momentum. 2 (K, - Kh) + W, Ro = b+ (M,/p) (11) Energy. d os 3. gel KW 3 (W, R) = - W, R* Cy (1 7b) Combining (10b) and (17b) leads to the requirement that a(ip) be constant and equal to Cr DSK (24) Since similarity requires that W,/W be constant, we may hereafter take a' constant in (5b) and (10b). Combining (5b) and (10b) leads to the result Self-Conveeting Flows 1+A, 1 Ie eo RueAy OF W/W, * (#2) 22) and, substituting (22) and (5b) into (11b) leads to the differential relation, Kp dR\w"'A)) — np dR K, - += = ort (23) ( | (e4 =) a’ dz which has the solution, a! _ R* Kos Ta °Z= x eee + const. (A; # 0) (24) where M u = eee W, Ro 0 and a! _ = , Kos — Kz =4n R + —#R + const. (A, = 0) al mn or 1 aK, Bed tn R + S2(R - 1) (24a) B Ro Substituting dR/dz derived from (23) into (5b) yields a relation between | and R, K Seta (25) WR aK and, finally, it may be shown that, W te 2 K __ (I+ Aj) sug ic Bone Seek (26) K, K, The type of motions which ensue from this theory in the case of strong circulation are seen to depend very much on the value of 335 Tulin and Shwartz A,, the constant appearing in the relation for circulation change, and which depends in part on the way in which vorticity is ingested into the vortex pair. In fact, the asymptotic behavior of the vortex pair changes radically as A; varies around the value unity. This is demonstrated in the table below. Asymptotic Behavior ( >> 1) aK pt AiKa) ak. aK, R(K/a'K,) R(K,/a'K,) For values of A, > 1, the velocity ratio is seen to decline, so that the strong circulation assumption must eventually become invalid. The case A, =1 yields results qualitatively similar to the weak circulation case. In the case where A,< 1, however, the velocity ratio increases to the asymptotic value shown and, most interesting, the added momentum coefficient (K, - K,¥) vanishes asymptotically, so that the motion becomes determined by volume, vorticity, and energy balances alone. Finally, in the case where A, << 1 (effective annihilation of ingested vorticity), then asympto- tically the motion ea, tae determined by volume and energy balances alone, yielding z~t III COMPARISON WITH EXPERIMENT (HOMOGENEOUS MEDIA) In Figs. 5 and 6 are shown data from actual experiments on two-dimensional vortex pair motions in homogeneous fluids. Most of the data shown were obtained in experiments carried out in our own laboratory. Suffice it here to show a schematic of the facility which was used, Fig. 7, and to show Table 1, in which the properties and characteristics of the experimental vortex pairs are listed. 336 Self-Convecting Flows LO Q O TYPICAL O EXPERIMENT fo) C) < 0.5 _o = O > O =| 5 O WR O O lu > O =) < V Zz lu a 2 W~R- 0.1 | 5 10 RADIUS, R/Ry Fig. 5. The vertical velocity vs. radius of a vortex- pair moving in a homogeneous fluid Most significant, we found in our experiments and from the data of Richards [1965] that the measured variation of vertical velocity and pair fadius (two-dimensions) conformed more closely to the law W~R™ or z~t'/2 than to the law derived in the past by others and which is based on complete similarity and momentum conservation; i.e., W~ R? or z~t'”, A test of the simple con- servation of momentum, W~ R™, using a typical trajectory is illustrated in Fig. 5 and, similarly, in Fig. 6 it is shown that the trajectories, so far as they have been observed experimentally, conform more closely to the asymptotic law derived earlier for the case of strong circulation, utilizing a small value of A, (Oi Ay <0 2). In the case of strong circulation, the radius grows ina linear fashion asymptotically, but the theory predicts that during the initial phases of the motion the quantity dR/dz is less than its asymptotic value. A similar behavior was observed in our experiments, see Fig. 4. The matching up of these observed trajectories with the theory offers an opportunity to determine some of the constants of the theory. For this purpose we assume to begin with that A, =0, since the comparison between observed and theoretical trajectories 35 RISE HEIGHT, 2/z, Fig. 6. Tulin and Shwartz THEORY: EXPERIMENTS: R ar UN 301 (z/z9) (2+D) (Wot) RUN 302 RUN 304 NOTE: D MAY BE REPLACED BY A RICHARDS (1965) D = 1 CORRESPONDS TO CONSERVATION OF MOMENTUM SOLUTION D = 0 CORRESPONDS TO CONSERVATION OF ENERGY SOLUTION WITH ZERO DISSIPATION es) 2 3 4 5 6 7 8 9 TIME, Wt/z, The rise of vortex-pairs in a homogeneous medium} comparison of experiment and theory 338 Self-Convecting Flows HOMOGENEOUS OR LINEARLY DENSITY-STRATIFIED FLUID 4g" APERTURE 0.75" WIDE Fig. 7. Experimental facility for studying vortex-pair motion suggests a small value. In this case, eK) EE al = ink + [R - 4] (24a) The trajectory according to Eq. (24a) with K,/p = 1/2 and aK /p = 0.38 is shown in Fig. 4. A fair fit with the experiments has been achieved, and noticeably better than is possible with any linear trajectory. The strong circulation theory thus explains the two important features of vortex pair behavior which cannot be explained by the usual theory of complete similarity. These features are: (i) the tendency for forced vortex Peajectorics (homogeneous flow) to more nearly follow the law z~ t”* rather than z ~ t'/3, and (ii) the ten- dency for the entrainment coefficient (dR/dz) to grow during the initial phase of the motion. These results suggest that in vortex 339 Tultin and Shwartz 24€000 *O- 9000 *O- TH000 *O- 12800°0O 00800 *O 94800 *O €L000 *O 21000 °O GLO00 *O {4000 °O 62000 *O 11000 °O 21000 °O GLO00 *O 29000 ‘O 11200 °O G9Z00 °O T6200 ‘0 €8z00°O 91200 °O GLZ00 ‘O 59200 *O 69200 °O TE900 *O SdTeg-xa do, pajyedaueyn ATaatst[ndwl jo sdajyowerzeg pue satydedotig ofystdaqyoereug T dTdvb 0200 °*O 0200 °O 0200 *O TEZO°O TEZO*O TEZO°O 0200 "0 0200 °0- 0200 *O 0200 *O 0200 *O 0200 °O 0200 *O 0200 *O 0200°O Tg00 *O TR00 “0 T800 *O Tg00 *O T800 *O T800 “0 Tg00 *O T800 *O LL10°O OH) OOOO. OO O49" O1O1O''O O1'O OO O'O1O Vom'en en eu en au en eu aulan au en one Quan auton @mtamray O'S Oy OO! OOO" ©; OOOO O1G!' ©: © ©: O71 OO (°34) T I jos fos 340 Self-Convecting Flows motions in homogeneous flows the internal velocity scale grows steadily relative to the translational (external) velocity, the ratio approaching a value considerably larger than unity, while at the same time, the virtual momentum coefficient associated with the vortex motion approaches the value zero. The data also suggest that vorticity ingested from around one half of the vortex pair is almost annihilated through mixing with vorticity ingested from the opposite side. IV. SIMPLIFIED THEORY (VORTEX PAIR MOTION IN STRATIFIED MEDIA) Convected masses in nature are often rising or falling ina medium of varying density, as in the case of a chimney plume pro- jected upwards into a stable atmosphere. The latter may be characterized by a characteristic time (the Vaissala period), 1/Jag, where a= - (1/pe)(dp,/dz) and P, is the potential density of the atmosphere. The same definition can be used to characterize any density stratified media. The motion of the convecting mass may also be characterized at any instant by the time, R/W. It is almost apparent that when the latter time is long in comparison to the Vaissala period that the effect of stratification will dominate, and conversely. That is, Stratification Effect of Gira ication Ry ag Dominates decreasing Ww increasing Vanishes _——— ete Quite clearly, too, as the motion proceeds in time, the ratio R/W increases continuously, so that stratification must eventually dominate. When this happens, the vertical motion of vortex-pairs may become oscillatory, and is accompanied by the collapse and horizontal spreading of the convected mass, as illustrated in Fig. 8. This behavior is, of course, not consistent with similarity either complete or of the kind assumed in the preceding section. It is sometimes desirable to be able to estimate the tra- jectory of a vortex pair while it is rising in a stratified media and particularly to predict the maximum height of rise and the time required to reach the maximum. For this purpose, we adopt here a simplified theory based essentially on the assumption of strong circulation and annihilation of ingested vorticity. In fact, the parti- cular assumptions adopted would apply if the velocity ratio, w, had already closely approached its limiting value. These assumptions are: (i) the motion is determined by conservation of volume, mass, and energy (neglecting vorticity and momentum); (ii) complete similarity (dR /dz = B, a constant). For further justification of these 341 Tulin and Shwartz uINn}pew popj}iesjs AyJsuep ATreoUuT[ e& uy ATed-xo0jz10A e jo Arojyoofery TeojdAy, *9 °3Tq ONIGVIUdS WesLV1 ° ¥/9°M 4 IWIL 04 oe 02 Ol 0 €0Z YJaWNN NN yz =z). LIPS ISdV1109 GNV ONISIY ONITIS 342 Self-Convecting Flows assumptions we shall depend finally upon a comparison between theoretical predictions and the results of systematic experiments. The energy balance is expressed as follows, 2 So) [tts + vy? + w*) +(p - pe) 8b] d& dn dt = - (Rate of Dissipation of Energy) (27) See Fig. 14 for nomenclature. For a self-similar, self-convecting flow, the dissipation of kinetic energy per unit volume which occurs due to the action of turbulent shear stresses must for dimensional reasons be of the form, Dissipation pw (28) Unit Volume R As a result, the energy balance, Eq. (27), for a self-con- vecting mass in a homogeneous medium of the same density takes the form, 2. 2+} 3 K WRT) 2g W Re (29) 2 dt DR where j = 0 (two-dimensional) j=1 (axisymmetrical) and K is a constant of virtual energy, defined by the identity, 2 2 2 ; iY eo dé dn dt = Swr } (30) and where Cp, is a dissipation coefficient. Making use of (29), together with the relationship R= Bz, it may be shown that the height of rise follows the law, 7 (2+ 1/2* Cp/BK) t (31) 343 Tultn and Shwartz in a medium of uniform density with Ap =0. The dissipation coef- ficient in nature, D= C)/ BK, may be determined by a comparison between theoretical trajectories such as given by (31) and obser- vations of vortex pair rise in homogeneous media. As shown in Fig. 6, such a comparison leads to the conclusion that D is quite small (D <"0,. 2). The trajectory given by (31) may be compared to the law which would apply if momentum were conserved, zeta t (32) which coincides with (31) only if D=1. The variance of observed trajectories from the momentum law (32) is clearly seen in Figs. 5 and 6. Volume conservation in a self-similar flow leads to a linear relation between the nominal radius of the mass and the height of rise from the virtual origin z = 0: R = Bz (33) Conservation of mass takes the form d 4! 2+) 1+] = | (3) aR p; | = 2)2nR’ p, WB (34) where p,, is the density of the surrounding fluid at any given height z and p,; is the average density within the rising mass, defined by 4\) 24) (3) mR |p, (z) - p,(z)] = { (Pp - Pe) d& (dn) dé (35) where the integration is taken over the entire volume of the rising mass. The formulation of conservation of energy is based upon (27) and (28) K 24) 2+} W>_ 24) de [2 PiW R™ +(e, - pleeR'” |= - Cori eR oy where W and R are the observed and measured gross properties of the rising mass while K and k are the coefficients of the virtual kinetic and potential energies, respectively, defined in two dimen- sions, e.g., by 344 Self-Convecting Flows 2_2 Kp, Se -{ £ (we +w*) dé ae (37) and k(p; - p,)zR° =f (Pp - Pe)(z +6) d& dg (38) In most practical instances where one is dealing with a mass of fluid convected through a homogeneous or stratified medium such as the ocean or the atmosphere, the difference between the densities of the convected and surrounding masses is yery small; that is Ap/p, << 1, being usually of the order of 10~, and therefore pj/pe can be taken as 1. This assumption, frequently referred to as the Boussinesq approximation, see Phillips [| 1966], will be used through- out the analysis presented herein. Using the Boussinesq approximation and the identity dz = w dt, the three conservation statements, Eqs. (33), (34) and (36), may be reduced to the following form in the case of a planar motion: R = 62, (39) dpi 4 24p _ 0 dz Zz or (40) d (Ap), 27Apy _ dz one alae gts 2 dw Cp 2, 2kg (Ap 4 _ i> +2(1 +e) W eal re az)z = 0 (41) where a=-(1/p,)(dpp/dz) and Ap = (p; - pe). Finally, explicit general solutions of (40) and (41) may be found. They are: oe = ((40) - Sele * ee 4) (ye ibe: Be enter ee rile eh a Wn (+ 2D) T¥D/2° 12D + A) 2° - bea] 2” (43) 345 Tultn and Shwartz where = (2k) = 298 (Ap A= (=) Grits anaY ite ane ee) (44) = (sR an = Wwe (45) C = —D D= BK (46) n= 2(1.+ D) (47) and ZS Z/fZg (48) The parameter G is a measure of the initial buoyancy of the convected mean relative to the initial momentum. G is taken as positive when a net buoyancy force is acting on the mass, i.e. 4p <0. S is a measure of the added buoyancy which would result from moving the convected mass a vertical distance zo througha stratified medium. Since fag is the frequency with which a finite volume of fluid of given density would oscillate in a stratified medium, often referred to as the Vaisala frequency, the parameter S can also be considered as the square of the ratio of the characteristic time of the convected motion, z )/W,, and the reciprocal of the Vaisala frequency. The maximum height of the rising mass, reached at the point where W = 0, is according to (43), given by the solution of the following: ee ee) eee) +B(rep72- T=) =° ee) It is of interest to consider certain special cases: 1. A mass rising in a homogeneous medium with the same density as itself; i.e., A=0O and B=0. Then -n/2 Gy)? Gs) (50) 346 Self-Convecting Flows or, (= =1+ (1 +3) wat (51) This result suggests how to estimate the dimensionless quantity n (or D) through the analysis of the trajectories of rising masses in this special case. 2. Amass with initial density difference rising ina homogeneous medium; i.e., B=0. Then (ie) = [) = tay]? + Eto)?" ae If A> 0O, then no maximum height is reached, but if A<0O, \/(1#2D) Z max -(- eet (53) Zo | A | The predicted rise of the mass as a function of time and the maximum rise of the mass for a range of values of A (<0) and D, as obtained from Eqs. (52) and (53), are presented in Figs. 9 and 10. These figures demonstrate clearly the effect of the (negative) initial buoyancy and energy dissipation parameters on the time history of an impulsively started rising mass moving througha uniform surrounding fluid of smaller density. 2a. The same case as above but for W,=0 and A>O. First of all, (43) may be rewritten: n “sw, (22) + eit (5°) ( basco on = or in this case es Fctbel GD Sl) | 6 3. A mass with no initial buoyancy rising in a stratified medium, i.e., A=0O. Then, 347 Tultn and Shwartz SOLUTION FOR B = 0 = «(ATS 0 A=-0.4 A= -1.0 RISE HEIGHT, z/zo TIME, Wot/ z, Fig. 9. The rise of an impulsively started heavy mass of fluid in uniform surroundings (B= 0), theory = oa (55) and the maximum rise of the mass, as a function of B, is obtained from Eq. (55) by setting W = 0. Approximate integrated solution for the height of the con- vected mass, z,; as a function of time can be readily obtained from Eq. (55) whenever D <<1 and B<> 1, the asymptotic behavior of the trajectory depends very much on the way in which vorticity from the shear layer is ingested into the vortex pair. In the case where the shear layer from opposite sides is ingested in such a way as to cause annihilation of the ingested vorticity, then the asymptotic trajectory is z~ t¥2, Under the same conditions, the velocity ratio, Ww, increases toward the asymptotic value K /Kp so that the virtual momentum coefficient tends to zero. As a result, the asymptotic motion assuming vorticity annihilation corresponds to a motion with complete similarity and with energy conservation. The ratio of growth of the pair radius with height is shown to increase, approaching a linear relation asymptotically. Systematic experiments have been carried out, and the results for rise versus time and radius versus height are compared with the theory. They lend strong support to the strong circulation theory and further suggest that ingested vorticity is to a large degree annihilated. Based on these findings for the case of homogeneous flows, a simplified theory is derived for the rising motion of vortex pairs in stratified media. The assumptions of the theory are: (i) the motion is determined by conservation of volume, mass, and energy (neg- lecting vorticity and momentum); (ii) complete similarity (dR/dz = By a constant). General laws of motion in stratified media have been derived and solutions given; particularly interesting cases are dis- cussed in detail. Motions in stratified media were shown to depend on four non-dimemsional parameters. Two of these depend upon the initial conditions of the motion and the stratification of the media. The other two are inherent in the details of the motion and had to be determined from experiments; one of these, the dissipation param- eter D = Cp/BK was found to be 0.2 while the other, the ratio of virtual kinetic and potential energy coefficients K/k was found to be 4. On the basis of these numbers it may be concluded that the dissi- pation rate is small and that the contribution of internal motions to the overall kinetic energy is large. The experiments confirmed the environmental scaling param- 356 Self-Convecting Flows eters, which were used to collapse data taken under differing con- ditions. Good agreement was found between predicted and observed trajectories. Particularly good agreement was found for the maxi- mum height of rise. The time required to reach maximum height was found to be inversely proportional to the Vaisala frequency, jag, and was given approximately by tmexY 2& = 1.8, in good agreement with the theory. In general the experiments confirmed the utility of the simplified theory for predictions of the motion of vortex pairs in stratified media. This theory has been utilized elsewhere for the prediction of the behavior of chimney plumes rising into a stable atmosphere, with very good agreement between the theory and full scale observations, Tulin and Schwartz [1970], and also with excellent correlation with experiments to the penetration of a density discontinuity by a turbulent vortex-pair, Birkhead, Shwartz, and Tulin [ 19691 « ACKNOWLEDGMENT This work was supported by the Naval Air Systems Command, the Air Programs Branch and the Fluid Dynamics Branch of the Office of Naval Research under Contract No. N00014-70-C-0345, which support is gratefully acknowledged. REFERENCES Birkhead, J. L., Shwartz, J., and Tulin, M. P., "Penetration of a Density Discontinuity by a Turbulent Vortex-Pair," HYDRO- NAUTICS, Incorporated Technical Report 231-21, December 1969. Lamb, H., Hydrodynamics, Dover Publications, N.Y., 1945. Morton, 5. R., laylor, G. I. and Turner, J. S:, “Turbulent Gravitational Convection from Maintained and Instantaneous Sources," Proc. of the Royal Society, A, Vol. 234, p. 1, 1956. Phillips, O. M., The Dynamics of the Upper Ocean, Cambridge University Press, Cambridge 1966. Richards, J. M., "Puff Motion in Unstratified Surroundings ," Je of Fluid Mechs, Vole 21, No. 1,.p. 97, 1965. Scorer, R. S., Natural Aerodynamics, Pergamon Press, 1958. Scorer, R. S., "Experiments on Convection of Isolated Masses of Buoyant Fluid," J. of Fluld Méch., ‘Vol, 2, No. 6,.p- 583, August 1957. Spreiter, J. R., and Sacks, A. H., "The Rolling Up of the Trailing Vortex Sheet and Its Effect on the Downwash Behind Wings," J. of Aero Sciences, Vol..18,:No..15 p. 21, January 1951. S5it Tulin and Shwartz Thomson, Sir. W., "On Vortex Atoms," Philosophical Magazine, Series 4, Vol. 34, No. 227, p. 15, July 1867. Tulin, M. P., and Shwartz, J., "Hydrodynamic Aspects of Waste Discharge," AIAA Paper No. 70-755, June 1970. Woodward, B., "The Motion in and Around Isolated Thermals," Quart. J. of the Royal Meteorological Society, Vol. 85, pe 144, 1959. C, n, & - CARTESIAN COORDINATES u,v, w - CORRESPONDING VELOCITIES CENTER OF RISING MASS AT TIME t ACTUAL ORIGIN OF MOTION VIRTUAL ORIGIN OF MOTION Fig. 14. Nomenclature 358 HYDRODYNAMICS IN THE OCEAN ENVIRONMENT Thursday, August 27, 1970 Morning Session Chairman: G. B. Whitham California Institute of Technology Page Radar Back-Scatter from the Sea Surface . 361 K. Hasselmann,M, Schieler, Universitat Hamburg Interaction Between Gravity Waves and Finite Turbulent Flow Fields 389 D. Savitsky, Stevens Institute of Technology Characteristics of Ship Boundary Layers 449 L. Landweber, University of Iowa Study of the Response of a Vibrating Plate Immersed in a Fluid 477 L. Maestrello, T. L. J. Linden, The Boeing Company 359 ioe at 1G - 1 . - ~ 5 -—- We y= re Ny : ; : a a a : yo ; ‘ oo pte! Was Preah rm _ ft . ‘ Phy oe ee Pies. AN, VArtes Mice {™ FPA \ 15a erp ators} Magi a Ma Vik. Ys, -id. SE pease dda (467, o T dls ~ on aye ws ** 3, F, Pity rgde SAirla P. socks tt Cie e ge A Paper fa. 10 > l itiee ‘Vite ; ; Fated al ‘ ia winrt m2 oy ee Aretg its ; ' 7 oy y. (ehx; ,« Gites Coyal Meltngyolagizal &,,-:& 1 aes y ‘> i ee rh SRY TMAMMOM AS AAASO BHT MI SOIMAVYOORGY - £1) ae OF@) VS revavA . veda’? “hAve®@ Gealasas ‘ : m ae a a ral . : r uw 48 20 Sesertled D> nolondseTl tq) om ildesti ‘ino lem ers. ; , it. | | ) me : oge% a - ' ar s(t % PGE : cia us eae tolis2-asae & Bang oth : ret) cof) .eeteivod. . ortismlawasH 5 ‘ 7 == eA oooh aby Retr 27) Gib ee nok ter £ i ' ; ablol% volt taskue VACMSeT le ejxdijen! enevosd vpletives. sh ee | ' y BS ~ Yrebsood 6142 to ec Halek : $3 Ryd eseyl att “adeebaal , ve bee SritetV¥ «6 6 San0cash. a bloft 6 ai beaut Mehal! sl oT .oflerees yusqatoD gniecd T Dae Tinene oo horde fig. Fie.” Wwjrehciaiues RADAR BACK-SCATTER FROM THE SEA SURFACE K. Hasselmann”™ and M. Schieler Instttut fuer Geophystk Untverstty of Hamburg ABSTRACT Doppler spectra of electromagnetic backscatter from the sea surface are interpreted in terms of general- ized Bragg models. The observed broadening of the spectra about the Bragg line is attributed to higher- order nonlinear processes. At conventional radar frequencies, good agreement with the measurements is achieved by an extension of the wave- facet inter- action model considered by Wright, Bass et al. and other workers. The correlation of wave slopes and orbital vélocities in the joint probability distribution of carrier-wave facets leads to significant differences between the Doppler spectra for vertical, horizontal and cross polarization. In the HF band, the Doppler broadening is interpreted in terms of quadratic wave- wave interactions. For the usual case that the electro- magnetic wave lengths are small compared with the principal wave lengths of the sea, the theoretical Doppler spectrum consists of the lowest-order Bragg line and superimposed images of the complete ocean wave fre- quency spectrum folded on either side of the Bragg line. Both wave-facet and wave-wave interaction models give promise of extracting significant information on the "state of the sea" from electromagnetic Doppler return at wave lengths short compared with the dominant wave lengths of the sea, * : Presently at Woods Oceanographic Institution. 361 Hasselmann and Schietler I. INTRODUCTION The development of numerical wave prediction methods in the past years [1, 2, 17] has increased the need for wave data ona synoptic scale, both as a reference for testing and improving the models and as real-time input for the computations. Synoptic wave data would also be of value for numerical weather forecasting by providing indirect information on surface winds in otherwise poorly covered areas of the oceans. The growing interest in electromag- netic backscatter from the sea surface stems largely from the potentiality of the method for furnishing sea-state data of this kind. Radar scatterometers in satellites could scan most of the world oceans in a few hours. Alternatively, large areas of the ocean can be sampled using HF stations on land. Following the pioneering work of Crombie [6] and others, Ward [22] has recently detected the backscattered return of ionospheric HF modes from relatively small, 100 km square patches of the sea surface at distances up to 3000 km. Unfortunately, both techniques suffer from wave length limi- tations. Cloud absorption and finite atenna size define an effective transmission window for satellite scatterometers in the conventional radar wave length range between a few fractions of a cm and about 50 cms. Backscatter measurements over long horizontal ranges are similarly restricted to ionospheric modes in the decameter band. In both cases, the electromagnetic wave lengths are consider- ably shorter than the principal components of the surface-wave spectrum, which normally lie in the range between 50 and 500 m. The bad wave length matching creates difficulties in relating the backscattered signals obtained by these methods to significant sea- state parameters. Scattering experiments in both the centimeter-decimeter and decameter bands have now clearly established the basic validity of the first-order (Bragg) wave-wave interaction theory. According to this model, the backscattered radiation arises from interactions with two gravity- wave components whose wavenumbers k9 are determined by the Bragg (resonance interaction) condition for constructive inter- ference, k?= +2k', where k' represents the horizontal wavenumber component of the incident radiation. For non-normal incidence, the wave lengths of the scattering and incident components are then of the same order, which implies that the scattering surface waves normally lie in the high-wavenumber, equilibrium range of the surface-wave spectrum. It appears therefore from first-order theory that backscatter measurements may yield a useful independent determination of Phillips' constant [15, 22], but do not contain sig- nificant information on the more interesting low-wavenumber part of the wave spectrum which contains most of the wave energy. Fortunately, the scattering measurements, while supporting the Bragg theory, also indicate that it should be regarded only as a first approximation. The Doppler spectra, in particular, exhibit 362 Radar Back-Seatter from the Sea Surface several features not predicted by the Bragg model. Generally, there is a marked dependence of the anomalies on sea state, sug- gesting that useful correlations between backscatter signatures and significant sea-state parameters may be discovered by extending the scattering theory to higher order. Two generalisations have been proposed: the wave- facet interaction model[ 23, 4, cf. also 3, 9, 10, 20, 24], in which the Bragg-scattering waves are superposed on longer carrier waves, and the higher-order, wave-wave interaction model originally inves- tigated by Rice [18]. The models have been applied hitherto mainly to the cross sections, which show only weak sea-state signatures. In the present paper, we consider their extension to the more strongly sea-state dependent Doppler spectra. In the cm-dm bands, good agreement with the observed Doppler spectra is obtained with the wave-facet interaction model, The Doppler spectra are found to be quasi-Gaussian and can be characterized to good approximation by the mean frequency and the frequency bandwidth. Both parameters depend on moments of the wave spectrum which are governed by the high-energy, low-wave- number range of the spectrum. They can therefore be used to obtain independent estimates of, say, the mean waveheight and period. The model allows only for electromagnetic interactions. Basically, the hydrodynamical modulation of short gravity waves by long carrier waves is of considerable interest, not only for the description of the surface wave field, but also for its energy balance. The interactions generally lead to an energy loss of the long waves at a rate which can be estimated from,the observed upwind-downwind asymmetry of the cross sections [11]!. However, because of the strong influence of white capping, the interactions cannot yet be described in sufficient detail to be included realistically in computa- tions of the Doppler spectra. Their effect on the Doppler bandwidth is probably negligible, but the mean Doppler frequency may be more strongly modified. The wave-facet interaction model is valid for electromagnetic wave lengths shorter than about 1 m. Thus it applies in the cm-dm radar band, but not in the dkm band. In the latter case, however, the Bragg theory can be generalised by straightforward extension of the wave-wave interaction analysis to higher order. The relevant TLonguet- Higgins [13] has shown that the momentum loss of short waves breaking on the crests of longer waves results in an energy transfer to the long waves. However, the gain in long-wave kinetic energy due to this process can be shown to be slightly less than the loss of potential energy arising from the simultaneous mass transfer between short and long waves. The net result of both processes is a weak attenuation of the long waves [ 11]. 363 Hasselmann and Schteler perturbation parameter of the expansion is given by the ratio of the amplitude of the interacting surface wave to the wave length of the incident radiation. In the first order analysis, the perturbation parameter is proportional to the slope of the scattering Bragg wave, which is small for all electromagnetic wave lengths. At second and higher order, however, the electromagnetic waves interact with longer surface waves of higher amplitude. In this case, the perturbation parameter remains small only if the electro- magnetic wave length is large compared with the amplitude of the entire wave field. This condition is satisfied by dkm waves, but not by cm-dm waves. The requirements for the wave-facet and wave-wave inter- action models are found to be mutually exclusive, so that the two expansions cannot be matched in a common region of validity. It is a fortunate coincidence that the theoretical wave length gap cor- responds to the gap between the two presently available techniques for measuring electromagnetic backscatter on a synoptic scale. The second order wave-wave interaction analysis yields a continuous Doppler spectrum superimposed on the first-order Bragg line. The continuum reduces to a particularly simple and useful form when the Bragg wave length is short compared with the wave lengths of the dominant surface waves -- the usual situation for ionospheric modes. In this case, the continuum is identical with the two-sided image of the surface-wave frequency spectrum, centered on the Bragg line as virtual frequency origin. The energy scale of the wave spectrum can be inferred from the observed energy of the Bragg line, independent of transmission or other cali- bration factors. Doppler side-band structures observed by Ward [ 22] and others are not inconsistent with this interpretation. However, most Doppler spectra published hitherto have been analysed from rather short records, so that the continuum is generally not well defined statistically. Longer records are needed to decide whether the one- dimensional frequency spectrum of the surface-wave field can indeed be detected in the Doppler spectrum of backscattered ionospheric modes above the inherent ionospheric noise. Il THE LOWEST-ORDER SCATTERING MODELS For electromagnetic waves short compared with the dominant waves of the sea, one might attempt to describe the scattered field by a specular reflexion model, in which the sea surface is repre- sented as an ensemble of locally plane, infinitesimal facets, each of which reflects the incident radiation according to the laws of geometric optics. The cross section o for the backscattered radiation (the backscattered energy per unit solid angle per unit surface area of the ocean) is then proportional to the number density of facet normals 364 Radar Back-Seatter from the Sea Surface pointing towards the source. As the distribution of normals ina random surface-wave field is approximately Gaussian, the depen- dence of log o on depression angle 9 is given by a parabola, with maximum at normal incidence (90° depression angle) and half-width typically of the order 10° (Fig. 1). CROSS SECTION Ow = Onn (o/ 90° 180° x x DOPPLER SPECTRUM ie) Wa =W, fo) W, Wy Fig. 1. Cross sections and Doppler spectra according to the specu- lar reflection and first-order Bragg scattering models (qualitative) Hasselmann and Sehteler The frequencies of the backscattered waves are shifted relative to the frequency of the incident radiation by the Doppler seid OF Nally 2k' * u induced by the facet motion, where K = (k', Vy is the wavenumber of the incident radiation and u the local orbital velocity of the waves. For an approximately “linear wave field, u is a Gaussian variable, and the Doppler spectrum also has a Gaussian shape. As the backscattered waves are reflected at normal incidence, it follows by symmetry that the cross sections and Doppler spectra are independent of polarisation. Vertical and horizontal polarisation are denoted in Fig. 1 by V and H, respectively, the first index referring to the incident field, the second to the backscattered field. The cross-polarised return VH and HV vanishes. Although applied successfully by Cox and Munk [5] to the analysis of sun glitter from the sea surface, the specular reflexion model fails to describe the observed electromagnetic backscatter at cm-dm and dkm wave lengths. It appears that for these wave lengths surface irregularities of length scale comparable with the radiation wave length cannot be neglected. Accordingly, recent models have been based on the Bragg scattering theory, in which these irregularities are regarded as the dominant scatterers. It is assumed in the Bragg model that the slopes of the scattering surface waves are small and that their wave lengths are comparable with those of the radiation field. The backscattered field can then be expanded in powers of the surface displacement. The first-order field is linear in the surface displacement and can therefore be constructed by superposition from the field scattered by a single gravity-wave component ¢ = Z exp {ik?- x - iugt}. This corresponds to the classical problem of refraction by a periodic lattice. The scattered field consists of two waves s = + whose horizontal wavenumbers and frequencies are given by the Bragg (resonant interaction) conditions k' + sk9 = k$ ic ie. ee (1) W; + sw = W. (The vertical wavenumber component 8 determining the scattering angle follows from the dispersion relation | w.| =c lie | » where < is the velocity of light). Hae scettcring (k* = k') occurs for the gravity- wave com- ponents k*= + 2k’. The (rae cattering cross section is accordingly of the form CA To T2B ae Tap (2) 366 Radar Back-Secatter from the Sea Surface where 08 = Tuer g(- 2sk') (2,8 = Vor He or = +) and F, (k) is the surface-wave spectrum, ngrmalised such that the mean square surface displacement ((?) = F,(k) dk. The cornered parentheses denote mean values. (The negative sign of the wave- number in the definition of ogg has been introduced so that c4g corresponds to a spectral line with positive Doppler shift, cf. Eq. (3).) Tag is a scattering coefficient obtained by expanding the electro- magnetic boundary conditions at the free surface [18]), T = 20} ee: (1- e)(e[1 +cos* 6] - cos” 8) VV (e sin O tye - cos’ 6) 2 2 T,,,= | “2b sin? e Bie c (sin 8 ie - cos? @)2 Tyy = Tuy = 9 where e€ is the dielectric constant of sea water. The normalized Doppler spectrum Xg@wg), defined by Jxag (ea) dwy= ogg where wq= ws - wj, is given according to (1) by two lines at the gravity-wave frequencies + wg, Xapled = Xapled) + Xagloa) with (3) X gla) = TaB(wWg - S Wg) Normally, one of the Bragg lines due to scattering from the surface wave component propagating in the downwind direction is very much stronger than the other line associated with the wave pro- pagating in the opposite, upwind direction. The general properties of the Bragg cross sections and Doppler spectra are indicated qualitatively in the right-hand panels of Fig. 1. In contrast to the specular reflexion model, there is a pronounced dependence on polarisation and appreciable backscatter at small and intermediate depression angles. The cross-polarised return again vanishes. 367 Hasselmann and Schteler L-BAND JULY 29,1965 A SEA STATE "A" © SEA STATE "B" x SEA STATE "C" — THEORETICAL (15 KNOTS) Ove (dB) og 10° 20° 30°40° 60° 90° DEPRESSION ANGLE Fig. 2. Theoretical and observed Bragg backscatter cross sections for vertically polarised 24 cm (L band) waves (from Wright [ 23]) Figure 2 shows a comparison by Wright [23]! of experimental and theoretical Bragg cross sections for vertically polarised cm-dm waves. The surface waves were represented by a Phillips’ spectrum F,(k) = (a/2)k-*5(), with a uniform half-plane angular spreading function, S(J) = 7m! for 0S || < 1/2, S(W) =0 for - 1/2 < |y|Snm. The constant @ was chosen to fit the observed cross sections, but is not inconsistent with other estimates from direct measurements of gravity-wave spectra (cf. also [15]). Shown in Fig. 3 are theoreti- cal and experimental cross section ratios oy, /o y Lhe agreement here is also very good, except for the shortest wave length (3.4 cm, 8910 MHz), where scattering by spray may be beginning to mask the T The theoretical cross sections shown in Figs. 2 and 3 were, in fact, computed for the wave~facet interaction model considered in the next section. However, the deviations from the first-order Bragg model are negligible. 368 Radar Baeck-Seatter from the Sea Surface X 1228 MHz,CROSSWIND 15 KNOTS © 8910 MHz,CROSSWIND 15 KNOTS — THEORETICAL DEPENDENCE FOR RMS TILT ANGLE AT 7.5° 20 /O;°, (dB) ° vv 5° 10° 20° 30°40° 60° 90° DEPRESSION ANGLE Fig. 3. Theoretical and observed ratios of Bragg backscatter cross sections for vertical and horizontal polarisation at wave lengths 24 cm (1228 MHz) and 3.4 cm (8910 MHz) (from Wright [ 24]) weak Bragg return for horizontal polarisation. Not predicted by first-order Bragg theory is the observed cross-polarised return, which is generally only slightly smaller than or comparable with the backscatter for horizontal polarisation; this can be explained by the wave-facet interaction model [ 23]. Although the observed cross sections oy, and o,, are in good agreement with theory, the Doppler spectra for these polar- isations point to limitations of the first-order model. In the cm- dm bands, the Bragg lines are found to be broadened into Gaussian shaped distributions with bandwidths of the same order as the Bragg frequency (cf. Fig. 4., from Valenzuela and Laing [20]). Earlier measurements by Hicks et al. [13] indicate that the mean frequencies of the distributions -- which were not measured by Valenzuela and Laing -- may also be considerably higher, by factors of the order 2to 4, than the theoretical Bragg frequency. 369 Hasselmann and Schteler “(loz | sujey] pue efenzueteA worl) “(ud F°¢) KX pue (UID 2 °9) Dd ‘(WD FZ) I ‘(uw OL) q spueq uy erzD0ds 1933e9SyDeq TaTddoq paestazejtod Aj{Teo}310A fo sotdurexy ZH OO} AININO3Z8S 7H OS AININO3ZYS ($1381930) S30NLINGWY 3AILV134 ($1381930) 3QNLIIdWY 3AILV138 S3AVM Y3L3W 9°0-¢0 (J19NV NOISS38d30 002) RDS (JT9NV NOISS3Yd30 02) AN "py “Sa 370 Radar Back-Seatter from the Sea Surface In the decameter bands, the observed broadening and shiit of the Bragg lines are much weaker. Instead, the Doppler spectra show pronounced side band structures (cf. Fig. 11, from Ward [22], and similar spectra in Crombie [6] and elsewhere). The basic difference in structure of the Doppler spectra observed in the cm-dm and dkm bands lends support to theoretical considerations calling for alternative expansion procedures in the two wave length ranges. Ill. THE WAVE-FACET INTERACTION MODEL In order to treat the scattering waves as small perturbations of a plane surface, it is assumed in the Bragg theory that the wave amplitudes are small compared with the wave length of the incident radiation. Ina strict sense, the expansion is valid if this condition is satisfied not only for the Bragg waves, but for the entire surface displacement. Thus the theory is not rigorously applicable to short electromagnetic waves of a few cm wave length, although the long surface waves of high amplitude which violate the expansion condition do not enter in the final scattering expressions. Various workers [e.g. 3, 4, 10, 20, 21, 23] have suggested that this formal short- coming may be remedied by dividing the surface-wave spectrum into two parts, a high-wavenumber scattering region, and the energy- containing region at low wavenumbers which defines the "sea." The "sea" is then treated as a random carrier wave which modulates the scattering by the superimposed Bragg waves. If the Bragg wave length 1/k' is short compared with a typical wave length 21/k° of the sea, the carrier wave may be represented locally as a plane facet, and the first-order scattering theory applied in the reference frame of the moving facet. The model involves additional conditions besides the two-scale assumption that it is possible to define a facet diameter D inter- mediate between the carrier and scattering wave length scales, (ki)! << D << (k°)| (4) The finite facet size implies an indeterminacy Ak = O(1/D) of the scattering wavenumber, which corresponds to an angular spread Ao = O{(K'Dsin @)"'} of the backscattered beam. The wave-facet interaction model is meaningful only if A® is small compared with the change in effective depression angle introduced by the facet slope 8{/dx = O(k°t) , where € is the carrier-wave amplitude. a requires k°4DK' sin 9 >> 1, or, since Dk* <<1, on account of 4\, K'f sin 0 = kyl >>> 1 (5) cog Hasselmann and Schieler Similarly, a wavenumber broadening Ak corresponds to a frequency broadening of the Bragg line of order Aw= (dwg/dk’) Ak = (wa/2k9) - Ak (ignoring capillary effects). The model assumes that this is small compared with the Doppler shift w,=- 2K'+ u induced by the facet velocity u. For u= O(weS), where weis a typical carrier-wave frequency, this requires DK! wk! /w, = DK! (k'k°)!"¢ >> 1 Substituting Dk*° << 1, this is equivalent to Ki c(i! [oy >>> 4 (6) Since k'/k° >> 1, the frequency condition (6) is less critical than the corresponding condition (5) for the angular resolution. The inequality (5) is normally fairly well satisfied at conventional radar wave lengths for surface-wave heights of order 1 m and higher (except for small depression angles, where the model breaks down, in any case because of shadowing effects). For electromagnetic wave lengths longer than about 1 m the inequality (5) is normally no longer valid, even though the two-scale inequality (4) may still apply. The total backscattered energy is obtained in the wave-facet interaction model by summing over the contributions from all scattering facets. Introducing a facet probability distribution p()) with respect to the five basic facet parameters = (A, o Nor Ngo Ngee) » where (AX, 2X53) = (u, » U5, U5) = facet velocity ee local long-wave orbital velocity), and (h4,X,) = (86 /8x, ,0C /8x,) = (n,,n,) = facet slope, the Doppler spectrum is given by Xap = { [ T3g5(wg - SW, - w¢)] p(d) dd (7) where oy , W, represent, respectively, the Bragg cross section and gravity-wave frequency in the facet reference frame. To the modulated Doppler spectrum (7) of the first-order Bragg field should be added the modulated spectrum of the zero'th order field reflected from a plane facet, as described by the specular reflexion model. However, this is important only near vertical incidence and will be ignored in the following. Experimentally, the probability distribution p(\) is found to be approximately Gaussian, in accordance with the theoretical distri- 372 Radar Back-Seatter from the Sea Surface bution for a random, linear gravity-wave field, p(M) = (20)? |c[ exp {- 5 Con 2,) (8) The covariance matrix Cjj can be evaluated from the surface-wave spectrum and the linear wave solutions, gk? /k g kk,/k | —— O —— g k,k,/k gkj/k ! ie ee ee oie ees ee (9) gk - WK, = wk, nn? 2 0 1 7 wk, k Ke 1 where Tykg/k= J F, (k)(kjkg/k) dk, etc. We note that the facet Doppler shift ws, is correlated not only with the facet velocity, but through the correlation (u,n,) also with the facet slope, (wai) = - 2k; (un) (10) For small wave slopes, the factor in square parentheses in Eq. (7) can be expanded in powers of nj. The integration can then be carried out explicitly for each term of the expansion yielding a solution of the form Xqpl@s) = Xqhee) + Xgglma) with 2 Xgl) = (alo > L- (wg stig) /2¢ 04) } 14 at Hace caer tal) (2m( ut) v2 where q®, qg®, ... are polynomials in (ws = 60.) of order 1,92; .<< | 9g in the facet Slope, g = damdfacsed | (2a), Aa + Meee (Sa) ca f Hasselmann and Schteler An 25.8 ~ ~s {5 (SH) says = S290 (2g) + ($e S88) The subscript 0 refers to values at n= 0. To lowest order, the Doppler spectra for vertical and hori- zontal polarisation are identical Gaussian distributions with mean frequency (w) = Swg and variance ( (w - day) =e) ="2 J Ky kgh Up Yq) + (ky) *(uy)t (14) (£,m= 1,2) The distribution represents an ensemble of Bragg lines of equal energies displaced by their appropriate facet Doppler frequencies Wee The higher-order corrections qi Bs qs ye ole represent dis- tortions of the Gaussian distribution due to the variations in energy of the Bragg lines associated with variations in the carrier-wave slope. These affect the shape of the Doppler spectrum through the correlation between facet slopes and facet Doppler frequency, Eq. (10). The degree of distortion depends on the depression angle and polarisation. In the cross-polarised case, the zero 'th and first- order terms disappear, since (T, wo = (8/8n; 5 Wo = 0, so that the Vv Doppler spectrum is non-Gaus sian already to lowest order. Computations of the Doppler spectrum were made for a Pierson-Moskowitz [16] spectrum using a half-plane cosine-to-the- fourth spreading factor, -4 4 ze ki exp {- B(w,/w)} for k, >0 Fy (k) = with @ = 0.0081, B= 0.74 and wo= g/U, where U is the wind velocity, aeeumed parallel to the x, axis. The same spreading factor was taken for both scattering and carrier waves. 374 Radar Back-Seatter from the Sea Surface For a Pierson-Moskowitz spectrum, ( wos) Satgeren (wn;) ~ U. The slope moments (njn;) diverge logarithmically at high wavenumbers. To obtain finite (njn;) , the "carrier-wave" spectrum was cut off at an upper wavenumber k /10. The exact position of the cut-off is not critical for the evaluation of (njnj) ’ and the slope moments themselves enter only rather weakly in the second-order term q§$ of the expansion (11). However, the existence of a divergence as such points to a conceptual difficulty of the wave-facet interaction model. It appears that for an asymptotic ic? spectrum the carrier-wave region of the spectrum cannot be rigorously separated from the Bragg-scattering region. Figure 5 shows the computed half-power bandwidths for the lowest-order Gaussian spectrum as a function of wave height. The values compare well with measurements by Valenzuela and Laing [20] < Deviations from the Gaussian form due to the higher-order corrections ay and q& are represented in Figs. 6 - 9 in terms of the mean frequency (w)/wg and the frequency bandwidth { (w - ( w) ie) / (ae) , normalised by their appropriate values for the zero'th order Gaussian spectrum. The strongest correction is found for the mean frequency, particularly for horizontal polarisation. The dependence on depres- silon-angle and polarisation, shown in Fig. 6 for U = 20 m/s, is found to be very similar at all wind speeds. The absolute values of the frequency shifts increase approximately linearly with wind speed, Fig. 7. Qualitatively, the polarisation and wind-speed dependence of the mean Doppler frequency are in agreement with measurements made by Hicks et al. [13] at low depression angles of about 5°, How- ever, the theory is not strictly applicable in this case on account of shadowing effects. The bandwidth corrections (Figs. 8 and 9) remain rather small for depression angles less than 45° and limited azimuth angles Ww relative to the wind. Larger deviations in the cross-wind direc- tions depend strongly on the spreading factors, which are rather un- certain for these angles. The experimental dependence of the Doppler bandwidth on radar frequency and polarisation [ 20] tends to be some- what larger and have a different trend than the corrections shown in Figs. 8 and 9. Valenzuela and Laing [ 20] suggest that these effects may be due partly to spray. To a fair approximation, the observed bandwidths can be represented for small and intermediate angles and @ by the zero'th order Gaussian bandwidth. Both the bandwidth and mean frequency vary significantly with wave height and can therefore be used for estimates of sea state. For the one-parametrical family of spectra considered in the present example, the two estimates are not independent. However, in general the mean square bandwidth ( (w- (w) *) ~ (we) (Eq. 14) and the mean 375 Fig. 5. Hasselmann and Schietler P-BAND a L-BAND 4 C-BAND o X- BAND OPEN POINTS (VERTICAL) CLOSED POINTS (HORIZONTAL) hp.b [m/s] 1 2 3 4 Comparison of theoretical half-power bandwidths (h.p.b.) for zero'thorder Gaussian spectrum with measurements by Valenzuela and Lang 20 . Doppler frequencies are in units of equivalent velocities Ug=w,/2k'. Theoretically, a Pierson-Moskowitz spectrum with cos*w spreading function yields 2 h.p.b. [m/s] = 1.06} £926 (4cos*p +1) +sin?ol'* (H,,[ m] Nes where the significant wave height Hy * Vena ee 0.209 u*/g. The computations were made for w = 09, 6 = 20°, 376 Radar Baeck-Seatter from the Sea Surface u=20m/s HH i (4) Wg 2 2 yz0° (4) a1) 1 0 1 15 30 45e /60 15 30 45 60 2 2 = (2) w=20° 1 1 15 30 45 60 15 30 45 _60 (4) (1) 15 30 45 60 1 30 45 60 5 2 2 (4) y =60° | SS Wy 0) (4) 15 1§ 30 45 60 30 45 60 2 1 (1) (4) (4) 15 30 45 60 Q -—e Fig. 6. Ratios of mean Doppler frequency (w) to Bragg frequency wg for the wave-facet interaction model at windspeed U = 20 m/s. The indices 1, 2, 3, 4 referto P, L, C and X bands, respectively. The computations include terms up to order q§ in the expansion (11). To this approximation, the cross polarised case yields (w) = wg S1f Hasselmann and Schieler Fig. 7. Dependence of (w)/wg on wind speed U for 06 = 30°, y= 0°, An approximately linear variation is found for all depression angles 9 and azimuth angles \W. 378 Radar Back-Seatter from the Sea Surface u=4m/s VV HH VH 2 2 2 =0? 4 ¥ 0 ‘ (4) , (4) tno RE 1) (1) 16 30 45 60 75 °®»+»15 30 45 60 75 45 "30. 46 160° 95 2 2 2 y=20° a) (4) a 1 1 1 (4) (1) Be A A Sle. 15 30 45 60 75 15 30 45 60 75 5 30 45 60 75 ae 42 2 3 V (3) 1 (4) : ‘ (1) (1) (1) y=40 4) 15 30 45 60 75 15 30 45 60 75 15 30 45 60 75 15 30 45 60 75 15 30 45 60 75 16 30 45 60 75 15 30 45 60 75 Fig. 8. Frequency variance of the Doppler spectrym computed to order q5, normalised by the variance (or) of the zero'th order Gaussian distribution (U = 4 m/s). 379 Hasselmann and Schtieler u=20 m/s 1S 30 45 60 75 45 60 75 , = = eS 15 30 45 60 75 15 30 45 60 75 30 45 60 A a F (1) Vv i) a. W =40° 45 60 75 45 60 75 15 30 45 60 75 (4) w=60° (4) 5 30 45 60 75 15 30 45 60 75 5 30 45 60 75 3 3 3 (1) 2 (1) (1) w=80° (4) (4) (4) 15 30 45 60 75 15 30 45 60 75 15 30 45 60 75 e@ — Fig. 9. Same as Fig. 8 with wind speed U = 20 m/s 380 Radar Back-Seatter from the Sea Surface product (w,n;) (Eq. 10), which is responsible for most of the mean- frequency variation, depend on differently weighted moments of the gravity-wave spectrum. The two Doppler parameters can therefore be used to obtain independent estimates of two sea-state parameters --for example, the mean wave height and mean wave period. IV. HIGHER-ORDER WAVE-WAVE INTERACTIONS For HF waves longer than about 10 m, the Bragg model can be generalised by straightforward extension of the wave-wave inter- action expansion to higher order. In this case, the perturbation parameter kjf is normally a small quantity even when ¢ is defined as the surface displacement of the complete wave field, and there is no need to consider the long waves of high amplitude separately. In fact, the wave-facet interaction model is not applicable for HF waves on account of the angular resolution condition (5). The in- equality kx{ << 1 and condition (5) are mutually exclusive, repre- senting a wave length gap between the wave-facet and higher-order wave-wave interaction models extending from a few fractions ofa meter to about 10 meters. At second order, the wave-wave interaction analysis yields scattered waves through interactions with pairs of gravity-wave somponents a,b satisfying the next-order Bragg conditions i a b= 4° i + og? + ok? = w togw, tow, =, (0,50, = +) (15) The second-order Doppler spectrum x? (w4) is obtained by summing over all pairs of surface waves yielding a backscattered component with the appropriate horizontal wavenumber k‘®= - k' and frequency w, = 0; + Wg, oy x?) (wg) = » iY TF, (k°)Fy (ik?) 8(coq- oy 4-004) dk (16) FD where k?=-o (2k! + oak") (Eq. 15). pi?) is a scattering function determfned by thé’ second-order coupling coefficients occurring in the expansion of the boundary conditions about the undisturbed plane surface (cf. Ref. (18)). (The polarisation indices are irrelevant for the following discussion and are ignored.) Te tenhiicant sea-state signatures are found only for the Doppler spectra and not the cross sections. On integrating Eq. (11) with respect to frequency the dependence on the moments ( we) and (wn; ) disappears, leaving only a weak sea-state dependence through the slope moments (njnj). 381 Hasselmann and Schteler The scattering function T'?) includes both electromagnetic and hydrodynamic interactions at the free surface. For wave lengths in the HF range and longer, the hydrodynamic interactions can probably be described to fair approximation by classical hydrody- namical theory, independent of the effects of wave breaking. Equation (16) represents the random-field expression of nonlinear effects such as nonsinusoidal wave forms!, nonlinear phase velocities, etc., that have been variously suggested as explanation of the observed side bands of HF Doppler spectra. In the limit of an incident wave short compared with the principal waves of the sea, the dominant interactions at finite de- pression angles are electromagnetic. The largest contributions to the integral in (16) arise in this case from interactions in which one of the gravity-wave components, say k°, lies near to the peak of the spectrum. Since k®° <> Wee 383 Hasselmann and Schieler A useful feature of the relation (17) is that it defines the surface-wave spectrum in absolute energy units independent of electromagnetic calibration factors, which are difficult to establish for long-range ionospheric mode propagation. Using Eqs. (3) and (2) to eliminate the surface-wave spectrum at the Bragg wavenumber, Eq. (17) becomes ll) mah (on Eg(wg - stg) + Eg(swg - wy) = ol =e (s = +) (18) where ¢'')8= Ix 129. (taa) ae is the energy of the first-order Bragg line. The ratio T'') /2T(2) can be determined from theory, and x8 and ef"s may be measured in arbitrary age units. In the relevant limit k°<x - wt) } riding on a long carrier wave fq = Ag exp {i(k® x- wet)} (whf{th is now, howeyer, assumed to satisfy the wave-wave interaction con- Aion Aoks << 1, rather than the wave-facet interaction condition (5)). For small slopes Agk* << 1, the principal effect of the carrier wave is presumably to alter the phase of the scattered field by raising and lowering the local mean reference surface of the short scattering waves!. Thus if the first-order backscattered wave in the absence of the carrier wave is of the form gl!) (1) 8 = Ci’ ASA, exp (i(k! + ak? )x - i(w, + opw,)t + ikyx,} I where Aj is the amplitude of the incident field, c' is im first- order coupling coefficient, and it + gpk? & - k', k§~ - kg the modulated scattered wave in the presence cot the Patiee wave will be given approximately by acikste ole (1 + 2ikle, Jo") =o + gi (19) 9 = Thus go = Cl) asaya; exp {- ik'x - i(wj; t+ wg)t - ikyxs} (20) with C® = 2ikic . Expressed in terms of a continuous energy spectrum, this is readily found to correspond to a scattering function ratio T A more detailed investigation indicates that slope effects can be ignored if k° << k,=k' sin 0. 384 Radar Back-Secatter from the Sea Surface = (ky) (21) For small depression angles (ks << ie), the effect of the carrier-wave slope becomes comparable with the phase shift induced by the vertical displacement, and the relations (20), (21) should be modified to include additional terms dependent on k®. However, this requires a more detailed investigation of the electromagnetic and hydrodynamic interactions. Examples of Doppler spectra obtained by Ward [ 22] from the sea echo of 21.840 MHz (14 m) waves at ranges near 3000 km are shown in Fig. 11. The analysis was based on short records of one minute duration, so that the continuum is poorly resolved statistically and individual spectra vary strongly. However, there is some indi- cation of two side-band structures appearing on either side ofa central Bragg peak. Theoretically, the Bragg line should lie at 0.48 Hz, which agrees well with the central peak of the first spectrum shown, but is somewhat to the left of the main peaks in the other cases. The displacement of the side lobes relative to the Bragg peak is of the order 0.1 Hz expected for typical ocean-wave frequencies. The ratio of the side-band energy e2) = if x (2) (wy) dwy to the energy e) of the Bragg line is given according to Eqs. (18) and (21) by ive, »2 (2) /e") = 4(kiy(¢ ) Ward estimates a depression angle of 12°, which yields e) 7" = 0.036 ((6[m] )*) The observed ratios of order unity correspond to root mean square wave heights of about 5 m, which appear rather high, but not im- possible. More plausible estimates of the wave height may have resulted from a more accurate determination of the scattering function ratio T By aia at small depression angles. Contamination of the observed spectra by ionospheric Doppler shifts may be an alternative explanation of the high ratios €(2) /é") - A spurious inter- action between the Bragg line and the low frequency ionospheric Doppler spectrum could also have been introduced in the present experiment by the data analysis, since the Doppler spectra appear to have been computed -- as is often done -- from the time series of Note added in proof: A detailed analysis has recently been carried out by D. E, Barrick "Dependence of Second-Order Doppler Side Bands in HF Sea Echo on Sea State," to appear in 1971 G-AP Internat. Symp. Digest. 385 Hasselmann and Schieler 4 TITS 1 jlsvecaty UU rere easbittH ith: 386 ges of 2700 km (18 ms) and (14 m) sea echo at ran Doppler spectra of 21.84 MHz 3000 km (20 ms), from Ward [ 22]. Figo 4. Radar Back-Secatter from the Sea Surface the signal phase (or phase cosine), which is nonlinearly related to the complex signal amplitude. More detailed investigations using longer time series are needed to decide. whether the ocean wave spectrum can be extracted from the Doppler spectrum of long range HF sea echo in the presence of unavoidable fonospheric noise. ACKNOWLEDGMENT This work was supported in part by the Office of Naval Research under Contract No. ONR N00014-69-C-0057. REFERENCES 14. Barnett, T. P., "Generation, dissipation and prediction of wind waves," J. Geophys. Res., 73, 513-534, 1968. 2. Barnett, T. P., Holland, C. H. Jr. and Yager, P., “General technique for wind-wave prediction with application to the S. China Sea," Westinghouse Res. Lab. Rep., June, 1969. 3. Barrick, D. E. and Peake, W. H., "A review of scattering from surfaces with different roughness scales," Radio Sci., 3, 865-868, 1968. 4. Bass, F. G., Fuks, I. M., Kalmykov, A. I., Ostrovsky, I. E., and Rosenberg, A. D., "Very high frequency radiowave scattering by a disturbed sea surface," IEEE Trans., AP-16, 554-568, 1968. 5. Cox, C. M. and Munk, W. H., "Measurement of the roughness of the sea surface from photographs of the sun's glitter," J. Opt. Soc. Am., 44, 838-850, 1954. 6. Crombie, D. P., "Doppler spectrum of sea echo at 13.56 mc/s," Nature, 175, 681-682, 1955. ai-Daley; J. ‘C., Ransone, J. Ts Jr, Burkett, J. A. and Duncan, J. R., "Sea-clutter measurements on four frequencies," Nav. Res. Lab. Rep. 6806, 1968. 8. Daley, J./C., Ransone; J. T.,Jre, Burkett, J. A. and Duncan, J. R., “Upwind-downwind-crosswind sea-clutter measure- ments," Nav. Res. Lab. Rep. 6881, 1969. 9. Ewing, G. C., ed., Oceanography from Space, Woods Hole Oceanogr. Inst., Ref. No. pe A0, 1965. 10. Guinard, N. W. and Daley, J. C., "An experimental study ofa sea clutter model," Proc. IEEE, 58, 543-550, 1970, 387 Lis 172 13. 14, 15. 16, 17. 18. 19. 20. 21. 22. 23. Hasselmann and Sehteler Hasselmann, K., "On the mass and momentum transfer between short gravity waves and larger-scale motions," J. Fluid Mech. , 50, 189, 1971. Hasselmann, K., "Determination of ocean wave spectra from Doppler radio return from the sea surface," Nature, 229, 16-17, 1971. Hicks, B. L., Knable, N., Kavaly, J. J., Newell, Grs-, Ruina, J. P. and Sherwin, C. W., "The spectrum of X-band radiation backscattered from the sea surface," J. Geophys. Res. , 65, 825-837, 1969. Longuet-Higgins, M. S., "A nonlinear mechanism for the genera- tion of sea waves," Proc. Roy. Soc. A. 311, 371-389, 1969. Munk, W. H. and Nierenberg, W. A., "High frequency radar sea return and the Phillips saturation constant," Nature, 224, 1285, 1969. Pierson, W. J. and Moskowitz, L., "A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii," J. Geophys. Res., 69, 5181-5190, 1964. ears Pierson, W. J., Tick, L. J. and Baer, L., "Computer based procedure for preparing global wave forecasts and w nd field analysis capable of using wave data obtained by a space craft," 6th Naval Hydrodynamic Symposium, Washington, Office of Naval Res., Washington, D. C., 1966. Rice, S. O., "Reflection of electromagnetic waves from slightly rough surfaces," Comm. Pure Appl. Math., 4, 351-378, 1951. Semenov, B., "An approximate calculation of scattering on the perturbed sea surface," IVUZ Radiofizika (USSR), 9, 876- 887, 1966. Valenzuela, G. R. and Laing, M. B., "Study of Doppler spectra of radar sea echo," J. Geophys. Res., 75, 551-563, 1970. Valenzuela, G. R., Laing, M. B. and Daley, J. C., "Ocean spectra for the high frequency waves from airborne radar measurements," 1970 (subm. to J. Mar. Res.). Ward, J. F., "Power spectra from ocean movements measured remotely by ionospheric radar backscatter," Nature, 223, 1325-1330, 1969. Wright, J. W., "A new model for sea clutter," IEEE Trans. AP-16, 217-2235 1968. 388 INTERACTION BETWEEN GRAVITY WAVES AND FINITE TURBULENT FLOW FIELDS Daniel Savitsky Stevens Instttute of Technology Hoboken, New Jersey ABSTRACT A laboratory study of the interaction of deep water gravity waves progressing into a turbulent flow field produced by a finite width grid towed in a wide tank showed wave height attenuation of nearly 90% in the grid wake and wave height amplifications of nearly 75% in the still water outside the wake. The transverse gradient of longitudinal flow in the wake was predom- inantly responsible for the large wave deformations and precluded an evaluation of direct turbulence effects. A simple, analytical solution using wave refraction, diffraction and superposition concepts is developed which qualitatively reproduces the measured results. I. INTRODUCTION As gravity waves progress from their source of origin, they encounter a wariety of ocean environments which may interfere with their ordered motion and, consequently, alter the amplitude and direction of the wave system. Although an extensive literature exists on the mechanism of wave generation and their subsequent propaga- tion through still water or a uniform flow, only recently has some attention been given to waves moving through a non-uniform flow -- and these have been restricted to relatively weak velocity gradients normal to the wave direction. In a realistic ocean environment, gravity waves may encounter regions of turbulent flow, particularly in the upper layers. These oceanic turbulent flow fields can be developed by various geophysical mechanisms. For example, the action of unsteady wind shear stresses exerted against the surface of the sea; the breaking of wave crests 389 Savittsky resulting in "splash turbulence" penetrating into the upper layers of the water; turbulent fields set up in intense currents; turbulence developed by high velocity, high Reynolds number flows in a tidal channel; ship wakes; etc. In each case, it is expected that wave attenuation will result from the interaction between the turbulent flow fields and wave motion. Such attenuation is of importance in develop- ing relatively "quiet" local areas in the sea for launching or recovery of small craft or submarines, or in tracing the progress of, say, one storm passing through the intensive turbulence of another storm. Phillips [1959] presents a theoretical study of the properties of waves on the free surface of a liquid in turbulent motion where the intensity of the turbulence is sufficiently small to preclude wave generation in itself and where the mean velocity of the flow is zero. There are two types of possible interaction, each of which results in the attenuation of the incident wave. One is an "eddy viscosity inter- action" in which wave energy is transferred from the wave motion through a stretching of the vortex filaments in the turbulence which tends to increase w*, the mean square vorticity associated with the turbulence itself, This straining process is of second order in wave height-length ratio and, hence, should be important for steep waves and when the turbulence scale is much less than that of the waves. The second type of interaction is a scattering phenomenon where random velocity fluctuations in the turbulence field will result in the convective distortion of the wave front, and produce a broad spectrum of scattered waves. This scattering effect is of first order in wave height-length ratio and, hence, predominates for waves of small slope. Phillips shows that, under typical conditions in the open sea, the attenuation from scattering will be greater than that from direct viscous dissipation for wave lengths greater than about 10 ft. An experimental study was undertaken at the Davidson Labora- tory, Stevens Institute of Technology, to investigate the interaction between mechanically generated progressive gravity waves and a controlled turbulence field developed by towing suitable grids ina towing tank. Since field measurements by Stewart and Grant [ 1962] supported the applicability of the Kolmogoroff hypothesis (that the statistical structure of turbulence has a universal form) to turbulence near the sea surface in the presence of waves, it was believed that grid-generated turbulence (known to satisfy the Kolmogoroff hypothesis) would indeed be representative of ocean turbulence on a model scale. Two experimental studies were undertaken. The first used a grid which spanned the width of a 12 ft wide towing tank and was towed in the direction of wave celerity at speeds less than the group velocity of the regular wave lengths generated by a plunger type wavemaker. In these studies, the test waves overtook and passed through the turbulence wake and grid. This so-called one-dimensional grid study was made in an attempt to develop a turbulent wake with uniform mean flow across any transverse section aft of the grid. Unfortunately, as will be subsequently discussed, a uniform flow field was not de- veloped near the outer edges of the grid wake and this seriously 390 Gravity Waves and Fintte Turbulent Flow Ftelds influenced the test results. The other series of experimental studies involved towing a 3-ft wide grid in a 75-ft wide towing tank. The in- tent of these tests was to allow any scattered wake system to be defracted outside the turbulence patch. However, the finite width grid also produced a pronounced longitudinal mean flow velocity gradient in transverse sections through the wake. Thus, in these latter tests, the generated waves were simultaneously subjected to three modification effects: (1) dissipation due to eddy viscosity; (2) scattering due to turbulent convective distortion of the wave front and (3) deformation of the wave due to mean flow velocity gradients. Measurements were made of the wave deformation in the wakes of both the one- and two-dimensional grids. An analysis of these results indicated that the velocity gradients in the wakes had a dominating effect on the wave deformation and thus, unfortunately, precluded a reliable evaluation of the possible dissipative or scatter- ing action of the turbulence field upon the incident wave. The studies are, nevertheless, of importance since they provide unique results, obtained under controlled laboratory conditions, describing the pro- nounced distortion of a deep water wave when encountering sharp current gradients, either naturally existing or artificially produced. It is shown that the wave distortion can be such as to provide locally areas of reduced wave motion which can be beneficial in launching or retrieving small craft or submersibles from a mother ship at sea. The experimental results are described in some detail and an elementary analytical model is developed which, using the combined mechanics of wave refraction, defraction and superposition, at least qualitatively reproduces the features of the test results and, perhaps more important, describes a possible physical mechanism respon- sible for the observed large wave deformations. These studies were supported by the Fluid Dynamics Branch of the Office of Naval Research, Department of the Navy, under Contract NR-062-254, Nonr263(36). They formed the basis for a dissertation submitted to the Graduate Division of the School of Engineering and Science in partial fulfillment of the requirements for the degree of Ph.D. at New York University. Il EXPERIMENTAL PROCEDURES Turbulence-generating grids have been used with great suc- cess in advancing the knowledge of turbulence in air flows, but have been used only occasionally in hydrodynamics -- particularly in towing tanks where a grid must be towed in quiet water to generate a turbulence field. Taylor [1935] has shown that disturbances generated in the wake of a grid transform rapidly into a quasi- isotropic turbulent field whether the grid is towed in quiet air or an airstream passes through the grid. 391 Savttsky In the present task, vertical turbulence grids of finite draft and two mesh sizes were towed at various constant speeds in Tank No. 2 and 3 of the Davidson Laboratory in a direction normal to the plane of the grid. Regular waves, generated by a plunger type wave- maker in quiet water, traveled in the same direction as the grid tow (with initial crest lines parallel to the grid) progressed through the turbulent wake and grid into quiet water beyond the grid. Waves of various constant length and height were generated such that the group velocity of each regular wave was greater than the grid veloc- ity. The wave lengths and water depth were such that deep water gravity waves were generated. Wave amplitudes were measured by resistance type wave wires which penetrated the fluid surface. Several of the wave wires were towed ahead and behind the grid (at the grid speed) while others were stationary and located both in and outside of the grid wake. The outputs of these wave probes were simultaneously recorded on a "Viscorder" oscillograph tape. The details of test procedure, grid characteristics, and test conditions for the one-dimensional and two-dimensional turbulence grid studies are described separately below. Common to both studies was the observation that, for the grid sizes and grid veloc- ities considered, the combination of physical grid and turbulent wake in smooth water did not produce a measurable wave system of its own. In fact, soon after passage of the grid and wake relative to a fixed point in the test tank, the water surface appeared unusually still. Further, the grid solidity was small enough that, when sta- tionary, it did not noticeably affect the wave forms which passed through the stationary grid. Neither was there a measurable wave reflection from, the grid, One-Dimensional Grid Studies The one-dimensional grid studies were conducted in Tank No. 3 of the Davidson Laboratory. This tank is 300 ft long, 12 ft wide and has a water depth of 6 ft. A plunger type wavemaker is located at one end of the tank and a slotted beach of 15° slope is located at the opposite end to absorb the wave energy with minimum reflection. Grid Characteristics: A turbulence grid 11.5 ft wide spanned the tank width, penetrated the water surface to a depth of 1.6 ft, was attached to a standard carriage and towed in a direction away from the wavemaker. Figure 1 shows the test setup. Two mesh sizes were tested; one had amesh M = 0.36 ft and was made of crossed square wooden slats 0.80 inches wide; the other hada mesh M = 0.71 ft and was made of crossed square wooden slats 1.60 inches wide. Thus, in both cases, the grid solidity was constant and equal to S= 0.40. The grid was towed at speeds of V = 1.0 and V= 1.7 ft/sec. The hydrodynamic drag and Reynolds No. of the grid (Re, = VM/w) for these conditions are: 392 Gravity Waves and Finite Turbulent Flow Fields PlzH Teuojsueurtq euo dn-yeg 3807, JF “81a 393 Savttsky Mesh Grid Velocity Drag Reynolds No. 0.36 £t 1 ft/sec 13.2 lbs 28,000 0.26 vee ff 37.9 49,000 ay | 4 ie yey 56,800 Oa71 ay 6 37.9 96,500 Measurements were initially made of the mean value of the longitudinal velocity (in the grid direction) of the grid wake at the centerline of the grid and at a depth of 0.80 feet below the water surface. At a distance of 10.0 ft aft of the grid, the wake velocity was 0.40 V and decreased slowly with distance aft of the grid -- at a distance 20.0 ft aft of the grid, the mean velocity of the wake was 0.36 V. In these initial tests, a straight line of confetti was sprinkled across the 12 ft width of the tank parallel to the plane of the grid. Visual observations of this reference line after grid passage showed that the confetti moved essentially in one straight line parallel to the grid, thus indicating a lack of noticeable velocity gradients -- at least on the free surface. An analysis of the wave distortion data in this wake yielded anomalous results (these will be discussed ina subsequent section) that could not be explained by the assumption of a uniform longitudinal mean flow through transverse sections in the grid wake. Hence, a detailed survey was then made of the mean flow at distances of 10 ft and 20 ft aft of the grid. These results are shown in Fig. 2 which presents a plot of longitudinal mean flow (V,) versus transverse distance from the grid centerline at a probe depth of 10 inches below the water surface. The wake velocities (V,) are normalized on the basis of grid speed (V). It is clearly seen that the mean flow in the wake is essentially constant for a distance of approximately 5 ft from the grid centerline but then rapidly decreases between this point and the tank wall. The significance of this local velocity gradient will be subsequently discussed. The wind tunnel results of Dryden [ 1937], who examined the turbulence aft of a rectangular grid having a mesh size M= 0.41 ft at a nearly similar Reynolds number, show that the turbulent veloc- ity fluctuations u' as a function of distance, X, aft of the grid are: V x = + aay) a Thus, for a distance 10 ft aft of the 0.36' mesh grid x/M = 27.7 and u' = V/37.7 or approximately 3% of the mean flow. At a distance of 20 mesh lengths aft of the grid, wind tunnel experiments have shown the establishment of quasi-isotropic turbulence. 394 Gravity Waves and Fintte Turbulent Flow Ftelds GRID WIDTH =1II.50; DRAFT=20'; MESH SIZE=5.4" (GRID TOWED IN [2 FT WIDE TANK) VELOCITY PROBE AT 10° DRAFT Vw=WAKE VELOCITY ; V=GRID VELOCITY 0.5 TANK 0.4 WALL | Vw 23 | V oe 10 FT AFT OF GRID | 0.1 | 20 FT AFT OF GRID ie) | 2 3 4 5 6 € y=LATERAL DISTANCE FROM GRID @ , FEET Fig. 2 Longitudinal Velocity Distribution in Grid Wake Wave Height Probes: Wave heights were measured by re- sistance type wave wires penetrating through the water surface. The position of the wave wires relative to the grid are shown in Fig. 1. It is seen that wave wires moving with the grid were located 12 ft ahead of and along the grid centerline; 13.25 ft and 16.50 ft aft of and along the grid centerline; and one located 13.25 ft aft and 4 ft transverse to the grid centerline. The wave wires 13.25 ft and 16.50 ft aft of the grid were used to obtain a measure of the ap- parent wave length in the turbulence field while the pair of wires 4 ft apart in the transverse plane 13.25 ft aft of the grid were used to measure any deformation of the wave crest line as it progressed through the turbulence. A stationary wave wire was located 60 ft forward of the wavemaker and was used to examine the regularity of the amplitude and period of the generated incident wave. 395 Savttsky A range of wave heights and lengths used in these tests were as follows: Wave Wave Wave Group Wave Length Period Celerity Velocity Height hj dt T, sec V2 ft/sec Vg, ft/sec His ff 20,0 0.025 32:20 1.60 0.05 2.0 0.625 3.20 1.60 0.10 5.0 0.763 a 93 fost 0.10 4.0 0.885 4.52 2.26 0.05 4.0 0.885 4; 52 2.26 0.10 6.0 1.080 5.55 2.18 0305 6.0 1.080 55 2.18 0.10 8.0 1.250 6. 40 52 20 0.04 8.0 1.250 6.40 eA) 0.09 Test Procedure: Several experimental procedures were used in these studies. In one group of tests, the grid was held stationary 70 ft forward of the wavemaker until several waves had passed through the grid. The grid was then towed and wave measure- ments were made with the moving wave wires. For certain runs, after approximately 50 - 60 ft of grid tow, the aft moving wave wire (16.50 ft aft) was released from the tow and remained stationary in the tank. Thus, wave height measurements were taken both at a fixed position relative to the moving grid and at a fixed position in the tank (variable position relative to the grid). The other test pro- cedure was to first tow the grid for a distance of approximately 50 ft which developed a turbulent wake and then start the waves which ran through the wake and overtook the moving grid. This technique avoided the possibility of a secondary wave formation as the incident wave ran through the moving grid. It was established that the results obtained with both test procedures were essentially similar. Two-Dimensional Grid Studies The two-dimensional grid studies were conducted in Tank No. 2 of the Davidson Laboratory. This tank is 75 ft square and has a water depth of 4.5 ft. A plunger mechanical type wavemaker spans one side of the tank and a sloping beach is installed on the opposite end to absorb the generated wave energy. Grid Characteristics: Two turbulence grids, one 3 ft wide and another 5.5 ft wide, were separately towed in a direction away from the wavemaker. The grid centerline was 17 ft from one edge of the tank. Figure 3 shows the test setup. As in the one-dimen- sional tests, two mesh sizes -- M = 0.36 ft and M= 0.71 ft -- were 396 Gravity Waves and Finite Turbulent Flow Fields pli3z [euotsueultp-omy dn-jos say, ¢ “BIg 3GIM ,€ giyS OSNIAON 3Y¥IM 3AVM AYVNOILVIS Of JYIM 3AVM ONIAONW OC 2 iS M3IA NV1d 2°ON ANVIL Hy stay ale, BH sas? oS HS3M 39uv7 .2'2 HS3W 11VWS NOISN3WIO GI¥9S a at T3A31 Y3SLVM | i f 191 397 ° Savittsky tested. The grids were constructed of crossed square wooden slats 0.80 inches wide. The solidity, towing speeds and Reynolds number of the grids were the same as for the one-dimensional tests pre- viously described. The hydrodynamic drag of the 0.36 ft mesh grid was 3.2 1bs and 9.25 lbs at tow speeds of 1.0 and 1.7 ft/sec re- spectively at a grid draft of 1.67 ft. Ata grid draft of 0.84 ft, the hydrodynamic drags for this grid were 1.5 1bs and 4. 33 lbs at 1.0 and 1.7 ft/sec. It is to be noticed that the grid drag increased as the square of the speed for all cases. Measurements were made of the mean values of the longitu- dinal velocities at depths of 0.80 and 0.40 ft in the grid wake across several transverse sections aft of the 3 ft wide, 0.36 ft mesh grid with a 1.67 and 0.80 ft draft. A plot of the ratio of wake velocity to grid velocity is given in Fig. 4. These velocity ratios were the same for both grid drafts and towing speeds. It is seen that there is a slow attenuation of velocity with distance aft of grid. Further, there is also a slow lateral spreading of the wake area. It is interesting to note that the wake velovities are all in the direction of grid tow. Surveys of the velocity field up to 5 ft from the grid centerline did not indicate a reverse flow. Visual observations did indicate a reverse flow along the bottom of the test tank. An empirical formulation was established to represent the wake velocity Vy as a function of distance, x, aft of the grid anda distance, y, measured from the grid centerline normal to the x- direction. The wake equation is given by: ay . = = [0.45 - 0.00745x] le Pukka oe aaa | where x and y are in units of feet. The above formulat was developed for use in the analysis of wave deformation for reguier waves running into a velocity gradi- ent. This analysis is presented in a subsequent section of this report. Wave Height Probes: The location of the wave amplitude measuring probes are shown in Fig. 3. It is seen that four probes were towed with the grid and 7 inches off its centerline; one 3 ft forward and three others 1 ft, 3 ft and 6 ft aft of the grid. In addi- tion, seven stationary probes were located in a transverse line normal to the direction of ,rid tow and at distance 38 ft and then 20 ft ahead of the wavemaker. It will be noted that one of the stationary wave wires was directly on the grid centerline. This installation was accomplished by mounting the probe on the floor of the tank and providing a slot through the grid which passed over 398 Gravity Waves and Finite Turbulent Flow Fields GRID WIDTH=36', DRAFT=20"; MESH SIZE =2.7" (GRID TOWED IN 75 FT WIDE TANK) VELOCITY PROBE AT IO" DRAFT y 8 V de .062 ) ; = -|0.45-0.00745 «| E MOORES GRID Vw =WAKE VELOCITY, FT/SEC V =GRID VELOCITY, FT/SEC ig X= 0.58 FT Vw X=20.5 FT DISTANCE AFT OF GRID, FEET ° EDGE OF MAXIMUM VELOCITY 052.0% ERO VELOCITY) EDGE OF GRID WAKE ( < Vw X=37.5 FT 20 BS 0....2,, 4, §6.);8,, 10 LATERAL DISTANCE FROM GRID ¢ , FEET Fig. 4 Longitudinal velocity distribution in grid wake 399 Savttsky the wave probe. Thus, the transverse probes covered an area from the grid centerline to a distance of nearly 9 ft outboard of the edge of the grid. Test Procedure: The range of wave heights and lengths were essentially Similar to those used in the one-dimensional tests. Also, the test procedures previously described were followed. The initial grid location was always 10 ft ahead of the wavemaker. After ap- proximately 60 ft of tow, the grid was stopped and the wavemaker continued in operation until the wave amplitudes recorded in the line of transverse wave probes were again equal to the incident wave amplitude. RESULTS OF EXPERIMENTAL INVESTIGATIONS Selected test results are first described to illustrate the general behavior of waves ina turbulent flow field. An elementary analysis of the results is developed in the subsequent section. One-Dimensional Grid Studies As previously discussed, the original intention of the one- dimensional grid study was to provide a turbulent wake with constant longitudinal mean flow in any transverse section through the wake. Regular waves would be passed through the wake and measurements made of the dissipative effects of grid-controlled turbulence on wave amplitude attenuation. It was expected that the deep water gravity waves would pass through the turbulence field with the crest lines always remaining parallel to the grid and that the wave amplitude would be essentially constant along a given crest line and decrease as the wave progressed further into the turbulent area. Under these circumstances, the amplitude attenuation would be due both to viscous dissipation and to "wave stretching" as it moved into a longi- tudinal current from an originally quiet area. This idealized situation did not develop but, rather, it was found that the wave crest lines were severely deformed; the wave amplitude was not constant across a given crest line; and, further, there were pronounced oscillations in the wave amplitude time history at each wave probe (whether moving or stationary) in the wake. In all cases, the control wave probe, which was fixed in quiet water aft of the turbulent wake, indicated a wave of constant amplitude and period continuously passing into the wake area. General Behavior: An example of typical wave amplitude oscillations recorded by both the moving and stationary wave wires along the grid centerline is given in Fig. 5. The test conditions represented are for a wave length of 4.0 ft and a wave height of 1.2 inches. The grid velocity was 1.7 ft/sec. The phase speed of the wave is 5.4 ft/sec while the average wake velocity is approxi- 400 Gravity Waves and Fintte Turbulent Flow Fields 598/45 Fy =A. we a Pek Plig [euOoTsusUITp 9UO - SoTIO4STY oUIT} oqord aAeM TedIdAT, gc ‘31q SGNO93S—- 3WIL oOo! O06 os Oz 09 OS Ob O£ 02 Ol ie) Q3ddO1S aiys 1YVLS dius JAVM TONLNOD AYWNO I LVS (b) JVM ae SVM SY3LN3 al ,49°1=L4vyd aid ,2'2 =HS3W G1y9 401 Savttsky mately 0.60 ft/sec. Trace No. 1 is for the moving wave probe located 13.25 ft. aft of the grid; trace No. 2 is for a moving wave wire located 66.5 ft aft of the grid; trace No. 3 represents the moving wave wire 12 ft ahead of the grid, and trace No. 4 is for the stationary control wave wire located approximately 20 ft aft of the start of the turbulent wake. The times of start-up and stop of the grid motion and the time of entry of the moving wave wires into the wake are also indicated on this figure. Perhaps the most notable feature on this typical test record is the pronounced oscillation of the measured wave amplitude at all but the stationary wave wire. It is seen that, for the specified test conditions, the measured wave amplitudes varied from nearly zero to values somewhat larger than the incident wave. Further, the time between successive minimum values is approximately 9-10 seconds for the waves in the wake but considerably longer, although not as clearly defined, for the wave probe ahead of the grid. For longer wave lengths, the wave ampli- tude variations were reduced and the apparent period between mini- mum values increased. A reduction in grid speed reduced the wave amplitude variation and increased the apparent period between mini- mum values. There was no discernible effect of grid mesh size on these general observations. It is to be noted from Fig. 5 that fluctuations in wave ampli- tude continued for a long time after the turbulence grid was stopped. This is, of course, due to the fact that the wake has a mean flow defined in Fig. 2 and, consequently, moves past the stationary grid. It is also interesting to note that wave deformation at wave wires 1 and 2 is first evident after approximately 3 seconds or, equivalently, after a wave crest has traveled nearly 5 ft into the wake. Specific Behavior: The envelopes of wave height (h) variation with time, normalized on the basis of incident wave height (h,), are plotted in Figs. 6 through 11 for a grid speed of approximately 1 ft/sec; a grid draft of 20 inches; and a mesh size of 2.7 inches. Data are presented for the 3, 4 and 8 ft wave lengths, each having a height of approximately 1 inch. The data for the 2 ft wave length are not pre- sented since the wave heights were most irregular even in the non- turbulent flow area. The data for the 6 ft long wave were not unlike those for the 4 and 8 ft test waves and, hence, are not included in this paper. Two companion plots are presented for each wave length. For example, the data for the 3 ft long wave are given in Figs. 6 and 7. The envelopes of the ratio h/h,; for the three moving wave wires are plotted in Fig. 6 along with the phase angle between wave crests at the centerline and at a point 4 ft outboard of the centerline at a longitudinal distance of 13.25 ft aft of the grid. A zero phase angle represents a crest line parallel to the grid. The complementary data plot for the 3 ft wave is given in Fig. 7 where, in addition to the envelopes of h/h;,, the apparent wave length is plotted at a longitudinal centerline position approximately 15 ft aft of the grid. This wave length is computed from the data obtained at the two centerline wave wires located at a distance of 13.25 ft and 16.5 ft 402 r te Turbulent Flow Fields tnt Gravtty Waves and F ses/ TT =A iL9°) = 7e4p ul°@ = Ysout pas Was e S 1 =X pli3 [euolsueultp-suo yim Sutaow soqoad aaem ye WStoy oaem jo odojeauq 9 *SIq SONODSS-LYVLS GIYd Y3L4V ANIL ool- 09 Ov ® Oo m Vv o (@) SQv37 (@) N3HM v v 002 JAILISOd ‘@LNIOd GNV VW, B LNIOd LV LS3YND JAWM oo¢e N33M138 JIONV 3SVHd =e yV e) Ovo = oso > O21 09"! 403 Savitsky ces/ TT =A 8 ,L9°R = wep ys*Z = ysom pws zt =*H Ug HN plas [euo}susultp-auo YyWM SuyAour soqoard oaem ye yystey eAem Jo odojeaumq 7 “3A SQNOD3S-LYVLS Gi¥d Y314V 3WIL os! og! Ov! 02! ool 08 09 Ov 02 Oo = 14 ‘8Y @ LNIOd QNV @ LNIOd N33ML398 HLON31 SJAVM 3AIL9354543 = 9X Q3ddO1lS Gly9d A WO a 02' 09"! 404 Gravtty Waves and Fintte Turbulent Flow Fields 208/33 OTT =A iL9 T= 4e4P ul °2 = Ysour pyas Pr dle Oe = iv =X plaid Teuotsusul{p-9uo YIJM SuyAaour saqoid aAem ye WYSToy eAeM JO odoTeAUmM gQ °BIq SQNOOD3S-LYVIS GI¥S Y314V 3WIL og! Ov! OZ! 00! o8 09 Ov 02 ie) Ovo a Q3ddO1S divs 02) o9' 405 Savitsky ses/ ToT =A iL2°F = erp ul°Z = Ysour pias iw°h= oi iv X plaid [euoTsuautp-suo yyM ButAou soqoid aAeMm ye WYBIoy oAeM jo odojeauq 6 *3I1q SGNOD3S-LYVLS GiY¥S ¥3140 SWIL og! Ov! Od! ool 08 09 Ov 02 0 14 ‘ay OvO0 Od 09"! 406 ses/IT T°T =A iL2°) = Wertp ul°2 = Ysour plas fe) Seer to ak plaid TeuoTsueul{p-suo Y4IYM Butaour saqoid sAeM ye }YSToOY oAeM Jo odoTeaAuM QT “BIT SONOO3S-1LYVLS GINS Y3914V AWIL ool- Vv os Vv g 09 Vv Vv Q3ddOLlS alud Ov'0 080 huyy 02 Gravity Waves and Fintte Turbulent Flow Fields Sa 091! 407 Savitsky ses/ZT°bT =A iL2°F = erp ul°Z = Ysour pyas Ooh ame id iS =X P]id TeuojJsueultp-suo YM BuyAaou soqoid oAeM ye WYS}TaYy SAM Jo odojTeAuW JT “sq SQNOO93S-LYVLS GINS Y3L4V SWIL OOo! 08 09 Ov 14 ‘ay Ov 0 Q3dd01S Gly 09’! 408 Gravity Waves and Finite Turbulent Flow Fields aft of the grid. Similar sets of plots are given in Figs. 8 and 9 for the 4 ft long wave and in Figs. 10 and 11 for the 8 ft long wave. The irregular, oscillatory behavior of the wave height envelope is apparent in all plots and decreases as the wave length increases. Further, the average wave heights at 13.25 and 16.5 ft aft of the grid continuously decreases with increasing time of grid travel (which corresponds to an increasing length of turbulent wake through which the wave travels). Repeat runs for otherwise identical test conditions did not produce identical time histories of wave height envelopes. This can be seen by comparing the time histories for a point 13.25 ft aft of the grid as shown in Figs. 6 and 7; 8 and 9; and 10 and 11. The time histories are much more nearly alike for the 8 ft long wave than for the 3 ft long wave. Crest Line Deformation: An examination of the phase relation (8) between the wave crest at a point 13.5 ft aft of and on the grid centerline and the wave crest at a point 4 ft transverse to this point indicates the first clear regularity to these one-dimensional test results. For the case of the 4 ft long wave (Fig. 6), it is seen that phase angle is zero, implying a crest line parallel to the grid for the first 20 seconds (20 ft of wake development) of grid travel. As time increases, the phase angle increases so that crest at the center- line precedes the wave crest 4 ft off the centerline. The phase angle increases nearly linearly with increasing time. For the 4 ft long wave (Fig. 8), a similar linear phase shift occurs except that the rate of phase shift is now somewhat slower. The phase shift for the 8 ft long test wave (Fig. 10) has a maximum value of only 45°. For this long wave, the centerline crest lags the outboard crest. A com- parison of the wave amplitude shows nearly similar values at both wave probes when the phase is 0° or 360° and maximum differences when the phase angle is 180°, Apparent Wave Length: The apparent wave length along the wake centerline was determined from an analysis of the time histories of the wave amplitudes at the two wave probes which were 3.25 ft apart (probes at distances of 13.25 ft and 16.50 ft aft of the grid). The apparent wave length generally increases with increasing time of gridtravel. The 3 ft long wave (Fig. 7) attains a value of approxi- mately 4 ft after 50 seconds of grid travel and then decreases to a value of 3.5 ft. The 4 ft test wave (Fig. 9) attains a value of nearly 6.5 ft after 100 seconds of grid travel while the 8 ft wave (Fig. 11) attains a value of approximately 11 ft after 90 seconds of travel. The wave lengthening is expected because of the longitudinal mean wake flow in the direction of wave celerity. Effect of Grid Velocity: Figures 12 and 13 present the wave height envelopes for the 3 ft and 4 ft long waves when the grid speed 409 Savitsky ses/HL T= A ,Le°P=Betp yL*Z=usourpyzs ,ypy="H ,€ =X pid Teuoysuswyp-ouo yzM SujAowu soqord oaem ye WYSJoy eAeM Jo odoTeauM 7] °31q SONOD3S-LYVLS GINS Y3L4V 3WIL oo1v 08 09 ool- og! O02! fo} 2 930‘e v Vv OY G3addO01S Qlus Vv 4 Vv 080 fuyy 02"! o9'! 410 Gravity Waves and Finite Turbulent Flow Fields ses/ITL*T =A iL2°T = 1Fe4p uL°? = ysour prag i2° b= "8 ib =X P]is Teuoj[susUIIp-sUO YUM BuyAour soqoid oAeM je JYSToY sAeM JO odoTeAUM ET “BIT SGNODSS-LYVIS GIY¥9 Y3145V SWIL og! Ov! Oz! ool 08 09 Ov 02 ie) ovo A G3ddO0LS divs 021 SAVM A = 09'1 A aus ,O'© = HLGIM giyS JO 14V 9 416 Gravity Waves and Fintte Turbulent Flow Fields O°) = H (9 =X plas Teuoysuewyp-7 Fo syxem ul SuzAaour seqoid aaem ye WYSToy oaeMm Jo odofeAauM PJ] *3tzT os OL .2'2= HS3W ,O2=143VH OE = HLOIM aius SGNO93S —3WIL O09 OS Ov O€ 93S/14 ODZS=A 93S/14 O1=A GQ3ddOlS gius oz ol fo) fe) alu 40 14v I oso i 7 alu 40 14v 9 Q3dd01S aud ae fe) aiu9 40 Lav I Oso , ll. yu 00’! giyS 4O 14V 9 417 Savttsky The envelope of wave height at the moving probes is given in Figs. 16 and 17 for wave lengths of 2 ft and 6 ft respectively. The wave probe positions are at distances of 1 ft and 6 ft aft of the grid and just off its centerline. The grid was 3 ft wide, had a draft of 1.7 ft and a mesh size of 2.7 inches. Figure 16 presents results for grid speeds of 1.0 and 1.6 ft/sec, while Fig. 17 is for speeds of 1.0 and 2.6 ft/sec. It is seen that there is a continuous reduction in wave height with time. For the 2 ft long wave and a grid speed of 1.0 ft/sec, the amplitude is reduced to nearly 10 per cent of its initial value after approximately 20 ft of gridtravel. It remains essentially at this value for the length of the test record which ex- tended for 30 seconds after the grid was stopped. The effect of increasing the speed of the grid from 1.0 to 1.6 ft/sec reduced the wave height to nearly 8% of its initial value. It is to be noted that there is a distinct absence of oscillations in these time histories. The results for the 6 ft long wave (Fig. 17) are essentially similar to those for the 2 ft long wave. At a grid speed of 1 ft/sec, the wave height is reduced to approximately 35 per cent of its initial value. When the grid speed was increased to 2.6 ft/sec, the wave height was reduced to 12 per cent of its initial value. An overwater photograph of the wave deformation for a typical two-dimensional test is shown in Fig. 18. The reduction in wave height along the centerline wake area and the amplification outside this area are clearly visible in this photograph. \* . ! Fig. 18 Typical wave deformation for two-dimensional grid 418 Gravtty Waves and Finite Turbulent Flow Fields Specific Results: To more clearly illustrate the modifications in wave height along a given crest line, transverse sections through the wake are plotted in Figs. 19 and 20 for a grid width of 3 ft, draft of 1.67 ft; mesh of 2.7" and speed of 1 ft/sec. The effect of a grid draft of 0.83 is given in Fig. 21. The results for the 2 ft wave are given in Fig. 19 while those for the 6 ft wave are given in Figs. 20 and 21. These data are obtained from simultaneous measurements of the recorded wave height at times corresponding to the indicated distances ahead of and behind the grid. A maximum phase shift of only 30° was discernible in the test records. It is seen that a substantial reduction in wave amplitude exists for a distance of nearly one grid width on either side of the center- line. The maximum wave height amplification occurs at approxi- mately two grid widths from the centerline and the wave amplitude appears to be unaffected at distances of approximately 4 grid widths from the centerline. This pattern exists for distances well aft of the grid. It is interesting to note that, as the wave passes through and ahead of the grid, where the wake does not exist, the deformed crest tends to return to its original uniform height. Again, it is seen that the 2 ft wave is much more attenuated and amplified com- pared to the 6 ft wave, all other conditions being equal. (Compare Figs. 19 and 20.) It is also interesting to note that reducing the grid draft from 1.67 ft to 0. 83 ft has a negligible effect on wave deformation for the 6 ft wave. (Compare Figs. 20 and 21.) TRANSVERSE SECTION 5 AHEAD OF GRID —— — TRANSVERSE SECTION 0.40 // 6 AFT OF GRID 7 — — — TRANSVERSE SECTION 32 X AFT OF GRID 4 6 8 10 l2 y= DISTANCE NORMAL TO GRID @ ~FEET Fig. 19 Height of wave oo line in transverse sections normal to grid’‘@.° X= 2", 1, grid width ="3", mesh ="2.7", draft = 1.67', V= ne ft/sec. 419 60 2 AFT OF GRID — —— TRANSVERSE SECTION 7 \ AFT OF GRID —— De VS GRID a ==> SESS = es, 2 h. YY, ' 080 4 m4 eee 040 we Go 2 4 6 8 10 12 y= DISTANCE NORMAL TO GRID q -FEET Fig. 20 Height of wave crest line in transverse sections normal to grid Go. Aw= 16; Hy= i eae grid width = 3', mesh,;= 22 fie draft = 1.67', V= 1 ft/sec. 2) FORWARD OF GRID 1.60 — —_ 2) AFT OF GRID ——— 4X AFT OF GRID TG, ea 1.20 a — GRID = LLL LLLLLA h hj hie. 0.80 ae See 040 fe) ) 2 4 6 8 10 12 y= DISTANCE NORMAL TO GRID ¢ -FEET Fig. 21 Height of wave crest line in transverse sections normal to Savttsky TRANSVERSE SECTION 2X AHEAD OF GRID —— — TRANSVERSE SECTION grid’'G. 4 ="6", H,= 1", grid width =3', mesh =(257"5 draft = 0,83", V = 1 ft/sec. 420 Gravtty Waves and Finite Turbulent Flow Ftelds Increasing the mesh size from 2.7 to 5.4 inches has a negligi- ble effect on the wave deformation. Increasing the width of the grid increased the area of wave attenuation and wave amplification. III. ANALYSIS AND DISCUSSION Viscous and Turbulent Effects The initial analysis of the experimental results was directed to relating the observed wave deformations to possible physical mechanisms associated with the turbulent and viscous nature of the grid-generated wake. It was found that the large changes in wave height could not be accounted for by these considerations -- parti- cularly in the two-dimensional grid studies. In this case, the square of the wave height was integrated along a crest line passing through the wake to obtain a measure of the energy in the deformed wave. This was compared with the energy in the incident wave for the same length of crest line. The results of this comparison are presented in Table 1 for wave and grid dimensions selected to be typically representative of the total test program. The length of integration, Y, along a given crest line was the distance between the grid center- line and the point where the wave height was again equal to the inci- dent wave height. The crest length thus includes both the attenuated and amplified wave height regions. For a given test condition, the integrations were carried out for several transverse sections through the wake, both ahead of and aft of the grid. It is seen from Table 1 that the integrated expression representative of wave energy pro- duces nearly similar results with and without the towed grid. In fact, for some test conditions, the integrated energy for waves in the presence of the grid results in values somewhat higher than for the case of no grid -- but this is attributed to experimental inac- curies. It thus appears that viscous dissipative effects were quite small and, although they most certainly existed, their magnitude could not be accurately detected because the limited number of transverse wave probes were inadequate to trace the unexpected large wave height deformation which developed along a crest line. A measure of the rms of the velocity fluctuations in the turbu- lent field yielded substantially the same values with or without waves passing through the wake. This was not surprising since the energy imparted by the grid to the fluid at a tow speed of 1 ft/sec was nearly an order of magnitude larger than the wave energy ina crestline length equal to the grid width. For the one-dimensional tests, it will be recalled that the wave height at a given position in the wake exhibited large fluctua- tions and was characterized by irregularities in the recorded time histories. These results could certainly not be accounted for by dissipative mechanisms in the turbulent field. It appeared then that for both the one and two-dimensional studies the principal 421 Savitsky TABLE. I Comparison of Wave Energy With and Without Towed Grid 0 y, ft. Y y 2 2 E,= J b ay E,g= HyY (with towed grid) (no grid) We; Guid" Transverse Section a *K oN ght Ts D M V6 Position Eng Ey 6.0 0.09 3.0 1.7 0.22 1.0 ft/sec 5d(F) 0.067 ft? 0.069 £t3 6X(A) 0.063 0.060 32X(A) 0.058 0.064 S200 l09 3.0 1.7 ,.02.22 . tae 2X(F) 0.081 0.082 2X(A) 0.081 0.083 7X(A) 0.081 0.079 2.0 =0:,09 12 340nl4s:7 00529 Akio 5X(A) 0.081 0.084 6X(A) 0.087 0.092 32X(A) 0.081 0.075 60) s0209' 5.5 1.7 0,22 ~ 1.0 2X(F) 0.093 0.100 2X(A) 0.087 0.099 4\(A) 0.087 0.099 22 02%:0) 097 <3. Oret.OS852.01A22)! 2120 5X(F) 0.081 0.098 6X(A) 0.072 0.078 23(A) 0.081 0.091 6.0 0-509, S20" 0. 85, U22- 1.0 2X(F) . 0.081 0.079 Z2XCA), 0, 0,70 0.076 4\(A) 0.070 0.078 *Wave and grid dimension in feet; L = grid width; D = grid immersion; M = grid mesh igs is transverse section forward of grid (ft) (A) is transverse section aft of grid (ft) 422 Gravity Waves and Fintte Turbulent Flow Fields mechanism of wave height deformation was due to a redistribution of energy along a crest line rather than to dissipative effects. In this regard, the results of Phillips [1959] were examined to determine possible convective distortions of the wave front resulting from scattering interference between wave and turbulence field. It was found that the observed results could not be accounted for by the turbulent scattering. If, in the present studies then, turbulence is assumed to have had a minor effect on wave deformation, it remained to examine the possible interference between the mean flow gradient in the wake and the incident wave. The velocity profiles for the longitudinal mean flow aft of the grids are plotted in Figs. 2 and 4 for the one- and two-dimensional studies, respectively. In both cases there is a relatively sharp velocity gradient between a region of constant wake velocity to zero velocity at the tank wall for the one-dimensional case and to zero velocity in the still water adjacent to the finite grid wake in the two-dimensional case. By application and superposition of elemental theories of wave refraction, defraction and interference, it was found that the observed results could be, at least, qualitatively reproduced and physical mechanisms described to account for the large wave deformations observed. A detailed analysis is first made of the two-dimensional tests since these results were free of possible wall reflection effects such as existed in the one-dimensional studies. Further, the 2-D analysis will provide the foundation for explaining the results of the 1-D studies which proved to be the more complex case. Wave Interaction with Finite Velocity Field Two-Dimensional Results. The longitudinal mean flow in the finite wake area aft of the grid is plotted in Fig. 4 and is quantified by an empirical formulation, Eq. (2). —— 16740.062x Vw =[0.45-0.00745x][e. ] Vi where Vw = mean value of longitudinal velocity in wake V = grid velocity x = distance aft of grid, ft y = distance normal to grid centerline, ft In the present analysis, regular waves in still deep water en- counter a variable current field Vy (x+y) moving in the same direc- tion as the waves. The waves are initially refracted by the current to an extent dependent upon the incident wave length, strength of current, and the velocity gradients in the wake. The orientation 423 Savttsky of the wave-wake system is shown in Fig. 22. For progressive deep water gravity waves in still water, the phase velocity of the wave, Co, is given by: Cr= g/k, (3) where g = acceleration of gravity, \,= wave length in still water, k, = wave number = 27/X., C, = wave velocity relative to still water. After the waves have run from still water into a current, the kine- matical condition that must be satisfied is that the wave period, T, remains constant while the wave length, i, velocity C, and height H, change. Given a current velocity V), the constancy of wave period is expressed as: 27 -k(C + Vy) = k(C, + Vu) (4) where the subscript, o, refers to the still water conditions. Thus: 2 ky C+ Vy .S kia LGastaVead (Cs For the present case Vy, = 0 so that: C_S Ny. 9 Co Co Co and 1 C= 5(Co + VG? + 4V\Co) (5) which is the wave speed relative to the water for waves progressing in the same direction as the current. More generally, for waves whose crest line is at an angle relative to the x-axis: 1 C= s (Cot Heng + 4V,C, cos 2) (6) where @ is the angle between a wave ray and the x-axis (Fig. 22). The wave velocity relative to the bottom C' is the vector sum of the wave speed relative to the water and the local current. 424 WAVE CRESTS Gravity Waves and Finite Turbulent Flow Ftelds DEFORMED WAVE CREST STILL WATER | FINITE CURRENT FIELD Vy =LONGITUDINAL CURRENT FIELD IN WAKE Co = WAVE VELOCITY, STILL WATER C = WAVE VELOCITY, IN CURRENT @ = ANGLE BETWEEN WAVE RAY AND X AXIS Fig. 22 Wave-wake system 425 Savttsky Grav ae (7) The wave length for waves progressing in the same direction as the current is thus: N -(4 +1 +4(V, om 2 te) (8) It can be seen that the effect of a following current is to increase the wave length relative to the water. The analytical solution for the refraction of waves traveling through a finite current field is obtained by application of Fermat's principle that waves will travel in a path such that the travel time is aminimum. Applying the method of calculus of variations will lead to a time history of the path of individual wave rays passing through the current. For the purposes of this analysis, it will be assumed that the wake properties do not vary with time. This is a reasonable assumption since it has been demonstrated that, for a grid-speed of 4 ft/sec, the mean wake-velocity is an order of magnitude less than the wave speeds. Mathematically, the problem is to determine the minimum time path of a given wave ray through a current region de- fined by a position dependent velocity vector. The magnitude and direction of the current are known as functions of position (Eq. 2). The magnitude of the wave crest velocity relative to the water is C, given by Eq. (6). The problem is to determine the path of a wave ray such as to minimize the time necessary to travel from point A to a point B. Analogous optimization problems for dynamic systems are described by Bryson and Ho [1969]. The equations of motion are: x(t) - V,(x,y) - C(x,y,) cos @ (9) - C(x,y,@) sina y(t) with initial conditions x(0) = x, (10) y(0) = yo and at end of computation x(t,) = xX, 426 Gravity Waves and Fintte Turbulent Flow Fields where ty is an unspecified terminal time of integration between points A and B. It is required to find a(t) and t; such that the above constraints are satisfied and that the performance index J(a) of elapsed time t, is a minimum expressed mathematically, " Ha) = § dt (11) 0) is a minimum. From the methods of calculus of variations, the Hamiltonian of the system is: H(x,y,%,d,,d) =it A(- bee (o; cos @) = dC sin a (12) where A, and X equations are: y are Lagrange multipliers. The Euler-Lagrange ° 0H Mme OFS - _ 86H Bp - _ 0H pee 3 - _ 0H Vite Ory ae 0H ~ Ba The terminal conditions are: dlt,) =0 (14) H(t.) = 6 Since the Hamiltonian is not an explicit function of time, Hl=0 and H is aconstant. Further, since H=0 at the terminal condition, then it follows that H=0 forall 0StX tye 427 Savttsky Evaluating 9H/8a from (12) and using the condition H = 0 leads to a determination of the Lagrange multipliers. C cos @ + SE sina i, So Cm aS Vic cos @ ce sin @) +c? (15) dC Ba Cos %- C sina $s Vc cos @ +ec sina) + (og y= The remaining differential equations are employed and, after extensive algebraic manipulations (which will not be reproduced herein), the following expression for a(t), the angular trajectory for minimum travel time, is obtained. 3 2 att) = [VC ce verge tc Be sina +C S¥w SE sin? a 2 +C B¥ a Sin meos'a HG pe coe. - ac 80 Se cos a 2 - ¢ Bly Se cost a - Ny eS sin a cos @ + C?5™ cosa 6? BE sina +20 902 sina +c°S Gpicos® a aV, aC dV, acy - 2 +26 Sw SE sina cosa + Suu( 52) sin e|/ Bop ae | (16) The partial derivatives of C and Vy contained in Eq. (16) are obtained from the definition of C and Vy, as follows: Cc =5 [Co + (C$ + 4V\C, cos a | V, (x.y) = (0.45 - 0.0074x) exp . (ae meee 428 Gravity Waves and Finite Turbulent Flow Fields Thus: c= —— VC, sina (Gc. + 4V,C, cos aye he = C, cos @ Cos + 4V0, cos oy oNw ac 2 - OV, by = C, cos @ (C, + 4V,C, cos a) oe 2 2 “1/2 er = Cy, sin a FV - (C, + 4V_C- cos @) -3/2 + 2C, cos (C,+4V,C, cos a) ] a°C av 2 Byda = C, sina Bye [- (C, + 4V,C, cos a) -3/2 + 2C, cos @ (Ge + 4VC, cos @) | anc _-_ vec a (C+ 4V.C ae ayes wo , — / >. | 20 —— WAVE RAY LINES pe AVE RAY LINES —\ | S | 24 = NS 2 | 28 OTE: (1) SHADED AREA REPRESENTS| GRID WAKE (FIG. 4) | (2) Yo =INITIAL RAY POSITION IN STILL WATER 32 36 WAVE CREST LINES 40 ) 4 8 12 16 20 24 28 LATERAL DISTANCE FROM WAKE ¢ , FEET Fig. 23 Computed wave refraction diagram X= 2!' V = 1 ft/sec 431 DISTANCE AFT OF GRID, FEET Savttsky GRID Yo =1.00 4.50 75 2.50 te 4.25 2.00 me ~~ af 3.75 = / 3.50 Ss 4 ———~ / / 8 12 pe J. / Na WAVE RAY unes | ——. NOTE: (1) SHADED AREA REPRESENTS GRID WAKE (FIG.4) (2) Yo =INITIAL RAY POSITION IN STILL WATER WAVE CREST LINES 16 20 24 28 LATERAL DISTANCE FROM WAKE @ , FEET Fig. 24 Computed wave refraction diagram h= 6' 432 V = 1 ft/sec Gravity Waves and Fintte Turbulent Flow Ftelds Crest Length Between © and 1 ft. In this region, the current field is essentially constant across any transverse section and in- creases slowly in the longitudinal direction. The effect on this finite wave is to increase the wave length in accordance with Eq. (8). For a mean flow of 0.30 ft/sec, the 2 ft wave length should increase by 20% while the 6 ft wave should increase by 10%. This is in reason- able agreement with the results in Figs. 23 and 24. Since the wave length increases, it is expected that the wave heights will decrease in order to maintain wave energy balance. Phillips [1969] accounts for this wave energy balance in treating the case of long-crested waves running into a current in which the surface velocity varies only longitudinally. His results, plotted on Fig. 3.6 of his work, show that for the present conditions a wave height attenuation of approximately 15% is expected for the 2 ft wave and approximately 8% for the 6 ft wave. This is consider- ably less than the experimentally attained values of 80% to 90% attenuation previously discussed. Thus, the refraction procedure alone does not account for the results observed in the vicinity of the wake centerline. It will later be shown that diffraction effects applied to this length of wave crest can indeed result in large local attenuations of wave height. Crest rss Soe Between 1 ft and 3.5 ft. The orthogonals for this crest length diverge rapidly in a direction which causes the local wave crest to be redirected out of the wake area into the still water. This finite crest length advances in a constant direction relative to axis system at a speed and wave length equal to the inci- dent wave. It then crosses the undeformed incident crest line at a distance 5 ft from the wake centerline. As a first order effect, it can be assumed that the wave heights decrease as the square root of the ratio of the initial wave ray separation to the separation at any subsequent position of the local crest. For the 2 ft long wave at a position 12 ft aft of the grid (28 ft into the wake), the wave height (Fig. 25) indicates that the average wave height for this local crest is approximately 30% of the incident wave height. The deflection of this local wave crest length into the area of the incident wave could account for the irregularity observed in the wave height time histories at fixed points between 7 and 12 ft from the wake centerline. Crest Length Between 3.5 and 5 ft. For this length of wave crest, it was seen that adjacent orthogonals converge and finally cross, resulting in a caustic curve [ Pierson 1951]. On the basis of simple theory, the wave became infinitely high on the caustic which, of course, is not the case. At present, quantitative analysis of the wave height at and beyond caustics must still be developed for the case where variable currents produce wave distortions. 433 Savttsky Crest Length Beyond 5 ft. This length of wave crest is always outside of the wake current and thus continuously progresses through still water with no alteration in wave length or speed. During its forward progress, it runs into the caustic area and deflected wave crest originating between 1 ft and 3.5 ft from the wake centerline. Application of Analytical Results Due to the omission of defraction effects for each wave crest segment, the refraction results discussed in this section are in themselves insufficient to represent the test results. They are nonetheless invaluable in forming the basis for providing qualitative information about the complex processes governing the interaction of a wave system with a finite current field. Reflection Effects: The results of the refraction analysis are first used to compute wave heights along crest lines in transverse sections through the wake. The detailed results presented herein are for a transverse section 12 ft aft of the grid and for a current field extending 40 ft aft of the grid (as shown in Figs. 23 and 24). Thus, the wave has progressed 28 ft into the wake at the time of com- putation. The grid was 3 ft wide; had a mesh of 2.7 inches; a draft of 1.67 ft; and a tow speed of 1 ft sec. Computations were made for 6 ft and 2 ft long waves. The experimental results for these conditions have already been presented in Figs. 19 and 20. Since the purpose of this computation is mainly to compare qualitatively the measured results with elemental analytical results, simplifying assumptions were introduced. First, it was assumed that the local wave height between adjacent orthogonals is inversely proportional to the square root of the distance between these adjacent ray s«,..hus:, Bs = (4) un where: separation distance between adjacent rays in still water se ° iT] = separation distance between adjacent rays on the de- formed crest line Hy= local wave height in still water = local wave height along deformed crest line This enabled wave heights to be constructed along a wave crest from the centerline to a distance approximately 6 feet from the centerline. Beyond this point, there is a superposition of the deflected wave seg- ment with the undisturbed length of the incident wave. In this area, 434 Gravity Waves and Finite Turbulent Flow Ftelds the two waves are combined in proper phase as indicated by the crest line plots in Figs. 23 and 24. The section of wave crest that develops into a caustic has not been included in this elemental con- struction. The results of this simple refraction analysis are plotted in Figs. 25 and 26 for the 6 ft and 2 ft wave lengths. It is seen that this procedure results in essentially unmodified crest heights just aft of the physical grid; then large reductions in wave height for areas transverse to the grid and, finally, increases in wave height in those areas where the deflected segment of the wave combines with the undeformed segment of the incident wave. The results of the refraction computations do not entirely agree with the experimental data -- particularly in the region of the grid wake where the test results show significant attenuations in wave height while the computed results show no wave height attenuation. Considering the variation of computed wave height along the crest line (Figs. 25 and 26), it is seen that there is a large increase in wave height for positions less than and greater than approximately 6 ft from the grid centerline. At this 6 ft point, the computed wave height is aminimum. These transverse gradients cannot remain in equilibrium and thus represent a source of energy flow along the wave crest from the regions of large wave height to the point of low wave height. This is a diffraction phenomenon which exists simul- taneously with refraction effects. A rigorous theoretical analysis of this problem appears to be extremely complex and is yet to be developed. For the purposes of the present study, a simplified analysis is developed which combines the results of elemental solu- tions of wave refraction, diffraction and superposition. Although not completely rigorous, this simplified approach is tenable and relatively easily applied. Diffraction Effects: As normally considered, wave diffraction occurs when part of a wave is "cut off" as it moves past an obstruc- tion such as a breakwater. The portion of the wave moving past the tip of the breakwater will be the source of a flow of energy in the direction essentially along the deformed wave crest and into the region in the lee of the structure. As explained by Wiegel, the "end" of the wave will act somewhat as a potential source and the wave in the lee of the breakwater will spread out with the amplitude decreas- ing exponentially along the deformed crest line. The mathematical solution of this phenomenon, which is taken from the theory of acoustic and light waves, is described by Penny and Price [ 1952] ; Johnson [ 1952] and Wiegel [1964]. The solutions for two basic diffraction phenomenon are presented by Wiegel: one is the case of a semi-infinite breakwater and the other is for the case of waves en- countering a single gap in a very long breakwater. The solution for both cases are presented by Wiegel in the form of contour plots of equal diffraction coefficient, K, defined as the ratio of the wave height in the area affected by diffraction to the wave height in the area unaffected by diffraction. For the case of the wave passing through a single gap, the solutions are presented for various ratios of wave length to gap width. 435 Savittsky NOTE: (1) TRANSVERSE SECTION 12 FT AFT OF GRID (2) WAKE LENGTH =40 FT Po eke / A EXPERIMENTAL DATA (FIG. 20) —— FROM REFRACTION COMPUTATIONS (FIG.24) ) 2 4 6 8 10 12 y= DISTANCE NORMAL TO GRID ¢ -FEET Fig. 25 Results of refraction computation versus measured crest height... ..= 6';. H,= £"3 . grid width = 3's: meshi=d2eq3 a@ratt =o6rtiecv = Wit/sec: 1.60 NOTE: (1) TRANSVERSE SECTION (2 FT AFT OF GRID A (2) WAKE LENGTH =40 FT / 1.20 / 4 / 4A—_A— — 080 / 7 ANC, \ A EXPERIMENTAL DATA (FIG. 20) 0.40 4 \ / = FROM REFRACTION COMPUTATIONS ‘\ (FIG. 24) te) 72 4 6 8 10 l2 y=DISTANCE NORMAL TO GRID @ -FEET Fig. 26 Results of refraction computation versus measured crest height. X= 2'; Hy= 1"; grid width = 3'; mesh = 2.7"; draft =1.67": V= 71 ft/sec. 436 Gravity Waves and Fintte Turbulent Flow Ftelds In applying these diffraction results to the present study, it has been assumed that the refraction phenomenon previously dis- cussed divides the wave crest into several segments which are separately diffracted as they pass through the grid wake. Specifi- cally, the segment of the wave crest just aft of the grid is assumed to behave as though it was a section of the wave which passed through a breakwater gap equal to the grid width. The justification for this analogy follows from the refraction results given on Figs. 25 and 26 where it is shown that, for a distance of approximately one-half the grid width on either side of the grid centerline, the wave height in the wave cannot be maintained at a constant height since just out- board of this segment the refraction analysis yields a small wave height. Thus, it appears reasonable to assume that diffraction effects will be developed and that this centerline segment of the wave will reduce in amplitude and spread transversely along the crest as it proceeds into the grid wake. The diffraction coefficients will be taken to be those corresponding to a wave at a breakwater gap as given on pages 188-189 of Wiegel. One other portion of the incident wave which appears to be modified by diffraction is that segment of the incident wave which is located 5 ft outboard of the grid centerline. From the wave refraction diagrams on Figs. 23 and 24, it is seen that wave rays and crest lines outboard of 5 ft are not influenced by the grid wake. Simple refraction considerations then result in a wave of constant amplitude along this length of the wave front. Again, this constant wave height cannot be maintained and a defraction process develops which causes a lateral spreading of the wave crest into the wake area with an attendant reduction in wave amplitude. This lateral flow of wave energy can be compared to the case of water passage past a semi- infinite breakwater, the solution for which is plotted on page 183 of Wiegel. Typical diffraction diagrams for the case of breakwater gap and semi-infinite barrier are given in Figs. 27 and 28 of this report. The computed results for these two diffraction processes are plotted in Figs. 29 and 30 for the 6 ft and 2 ft wave lengths respectively. Again, the computations are made for transverse section approximately 28 ft into the grid wake. For the 6 ft wave, the ratio of effective "gap width" to wave length is 3/6 = 0.50; for the 2 ft wave, the ratio is 3/2 = 1.50. It is seen that the initial constant height wave segment between the grid centerline and 1.5 ft outboard is diffracted to approximately 0.30 of this height and is spread laterally to a distance nearly 12 ft from the grid centerline. Considering the diffraction of the entire wave segment initially 5 ft outboard of the grid centerline, it is seen that this section is spread inboard to the grid centerline with a corresponding reduction in wave height at approximately 0.30 of its initial height. It is seen that, for this wave segment, the attenuation of wave height as it spreads to the centerline is much more rapid for the 2 ft wave than for the 6 ft wave. 437 Savttsky x/Xd 10 Oo I 2 4 6 8 10 12 14 16 18 20 Fig. 27 Diffraction of waves at breakwater gap contours of equal defraction coefficient [ Johnson 1952] Neclistnlich al eee very? pbluay pal ae ay yVAVAN ovale bhepigeey SAMA tae \ y\ ATT z/L Fig. 28 Diffraction of waves passing semi-infinite breakwater [ Penny and Price, 1952] 438 Gravity Waves and Finite Turbulent Flow Fields oes/i yt = A ul9*F = 3FeIp ul°? = ysour 1€ = UIPTM prado, g = “H yUSIoy JSeIO poanseow snsi9A poynduroy 67 °31T 14 Ob=HLON37 JVM (2) aiy9 40 13V Lidl NOLLDAS SSYSASNVUYL (1) -3SLON 14‘ 3 WOUMS JONVISIG ‘2 Vv LNVLIANS3Y Vv aguvogino 13S°¢ ONV 14 S' N33ML39 LNIW9SS JAVM JO NOI LOVYSSY QuUVOSLNO 143 S°1 OGNV 9D N3J3ML39 INSW935S 3AVM JO NOILIVYSSIG dD 4O GuUVOsLNO 14S WOUS INSW93S 3AVM JYILNS JO NOILIVYSSIG (O02 91d) VLVO TWLNSWIYN3dX3 <0 M" RA Ovo 02"! 439 Savitsky v! 7 .es/IFT =A ” él / / a 14 Ov=HLONST JVM (2) uL9°T = 3Fetp uL°Z = sour ,€=yIpIM pws yp="H ,Z=X JYWsley 4Sei1d porinseosw snsiz9A payndwioy OF °31q7 Oovo- ~ ‘ ¢ ms \ 14‘ 4 WOYS JONVLSIO ‘Z 6% Ol 8 7 9 v A O 7 x y <=—_ ey a == ‘\“ & a“ wT” —_— << oo owe es ies ae — «=e ase a= Fs Ovo giud 4O Lav 1421 as NOILOSS JSYSASNVUYL (1) :3LON na : e g Vv Ps 080 Vv = VZZZZZZZL2 aiyd Vv INVLINS3Y 02'I auvosg ino 14S’€ GNV 14S' N33M1L38 LNSW93S 3AWM 4O NOILOVYES3IY — — 9 Quvosino 14S) ONV > N33ML39 LNIW93S JAVM JO NOILIVYSSIG ——— Vv > 40 Guvogino 13S WOYS — 991 LNAW93S SAVM AMILN| JO NOILIVUSSIG (O02 914) VLVO IWLNSWIY3ad X3 V 440 Gravity Waves and Fintte Turbulent Flow Fields Superposition of Elemental Results The results of the refraction and diffraction results have been superposed in order to provide an analytical estimate of the wave height distribution along a crest line as it progresses through the grid wake. The computations were carried out for a transverse section 12 ft aft of the grid for a grid wake 40 ft long. These are identical to the refraction calculations previously described. The following procedure is used in this superposition of elemental results: a) The initial crest length between the grid centerline and 1.5 ft outboard is diffracted by the breakwater gap technique as plotted on Figs. 29 and 30. b) The initial crest length 5 ft outboard of the grid centerline is diffracted by the technique of wave passage past a semi-infinite barrier as plotted on Figs. 29 and 30. c) The segment of wave length initially between 1.5 ft and 3.5 ft is refracted by the orthogonal method and the wave heights are obtained by Eq. (17). The orientation (or phase) of this wave crest segment to the transverse section 12 ft aft of the grid is ob- tained from the computed refraction diagrams such as given in Figs. 23 and 24. The resultant wave heights for this segment are plotted on Figs. 29 and 30. d) The segment of the wave crest between 3.5 ft and 5.0 ft outboard has been neglected in this simplified procedure since it develops into a caustic line. e) The results of (a), (b), (c) and (d) are superposed to ob- tain the final wave height distribution. The results of the above procedure are presented in Figs. 29 and 30 for the 6 ft and 2 ft wave respectively and compared with the experimental data. It is seen that agreement between computed and measured results is qualitatively acceptable for the 2 ft wave length and is good for the 6 ft wave length. It appears then that the physical processes responsible for the observed deformation of waves pro- gressing into a finite current field have been established. It is strongly recommended that further studies of this problem be di- rected towards the development of a unified, rigorous theory which can be used to quantify this interesting wave-current interaction phenomenon. One-Dimensional Results No similar detailed analysis has been made of the one-dimen- sional results previously described. It does appear, however, that the transverse gradient in the longitudinal wake velocity existing at the outer edges produces a local wave refraction. This refracted wave segment must then be reflected from the tank walls and prog- ress across the wake, running into similarly refracted wave seg- ments from the opposite wall. These continuously crossing wave 441 Savttsky segments passing over the incident wave may develop distortions in wave height time histories such as observed in the experiments. The wave height irregularity at any point in the wake thus precludes a reliable evaluation of the dissipative effects of the grid-produced turbulence since only two wave probes were used in this study. IV. RECOMMENDATIONS FOR FURTHER STUDIES The original objective of the present study was to investigate experimentally the interaction between gravity waves and turbulence fields generated in the wake of a towed grid. Unfortunately, the longitudinal mean flow velocity gradients in the wake had a dominating effect on wave deformation and thus precluded a direct evaluation of turbulence effects alone. Although, ina realistic ocean environment, turbulence fields can be generated by, and exist simultaneously with, velocity gradients in ocean currents, it is nevertheless of fundamental scientific int¢rest to study separately the effects of turbulence fields with no mean flow interacting with gravity waves. The results of such an elemental turbulence study can then be combined with velocity gradients to represent wave passage through realistic ocean currents. Also, the results can be used alone to study the wave interaction with isolated turbulence fields such as exist, for example, in regions of "splash" turbulence developed by breaking waves. It is thus recommended that the present study be continued but with an experimental apparatus designated to produce localized turbulence areas with no mean flow. The experimental procedure should be capable of generating turbulence fields of controlled eddy size, turbulence intensities, depth of penetration below the free surface, and length and width of turbulence patch. It is further recommended that the turbulence generator be capable of developing vortices with either a horizontal axis or a vertical axis or a combina- tion of both. The control of the vortex direction will be important in the study of the eddy viscosity interaction in which energy is transferred from the wave motion to the turbulence. As discussed by Phillips [ 1959] , the passage of the wave results in straining the elements of the fluid near the surface in a manner periodic intime. The mean strain per cycle of the incident wave is of second order, namely (z7/)® , where a is the amplitude and the wave length of the incident wave. The wave motion thus provides a mechanism for stretching the vortex lines that operates in addition to the stretch- ing inherent in the turbulence itself, and so tends to increase w*, the mean square vorticity associated with the turbulence. It is ex- pected that this possible mechanism for transfer of wave energy will be for waves interacting with vertical vortex fields. In this case, the vertical velocity gradient in the long-crested wave stretches the vertical vortices in the turbulence field, but should not effect the 442 Gravity Waves and Fintte Turbulent Flow Ftelds horizontal vortices. An experimental setup designed to control the direction of the turbulent vortices can be most instructive in under- standing this dissipative process. The experimental procedure proposed to develop controlled turbulence fields with no mean flow is to sinusoidally oscillate a series of grids in a physically confined area in still water. The barrier confining the turbulent field can be constructed of four thin, vertical plates housing a rectangular box penetrating through the water surface to a depth below the lower ends of the oscillating grids. After oscillating the grids, the rectangular barrier can be lifted above the water surface just as the waves approach so that there is an interaction between waves and turbulence. The dimensions of this rectangular container can be varied to represent various sizes of turbulence areas. The use of an oscillating grid in a confined area has been investigated by Murray [ 1968] in his laboratory studies of horizontal turbulent diffusion. In his work, the grids which filled a 50 cm wide channel were composed of several rods 1 cm in diameter and 5cm apart. The array consisted of 3 grids, each 30 cm apart, and had a stroke of 40 cm. The following con- clusions concerning the generated turbulence are described by Murray. 1. There is no mean flow within the confined area of turbu- lence generation. This is precisely the objective of the proposed experimental procedure. 2. The oscillating array of grids produces turbulence fields which are essentially homogeneous and stationary. 3. The turbulent velocity distributions are Gaussian. 4, Taylor's statistical theory of turbulence effectively de- scribes the variance, scale time, and scale length of the generated turbulence field. In summary then, the proposed turbulence stimulation tech- nique appears to be adequate for generating controlled and mathe- matically definable localized turbulence fields. The experimental procedure will consist in mounting the turbulence generator over a section of the 75 ft square tank away from the side walls. The grid array would be oscillated to generate the turbulence field and mechanically generated gravity waves would approach this field, Just prior to the waves reaching the area of turbulence, the rectangular barrier surrounding the oscillating grids would be lifted clear of the water surface so that the waves would interact only with the turbulence. It is expected that the lateral diffusion of the turbulence area will be very slow compared to group velocity of the gravity waves so that, at least for the pas- sage of several wave lengths, the turbulence properties may be assumed to be stationary. 443 Savitsky Parametric variations in this study will include: 1. length and width dimensions of turbulence area; 2. depth of turbulence area below water surface; 3. spacing of vertical oscillating rods alone to investigate the interaction between vertically oriented vortices and gravity waves; 4. spacing of horizontal oscillating rods to investigate the interaction between horizontally oriented vortices and gravity waves; 5. combination of (3) and (4) to construct a grid having a rectangular mesh of varying dimensions to provide for various scales of two-dimensional turbulence; 6. vary speed of grid oscillation to obtain various levels of turbulence intensity; 7. vary length and height of gravity waves. Measurements should be made of the wave-height time history at various locations both inside and outside of the turbulence patch. A spectral analysis of these wave height time histories should be carried out to determine the extent of wave scattering due to the presence of turbulence. Further, a hot film probe should be slowly towed through the turbulence area to characterize its statisti- cal properties with and without the presence of passing waves. It is believed that the suggested experimental procedure is practical and can provide data necessary for basic studies of gravity waves interacting with local turbulence areas. V. CONCLUSIONS An experimental study was undertaken to investigate the interaction between deep water gravity waves progressing into a turbulent flow field generated by a finite width grid moving in the wave direction in a large towing tank. It was found that the lateral gradient of the mean longitudinal flow in the wake had predominant influence on wave deformation and precluded an evaluation of the direct effect of turbulence. The presence of the velocity gradients resulted in combined refraction, diffraction and interference between finite and adjacent segments of the incident crest line. Their combined effects were to reduce the wave heights in the wake area to approximately 10% of their original value. The wave heights outside of the wake were increased to values 75% larger than their original value. 444 Gravity Waves and Finite Turbulent Flow Fields An elementary analysis was performed of the refraction of waves entering a finite current field. A combination of these results with simple diffraction considerations qualitatively reproduced the measured crest line deformations. A unified theoretical study of this complex problem is required to provide quantitative results. Recommendations for further investigation of wave interaction with turbulence field with no mean flow are made. It appears that the present results may be useful in develop- ing full-scale procedures for local "quieting" of the deep water waves behind support ship for retrieving or launching submersibles or landing craft in a following sea. ACKNOWLEDGMENTS The author wishes to express his appreciation to Dr. R. Hires of Stevens Institute of Technology for valuable discussions and tech- nical advice rendered during the course of this study. He is also indebted to Professors W. J. Pierson, Jr. and G. Neumann of New York University for their continued encouragement and helpful suggestions throughout the study. Professor Eric S. Posmentier of New York University is thanked for his thorough review of the dissertation. REFERENCES Bryson, A. E., Jr. and Ho, Hu-Chi, Applied Optimal Control Theory, Blaisdell Publishing Co., 1969. Johnson, J. W., "Generalized wave refraction diagrams," Proc. Second Conf. Coastal Eng., Berkeley, Calif., 1952. Johnson, J. W., "Generalized wave diffraction diagrams," Proc. Second Conf. Coastal Eng., Berkeley, Calif., the Engineering Foundation on Wave Research, pp. 6-23, 1952. Murray, S. P., "Simulation of horizontal turbulent diffusion of particles under waves," Coastal Engineering Proceedings of Eleventh Conference, London, England, Vol. 1, pp. 446- 466, Sept. 1968. Penny, W. G. and Price, A. T., "The diffraction theory of sea waves by breakwaters and the shelter afforded by break- waters," Phil. Trans. Roy. Soc. (London) Sec. A, 244, pp. 236-53, March 1952. 445 Savttsky Phillips, O. M., "The scattering of gravity waves by turbulence," J. Fluid Mech., Vol. 5, part 2, pp. 177-192, 1959. Phillips, O. M., The Dynamics of the Upper Ocean, Cambridge University Press. Pierson, W. J., Jr., "The interpretation of crossed orthogonals in wave refraction problems," U.S. Army, Corps of Engineers, Beach Erosion Board, Tech. Report No. 21, January 1951. Stewart, R. W. and Grant, H. L., "Determination of the rate of dissipation of turbulent energy near the sea surface in the presence of waves," J. Geophysical Res., Vol. 67, No. 8, pp. 3177-3180. Taylor, G. I., Statistical Theory of Turbulence) /Partst(=.IW, Proc. Royal Society A; CST Ppp. 421-428. Wiegel, R. L., Oceanographical Engineering, Prentice Hall, Inc., 1964. x ok 3 oi * DISCUSSION Dr. N. Hogben ; National Physical Laboratory, Shtp Divtston Feltham, Middlesex, England Dr. Savitsky has undertaken a very interesting investigation of the effect of turbulence on waves in an oceanographic context. His main finding is that waves can be dramatically attenuated by turbulence from travelling grids. He explains this in terms of refraction and diffraction and comments on the potential use for quieting sea waves. Whilst listening to his presentation it occurred to me that his findings may also have an important bearing on the understanding of wavemaking by ships. It is a common experience that the wave system originating from the stern region of a ship tends to have much smaller amplitudes than would be predicted from the usual theories. I would be glad if Dr. Savitsky could comment on whether this suppression of wavemaking by ship sterns may be at least partly explained in terms of a refraction and diffraction analysis such as he has described in the paper, applied to the interaction between the vorticity and turbulence in the boundary layer and wake and the stern wave system. * * * % * 446 Gravity Waves and Finite Turbulent Flow Fields REPLY TO DISCUSSION Daniel Savitsky Stevens Institute of Technology Hoboken, New Jersey It may be possible for ship wakes to locally attenuate the wave generated by the afterbody. If the mechanism described in the paper is applicable, it would necessarily require that wave amplitudes be larger at some distance transverse to the ship wake. This, of course, follows from considerations of preserving wave energy. Much further study of afterbody generated waves would be necessary to determine the association of ship-wave attenuation with the mechanism described in the present paper. 447 CHARACTERISTICS OF SHIP BOUNDARY LAYERS L. Landweber Untverstty of Iowa lowa, City, JTowa I. INTRODUCTION When I accepted the invitation to lecture on ship boundary layers, my original plan was threefold: a) to review three-dimen- sional boundary-layer theory, b) to discuss the few available appli- cations of the theory to ship forms, and c) to present certain un- published results on ship boundary layers that have been reported in several theses at the University of lowa. In the course of attempting to "catch-up" on the literature on three-dimensional boundary layers, so that I could pretend to be an authority on the subject, I encountered so many excellent review articles, that it became apparent that a review-of-reviews was hardly likely to match the immortality achieved in its category by the "song-of- songs." Rather it seemed to be more useful and interesting to examine the validity and applicability to ship forms of the assump- tions of existing methods for computing three-dimensional boundary layers, and to suggest and partly to implement certain approaches which appear to be better suited to the ship problem, Some of the common assumptions of three-dimensional boundary-layer theory are the following: 1. Assumption of small cross-flow -- that the direction of flow within the boundary layer deviates by only a small angle from the direction of the streamline at the outer edge of the boundary layer. 2. Assumption of methods of calculating two-dimensional boundary layers for determining the velocity component parallel to the outer streamline, even when the small cross-flow assumption is avoided. 3. The assumption of monotonic cross flow -- that as the wall is approached from the outer streamline, and angle of deviation of the boundary-layer streamlines increases monotonically up toa certain value at a small distance from the wall, beyond which it 449 Landweber remains nearly constant. 4. The assumption that three-dimensional boundary-layer problems are best treated with equations in streamline coordinates. A stimulating article by Lighthill [1] on the fundamental significance of vorticity in a boundary layer initiated the development of a proposed method for treating ship boundary-layer problems. This will be presented in two sections; a first in which the vortex sheet on the ship hull, which generates the irrotational flow about it, is determined; a second in which the vorticity equations for a three- dimensional boundary layer, in terms of a triply orthogonal coordi- nate system, are derived. The significance of the first part is that it furnishes the initial values for the second. So there will be no "review"; but it still seems desirable to touch upon the ship boundary-layer treatments of Lin and Hall [2], Webster and Huang [3], and Uberoi [4], and the contributions in the theses of Pavamani[ 5], Chow [6], and Tzou[7]. II NATURE OF THE SHIP PROBLEM In comparison with other three-dimensional boundary-layer problems, that for the ship is much more complex because of the presence of a free surface at which the body is moving partly im- mersed. Some ship boundary-layer problems will now be described. 1. The first step in a boundary-layer calculation, the deter- mination of the irrotational flow outside the boundary layer (the outer flow) is a difficult problem. Solutions employing linearized free- surface boundary conditions and thin-ship theory furnish inadequate approximations. The development of more accurate methods of cal- culating the irrotational flow about ship forms is a current research problem [ 8]. 2. At Froude numbers sufficiently low so that the free sur- face may be treated as a rigid plane, (zero-Froude-number case), the three-dimensional flow about the double model, obtained by reflecting the immersed portion in this plane, is of considerable interest. Methods of computing the irrotational flow for this case are available [8, 9]. Calculation of the viscous drag for this case, and its ratio to the frictional resistance of a flat plate of the same length, wetted area and Reynolds number, would yield the so-called form factor of the hull form which is required in one method of predicting ship resistance on the basis of model tests Pat. 3, The three-dimensional boundary layer is very sensitive to the shape of the bow. The nature of the boundary layer near the forefoot, which determines whether or not bilge vortices will be generated, can also be studied at zero Froude number. Bows 450 Charactertsttes of Ship Boundary Layers frequently are designed with zones of reversing curvature, at which boundary-layer profiles with S-shaped cross-flows may occur, 4, At higher Froude numbers the boundary layer will lie over a hull surface area which depends upon the equilibrium trim and draft and the surface-wave profile along the hull at that Froude number. The curvature of the outer streamlines at the free surface strongly affects the cross-flow components of the boundary-layer velocity profiles [6,7], an effect which is completely ignored in boundary-layer studies at zero Froude number. 5. Near the stern the boundary layer thickness becomes of the same order of magnitude as the radii of curvature, and the methods of thin boundary-layer theory cannot be used without modi- fication. A detailed study of boundary-layer characteristics in this region is desirable in connection with the development of improved rational methods of computing the viscous drag, and the design of stern appendages from the point of view of strength and cavitation. Of course, if such a calculation could be extended into the near wake, it would be a great boon to the propeller designer, 6. The draft and trim of a ship may vary greatly, depending upon its cargo. It operates at various speeds or Froude numbers, and if model tests are involved, the effect of the scale or Reynolds number would be of interest. Since the flow pattern would vary with each of these four parameters, one may wish to calculate the boundary layer for many combinations of parametric values, III. SHIP BOUNDARY-LAYER CALCULATIONS Ship boundary layers at zero Froude number have been calcu- lated by Uberoi[4]. To determine the outer irrotational flow he introduced a distribution of n discrete sources lying within but close to the hull and determined their strengths by solving simul- taneously n linear equations, obtained by satisfying the boundary condition on the hull at n points. This source distribution was then used to calculate the streamlines. For calculating the boundary layer, the flow was treated as two-dimensional along each streamline, and the momentum thickness and shape parameter determined by an available two-dimensional semi-empirical procedure [10]. A better approximation could have been obtained with little additional effort had one of the available three-dimensional boundary-layer procedures assuming small cross- flow been used [11], since these would have taken into account the important three-dimensional property of the spreading of streamlines. Nevertheless, since the spreading of the streamlines is small except near the bow and stern, the results should furnish a useful approxi- mation. 451 Landweber Finally, to determine the viscous drag, an empirical formula relating the shape parameter H, with the outer velocity U, at the tail (designated by the subscript t) and the velocity of the uniform stream at infinity, U,, Geode ttent Won is assumed, as well as that the equations of thin boundary-layer theory may be integrated to the very tail, a dubious assumption. Since this empirical relation is unlikely to be universally valid, the fore- going procedure, which is that usually employed to compute viscous drag, emphasizes the need for additional research on the character- istics of the thick boundary layer near the stern. An approximate method for computing the boundary layer on a ship form at a nonzero Froude number has been developed and applied by Webster and Huang [3]. Guilloton's theory of ship wave resistance [12] as presented by Korvin-Kroukovsky [ 13] furnishes tables from which the outer flow can be determined along three streamlines on the hull. The boundary layer along these streamlines is then computed by a small cross-flow method employing streamline coordinates, due to Cooke [14]. This method has been applied to two Series-60 forms of 0.60 and 0.80 block coefficients, over a range of Froude and Reynolds numbers. Although the assumption of small cross-flow is basic to the method, it was nevertheless applied to estimate the locations of separation points on these streamlines on the basis of Cooke's criterion that separation occurs when the cross- flow is 90°. Smith's comparative study of five different methods of com- puting a turbulent three-dimensional boundary [11] indicates that a method which does not assume small cross-flow, and which employs a three-dimensional extension of Head's entrainment hypothesis [ 15] for the variation of the streamwise shape parameter gives better predictions of the cross-flow than methods which assume small cross- flow, and a constant value of the shape parameter. All five methods, however, yielded values of the momentum thickness in poor agree- ment with experimental results. Smith conjectures that this failure is probably due to the adoption of empirical relations for the shear stress from two-dimensional theory. These results of Smith indicate that the Webster-Huang pro- cedure for calculating separation points could be improved considerably by the adoption of the best of the five methods. None of the methods, however, can be used reliably to calculate the viscous drag. Lin and Hall [ 2] also employ streamline coordinates and the small cross-flow assumption in computing the boundary layer on a ship form. As in the method of Cooke [14], the momentum integral 452 Characteristics of Ship Boundary Layers equation in the streamwise direction becomes a differential equation for momentum thickness after assuming a power law of variation for the streamwise velocity profile and a semi-empirical relation from two-dimensional boundary-layer theory between the shear-stress coefficient and the momentum-thickness Reynolds number. An additional assumption, that the cross-flow angle varies as the square of the distance from the outer border of the boundary layer,:is intro- duced to determine the cross-flow, Finally a new auxiliary relation between the shape parameter and the momentum thickness is derived by combining the streamwise momentum and energy integral equations and introducing one more assumption, another semi-empirical rela- tion between the dissipation coefficient and the momentum thickness Reynolds number, also borrowed from two-dimensional theory. Each of the five assumptions of the method used by Lin and Hall is of doubtful validity for a ship boundary layer. Boundary- layer data on ship forms, which are discussed in subsequent sections, indicate that the cross-flow is not everywhere small, that the two- dimensional relations are not generally valid in a three-dimensional boundary layer, that a power law is not a good approximation for the streamwise velocity profiles, and that the cross-flow angle cannot obey a quadratic relation. Finally, a paper due to Gadd [16], which the author has not yet seen, should be mentioned. He determines the outer potential flow, taking wavemaking into account, and applies this to calculate the boundary layer on an equivalent body of revolution, neglecting cross-flow. In referring to this paper, Shearer and Steel [17] remark that "Gadd has recently applied a three-dimensional boundary-layer theory to the pressure distributions obtained using the Hess and Smith method, taking account of the free surface, to give friction distribu- tions which agree very well with measured values. Comparison of this theory with some of the experimental values detailed herein (in [17]) are given in... " (in[16]). IV. BOUNDARY-LAYER DATA FOR SHIP FORMS It has been indicated that the relations for the shear stress used in calculating two-dimensional boundary layers may not be valid for a three-dimensional boundary layer. In order to investigate the applicability to ship forms of these and other empirical relations that have been proposed, it would be desirable to have a set of data, in- cluding pressure distributions, mean velocity profiles for both the streamwise and cross-flow directions, and shear stresses, for some shiplike forms. Full scale boundary-layer measurements ona 210-foot ship, the USS Timmerman, have been reported by Sayre and Duerr [18]. Mean velocity profiles are given for four points along the hull, at speeds of 5, 10, 15 and 20 knots. The measured boundary-layer 453 Landweber thicknesses are in poor agreement with values computed from a for- mula for two-dimensional flow on a smooth flat plate. Although no other analysis was attempted, these data offer an opportunity to test procedures for computing a three-dimensional boundary layer, e.g., by the suggested modification of the method of Webster and Huang. Some boundary-layer measurements on a 70-meter research vessel "Meteor" [19] and a 1:30-scale double model in a wind tunnel [ 20] have been reported by Wieghardt. The full-scale measurements, taken at a point 40 per cent of the draft from the free surface and 40 meters from the bow, yielded a value of the shear stress approxi- mately equal to that for a flat plate, but a definitely lower value of the momentum thickness. The results for the boundary layer at the corresponding point on the model were consistent with the full scale measurements in spite of the neglect of free surface effects in the wind-tunnel tests. Several phenomena peculiar to ship boundary layers were displayed by the model study. One of these is the unusual shape of the boundary layer (vorticity-containing region) around the girth of a fore-ship section, showing bumps at the sides and a great increase in thickness at the keel, attributed by Wieghardt to secondary flow (i.e., large cross flow) initiated near the bow. The shear-stress coefficient at midship section was nearly constant at about Cy =,0'.,0885, but decreased to 0.0025 as the keel was approached, and then increased rapidly to 0.0039 at the keel. The momentum thickness 6 varied even more, from a mean value of @/x = 0.0013 down the sides, in- creasing to a maximum of 0.0028 as the keel is approached, then falling to 0.0018 at the keel. These results indicate that, at least near the keel, the two-dimensional shear stress formulas frequently assumed in computing three-dimensional boundary layers, are very inaccurate. Wieghardt concludes that "much more experimental knowledge about the flow in ship boundary layers, including secondary flows and trailing vortices is needed for semi-empirical calculation methods for such three-dimensional boundary layers ..." A project to obtain full-scale measurement of ship boundary layers is under way in Japan, and some resuits of this work were reported at the 12th International Towing Tank Conference in Rome [21]. The unusual shapes of certain velocity profiles astern of the parallel middle body were attributed to the presence of vortices separated from the hull. Clearly these profiles could not be repre- sented by a power law. On one ship an array of five longitudinal vortices was observed in the wake, of which one pair originated at the bow, another pair was shed astern of amidships, and the fifth was due to the propeller. A recent paper by Shearer and Steel [17] is noteworthy in that it presents the results of shear-stress and pressure surveys on two ship models at a particular Froude number. The effect of the Froude number on the shear-stress coefficient C; was found to be small except at the uppermost measurement locations along a water- line at 25 per cent of the draft from the free surface, for which the 454 Characteristtes of Shtp Boundary Layers curve of C, against longitudinal distance along the waterline undulated 180° out of phase with the wave profile. The most interesting feature of the C, curves for various waterlines is their large variation along the waterline even at depths where the free-surface effect should be negligible, in agreement with Wieghardt's results. Furthermore, the variation was found to be sensitive to the shape of the bow. This again indicates that one is not free to assume a simple formula for the shear stress in calculating a three-dimensional boundary layer. The boundary layer on an ellipsoid with axis ratios 20:4:1 and the incident flow in the direction of the longest axis was investigated in a wind tunnel by Pavamani [5]. He measured the distribution of both pressure and shear stress, the velocity profiles, as well as the flow directions in the boundary layer. With the equipment used it was not possible to probe the boundary layer in the regions of largest curvature. It was found however that the shear stress in a transverse section increased in the direction of increasing curvature. Two shear-stress formulas that are used in computing three- dimensional turbulent boundary layers are one due to Young, -0.2 c,= 4% = 0.0176 (Y) 2 74 pU and another due to Ludwieg-Tillmann, 0.678H tae C,=0.246x10° (S2) Here 7 is the shear stress at the wall, p is the mass density of the fluid, U is the velocity at the outer edge of the boundary layer, @ is the boundary-layer momentum thickness computed for the streamwise component of the velocity, v is the kinematic viscosity, and H is the shape parameter of the boundary-layer velocity profile. Although not done by Pavamani, his data can be used to compare the predictions by these formulas with his shear-stress measurements by Preston's method. Comparisons at two points in the midsection, one on the centerline and the other in the vicinity of the edge, and given in the following table. COMPARISON OF MEASURED AND COMPUTED SHEAR STRESS AT MIDSECTION OF 20:4:1 ELLIPSOID (R, = 109) Measured Young Ludwieg-Tillmann 0.00330 0.00380 0.00283 0.00466 0.00444 0.00318 at centerline near the edge 455 Landweber These results for only two points already indicate that neither of the above formulas gives good agreement, although Young's seems to be preferable. It is planned to continue the analysis of Pavamani's data with the aims of representing his shear-stress data by an alternative formula, and to compare his measurements with computed values of the boundary-layer characteristics. V. SHIP BOUNDARY-LAYER PHENOMENA At a ship's bow certain streamlines of the outer flow pass downwards along a side, turn around the bilge, and continue along the underside of the hull. Because of the large curvature at the bilges, the cross-flow angle in the boundary layer may become large and the resulting secondary flow has been observed to roll-up into a pair of so-called bilge vortices [22]. Clearly the small cross-flow assumption is not suitable for treating this phenomenon. It has been observed that these bilge vortices can be eliminated by attaching a large bulb to the bow [ 23]. A possible explanation of this effect is that the curvature reversals as an outer streamline passes from the bulb to the bow, and then around a bilge result in an S-shaped velocity profile, i.e., one in which the sign of the cross- flow angle changes in passing from the outer limit of the boundary layer to the wall. In any case, since bows are frequently designed so that streamlines would undergo changes in the sign of the curva- ture, S-shaped velocity profiles would occur, so that the assumption of monotonically varying cross-flow angle, and in particular its fre- quently assumed quadratic variation, would be improper. The surface wave profile along the hull affects the boundary layer in two ways, as has been shown by Chow[6]. Climbing from a wave trough to a crest is equivalent to passing through a region of adverse pressure gradient. If the free-surface slope is large enough and continues long enough, separation will ensue near the free sur- face. Secondly, the curvature at a surface-wave crest along the hull tends to generate a secondary flow. Chow[6] has attributed a second zone of separation at some distance beneath the free surface to this phenomenon. The conjectured mechanism of the effect of a surface wave on a boundary layer was confirmed by Tzou[7]. He simulated the free surface by a sinusoidal ceiling in a wind tunnel, and observed and photographed the flow directions in the boundary layer ofa vertical ogival strut, as indicated by an array of fine threads sup- ported at various distances from the wall. He also verified the effect by solving the Navier-Stokes equations and the equation of continuity numerically, by a combination of a finite-difference method together with the Blasius solution for a flat plate, for a simplified model of his experiment. These results indicate once again the unsuitability of the small cross-flow assumption for ship boundary layers. 456 Characteristics of Shtp Boundary Layers VI. THE COORDINATE SYSTEM A set of mutually orthogonal lines on a surface S can be selected in infinitely many ways. Such a net, together with the distance along the normal to S form a system of space coordinates which, in general, are triply orthogonal only on S. Although a non- orthogonal system of space coordinates is usually an awkward choice in formulating the Navier-Stokes equations, when these equations are simplified in accordance with the usual assumptions of thin boundary-layer theory, Squire [| 24] has shown that the boundary-layer equations are identical in form with that for a fully orthogonal system. When the third coordinate is the distance ¢ along the normal to S, the surfaces © = const. are, for obvious reasons, said to be parallel to S. It is shown in texts on differential geometry that the lines of principal curvature, and only these lines, have the property that the surface normals along them generate developable surfaces C = const. and = const., and that these, together with the parallel surfaces (€ = const., form a mutually orthogonal family. For this reason Howarth [ 25] and Landweber [ 26] employed the lines of principal curvature as surface coordinates in formulating the equations of motion. Nevertheless, according to Crabtree, et al., [27], "this is an undesirable restriction," a feeling that seems to be shared by most of the contributors to the subject of three-dimensional boundary layers. Preferred is the streamline-coordinate system, although geodesics and rectangular coordinates have also been used. Only Howarth [ 28] has adopted the lines of principal curvature for the coordinate system in his treatment of the three-dimensional boundary layer near a stagnation point. There are two good reasons for using streamline coordinates. One is that, in the cases to which they have been applied, the inviscid- flow streamlines could be readily obtained; the other is that practical, approximate methods of solving the boundary-layer equations, em- ploying techniques developed for two-dimensional boundary layers, are available for the equations in streamline coordinates. The simplest of these methods are based on the assumption of small cross-flow in the boundary layer. According to Smith[ 11], however, who applied five of these methods to compute the boundary layer on a yawed wing, none of these was found to be completely satisfactory, as has already been indicated. For the case of present interest, the boundary layer ona ship form, the first of the aforementioned reasons does not apply. Calculation of the velocity distribution and the streamlines on a ship form at the particular Froude number is a task of the same order of difficulty as that of solving the three-dimensional boundary-layer equations. For the zero-Froude-number case, methods are available for computing the potential flow [8,9]; at nonzero Froude numbers an approximate method due to Guilloton [12,13] furnishes tables for the calculation of three streamlines along a ship hull. 457 Landweber Another consideration is that the streamline pattern on a ship form is a function of four parameters, the Froude number, the Reynolds number, the trim angle and the draft-length ratio. Thus, if streamline coordinate were to be used, it would be necessary to calculate a great many coordinate systems. It appears to be more practical to select a unique coordinate system which depends only upon the geometry of hull and is independent of the above four parameters. If it sufficed to study thin boundary layers, there would be a free choice of orthogonal surface coordinates on the hull surface. But the boundary layer near the stern cannot be considered thin, and a continuation of the boundary-layer calculations into this region could not be undertaken with the equations for an orthogonal coordi- nate system unless the surface coordinates had been selected to be lines of principal curvature. VII. DETERMINATION OF LINES OF PRINCIPAL CURVATURE First suppose that the equation of the surface S is given by F(x,y5z) = 0 (1) where (x,y,Z) _a: are the rectangular Cartesian coordinates of a point P on S Let ds=idx +tjdy +kdz denote a vector element of arc along one of the lines of principal curvature, where i, j, k are unit vectors along the x,y,z axes. Then grad F=VF=iF, +jF, tkF, (2) is a vector along the normal at P and dVF = ds - VVF (3) is the change in this vector along the normal in moving an increment ‘ds from P to P' along a line of principal curvature. It can be shown [ 29] that the normals to S at P and P' intersect if and only if ds is an element of arc of a line of principal curvature. This implies that the vectors ds, VF and ds*« VVF are coplanar, and hence that ds-« VE X(ds'* VVF) =0. (4) 458 Charactertstics of Ship Boundary Layers Also the condition that ds be normalto VF is ds > VF =0 (5) Equations (4) and (5) are the differential equations of the lines of principal curvature. In terms of their components, (5) becomes F dx + F dy + F dz =0 (6) and from (4) we obtain (FF, - FF ,,)(dx) HER, - FF )(dy) H(E FY, - FF )(dz)’ t Care aa - ee a EK. - FF) dy dz 1 (EE - BP yy ze FF y - FF 2) dz dx +(FF -FF +FF_ -FF_ )dxdy=0. (7) z Xx z yy y yz x = XZ Because of the quadratic nature of (7), the simultaneous solution of (6) and (7) yields a pair of solutions for (dx, dy, dz), which can be shown to be orthogonal. Thus, from an initial point P, one can calculate the lines of principal curvature in step-by-step fashion. If the equation of the surface is given in the form y = f(x,z) (8) where x is directed from bow to stern, z is positive upwards, and the plane y = 0 is the vertical plane of symmetry, then (6) and (7) can be combined into the differential equation of the projection of the lines of principal curvature on the plane of symmetry, 2 [ pqt - s(1 +q?)] (<2) +[ (1 +p%t - (1 +¢2)r] @ +[(1+p?)s - pqr] = 0 where t=f (10) and the principal radii of curvature p are given by 459 Landweber (rt - s*)p2 + K[t(1 +p) + r(1 +q2) - 2pqs]p + K* = 0 (11) where ali K =[ t+ p® #ge] Other relations between the geometric parameters of the orthogonal coordinate system based on the lines of principal curva- ture are given in [ 26]. VIII. EQUATIONS OF VORTICITY IN A BOUNDARY LAYER Lighthill [1] makes a convincing case for the primary im- portance of vorticity in a boundary layer. If the vorticity is known, the velocity field can be calculated by the Biot-Savart law. Secondly, vorticity is diffused and convected more gradually than other fluid properties and hence is more readily determinable numerically. From the mathematical point of view, Lighthill implies that it is easier to solve the diffusion equation for vorticity than the boundary- layer momentum equations governed by an outer irrotational flow. Sherman [ 30] has also been impressed by Lighthill's views, and has contributed a more mathematical discussion of "sources of vorticity." Neither he nor Lighthill, however, have formulated the vorticity equations for a three-dimensional boundary layer. This will now be undertaken. The Navier-Stokes equations for an incompressible fluid may be written in the vector form Wy xGtgrad(4¥-v+2 + gz) =-v cud (12) at 2 p where Vv is the velocity at a point ot the fluid, w = curl v is the vorticity, t denotes time, p is the pressure, p the mass density, g the acceleration of gravity, z is a vertical coordinate, positive upwards, and v is the kinematic viscosity. An immediate conse- quence of (12), obtained by applying the nonslip condition at the wall surface S, is grad ‘S + gz ) =-vecurlw on § (13) which relates the vorticity at the wall to the pressure gradients of the flow outside the boundary layer. By taking the curl of the mem- bers of Eq. (12) we obtain the Helmholtz vorticity diffusion equation 460 Characteristics of Ship Boundary Layers dw - - — 3 7 curl (v Xw) - v curl curl w (14) a form from which the pressure gradient has been eliminated. The velocity, however, still appears. In rectangular coordinates we would have curl curl w= VX(VX0)=VV° o- Voz - Vo since V°* w = 0, and (14) could be written in the form af _¢ Qu,,9u,, du_ a ab _ ae at — —— + == SSe i == —— at ox | dy dz ox dy oz Ox an _¢ 4,247 a¥_ yan yan _ any on Bee ha by ie oe ay (eye a) 2 06 _~ Iw Ow Ow | OO eo OG ag Be ae | by Set bs Oy Nios. oe We wish to obtain the equivalent set of equations for a three-dimen- sional boundary layer, employing a triply orthogonal coordinate system (2,8, y), where hda and h,dB are elements of arc along the lines of principal curvature on S, and y is distance along the nor- mal, with y=0 on S. TetG.; Cos €, denote unit vectors in the directions of in- creasing @, 6, y. Put veeutevtew, w=eé ten tet. | 2 3 | From w= curl v we have in this system of coordinates, with h, =a, 1 fow _ O(hev) 1 Ow 8v_ v dhe 6 h, Lop Oy ] h, 068 Oy h, dy 178 dw]_ du 1 dw, u Oh =e —— = — SS — ieee peel eset! IF 1= t Lay 1) - Gal By” h 8a” h, By | Mat ihe p dv 1 du = aloe th) 3p |= 5 oa” he OB 461 Landweber Put h (2, 6,.0)%= H, on AC 6,0) H,; let K; K_ be the curvatures in the plane tangent to °S of the arcs @= const. and 6 = const.; let K,) h,= H(i +Ky), h,= H(t + Ky) whence 8h __Ky 4 Bhp _ Ka 1 h, oy ey he oy fy and, also from [ 26] Hence the expressions for &,n,G become potest (sOieroiy Ove Kav h, 0p dy ft Ky we see that This indicates that the vorticity lines and the skin-friction lines S form an orthogonal net, as is well known. 462 K, be the principal curvatures of the surface S corresponding to the directions of increasing @ and 8. Then we have [ 26] (16) (17) (18) (19) (20) (21) (22) (23) (24) on Charactertsttces of Ship Boundary Layers Since in a boundary layer w is small in comparison with u and v, and derivatives with respect to @ and B are small in com- parison with derivatives with respect to y, we are justified in omitting the derivatives with respect to @ and f in (19), (20) and (21). Ina thick boundary layer it may be necessary to retain the terms K,y and Ky in the denominators of (19), (20), and (21), but we shall neglect these terms in the present treatment. Thus the expressions for the vorticity components in a boundary layer become _ Ov E =- By = Kv (25) du =- — + n ay K, (26) ¢=Kv-Kwu. (27) Near the wallthe y derivatives are dominant so that the expression for the vorticity remains that given by (22). Farther into the boundary layer, however, the terms Kv, K,u, K.u, and K,v may become appreciable when the curvatures are large, as at the bilges of a ship form. When € and n are known, the corresponding values of u and v, obtained by integrating the differential equations (25) and (26), are given by “Kix?” K.y wee ol ne > dy (28) 0 EY" se ve-e 2 te” ay (29) ie) somewhat more simply than by the Biot-Savart law. We can now obtain the components of curl w in the boundary layer by replacing u, v, w by §,7, © inthe right members of (25), (26), and (27). Thus we obtain curlw-+ é€, = - ri K,n (30) ae ot9 curl we, 5 = + K§ (35) Gurl wsttege uk 1.5 K§ (32) Landweber and similarly, from (19), (20), and (21), curl (v X w) = ele op (un > WE) - wy (w& - ub) - K,(w6 - ut) | 2 +e, e (vo, 0) cee Z (un - v§) + K(v6 - om ae [= aa (w§ - uf) ee (vG - wn) + K\(w6 - ul) - Klvg - wn) | (33) For the components of curl curl w inthe boundary layer we obtain, neglecting small terms, 2 Pe ee 0g curl curl w « e, =e By2 = (K, F K,) dy (34) curl curl w- e -.2'n_ K ie (35) 2 dy2 3 4° Oy aan ese ag Sioa curl curl w ae = K By aie Ko By a K,K,§ ar K,K,n ° (36) Substituting these results into (14) yields the vorticity equations 2 ee =5 a (un-vé) _ Fy (we-ut) a K,(w§-ub) + [S54 (KtK)) = | (37) 2 an _ 9 _ ee ee) = _ an an at zo OY (vo wn) h, oa (un vé) a K{vS wn) Ee + (K, +K,) 5 | (38) Be = a (wE-Ue) ~ He gp (ve-wn) + K(wE ub) - Kvo-wn). (39) Here u and v are given in terms of the vorticity by (28) and (29); w can then be obtained from the continuity equation. In order to start the calculation, conditions at time t = 0 are required. This may be taken to be the vortex sheet for irro- tational flow about the hull in a uniform stream, since this gives the initial vorticity distribution when the body is impulsively accelerated from rest to its constant speed. A procedure for determining this vortex sheet is developed in the following section. 464 Charactertsttes of Ship Boundary Layers IX. INTEGRAL EQUATION FOR A VORTEX SHEET FOR IRROTA- TIONAL FLOW ABOUT A THREE-DIMENSIONAL FORM A three-dimensional form bounded by a surface S is im- mersed ina uniform stream of velocity U inthe positive x-direc- tion, of unit vector i. We shall suppose that the fluid is inviscid and incompressible. Let us assume that the disturbance of the flow due to the body may be represented by a vortex sheet of strength ‘Y= yo where o isa unit vector tangent to the surface S such that the fluid within the body is at rest. In crossing S in the direction of its outward normal, designated by the unit vector n, there is a discontinuity in the tan- gential component of the velocity of the fluid, of magnitude y, in the direction with unit vector B= ox ne (40) By continuity, since the fluid on_the interior side of S is at rest, the velocity components inthe o and n directions at the exterior side of S must also vanish, and hence the velocity at the exterior side of S is given by _ u=yoXn=y xu. (41) Since, a priori, the mutually orthogonal directions of the streamlines, %S, and of the vortex lines, o, are unknown, it is necessary to introduce a set of orthogonal, curvilinear coordinate lines,on S, € = const. and 1 = const. _Denote unit vectors in the directions of increasing § and » by e, and e,, with sense such that e Xe,=n. Put Vee +e u=ze +e VS Ghee. We Ute ve (42) Then, by (41), we have w= Yor v=-y¥,- (43) An integral equation for the vorticity vector ‘y can be derived from the condition that the contributions to the velocity on the interior side of a point P of S must sumto zero. This gives 2 2 —_— Yp% mp = U (44) — 1 ( YaX Vp (s+) a8, + s 1 4m Js PQ in which the integral, obtained from the Biot-Savart law, represents 465 Landweber the velocity at P induced by vortex elements at points Q of S, and, by (41), the negative of the second term is the contribution from the local vortex element y,. Here r,, is the length of the chord joining the points P and Q of S and Vp denotes the gradient with respect to the coordinates of P. The integral in (44) is not suitable for numerical evaluation in the given form because rpg which goes to zero as Q approaches P, occurs in the denominator of the integrand. This singularity can be eliminated, however, in the following manner. First take the cross-product of (44) by iis to obtain 1 - 1 = fo ie eee RA cae) Xn, dS, +5 (y,Xn,) Xn, = Ui Xn, (45) Since, in the neighborhood of P, both ap and Volt /tpq = Tep/ Tay lie very nearly in the tangent plane at Q, their cross-product is very nearly parallel to no, and hence the integrand of (45) is proportional to the angle between n, and n, or r,o/R, where R is the radius of curvature of the"arc,of 75 subtended by the chord PQ.” Thus the order of the singularity of the integrand of (45) has been reduced to that of 1/r,.. In order to eliminate this singularity, consider the relation be 1 nasty 1 are { [ 3% Vo (s+) ] xB, = Ye + HMa(st) - We * Yo(=*) Q PQ he 1 SS yen Re a7 (4) (46) Bel! Nea since Ae ° a. = 0. Also we may write = 1 — 1 - 4 np * Vo(=—) dSg= \ | p+ Yo(=— a era) dS, + 2m 5 TPQ S PO PQ (47) since f, Hg+ Vg (t/rpQ dS, is the flux through S due to a sink of th at? i. Applying (46) and (47), and noting that unit streng 1 4 eine vag) Pt Q lpg we obtain from (45), 466 Characteristics of Ship Boundary Layers The singularity has been removed from the first integral in (48) because a factor proportional to tpg is contained in — Yo '= Ve Zep (Ver The second integrand is also singularity-free at P since = { no° © | ° PS Pe 2 —_— oe Vp ( ) am) REA and n ° V. (=—) a é Q P Teg 2Rreo Thus we see that (n, + n)) ° V50eyr..) is regular at P. A procedure for obtaining a numerical solution of the integral equation (48) consists of replacing the integrals by quadrature for- mulas to obtain sets of linear equations. Expressing y in terms of e, and e, as in (42), for each of n points P the quadrature for- mula yields a linear equation in the unknown values of u and v at n points Q. This gives n vector equations or, resolving in the directions e, and €, at P, 2n scalar equations in 2n unknowns. When @Q coincides with P, the integrand is set equal to zero. hs Taking the scalar products of the members of (48) by e,, and IP €op, we obtain the pair of scalar equations er as rl _ es AF { CYo'> Vp) x Vp (=) - e,, dS, t a | (np tng) °V, (=) dSg PQ “PQ =4nUi +e, (49) ie cs he Nc, ae { Wo- We XV (z-) *® ds tye) & +h “Veli -) ds a Gaur P=. ara Sea ke ee oe Peo Q = 4nUi + eop- (50) In applying these equations, one needs to express Yg and ng in 467 Landweber terms of the unit vectors aes: Cre and Tips This requires that the direction cosines of C19? Cogs and No relative to C\p» Cap, and np be calculated for each combination of P and Q; i.e., $n(nt1) tables of direction cosines. Furthermore, if (x,, y,, z,) and (Xg, y Yq: 2 Q) are the coordinates of P and Q ina rectangular Cartecian coordinate system with unit vectors ie Vs k, then Ean = I(x, rica as ay, Set Ke eee) and the expression of Vp(1/rpqd = Tpo/Tpg in terms of €,p, €pp and np requires that n tables of direction cosines of the latter set of vectors with, respect;to.the .i,;j, k system also be obtained. These direction cosines and the components of rpg can be readily deter- mined if the equations of the surface are given in the form <= (Ee .), y= GE sn); z = H(€,n). (51) A procedure for solving (49) and (50) by iteration is suggested by the following modifications: = = { = _ Hl if (Yo ~ Yo) x Vp (+) ip dSq . Up nel i (np x Ny) T Vp (=) dSg s PQ ) PQ = 4nrUi e ip (52) { (a - Yen X Yo( s+) * Sep 45g Pein a, toa: Ye (=e ) a8, S PQ PQ = 4nUi + €5p. (53) For ship forms the foregoing procedure can be used to deter- mine the velocity and vorticity distributions and the streamlines and vortex lines on a double ship model at zero Froude number, At non- zero Froude numbers, a similar pair of integral equations can be derived, but these would be considerably more complicated because of the contributions of the wave potential to the velocity on the body surface S. X. CONCLUSIONS It has been indicated, on the basis of the limited available boundary-layer data on actual ships and ship models, that the various integral methods, with or without the small cross-flow assumption, and employing streamline coordinates, are of dubious applicability to ship forms because three additional assumptions concerning the 468 Characteristics of Ship Boundary Layers velocity profile, the cross-flow angle, and the shear-stress coefficient are not in accord with these data. If the energy integral equation is also used to obtain an auxiliary equation, then an additional assump- tion concerning the dissipation coefficient comes into question. Two significant ship boundary-layer phenomena, the generation of secondary flows and possibly of vortices at the bilges near the bow and at a wave crest along the hull, indicate that cross-flow angles may become large, so that the small cross-flow assumption would be inappropriate. The possibility that the cross-flow may change in sense and that the velocity profiles may become S-shaped both at the bow and along the wave profile on the hull must also be taken into account. Lines of principal curvature are recommended as the basis of the orthogonal coordinate system for treating ship boundary layers because, in contrast with alternative choices, this system remains orthogonal even in the thick boundary layer at the stern, and because, unlike the streamline coordinates, the former system does not change as the draft, trim, and the Froude and Reynolds numbers are varied. For this reason, equations for determining the lines of principal curvature have been included. Since integral methods seem to be wedded to the use of stream- line coordinates, the recommendation that these be replaced by the lines of principal curvature implies that a differential method must be adopted. One such method, based on the work of Bradshaw, Ferriss and Atwell [31] for a two-dimensional boundary layer, has been extended to the case of a three-dimensional surface by Nash [ 32]. An alternative approach based on determining the vorticity in the boundary layer, strongly promoted by Lighthill [1] , motivated the derivations of the vorticity equations in principal-curvature coordi- nates and the integral equations of a vortex sheet for irrotational flow about a three-dimensional form. Considerable further develop- ment is required for application of these vorticity equations to a tur- bulent three-dimensional boundary layer. Lastly it should be remarked that presently we cannot deter- mine the outer flow about a ship form with sufficient accuracy for reliable boundary-layer calculations due to a combination of errors due to linearization of the free-surface boundary conditions, approxi- mate satisfaction of the hull boundary condition, and the effects of viscosity on the wave making. In comparison with the outer-flow approximation for the flow about a body without a free surface, the effects of viscosity are experienced much farther upstream along the body because of the phenomenon of interference between waves generated near the bow and stern. Because of the strong interaction between the outer flow and that in the boundary layer and wake, it appears to be necessary to develop an iteration procedure, alter- nating between these regions, which hopefully would converge toa solution for the flow about a ship form. 469 Landweber ACKNOWLEDGMENT This study was supported by the Office of Naval Research, under contract Nonr i611-(07). REFERENCES [41] Lighthill, M. J., Chapter II of Laminar Boundary Layers, editor L. Rosenhead, Oxford, Clarendon Press, 1963. [2] Lin, J. D. and Hall, R. S., "A Study of the Flow Past a Ship- Like Body," Univ. of Conn., Civil Engineering Depart- ment, Report No. CE 66-7, November 1966. [3] Webster, W. C. and Huang, T. T., "Study of the Boundary Layer on Ship Forms," Hydronautics, Inc., Tech. Report 608-1. [4] Uberoi, S. B. S., "Viscous Resistance of Ships and Ship Models," Hydro-Og Aerodynamisk Laboratorium Report No. Hy-13, September 1969. [5] Pavamani, F. S. A., "Three-Dimensional Turbulent Boundary Layer," M.S. Thesis, The Univ. of lowa, August 1960. [6] Chow, S.-K., "Free-Surface Effects on Boundary-Layer Separation on Vertical Struts," Ph.D. Dissertation, The Univ. of Iowa, June 1967. [7] Tzou, T.-S., "Secondary Flow Near a Simulated Free Surface," M.S. Thesis, The Univ. of Iowa, June 1966. [8] Landweber, L. and Macagno, M., "Irrotational Flow About Ship Forms," The Univ. of Iowa, IIHR Report No. 123, December 1969. [9] Hess, J. L. and Smith, A. M. O., "Calculation of Potential Flow About Arbitrary Bodies," Progress in Aeronautical Sciences, Vol. 8, Pergamon Press, New York, 1966. [10] Thompson, B. G., "A Critical Review of Existing Methods of Calculating the Turbulent Boundary Layer," A.R.C. R. & M., August 1964. [11] Smith, P. D., "Calculation Methods for Three-Dimensional Turbulent Boundary Layers," A.R.C. R. & M. Now 3523, December 1966. 470 Characteristics of Shtp Boundary Layers [12] Guilloton, R., "Potential Theory of Wave Resistance of Ships with Tables for its Calculation," Trans, Soc. Naval Arch. & Marine Engrs., Vol. 59, 1951, [13] Korvin-Kroukovsky, B. V., and Jacobs, W. R., "Calculation of the Wave Profile and Wave Making Resistance of Ships of Normal Commercial Form by Guilloton's Method and Comparison with Experimental Data," Soc. of Naval Arch. & Marine Engrs., Tech. & Res. Bulletin No. 1-16, December 1954. [14] Cooke, J. C., "A Calculation Method for Three-Dimensional Turbulent Boundary Layers," A.R.C. Re. & M. No. 3199, October 1958. [15] Head, M. R., "Entrainment in the Turbulent Boundary Layer," A.R.C. R. & M. No. 3152, September 1958, [16] Gadd, G. E., "The Approximate Calculation of Turbulent Boundary Layer Development on Ship Hulls," Paper W5 (1970) published by RINA for written discussion. [17] Shearer, J. R. and Steel, B. N., "Some Aspects of the Re- sistance of Full Ship Forms," Paper W4 (1970) published by RINA for written discussion, [18] Sayre, C. L. Jr., and Duerr, R. L., "Boundary-Layer Investigation of USS Timmerman," David Taylor Model Basin, Hydromechanics Laboratory R. & De Report 1170, August 1960. [19] Wieghardt, K., "Boundary Layer Tests on the Meteor," Jahrbuch der Schiffbautechnischen Gesellschaft, 1968. [20] Wieghardt, K., "Boundary Layer Measurements on a Double Model," Proc. 12th International Towing Tank Conference, Rome 1969. [21] Executive Committee for the Project "Measurements of Boundary Layers of Ships," Proc. 12th International Towing Tank Conference, Rome 1969. [22] Tatinclaux, J. C., "Experimental Investigation of the Drag Induced by Bilge Vortices," Schiffstechnik, Bd. 17, May 1970. [23] Takahei, T., "Investigations on the Flow Around the Entrances of Full Hull Forms," Proc, 11th International Towing Tank Conference, Tokyo 1966. 471 [ 24] [ 25] [ 26] [ 27] [ 28] [ 29] [ 30] [31] [ 32] Landweber Squire, L. C., "The Three-Dimensional Boundary-Layer Equations and Some Power Series Solutions," A.R.C. RR. &'M. 3006, 1957. Howarth, L., "The Boundary Layer in Three Dimensional Flow -- Part I, Derivation of the Equations for Flow Along a General Curved Surface," Phil. Mag., Ser. 7, Vol. 42, 1951. Landweber, L., "Appendix A. Equations in Curvilinear Ortho- gonal Coordinates," Advanced Mechanics of Fluids, Edited by H. Rouse, John Wiley & Sons, New York, 1959. Crabtree, L. F., Kiichemann, D., and Sowerby, L., "Three Dimensional Boundary Layers," Chapter VIII, p. 415 of Laminar Boundary Layers, Edited by L. Rosenhead, Oxford University Press, 1963. Howarth, L., "The Boundary Layer in Three Dimensional Flow -- Part IL, The Flow Near a Stagnation Point," Phil. Mag. Ser. 7, Vol. 42, 1951. Smith, C., An Elementary Treatise on Solid Geometry, Macmillan & Company, London, 1891. Sherman, F,. S., "Introduction to Three-Dimensional Boundary Layers," Rand Corporation Memorandum RM-4843-PR, April 1968. Bradshaw, P., Ferriss, D. H., and Atwell, N. P., "Calcula- tion of Boundary Layer Development Using the Turbulent Energy Equation," J. Fluid Mech., Vol. 28, 1967. Nash, J. F., "The Calculation of Three-Dimensional Turbulent Boundary Layers in Incompressible Flow," J. Fluid Mech. Vol. 37, Part 4, 1969. 472 Charactertstics of Ship Boundary Layers DISCUSSION Dr..N. Hogben National Physical Laboratory, Shtp Diviston Feltham, Middlesex, England This paper performs a valuable service in laying the founda- tions for a new method of calculating ship boundary layer properties in terms of the vorticity field. A verdict on the merit of this ap- proach must await the results of actual computations and comparison with experiment. The purpose of this contribution is to supplement the review of experimental data given in the paper by drawing atten- tion to work which I reported in 1964 (Ref. (*)). It comprised boundary layer explorations covering the surface of a model with mathematical form as defined by Wigley (Ref. (**)) having parabolic waterlines and section shapes. Measurements were made at 5 speeds in the range 0.16 < F,< 0.32 and it is of interest that features noted by Wieghardt as cited in the present paper were also observed. In particular considerable stretching of the boundary layer below the keel, likewise attributed to secondary flow effects, and bumps on the velocity profiles attributed to trailing vortices, were found. REFERENCES (*) Hogben, N., "Record of a boundary layer exploration ona mathematical ship model," Ship Division NPL Report No. 52, July 1964. (**) Wigley, W. C. S., "Calculated and measured wave resistance of a series of forms defined algebraically," Trans. R.I.N.A., 1942. 473 Landweber DISCUSSION H. Lackenby The Brittsh Ship Research Association Northumberland, England I certainly agree with Professor Landweber's plea for a more detailed study of the relatively thick boundary layers in way of ship sterns especially in very full tanker forms. In Ref. 17 (Shearer and Steele) it is apparent that some of the models of the full tanker forms considered were suffering from gross separation at the stern. There is little doubt that this accounts for the viscous pressure resistance being as high as 30% of the total. Incidentally, in this reference the corresponding wave making re- sistance or gravitational component was stated to be only 3 to 5% of the total. Reference is made in Professor Landweber's paper to con- ditions which bring about separation but not, as far as I can ascer- tain, to methods of calculating the shear stress after separation and the added pressure resistance which this brings about. I would be glad if the Author would care to comment on the development of calculation methods for these conditions. I would also mention that in some of the latest full tanker forms there is little doubt that separation is taking place on these ships at sea and the situation is having to be accepted. The viscous component of ship resistance has always been an important one but owing to the developments in tanker forms it is becoming even more important than it was in this class of ship. Professor Landweber's paper is, therefore, of particular interest and importance at this time and I look forward to further develop- ments in his approach to the problem. 474 Characteristics of Ship Boundary Layers REPLY TO DISCUSSION L. Landweber Untverstty of Iowa Ttowa City, Lowa Mr. Lackenby emphasizes the importance of boundary-layer studies on tanker forms. Since the resistance of a tanker is mainly viscous, and the power-wasting phenomena of bilge-vortex formation and stern separation are viscous in origin, it is clear that such studies are needed, if only to develop designs which avoid their occurrence. For the particular tanker form to which Mr. Lackenby refers, the wavemaking resistance was stated to be only 3 to 5 per cent of the total. Our experience has been that the wave resistance of a tanker model may be about 10 per cent of the total, and that ifa proper bow bulb is fitted, it may be reduced to about 6 per cent. Since others have found that bulbs on tanker models reduce viscous, rather than wave resistance, this indicates that the problem remains to be resolved. I would like to thank Dr. Hogben for reminding us of his important 1964 boundary-layer paper, one of the very few available studies of ship boundary layers. Explorations of this kind on other ship forms are urgently needed to guide the development of methods of computing ship boundary layers. 475 ~ | it Sayed yunbr dh peag ho ont rebeesenendd woreda aA 2S eh Mahl, mn “ Sete y vein | li. OP SATO typ ene tiga “RwWol 422 sot . Ss : ; f aT) on Ha ar i “u ‘er an eryete yt abi! Lo eoda boii sdy c® seni ‘ha watt ad * porate Hidihe: © | odes? oe Fo She eee ea! 25518" mt ey] ian nom tbe BOE RET | ; Kip PT Cif « seild 20 SPSITIC nerng a hides rey ow oO ¥ itd te he voogiy 3 fon 4 ae cho lover of : hiiehfo® 2 eG Otnlvy See hi idl gay ' rr has o PY oe Pett: 2. 0% a Aoi é wot wt oh fay" Mo yiaé 3/7 bebe eed ; i ayy? < 4 _ of i t that fia f ie | 7 Ot ars 7 ¥ a ar ee AIT t ~ seevi SI Fohos Ne? LAM ASIAw dt neh Aho st CSS BR TKG" Sah TO Ois> TSC OWE yh wd SP heres Ge Ssh tirleot gihae & To-sonesaieo: evaw adi jadi nosd ii so asizsaKe 200 AU Jan? Bas igior al ta Spe cag 7): to ce oS yest lebo 84 dep Piteds Ul hbavbes Sei Ver ic JhSECY'S! ape Ape peUGuuty SouDsY BISors Mis so silo? ted bata) ‘sve ri bit eft sr apeere om ait! e538 ot bn ejitt sank ei eds SERGI | ee . 1 OS ti i. iy & Py ay hav | ty ‘ i ‘\Vamreeape Le RAE TO aki RribA Nes ot wedGbth GF Ina dh OY oath ee ies v 2 WO YT se wit Uy ‘site WA BRS TS yal> iB) rit Eels awry “wteete tte Biehl elti Ee sxaAVarolge® snow at es “Fh hd ee idem f “Se FOURS eel odd ablisewe of bubese vVononty ope ee Aroves + tsbauod cide gaitoge oh ol isaten’ ehh & one may « RYU Ls ’ Like @ ni tanks > tg rr r Co lai Lae ed sia | rie : As Class FoOteaee? bac é Gi va rticcler Mn ‘ ee Seuss tyres: at trial 5 Cite, Ma faye t3 further de PHCRLA THe Oy nar STUDY OF THE RESPONSE OF A VIBRATING PLATE IMMERSED IN A FLUID L. Maestrello WASA Langley Research Center Hampton, Virgtnta and Peele wo inden European SpaceOperattons Center Darmstadt, Germany I. INTRODUCTION A large aircraft in supersonic flight undergoes large variations in flow field over its surface. This paper is concerned with studying the response of a structure excited by convected turbulence at nearly zero pressure gradient and by shock-boundary layer interaction, with the inclusion of the coupling due to the acoustic field on each side of a panel. Shock waves on thin-walled structures can impose severe loading problems, the most common of which is the self- induced oscillation which is generated by an oscillating shock. The shock wave can easily couple with the forcing frequency present in the environment, including panel resonances. From interior noise point of view, the upper region of the airplane fuselage is considered the principal noise radiator. The aerodynamics in this region are known from the Prandtl- Mayer relation, and further downstream by shock-boundary interaction. In addition, the fuselage skin experiences traveling shock waves which run up and down the skin during the acceleration period, which might last twenty minutes for a Mach 3 airplane. In supersonic flight, the vibration of the surface is influenced by the back pressure resulting from the radiation of sound on both sides of the surface, so that, the surface motion and radiation are coupled phenomena. The interior noise level is determined by skin panel vibrations. For radiation below the critical frequency, the major source of sound arises from the interaction of the bending wave with the discontinuity of the boundary. Above the critical frequency, 477 Maestrello and Ltnden the action of discontinuities like tear stoppers, etc., have little effect on altering sound pressure level, since the sound radiation by the panel is in the form of Mach wave radiation. The experiment described in this paper indicates that some simplifications in the model can be made, viz. (1) that there is no significant interaction between the plate and the aerodynamic forces on the plate; and (2) that the panel displacement is small in compari- son to its thickness so that thin plate theory may be used. The plate is, however, acoustically coupled to the external flow field and the internal cavity. Lyamshev [1968] has solved a similar problem for a com- plex structure. Dowell [1969] computed the transient, non-linear response of a simply supported plate coupled to an external flow field and a cavity. Dzygadlo [1967] presented a linear analysis allowing mutual interaction between the plate and the external flow. Fahy and Pretlove [1967] have computed a first order approximation to the acoustic coupling of a flexible duct wall to the flow field through the duct. Maidanek [ 1966] considers an infinite, orthotropic plate coupled acoustically to an external flow field. Numerous other in- vestigations have been reported on acoustically coupled structures with varying degrees of approximation, Irgens and Brand [ 1968], White and Cottis [| 1968] , Strawderman [1967], Creighton [ 1970], Ffowcs- Williams [1966], Crighton and Ffowcs- Williams [1969], Peek [1969], Feit [1966], Lapin [1967], Pal'tov and Pupyrev 1967]. II. MEASUREMENTS a) The Experimental Arrangement The flow investigated was the sidewall boundary layer of the Jet Propulsion Laboratory 20-inch supersonic wind tunnel; the shock was induced by a 30° wedge mounted outside the boundary layer, off-center and on the same side that the measurements were made. This was done to offset the position of the reflected shock from the opposite wall. The position of the shock was determined by observing the displacement of a line of tufts, and by a static pressure survey. For zero pressure gradient, detail of flow field and panel response has been previously reported by Maestrello [ 1968]. The experiment was arranged to perform three basic measure- ments: mean velocity profile ahead of the shock with static pressure distribution across the shock, wall pressure fluctuations and measure- ment of displacement response of a simple panel structure. The titanium test panel measure 12 X 6 X 0.062 inches and was brazed on all four sides of a 3/4 inch X 3/4 inch titanium frame. The brazing Uap isuportt less % TI-6AL-4V Titanium alloy containing 6% aluminum, 4% vanadium, 90% titanium. 478 Response of a Vibrating Plate in a Fluid was intended to simulate the clamped edge condition. The panel formed most of one wall of a rigid cavity measuring 14 X 8 X 6.6 inches. The other surface of the panel was exposed to the flow field. The pressure differential across the panel was variable. The experi- ment was conducted at two pressure differentials, viz. 0.06 and 14 psi; the latter corresponds to the actual differential between wind tunnel pressure and local ambient. The side wall of the tunnel was modified to accommodate two identical, rigid, steel plates, which supported the necessary instrumentation. One plate contained an array of holes in which pressure transducers were mounted. The pressure transducers were mounted on the center-line of the tunnel in the streamwise direction at the same locations where the mean static pressure measurements were made. Two types of pressure transducers were used; one, the conventional lead zirconate titanate type made by Atlantic Research, the other a capacitance type made by Photocon Corporation with sensitive diameters of 0.06 inch and 0.09 inch respectively. Correction due to finite size transducers was made adopting the Corcos [1963] approach. The panel displacement was measured with Photocon capacitance, displacement transducers mounted on brackets which could slide along a bar and could be set precisely by means of a screw mechanism. The output of both pressure transducers and displacement transducer were recorded on Ampex FR-1800H 14-channel tape, recorded in the FM mode. Four channels were used for simultane- ously recording data for correlation measurements. The maximum dynamic range was obtained by splitting each data channel into two tape tracks through phase matched filters to separate the lower and higher frequencies. b) The Wall Pressure Field Measurements indicated that the flow field in front of the shock closely approximated the properties of equilibrium of an adi- abatic flat-plate boundary layer | Maestrello 1968]. The flow in front of the shock has the following characteristics: Mach number Me, = 3.03, free stream velocity Ue = 2,100 ft/sec, total tempera- ture T, = 5679R, boundary layer thickness 6 = 1.37 inch, bound- ary layer displacement thickness 5* = 0.445 inch, momentym thick- ness = 0.083 inch, Reynolds number R = Ue5/U = 4,87 X 10", skin friction coefficient Cr = 1.27 X10~, and C; Rg =39.8 Coles param- eter [ Coles 1964]. The pressure ratio across the shock is a well defined function of Mach number, for a 15° half-cone angle, the pressure ratio is approximately 8.5. Experimental results show, however, that this ratio is considerably smaller (Ap = 2.3). It is postulated that inter- action with an expansion wave originating at the base of the wedge is responsible for lowering the pressure differential and producing an 479 Maestrello and Linden Psd Ps 0 1,0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 x/§ Fig. 1. Static Pressure Fluctuations and Mean Pressure Distribution Downstream of the Shock effective decay downstream, Fig. 1. In the present case, the wedge angle induces a shock in the boundary layer large enough to cause a separation: farther downstream, the flow becomes reattached and goes back to the flat plate condition. This transition takes place within a few boundary layer thicknesses. Downstream of the shock, the ratio of the mean pressure distribution p,g/p, and the ratio of the rms pressure fluctuation Piq/P, vary with a consistent relationship and both reach a maximum at x/5= 2.3, where subscripts s and sd, mean upstream and downstream of the shock, respectively, Fig. 1. Beyond x/6= 6 the effect of the shock on the static pressure vanishes. Kistler [1963] indicates a similar behavior between mean and fluctuating pressure in the spearated region ahead of a forward-facing step at the same Mach number and upstream Reynolds number. The differ- ences in the flow geometry only alter the magnitude of the pressure, in that the ratio of the mean pressure to the fluctuating pressure Pq/, Pa = 14 in the present experiment while Kistler found that Py/ Pig * 32. The normalized power spectral density measured upstream and downstream of the shock are shown in Fig. 2. The spectra are normalized by requiring Jno) d=i1 in order to demonstrate the deviation from the zero pressure gradient ease. For the spectra just downstream of the shock more energy is concentrated ina narrow low frequency band while further downstream at x/6= 4, the energy is distributed over a much broader bandwidth and approaches the shape and level of the spectrum taken upstream of 480 Response of a Vtbrating Plate in a Fluid x 1.0 OP. \ 1 (w)U,/5 7 <-— Zero Pressure Gradient Spectrum 0.01 0.10 1.0 10 100 Fig. 2.. Power Spectral Density of the Wall Pressure Fluctuation the shock. The normalized power spectral density found upstream of the shock corresponds to the zero pressure gradient, and peaks at w5/Ue™= 2 while downstream the spectral density is modified in the region below the peak. It is significant that by altering the local flow conditions, only the low frequency ends of the spectra are appreciably affected. It is noticed that the pressure flucutation measurements at x/5= 0 where the shock impinges show a notice- able deviation from the general pattern in the higher frequencies. This is attributed to an intermittant signal superimposed on the regular pressure signal as seen on the oscilloscope. It is possibly due to the characteristic fanning of the shock as it goes through the boundary layer. Measurements of the cross-correlation are shown in Fig. 3. The cross-correlation characteristics are a function of position downstream of the shock. The cross-correlation between positions x/§ = 0.33 and x/6 = 3.80, the farthest apart, has characteristics similar to those found at zero pressure gradient boundary layer in that the ratio between the convection velocity and the freestream velocity Uc/Ue = 0.72 and that the correlation between those two points is still significant. The cross-correlation of the shortest distance between x/6 = 0.33 and x/&=0.75, shows that the con- 481 Maestrello and Linden Fig. 3. Longitudinal Cross-Correlation of the Wall Pressure vection velocity is very low Uc/Ue = 0.13 and the correlation is very weak. The correlation between x /6 = 0555 and x/5 = 0:25, where x/5 = 2.25 corresponds to the maximum static pressure ratio is negative. The shock induces the boundary layer to separate and the recirculation within the separation region permits the sign of the pressure to change. Kistler argued that the fluctuating pressure in the separated region arises from the combined action of the turbu- lent shear layer and the recirculating flow. The picture, however, is not yet clear enough to develop a model for time dependent loading, since the geometry of the separated region is the primary variable in estimating the pressure amplitude and resulting phase. No measurement of the lateral cross-correlation was made during the test; however, for the purpose of computing the response of the panel, it is assumed that the pressure decays similarly to that in the case of zero pressure gradient e 7/92 where a = 0.26 and 7 is the spatial separation [ Maestrello 1968]. This choice overestimates the lateral cross-correlation, since the flow field is far from being homogeneous. However, the overestimation may not bé exceéded by a factor of Z. 482 | Response of a Vtbrating Plate in a Fluid c) The Panel Response Field Measurements were made of the power spectral density and cross-correlation of the displacement. Typical results are shown in Figs. 4 and 5 for a pressure differential of 14 psi. The static deflection of the panel was 0.06 inches at the center, and the dynamic deflection was small in comparison with its thickness. The displacement spectral density at the center of the panel show pronounced spikes, the lowest frequency of which corresponds to the lowest mode of the panel. The accuracy beyond a frequency of 3100 Hz was poor due to the spatial resolution of the capacitance transducer, and therefore the spectrum beyond 3100 Hz was ignored. Space-time correlation measurements were made along the panel centerline from x =x'=3 in. y =y'=3 in. at one-inch inter- vals up to a maximum separation of 6 in. The correlogram indicates a convected feature with a phase velocity + Ucp = 770 ft/sec. This convection velocity corresponds to that found in the previous experi- ment using the same arrangements, except that no shock was present [ Maestrello 1968]. f = 556 N = a> 996 is] = 3 750 x=x'=0.5 ft. = =y'=0,25 ft. z ss yay a 10 M_ = 3,03 FB e i Press, Diff. Ap = 14 psi pe 1300 ey { a * 1664 FA 10 1930 4 A 2720 rat ey 2340 10 3150 10724 10° eae 1 N L 1 ! N ! oer ame wee ee 700 900 1100 1300 1500 1700 1900 2100 2300 2500 2700 2900 3100 3300 FREQUENCY Hz Fig. 4. Displacement Spectral Density 483 Maestrello and Linden In comparing the results of the present and previous experi- ments, it is concluded that the sign change of the convection velocity is attributed to the presence of the shock. Furthermore, the cross- correlation of the wall pressure also reflects a phase change fora separation of 2.5 inches, which is in the same location as the phase change which occurs for the displacement correlation in Fig. 5. III ANALYSIS OF ACOUSTICALLY COUPLED PANELS a) Two-dimensional Finite Panel The vibration of the panel is induced by an arbitrary, external pressure field F. It is assumed that the panel motion does not interact with the turbulent boundary layer, i.e., the forcing field is not altered by the plate motion. However, the panel is acoustically coupled to the fluid on both sides of the panel. The equation of motion for an harmonic component of the dis- placement, W, of a thin panel with a force, F, and a pressure differential, pp. - pj t 6p acting upon it, obeys the equation BAW - ppw W =F tp, -p, + 6p (1) where the bending stiffness, B, may include hysteretic damping, and where pp is the mass per unit area of the panel, w is the angular frequency, py is the acoustic pressure on the streamside of of the panel, p, is the acoustic pressure below the panel and dp is the static pressure differential. The perturbation pressures, p, and pp, are related to the velocity potentials, which satisfy time-independent wave equations in the appropriate regions. In solving these equations one uses a boundary condition which relates the potentials to the panel displace- ment. These relationships may be made more obvious through the use of Green's theorem. Thus, it is required to solve a system of three coupled partial differential equations, the first of which is not separable for the clamped edge boundary condition. Pp; and py may be found directly as function of W. Thus, consider first the cavity. The acoustic velocity potential, gy, satis- fies the Helmholtz equation Ag +k’ =0 (2) with boundary condition 89/8n = 0 on all walls except on the plate where 09/8n = - iwW. The Greens function, g, for a cavity with hard walls satisfies 484 Response of a Vtbrating Plate in a Fluid pa ie Tg | o AER ies - 0 ee ee a ee | See ae %= xX’ = 0.25 FT Y = Y’ = 0.25 FT PRESS. DIFF. Ap = 14 psi 400 Hz HI PASS FILTER M,= 3.03 UPSTREAM CORRELATION | POSITION OF SHOCK IMPINGE MEN T ! DOWNSTREAM CORRELATION R(E 9,7) Fig. 5. Broad Band Space Time Correlation of the Panel Displace- ment Along the Center from x = x' = 0.25 ft, y=y' = 0.25 ft. 485 Maestrello and Linden the equation Ag(r|r) + keg (3 | x") = 4n6(r - r') (3) and is given by Morse and Feshbach, Vol. II [1953] ' mx m1rx n n Cc Oslo leone re cos ac os ae Be De g(r/r') = aabe » » CS) =" = c Kinn Sin. (Kmnd) m=0 n=0 cos kyj,z cos k,,(z'+d) a ae Ae Xx (4) cos k,j2' cos kantZ +4) z Q o@)= 2 T eet at (5a) Now using the boundary conditions, this becomes — iw we — wea sea =, g(r) =- = | a(n 71) W(r))'d x (5b) plate The pressure p, is related to g by Pp, = - iwp.¢e where pc, is the mass density of the fluid in the cavity. To compute p,, it will be more convenient to operate with the differential equation. Let the acoustic velocity potential in the flow 486 Response of a Vibrating Plate in a Flutd field be denoted by WW By applying the Fourier transform on the (x,y) coordinates, one gets the ordinary differential equation 2 aA POP 2) + Pila,p,2) = 0 (6) dz where 2 2 2 2 2 GC =k +(M - 1)@ - 2kMa - £B @ Ao oxy) + Aa W(x, yz) -{ ( da dp Suen w(a,B,z) “0 “-09 k=w/c, M is the flow Mach number and c the speed of sound in the region above the plate. Only the positive exponential solution to Eq. (6) is chosen, since it is the solution representing outgoing waves. Thus, b(e,B,z) = A(a,p)e™ (7a) The boundary condition, arising from the continuity of normal dis- placement is du(a,B,z) - . ear dz 520 =O where the differential operator L=k+t+FiM — Ox Thus, Aa ~~ . H(a,B,z)=- 5M eM (7) Now, since nN a 1) Es ' ' LW -{ { dx' dy'e lide LW! sy) then 487 Maestrello and Linden a b W(x,y,2) = - a hs dx' dy' G(x,y,z|x',y',0)L W(x',y') (8) Ther 23 © 00 ila(x-x') +B(y-y')+C(z-z')] a(F | =f if SO ee, a (9) =00 ““00 which is found in Appendix A to be for supersonic flow, where ie(Muty/ ue -R*) e 271i me M’-1 an - R° G(r | x") = 0 outside the Mach cone and for subsonic flow, ix (Mu+yue+R® ) Gi bel ot (10) Vvi-M2 vut+ R? YN dieept inithis case, K = efyl- Mo hand aasi(x-at) (=e = aha had been evaluated as a (k - aM)W then Eq. (8) would read a b xe U(x,y,z) = - =|) y dx' dy' W(x',y")L G(x, y,z|x',y',0)(11) TT * This equation is formally correct if L G is interpreted as a distri- bution, which is to say that one partially integrates to obtain Eq. (8). Now using Eq. (8) Pp(x,y +2) at ip cLi(x,y,z) OY aa 2 = leae’ ( dx' dy' G(x,y,z|x',y',0){[L| Wlx',y’) 4% Yo Yo (12) 488 Response of a Vibrating Plate in a Fluid where fp is the density of the fluid above the plate, and where partial integration has been utilized. Had Eq. (11) been used instead, Eq. (12) would read e 2 1P9C Po(x sy» Z) = wee 40 re) a b 2 { dx' dy' W(x,y)|L | G(x,y,z|x',y',0) (13) ié) which is reducible to Eq. (12) by partial integration. Thus, super- sonic flow does not present any especial difficulty aside from the fact that G is singular all along the Mach cone, and this is an integrable singularity. Inserting the expressions for p, and p, into Eq. (1) results in a single partial integro-differential equation to solve, viz., a b 2 2 Es ipoc? ( a BAW - ppw W = F + 6p es A , G(x,y,|x',y")|L| wW(x',y") dx'dy' 2 + reas if B(XosVo|xXbrve) W(x, y) dx dy (14) plate where the subscript c refers to the cavity, thus x, =x + 7? u < + Ye Equation (14) presents a formidable computational problem. The Green's function g is known as an infinite series which is slow to converge (1/n) thus compounding the difficulty by an increasing num- ber of necessary operations to maintain a given accuracy. An alternative to solving Eq. (14) is to convert it to an integral equation for its Fourier amplitudes and to solve the resulting equation. The advantage is that this equation is simpler (though it is a singular integral equation). The following notation shall be employed: 489 Maestrello and Linden The result of applying the Fourier transform to Eq. (1) is 2+ Qa a a ve BA W- pw W=f + Po - py (15) where f=F + 6) The first term in (15) may be evaluated using Green's theorem; thus, —> <=> -iK-r 2a 4a — rs] 2 AB WEIOWIF OF Grete (Aw +K'w)| (16a) Now, for a plate clamped on its edge to a rigid, plane support, the following boundary conditions hold, on edge where s is in the direction of the edge, i.e., the tangent. Thus, 22 4° Aw=KWtdAl Ww] (16b) where If more general boundary conditions are to be considered (e.g. elastic foundation) the above expression must be replaced by the right side of 16a. From the Eq. (7b) and the equation prior to Eq. (12) it is found that E 2 eel p, = Pec en MY Wik) (17) From Eq. (5b) it is found that 490 Response of a Vibrating Plate in a Fluid 2 ~ W —~ —_ —_ — P, = - So { dr, g(K | r,) W( rg) plate e - “a[w] where g(k| r') = = { dre g(r |r') (18) plate Substituting these results into Eq. (15) gives BT(K)W(k) + BA[W] - Q[w] =£(K) (19) where oe 2 ear a 1(K) = K*- be _ oc eM) (19a) Let Wp denote the finite Fourier transform of a beam eigenfunction, 9,2 It follows from the Fourier representation that the Wn, form an orthogonal set on the infinite interval. Thus, expanding W as w(K) -) Winn'im(%2) Up (Bb) (20) m,n or alternatively, W as W(r) = » Wmn9m(= ) Pn (Z) (21) and introducing these expressions into (19) and subsequently utilizing the orthogonality, gives Win * > Dace as enn (22) 491 Maestrello and Linden and Pas ={ § ak YadodbolP) (pal, (2) o,(Z)| om Oo t(K) - a[¢, (2) « ()]) The computation of the integral Iimnr, may be simplified by deforming the contour on the @-plane. Due to the manner in which the Fourier transform was chosen, the integrand, except the term T(k), is single-valued and analytic in the lower half-plane. The contour will, thus, be deformed in this half-plane. This deformation is determined by the analytic properties of the function T(k), Eq. (19a). 2 T(a,B) = (a2 + p2)? - Se ee Bo V2 + (M2 - 1)a? - 2kMa - & This function is two-sheeted with square-root type branch points at kM + Vk* + (M? - 1)p? pba ESI Ee M* - 1 The sheet associated with the positive value of the square root will be termed the physical sheet, since it corresponds to outgoing radi- ation. The function has ten zeros on the two sheets, four zeros on each sheet with the same values, corresponding to resonances of the plate and the other two zeros are located near the branch points on one of the two sheets, independent of each other. It is convenient to make the following substitutions 2 = Po and 4 = Pp© oy Milsias The Eq. (19a) may be written 492 Response of a Vibrating Plate ina Flutd aM i T(a,B) = (a + B*)* Bec a k2 + (m2 - 1)a2 - 2kMa - Bp? In the present case,p is a small number (= 0.0015) so that approxi- mate values of the zeros may be found, expressed as a power series in p. To the second order these zeros are [xe + (M2 - 1)B7(Az* + 2A,°B +B - y*)” where + kM+ yk? +(M?- 1)p° ay = eee ee M2 - 1 (£), (4), > A(4), (4), (ee (“en (22) J i 4A (p2+A2 )[x° +(M?-1) A> -2kMA eras (+), (+). (+), (+) (+) (+). (+) (+) where = / 2 2 Avs), (£)g 7 i aE (*),¥ The last four zeros exist on both sheets. The location of the first two zeros may be distinguished into three possibilities: when ve— A, then a, and a; are respectively on the unphysical and physical sheets; as y is increased such that A, (Rosenblatt writes this as X»,(w) to explicitly indicate that it is a function defined on a sample space) with a cross-correlation defined as R( 2 , tr’; t") = (Xo XTh > (24) where ( ) is the expectation operator, i.e., R is defined through the ensemble average. Because the process is homogeneous and stationary, R(r ,t;r',t') = R(t - r', t-t') or R(r - r',t-t!) dr dt = (dM (xz*,) dM (Xm 4, )) where dM (x> +) is the Stieljes measure of the process. The pro- cedure may be simply stated as the problem of finding a Fredholm expansion of R and subsequently representing X,, by such as expansion. Such an expansion is provided by the eigenfunctions and eigenvalues of the integral equation (e8) w(t ,t) = NY R(r-r',t-t')y(r',t") dr! dt (25) -© The spectrum, is of course, continuous. The eigenfunctions are plane waves and the eigenvalues the inverse of the power spectral density as can be seen by applying the Fourier transform. Thus the desired expansion for R is R(T-T",t- ay ay fe eee Pt Rik aids Now let @ ir - wt) re - = e dM (X=, ) (26) ° (2m)P VR(K ,w) “© fs It follows from (24) and (26) that (Zgu2ZFw) = 6K - K')5(w- a") 496 Response of a Vtbrating Plate tn a Fluid since (dM(X—,)dM (X~,)) = R(r-r',t-t') dr dt. Thus the Zr, are independent random variables with unit variance and with zéro mean if EX, ,=0. The transform of (26) is, 00 —> —> —> -i(Ker-wt) > Xo, - Z JR(K,w)e dK dw (27) ’ Kw oO A simple calculation reveals that (27) satisfies (24). These results will now be applied to the plate displacement. Thus, the cross-power spectral density (CPSD) of the plate response is given by (asterisk denotes complex conjugate) (ww) =D om (S) en) (EZ) eC) (Wan Wie") (28) m,n,r,s Now if the solution to Eq. (22) is represented as Winn o » Ymnij Pj i,j then ( Winn Wes ) ~ » Ynniy Yren! § 2ij Pkt > ijkl Barther, oo ,0 ve os (5 ©) -( dK dK' Seeyeeeeee abbr (BY) (5) ery -00 “-00 The force F is now identified with Kooy so that (FER) = [RE wa Kol} (Zu Z 0) * —_ A —_. VA > _. [R (K a) RK, )] * 5K - KY) In summary then = — * x (Winn Wes.) = y Vinny Yrek ) - am Vea (bby Bie) Meee ie nk [T(K) | (29) a77 Maestrello and Linden Analogous to (28), the expression for the CPSD of P, can be written in terms of E Wmn Ws from (29). The same can ailize be done for the radiation. b) One Dimensional Model To simplify the computations we assume that the transverse plate dimension is very large and that no flexural waves propagate in the transverse direction. With these simplifications Eq. (22) may be written w, + > Paw ue, (30) where 00 ur (Ka)A Pm = r 2 ( dK [ (=) | (31) am -© T(K) where 4 K NAP ipy (1 - T(K) = K* = “i aN et ae ae Vk? + (M2 - 1)K? - 2kKKM and 00 1 a -\ ak Yol(aK)F(K) (32) Bon T(K) The circumvention of the branch points in the above expressions will be described shortly. If Gmn denotes the inverse matrix to 6mn +t Im then the solution to (30) is Wa = » Gam?m m Now performing the ensemble averages as before gives (WnWa) =) Gurl rs) Gr (33) r,s but 498 Response of a Vtbrattng Plate in a Fluid (a Hel2K) (yc Yala) Ket (O45) -{ ak Sea io dK Fieey Waele (K')) To make the discussion concrete let * (F(K)F (K')) = P(a)6(K! - (K - 3)) which corresponds to a spatially uncorrelated pressure field with convection velocity U, and power spectrum P(w). Thus aw U;, TK (Ke >) Cc UF (aK) uglaK - m 00 (4,0) = Pt) | ax (34) -00 The major contribution to the integral for In, Eq. (31), comes about when the peak of Up is close to the peaks of 1/f(K); since Wn is a highly oscillatory function (period = 27/a) with a peak at x, /a and de- caying with the distance from this point and 1/Y(K) is a non-oscilla- tory function with peaks whenever K equals the real part of the poles which are roughly located at y times the four roots of unit and the trapped wave poles near the branch points (k/(1+M) )(k/(M-1)). But for the frequencies we are considering (up to 3000 Hz) only the pole near k/(1+M) lies on the physical sheet. For this frequency range the trapped wave pole is bounded by 0 and 0.15 and the pole near y by 0 and 0.6. Now, the peak of py is given by y,,/a which is numer- ically (see Appendix B) 0.155, 0.26, 0.36, 0.465, 0.57, 0.67, ... and the period is 0.206. The height of the peaks decays roughly as (1 /Ka)> is 0.05 of the first. The function 1 /T(K) also has peaks that decay in the same manner. Thus, the infinite matrix Ip, has appreci- ably non-zero entries only in the upper left-hand corner. Consequently, we need only compute Imn for the first 4 or 5 modes, say, and then invert this matrix + I to obtain the upper left square matrix of order 4or 5 of Gmp and thus for higher modes Grn = 5mn So that * X (Wn Wn) = (bmPn > for other than the first few modes. The contour for the integral in (34), Fig. 8, is similar to one 499 Maestrello and Ltnden K-Plane K - Plane | e S|\7r eee wer wmr neem me ew KH Fig. 9. Deformation of Contour in Fig. 8 described earlier but the term Y*(K - (k/m)) introduces further poles and branch cuts. The contour and its deformation are shown in the Figs. 8 and 9. In Fig. 9 only the contours for that part of the integrand which is analytic in the upper half-plane are shown. Wm(K) may be written Um(K) = Sp (K) + T,(K)e 8° where H 3 3 4 4 (aK) - Xm and 500 Response of a Vibrating Plate in a Fluid 5 Tm(K) = Ney [bn + ia, aK) ( cos Xm Eg ae) 2 2 2 (ak)* - x2 (ak)? +x, a - (amXm - iaK) (8 Eis ee :)| (aK) - Xm (aK) + xq with @, and Xm defined in Appendix B. Now dm(K)Yy(K - +) = LOK) + U(K) c where -i(K-—)a ae Ww (a) Uc EK) = sm (K) (s,(K - U,) + T,(K - U,/° ) and i(aw/Uc) i(k-7)e U(K) =e T,, (K) (s,(K -T) TAK = Tw, Uc ) Thus, we have [ee] (mon) = P(u) f dK ee c The above expression is evaluated as a sum of residue contributions plus branch--cut integrals. 5 1 * L(z}) aoe (o,¢,) = 2mi Res | —_Ha)__| +I +I P(w) 2 T(2, )r* (2, - ) | 2 U, l2 ap Anil dq nes |e) | tit, os 252) (r(z)T (z - we) ET U, j=9 where the z, are identified in the above figure. 5014 Maestrello and Linden Fig. 10. Location of Poles and Branch Cuts As indicated previously, zg and Zg are not on the physical sheet for the range of frequencies considered (up to 14,000 Hz). k a geremale acme a “0 (aoa Get iy) | "(ar i ty) ter) where T*((k/M +1) 5 iy) is the expression following Eq. (31),the plus being included only to explicate that it is the positive side of the square root branchcut. Y7((k/M+1) + iy) is obtained from T* by replacing pp by - wp. She UGa—a tt ty) 1 1 c(i L (477 - iy) 1 4 dy oe le os (Wo ai) txt: iy) TGS + iy) ae Met) | ot 0 ol eee aed, "(art - iy) "(art - iy) It is felt, that the effect of the pole at z3 cancels the contribution of I; and similarly the residue at z,, cancels the contribution of I,. 502 Response of a Vibrating Plate in a Fluid Thus only I, and , will contribute. Therefore 5 4 * : U(z) Pay 6 omen = 2ni_) res |__| +1 2 ) er z= 2; T(z)t*(z - wT) ° j43 : aN: L(z) | + 2ni Res) ee a i: 2, Reel ate ys) j79 S By the same argument which eliminated I, and Iz we may replace I, and I, by pretending that z, and zg are on the physical sheet and thus ; U(z) I,=- 2ri Res 4 ———— 5 2726 are = ‘ Cc and : L(z) Ig=- 2mi Res 4,—— 4 ‘i 2=2q Sere = =} 4 To evaluate the residues it is necessary to determine T(z) = 423 + ipy(32M7k* + 2°M* - 2°M® - 3kM°2* - kz - k°M) vk* + (M® - 1)z* - 2kMz° The following relationships among the poles are valid * * Za = yt 27 Zs= a + Ze z= at Zs Uc Ue Ue 7) * W * Z10 aie or Zy = tie ce *4 4 ipy*(1 ri Sa) Zz iy + Vice + (M2 - Hee z 2kMz 503 Maestrello and Linden So T(z y= Tite) Thus bipil pee aa |e? te: U Similarly ' eee ee Lim | - i | Ag (z,) cei, T(z - zz, L7"(z, - a) 4 * U(z,) U(ze) : bd 1k SUNS ES EN e 2 EAE ne ZaTP (a) ‘ Smt) T'(z,)T (z, - a T'(2,)T"(z» - Tu (a) * Ww *K 2 U(5- + z2) . U(—- + 2g) -s * SAT ae li +z 7)T (z,) si +2,)T (zg) ‘ Uae +P Zg) Tg #zJt7 (zg) P L(z7) P L (za) : Tz)T (z_- u,) T'(z,)T (zy - wr, ) 504 Response of a Vtbrating Plate in a Fluid EE e3)) V2 425) ete eo Oe ee Tee 2 xs * Ta + 2))T (z.,) Ta + Z2)T (zo) ____ Lz) T'(zg)T "(2g - a) In a similar, though simpler fashion, we may evaluate the integral in Eq. (3.2). Al m(=)] = Am(K) + Byik)e where 2 ZN X: [ COS, Xm CO. Xm fe ] ==, (| AM = AM Am(K) a es Xm sin Xm + sh Xm Boag and 2 ° Bn(K) = - SSm (Xm) = sh x, + sinx,chx,,) - iKa sin x,sh | sin Xm + sh Xm * Thus , Al $m] may be decomposed into factors analytic in upper, and lower half-planes, respectively. Denoting these terms by G and G we have, ” GtiK) + G"(K) 1 ae - an aa dK -00 where G"(K) = Am(K) (Sn(K) + Tr (K)e"") and G"(K) = Bm(K)(Tn(K) + Sp(K)e) The contour of integration is indicated below. 505 Maestrello and Linden Fig. 11. Contour-for I, Eq. (31) The contour is deformed in the appropriate half-planes and branch cuts are replaced by poles as discussed previously. Thus, Tat Tmo = Res {Sc} + nae {S0P} ae Moma vaeicnd We now invert the matrix 6mn+Imn- Let us denote this inverse by Gij - Then * ee Co “,) = Ginp < %%q) Gon where the tilde denotes Hermitian conjugate and sum of p and q is implied. The final step, then, becomes the diagonalization of the CPSD matrix ( Wm Wa) thus giving the PSD for the actual degrees of freedom of the system. c) Acoustic Power, Power Radiated In computing the power radiated, we make use of the unitary matrix Ujj which diagonalized the CPSD matrix. The average power radiated, to the far field, P, is given by P~ (Im jy" Set) 506 Response of a Vtbrating Plate tn a Fluid With the use of the asymptotic form of the Hankel function, the asymptotic form of § may be computed. This form is ; ke 1- M@sin®@ + Mcos @) A(8)e lig w(r, scrap oe Jonikr V1 Evie sin* @ where it G a —— -M) A(@) -\ dx'e W(x!) 0 The acoustic velocity in the radial direction u, is given by a, = S¥ (r, 6) _ ik(yi - M® sin® @+ M cos 6) sin* 0+ M cos Oy ie r) Mo 4 The pressure, p, a microphone in the far-field would measure is given by ipo oa oe BP ox cL pc ikM? EET Rs Mo=4 iT] _ ikp(r, ®) Me ol The average radiated intensity is given by the expectation value 1 * = Re(pu ) mM iT] 2 - ; ik(¥i- M sin 6+ aM coe 5 Re(wy') 2 2 2 > pck (y1 ait arn + M cos 8) Re (yy*) Now, Maestrello and Linden * 4 * Cb) Se (ALBA (8) (2mkr)) 1 - M* sin* 0 where — (xx) cos 8 7 | |-M@ sin se (.A(@)A7(0)) - ax dx'e ( W(x!) W*(x)) 0 ) and where (W(x!) W" (x) = > (Win War) Pun 21) q (20) m on *. Let U;; denote the unitary matrix which diagonalizes (W,,W,); LCs 5 the transformation to the actual degrees of freedom. Further, let \m denote the power in the m!" degree of freedom, then we have ( w(x") W(x) = yi 9; (x') Uni (5); Uaj _(x) i,j,m,n The integrals have already been encountered, they are simply the finite Fourier transform of the g(x) so that, (A(0)A"(8)) = », Wi (2) Usa i 8ij Uny Yj(-2) i, j,mn where ik cos 6 _M yi - M2 sin@ 6 Z = T.f 2 feo. Gop 2 ee Ms = 1 508 Response of a Vibrating Plate in a Fluid REFERENCES Corcos, G. M., "Resolution of Pressure in Turbulence," J. Acoust. Soc'.'Am:'; ‘Vol. '3'5, 'N2;°1963. Crighton, D. G., "Radiation from Turbulence Near a Composite Flexible Boundary," PRDC. Rog. Soc. London A314, 153- 173, 1970. Crighton, D. G. and Flowcs-Williams, J. E., "Real Space-Time Green's Functions Applied to Plate Vibration Induced by Turbulent Flow," J. Fluid Mech., Vol. 38, Part 2, pp. 305- 343, 1969. Dolgova, I. I., "Sound Radiation from a Boundary Layer," Soviet Physics Acoustics, Vol. 15, No. 1, July-Sept. 1969. Dowell, E. H., “Transmission of Noise from a Turbulent Boundary Layer Through a Flexible Plate Into a Closed Cavity," J. Acoust. Soc. Am., Vol. 46, 1969. Dzygadlo, Z., "Forced Vibration of Plate of Finite Length in Plane Supersonic Flow," Proc. Vibration Problem, Warsaw, 1.8, 1967. Fahy, A. J. Petlove, "Acoustic Forces on a Flexible Panel Which is Part of Duct Carrying Airflow, " J. Sound and Vibration, Vol. 5, 1967. Feit, D., "Pressure Radiated by a Point-Excited Elastic Plate," J. Acoust. Soc., Vol. 40, No. 6, pp. 1489-1494, 1966. Flowcs-Williams, J. E., "The Influence of Simple Supports on the Radiation From Turbulent Flow Near a Plane Compliant Surface," J. Fluid Mech., Vol. 26, Part 4, pp. 641-649, 1966. Lapin, A. D., "Radiation of Sound by a Vibrating Non- Uniform Wall," Soviet Physics - Acoustics, Vol. 13, No. 1, pp. 55-58, July-Sept. 1967. Lyamshev, L. M., "On the Sound Radiation Theory From Turbulent Flow Near an Elastic Inhomogeneous Plate," The 6th Inter- national Congress of Acoustics, Tokyo, 1968. Maestrello, L., "Radiation From and Panel Response to Supersonic Turbulent Boundary Layer," J. Sound Vibration, Vol. 10, Zant I O90 509 Maestrello and Linden Magnus, W. and Oberhettinger, F., Formulas and Theorems for the Special Functions of Mathematical Physics, Chelsea, 1961. Morse, P. and Feshback, H., Methods of Theoretical Physics, Chap. 11, McGraw Hill, New York, 1953. Pol'tov, V. A. and Pupyzev, V. A., "Vibration and Sound Radiation of a Plate Under Random Loading, Soviet Physics - Acoustics, Vol. 13, No. 2, pp. 210-214, Oct.-Dec. 1967. Rosenblatt, M., Random Processes, Chapter VIII, Oxford University Press, New York, 1962. Strawderman, W. A., "The Acoustic Field in a Closed Space Behind a Rectangular Simply Supported Plate Excited by Boundary Layer Turbulence, USL Report No. 827, 1967. White, P. D. and Cottis, M. G., "Acoustic Radiation From a Plate -- Rib System Excited by Boundary Layer Turbulence, Measurement Analysis Corporation 702-06, 1968. APPENDIX A If we make the following changes in variables = ¥(M* - 1) a@- KM & K= M2 - 4 See M* - 4 then expression for the Green's function in Eq. (9) becomes | Qixmu : aieueB- y')+z,/ €2 -(x? + B?)] G(x,y,z|x',y',0) ari ie d dp = : VM? - 14 40 “00 Je? - (x? +p? @iMu fl, or age i[Bly-y' +2] (€2+x2)- B2) 2 os K*) = B? 510 Response of a Vtbrating Plate in a Fluid (see M+F, Vol. 1, Page 823) ieMu 00 el dé e*" rt”) (RYE - K*) s -@ R? = (y - y')* +22 The contour of integration for the above integral is shown below. & - Plane Branch Cut i) ; I(u) -{ dé Ht” (nye? — Ky -0O Define By considering the asymptotic form of the Hankel function, the above integral is seen to be convergent in the upper half-plane for values of u and R _ such that u>R that is, the region inside the Mach cone. The contour may be deformed to be a contour along the branch cut as shown below. €- plane Thus Maestrello and Linden He) «| ( (eles “Hl (VE? - ) a€ K foe) below cut above cut It oo . J 2{ ely (RVER - K*) ab We thus get (see Magnus and Oberhettinger, p. 179) in/ju?-R? I(u) = 2i— u2 - R2 so that in(Mut,/ u2-R2 2Tri e! ( : ) GCx:| 2) eee Be a Meee fue Bo elk MR (cos O+sin @) Me = 1 R sin 0 APPENDIX B Lr = ‘a cos — or (= ) dx fe) (oe NeeC deer E 9, (=) See = she te (cos “= - if a a r a 412 [o @) ise) 4 \e ee Rye one “i - be alr, (RYE - IC) aE Response: Of a Vibrating Plate in a Fluid Set = aoa fo (cn [Ce B2)»-BECGS)] +anf22C54)]) + coofZE (%8)] ~coo[(A-B2) aE (858)]} + Spm coe ESEC)] -coel te BD ECT +o (ain [(EeoBt)a +22 (28) tn [BE@2))} eel Sie cos [22 (25%)| (ch x, tash x ' + ayParmp oie RECE] -otoREEFA)] (ane, ta.cnah as where = Gos K= ch ik r sin K, t sh K, and the normalization N, is given by [ Dzygadlo 1967] -2 L 1 N, = aK, (sh 2K, - sin 2K,) 2 ee K, COS K, - sin Kr ch K,) Ar ° 2K (ch 2K, - cos 2K, - 4 sin kK, sh K,) + ar ag +3 (sin 2x, + sh 2x,) = = (sin K, ch K, + sh kK, cos K,)| and the eigenvalues xk, are the roots of the equation cos K, cosh K, = 1 Maestrello and Linden and are approximately given by ul K, = 4.730 K, = 7.853 K, = 7) (2n “+ 1) Jns is the same as Ings with a and a, replaced by b and kh, respectively. APPENDIX C 3 2 na aaa Ow > —- dw -iK-r Atw] =§ sues BS ae e | Now A[en(2) ea(E)] = Otn.0.2,8,a,0 - Q(n,m,f,@,b,a) where g is defined in Appendix A and O(m,n,@,8,a,b) 2 . : i = 208) [B- tnp+ oPaen us iP ate] J, oe enC§) The integral may be found by appropriately combining the following four integrals i(x_ + @a) - i(x_- aa) e - | i(aa-«,) i(aa+x,) ale oA Ke A | 2 aa - K, aat K, ea n he =} xs —] |x O_ Q x Qu * I a i(aa+x«,) i(aa-«,) x iax _ ee -i +2£ -i - cos Kas e dx= “Stl aa hk. = Ee K, Response of a Vibrating Plate in a Fluid (aati x,) i(aa-ix,) ‘ che Zeta ai | s =i Ue = aa PEK, aa - LK, ; i(aatix,) (aa-ik,) ‘ Ohi a | Serie Shee hearst -1] na Zi aa tik, aa - ik, and where a, is given in Appendix B. where and J+ APPENDIX D (a —-> svar aa a Ql w] -aiPs ( d r,/g(k/r,) W( r,) 00 QL o(x/algsly/b)] = wp, » Emén amiaae)in(Bbn)Emeins Kn Sin Kinae is given in Appendix B. HYDRODYNAMICS IN THE OCEAN ENVIRONMENT Thursday, August 27, 1970 Afternoon Session Ghairman: Ts Inui University of Tokyo, Tokyo, Japan Page Recent Research on Ship Waves 519 J. N. Newman, Massachusetts Institute of Technology Variational Approaches to Steady Ship Wave Problems 547 M. Bessho, The Defense Academy, Yokosuka, Japan Wavemaking Resistance of Ships with Transom Stern oye: B. Yim, Naval Ship Research and Development Center Bow Waves Before Blunt Ships and Other Non-Linear Ship Wave Problems 607 G. Dagan, Technion-Israel Institute of Technology, Haifa, Israel, and M. P. Tulin, Hydronautics, Inc. Shallow Water Problems in Ship Hydrodynamics 627 BE. O, Tuck and P. J. Taylor, University of Adelaide, Adelaide, South Australia BAT RECENT RESEARCH ON SHIP WAVES ee rm she) de Nig Newman 2 - 3 -\ ° ai had Massachusetts Instttute of Technology Cambrtdge, Massachusetts ABSTRACT This paper is concerned with various aspects of the far-field wave pattern generated by a ship or other moving body in steady translation. Section II contains a brief derivation of the classical Kelvin wave pattern, based upon linear inviscid wave theory. In Section III full-scale aerial photographs are presented for the waves generated by the Ferry Boat UNCATENA, and compared both with the theory and with photographic observations of a small scale model of the same vessel. In Section IV we discuss a towing tank experiment, designed to compare the waves generated by a "wavy wall" with the nonlinear theory for this hull form. Finally, in Section V, a third-order solution is out- lined for the Kelvin wave system, which indicates the occurrence of a nonlinear instability on the cusp line. I. INTRODUCTION Ship waves are intriguing from several viewpoints. To the observer of a moving vessel, either layman or professional, they are a fascinating pattern on the free surface -- in otherwise calm water one might, in fact, regard them as a thing of beauty. To the naval architect they are of primary interest as a source of energy radiation, and hence of wave drag on the vessel. More generally, to ship hydrodynamicists of all disciplines, they are a source of complication, interacting with, and affecting the boundary conditions of, such fields as viscous drag, propeller-hull interactions, sea- keeping, and maneuverability. To anaval vessel ship waves are a possible source of detection, visible for hundreds of wavelengths or ship lengths downstream and to each side of the vessel's track. Similarly, to the operators of all types of vessels, they are a source of damage to property and of personal injury to persons, in or be- neath the water surface, or on the shoreline. And, to the theoretical hydrodynamicist, they are the source of seemingly endless challenges, both analytical and computational. 5.9 Newman The classical analysis of ship waves using linearized inviscid wave theory, originated in the nineteenth century by Kelvin and Mitchell, has enabled us to understand and predict the qualitative features of ship waves. The period since 1900 has seen a wide vari- ety of refinements and applications of this basic model, and the present unsatisfactory state-of-the-art in no way detracts from the dedicated contributions of Havelock and others, some of whom are in this audience, who labored with ship-wave theory before it was facilitated by digital computers and a more widespread understand- ing of the methods of mathematical physics. As in most other branches of ship hydrodynamics, our present knowledge of ship waves is sufficient only for a qualitative under- standing of the phenomena, and does not permit quantitative predic- tions with the accuracy required by most engineering situations. In the past decade many ambitious scientists and engineers have, there- fore, abandoned the assumptions of linearization, or of an inviscid fluid. Some of these attempts at fundamental improvements of the Kelvin-Michell approach were summarized by the author in a Panel Report (Newman [ 1968]) to the Seventh Symposium on Naval Hydro- dynamics. The present paper is not intended to cover such a broad range of contributions, but only to report on my own recent and very limited activities in this area, both experimental and theoretical. In the experimental domain, photographic observations have been made of the Ferry Boat UNCATENA and of a small scale model of the same vessel, in order to verify the Kelvin wave pattern prediction and to search for variations in the wave system resulting from scale effects. In addition, a series of experiments have been made in a towing tank with the objective of confirming the striking nonlinear phase-jumps predicted by Howe [1967, 1968]. Finally, in an-extensive theoretical investigation, which is reported upon in more detail elsewhere (Newman [ 1971]), we consider the possibility of third-order nonlinear resonant interactions in ship waves, motivated by the importance of these interactions in the field of ocean waves (Phillips [1966]). For the sake of completeness, we shall first give a brief outline of the classical Kelvin ship-wave system. Il.’ THE KELVIN SHIP=“WAVE PATTERN Disregarding the local effects close to a ship hull, we can assume that ship waves are a distribution, in wavenumber space, of individual plane water waves. Generally speaking, these are ob- served to be of small amplitude relative to their wavelength, and the relevant Reynolds numbers are of order 10®to 10°, so that we are led to a linear potential-flow model. The individual plane wave system can then be described by the free-surface elevation ik(x,cos O+y, sin@) -iwt C (xosy,) = Ae (1) 520 Recent Research on Shtp Waves where A is the wave amplitude, k is the wavenumber (20/d, if X is the wavelength), 9@ the direction of propagation of the wave with respect to variations of time t. The kinematic and dynamic properties of the wave motion can be readily determined from the velocity poten- tial, which differs from the above only by a factor (sip jue © if the fluid depth is large, the vertical z,-axis is positive upwards with z,=0 the plane of the undisturbed free surface, and the (X55 Vo» Zo) coordinates are fixed with respect to the bulk of the fluid volume. Finally, with the above restrictions, the frequency w and wavenumber k obey the dispersion relation k= w*/g. The most general distribution of these elementary plane waves is obtained by integrating over all scalar wavenumbers k (or fre- quencies w) and wave directions 90, so that Ik(x, cos @+y, sin @)- iwt e (00) 27 E(x 5 Vo) -{ ax f dé A(k,8) ° (2) (@) 0 However, steady-state ship waves are independent of time, when viewed from a moving coordinate system which translates with the ship, say with velocity V inthe +x direction, and this condition restricts the frequency, or wavenumber, of the contributions to the integrand in (2). If (x,y) denote moving coordinates with Xg= xt Vt Yo Y? then by direct substitution ik(x cos O+y sin@)+it(kV cos O-w) 00 27 t(x,y) = l ax { d® A(k, O)e (3) fe) 0 This integral will depend on time, for arbitrary amplitude functions A(k,9), unless w = kV cos 8 (4) 521 Newman or from the dispersion relation, kee (g/V?) sec’ 0, (5) Finally, if the ship's velocity V is positive, it follows from (4) (or from an obvious physical argument) that the wave direction ® must lie in the interval - 1/2 S05 1n/2. Thus we arrive at the "free- wave" description of the ship-wave system i(g/V°) sec” @(x cos @+y sin@) 11/2 Heme -{ de A(o)e (6) -7/2 which is the starting point for many analyses of wave resistance. Kelvin's ship-wave pattern may be obtained from (6) by noting that if the polar radius R = (x? + y?)!/2 is large compared to the typical wavelength 271V?/g, then from the method of stationary phase the dominant contributions to (6) will arise from those angles @ where the phase function (g/V’) sec’ 9 (x cos 8 + y sin 8) (7) is stationary, or & sec’ Q (x cos 8 ty sin 8) = 0. (8) Carrying out the indicated differentiation, it follows that x tan @ + y(2 sec*@ - 1) =0, or that the significant ship waves will be situated at points such that sin 8 cos 9 - |y/x| Se eT a ma (9) The behavior of this function is indicated in Fig. 1, and the essential features of a Kelvin system are immediately clear: -/2 1. The waves are confined to a sector ly/x|< 8 * tan 19°28". 2. On the boundaries of this sector, or cusp line, the waves are oriented at an angle |0| = cot! 2V@ > 35016", 5iZz Recent Research on Shtp Waves In the interior of the sector, two distinct wave systems Sis will occur, the diverging (|@| > 35°) and transverse (|@| < 35°) systems. | x <= --+4y= tan/2°28" ~ — Nbcos® aN O= 35%" Vx 2-cos*@ ae O=-35°le' 0 0 Yne~tan 13°28 Fig. 1 Plot of y x, from Eq. (9), as a function of the wave direction 0. Finally, if the loci of a given wave crest are plotted, by requiring that the wave phase (7) be constant, while (9) is satisfied, the familiar Kelvin wave pattern shown in Fig. 2 is obtained. (The phase difference of 1/2 between the two wave systems along the cusp line is not explainable from the above simplified argument, but is a natural consequence of the method of stationary phase. Fig. 2 is reproduced from Lunde [ 1951].) Transverse Wave Crest Fig. 2 Wave crests of the Kelvin wave system (from Lunde [ 1951] ) The amplitude of the individual elements in the Kelvin wave system will vary, in accordance with the function A(®@) or the "free- wave spectrum" of the vessel. Moreover, the wave amplitude will vary as R for large R, in consequence of the radial spreading 523 Newman of wave energy. This result follows, too, from the stationary phase approximation, wien in addition, tells us that the attenuation rate is changed to R"’3 on the cusp line. A uniformly valid expansion near the cusp line has been obtained by Peters [1949] and by Ursell [1960]. The latter work includes numerical computations. Ill PHOTOGRAPHIC OBSERVATIONS OF SHIP WAVES Kelvin's ship-wave pattern, as developed in the preceding section or as originally developed by Lord Kelvin using an initial- value approach, is well known and widely accepted, since the final results are consistent with our observations of ship waves. Never- theless, and perhaps to the surprise of many, the author knows of no definitive experimental confirmation of the Kelvin pattern. Aerial photographs are generally of the near-field (e.g., Guilloton [ 1960]; Inui [ 1962]) or from oblique angles (Wehausen and Laitone [ 1960], Fig. 23). Stoker's "Water Waves" contains striking high-altitude photographs which are from directly above the vessels, but during turning maneuvers or while in convoys. One exception to the above may be a special volume™ on aerial reconnaissance prepared during World War II, but this is not generally available, nor has it been used at all for a comparison with theory. Two years ago I had the opportunity to obtain aerial photo- graphs of the Ferry Boat M. V. UNCATENA. This vessel is 147 feet long by 28 feet waterline beam and 9 feet draft, displaces 400 tons, and operates at a speed of 153 knots between Woods Hole and Martha's Vineyard, Massachusetts. Propulsion is from three propellers, turning at 1,200;,1,000, and 1,200 rpm. The water depth injgiie area where photographs were made ranges from 50 feet to 80 feet, with depths of 70 feet predominant. Originally, this vessel was chosen for observation because of the severe wave systems which it generated, in consequence of its high (0.38) Froude number. Pre- liminary observations from a surface vessel indicated the most severe waves to be substantially shorter than those which are predicted on the cusp line of the Kelvin wave system, but it is apparent from the subsequent photographic observations that these waves, in fact, are located inside the 19°28' boundary of the cusp line. Three of the photographs obtained are shown here as Figs. 3-5. These were made with a Hasselblad 2-1/4 X 2-1/4 inch camera and a wide-angle lens of 38 mm focal length. Figures 3 and 4 were taken consecutively, looking directly downward from an altitude of 1,000 feet. Figure 5 is an oblique shot from 1,600 feet. As is clear oe these photographs, the predominant waves generated are a portion of the diverging wave system, lying inside of the 195° cusp line. In Figs. 3 and 4 we have drawn in boundary lines + 19°28' * "Speed of Shipping," revised edition, Central Interpretation Unit, revised November 1943, 524 Recent Research on Shtp Waves * 7 eas ORM 4 oe Fig. 3 Aerial view of the UNCATENA wave system with 19°28' boundary lines and typical 35916' cusp- crest tangent lines superimposed Newman 2 ; ” > 3 further down- boundary lines and tangent lines superimposed > stream, with the 19°28' Fig. 4 Aerial view of the UNCATENA typical 35°16" 526 Recent Research on Shtp Waves Fig. 5 Aerial oblique view of the UNCATENA from ahead of the bow from the center of the wake, which, in principle, should lie on the cusp line or boundary of the wave system. The apex of this center is to some extent arbitrary, since, without further knowledge of the amplitude function A(6), as it appears in Eq. (6), for this particular vessel, the location of the ship's bow relative to the origin of the (x,y) coordinates is arbitrary. It is clear, in fact, from Fig. 3 that the apex of the UNCATENA cusp lines is somewhat ahead of its bow, by a distance of about one ship length. This conclusion is consistent with other observations of ship waves (e.g., Gadd [1969]), and we emphasize here that, in principle, there is no contradiction between this observation and the linear Kelvin prediction. Also shown in Figs. 3 and 4 is a typical pair of 35°16' angles, which should be tangent to the wave crests on the cusp lines, Both the 19°28' boundary angle and the 35°16' wave crest angle are sub- stantially confirmed by these observations, within the accuracy obtainable from these photographs. In fact, we have not observed any phenomena in these tests which are inconsistent with the linear Kelvin description of the far-field waves, in spite of the obviously nonlinear near-field disturbance associated with this vessel, espe- cially at its bow. There is also no noticeable effect on the waves from the ship's viscous wake and propeller wake region, in spite of the persistence of this wake far-downstream. (However, the latter wake effects may be expected to affect the transverse waves near the centerline; this effect could not be detected here because of Newman the relatively weak transverse wave system of this vessel, and the fact that aerial photographs from directly above tend to emphasize the shorter, and hence steeper, diverging waves.) As can be seen in all three photographs, but most noticeably in the downstream portion of Fig. 5, the diverging wave system includes three discrete groups of waves, separated by relatively calm regions or "nodes." The observation of three wave groups is also noted by Wehausen in the text adjoining Fig. 23 of Wehausen and Laitone [1960]. The explanation of this phenomenon is somewhat controversial. One possibility is in terms of the conventional "bow" and "stern" waves, and possibly others associated with the shoulder or knuckle of the vessel. My own view is that, while this synthesis is applicable near the ship, it is inappropriate in the far-field where the stationary phase approximation developed in the previous section is valid. Indeed, if two discrete "bow" and "stern" wave systems are superposed, in the Kelvin stationary phase approximation, the result is only one wave system, provided that the observation distance downstream is large compared to the separation distance between the two disturbances. There will, of course, be interference effects, with regions of reinforcement and other regions of cancellation, just as in the simpler radiation and diffraction patterns which we associ- ate with nondispersive wave problems. In the present context these will be introduced via the amplitude function A(@), and my own view is that the nodal regions between the three observed wave systems correspond to zeros of the function A(9) for this particular vessel. Indeed, it is not difficult to perceive, in some parts of these photo- graphs, a phase difference of 180° across the nodal regions. In principle, there should be additional nodal lines and discrete diverg- ing waves in the interior portion of the wave system, but these will correspond to relatively short waves which are not so strongly generated by this vessel, and more quickly attenuated by viscous and other effects. As one possible measure of the validity of Froude's hypothesis, that ship-wave effects are dependent only on the Froude number and the hull form, a series of photographs have also been made with a scale model of the UNCATENA. For this purpose a six-foot (scale ratio of 24) fiberglass model was constructed, and equipped with a single battery-powered electric motor and screw propeller, anda radio-controlled rudder system. Figures 6 and 7 show the model, as fitted with its single propeller and rudder, with an antenna mast for the radio control receiver. Test runs were made with this model on the Charles River, with photographic observations from the Boston University Bridge at a height of approximately 50 feet. These photo- graphs were made with a Minolta 35 mm camera and 28 mm focal- length lens. Figures 8 - 10 show the model wave system from various oblique angles. Unfortunately, no observations could be made from directly above, so that measurements of the wave angles are not obtainable for the model scale. Figures 8 - 11 show indeed that close to the model discrete bow and stern wave systems are 528 Recent Research on Ship Waves Fig. 6 UNCATENA model Fig. 7 UNCATENA model bow view side view distinguishable, whereas further downstream these blend into a single diverging wave pattern with several nodal regions. It is apparent that, on the model scale, more than three wave groups can be distinguished far downstream; in Fig. 10 four or possibly five dis- crete groups can be noted. Since viscous attenuation is stronger at the smaller Reynolds numbers corresponding to the model scale, it must be presumed that the attenuation of the short diverging waves in the full-scale tests is due to other effects, such as the higher level of ambient waves and turbulence in the full-scale flow. Another noticeable difference is the obvious presence of transverse waves in the model tests, especially in Fig. 9 which is from approximately the same viewing angle as the full-scale photograph shown in Fig. 5. Fig. 8 UNCATENA model and wave system (compare with Fig. 5) 529 Newman Figs 9 UNCATENA model and wave system aerial view SONG Fig. 10 UNCATENA model and wave system 530 Recent Research on Shtp Waves This infers a significant difference between the transverse wave amplitudes for the model and full-scale, which could have important ramifications on the predictions of wave resistance from conventional model testing, but this tentative conclusion may be biased by minor differences in camera angles or lighting™, and it is felt that a quanti- tative measurement of the transverse wave amplitudes for the model and full-scale vessel should be made, with wave buoys or stereo photographs. IV. TANK TESTS OF A WAVY WALL In perhaps the only truly nonlinear analysis of ship waves carried out to date, Howe [ 1967, 1968] has considered the waves generated by a "wavy wall" or ship hull form consisting of a slowly damped sine wave. This geometrical form generates preferentially only one wave system. By suitable choice of the hull wavelength and velocity, a diverging wave system can be generated which, according to Howe's theory and based on the analysis of slowly vary- ing finite amplitude waves as originally developed by Whitham [1965], will become unstable. The most striking feature of Howe's compu- tations, resulting from this instability, is the occurrence of a shock or "phase-jump" across which there is an abrupt change in phase and wavenumber. Figure 11 is reproduced from Howe [ 1968], and shows the calculated wave system and the region where a phase-jump is predicted. Also shown, on the abscissa and with an exaggerated scale, are the waterlines of the hull form. Fig. 11 Cross sections of the free surface perpendicular to the phase-jump. The broken line segments indicate a possible form for the free surface in the neighborhood of the phase- jump (from Howe [ 1968]). *In color slides shown during the oral presentation of this paper, some weak transverse waves can be noted in the full-scale tests. 531 Newman It should be noted that Howe's choice of a specific problem to which to apply the Whitham technique was based largely on the rela- tive ease of veryifying the results with a suitable experiment. We therefore set out to conduct such an experiment in the MIT Ship Model Towing Tank. For this purpose a model was constructed of Formica plastic laminate, bent to conform to Howe's damped sine wave, with fiberglass and polyester resin reinforcement and fairing of the back side of the Formica. The model was 10.4 feet long, by 1.5 feet vertical depth, and was immersed to a wetted draft of 1.0 feet. This "model" was fitted to the towing carriage in an off-center position to maximize the effective width of the tank and minimize reflections from the tank walls. The tank width is 8.4 feet, and the model was set up to give a separation of 5.5 feet between the wavy side and the facing tank wall. Tests were carried out at a speed of 4 feet per second, and the wave system was observed visually, photographically, and with a pair of wave probes which were placed at varying distances from the model to obtain a total of sixteen longitudinal wave records. Figures 12 and 13 show the model in operation, and the re- sulting wave system. In no case was a phase-jump observed, in the region where it was anticipated. One can discern a somewhat irregu- lar local effect along a longitudinal line about one foot from the tank wall, but this phenomena extends to the front of the wave group, is parallel to the longitudinal axis, and, moreover, originates further away from the model than the predicted phase-jumps. This discrep- ancy is unexplained, although D. J. Benney (private communication) has pointed out that the existence of phase-jumps can be questioned, Fig. 12 Photograph of the wavy wall and wave pattern looking down- stream 532 Recent Research on Shtp Waves Fig. 13 Photograph of the wavy wall and wave pattern looking upstream 533 Newman in principle, on the grounds that its existence violates the preassumed condition of a slowly varying wave system which is the basis for Howe's work. V. THIRD-ORDER INTERACTIONS IN KELVIN WAVE SYSTEMS One of the fundamental properties of a linear boundary-value problem is the principle of superposition; thus, for example, Kelvin's ship-wave pattern, although originally derived for a single "pressure point,” is valid for any distribution of singularities and hence for arbitrary ship hulls. But as soon as the assumption of linearity is discarded, the possibilities for nonlinear interactions, among the previously independent components of the solution, must all be examined. In water-wave theory it was shown ten years ago by Phillips (cf. Phillips [1966]) that for deep water gravity waves the second-order interactions are relatively uninteresting, but when third-order effects are included it is possible for "resonant" inter- actions to occur. Thus two or three primary waves can interact, over large scales of time and distance, so as to transfer a substantial portion of their energy into a completely new wave system of a differ- ing wavenumber. This striking result has been confirmed by others, both theoretically and experimentally, and can be regarded as well established. Motivated by the occurrence of third-order interactions in ocean wave systems, and by the striking nonlinear effects obtained for a special case of the ship-wave problem by Howe [ 1967, 1968] as noted in the previous section, I have studied the third-order per- turbation solution of the Kelvin wave problem. The details of this investigation are "messy," to say the least, and will be presented in a separate paper (Newman [1971]|), but I shall briefly describe the technique employed and the form of the results. First, as a pre- liminary approach to this problem, we may examine the possibility that, at any point in the Kelvin wave field, the transverse and diverg- ing waves are such as to satisfy the criteria developed by Phillips for resonance between two primary waves. It is not difficult to show, in fact, that the wavenumbers of the diverging and transverse waves are not resonant, except possibly on or near the cusp line, where the simple stationary phase results are invalid. To develop a complete solution of the ship-wave problem valid to third-order would be a formidable task; local effects near the hull, and nonlinearities associated with the boundary condition on the hull would have to be included, and the possibility of a breaking wave near the bow would raise fundamental questions of validity of the solution. Instead, we focus on nonlinearities associated only with wave propa- gation on the free surface, and taking place slowly over scales of many wavelengths, so that local effects and hull nonlinearities can both be neglected. The first-order linearized velocity potential must satisfy the familiar free-surface condition 534 Recent Research on Shtp Waves 2 8, ae P isk = 0 a z= 0 (10) where subscripts denote partial derivatives. The notation and co- ordinate system are as defined in Section II. By suitably non- dimensionalizing the coordinates, we may replace (10) by the con- dition >, + $),,=0 on 7a Ole (11) The general solution of this free-surface condition and of Laplace's equation, not including local effects near the disturbance, is (cf. Eq. (6)) 2 is | do £(0) ef? *k'x (12) [@) where k= (g/V*) sec* 8, k=(kcos 0, k sin 0), and x= (x,y). (In Section II the wave eae were restricted to the sector - 1/2<@< 1/2. Here we allow all values of @ in the integrand of (12), in order to avoid taking the real part of the complex exponential; Eq. (12) will be real if £(6) = f (w- 0), and to avoid difficulties with the radiation condition we shall assume that (12) holds only if x is large and negative.) The second-order free-surface condition, analogous to (11), is az Poxx = 2V >, V Ox - Pix Pi 27 a Pixxz)e (13) By inserting the first-order solution (12) for 9, in (13), and replac- ing products by repeated integrals, it follows that a particular solu- tion of (13) will be 27 21 k z+ik *x bo= J) do i do, £8) £(0,) W(0,,O,)e Ge enle (14) where hae ig = k(8,) + k(®,). The weight function W is an algebraic function, determined by the various derivatives in (13), and it can be shown that this function contains only removable singularities. Thus by repeated application of the method of stationary phase, $)= O(R"!) for large distances R from the disturbance, and this second-order potential will be 535 Newman masked by the first-order potential 9, = te )e Extending these results to third-order involves straightfor- ward but tedious analysis. The third-order free-surface condition is analogous to (13), but involves more terms on the right-hand side: 1 3 if P34 ate 2V6, ° vive,)° : 26 (Vo, ; hate Te y Vo) + 2(V >, , V bo. 7. ee ° Vo) l - b(doz7+ doxxz) (15) A particular solution, analogous to (14), is given by the triple integral 2a 2r 2a eT os 123% 123) = d3 = i do, if de, \, de, £(0, ) £(0,) £(05) W(0, »9,,6,)e (16) where Ky = RO) TKO), F k(0,). The weight function W(0, 8., 9.) is singular at points where its denominator vanishes or where 2 Kio, — (sec 9, t+ sec 6, + sec 8,) = 0 (17) and it is necessary, therefore, to study the roots of this equation. It can be shown that the strongest singularities occur along the cusp line; for example, at the point The integral (16) is improper at these points and we, therefore, conclude, as in many linear wave problems, that a steady-state solution cannot be assumed _a priori, but must be derived as the appropriate limit of an initial value problem. An expedient initial value problem is obtained by regarding the right-hand side of (15) as a pseudo-pressure distribution, im- posed on the free surface from an initial state of rest, and then looking for the steady-state limit which results. To avoid unneces- sary algebra we rewrite (15) in the unsteady form 536 Recent Research on Shtp Waves $5, us % xy Pe 2b sys t 344 =e! P(x,y) (18) where P denotes the right-hand side of (15). A solution correspond- ing to (16) is readily obtained, and in the limit € ~ 0 we find that the only modification is to replace Eq. (17) for the roots of the denominator of W by the new equation 3 Kio + (€ 70) sec @))" = 0. (19) jr Thus the singularities in (16) become slightly complex, and for e > 0 the integrals in (16) are proper. Finally, the behavior of (16) as € ~ 0 must be examined. Here the algebraic details are critical, since cancellation occurs between many of the leading-order terms. In view of the numerous possibilities for error, the following surprising result must be re- garded as tentative, and I would hope that it will be verified inde- pendently by others who are willing to tackle the algebra involved. As €—0, Eq. (16) predicts waves on the cusp line of the same form (0 = 35°) as the first-order solution, but with an ampli- tude which tends to infinity logarithmically in €. Thus there can be no steady-state solution of the third-order initial value problem, as posed in Eq. (18) and, presumably, in the more general case of a "steady" moving disturbance initiated from a state of rest. Ulti- mately, as in the analogous case for ocean waves, the logarithmic growth rate will be modified by further nonlinearities, but, never- theless, we must conclude that significant amounts of energy can be exchanged, through nonlinear processes, in the region of the cusp line, among adjacent wavenumbers. VI. CONCLUSIONS AND RECOMMENDATIONS The caption above is the standard one for theses and reports. In this paper we have obviously raised more questions than we have answered. The observations of the UNCATENA show that Kelvin's wave patterns are confirmed, even for a highly nonlinear near-field, provided the observation point is sufficiently far downstream (much further than is possible in a conventional towing tank). But are the differences noted in the photographs of the full-scale vessel and the 1/24th-scale model due to differences in photographic conditions and experimental errors, or are the transverse waves (and very short diverging waves) of substantially larger amplitude on the model scale? Here we would emphatically recommend further experiments in which the wave heights can be measured quantitatively, both for the full- scale vessel and for its model. This task can be simplified if only 537 Newman the transverse waves are examined, but the difficulties of carrying out full-scale measurements in an ambient wave system are well known. (In spite of their appearance to the contrary, Figs. 3 - 5 were made early in the morning of a relatively calm day to ensure little ambient wave motion. The low rising sun in the model photo- graphs is explained similarly !) Turning to the wavy wall tests described in Section LV, we have attempted, without success, to verify a theoretical anomaly -- Howe's phase-jumps. It is an open question whether this failure is due to experimental error (an obvious possibility is the effective modification of the wavy wall shape due to viscous effects), or if, in fact, the phase discontinuity obtained by Howe is a consequence of pushing Whitham's slowly varying finite amplitude technique too far, with a solution which is not always slowly varying but contains local "singularities" or "shocks." It is not likely that this question can be answered by further experimental work, unless possibly a viscous correction can be incorporated in the model's shape (to allow for a turbulent boundary layer in the presence of a slowly varying sinu- soidal pressure gradient !). Finally, in Section V, we have outlined a nonlinear analysis of the Kelvin wave system which predicts an instability along the cusp line, but for which (unlike Howe's instability) the conclusion depends critically on a delicate avoidance of algebraic errors. Having thus gone out on a limb, I can only express the hope that independent verification is soon forthcoming. VII. ACKNOWLEDGMENTS The three experiments described here were carried out with the generous assistance of many persons. The full-scale photographs of the UNCATENA were made by Mr. F. Claude Ronnie of the Woods Hole Oceanographic Institution; the Woods Hole Oceanographic Insti- tution also furnished the aircraft for these tests and the services of Mr. Robert Weeks as pilot. The subsequent experiments were per- formed by several MIT graduate students: David MacPherson and William McCreight built the fiberglass mold and model of the UNCATENA; Albert Bradley kindly loaned his radio control system; and the model completion and testing was carried out by Charles Flagg and Nan King, with photographs made by Ronald Walrod. Messrs. Flagg and King also built and tested the wavy wall model, and photo- graphs of this test were again provided by Mr. Walrod. The propeller for the UNCATENA model was kindly loaned by Professor Daniel Savitsky of the Davidson Laboratory, Stevens Institute of Technology, and the wave probes used for the wavy wall test by Professor Jerome Milgram of MIT. 538 Recent Research on Ship Waves REFERENCES Gadd, G., "Ship Wavemaking in Theory and Practice," Trans. Royal Inst. Nav. Archs., Vol. 111, 4, pp. 487-506, 1969. Guilloton, R. S., "The Waves Generated by a Moving Body," Trans. Inst. Nav. Archs., Vol. 102, 2, pp. 157-174, 1960. Howe, M. S., "Non-Linear Theory of Open-Channel Steady Flow past a Solid Surface of Finite-Wave-Group Shape," J. Fluid Mech., Vol. 30, 3, pp. 497-512, 1967. Howe, M. S., "Phase Jumps," J. Fluid Mech., Vol. 32, 4, pp. 779-790, 1968. Inui, T., "Wave-Making Resistance of Ships," Trans. Soc. Nav. Archs. and Mar. Engs., Vol. 70, pp. 283-353, 1962. Lunde, J. K., "On the Linearized Theory of Wave Resistance for Displacement Ships in Steady and Accelerated Motion," Trans. Soc. Nav. Archs. and Mar. Engs., Vol. 59, pp. 25-685, 1951. Newman, J. N., "Panel Report -- Nonlinear and Viscous Effects in Wave Resistance," Seventh Symposium on Naval Hydro- dynamics, Rome, 1968. Newman, J. N., "Third-Order Interactions in Kelvin Ship- Wave Systems," J. of Ship Research, Vol. 15, 1, pp. 1-10, 1971. Peters, A. S., "A New Treatment of the Ship Wave Problem," Communs. Pure and Appl. Math., Vol. 2, pp. 123-148, 1949. Phillips, O. M., "The Dynamics of the Upper Ocean," Cambridge University Press, 1966. Ursell, F., "On Kelvin's Ship-Wave Pattern," J. Fluid Mech., Vol. 8, 3, pp. 418-431, 1960. Wehausen, J. V., and Laitone, E. V., "Surface Waves," Handbuch der Physik, Vol. IX, Springer-Verlag, 1960. Whitham, G. B., "A General Approach to Linear and Non-Linear Dispersive Waves Using a Lagrangian," J. Fluid Mech., Vol. 22, pp. 273-284, 1965. 559 Newman DISCUSSION SOME DEVELOPMENTS IN SHIP WAVE PATTERN RESEARCH N. Hogben National Phystcal Laboratory, Shtp Diviston Feltham, Middlesex, England I. INTRODUCTION This note briefly reports two developments in ship wave pattern research. The first concerns progress in application of the 'Equivalent Source Array' concept described in Ref. 1; the second is the development of a fully automated system of recording and analyzing the waves, a more detailed account of which is being prepared asi /Refs<2. II. EQUIVALENT SOURCE ARRAYS An 'Equivalent Source Array' means for the present purpose, a source distribution which according to linear theory would generate a given wave pattern. It can be used for evaluating and interpreting the correlation between wave theory and experiment and also for predicting the effects of wavemaking on changing tank width and depth. In Ref. 1, this concept was invoked to interpret wave pattern measurements behind 2 series of 3 models with parabolic hull forms and systematically varied beam. More recently, a similar investi- gation has been made of the wave patterns behind 3 trawler type hull forms (aparentand 2 derivatives) tested by Everest (Ref. 3). In the case of these more realistic models, the 'equivalent source arrays’ were found to vary significantly with speed as indicated by the sample results for one of the models shown in Fig. 1. It may be seen that sources and sinks appear at a distance ahead of the bow which in- creased with increasing Froude number. This effective lengthening of the array may be explained in terms of 2nd order increases of wave phase velocity due to nonlinearity of the waves generated in the bow regions. III. AUTOMATED RECORDING AND ANALYSIS A prototype for a fully automated recording and analysis system has now been developed and operated. It comprises a station- ary array of 4 capacitance probes with paper tape output and a com- 540 Recent Research on Shtp Waves O68% 130OW sA@izre ad.1no0s JusTeAINby ey BIg 541 Newman MODEL 4890 —— SOURCES — — —)— — EXPERIMENT a | 0-25 030 035 0-40 0-45 0sO0 0:55 Fig. 1b Wave pattern resistance puter program which analyses the tapes as punched by the recording digitizer. The probes themselves are as described in Ref. 4. The computer program uses the Matrix method of analysis developed with this application in view and described in Refs. 4 and 5. It works by a least square fitting of an appropriate function to the wave sur- face defined over a suitable grid of positions. Some sample results for a 20 foot model of one of the trawler type models tested by Everest (Ref. 3), are shown in Figs. 2 and 3. Fig. 2 is a copy of part of a computer output. At the top a tabulation defining the wave ordinates Z(X,,Y,) 'as measured! by listing R, Xp Yr and Z (R isaserial number, Xp, Yp are longitudinal and transverse coordinates respectively in feet, and Z is the wave 542 Recent Research on Shtp Waves R X Y Zz 1 25.218) + (8.0005 st 1643 2 + 25.436 + 6.000 + 0.244 B “+ 2'5.653 + 4.000 + 0.722 4, 4°25.871 + 1.333 + 0.893 5 + 26.307 + 6.000 + 0.7 6 + 26.524 + 4.000 tt 26.042. 95 1.2 PARE WAC 22 0.739 146 + 5%. + 1.333 - 0.623 147 + 60.058 + 8.000 - 0.538 148 + 60.275 + 6.000 - 0.804 149 + 60.493 + 4.000 + 0.353 150° + 60.711) + 1.333 = fit CASE 535.100 SPEED 8.710 N THETA AN By 100DC AL F. Fo 0 00.00 0.695 -1,112 0.0608 08.481 -0.0371 0.0232 1 32.09 -0.460 -O.111 0.0051 10.019 -0.0030 -0.0124 2 45.87 -0. 636 0. 683 0.0235 12.191 0.0319 -0.0298 3 53.12 -0.127 0.218 0.0019 14.144 0.0166 -0.0097 4 57.70 0.005 -0.277 0.0024 15.887 -0.0334 0.0006 5 60.92 -0.028 -0.397 0.0050 17.468 -0.0748 -0.0052 6 63.35 -0.048 -0.217 0.0016 18.923 -0.0635 -0.0139 7 65.25 -0.137 -0.021 0.0006 20.276 -0.0093 -0.0620 8 66.80 0.056 0.145 0.0008 21.546 0.1011 0.0387 9 68.09 0.076 0.068 0.0003 22.747 0.0728 0.0812 10 69.18 0.185 0.038 0.0012 23.888 0.0621 0.3033 11 70.13 -0.068 0.047 0.0002 24.977 0.1174 -0.1708 12 70.96 0.003 -0.082 0.0002 26.021 -0.3172 0.0100 13 71.68 -0.074 0.063 0.0003 Zils O25 0.3709 -0.4387 44 72.35 -0.021 0.062 0.0001 27.993 0.5631 -0.1856 [ This is to be compared with 100C W=0.09375 obtained by 100CW= 0.1041 Everest with 15 foot model of the same form (Ref. 4).] R x Y CZ DZ 1 + 25.218 + 8.000 + 1.661 + 0.018 2 + 25.436 + 6.000 + 0.194 - 0.051 3 + 25.453 + 4.000 + 0.764 + 0.042 1,333 + 0.852 - 0.041 + pyar 59.840 + 1.553 Uli 478 147 + 60.058 + 8.000 - 0.502 + 0.036 148 + 50.275 + 6.000 + 0.766 - 0.038 149 + 60.493 + 4.000 + 0.607 + 0.254 150) #_ 60.7145 F 15333 = OF7693 + 0.418 RMS RESID 00.163 RMS Z 01.080 Fig. 2 Sample computer output 543 Newman elevation in inches). In the middle is a tabulation of wave spectrum parameters accompanied by the resulting wave resistance coefficient 100 Cw. The first 4 columns define the amplitudes and resistance contributions of the various wave modes in notation which corresponds to that used for example in Ref. 4. The last 3 columns define functions used for computing ‘equivalent source arrays' in notation explained in Ref. 1. It may be seen that the resistance coefficient 100 Cy checks reasonably well with a result obtained by Everest for a smaller model of the same form, using manual pointers on transverse cuts analyzed by the method of Eggers (Ref. 6). At the bottom is a tabulation defining the wave ordinates CZ (Xp-° Yr) 'as fitted' in the same format as the 'as measured’ results but with an extra column listing the difference between measurement and fit. Fig. 3 shows a sample profile plotted from the computer output. =o AS MEASURED: —_§\f--——_ AS. FITTED’ 3 PROFILE AT y=|-333 FEET FROM TANK CENTRELINE Fig! 3 “Profile at» y ="1.533 feet from tank centerline 544 Recent Research on Ship Waves ACKNOWLEDGMENTS Appreciation is expressed for the contributions of Mr. B. Garner and Mr. H. G. Loe in developing the automated wave recording system and of Mr. E. J. Neville and Mr. M. Wilsdon in conducting the experiments. REFERENCES Everest, J. T. and Hogben, N., "An experimental study of the effect of beam variation and shallow water on 'thin ship’ wave predictions ," Trans. RINA paper W11 (1969) issued for written discussion. Hogben, N., "Automated analysis of wave patterns behind towed models," in preparation as Ship Division Report No. 143, Everest, J. T., "Some comments on the performance in calm water of a single hull trawler form and corresponding catamaran ships made up from symmetrical and asymmetrical hulls," NPL Ship Division Report No. 129, February 1969. Gadd, G. E. and Hogben, N., "The determination of wave re- sistance from measurements of the wave pattern," NPL Ship Division Report No. 70, November 1965. Hogben, N., "The computing of wave resistance from a wave pattern by a matrix method," NPL Ship Division Report No. 56, October 1964. Eggers, K. W. H., "Uber die ermittlung des wellenwiderstandes eines schiffsmodells durch analyse seines wellensystems," Schiffstechnik Vol. 9, part 46, p. 79, 1962. ssh 54 £ BLANK S47] FOLLOWS Sa ks Co PD lh TE meget | | ey OPAL. Wf ty ie t i awamien POU eee: ah is ; i : s ee Ay ety a ue eee i me B i ‘ 5 Fendi Mi f fed 4 pol a) re t aae ie v age ij F Oe i Ah eee | Oy Wh ont Bach al heat, 8 oh SS Baa ae rie AG) C1 oe ; j APR AG m i he ba, r ¥, , | a » : +“, v hey 7 m bes i, oe ; “ _ a : y y" t 1” ssl p29) dh ee eh ath vehiteih Nad NIA Ae oigod ta Be a ; a" By a ae Wy (i? ny i H Lay ey) iy wet i renee ! tJ 5 iF) ay | he axa kbp ‘ ne ihe / 4 om t : i PS *. ia i” , 7 ‘} (AA Teas i ; " ie y guy ‘ g yes * ey 4 pe" | i PE oA yy ale, ar | Me Pat : ' Nie 1. os ¥ . etiegg Te CMM ERIN Se) watts! ree Siberadt ho) OW: gf big Ps 4 perky Ud LLY oe J Lite Ws tect eren bit AY Regio, ry Oa y eh v7 . wird irae rigs J Dd\een MA La ry ’ . MVA Set LN o RWOLIOA ANG VARIATIONAL APPROACHES TO STEADY SHIP WAVE PROBLEMS Masatoshi Bessho The Defense Academy Yokosuka, Japan INTRODUCTION Although there have been many fruitful. engineering applica- tions of the theory of the wave-making resistance of ships, it is still not possible to completely explain the wave resistance of the usual surface-piercing ships. The so-called order theory gives us insight into the structure and composition of our approximate theory; however, we do not yet have a consistent and practical theory which is univer- sally acceptable. The author has speculated on what would be the best approxi- mation to our boundary value problem. In this connection, is there a useful principle which corresponds to the Rayleigh- Ritz principle in the theory of elasticity? The present paper will provide a partial answer. Our first aim is to introduce a variational principle which corresponds to the linearized boundary value problem. This is accomplished by introducing Flax's expression from wing theory. [ 6] Our second aim is to find an alternate expression which will enable us to treat blunt bodies, since Flax's method is useful only for thin wings. Gauss' variational expression [ 24,25] for the boundary problem of a harmonic function is introduced for this pur- pose. This is shown to be equivalent to extremizing the Lagrangian or kinetic potential. The resulting dynamical interpretation of the boundary value problem is similar to the approaches of many other authors who have studied free surface problems by using the Lagrangian [| 3,12,13,14]. I. FLAX'S VARIATIONAL PRINCIPLE The variational principle introduced by A. H. Flax in wing theory [6] may be directly applied to our problem. Those unfamiliar with this principle are directed to Appendix A. 547 Bessho If the Kutta-Joukowski condition |6,7,8] is satisfied at the trailing edge, we have the reciprocity relation ‘i pw dx ay= | { pw dx dy (f.1) S S by (A. 8) and (A.24), where p is the pressure, w is the vertical velocity component, and tildas denote reverse flow quantities. The integration is over the wetted portion of the ship hull S. Let C(x,y) be the free surface elevation. The variation of the integral 1299 [tw - Be, - Bul ax ay (1.2) S due to variations of p and p takes the form 51: = iy [ 6p(G, -w) - 6p(%, + w)] dx dy. (1.3) S Since the variations 6p and 6p are arbitrary, the pressure which extremizes the integral I is equivalent to the solution of the boundary value problem (A.25) and (A.26); that is, the problem for the pertur- bation potential $ with the conditions 6, = -we ¢, (1. 4) t= W= -4, x on the free surface. The stationary value of I is the drag; namely, [1] = SS pb, dx dy, (1.5) where p, denotes the correct solution. [6,24,26] Thus, the bound- ary value problem is converted to a variational problem, the solution of which is suggested by various methods of approximation. |[ 6] If we introduce the error integral, * ms E -{f (p - p,)(® - &,) dx dy, (1.6) S we see from (1.1), (1.4), and (1.5) that 548 Variattonal Approaches to Steady Shtp Wave Problems Deer ee (1.7) Therefore, Flax's principle produces an approximate solution which makes the error integral (1.6) stationary. [23] This method suggests powerful means for obtaining approxi- mate solutions, but unfortunately it has been applied only to thin hydroplanes and wings. [ 7] Il. GAUSS' VARIATIONAL PRINCIPLE In this section, we assume there is no free surface. Then the velocity potential has the following representations for the source- sink and doublet distributions: o(P)= 355 J) aE 45(2), $2 04eteee Veet) and oP) = a5) Hi (Q) & AES 4512), PeO; 422,600 ee) Here, quantities with the suffix zero stand for the correct solutions while those with other suffices are not necessarily correct. For these potentials we have the following reciprocity relations: \{ oo, dS -\{ To, as; (223) S S i Poo) dS = pF b,O,,dS, (2. 4) Ss Ss ia $15, dS -{f $6), dS. (2.5) Ss Ss Gaus s's variational principle for the Dirichlet problem states that if we consider the functional and G =3/ (b - 2f)o0 dS, (2. 6) S where 549 Bessho f=%, is givenon S, (Zea) then the function which gives the maximum value to G is the solu- tion of the Dirichlet problem. [9,10] This is easily verified by making use of the reciprocity (2.3). In the same way, we may construct a variational principle for the Newmann problem as follows: Let us consider the extremum problem for the functional = ant (6, - 2f,)y dS, (2.8) $ where f= Oo, 18° givenron (Si (2.9) This problem is seen to be equivalent to the present boundary value problem by making use of (2.4). Alternately, we may construct a variational problem by making use of (2.5); namely, by introducing the functional J= an $(2f, - ,) ds, (2, 10) and taking the variation, we have 6J = a S4(f, - $,) dS. (2.44) S From this we see the equivalence to the boundary value problem. [ 24,25] Now, since G0, eraas= $06 vaive, ar ety where D is the entire water domain and d7 is a volume element, a natural measure of the error of an approximate solution 96 is ey [Vib - 9] ar, (2.13) E 550 Vartattonal Approaches to Steady Shtp Wave Problems which becomes E= Ane (o i bo) (>, - bo, dS = J a J; (2. 14) by Green's theorem. Here, Jo -3(( Mobo, dS (2, 15) S is the correct value. We see clearly that SE = - 6J. (2.16) Since E is non-negative, we have the inequality [ 10] t,o. (2347) It is well-known that among all functions $ having a finite energy integral, 2 r=3((( [Vo] drt, (2.18) D and a given normal derivative on S, the one which minimizes T is a harmonic function [1,4]. Accordingly, if we solve this minimiza- tion problem, say by the relaxation method, we have the inequality ea en eles (2.19) fe) This is the dual of (2.17) and we now have the variational problem (2.7) as an involutory transformation of the latter minimization prob- lem. (See, for example, the textbook on variational calculus [11] .) Ill. A VARIATIONAL PROBLEM FOR THE LAGRANGIAN The preceding principle can be easily extended to flow ina gravitational field. Let us consider the functional L=T-V, (3.4) where Bessho A annie Ive] dt (3. 2) and v= 8 O° dxidy (3.3) F are the total kinetic and potential energies, respectively. L is just the kinetic potential or Lagrangian. [5] Assume that the function @ has a given normal derivative dy=-xXy on S and F. (3. 4) Taking the variation of L, we have 6L= any $V 8h dT al $56, dS + si [ 656, + {(374)* - gt}6v] ds. Making use of (3.4), which is also true for the new deflected free surface, we have ai=-\|( $v so dt + an pév dS, [3,14] (3.5) D where p/p = - 6, - 3(Vd) - gh. (3. 6) Hence, if the pressure at the free surface vanishes, the stationary value of L will be attained when 6¢ is harmonic. This is just an extension of Kelvin's minimum energy principle. [1,4] On the other hand, if 6¢ is harmonic, then the stationary value of L is attained when the free surface pressure is constant and zero. The latter is an extension of Riabouchinsky's principle of minimum added mass. [3,14] The variational problem can be transformed so that the con- straint condition is converted to a natural condition. Let us adda term which is zero at the stationary point. Consider the functional baz Variattonal Approaches to Steady Ship Wave Problems P=T- v-\ (xv + o,) dS. (3. 7) S+F Assume that $ is harmonic and, for simplicity, assume that the integral over an inspection surface at infinity vanishes. Making use of Green's theorem we have p= - SVG Cont uvorl ar - 89 0? ax ay = ait p dt + Const, (3. 8) p D where Const = §'( H? ax dy -3{\ z* dx dy, (3.9) S and H is the depth to the bottom. Taking the variation, we have 4 6P = SS. pév dS - SS. 5o($, + x,) dS. (3. 10) Therefore, when p=0 on F and $,+x,=0 on S and F, (3.41) P is stationary. This result was first given by J. C. Luke [12,13], who pointed out that the volume integral of the pressure is equivalent to the Lagrangian. Furthermore, we may write (3.8) as P=M-H, (3.12) whe re H= Tt V (3.13) and 553 Bessho M = - (Ah oy dT. (3.14) M is the total momentum of the system in the x-direction and becomes equal to twice T, M = 2T = ae éx, dS -{{ oo, dS, (3.15) StF S+F when 96 satisfies the boundary condition. Hence, 6P=0 means that 6H = 6M. (3.16) That is, when the variation of the total energy equals that of the total momentum in the x-direction, the potential satisfies the boundary conditions (3.11). For purposes of application, it may be convenient to write P as P=- ({ (x, + 46,) dS - g(( OF se doy. (3.47) S+F F This principle is applied to a regular, two-dimensional wave- train in Appendix B. In general, there is some difficulty in the appli- cation of this theory since the integrals P and L may not be finite. This is because the kinetic energy exceeds the potential energy for a finite amplitude wave. [1,2,4] One way to bypass this difficulty may be to assume a flow model like the Riabouchinsky model [3] in cavitation theory (see Fig. 2); however, this may be impossible in the three-dimensional case. Another way may be to introduce Rayleigh's friction coef- ficient so that the waves far downstream will die out. In any case, there are still some problems which make us hesitant to begin the actual numerical computations. Finally, let us consider the linearization of the free surface condition. Neglecting higher order terms in the integral over the water's surface and assuming that go(x,y) = - 6,(x,y,0), (3.18) we have 554 Vartattonal Approaches to Steady Shtp Wave Problems (a) (b) L: SUFFICIENTLY LARGE DISTANCE Fig. 2. Riabouchinsky Models P=P-.+P,, where Poe Sf. (x, + $4,) dS, and Pe = - 4). (xx + goz) dS. Accordingly, if we set xx + go, = 0 on F, which is just the dynamic boundary condition, then and we are left with a variational calculus problem for Py ° 555 (3.19) (3. 20) (3521) (322) (3523) Bessho IV. THE LINEARIZED PROBLEM The variational problem for Pg (3.20) is not satisfactory since there is no reciprocity relation for this form. We must intro- duce the reversed flow potential as was done for Flax's principle. Let us consider the integral Tete e) = Ll" Go.4) = - Sy Vo,Vo, dT - aS C05 dx dy.44o8) Assuming that 9, and $5 are harmonic and satisfy the free sur- face condition, we have, by Green's theorem, L*($,,) =-2 \h, $140, 4S =-% Mi ox, ds, (4. 2) where S is the surface of a submerged body. This is the recipro- city theorem for a submerged body. [8] If $,= - $,, then 1*(,) = +\ i bb, dS = L(, 4), (4. 3) where L($,9) = 3 aa (74) dr - g(( 1 dx dy. (4. 4) * L_ is called the modified Lagrangian integral [5]. Note that L(¢, 9) has a finite value in the linearized case but not in the finite amplitude case. If S is the wetted part of a surface-piercing body which is under the waterline before the free surface is disturbed, there is an additional term from the surface integral. [15,16,19,20,21] The reciprocity theorem, in this case, is oa ‘f bib dy - a\t $192, dS rf $6, dy -2 i $,6,, dS. (4.5) L*(4,,6,) 556 Vartattonal Approaches to Steady Shtp Wave Problems When 4, = 9, $52 @ and ¢$,=-+x,, Tae becomes 1 (6, 8) Se \ oLx, dS + aft ox, dS, (4. 6) where n is the inner normal to the waterline curve L inthe hori- zontal plane. Thus, the first term in the right-hand side of (4.6) is the correction for the change of the wetted surface S. [16] This is justified, on the one hand, by the dynamical meaning of the Lagrangian and, on the other hand, by the linearization procedure of the pre- ceding section. For the case of a pressure distribution over the water surface, we may integrate (4.5) by parts and make use of the formulas in Appendix A. This results in the expression -2 ae 15 dx dy = af. $.,6, dx dy i : 1 x a4). [P, + pgt|c, dx dy = 75, [P, + pgt.|t, dx dy. (4. 7) 1*($,, 90) Thus, the reciprocity becomes [ 8] S (pice) = taxdy= tll 52 dx dy (4.8) P, »Po 2p s P, 2 yy. 2p S 2 | where £*(p,.B,) = 116.4) - all tt, dx dy. (4.9) Making use of these reciprocities, we may easily show the equivalence of the boundary value problem to the variational problem for the functional I”, where i a [ $6, - (@ - $)x,] as, (4. 10) for a submerged body, and * I =- a [pe "(b= pit de dy, (4,11) Puvs for a pressure distribution. [ 24,26] 557 Bessho Alternate representations for these integrals are To sd (bordel eD (bos Ooh): (4, 12) 2K 2k ~ * ~ ~ I = (po,Po) - £ (p - pos Pp - Po), (4.13) where the suffix zero stands for the correct solution. These for- mulas show that the variational principle extremizes the Lagrangian of the error and that the stationary values are just given by the Lagrangian. The difficulty arises in the case of a surface-piercing body. From (4.12), the functional to be extremized is * I = - 16,4) +a ($£0- $f) dy + =e) ($ - $)x,dS. (4.14) Taking the variation, we have the boundary conditions equivalent to this variational problem, >y= - glo $,= 8% on L, (4.15) by= - Ove -Xy,y on Se. (4.16) But we have no knowledge of the surface elevation on L, a priori, as this problem may be indeterminant. [17,23] We must remember here that the solution is unique only when the detachment points are fixed by the theory of cavitation. [3,14] This difficulty may be avoided by introducing a homogeneous solution for the two-dimensional, linearized case (see Appendix C). For the present case, we might proceed as follows: Let us consider the difference between a surface piercing body and the limiting case of a submerged body moving very close to the free surface as in Fig. 3. [23] The boundary condition on the water surface above the submerged body must be 9, = 0, but since the top is also the free surface, this is equivalent to >, = &x(x,y) =O on aoe (4,17) or integrating, we have o,(x,y,0) = - g€(x,y) = Const = func (y) on FF. (4.18) 558 Variattonal Approaches to Steady Shtp Wave Problems (a) SUBMERGED OT ae \ YU) (b) VERYSLIGHTLY SUBMERGED (c) SURFACE PIERCING Fig. 3. Slightly Submerged Ship This formula shows that there may be a thin layer of uniform flow over the top of the submerged body. When this layer moves with the body, $x(x,y,0) = - go(x,y)=-1 on F, (4.19) and we clearly have the case of a surface-piercing body. On the other hand, the boundary value problem of a submerged body is equivalent to the variational problem (4.10). After solving this problem, we may calculate the surface elevation over the top water plane by (4.18), but it will differ from (4.19), in general. In this case, it might be necessary to introduce another potential which satisfies condition (4.19), in addition to the above potential. This procedure may not be practical because the treatment of the top water plane is difficult. In this case, it would be more convenient to consider the follow- ing two boundary value problems: Let us split the velocity potential into two parts, d= 6, +, (4, 20) with boundary conditions 559 Bessho >, = 0 on LL (4. 21) >, =- Xy on S to = Lo = Leo, given on L (4. 22) $9,=9 on S The corresponding functionals are of the form (4.10) for 9), and of the form (4.14), without the third term on the right-hand side, for Oo. For the present case, 4) must be equal to 1/g by (4.19); however, in general, it will be arbitrary and, perhaps, a constant oa form (4.18). ¢$. is called the homogeneous solution. {18,22, 26 Finally, it should be noticed that the Lagrangian is closely related to the far-field potential. For a submerged body, we have, from the boundary conditions, (A.9), (A.41), and (4.3), B : i xxy dS + 21($, 9) if) 2L(>,9) + V, (4. 23) where V is the displaced volume. For a surface-piercing body, interpreting condition (A.10) as a correction for the real wetted surface, we have iv xg,as +i x, dy =V, (4, 24) s EYL where V is the displacement volume under the still waterline. Therefore, we can write (A.1i1) as B=V +21 (9,6, + $), (4, 25) where 9, and $9 are defined by (4.21) and (4.22), with C,5= Wierd For a pressure distribution, we have, from (A.18) and (4.8), B = 2p "(p,P,)s (4. 26) 560 Vartational Approaches to Steady Shtp Wave Problems where pz, is a homogeneous solution, as is 5, and to= tg Since B is also a measure of the total lift, this formula shows that the homogeneous solution for the constant surface elevation influences the lift, as we have easily verified by the reciprocity (4.8). [ 26] It should be noticed that, in this case, the condition A=0 in (A.18) insures the continuity of the planing hull. Kotchin's function (A.17) is also given in the form a) wo-- $f where ‘by has the boundary values 6.x, dS =| o.6 dy + 2L.°(6,%), (4. 27) L Pay = ~ Per ede Ss and (4. 28) ~ gt, = ee mS Pay on L @qis called the diffraction potential. [23,26] Here, the second term of (4.27) may be omitted as in (4.25). For a submerged body, there is no integration along L and H may be written as H(6) = - ae (be + dy)x, dS. (4. 29) Finally, for a pressure distribution, H(0) = 2p£"(p, py) (4. 30) where a oe (4.31) d g ex V. CONCLUSION We have presented two variational principles for the boundary value problem associated with the waves of a ship advancing at a constant speed: The first is Flax's principle, which makes use of the stationary character of the drag. This principle is useful only for 561 Bessho planing boats or for submerged thin wings. [6,24] The second is based on Gausz's principle, which converts the boundary value prob- lem to a variational problem. This method is shown to be an exten- sion of Riabouchinsky's principle of minimum virtual mass. [3,24] The latter principle is based on the stationary character of the Lagrangian and has recently been used by Luke, in a more general form, to study water wave dispersion problems. [3,12,13] We also have analogous principles for light and sound wave diffraction and for the radiation of energy due to the heaving, swaying, and rolling oscillations of ships. [25,27,28,29,30] The variational principles emphasize the dynamical meaning of the boundary value problems and permit us to solve them approxi- mately by the Rayleigh- Ritz-Galerkin procedure. [6,28,29] How- ever, when we try to apply these principles to our problem, there are two difficulties: The first is that our system is not conservative because of the trailing wave. This may be bypassed by introducing an artificial model, as in Fig. 2, or by introducing a reversed flow for the linearized case. The second difficulty is for the surface-piercing body, in which case the wave profile is not known, a priori, even in the linearized case. This difficulty may be avoided by introducing homogeneous solutions [27] which appear in the case of a surface pressure distri- bution. [ 26] Finally, although a variational method does not necessarily represent a new method of analysis, it does suggest new methods of approximation. For this reason, it may be useful, especially for engineering purposes. Rk FERENCES i. Lamb, H., Hydrodynamics, 6th Ed., Cambridge University Press, 1932. 2. Wehausen, J. V. and Laitone, E. V.,"Surface Waves,’ Handbuch der Physik Bd. 9, Springer and Co., 196U. 3. Gilbarg, D., "Jets and Cavities," Handbuch der Physik Bd. 9, Springer and Co., 1960. 4, Milne-Thomson, L. M., Theoretical Hydrodynamics, 4th ed., Macmillan and Co., 1962. 562 i 5 i 13. 14, 15s 16. re 18. 19. Vartattonal Approaches to Steady Ship Wave Problems Morse, P. M., and Feshbach, H., Methods of Theoretical Physics, in 2 volumes, student ed., McGraw Hill and Co., 1953. Flax, A. H., "General Reverse Flow and Variational Theorems in Lifting-Surface Theory,"J. Aeronaut. Sci., vol. 19, 1952, Ursell, F., and Ward, G. N.; "On Some General Theorems in the Linearized Theory of Compressible Flow," Q. J. Math. and Mechs, vol. 3,.1950. Hanaoka, T., "On the Reverse Flow Theorem Concerning Wave- Making Theory," Proc. 9th Japan Nat. Congress for Appl. Mech. ? 1959. Frostman, O., Potential d'Equilibre et Capacité des Ensembles, Lund Univ., 1935. Inoue, M., Theory of Potential, Kyoritsu, Tokyo, 1952 (Japanese). Hayashi, T., and Mura, T., Variational Calculus, Corona, Tokyo, 1958 (Japanese). Luke, J. C., "A Variational Principle for a Fluid with a Free Surface," J. Fluid Mech., vol. 27, 1967. Lighthill, M. J.,"Application of Variational Methods in the Non-Linear Theory of Dispersive Wave Propagation," Proc. IUTAMSymposia, Vienna, 1966. Garabedian, P. R. and Spencer, D. C., "Extremal Methods in Cavitational Flow," J. Ratl. Mech. Anal., vol. 1, 1952. Wehausen, J. V., "An Approach to Thin Ship Theory," Proc. Int. Semi. on Theor. Wave-Resistance, Michigan, 1963. Yim, B., "Higher Order Wave Theory of Ships," J. Ship Res., September, 1968, Kotik, J., and Morgan, R., "The Uniqueness Problem for Wave Resistance Calculated from Singularity Distributions Which are Exact at Zero Froude Number," J.S.R., March 1969. Van Dyke, M., Perturbation Methods in Fluid Mechanics, Academic Press, 1964, Bessho, M., "On the Theory of the Wave-Resistance," Je.Zosen Kyokai, vol. 105, 1959, 20. 21. 226 23. 24. 25s 26. 27. 28. 29. 30. Bessho Bessho, M., "On the Theory of the Wave-Resistance," (2nd Rep.), AA ee vol. 106, 1960, Bessho, M., "On the Formula of Wave-Making Force Acting on BWOnips | de LieKke Vole 1101960, Bessho, M. and Mizuno, T., "On Wave-Making Resistance of Half Immersed Circular Cylinder and Vertical Plate," Rept. of Defense Academy (Japanese), vol. 1, 1963. Bessho, M., "On the Boundary Value Problem in the Theory of Wave- Making Resistance," Memo. Defense Academy, vol. 6, 1967. Bessho, M., "Gauss' Variational Principle in Boundary Value Problems," Read at Sea-Keeping Sub Com. of Japan tow. Tank Comm., October 1967. Bessho, M., "On Boundary Value Problems of an Oscillating Body Floating on Water," M.D.A., vol. 8, 1968. Bessho, M. and Nomura, K., "A Contributiion to the Theory of Two-Dimensional Hydroplaning," M.D.A., vol. 10 (in print). Miles, J. and Gilbert, F., "Scattering of Gravity Waves by a Circular Dock," J.F.M., vol. 34, 1968. Mizuno, T., "On Swaying Motion of Some Surface-Piercing Bodies," M.D.A., vol. 9, 1969. Mizuno, T., "On Sway and Roll Motion of Some Surface- Piercing Bodies," read at Spring Meeting of Jap. Soc. Nav. Arch. -°1970% Isshiki, H., "Variational Principles Associated with Surface Ship Motions ," read at Korea-Japan Seminar on Ship Hydro- dynamics, Seoul, 1970. 564 Vartattonal Approaches to Steady Shtp Wave Problems APPENDIX A The Linearized Velocity Potential [2,23] Let us consider the flow of water around a ship S, taking the coordinate system as in Fig. 1 and the velocity of the stream at up- stream infinity to be unity. x We Vee tir amtevs (UNDISTURBED) ) ow, ss ae a —_ yf Fig. 1. Coordinate System The pressure p(x,y) on the water surface is given by = (x,y) = - $,(x,y,0) - g&(x,y), (A. 1) in the linearized theory, where p is the water density; g, the gravity constant; (€, the surface elevation, and 9, the perturbation potential (d@= -udx-vdy-wdz). The suffix stands for differ- entiation. The kinematic condition on the water surface is $(x,y,0) = € (x,y). (A. 2) Since the pressure on the free surface is constant, the potential must satisfy the condition $,,(x»y 29) + go(x,y,0) = 0. (A. 3) A solution which has a source singularity at a point Q and Bessho satisfies the above water surface condition can be expressed as t tl - & lim (P,Q) ” (pO) My +0 T fo) k(z4z') +ik(@+ @') , e dk dé i fre ne co “Tr k cos*6 - g tyul cos 8 (A. 4) where P= (x,y,z), Q= (x!,y'sz'), C= (Ov laee )s r(P,Q) = PQ and w=xcos@+ysin®@, w'=x'cos@+y'sin 0. Hereafter, we will call this the fundamental singularity. This solution approaches the following values. asymptotically: 4 1 1 (P.O) eae eo) tea FY (A. 5) x>>x! 2 f x sec O{(z+2') +i(@+")} 2 S(P,Q) ame E tm ec“0 dé. (A.6) x<) = -4 Sf. Vp Vb, dx dy - ral CS, dx, (C. 5) F we have, directly, the reciprocity L(y, 09) = sc ,) (C. 6) i Yi bou ds - Yet, ds s s In particular, from (C.3), L(y.) = - 4 \. Uso, AS = - 4 \ zp, dS = aN Yolo, IS =4 I Tue) dx dy - ral to dx = L(bo, Ho). (C. 7) J dp F The variational problem with the function byt Bessho I” = L"(Yorto) - L(h - Gord - Ye) 1 rw ~~ ~~ = a) (Web, - Yoby - Yoh) dS. (C. 8) Ss is equivalent to the boundary value problem for yw Here, the boundary values, Wo and wo, are given by (C.3). Since a stream function has an arbitrary constant, we should also consider the modified problem with boundary conditions Wo=- Wo = C: constant on S, (C.9) which is the homogeneous problem. [ 22] If condition (C.9) holds, the surface elevation at the fore and aft ends is C (instead of zero for the condition (C.3)), but the x- component of the velocity at the same points is-(1 +gc), by (C.2). Hence, the water flows in and out the body unless C=-1/g. Thus an adequate condition for a surface piercing body is w=-z-— on S&S. (C. 10) Throughout this section, we have treated a class of functions yw and { which are finite and continuous everywhere. As long as the integrals considered exist, the method may be applied with some minor changes to other classes of functions. The question of the uniqueness of solutions will be left to the future. 572 WAVEMAKING RESISTANCE OF SHIPS WITH TRANSOM STERN Be. Yim Naval Shtp Research and Development Center Washington, D.C. ABSTRACT The wave resistance of ships, having transom sterns and bow bulbs, is analyzed by an indirect method. The total wave resistance of a combination of singu- larity distributions for such ships is minimized with various parameters of the bulb and the transom stern at each Froude number. The effect of free surface on the body streamlines near the stern is also analyzed. INTRODUCTION There has been an increasing interest in ships having transom sterns ,not only for high-speed ships like destroyers but also for cargo vessels, because of the advantages of more cargo room as well as modern improvements in techniques of loading and unloading cargoes over the stern. Recently a theoretical study was made by this author [1], and a mathematical model was suggested in view of applying a higher-order ship-wave theory [2]. In this paper, we shall look at the problem again from a different angle, i.e. , approaching the problem in a more practical manner. There are usually two approaches to the ship hydrodynamics problems, direct and indirect. In the direct approach one starts with a given ship and finds a corresponding mathematical representa- tion for it, then, calculates physical quantities, and verifies the results by experiments. On the other hand, one could start with a mathematical model such as a singularity distribution, and calculate the physical quantities of the model, find the corresponding ship form, check the behavior with the experimental results, and utilize the knowledge in designing practical ships. This is an indirect approach. Both approaches are useful in the development of ship science. In Reference 1 by a direct approach, it was found that the transom stern might be represented by a sink line along the stern, having a strength 519 Yim proportional to the stern draft. Now, we would like to use the in- direct approach to the transom stern theory. Noting that a sink line on the free surface is tantamount to the constant pressure distribution ahead of the line and that it produces a negative cosine regular wave and a depression in the free surface immediately behind the sink line, we expect to get streamlines similar to the transom-stern ship from a combination of a normal ship-singularity distribution and the tran- som sink. First the free-surface streamline due to a two-dimensional sink line is plotted to establish the validity of this model, which will be used later for plotting the streamline near the transom. Then, several simple original ship singularities are considered so that basic ship models can be modified to those having transom sterns. Since the transom sink is supposed to behave like a stern bulb [ 1] to cancel stern waves, a bow bulb [ 3,4] made of a source is also considered together with the stern sink to cancel bow waves as well as to supply source strength which helps form a closed body. Thus, optimum strengths for the bow bulb source together with the transom stern sink are calculated to minimize the total wave resistance. The wave resistances with and without the bow bulb and the transom stern are calculated. The Sretensky formula for wave resistance is used since it is much simpler to program in the high- speed computing machine than the Havelock formula. Finally, approximate waveforms near the stern are investigated, which will help in designing a good afterbody near the transom stern. This is part of a project in which the ultimate goal is to under- stand more the physical meaning of a transom stern in the wavemaking resistance of a ship; to obtain better design criteria for ships with transom sterns; to find out the possibility of an improved ship design with the gained knowledge, and, hopefully, to design a good ship with a transom stern, making full use of high-speed computers as well as testing the model in a towing tank. Although this is a small part of ship-designing problems, it is not easy to complete ina short time. At this stage, it is merely hoped that this paper will achieve several objectives: (1) to validate the mathematical model of a ship having a transom stern as a stepping stone to analytical investigation of transom sterns, (2) to determine the practical ranges of parameters within which the application of bulbous bows and transom sterns would be beneficial, and (3) to initiate a computational procedure which would be used for an overall design program using a high-speed digital computer. 574 Wavemaking Reststance of Ships wtth Transom Stern Il. A SINK ON THE TWO-DIMENSIONAL FREE SURFACE Lamb [5] showed a formula for the free-surface shape due to a point sink with the strength M located at x =0, y = 0, moving with constant speed U to -x direction on the free surface, where +x is on the mean free surface pointing right, and ty points vertically upward. The wave height is mx Z where Gee 720 -mx Ky | fe Cin {3 - Si kx) cos kpx + Ci k,x sin kox} 3 m* +k-e zZ in sd>"0 (2) oe es oat (3) 6 Ge geen eo (4) y = 0.577215665 (5) and g is the acceleration of gravity. If this is compared with the wave height due to the distribution of constant pressure p, on x<0O, it can be easily seen that (6) holds. Thus, from the Bernoulli equation the form of the free surface in x<0O can be given by 00 mx ee, k : n=-Bo(i+ kal —z—z dm) , in x<0O (7) Pg 0 EK, The wave height for x= 0 is plotted in Fig. 1 and it clearly shows 575 Yim Fig. 1. Stem Waveform by a Sink Line Transform Stem the appropriateness of this flow model in the vicinity of the transom stern. The integral term is the local disturbance which dies down rapidly with aie and the expression of n in x <0 can be inter- preted as the body streamline of the half body which is formed by a sink on the free surface with a total flux equal to ut.»= upo/(pg) = 27M. This kind of two-dimensional half body was treated by Afremov[ 6] in investigating the flow near a transom stern, thus obtaining pressure distribution of the two-dimensional half body, y=-me, in’ x=0 (8) with the parameter a=0O. He obtained a sharp rise of pressure near the edge of the transom stern where the pressure is zero, the value of atmospheric pressure. This sudden rise of pressure at the stern can also be observed from experiments of planning ships [7]. However, the application of the two-dimensional analyses is generally valid only near the transom stern. In addition, in designing the afterbody near the transom stern, the investigation of hydrodyna- mic interactions between other parts of ship hull and transom stern is important, especially the superposition of ship wave systems from bow, stern, and any discontinuities of the hull; this will be discussed later. Ill SHIPS WITH TRANSOM STERNS AND BOW BULBS It seems reasonable to say that the general representation of a ship with a transom stern and a bow bulb may be achieved by com- bining singularity distributions: a source distribution along a given base surface with either point or line doublets or sources for the bow bulb, and a line sink distribution for the transom stern along a line 576 Wavemaking Reststance of Ships with Transom Stern that is on the free surface, at the stern, and perpendicular to the ship centerplane. In the previous section it is known that the transom stern can be represented by a sink line at the stern, it is therefore natural to deduce that the sink could contribute in cancelling stern waves [1]. Since a moving point sink produces negative cosine regular waves behind, the proper main hull should have a hull shape that produces positive cosine stern waves. To investigate the best shape of transom sterns, in a simple way, we have chosen two simple bare- hull forms, represented in the Michell sense by the following equa- tions: B -10O, Hs =. a7 (17) is the maximum stern draft; and @ is a parameter to be determined for the total optimum wave resistance. Following Ref. 1, the source distribution for the transom stern is taken as He = o, ron) ~ $+ ay ) (18) although actual shape of the transom stern corresponding to this source distribution will be found later by streamline plotting. In addition, a point source located at bow stern, x=-1, y=0,2=42, with the strength o, is considered as a source type of bulb that con- tributes to form a closed body with the sink distribution at the stern as well as to reduce bow waves. IV. WAVE RESISTANCE The wave resistance due to the source distributions mentioned previously can be written according to Sretensky [8] @ 2 {ont 2b 2p ie pesca 35 —= » En eo (P +) (19) SoU L mio 7A oon | where gL 1 k= 2S Ue Fe 578 Wavemaking Reststance of Ships with Transom Stern 1 when sagt — (0, a (20) ts 2 when m2 i _ {i 1 47™ be /5ts f1+(5=) (24) Lw = tank width where the ship is tested, Pee) -{ o(x,y »z) exp} kb(zb tix) t+i2ty =| dS(x,y,z) (22) s ‘Substituting in Eq. (22) all the source distributions for the bare hulls, the transom stern, and the bow bulb, and performing the integration, we obtain TT B, = _2 0 __2Be_\ 4 sin (kb)Ba] , P= E -cos (xb) ace 7: mer + SL E (23) | m P,= ——, {1 -cos(B,7— (24) Ww P, = cos (kb) exp (kz, b’) (25) _ wB, /(2L) 2Be2 Bo Bo Q, = [p= (kb) Pee = | + nmkbL +cos (kb) oa E (26) Q, — (0) (27) Q, = - sin (kb) exp (kz,b’) (28) where subscripts 1, 2, and 3 correspond to the bare hull, transom sink, and bow bulb, respectively, E= |t- exp (KO TH Se (29) 'U 1 = P, +eP, +*)P, (30) oat ed Yim and Q=Q, taQ, +2, (31) with the functions P and Q evaluated and substituted into Eq. (19), the wave resistance may be minimized with respect to such parameters as @ and o,- V. OPTIMUM TRANSOM STERN AND BOW BULB A usual technique is employed to obtain the optimum values of @ and oc, for given k, B,/L, B /L,2,, and H/L. Namely, we solve two linear BGG! eancees pean he in @ and o, 8 =e ae ee Sat a ZS = 2a(P; + Ob) + 20,(P,P, + Q,Q,) + 2(P,P, + O,Q,) = 0 (32) aR a Bey = 2UlPP, + Q,0)) + 20,(P; Pp? +04 +2(P P, +QQ)=0 for @ and Op? where = 2 _ 16n°k 2b m=O etc. from the formula of wave resistance. Cases of B, = 0, B, #0; B, #0, B,=0 and B,#0, B,# O were calculated for each Froude number F,, which will be shown later. VI. SLENDER BODY THEORY For the case of B,=0, the cross sectional area curve is a cusp at both ends. Thus a slender body theory can easily be applied here. The result will be only the’ change of Diced | (34) and al = (35) in P, and Q, in Eqs. (23) and (26) of the wave resistance of the previously described Michell ship, where A is the area of the mid- 580 Wavemaking Reststance of Ships with Transom Stern ship section. The slender ship approximation is useful for the ship with a transom stern because this may give better chances to represent the ship shape near the transom than Michell approximation. Yet for the case of low Froude numbers it is well known that the usual slender ship theory is also very poor. To improve this situation, the slender ship theory can be modified further from that developed by Maruo [11]. In the equations of wave resistance Eqs. (19) through (22), by consecutive applications of integration by part to Eq. (22) P +iQ -{ o exp | kb(zb + ix) + i2my = ds Ss 1 = 15 Se o(0,y,z) exp (eebe + i2ny =) dc Ae o(-1,y,z) exp | kb(zb = 4) erp i2ny —| dc fax eli a a o exp (kzb* Tarai Bde c(x) ~ S m tis |S, o(0,y,z) exp (kzb + i2ny ail dc “i o(-1,y,z) exp | kb(zb - i) + i2ny mt dc c(-1) be 2 . x=0 =m 2.6 o exp (kzb + i2ry —) ac{ ik kb = c(x) A Soe 1 22 ikxh ae (" 2 m - \ dx e'** a, \ o exp (kzb + i2ty ee) ad (36) c(x) where dS = dx de (37) The last integral can be approximated by 5814 Yim 1 ie ikxb n 3 dx e M*"(x) kb ¥¢,, where M(x) -/ o(x,y,z) de (38) c(x) This reduces the influences of the line singularity approximation of ship hull singularities in two ways, (1) by the factor 1 /b*, which corresponds to cos°@ inthe Havelock wave resistance formula, (2) by the factor 1/k® which is the smaller if the Froude number is the smaller. Of course, the number of terms have been increased in the singularities along c(0), which is the intersection of the free surface and the transom stern, and c(-1), which is the straight bow stem line. VII. STREAMLINES To establish the validity of the mathematical model of the ship with the transom stern, the double model[9,10] scheme is inadequate because free-surface waves play a vital role in the flow field of a transom stern. Fortunately a slender ship model [ 11 ,12] gives an easy repre- sentation of the wave height along the train of ship. The wave height is Q and k Bt g(x) +y,>0) pet 4 o(x,y,zZ) ds Ss w © eitw x Re | secto | —__¢ 7 __at dé (40) _W o t-ksec*@-ipsec® where = (x, = x) cos 0+ (y, - y) sin 0 (41) and S is the ship surface. The inner double integral was treated by Havelock and was represented by both a Bessel and a Struve functions. [13,14] We will consider, for simplicity, a source distribution which becomes 582 Wavemaking Resistance of Ships with Transom Stern zero at both ends of a ship such as we considered in Eq. (11), although this is not a basic necessity for evaluating physical quantities, namely, o(0,y,z) = o(-1,y,z) = 0 By making use of this assumption we integrate the wave-height Eq. (39) by parts x C(x, sy)) ae x (x) sy; 0) 4 ti -imxcos@ qd preeie pectpe | | dade doe = - = e dx a> odc rel eee wVe-| o(x) 0 t-ksec’O - ip secé ’ Gis . Re ("f° ey 8) eee ee aed (42) 0 t-ksec *o - in sec where w, = x, cos 8 + (y,; - y) sin 0 (43) 0 itx cos 8 d 2. tliw) +z) I(x,,y,3t,9) = - e dx aS o sec Oe dc (44) = * JYo(x) Integrating I by parts with respect to x, we obtain 1 d tiiw) + 2) I(x, ,y,3t» 9) = ee o(0,y,z) gece ble : dc A ¢(0) : & A Cites exp |ti(x, FH cos 0 +y,-y sin@)ttzbdc (8) -it 8 2 J axe ay {x,y ,z)sec® 0 el(l'*2)| ac (45) = dx “¢(x) For simplicity o,(0,y,z) and o x(-1,y,z) are assumed to be uni- formly distributed , respectively, along the stern on the free surface and along the bow stern vertically. For investigation of the flow field near the transom stern, the contribution of o,(0 Vee) tO tue wave height may be approximated by two-dimens tonal values; the contribution of o,(-1,y,z) may be approximated by the stationary phase; and the contribution from the last integral of I(%, yit, 9). may be approximated by a slender body theory, and will be investigated first, here. 583 Yim Now let us investigate the last (third) term of the above inte- grallI. From the slender body theory 2 ee a o(x,y,z) exp {ti(x, cos © + y, ~y sin 6) ttz} dc 5 d°M(x) = awe z exp {ti(x, cos © + y, sin 6)} (46) x With this approximation, we go back to the wave-height Eq. (42) and consider the values only on y, = 0 ti(x)-x) cos @ tabs,.0) = - B60" axe mttay (" f sectee ST at ae (an ai O it(t-ksec a= ip sec 8) Changing the contour of integration with respect to t in a complex plane such as Havelock [13] used, we have i sels ac mtey seco ae | GEN sesame) oo £5(2, ,0) ae Ky T +f dx M(x) { 2m sec 0 cos (kx, -x sec 0) dO -| -7 (0) T Z a) dx M"(x) [{ 2 log |x,-x| - Yo(k|x,-x|)} X sign (x,-x) + Hy(lex, =) | cee. T ; + dx M"(x}Y,(k|x,-x|) (48) where Y is the Neumann function, H is the Struve function, and sign (x,-x) is +1 for x,-x>0 and -1 for x,-x<0. When only the integral of the first term of I in Eq. (45) is taken, we may approximate the wave height as follows, for x, <0: 584 Wavemaking Resistance of Ships wtth Transom Stern B/2 it(x,cos @- y sin @) Re 39 J t y _ Reo Ace ( ay J do a ie BO ea se C B/2 it(t - ksec“O - in sec 6) ' é @ ave ee it(x cos @- y sin @) = dx R x { d ‘ ao { CSS SS SSS f. * aU -B/2 _ = Oo : t-ksec“@ -ipsec 6 = x \, dx Re oo "ay of at | ce ee -B/2 t-ksec@ - ip sec ® eit(x cos 9- y sin 4), J x, B/2 x 1/2 my ax ot | dy ~ Eu ox lim ao { dt fe) B/2 Tt ¥,; > 0-1/2 0) x genes oir (ty, sin 6) +J sin @(t - ksec*6 - ip sec 6) ao] e* oO itx Vx? +B2/4 +B/2 o» e'* dt - a\ dx log UX tB/4+B/2 4 4 dx +J » XRD eS, > eet |x| (49) Note that t= 0 is not a singularity because of the zero of sin {t(x, cos ® - y sin 0)} in the numerator of the real integrand. The integrand J is defined as follows: 8/2 = ee a (a Z ne of eee iwy sin® al 7? o it(t - ksec’® - ip sec 6) B/2 = Re ay [ a"? at sec of = 1 \ -ity sin @ ity sin @ e te ee ) t-ksec*@ - in sec 0 + o wv ik ae Re =a lim an( d@ cos 8 |! $ sec 8 sin@ TY B--0 o ik sin 8 20,7 /(k°U) (50) 585 Ytm Letting x . _ al dt ies pen er Se re) 0 a we have 00 ™ 1 I -\ = dm in ‘sx< 0 | 0 me +k? 1 ee sis con koe Cie a iioc % Zak ii ee! | 1 1 igs * and _ 20 30S us I, = = cos kx, - EF - % U( 5 - Silex,) cos kx, + Cikx, sin kx,} in x, >0 Therefore, for x, < 0 x | Oy , op ru | 4x1 (log [x,| - 1) - a) dx log (jx? + B2/4 + B/2) 0 +2 (( 2 - sikx,) cos kx, + Cikx, sin kx}] (52a) and for x, > 0 x I C a [4x,(log x, - 1) + sf dx log (¥x* + B“/4 + B/2) 8T 4n 4 T Lis cos kx, - erie es - Sikx,) cos kx, + Cikx, sin kx,}] (52b) The wave height related to the second term of I can be approxi- mated by the method of stationary phase, [5] neglecting the local disturbance: / -1,0,0 u Co=- 4 a ie een a} cos (kx +7) (53) 586 Wavemaking Reststance of Shtps wtth Transom Stern The wave height near the stern due to a point source at the bow can be approximated also by the method of stationary phase: tp = 4k exp (kz, ) / a cos (kx, + z) (54) The wave height near the stern due to the transom stern sink, approximated by the two-dimensional one, was given earlier, say ¢.. As a result, the total sum of ¢,, Cor Sp, and C€, would repre- sent an approximate wave height in the wake of the ship near the transom stern. The streamlines on the body near the stern can be obtained by considering the pressure which is given by the singularity representation as was done in Eqs. (6) and (7). The streamlines near the bow may be obtained by a double model approximation. The inte- grated scheme to produce approximate body streamlines will be pro- grammed for a high-speed computer in the near future. VIII. NUMERICAL RESULTS AND DISCUSSIONS The optimal strength of singularities for the transom stern and for the bow bulb are shown in Figs. 2, 4, and 6. The former is shown in terms of the deadrise angle a of the afterbody near the stern. The latter is shown in terms of the radius of the correspond- ing half body produced by the point source located in the infinite medium. These are all functions of Froude numbers for the given hull shapes. The wave resistance at each Froude number is computed for the given hull with the transom stern and the bow bulb optimal at a given Froude number} see Figs. 3, 5, 7, and 8. 587 =z 2 = a -|4 eee 1 0 0.2 Fig. 2. 0.3 Yim oli Fn 0.8 0.6 = = Ct rad 0.2 0.4 0.5 Optimal Values of Bulb Size and Transom Dead Rise Angle for a Cusped Cosine Ship Using the Michell Ship Theory 588 Wavemaking Reststance of Ships wtth Transom Stern 2.5 By T 70.12, B, =0 0.05, 2 = 0.035 2.0 1 oom Ys 15 WITHOUT BULB a> WITHOUT TRANSOM NN Qin — De [a4 x ~o = 1.0 ; OPTIMUM AT Fn =0.4 WITH BULB es WITHOUT TRANSOM 0.25 0.3 (4 0.5 “Pp 0.35 0.2 0.3 0.4 0.5 Fn Fig. 3. Wave Resistance of Cusped Cosine Ships with Optimum Bows and Transom Stems, using the Michell Ship Theory 589 (r2/A) x 100 Fig. 4. Yim Optimal Values of Bulb Size and Transom Deadrise Angle for a Cusped Cosine Ship, using the Slender Body Theory 590 Qt rad Wavemaking Resistance of Ships wtth Transom Stern 2.5 = 0.006 L 2, =-0.035 WITHOUT BULB 2.0 WITHOUT TRANSOM OPTIMUM AT Fn=0.4 — 1.5 NS wr > Qin ~~” = [= 4 =x oO 2 1.0 0.5 y, 0.2 0.3 0.4 0.5 Fn Fig. 5. Wave Resistance of Cusped Cosine Ships with Optimum Bulbous Bows and Transom Sterns, using the Slender Ship Theory Yim (r/L)" x 10° NR Fig. 6. Optimal Values of Bulb Size and Transom Deadrise Angle for Cusped Cosine-Parabolic Ships B, = Bo, B,/L = 0.06, H/L = 0.05, Z, = - 0.035 592 Wavemaking Reststance of Ships wtth Transom Stern 2.0 PARABOLIC CUSPED COSINE OPTIMUM BOW BULB +TRANSOM STERN AT Fn =.325,AT Fn =375 1.0 PARABOLIC Bas 0, Bo= 0.12 10° x r/( 2v'.’) 0.5 PARABOLIC+SINE, B, = By = 0.06 0.2 0.3 0.4 0.5 Fn Fig. 7. Wave Resistance of Ships of Cusped Cosine and Parabolic Waterlines with and without Bulb and Transom Sterns (x, = 0) 593 Yim 2.0 B, By — = — = 0.06 L L bile p= 0.035 r70.05 71 WITHOUT BULB AND TRANSOM STERN OPTIMUM AT Fn=0.375 Fn=0.325 0.5 0.2 0.3 0.4 0.5 Fn Fig. 8 Wave Resistance of Ships of Cusped Cosine and Parabolic Waterlines with and without Bulbous Transom Stern (x, = 0.05) 594 Wavemaking Reststance of Ships wtth Transom Stern For a slender body model, Eqs. (11) and (35) are used for the bare-hull source distribution, which is called a cusped cosine ship here. For a Michell thin ship model, computations are per- formed for Eq. (11) for cusped cosine and parabolic ships. For the combined bare-hull source distribution, the influence of the location of transom stern xg, is shown in Figs. 6 through 8. It can be under- stood that there is an optimal location for the minimum wave resis- tance as in the case of a bulbous bow; however, it is not computed here. It is interesting and reasonable to see that the optimum size of the bulb becomes the smaller for the large Froude numbers over 0.4, and eventually the strength becomes negative at F,>0.5. In other words, for a large Froude number, a ship behaves like a single point doublet far behind the ship so that the only way to reduce the wave height is to reduce the ship volume. Indeed it is possible to take advantage of the transom stern as well as the bulbous bow to reduce wave resistance in the Froude number range F,< 0.5 by a proper combination of the ship hull shape and the transom stern and the bow bulb. For the case ofa high-speed ship such as a planing boat, there is no alternative to evade the detrimental cavitation without having the full separation occur at the transom stern, whether it is beneficial to the wave resistance or not. The numerical results of streamlines are not given in the present paper because of their complexity. The approximate method of computation of the streamlines near the stern is shown in the previous section. When the ship draft is fairly large, compared with the wavelength, the ship shape from the singularities can be approximately computed from the double model. However, fora transom stern, the free surface follows immediately behind the usually shallow drafted afterbody. Thus, the modified slender body theory used in the previous sections, combined with a double model approach to the forward part of ship hull seems to be promising. Some imaginative approximate configurations from the concerned source distributions are shown in Fig. 9. Last, but not least, the importance of experiments on the design of ships with transom sterns should be emphasized. There are very few experimental results available [15]. However, this has to be done in close coordination with the theory so as not to grope in the dark. The theory is now on a solid foundation. More time and effort are needed to achieve experimentally usable and complete results on ships with transom stern. In the future, the author hopes to finish a systematic computer program for designing ships with bulbous bow and transom stern that includes information about the wave resistance, the bulb size, the transom-stern draft, and the main hull shape. 595 Yim ae A Cusped Cosine Ship with Bulb and Transom Stern 1 = A Cusped Cosine Ship with Bulb and Transom Stern 2 eee er A Cusped Cosine - Parabolic with Bow Bulb and Transom Stern (x # 0) Fig. 9. Imaginative Diagram for Ships with Bulb and Transom Stern ACKNOW LEDGMENT This work was carried out under the General Hydrodynamic Research Program of the Naval Ship Research and Development Center, The author expresses his thanks to Mr. J. B. Hadler, Head, Ship Powering Division, NSRDC, for his encouragement in numerous discussions. Thanks are also due to Dr. P. C. Pien for his valuable advice, Mr. H. M. Cheng for reviewing the manuscript and his help in editing, and Mrs. L. Greenbaum for her patient effort in the preparation of the manuscript. REFERENCES 1. Yim, B., "Analyses of Waves and the Wave Resistance due to Transom-Stern Ships," Journal of Ship Research, Vol. 13, No. 2, June 1969, 2. Yim, B., "Higher Order Wave Theory of Ships," Journal of Ship Research, Vol. 12, No. 3, Sept. 1968. 3. Maruo, H., "Problems Relating to the Ship Form of Minimum Wave Resistance," Proceedings of Fifth Symposium on Naval Hydrodynamics, ONR, Department of the Navy, 1964. 596 Te Wavemaking Reststance of Ships wtth Transom Stern Yim, B., "Some Recent Developments in Theory of Bulbous Ships," Proceedings of Fifth Symposium on Naval Hydrody~ namics, ONR, Department of the Navy, 1964. Lamb, H., "Hydrodynamics," Cambridge University Press, Cambridge, England, 1932, Sixth Edition. Afremov, A. Sh., "Sbornik Statey po Gidromekhanika 1 Dinamike Sudna," USSR, L967, PPe 130-146, Sottorf, W., "Experiments with Planing Surfaces," Tech. Memo. No. 739, NACA, 1934, 8. Sretensky, L. N., "On the Wave-Making Resistance of a Ship Moving Along in a Canal," Philosophical Magazine, Vol. 22, Seventh Series, 1936. 9. Inui, T., "60th Anniversary Series, Vol. 2," The Society of Naval Architecture of Japan, 1957. 10. Pien, P. C., and Strom-Tejsen, J., "A Hull Form Design Pro- cedure for High Speed Displacement Ships," Transaction of The Society of Naval Architects and Marine Engineers, 1968. 11. Maruo, H., "Calculation of the Wave Resistance of Ships, the Draught of Which is as Small as the Beam," Journal of the Society of Naval Architects of Japan, 1962. 12. Tuck, E. O., "The Steady Motion of a Slender Ship," Ph.D. Thesis, Cambridge, 1963. 13. Havelock, T. H., "Ship Waves: the Calculation of Wave Pro- files," Proc. of the Royal Society, A, Vol. 135. 14. Wigley, W. C. S., "Ship Wave Resistance," Trans. N.E. Coast Inst. Engineers and Shipbuilders, Vol. 47, pp. 153- 196, 1931. 15. Michelsen, F. C., Moss, J. L., Young, B. J., "Some Aspects of Hydrodynamic Design of High Speed Merchant Ships," Trans. of SNAME, Vol. 76, 1968. LIST OF SYMBOLS A Midship section area b Defined by Eq. (21) B Ship beam BB, Ship beams associated with two hull forms given by Eqs. (9) and (10), respectively 597 oe Pars Se ober ee eG “amet OD DY 6,620,503 by Yim Cosine function defined by Eq. (4) Domain defined by Eq. (12) Defined by Eq. (29) U /(gL) Acceleration of gravity Draft of ship Stern draft Froude number, Ve Expression defined by Eq. (44) Expression given by Eq. (51) Expression defined by Eq. (50) Lg/u* Ship length Source strength Pressure Expression defined by Eq. (22) Expressions defined by Eqs. (23), (24) and(25), respectively Expression defined by Eq. (22) Expressions defined by Eqs. (26), (27) and (28), respectively Bulb radius Wave resistance Ship surface Sine function defined by Eq. (3) Velocity at x - @ Ship speed Tank width Rectangular coordinates The x coordinate of the location of transom stern sink Ship hull forms given by Eqs. (9) and (10), respectively The z coordinate of the location of the point source for bulb Dead rise angle of transom stern defined by Eq. (16) Wave height Two-dimensional wave height Wave heights due to the first, second and third term of I in Eq. (45) Wave height due to bulb source 598 Wavemaking Reststance of Ships wtth Transom Stern Gs Wave height due to transom stern sink o Source strength for ship hull |S Source strength given by Eqs. (11) and (12), respectively oT, Source strength for bow bulb 0; Source strength for transom stern p Density of water Y Quantity given by Eq. (5) DISCUSSION Georg P. Weinblum Institut fur Sehtffbau Hamburg, Germany Some general remarks may be permitted, especially from the point of view of application. So far investigations of more practical character deal pre- ferably with the bow wave formations, while linearized wave re- sistance theory treats with equal love the forebody and the afterbody of displacement ships. Few experiments only have been conducted to check, ceteris paribus, the advantage of form symmetry with respect to the midship section in real fluid. Such tests have been performed by the present writer with bulbous forms by comparing simplified ship hulls: a) without a bulb (naked hull), b) a bulbous bow only, c) a stern bow only, d) bulbs symmetrically arranged at bow and stern. These experiments are useful in the present context, starting from the author's and my personal viewpoint, that in ideal fluid a similarity can be reached in wave effects due to a transom stern and a bulbous bow (because of a similarity in form representation by dipole arrangements). The sketch annexed shows an impressive improvement by the symmetrical bulbous ship design (d), and this indicates, that the combination bulbous bow + transom stern should be useful as shown theoretically by the author. 599 Yim The application of linearized wave resistance theory to slow full ships following our present state of knowledge overstrains this theory heavily. This theory should not be discarded, however, completely as long as it is used as a heuristic principle only, i.e. as means to look for solutions which must be checked experimen- tally. It is recommended to use in this sense several earlier inter- esting papers published by the author. Considering the present critical attitude towards linearized wave resistance theory in general, I wish to state that its use (including perhaps some correcting "im- provements" for practical purpose still can be highly recommended in case of medium or especially high Froude numbers. It should be remembered that in the range of the large wave resistance hump values computed by Michell's theory may differ by an amount only from experimental results which corresponds to the scatter of the latter derived from models in different scale. Therefore I welcome the author's present second approach on the subject of transom sterns although the correlation between form and generating singularities (so far rather indicated than carried out) may still require further studies. With regard to the author's statement about lack of systematic experimental evidence it is suggested to look into report No. 167 Institut fuer Schiffbau Hamburg and to check if something useful can be found there. 600 Wavemaking Resistance of Ships wtth Transom Stern DISCUSSION S. D. Sharma and L, J. Doctors Untverstty of Michtgan Ann Arbor, Michigan In an oral discussion at the Symposium Professor Maruo and the first-named discusser challenged the validity of the author's Fig. 1 because they felt that the wave profile should have been dis- continuous at the location of the line sink or the pressure step, x=0,. Dr. Yim insisted that his figure was correct, arguing that a similar curve is shown in Lamb's "Hydrodynamics," p. 405, In the meantime, we have examined the problem more closely and arrived at the following conclusions. Let us examine the case of the pressure step first. Consider a two-dimensional pressure distribution, p(x) = po{1 + sgn(x)} /2, (D1) on the mean free surface, z=0, moving steadily with speed U along the direction of Ox. The resulting motion can be described by a velocity potential ¢(x,z) subject to the conditions ,,(x»2z) + ,,(x,z) = 0, (D2) p(x) - pU¢,(x,0) + pgo(x) + ppUd(x,0) = 0, (D3) US (x) + o,(x,0) = 0, (D4) $, 2(*»-00) = 0, (D5) where z = ((x) describes the free surface elevation and the limit . — +0 is understood as usual. It is easy to verify that the solution is ” exp (ikx +kz) 2 (x,y) = - (eo/now) | ae aa dk, ky = g/U ’ (D6) @ G(x} = - (p,/pg) {1 tsgn(x)}/2 - p,/no8) | ae dk. (D7) 601 Yim The limit » — +0 then leads to the following real expression 00 pgt(x)/p, = -{1 +sgn(x)} /2 - Senta) { exp (-wlkgx|) i ) 1 + w* - {1 - sgn (x)} cos (k,x), (D8) which is indeed continuous at x =0 although the wave slope E. becomes infinite at that point. This is evident in the accompanying Figure (see Curve 1), but does not show up on the scale used by the author in his Fig. 1. ___. _ _ Curve2: Wave profile for a line sink, sk 2TIm | ~~ ~ Curve 1: Wave profile for a pressure step, “p~ ° gx/U? > Fig. Di. Comparison of Wave Profiles for a Line Sink and a Pressure Step On the other hand, one can also approach the problem as the limiting case of a submerged line source as the submergence tends to zero, The velocity potential for a line source of strength m (that is, output 2mm per unit length of line) on the line x =0, 25 - f is found to be 2 2 i - x + (ztf) exp {k(z-f + ix)} 602 Wavemaking Reststance of Ships with Transom Stern and the wave profile C(x) = Ud,(x,0)/g, (D10) now becomes © -Wikox! yds _ sgn (x) e {cos (wkof) + w sin (wkof) } - US (x) /2mm =. sane) | See dw -k.f - {1 - sgn (x)} e Ka cos (kof). (D1i1) It is obvious that for f= 0 the wave profile of the line source be- comes discontinuous at x=0. For any nonzero value of f the profile remains theoretically continuous at x = 0. However, for all practical purposes it is discontinuous in the limit f - 0 as shown in the accompanying figure (see Curve 2) for kof = 0.00001. If we assume p,/pg = - 2mm/U, then the potentials of the pressure step (D6) and the line source (D9) become identical in the limit f—- 0. But the wave profiles differ by the first term of (D8). Incidentally, the author's Eq. (1) differs from our (Di1) by a factor of 2. But his relation (6) seems to have a compensating error of factor 1/2 so that his wave profile (7) does agree with our (D8). We have not investigated what effect this discrepancy has on the author's further calculations of wave resistance. But we did notice an obvious slip in Eq. (18) for the strength of the line sink repre- senting the transom stern. If o, is regarded as a line density, apparently a factor L is missing on the R.H.S. On the other hand, if o, is interpreted as a surface density, then the R.H.S. should contain the Delta function 6(0) as a factor. We also find the idea of using a line sink to represent the transom stern rather unconvincing. The line sink would tend to force the flow around the corner of the transom, which in practice occurs only at low speeds, but in a highly viscous manner not tractable by ideal fluid theory. The case of real interest is the one at high speeds where the flow separates smoothly from the transom. In this regime, we feel that the line sink should not be used so that the excess sources in the hull can produce a semi- infinite half-body. We would appreciate the author's comments on this point. Notwithstanding minor differences of opinion, we wish to congratulate the author on his imaginative approach to a very inter- esting problem. 603 Yim REPLY TO DISCUSSION B. Yim Naval Ship Research and Development Center Washington, D.C. The author would like to acknowledge Prof. Weinblum's encouragement. The author fully agrees with him on everything he mentioned. As is indicated in the text, the model of the transom stern assumes the linear free-surface condition although, in practice, very often nonlinear phenomena, e.g., a rooster-tail or cavity collapse, do occur. Therefore, this point also needs care, in addition to the error in Michell's thin ship theory or the slender ship theory. REPLY TO DISCUSSION B. Yim Naval Ship Research and Development Center Washington, D.C. The author sincerely appreciates the deep interest shown by Drs. Sharma and Doctors regarding his paper. About the validity of Fig. 1, the author will attempt to make a detailed explanation. First, the author would like to point out that the discussers agree by their Eqs. (D6) and (D9) that, with P, /Pg = - 2mm/U, the potential due to a point sink located on the free surface at x =0 and the potential due to the corresponding uniform pressure distribution along the free surface from x=-ooto x=0 are identical everywhere in the flow field and on the boundary. This fact has long been known. Thus, velocities of the two cases are identical everywhere, and the wave heights of the two cases are identical from the relation (Di0). Namely, one problem with the given pressure distribution is in fact the same problem with the properly given source distribution as in many fluid mechanics problems. Admitting this fact, it is impossible to claim that the representation by pressure gives 604 Wavemaking Resistance of Ships wtth Transom Stern the smooth boundary and that the representation by source gives the discontinuous boundary. To elaborate a little more, the discussers did not notice that the boundary ahead of the location of sink or at x<0, z=0 is no longer a free surface but has become a part of body boundary formed by the sink flow field, where the pressure is a constant different from zero. This may be understood better if we consider another identity of potentials due to a point sink on x = 0, z=0 and a uniform distribution of doublet in - x direction on -o%.9 Vie P ybod wha * e A ie “ro iw) t2, & XO 27 BO Apeersot, os MING Set do Sides) 95 ul , gases gpa Wt ‘tony antwollor og eh dao < ¥RAbE! Cr ridlaieta pom vilbanon ayo otsow bo «hit ofgeedie eu ‘gaicma al & lo Loqaq e' todos tel ,atlunod Eas veo off ddlees > wae bye il ( ir i rr BOW WAVES BEFORE BLUNT SHIPS AND OTHER NON-LINEAR SHIP WAVE PROBLEMS Gedeon Dagan Technton-Israel Institute of Technology Hatfa, Israel and Marshall P. Tulin Hydronauttiecs, Incorporated Laurel, Maryland NOTATION a Draft at bow of a completely blunt shape (dimensionless) b, b Outer and inner coordinates of point B in the t and @ planes al aah a ah He ee Arbitrary constants e.. »€,..d), rd,. | | t I Cp Drag coefficient D'! Drag force; D = D'/opu*t! f' = »' + iy' Complex potential; F = f'/g'/2T'3/2, £ = f'g/u'>; t = £'/u'T! Fr, = U'/(gL')'”? Length Froude number Fr,= u'/(gT')”2 Draft Froude number h'(x') Function describing the body shape; H = h'/T'; b= he A,' Forebody length; £, = £,' g/u'® ti Characteristic length; £ = £'g/u'? Nea / 1 Dimensionless free-surface elevation P = p'/pu'? Dimensionless pressure t' Jet thickness; t = t'g/u'*; ee th/T! Tr Draft wisu! + iy’ Complex velocity; w= w=w!'/U'; W= w'/(gt')'# 607 Dagan and Tultn U' Velocity at infinity Ze x! tay’ Complex variable; z = z'g/U"; 2 =2'/T'; 22 T! 8 Angle at bow 5,2 55, A, Gauge functions e= t'c/u'* Small parameter to=i6 + i Auxiliary variable; @ = ¢/e n' Free-surface elevation; 7 = n'g/U"; n= n'/T'; Q = £n(1/w) N= 7/7 Logarithm of complex velocity = T + i@ I. INTRODUCTION The conventional linearized theory of ship waves is based on a first-order perturbation expansion in which the length Froude num- ber is of order one, while the beam Froude number (thin ships) and/or the draft Froude number (slender of flat ships) tend to infinity. While the theory is in fair agreement with laboratory results in the case of schematical fine shapes (e.g. Weinblum et al. [1952]), it is of a qualitative value at best in the case of actual hulls. To improve the accuracy of the linearized solutions, second order nonlinear effects have been considered, either in the free-surface condition or in the body condition (e.g. Tuck [1965] , Eggers [1966]). A different nonlinear effect, overlooked until recently for the case of displacement ships, is that associated with the bow bluntness. It is well known from the theory of inviscid flow past airfoils or slender bodies (Van Dyke [ 1957]) that the linearized solution is singular near a blunt nose in the stagnation region. The singularity may be removed by an inner expansion in which the length scale is a local one associated with the nose bluntness. In the case of a free-surface flow with gravity the phenomenon is more complex. The pressure rise in the stagnation region is associated with the free-surface rise and the formation of a breaking wave or spray and the existence of a genuine bow drag. The inner expansion of the Bernoulli equation shows that the inertial nonlinear terms become more important than the free gravity term, for sufficiently high local Froude numbers. The bow nonlinear effects have been recognized a long time ago in the case of planing plates (Wagner [1932]), but they have been always associated with a relatively high Fr,, such that the lift /buoy - ancy ratio is of order one. Here we are primarily interested in the case of displacement ships which move at a small Fr, and the hull 608 Bow Waves and other Non-Linear Shtp Wave Problems position beneath the unperturbed level is practically independent of Fr,. Nonlinear inertial effects may be important nevertheless near a blunt bow. A systematical experimental confirmation of the role played by the bow bluntness has been provided recently by Baba [1969]. From towing-tank tests with three geosims of a tanker (Cp = 0.77) it was found that in ballast conditions ata Fr,* 1.2 a breaking wave appears before the bow. At the maximum Fr, tested (Fr, = 0.24, Fry= 1.7, Fig. 1a) the energy dissipated in the breaking wave con- tributed 18 per cent of the total resistance, while the energy radiated by waves gave only 6 per cent. Baba has suggested a two-dimensional representation of the breaking wave of this experiment, as if it were uniform and normal to the bow (Fig. 1b), and has estimated equivalent length as half the beam. The drag coefficient per unit length, corresponding to a two-dimensional flow across the breaking wave is Cp=D'/0.5 pU"T' = 0.08 for Fr,=1.7. Sharma [1969] has indicated a larger breaking wave resistance for a higher block coefficient tanker and has suggested that the bow bulbs main effect is to reduce the breaking wave resistance. With the development of large tankers, as well as large and rapid cargo ships, the study of the bow free-surface nonlinear effect becomes particularly important. We present here some of the results of our last year's studies, which are reported in detail in two reports (Dagan and Tulin [1969, 1970]). In this first stage we have attacked the two-dimensional prob- lem of free-surface flow past a blunt body of semi-infinite length. The two-dimensional study is a necessary step in the development of a theory for three-dimensional bows since it provides a valuable gain in insight at the expense of relatively simple computations. Moreover, it gives an estimate of the bow drag of flat ships and opens the way to more realistic computations by further approximations. Taking the length as semi-infinite is very useful from a mathematical point of view and it is equivalent to the limit Fr, ~ 0. This assumption is entirely justified for the small Fr, considered here and for determining the bow flow, which is not sensibly in- fluenced by the trailing edge condition. Il. THE FREE SURFACE STABILITY (SMALL Fr, EXPANSION) We consider the two-dimensional gravity flow past the body of Fig. 2. The box-like shape has been adopted for the sake of com- putational simplicity, but the method can be easily extended to any other shape. When Fry is small the free-surface is smooth. We assume that breaking wave inception is related to the instability of the free- surface. According to Taylor's criterion (Taylor [1950]), the 609 Dagan and Tultin (a) (VIEW FROM FRONT) “KT bh WAVE (UNIFORM) Fig. 1. Baba's [1969] experimental results. (a) Breaking wave before a tanker; (b) Baba's two-dimensional representa- tion of the breaking wave. > SHVUEUI (0) ‘b) “ A 01) B(+1) ae me (c) Fig. 2. Small Fr, flow past a box-like shape body. (a) The physical plane; (b) the linearized dimensionless physical variables; (c) the auxiliary ¢ plane. 610 Bow Waves and other Non-Linear Shtp Wave Problems free-surface becomes unstable when the normal acceleration vanishes. In our case this occurs when the centrifugal effect related to the free-surface curvature offsets the gravity acceleration. Since we expect the free-surface to become steep as Fry increases, there must be a critical Fr, characterizing instability. The gravity free-surface problem is, however, nonlinear, To linearize it we consider a small Fr, perturbation expansion, i.e. and expansion for a state near rest. Referring the variables tor (Pig: 2)? and (gT ')'/2 and expanding as follows: F(Z)= @ +iW = Fr,F(Z) + Fr °F,(Z) +... (1) W(Z)= U - iv = Fr,W (Z) + Fr 3w,(Z) +... (2) N(X) = Fr/°N (X) + Fr,4N,(X) +... (3) we obtain from the exact free surface and body boundary conditions the following equations: at first order (Fig. 2b) Vv, =0 (ASBA) (4) wW,=1 (X — - o) (5) i.e. a flow beneath a rigid wall replacing the free-surface at its unperturbed elevation. In addition Napii-u,). (<0, ¥ =6) second order W, = - UN, (AS, X<0, Y= 0) (7) U, = 0 (SBA, X>0, Y=H(x)) (8) i.e. a flow generated by a source distribution along the degenerated free-surface, and N,= 4 0,0, Der Onna p) (9) It is easy to ascertain that Wo is zero at infinity such that 6i1 Dagan and Tulin the total source flux is zero. Similarly N, is zero at both the origin and infinity. In fact the first order solution gives the exact values of N at infinity and at the stagnation point, the higher order approxi- mations correcting only the free-surface shape between these two anchor points. The above expansion is consistent and hopefully uniformly convergent. It differs from that suggested by Ogilvie [1968] who has kept terms of different order in the same equation in order to obtain waves far behind a submerged body. The solution of the first order approximation for the box-like shape body (Fig. 2a) is obtained in terms of the auxiliary variable C as 72 W, = ($3) F,=t/n (10) where the mapping of the linearized Z plane (Fig. 2b) onto ¢ (Fig. 2c) is given by z= (e? - 1% +2 tn [(e? - 1)? - 2) (14) T Hence, by Eq. (6) we have N, = tes For the second order approximation (Eq. (7)) we get /2 y, = 64H one a7 (SiS Sad rip. =O) (12) W, given along the ¢ real axis (Eqs. (8) and (12)), leads by a Cauchy integral to { mr dd &, (6) a T Re \; We () (2p © 1 ‘ork (ee eee ay/2 -i\2; +(g44) BEE aS U2 and No as functions of €, are easily found from Eqs. (143) and (9) (for details see Dagan and Tulin [1969]). The shape of the free-surface at second order is given in Fig. 3. As expected, the profile becomes steep as Fr, increases. 612 Bow Waves and other Non-Linear Ship Wave Problems VV l 0.25 X=x'/T N=n'/T' 0.20 N (Xx) = Fr? N, X)+ Fr# No (x) +... ” 0.15 ae een (xy : ie = Z Ti Fr? 0.10 T=9.5 ~~ 1.0 == Gp : 0. Pas) S 3.0) 2.0 Fig. 3. The free-surface shape in front of a rectangular body The dimensionless pressure gradient component normal to the free-surface is proportional to 2 2 a{ tard, \ © * 0, p= 0) (20) where 615 Dagan and Tulin -0 and cs a= be | w, do (21) -0 at second order é3= «? Re (Siz + iw,) = - Re[ (w, + 26,) S| (E<0, p= 0) (22) Im w,=-5Imw,” (6>0, >= 3) (23) We determine now the first order solution by replacing the body along § >0 by an unknown pressure distribution (equivalent to a vortex distribution, see Stoker [1957]) of strength g,(§). The function k \() satisfying Eq. (19) and the radiation conditiea becomes peo -v) Keg) en BILL Vi] go) (25) Eq. (20) becomes now @ . =) Re e*”) Fi [i(é - v)] g(v) dv = - (6) (26) (@) The integral Eq. (26), with a displacement kernel, may be solved by the Wiener-Hopf technique. The Fourier transform of Eq. (27) reads + 1 - + M(A)G, (A) = —— [N, (4) + H,'(\)] (27) l > ! | where M, G,; H,, and N, are the transforms of the kernel, 8)? h, and the freee paises profile, respectively. The kernel's transform has been factorized by Carrier et al. 616 Bow Waves and other Non-Linear Shtp Wave Problems [1967] 1 1 (20)'* 1+[a| M(A) = =[M()]-M(\)] (28) The separation of Eq. (27) can now be accomplished, pro- vided that we select a given body shape h,(x). We limit ourselves here to the case of the completely blunt shape of Fig. 4d. The forebody length &£, is of order €, such that at the limit € ~ 0 the bow degenerates at first order into a point singularity at the origin. Any shape with the same length scale of the forebody will yield the same first order body condition. With (Fig. 4e) (20) 1 1 1-a 1 mes (2m) In ie (2m)”? 5 ES A (29) we obtain from the separation of Eq. (27) eu Le ed { 1 1 : 6°00 = a X- ba) geze| een actor GP 89 where the last term, representing eigensolutions, results from the application of Liouville's theorem, cj, being arbitrary. Equation (30) cannot be inverted exactly, because of the integral appearing in M*(\), but the inversion can be carried out for large \ by expanding M*(A). After carrying out this process (see Dagan and Tulin [1970]) we arrive at the following expression for g,(6) in the vicinity of the origin n a Be eae GE’ eyes Soe (E> 0) (3 1) | (né)/2 » plane where d,, are related in a unique manner to Cy, + | 617 Dagan and Tultin From Eq. (31) we obtain n BOM a V2 ans w, (6) = ey + OVC TSA art )ar 2, vi Pe (C0) (32) which is the central part of our analysis. The expression of the second order solution, satisfying Eqs. (22), (23), and (32), was found to be at do, w(t) = sap + Olln £) + aa (33) Summarizing the results for the outer expansion, we have, with the estimate t = O(e?) €a_ 4 bd, Pe cat bs + €2 ide, we (me) pir3/e ont pivsre + €O(t' fn f) (34) : 1/2 e\. 2 Zi Gae € [ai + 2a(£) J+e) cine +S-fine 1 t 52} 2 ter) vie (35) ' er, and eo, being again constants related to Cis Coe The velocity has the familiar square root singularity at first order and a source singularity at second order. The free-surface is continuous and attached to the bottom at first order, while at second order it rises at infinity. The eigensolutions of the problem, which represent in fact the linearized solutions of a free-surface flow past a flat horizontal plate, as well as the flow details near the bow will be subsequently determined with the aid of an inner solution. It is worthwhile to mention here that only at second order are the details of the adopted model (i.e. the jet) manifested in the solution. Any other model attached to the bow will produce an identical first order solution. 618 Bow Waves and other Non-Linear Shtp Wave Problems (b). The Inner Expansion and Its Matching with the Outer Solution We stretch now the coordinates and adopt the following inner variables, t = t/e; ee: z= 2/€; t= t/e; b = b/e (36) and expand the function Q = £n(1/w) = 7 +i@ ina perturbation series ~ Q Q, + A,(e)Q, +... (37) For the body of Fig. 4d we obtain from the inner expansion of the exact equations the boundary conditions for %, specified in Fig. 4f, which represent a nonlinear free-surface flow without gravity. The conditions at infinity are provided by the matching with the outer expansion. Only in the case of the straight bow of Fig. 4e are the inner conditions so simple. Ina general case we have to solve an integral equation for 6 [ Wu, 1967] orto start with a given 6(€). The solution of Wo is readily found in the form B/t ~ = ~ 72 7 ~ (/2 Ale am NG d... =. Oem oad exp ( ) Tae ) (38) (Eye ete sale th eb where the exponential represents the eigensolutions of the problem, ce being arbitrary constants. Expanding W, for large { we obtain ~ d posh He37.6(89) 654 Ge ae p/n) + 2[ (E /m)? - b/w] ae is t z=( Lt/mb af =F - l(t /n)® - pB/u] + 2 /a)”® - pb/l® tn? Wo t jp td Nie fee (72 -cénG-ait2dt +... (40) Before proceeding to the matching we rule out the eigen- solutions appearing in Eqs. (38), (39) and (40) because they lead either to an infinite velocity in the jet or to an infinite jet thickness, depending on whether doi are positive or negative. The matching of wo and z (Eqs. (39) and (40)) with w and z (Eqs. (34) and (35)) now gives 619 Dagan and Tulin we ea eee eat b= Se Oy da, =O (41) and both inner and outer solutions are uniquely determined. Our estimates of t and b are confirmed, Eq. (41) showing that t = O(e2) and b = O(e3/4). The nonlinear character of the prob- lem is manifest in the inner expansion. We have now a uniform solution which can be written by adding the inner and outer solutions and subtracting their common part. (c) The Bow Drag The horizontal force acting on the bow is found by the pressure integration in the inner zone (Fig. 4f) J J 0 ~ B=! p dy = 5 Im (+89 dz=tim! (2+%)(1- 5) dt B B b w 1 Since w is analytical in the lower { half plane the integra- tion along BJ may be replaced by integration at infinity (at H) and around the origin J. After expanding 1/w, +w. at infinity and near ae 0 the origin, we get for D D fae (1 + cos B /m) (43) The same result, excepting the cos B/m term, may be obtained directly from the first order outer expansion. To roughly compare the result of Eq. (43), with Baba's findings, lets assume that the bow is completely blunt with B= 1/2 and,.a-=,l...For’¢€ = Wy ures = 0.34 we have Gu= 2Di= 0.34 (44) which is roughly four times larger than the value estimated by Baba. At this stage it is difficult to find which of the following factors explain this discrepancy: The asymptotic character of the solution, the lack of details on the bow shape or, may be the most important, the crude representation by Baba of a three-dimensional flow by a two-dimensional equivalent (Fig. ta). Future experiments 620 Bow Waves and other Non-Linear Shtp Wave Problems and theoretical developments will give the answer to this question. The method presented here is applicable to other bow shapes, like blunt round bows. In this latter case the bow drag appears at higher order than in the completely blunt case. The extension to other shapes, as well as to three-dimensional bodies is left for future studies. IV. CONCLUSIONS Theoretical models of breaking wave inception and of a free- surface bow drag have been derived for the case of a two-dimensional gravity free-surface flow past a blunt body. In both cases the effects are nonlinear and are related to the important role played by the inertial term of the Bernoulli equation in the vicinity of the bow. The results are of the order of magnitude of those found by Baba [ 1969] , but an improved verification has to be done by carrying out two-dimensional experiments. The theory presented here may be extended, with additional approximations, to three-dimensional flows. ACKNOWLEDGMENT The present work has been supported by the Office of Naval Research under Contract No. Nonr-3349(00), NR 062-266 with HYDRONAUTICS, Incorporated. REFERENCES Baba, E., "Study on Separation of Ship Resistance Components ," Mitsubishi Tech. Bul. No. 59, pp. 16, 1969. Carrier, F. G., Krook, M., and Pearson, C. E., Functions of a Complex Variable, McGraw-Hill, pp. 438, 1966. Dagan, G., and Tulin, M. P., "Bow Waves Before Blunt Ships," HYDRONAUTICS, Incorporated Technical Report 117-14,1969. Dagan, G., and Tulin, M. P., "The Free Surface Bow Drag of a Two-Dimensional Blunt Body," HYDRONAUTICS, Incorporated Technical Report 117-17, 1970. Eggers, K. W. H., "On Second Order Contributions to Ship Waves and Wave Resistance," Proc. 6th Symp. of Naval Hydro- dynamics, 1966. 621 Dagan and Tultin Ogilvie, T. F., "Wave Resistances: The Low Speed Limit," The Univ. of Michigan, Dept. of Naval Arch., Rep. No. 002, pp. 29, 1968. Stoker, J. J., Water Waves, Wiley, New York, 1957. Taylor, G. I., "The Instability of Liquid Surfaces' When Accelerated in a Direction Perpendicular to Their Plane, I," Proc. Roy. Soc. ; London, A, 201, pp. 192, 1950. Tuck, E. O., "A Systematic Asymptotic Procedure for Slender Ships ," de Ship Res. ; Vol. 8, No. 1s, pp. 1.523, 1965. Tulin, M. P., "Supercavitating Flows -- Small Perturbation Theory," Proc. of the Internat. Symp. on the Application of the Theory of Functions in Continuous Mechanics, 2nd Vol., pp. 403- 439, 1965. Wagner, H., "Uber Gleitvorgange an der Oberflache von Flussig- keiten,". Zammi,. Vol.. 12, No. 4, pp. 193-216; 1932. Weinblum, G. P., Kendrik, J. J. and Todd, M. A., "Investigation of Wave Effects Produced by a Thin Body," David Taylor Model Basin Report No. 840, pp. 19, 1952. Wu, T. Y., "A Singular Perturbation Theory for Nonlinear Free- Surface Flow Problems," Int. Shipbldg. Prog. Vol. 14, No. 151, pp. 88-97, 1967. * * F RF FR DISCUSSION L. van Wijngaarden Twente Institute of Technology Enschede, The Netherlands I would like to ask a question about the authors' interpretation of Taylor's instability. Th -y use as a criterion for marginal sta- bility that the normal component of the pressure gradient at the interface between gas and fluid vanishes. This is indeed the case for a plane interface. 622 Bow Waves and other Non-Linear Shtp Wave Problems For a plane interface and y Vv variables indicated in Fig. 1 we pave fluid dv __ ap interface Pp a e ot oy gas For negligible gas density Taylor's Figs 4 result is: : Ov . Op stable: SE = 0; En Ors instable: UNE > 0s Sp <0. Ov oy marginal Op _ 0 stability: oy The question is whether this criterion (ap /8y = 0) holds also for more general interfaces. As an example consider the spherically symmetric implosion of an empty gas bubble (Fig. 2). The equation for the radius R is 2 fluid RR +32R =-- Po 2 p where Pg is the pressure far away in the fluid, pw > 0. From this relation ee o2 RR= -Pm_>R' 0) with the water, maintaining constant displacement. If at the same time we restrict attention to the case when the canal is constant in section area, and in the region of interest at the free surface has a width W independent of Z (locally vertical sides), then we have from (2.3) that (U* - 2gZ (A, t WZ - S) = UA, (2. 4) 6314 Tuck and Taylor where S is now constant. On squaring, (2.4) gives a cubic equation which may be solved directly for Z. Alternatively, following Constantine [ 1961], we may treat the problem in an inverse manner, solving for the speed as a function of Z and obtaining in non-dimen- sional form |e me [calcd 3 (2.5) {= ("d= where F = U/Vgh (2.6) d= "2h (257) and s=S/A,, (2.8) with h= A,/W (2.9) as the mean depth. Constantine [| 1961] discusses the nature of the flow predicted by (2.5) and presents curves of F against d. Equation (2.5) permits only a restricted range of Froude numbers F for any given blockage coefficient s, namely O, 9d _ Be ot pee -~ h <2 < 0, (3:52) O19 2 Oulart ae, (3,3) the linearized free surface condition 2 C) 290 Bye TU ge H Oo on =O, pair and the linearized hull boundary condition 3 +2 AUb Gd eeon. 'y=80,. (3.5) Both equations (3.4), (3.5) are linearized on the basis that the ship is thin, i.e. that its slope b'(x) is everywhere small, so that $ and its derivatives are small, as is the free surface elevation. We now apply the assumption that the depth h is small. The corresponding approximate equations may be obtained formally by stretching the z-coordinate with respect to h, then carrying out an asymptotic expansion in terms of the small parameter h/L, see Wehausen and Laitone [1960]. However, the leading terms are easily obtained by simply expanding @$ ina Taylor series with re- spect to z, about the bottom value z= -h, i.e. (x,y, Z) = (x,y, -h) a (z th) 6,(x,y ,-h) a (2th) ,.(x,y 5h) Trewin 6 (35 6) The second term in the expansion (3.6) vanishes by (3.3), and we use (3.1) to express $,, in terms of $,, and $y, , writing 635 Tuck and Taylor (x,y,z) = o(x,y,-h) - 3(z th V7b(x,y,-h) +... (3.7) 2 where V = (a°/ax°) + (a /ay*). On substitution in (3.4) we obtain immediately to leading order in h the equation - ghV°o(x,y,-h) + U2, (x,y,-h) = 0, or 2 ra (1 - F) Fa + Fa] olx,y,-n) = 0, (3.8) where F = U/ygh. Equation (3.8) is formally identical to the equation describing linearized aerodynamics in a two-dimensional flow of a compressible fluid, with the Froude number F playing the role of the Mach number (see e.g. Sedov [1965]). Indeed, the problem of solving (3.8) subject to (3.5) is identical to that for subsonic (F < 1) or supersonic (F > 1) flow over a non-lifting wing of thickness b(x), and we may use directly the results obtained in aerodynamics. Of course Michell was not so fortunate, and we should say that aerodynamicists could have used Michell's results, the first solution of any boundary-value prob- lem for a non-trivial general boundary. The character of Eq. (3.8) is different according as F< 1 when it is elliptic and F >1 when it is hyperbolic, and different mathematical properties and solution techniques apply in these two cases. Here we quote only the final result for the hydrodynamic part of the pressure distribution over the body surface, namely - pu? (C” v\(é) aé 2mj1- F* 4-0 x= 6 p= (3.9) itt <4 2 eee on Ei ce ty 2m F - 1 the bar denoting a Cauchy principal value. Note that the pressure given. by (3.9) is a function of x only. The z-dependence has been neglected as part of the shallow-water approximation and there is no y-dependence because of the thin-ship approximation. The complete pressure distribution is obtained by adding to (3.9) the hydrostatic pressure. The only possible force on this cylindrical body is in the x- direction, and there is no net moment. Michell found by integration 636 Shallow Water Problems in Shtp Hydrodynamics of p times the slope b'(x) that the net force (wave resistance) is Oy “Pit. Palate No doubt Michell was disappointed in his conclusion of zero wave resistance in the more important sub-critical regime, and indeed this conclusion may have contributed to the neglect of his shallow-water results. However, we can expect no other result from the present theory, which lacks a dissipation mechanism in the sub-critical regime to leading order. This feature it has in common with linearized aerodynamics. However, in aerodynamics the drag vanishes even according to nonlinear theory for Mach numbers every- where less than unity, whereas in the present water-wave problem it is only to leading order that the wave-making dissipation mecha- nism disappears. No second-order calculations seem to have been carried out to find the non-zero subcritical wave-resistance, and this is a problem which merits attention. Michell's analysis for a wall-sided "ship" was extended to ships of arbitrary cross-section by Tuck [ 1966]. In this case we can expect to predict a squat effect, and, although the analysis in the 1966 paper is rather complicated, the main conclusion is quite simple. By the method of matched asymptotic expansions (Van Dyke, [1964]), Tuck showed essentially that Michell's result (3.9) for the pressure still holds, providing we interpret the function b(x) as the mean thickness of the ship at station x, averaged over the full depth of the water, i.e. set “bis =< s(x) (3.11) where S(x) is, as in Section 2, the cross-sectional area of the ship at station x. Thus, for example, we obtain again Michell's wave resistance formula (3.10) but with (3.11) used to rewrite it in terms OE. FO) On the other hand, the modified geometry of the ship does now allow non-zero vertical-plane forces and moments, and we find an upward heaving force 637 Tuck and Taylor v2 £ £ ue) ax ( dEB'(xjS'(E)log |x-£|, F241 (eee 2thy 1 - £ -f 4 S'(x) BGe)-dx3, F > 1 (32:43) -f ee ip 2h/ F*- 4 and a bow-up pitching moment ig £ L - —— ax f dé (xB(x)) *S'(S) log |x-€ | ’ F< 1, jal 2th: 1i-F -f -f (3.14) S'(x)xB(x) dx, F> 1, (3515) Suc 2hyF -1 -2 where B(x) is the width of the ship at the waterline at station x. In fact the force written down in (3.12) is invariably negative at sub- critical speeds so that a sinkage is to be expected rather than a lift. Tuck [ 1966] also gives formulae for the actual sinkage and trim displacements of the ship in response to these forces, assuming equilibrium with hydrostatic restoring forces, and provides some computed results which are in reasonable quantitative and excellent qualitative agreement with experiments of Graff et al. [1964]. There is a need for more experiments, especially in the very low water depth range, but it would appear from the comparisons so far made that the theory is quantitatively accurate so long as the depth is less than about one eighth of a ship length, and the Froude number based on depth is less than about 0.7. It may be worth observing here that the integrals in (3.12) - (3.15) are fairly insensitive to the shape of the section curves B(x), S(x). For instance, the ratio ° fi ax fi a6 B'(x)S'(§) log |x-6| ve n n iy B(x) dx = hi S(x) dx (3.16) is nearly an absolute dimensionless constant, taking values between 2.0 and 2.4 over a very wide range of B(x), S(x) curve shapes, including actual ships and mathematically defined curves. Thus a nearly universal approximation to the subcritical vertical force is 638 Shallow Water Problems tin Shtp Hydrodynamtcs ou? Q Q ee rf B(x) ax f S(x) dx (3.17) : 2mhVi - Fs? J -¢ -t with a fixed value of \. From this follows a similar approximation to the actual sinkage, say a displacement of 6 downwards, where re J) Sx) dx aa poe ee (3216) 6=2- Ti or Finally, introducing the displaced volume £ a { S(x) dx (3.19) and making a further assumption (justified in most practical situations) that F << 1, we have z nae Wy a eh ie (3. 20) where L = 2£ is the ship length. In practical terms, if 6, L and h are infeet, V _ in cubic feet and U in knots, and if we insert reasonable (conservative) values for X and g, (3.20) implies 6=0.13572- (3321) We put forward this formula (3.21) quite seriously for practical use by anyone interested in a quick estimate of squat in a wide expanse of shallow water. One should note the quadratic dependence on forward speed, the inverse dependence on water depth, the proportionality to displacement (at fixed length) and inverse square dependence on length (at fixed displacement). In a subsequent paper, Tuck [1967] extended the 1966 work to the case where the ship is moving along the center of a rectangular channel of width w, considering only the sub-critical case. The assumption made was that w is comparable with the ship length L; however the results obtained were uniformly valid, in the sense that the infinite- width results were reproduced as w/L — o, while as w/L— 0 we obtain predictions which may also be obtained by ele- mentary (linear) one-dimensional theory as in Section 2. An inter- 639 Tuck and Taylor esting mathematical feature of this small width limit is that the singu- larity at F = 1 becomes stronger as w/L— 0, changing from in- verse square root (e.g. (3.12)) to inverse first power (e.g. (2.15)). Another conclusion in the 1967 paper was that the ratio between the sinkage at width w and that at infinite width was almost independ- ent of the shape, size or speed of the ship, depending only on the parameter (w/L)yi-F*. Thus, starting with any estimate (even (3.21) !) of the infinite width sinkage, we may further estimate the effect of finite width by use of the universal curve given in the 1967 paper. For example, at low values of F, a channel width of two ship lengths increases the sinkage by 10%, one ship length by 33%, over the infinite width values. For channel widths less than one ship length a one-dimensional theory as in Section 1 is sufficiently accurate and probably to be preferred. IV. THREE-DIMENSIONAL THEORY OF SQUAT IN INFINITE WIDTH, FINITE DEPTH We begin the present section by presenting the solution! Suppose S(x) is the cross-section area curve of a slender ship moving at velocity U in water of finite constant depth h, and let 4 S*(k) = \ S(x) e!®* ax, (4.1) -£ Then consider ; 00 , : 00 -idy Bea oe dk kS"(k) cin ( d= 4 q -00 - 00 : [e* 4 eo cosh QZ k* cosh q(z th) ] (4, 2) sinh qh(k® - Kq tanh qh) : sinh gh i where K=g/U* and q= (k* +22, Although the expression (4.2) is extremely complicated, it has the following properties, easily checked: 1 Gd 0. tk eer = Ayo Jace, (4,3) Gay OG) Oz = Oh om tz =. in, (4. 4) (iii) °k'(8d/dz) +(0'6/dx = 0 “on z=0, y #0, (4.5) 640 Shallow Water Problems tn Shtp Hydrodynamics (iv) o-4 S (x) dog 2's f(x), P O(e tos +) as’ r= 0. (4. 6) The physical interpretation of $ is as follows. The contribution from the first term "e%" inside the square brackets is just the potential of a line distribution of sources, of strength proportional to S'(x), in a fluid which extends to infinity in all directions. The contribution from the second term inside the square brackets cor- rects for the presence of a bottom wall at z= - h, while the last term in the square brackets corrects for the presence of a free surface at z=0. The last property (4.6) indicates that the given solution (4. 2) can serve as an outer approximation (see Tuck [ 1964]) and will match an inner approximation which satisfies the correct boundary condition on a slender hull surface. Thus (4.2) gives the disturbance potential for flow around a slender ship in finite depth of water, no shallowness assumptions having been made. The function f(x) in (4.6) is of crucial importance, and may clearly be considered in three parts, arising from the three terms in the square brackets in (4.2). Let us write f(x) = fop(x) + g(x) (4.7) where L US'(x) Aa? U S'(x) - S'(E) ns - Pa) = log 4(z x ) a i: dé aaa (4. 8) (Tuck and von Kerczek [ 1968]) is the corresponding function for the double-body flow in an infinite fluid (no bottom or free surface), while g(x) is the contribution from the second two terms in the square bracket of (4.2), and takes the value . 00 ae g(x) = ae ak ks Hic) tK* A* (1) (4.9) where eee ee (4.10) k“ - Kq tanh qh In the integral (4.10), if Kh< 1 there is a pole on the real q-axis at q = 4q,(k), where 641 Tuck and Taylor 2 k = Kq, tanh qoh (4.11) and this pole must be avoided by passing beneath it in order that the waves are behind the ship. Thus the real part of A*(k) may be written as a Cauchy principal value integral, which can be evaluated by standard numeri- ( cal quadratures,whereas the imaginary part of A‘“(k) can be obtained from the residue at the pole, and we have wi do So , xh <1 JA “(k) = (4.12) 0. Rae's s Once A* is determined, g(x) follows by further numerical quad- ratures from (4.9), if actual numerical values of g(x) are required. However, our main aim is to find the forces on the ship, which follow from g(x) via the pressure distribution, given by P(x,y»Z) = Polx,y,Z) - pUg'(x) (4.13) where Pog (Xs ¥ > Z) is the pressure on a double body in an infinite fluid. Hence the vertical force is £ F, = F, - oul , dx B(x)g'(x) 20° —— = FS - eu dk k?S"(k) B*(k) A*(k), (4.14) -00 and the trim moment is L F =F +pul dx xB(x) g'(x) 2 eee Fe +h, \ dk k?S*(k) xB*(k) A*(K), (4.15) -00 where F%, Fo are the corresponding quantities for the "submerged" half of the infinite fluid double body, B(x) is again the waterplane 642 Shallow Water Problems in Shtp Hydrodynamtes width curve, B*(k), xB “(k) are Fourier transforms (cf. (4.1)) of B(x), xB(x) respectively, and a bar denotes a complex conjugate. The quantities Ber He must be computed separately, e.g. by computer programs such as those of Hess and Smith [ 1964] or Tuck and von Kerczek [1968]. Alternatively, one may estimate them experimentally. It is important to note that F%, F® are independent of water depth and of Froude number; indeed, when divided by euU- these are constants which are a property of the hull geometry alone. An interesting special case isa hip with fore-and-aft sym- metry, where (neglecting viscosity) FS = 0. In addition, since S(-x) = S(x) and -xB(-x) = -xB(x), s* is real and even with respect to k whereas xB” is imaginary and odd. Asa result, only the imaginary part (4.12) of A*(k) contributes to the integral in (4.15), and we have ish Ga Bie ae \ dk k°S*9(xB*)9A* (4.16) OF = (Kh) = <1 7 (4.17) (ee) ae dapack S*(K)IxB"(k), F> 1. 27r 2 2 (@) do = k Similar, but more complicated, results are obtained for F on he and ¥F.- es when the ship does not possess fore-and-aft symmetry. Evaluation of the supercritical trim moment F,; from (4.17) re- quires only a single numerical quadrature (apart from the prior estimation of the Fourier transforms S*, xB*). However, in the general case an additional numerical quadrature is needed to deter- mine the real part of A *(k) from (4.10). The shallow-water limit of the above finite-depth results corresponds to letting kh ~ 0, i.e. we let the depth tend to zero relative to a typical effective wavelength 2n/k. In particular, from (4.11) we have qgh > 0 as kh—O and hence k—~ykKkhq, or do—~ Fk. Thus for F >1, (4.17) gives for ships with fore-and-aft symmetry that poe Pe ES *k *k = dk kS* 9xB" (4.18) 2 Q on, eee S'(x)xB(x) dx, (4.19) 643 Tuck and Taylor in agreement with (3.15). It is quite straightforward to show ina similar manner that all of the shallow-water results of Section 3 are reproduced in the corresponding limit, even when fore-and-aft symmetry is not assumed. In carrying out this shallow-water limit, one may wonder what happens to the "double-body" terms F® and F2. The answer is that they are of course quite independent of water depth and hence nothing happens to them, and in principle they remain in the formulae, However, when the depth is small the shallow-water terms formally dominate the total expression for F, or F,, so that Bg and ee may be neglected. In Fig. 2 we present computed finite-depth sub-critical sink- age and super-critical trim for a ship with parabolic waterline and section-area curves, a length of 600 ft, beam of 60 ft, draft of 20 ft, and block coefficient 0.533. This geometry and size was chosen for analytical convenience, but is not unlike a destroyer hull. The super- critical trim was calculated directly from (4.17) by a single numeri- cal quadrature (since this hull has fore-and-aft symmetry), whereas the sub-critical sinkage required an extra numerical integration of (4.10), and, furthermore, required separate estimation of the infinite- fluid contribution td in (4.14). DEPTH = 30 FT. SINKAGE (FT.) TRIM (DEGREES) A O oo SHALLOW 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 FROUDE NUMBER BASED ON DEPTH Fig. 2 Finite depth squat for a ship 600 ft long and 20 ft draft 644 Shallow Water Problems tn Ship Hydrodynamics Instead of using sophisticated numerical techniques for F®, in the present case we estimated ro by assuming that the under- water hull could be approximated by an equivalent spheroid, with the same length and displacement. If we define the slenderness e€ by {2 V € = (ee. (4. 20) which is equal to the beam/length ratio of the equivalent spheroid, the exact infinite fluid force on the submerged half of the spheroid can be obtained from the formula given by Havelock [ 1939] in the form h FO: -pu'- ( B(x) dx + C_(e) (4, 21) where -2 2 mee S log == ) ‘ (4. 22) 2y 1-2 {=./4-«2 In fact, since € is generally small, an adequate slender -body approxi- mation to (4.22) is bole 2 C.(€) => tap tt - 3€ ee) & 2 jo! es Cele) = - € “(log 5€ +5) + 3 + O(e4log €). (4. 23) The result (4.22) is of course in exact agreement with Havelock's [| 1939] more general formula for ellipsoids whose sections are not circular. On the other hand (4.23) is within 10% of computa- tions based on Havelock's formula for general ellipsoids, providing € < 0.2 and the half-beam/draft ratio of the general ellipsoid lies between 0.65 and 7.0, a range of parameters which includes the usual ship dimensions. Some preliminary numerical computations using the theory of Tuck and Von Kerczek [ 1968] have shown that (4.23) is a good estimate for non-ellipsoidal geometries, while Havelock [ 1939] himself made satisfactory comparisons between his ellipsoid estimates and experiments of Horn [ 1937] on actual ship models, so that there are grounds for believing that (4. 21) subject to (4.23) gives a useful prediction of the infinite fluid zero Froude number sinkage force. The finite depth computations were carried out for water depths of 100, 60 and 30 feet. The results for the smallest of these depths are in very close agreement with the shallow-water theory of Section 3, shown dashed on Fig. 2, over the complete range of Froude numbers shown. This indicates that a water depth/ship length 645 Tuck and Taylor ratio of 1 in 20 is quite adequately shallow for the use of shallow- water theory. But the results at twice this depth are also in reason- ably good agreement with shallow-water theory, the latter theory underestimating sinkage by about 20%. The corresponding under- estimate at 100 ft depth is about 40%, so that one should consider a water-depth/ship length ratio of 1 in 6 as too great to use shallow- water theory for sinkage. However, it must be pointed out that the difference between the finite depth and shallow water predictions of sinkage is pre- dominantly due to the influence of the term F® in (4.14). If this (positive) term is left out of (4.14) the finite depth computations at all depths merely oscillate about a mean which is quite close to the shallow-water curve. These oscillations are clearly visible in Fig. 2; they are quite similar to the humps and hollows in theoretical wave resistance curves, and have the same explanation, as an inter- ference effect. One may thus speculate that, by ignoring these oscil- lations, we may obtain a useful empirical scheme for computing finite depth sinkage by adding the shallow water estimate (e.g. (3.21)) to the infinite fluid zero Froude number estimate computed by (4. 23). Further work needs to be done to test this suggestion, which is of some significance since computations based on the theory of the pres- ent section are too complicated and expensive of computer time for general use. No direct comparisons of the finite depth computations with experiment have yet been made, but the differences between the finite- depth and shallow-water results appear to be in the right direction to explain most of the discrepancies already noted by Tuck [ 1966] between shallow-water theory and the experiments of Graff et al. [1964]. In particular, more detailed computations for a depth of 100 ft, (h/L = 0.167) show that the peaks in the sinkage and trim curves occur at about the right Froude numbers, 0.94 and 0.98 respectively, and that the trim starts to become significant at a Froude number as low as 0.8. V. THE ACOUSTIC ANALOGY FOR UNSTEADY LATERAL FLOW In the remainder of this paper we shall be concerned with a very special aspect of the problem of ship motions in shallow water, namely computation of the exciting force on a stationary ship under the influence of regular beam seas. A more general formulation and partial solution of the problem of ship motions in shallow water is given by Tuck [1970]. Most other work on ship motions in shallow water, e.g. Freakes and Keay [1966], Kim [ 1968], concerns finite depth rather than shallow water. An exception is a number of papers by Wilson (e.g. [ 1959]) on responses of moored ships in harbors, but no account is taken there of ship geometry. Mention must also be made of the thesis by Ogilvie [1960], in which the shallow-water asymptotic expansion was developed rigorously for a class of two- 646 Shallow Water Problems tn Shtp Hydrodynamics dimensional diffraction problems. We now suppose that, except for the scattering effect of the ship, the flow field is described by an incident plane wave moving in the y-direction, with wavelength 21/k, frequency o, and amplitude A, where k and o are related by the shallow-water dispersion relation o =7ghk. (5.1) The potential of this wave will be taken to be the real part of beiet F where >= A en: (5.2) 0 h ik * : In fact, (5.1) and (5.2) are of course already approximations to the exact formula (e.g. Wehausen and Laitone [ 1960]) for small-ampli- tude waves in finite depth h; for instance the exact expression for do is that given by (5.2) multiplied by cosh k(z +h) /cosh kh, which tends to unity as kz —~ 0. This incident wave is modified by the presence of the ship. We suppose that the total field is then the real part of (od) + d)e'*, where = $(x,y,z) is the disturbance potential, which is to be found. The exact equations satisfied by » are (3.2), (3.3), an unsteady free surface condition analogous to (3.4), namely gse-o%}=0 on 2=0 (522) and a boundary condition on the ship's hull of the form Fa (bo + 4) = 0 (5.4) ony 2 i where 8/8n denotes differentiation normal to the hull. We construct first an outer shallow-water approximation to @ in the same way as in Section 3, i.e. by expanding in a Taylor series’ with respect to (z th). Equation (3.7) still applies, but on substitution of (3.7) in (5.3) we now find 2 2 647 Tuck and Taylor i.e. (x,y,-h) satisfies the Helmholtz equation Vb +k} = 0 (5.5) in the (x,y) plane. The Helmholtz equation is of course simply the "reduced" wave equation, and so applies to any scalar wave problem in two dimensions, for sinusoidal time dependence. In particular, it describes linear acoustics in two dimensions (e.g. Morse [ 1948]), and many results obtained in solving acoustic problems may be utilized. For example, we may treat immediately the scattering of a thin cylindrical ship, as in Michell's model of Section 3, which extends from top to bottom of the water. An important difference from the theory of Section 3 is that, even in the limit as the thick- ness tends to zero, the thin "ship" is capable of scattering beam waves. Thus, to leading order, the problem is independent of thickness, and reduces to acoustic scattering by a ribbon or strip of zero thickness placed broadside on to the waves, with a "hard" boundary condition ao A [2 = constant on y = 0,, [ao {<0 (5.6) The exact solution can be written down as a series of Mathieu functions (Morse and Rubenstein [1938]). Results for the scattering cross section, the far-field polar diagram and the force on the strip can be computed from this series, but only with some difficulty, especially for high frequency. Alternatively, integral equation for- mulations of the problem can lead to useful high and low frequency asymptotic solutions (Hénl, Maue and Westphal [1961] ) or even to efficient numerical solutions (Taylor [1971]). Such numerical results are included with the discussion of the general case in Section 7. Once again this idealized ship is deficient from the practical point of view. In particular, it allows no account to be taken of flow beneath the keel of the ship. In any situation of real interest, wave energy is not only scattered, diffracting around the ends of the ship, but also transmitted underneath the ship if there is any reasonable amount of clearance. The most interesting situation is that which applies when the amounts of disturbance scattered and transmitted are of the same order of magnitude; we shall see later that this is true for draft/water depth ratios in the range 0.5 to 0.95. We shall retain the approximation that the ship is thin, and hence slender, since it must have small draft. However, the 648 Shallow Water Problems in Ship Hydrodynamics possibility of fluid passing underneath the ship means that we must replace the "hard" boundary condition (5.6) by a more general con- dition, expressing in effect a relationship between the velocity (8¢/dy) + AVg/h of fluid passing "through" (i.e. under) the strip y = 04, |x|< £ and the pressure difference (proportional to potential difference) across the strip, which causes this underflow. Thus we write C) a tA [R= FPS on y = 0s, acl ek, (5.7) where P = P(x) is the "porosity" of the ship section at station x. If the ship is actually touching the bottom, then P=0 and (5.7) reduces to (5.6); at the other extreme, if there is substantial clearance, P—- oo andthe jump in potential $ across the strip tends to zero, leading as expected to zero force on the ship. In the following section we indicate how to obtain the porosity P(x) for any given ship and sea bottom geometrical configuration. The problem of solving (5.5) subject to (5.7) is then identical to that for acoustic scattering by a "semi-soft" or porous ribbon with finite acoustic impedance, see e.g. Morse [1948]. However, no general procedure seems to be available in the acoustic literature for solving this type of problem, and we present in Section 7 a numerical approach based on an integral equation formulation. It should be remarked that as k—~ 0 the present problem reduces to uniform steady streaming flow across the ship, the free surface being replaced by a rigid wall. This problem was discussed by Newman [ 1969] , who presented solutions for the added mass of the ship in such a flow. The present theory can be considered a generalization of Newman's theory to allow for waves, and gives results which agree with Newman's in the limit as ki ~ 0, i.e. as the waves become long compared with the length of the ship. VI. THE DETERMINATION OF THE EFFECTIVE POROSITY The problem formulated in the previous section is to be inter- preted as an outer problem, which provides a solution for the scattered field everywhere except within a beam or two of the center plane y = 0 of the ship. In this latter region, the outer solution must match an inner approximation which describes the detailed flow field beneath the hull. This flow can easily be shown (Tuck [ 1970] , Newman [ 1969]) to be locally two-dimensional in the (x,z) plane, and to satisfy the two-dimensional Laplace equation S$ + SF =0 (6.1) 649 Tuck and Taylor in that plane. Furthermore, the free surface condition reduces toa rigid- wall boundary condition SP = 0 or z=0. (6.2) Thus the inner problem is identical to that treated by Newman [ 1969], who assumed that (6.2) was valid everywhere. The boundary condition at "infinity" for this inner solution is that the inner solution should match the behavior of the outer solution in a common domain of validity, say many beams away from y=0, but not so far away that y is as large as £ or 2m/k. In effect, this simply means that the inner boundary condition (5.7) for the outer solution becomes the outer boundary condition for the inner solution. Thus the inner approximation to the disturbance potential ® must satisfy fia [k= Ps neha aries (6.3) which is satisfied if $ is asymptotically independent of y, i.e. p+ Avg/h as y->+o (6.4) implying LL pig as yi (0. (6.5) The boundary condition on the hull is (5.4) but where now 8/8n denotes differentiation normal to the hull cross section IT at station x, and where, since ky is small in the inner region, we may replace the incident wave 9, by b—- A JE (xz ty), (6.6) i.e. by an incident stream of speed Ayg/h. Then (5.4) becomes Sat ogy (ey ae A Fon on Fr. (6.7) Thus the inner approximation to $ is the potential for flow due to 650 Shallow Water Problems in Ship Hydrodynamics motion of the section I as if it were an infinitely long rigid cylinder moving in the y-direction with velocity Ayg/h, the fluid at y =+ 00 being at rest. The problem specified by (6.1) subject to (6.2), the bottom condition (3.3), the hull condition (6.7) and the rest condition (6.5) is a classical Neumann boundary value problem, and @ is deter- mined by these conditions apart from an additive constant. If in addition we prescribe the natural symmetry condition that o is an odd function of y, in conformity with (6.4), then $ is uniquely determined, and the actual limit of ¢ as y—~+0o must, via (6.4), provide a determination of P. A number of techniques are available for solving this two-di- mensional boundary-value problem. If the section is sufficiently sim- ple (e.g. rectangular) a solution may be found by conformal mapping methods (Flagg and Newman [1971]). For actual and quite general ship sections we have (Taylor [1971]) developed a computer program based on the methods used by Frank [1967] for a similar problem. In this method one represents the flow by a distribution of sources around the section I, with variable but unknown density. These sources individually satisfy the "free" surface and bottom conditions (6.2) (3.3), but not, of course the hull boundary condition (6.7) on IT. We now attempt to choose the source density function in order to satisfy (6.7), thereby obtaining an integral equation from which the source density is in principle obtainable. Since analytic solutions for general TI are out of the question, we adopt a numerical approach in which [I is first approxi- mated by a set of straight line segments, on each of which we assume the source density to be constant. The integral equation then reduces to a set of linear algebraic equations for these unknown constant source strengths, and this set of equations is solved by direct matrix inversion. It is convenient to define a blockage coefficient Cx) = pry: (6.8) as the inverse of the porosity. With this definition, we see from (6.4) that to obtain C(x) we merely divide the potential by minus the speed Ayg/h of the motion of the section I and take the limit as y—~> too. Thus, if our aim is solely to determine C(x) or P(x), we may, without loss of generality, take A=-vh/g for the purpose of the present section only, and identify C as the limit of $ as y —~ + oo, a quantity which is readily evaluated from the numerical solution for the generating source strengths. 651 Tuck and Taylor The program has been tested by comparison with Flagg and Newman's [ 1971] computations for a rectangular section, and gives good agreement over the range of dimension of interest. For instance, with a rectangle of total width 0.25 and a (submerged) draft 0.1 in water of depth 0.125, Flagg and Newman's [ 1971] computations give C = 0.598, while our program with 24 segments on the bottom of the rectangle and 12 segments on each side gives C = 0.603. Although this accuracy (1%) is already very good in the present application, it can easily be and has been improved by use of a larger number of segments, especially in the neighborhood of the corners. Another check is by means of asymptotic estimates for small clearances (Taylor[1971]). A formula which is valid for arbitrary sections, providing they have substantially vertical sides 2b units apart and a substantially flat bottom c (<t) $x y) $(x,y »z3€) $j (x,y »Z) G(x,y,z3€) }i(x,y »z) X (x,y,z) bj (x,y »Z) W(x,y, Zz ,t) Wj(x,y»z) WwW Q: Gciyss) terms in a far-field expansion of €(x,y;e) (cf. ZA(Xsy3€) ) function mapping the complex variable z onto an auxilliary (¢) plane (in Section 5) steady part of free-surface deflection in ship-motion problem free-surface deflection in problems in two dimensions part of free-surface deflection in low-speed problem in two dimensions (see (5-7) ) angle variable in cylindrical coordinate system time-dependent part of free-surface deflection in ship- motion problem g/U*, a wave number in steady-motion problems density of dipoles on a line (see (2-40) ) density of dipoles on a line (see (2-40) ) density of dipoles on a surface w*/g, a wave number in oscillation problems displacement in the j-th mode of motion (see Section 2232) water density density of sources on a line density of sources on a surface velocity potential (the arguments may vary, but 6 generally denotes the complete potential function ina problem) in Section 5.4, potential for the problem in which the free surface is replaced by a rigid wall terms in a far-field expansion of $(x,y,z;e) normalized potential functions (see (2-73), (3-28) ) terms in a near-field expansion of (x,y, ze) normalized potential functions (see (3-44) ) velocity potential for the perturbation of a unit-strength incident stream by a slender ship normalized potential functions (see (276) ) time-dependent part of velocity potential in ship-motion problem (with forward speed) normalized potential functions (see (3-45) ) radian frequency of sinusoidal oscillations normalized potential functions (see (3-46) ) 665 Ogilvie MISCELLANEOUS CONVENTIONS 1) —— Potential: The velocity is always the positive gradient of the potential function. 2) Coordinates and Orientation: In problems involving a steady ncident flow, that flow is always inthe positive x direction. The vertical axis is the y axis in 2-D problems, the z axis in 3-D problems. 3) Time Dependence: In problems of sinusoidal oscillation, the time aependen ce is always in the form of the exponential function, e!#f, In such problems, the real part only is intended to be used, but we do not indicate this explicitly in general. 4) Fourier Transforms: These are denoted by an asterisk. For example, foe) @ * A o (k) -{ dx es (x) : o(x) = + dk ae oe (ils -@ ue -@ tok sak Gee ~i(kx +f y) 6 (k,£;z) -{ iY dx dy e o(x,y,Z)- -00 “-00 5) Principal- Value Integrals: These are denoted by a bar through the integral sign: 6) Order Notation: There are three symbols used: O.0,”. a) "“y = O(x)" means: |y/x|< M as x0, where M is a constant not depending on x. b) "y = 0(x)" means: ly/x| > 0 as x0. y ~ £(x) " means ly - £(x) | = o(f(x)) as x0. 666 Singular Perturbation Problems in Ship Hydrodynamics I. INTRODUCTION This paper is a survey of a group of ship hydrodynamics problems that have certain solution methods in common. The problems are all formulated as perturbation problems, that is, the phenomena under study involve small disturbances from a basic state that can be described adequately without any special difficulties. The methods of solution make explicit use of the fact that the disturbances of the basic state are small. Mathematically, this is formalized by the introduction of one or more small param- eters which serve as measures of the smallness of various quantities. The solutions obtained will generally be more nearly valid for small values of the parameter(s). However, the problems will also be characterized by the fact that they are ill-posed in the limit as the small parameter/(s) approaches zero. Thus, we call them singular perturbation prob- lems. Special techniques are needed for treating such problems, and we have two which are especially valuable: 4) The Method of Matched Asymptotic Expansions, and 2) The Method of Multiple-Scale Expansions. The first has a well-developed literature, and it has been made particularly accessible to engineers by Van Dyke [1964]. The second, which has a longer history, is perhaps less well-known, but we now have a textbook treatment of it too, thanks to Cole [ 1968]. Because of the availability of such books, my treatment of the methods in general will be extremely terse. The necessity for treating ship hydrodynamics problems as perturbation problems arises most often in the incredible difficulty of handling the boundary condition which must be satisfied at the free surface. Even after neglecting viscosity, surface tension, com- pressibility, the motion of the air above, and a host of lesser matters, one can still make little progress toward solving free- surface problems unless one assumes that disturbances are small -- in some sense. Historically, it has commonly been assumed that the boundary conditions may be linearized; in fact, this has so commonly been assumed that many writers hardly mention the fact, let alone try to justify it. The two methods emphasized in this paper can also be applied to problems involving an infinite fluid. In fact, neither method was applied specifically to free-surface problems until quite recent times. Section 2 of this paper is devoted to several infinite-fluid problems. My justification, quite frankly, is almost entirely on didactic grounds. The methods can be made much clearer in these simpler problems, and so I include them here, although in some 667 Ogilvie cases the infinite-fluid problems can be treated adequately by more elementary methods. Most of the material in this paper has appeared in print elsewhere. My intention has been to present a coherent account of the treatment of singular perturbation problems in ship hydrody»* namics, and so I have reworked solutions by other people and put them into a common notation and a common format. In some cases, I have made conscious decisions to follow certain routes and to ignore others. I am sure that I have made many such decisions unconsciously too. I have tried to give credit where it is due, but Iam also sure that I have committed some sins of omission in the references. I apologize to those whom I may have slighted in this way. 1.1. Nature of the Problems and Their Solutions We never really derive the perturbation solution of the exact™ problem; we derive, at best, an exact solution of a perturbation problem. That is, we formulate an exact boundary-value problem, simplify the problem, solve the simplified version, and then hope that that solution is an approximation to the solution of the exact problem, Thus, there will almost always be open questions about the validity of our solutions, and these questions can only be resolved through comparisons with exact solutions and experiments. We can have little hope of being rigorous. In fact, it is difficult to provide completely convincing arguments for doing some of the things that we do; in many cases, our approach is justified by the fact that it works! Much progress has been made in this field by people who try approaches "to see what will happen." This does not imply that we shoot in the dark. It does sug- gest that we often depend more on intuition (or experience, which is the same thing) than on mathematical logic in deciding how to solve problems. The small disturbance assumptions by which free-surface problems have traditionally been linearized must have been tried first on this basis. The predictions which result from making such assumptions agree fairly well with observations of nature, and so we are encouraged to go on making the same assumptions in new prob- lems. We may expect to be successful sometimes. There are also open questions about the uniqueness of solutions. Engineers do not often worry about such matters, but they should certainly be aware of certain situations in which the dangers of ae Vissi ae "Exact" means only that nonlinear boundary conditions are treated exactly. I neglect viscosity, surface tension, compressibility, etc., and still call the problem "exact." 668 Singular Perturbation Problems in Ship Hydrodynamics non-uniqueness are especially great. The history of the study of free-surface problems provides numerous examples of invalid solu- tions being published by authors who were not sufficiently careful on this score. We have learned to be careful about imposing a radia- tion condition when necessary, although newcomers to the field are still occasionally trapped. ~ Questions about stability of our solu- tions are not so well appreciated, but of course solution stability is just one aspect of solution uniqueness. A particularly startling example has been pointed out in recent years by Benjamin and Feir [1967]: Ordinary sinusoidal waves in deep water are unstable. This has now been demonstrated both theoretically and experimentally. It comes as no great surprise to those experimenters who had tried to generate high-purity sinusoidal waves for ship-motions experi- ments, but it was certainly quite a surprise to the theorists, who apparently did not suspect any such phenomenon before its discovery by Benjamin and Feir. Since we shall be considering small-perturbation problems, we may expect the solutions to appear in the form of series expres- sions (not necessarily power series!) Often, we are content to obtain one term in sucha series. Practically never do we face the question of whether the series converges. In fact, we usually just hope that the series has some validity, at least in an asymptotic sense. The question will arise from time to time, "How small must the small parameter be in order that a one- (or two- or three- or n-) term expansion give valid predictions?" In ship-hydrodynamics problems, it is quite safe to assert that the only answer to sucha question must be based on experimental evidence. In fact, even in simple problems, the knowledge of a few terms is not likely to help much with this question. For example, suppose that one tries to solve the simple differential equation: y"(x) + y(x) = 0, by means of a series of odd powers of x. How does one know that a two-term approximation is accurate to within one per cent even if x is as large as unity? One might compute the third term, of course, and compare it with the second term, hoping to guess what the effect of further terms would be. If it were too difficult to compute that third term, one could only hope that the solution had some validity, and perhaps one would try to find some experimental evidence on which to hazard a guess about validity. So it is in our ship-hydro- * Within the last few years, a leading German journal published an article on wave resistance in water of finite depth, in which it was concluded that a body had identically zero resistance if it were symmetrical fore and aft. The author was, I believe, primarily a numerical analyst, not familiar with the pitfalls of free-surface problems. He did not impose a numerical condition equivalent to a radiation condition. (This is one reference that I intentionally omit. ) 669 Ogilvie dynamics problems. It will be necessary to discuss this point further at an appropriate place. A related question concerns the precise definition of the small parameters that we use to formulate the approximate problems. In this paper I avoid defining the small parameter quantitatively. It is usually unnecessary and it is dangerous. I shall return to this point also. 1.2. Matched Asymptotic Expansions For most of our problems, the approach advocated by Van Dyke [1964] is entirely adequate. I shall assume that the reader is familiar with (or has access to) Van Dyke's book. Only a few definitions and concepts will be mentioned here. Perhaps the simplest problem that demonstrates the applica- bility of the method of matched asymptotic expansions is the following: Find the solution of the differential equation, ey + 2y ty =0, subject to the initial conditions: y(0)=1; y(0) =0. The parameter e€ is to be considered small, and, in fact, we want to know how the solution of this problem behaves as € ~0. Now, if we set € = 0, the order of the differential equation is reduced, and two initial conditions cannot be satisfied. Therefore, one cannot obtain a series expansion for the solution by a simple iteration scheme which starts with the solution for the limit case, € =0. The exact solution for this problem is: apie a Pst y (t) = =_P28 ___ Pie Pine ’ where ae) N mm 670 Singular Perturbation Problems in Ship Hydrodynamics If we consider that t = O(1) as € —~ 0, then the following approxi- mation is valid for y(t): y(t)~e Mh +E (2-0 4S Slee ee et. This approximation could be obtained step-by-step, iteratively: 2¥n tT Yn = - €Yp-1 » where y(t) ~ y,(t). However, it is not uniformly valid at t =0, and the constants cannot be determined. On the other hand, we could consider that t = O(e€) as € ~ 0 and rearrange the exact solution accordingly. This is most easily done if we set t = €T and rewrite everything in terms of T. The approximation for y(t) is then: 2 yi 1 te(Z-Fy +g -F+ Bye... € we te er + g) ted This approximation could be obtained completely from the differ- ential equation by an iteration scheme in which we let y(t)~ 7, Y,(Ts€), the individual terms satisfying the equation: YM(7) + 2¥' (7) = - €¥,, (7) = [ ¥ = a¥/d7] and the conditions: ¥,(0)= 4; ¥,(0)=0, n>14; ¥(0)=0, n21 However, this solution is not uniformly valid for 7 00; in fact, one would hardly suspect that it represents a solution decaying ex- ponentially with time. The difficulty arises because the problem is characterized by two time scales, 1/p, and 1/p, , and the two are grossly differ- ent. One of the two exponentials in the exact solution decays very rapidly and the other decays at a moderate rate. The contrast in these two time scales, along with the fact that each has its dominant effect in a distinct range of time, allows us to apply the method of matched asymptotic expansions to this problem. The Van Dyke prescription for doing this is as follows: 671 Ogilvte Define the n term outer expansion of y(t) as [y,(t) +... + yn(t)] ; define the m term inner expansion of y(t) as [Y,(7) +... + Ya(7)]. Inthe n term outer expansion, substitute t= €7T and rearrange the result into a series ordered according to €; truncate this expression after m terms, which gives the m term inner expansion of the n term outer expansion. Similarly, in the m term inner expansion, substitute 7 =t/eé and rearrange the result into a series ordered according to €; truncate this expression after n terms, which gives the n term outer expan- sion of the m term inner expansion. The matching rule states that: The m term inner expansion of the n term outer expansion = the _n_ term outer expansion of the m_ term inner expan- sion. In the example discussed in the previous paragraphs, the outer solution could not be obtained by a simple iteration scheme. The matching principle can now be used to determine the constants in the outer solution, and so an iteration scheme is now available, requiring, however, that inner and outer expansions be obtained simultaneously. In the example, the inner solution could be ob- tained completely and independently of the outer, but this is an accident which occurred because of the simple nature of the prob- lem above. Ordinarily, in cases in which one might consider using the method of matched asymptotic expansions, one must proceed step-by-step to find first a term in one expansion, then a term in the other expansion, and so on. It is worthwhile to be fairly precise about certain definitions. We use the equivalence sign, "~," frequently. For example, we write: N d(x, y,Z3€) ~ > b,(x,y »Z35€) n=O This means that: N | - . o,| = o(d) as e —~ 0 for fixed values of (x,y,z). n=O Also, it implies that py = 0(¢,) as € 7 0. The qualification that (x,y,z) should be fixed is very important. In the example above, we would have the equivalent statement for the outer expansion: N | y (tse) - , yp(tse) | = oly,) as a>, 0 for fixed t, n=l and, for the inner expansion: 672 Singular Perturbatton Problems in Shtp Hydrodynamics N ly(tse) - » ¥itrie) p= o(Yy) as e—~0 for fixed T. n=l In the latter, we evaluate the difference on the left-hand side for smaller and smaller values of t (= €T) as e€ —~ 0; in other words we restrict the range of t more and more as e~0O. This is in contrast to the interpretation of the outer expansion, in which we simply fix t at any value while we let € ~ 0. In even more physi- cal terms, we may say that the inner expansion describes the solu- tion during the time when the e?!t term is Maree rapidly, and the outer expansion describes the solution when the e it term has effectively reached zero and the eP2! term is varying significantly. This separation into two distinct regimes is characteristic of prob- lems in which we apply the method of matched asymptotic expan- sions. Of course, the real key to the success of the method is in the procedure by which the two aspects of the solution are matched to eachother. After all, they do represent just two aspects of the same solution. Usually, we insist that our asymptotic expansions be consistent. A precise definition of this term is awkward, but per- haps it is clear if we state that each term in sucha series depends on € in asimple way that cannot be broken down into simpler terms of different orders of magnitude. For example, the following two series are equal: 2 3 1 1 2 ee! = = _ = 1 [ie ite ee” Fo] ise ae Way ae 1 f.2 5 ih 3 ae ra tee Te ] : ey ae se a= ea Ts ] 1 3 + [ie + arate ic i On the right-hand side, let: il il 2 1.2 fle)=itszetze tee Taereuel 3p Pa f (e) = = fp(e), for mo 0% Then we can write: 673 Ogilvte N us N > é€! ie >. f,(e) as e= 0. n=O n=O These happen to be convergent series (if € <1), but we can inter- pret them as asymptotic series just as well. The series on the left is "consistent"; the one on the right is not, because individual terms have their own e€ substructure. The striving for consistency can become a religion, but it is not a reliable faith. Consistency (or the lack of it) tells us nothing about the relative accuracy of otherwise equivalent asymptotic expansions. In fact, we could define a third asymptotic series with terms given by: BOE) = 1/(1-€) ; g,(€) = (9) for n.> 0} This series is grossly inconsistent, but one term gives the exact answer for the sum of the previous series! Occasionally one can make educated guesses about such things, replacing a few consistently arranged terms by a simple, inconsistent expression having much greater accuracy in practical computations. Mathematically, these different asymptotic series are equivalent, and, if € is small enough, they will all give the same numerical results. But we want in practice to be able to use values of € that are sometimes not "small enough. " We shall work with consistent series, for the most part, in spite of such possibilities of improvement through the use of incon- sistent series. Most newcomers to this field of analysis find that there is a considerable element of art in the application of the method of matched asymptotic expansions, and I personally consider that the improvement of the expansions through the development of inconsistent expansions is the highest form of this art. Except in one respect, I do not intend to pursue the possibilities of inconsistent expansions in this paper. The exception that I make is the following: Many singular perturbation problems lead to asymptotic-expansion solutions of the form: N n » » anm€ (log ey n=0 m=O where a,,, does not dependon €. We can, of course, write this out in a long string of terms quite consistently arranged. However, my practice will be to treat the sum: 674 Stngular Perturbation Problems in Shtp Hydrodynamics n €” > anm(log €)" m=0 h,(e) as a single term (albeit inconsistent) in the series Dy bal €) « An alternative way of describing this practice is to say that I consider log € = O(i) as € ~ 0! I have encountered some practical prob- lems which could apparently not be solved by the Van Dyke matching principle unless treated in this way, and I have never seen or heard of a problem in which this practice led to difficulties. There are some good arguments for proceeding in this way, but I know of no proof that either way is the correct way. (Some of my colleagues will call this a cheap trick, rather than a higher expression of an art form.) The classical example in physics of this kind of mathematical problem is the boundary layer first described by Prandtl in 1904. The thickness of the boundary layer becomes smaller and smaller as the small parameter, 1/VR approaches zero (R is the Reynolds num- ber), but the presence of the boundary layer cannot be neglected, because then the governing differential equation becomes lower order, and the body boundary conditions cannot all be satisfied. Unfor- tunately, Prandtl did not realize the generality of the analysis which he introduced into the viscous-fluid problem, and, lacking the modern formalism for treating such problems, he could not obtain higher-order approximations. Perhaps I should include a discussion of Prandtl's problem in this paper, since it might be considered as a "singular perturba- tion problem in ship hydrodynamics." However, I shall not do this, for several reasons. Van Dyke's coverage of the problem is excel- lent, I think. Also, the analysis concerns only laminar boundary layers, and they are really of quite limited interest in ship hydro- dynamics. Finally, the formal procedure breaks down completely at the leading edge of a body, and the singularities that occur there cause major difficulties in all attempts to use the formalism to obtain higher-order approximations. One final point should be emphasized, even at the risk of insulting the intelligence of readers who have read this far. When- ever we write, "e ~ 0," we are implying the existence of a se- quence of physical problems in which the geometry of some funda- mental parameter varies. For example, in Prandtl's boundary- layer problem, we may consider that viscosity changes as e=1/¥R—0. Inthe simple ordinary-differential-equation example presented above, we may think of a spring-mass system in which the mass is changed systematically from one experiment to the next. Later, when we treat slender-body theory, we consider a sequence of problems in which the body changes eachtime. The theory always implies the possible existence of such a series of problems, and the quality of the predictions improves as the problem more nearly fits 675 Ogtlvte the limit case. Thus, we shall be able to apply the results of slender-body theory to bodies which are not especially slender. In such cases, we may expect that the predictions will be less accurate than the predictions that we woutd make for a much more slender body. But we never know a priori how slender the body must be for a certain accuracy to be realized, and it would be wrong to assert that the theory applies only to needle-like bodies. All that we can say is that it would be more accurate for such bodies than for not- so-slender bodies. 1.3. Multiple-Scale Expansions In the problems of the previous section, we had two greatly contrasting scales for the independent variable. The fact that enabled us to obtain two separate expansions was that each of the scales dominated the behavior of the solution in a particular region of space or aparticular period of time. The major practical concern was to ensure that the separate expansions matched, because they really represented just different aspects of the same solution. The present section is devoted to problems in which there are again two greatly contrasting scales. However, in these prob- lems, it will not be possible to isolate the effects of each scale into a more or less distinct region of space or time. The effects of the two scales mingle together completely. However, we may still expect to be able to identify these effects somehow, just because the two scales are so different. There are classical problems of this kind, the most famous being related to nonlinear effects on certain periodic phenomena. Cole [1968] discusses a number of these problems. Perhaps the simplest example of all is alinear one: Find approximate solutions for small € inthe problem of a linear oscillator with very small damping, where the differential equation might be written: y t2ey ty =0. To be specific, let the solution satisfy the initial conditions: y(0) = 1 and y(0) =0. Physically, we expect that the system will oscillate with gradually decreasing amplitude. It would be desirable if the approximate solution at least did not contradict this expectation. We might try representing y(t;€) by an asymptotic expansion with respect to €: y(t;e) ~ 7, ypltse). We would find immediately that the first term in this expansion is just: yg(t;e) = cos t. This seems quite reasonable, since it represents a steady oscillation at the frequency approximate to the undamped oscillator. The second term in the expansions would be obtained from: 676 Stngular Perturbation Problems tn Ship Hydrodynamics y + y, == 2€Y5 = 2€ sin Gs with y, (0) yt (0) = 0. It is impossible to obtain a steady-state particular solution of this problem. In fact, the solution is: y, (tse) = e[ sin t - t cos t]. Thus, we obtain an expansion in which the second term grows linearly with time. One might expect that succeeding terms will grow even faster. This expansion is correct, and, for small values t, it could be used for numerical predictions. But we would certainly prefer to obtain an expansion which is uniformly’ valid, even for very large t. The exact solution is easily found, of course. It is: y(tse) = en [ cos Vi-e*)t trae sin y(1-e°)t] . The approximate solution becomes worse and worse with increasing t because the frequency is wrong and because the exponential factor is expanded in a power series in t. If we watch the oscillating mass on atime scale appropriate to the period of the oscillation, we do not see the exponential decay and the slight shift of frequency caused by the damping. On the other hand, if we watch for a very long time, the effects of damping accumulate gradually. Thus, the effects of the "slow-time" scale, 1/e, persist throughout the history of the motion as observed on a real-time scale, but these effects never occur suddenly. It is this fact which enables us to separate them out of the real-time problem. There seems to be less reliable formalism available for handling such problems than in the case of the method of matched asymptotic expansions. More is left to the insight and ingenuity of the individual problem solver. In the example discussed above, the procedure is fairly clear: Expand y(tje) in a series such as this: y(ts€) ~ yolt, Tse) Fy (t,73€) +... , where we define: “~ t. = €t 3 tam FE (rs€) Fete) tees * Strictly speaking, the series really is uniformly valid except at t =o, Ogilvie and the functions f, are to be determined in such a way that the approximation is uniformly valid for all t. In treating this parti- cular problem, Cole immediately assumes that t~ 7 and further that t/7 = 1 + O(e2). These extra assumptions speed the solution considerably, but it is not clear how one would know to make them if the exact solution were not available. The exact solution takes the form: y(t;e) = at [iss Pur (¢ /7) sin T], in terms of the new variables. (The factor (t /T) does not depend on t.) Here it is clear how the two time scales enter into the solution as well as the problem. One may expect the relationship between t and 7 to be equivalent to the expansion of the quantity (1-e€2). The reader is referred to Cole's book for further discus- sion of the solution of such problems. One problem that will be discussed later is a close relative of the classical problems mentioned above. The solution by Salvesen [1969] of the higher-order problem of the wave resistance of a sub- merged body leads to a situation in which the first approximation is periodic downstream and that period is modified in the third-order approximation. (Otherwise the waves downstream in the higher approximation would grow larger and larger, without limit.) A similar problem involves the oscillation of a body on the free sur- face, in which the wave length of the radiated waves must be modified in the third approximation. For example, see Lee [1968]. A quite different application of this method is the problem of very low speed motion of a body under or ona free surface. The simplest such case has been discussed by Ogilvie [1968]. Fora translating submerged body, there are two kinds of length scales: length scales associated with body dimensions and submergence, and the length scale U Sas which is associated with the presence of the free surface. Presumably, the latter has effects primarily near the free surface, ina "boundary layer" with thickness which varies with u*/g as that variable approaches zero. But the effects of the body dimensions are also important near the free surface (or at least near a part of it). Thus the effects of the two length scales cannot be separated into distinct regions. A brief discussion of this problem appears in Section 5.42 of the present paper. There may be many other problems of ship hydrodynamics in which this approach would be valuable. For example, many authors have obtained approximate solutions of problems involving submerged bodies by alternately satisfying a body boundary condition, then the free-surface condition, then again the body condition, etc. At each stage, when one condition is being satisfied, the other is being violated, but it is assumed that the errors become smaller and 678 Stngular Perturbation Problems in Shtp Hydrodynamics smaller with each iteration. Such a procedure is discussed, for example, by Wehausen and Laitone [1960], who point out the use- fulness of Kochin functions in such procedures. However, there is often a question about the precise nature of such expansions. In the first approximation, for example, the effects of the free-surface are likely to drop off exponentially with distance from the surface. This makes it inappropriate to treat depth of submergence as a large parameter in the usual manner, because exponentially small orders of magnitude are either trivial or exceedingly difficult to handle. I do not believe that anyone has yet shown how to treat this problem systematically. II. INFINITE-FLUID PROBLEMS It is mainly the presence of the free surface in our problems that forces us to seek ever more sophisticated methods of approxi- mation. However, the nature of the approximations can often be appreciated more easily by applying those methods to infinite-fluid problems. In this section, I discuss a number of problems that are geometrically similar to the ship problems that are my real concern. In some cases, it must be realized that the methods used here are not necessarily the best methods for the infinite-fluid problems. However, without the complications which accompany the presence of the free surface, one can better understand the significance of the coordinate distortions, the repeated re-ordering of series, and the matching of expansions. The reader who feels comfortable with matched asymptotic expansions is invited to skip this chapter. Zed) hin Body A "thin body" has one dimension which is characteristically much smaller than the other dimensions. In aerodynamics, the common example is the "thin wing," and, in ship hydrodynamics, one frequently treats a ship as if it were thin. In such problems, the incident flow is usually assumed to approach the body approxi- mately edge-on, and so the thinness assumption allows one to linearize the flow problem. In this section, thin-body problems are treated by the method of matched asymptotic expansions. This is not the way thin-body problems are normally attacked, and, in fact, I do not recall ever having heard of suchatreatment. At the outset, I must point out that there are good reasons why this has been the case. If the body is symmetrical about a plane parallel to the direction of the incident flow, one does not need inner and outer expansions for solving the problem. And if the body lacks such symmetry, the lowest-order problem cannot be solved analytically, and so the method of matched 679 Ogtlvte asymptotic expansions does not offer the possibility that one may be able to obtain higher-order approximations. In fact, the problem of a thin body in an infinite fluid is not a genuine singular perturbation problem (although it may contain some sub-problems that are singular, such as the flow around the leading edge of an airfoil). However, I believe that the problem of a thin ship is singular; I shall discuss this in Section 4. There has been a considerable amount of misunderstanding as to what consti- tutes the near field and what constitutes the far field in the thin-ship wave-resistance problem, and the rectification of such misunder- standing requires a careful statement of the problem. It is conceivable that this interpretation of the thin-ship problem may be useful in formulating a rational mathematical idealization of the maneuvering-ship problem. For convenience, I separate the thin-body problem into two parts: a) the symmetrical-body problem, and b) the problem ofa body of zero thickness. To treat an arbitrary thin body, with both thickness and camber, one should certainly consider both aspects at once. It is not really difficult to do this, and indeed the problem of an unsymmetrical body of zero thickness actually involves thick- ness effects (at higher orders of magnitude than in the symmetrical- body problem). I have kept the problems separate here only for clarity in discussing certain phenomena that occur. 2.11. Symmetrical Body (Thickness Effects). Let the body be defined by the equation: +th(x,z3e) for (x,0,z) in H>, a (2-1) 0 for (x,0,z) not in H, where | is the part of the y = 0 plane which is inside the body. (It is the centerplane if the body is a ship.) The "thinness" of the body is expressed by writing: h(x, z;€) = €H(x,z), (2-2) where € is asmall parameter and H(x,z) is independent of e€. The body is immersed in an infinite fluid which is streaming past it with a speed U inthe positive x direction. The flow, in the absence of the body, can be described by the velocity potential: Ux. It will sometimes be convenient to say that the body is defined by the equation: y = + h(x,z;e), implying that the function h(x,z;€) is identically zero if (x,0,z) is not in H. Also, note that we shall 680 Singular Perturbatton Problems in Ship Hydrodynamics frequently drop the explicit mention of the € dependence. As € ~ 0, the body shrinks down to a sheet of zero thickness aligned with the incident flow. Thus, the first term in an asymptotic expansion of the velocity potential in the far field is just the incident- stream potential. In general, let the far-field expansion be expressed as follows: N (x,y ,Z;€) ~ » $(x,y,z;€), where $,) = o(¢) as n=O e—>0 for fixed (x,y,z). (2-3) Then we have: $y(xsy»Z5€) = Ux. (2-4) The far field is the entire space except the y =0 plane. Since the potential $(x,y,z;€) satisfies the Laplace equation through- out the fluid domain, the individual terms in the above expansion satisfy the Laplace equation in the far field: on. t Pnyy t Pis= 0 for ly|> 0. (2-5) At infinity, we expect (on physical grounds) that: V(o - Ux) > 0. (2-6) Therefore, for n>0O, every $, must be singular onthe y = 0 plane or be a constant throughout space. The latter would be too trivial a result to consider, and so we assume that 6, is indeed singular on the y = 0 plane. But what kind of singularities will be needed? Because of the symmetry of the problem, it is not difficult to show that a sheet of sources will suffice. One can use Green's theorem to show this. Alternatively, one can use transform methods for solving the Laplace equation, which is practically equivalent to solving by separation of variables. Whatever method is used, the result is the same} $(x,y,z3€) has a representation: ve og (& ,o3€) d& do (2-7) 1 0 ( rV¥ 2Z3€) = - it ’ OMX ry + 40 J Yo [ tx-é)" ty" 2 (z-t)"] 172 where o,(x,z;€) is an unknown source-density function. The outer 681 Ogilvie expansion is just the sum of these: n(SsG3€) d& dt ee ere eee [ (x-8) yiot (ze oy! This is the most general possible outer expansion for this problem. It will be necessary presently to know the inner expansion of the above outer expansion. To find it, define an inner variable: y'=y/e, (2-9) substitute for y in the outer expansion, and re-order the resulting expression with respect to €. A direct approach to this process is difficult, but the following method, in four steps, allows us to obtain the desired results to any number of terms ina fairly simple way: 1) Take the Fourier transform of 4, with respect to x: 2 © -ikx ) 1 * dx e > (ky ,2;€) =-— dG o. (k;G3e) te ns i [x ty? + (2-67)? co SV ag ot tastse) Ky [kl vy? + 2-01) -@ where - 0 is the modified Bessel function usually denoted this way, and of *(k;z;€) is the Fourier transform of the function o,(x,Z5€)- The convolution theorem was used in the first step above. 2) Take the Fourier Transform next with respect to z: 40% 1 ** a -i 2 2,V2 >, (ksy;m;e) = - san (mie) | dze ™ KO lKILy +z iv ) -@ * 2 _ 21/2 Uk, mje) _-fk+m) - ca lea , 21" +m ) where o mie, mje) is the double transform of oa,(x,z;€). 3) Substitute: y = €Y and expand the exponential function into a power series: 682 Singular Perturbation Problems in Shtp Hydrodynamtes agin €) f- -2eG> 2 2 Tn »™M,; oe 2,2 [1 top Pv tk soak ** an (iyim;€) = = ——> 2(k +m ) 4_4 2 + Fe Vile +m) +...] +5 o. (ky mje) [e |y| + e°| ¥ |? (k* + m?) 5 Sruie 22 tore ly|(k +m) sane P 4) Note that: eK : Kk KI _on (Ksmje) , (k;0;mj;e) = a, (k,m;e) . (2-10) 2(k? oO m_?)!/2 *%* ; P Also, we observe that, if f (k,m) is the Fourier transform of f(x,z), then (k? + m*)f**(k,m) is the Fourier transform of - (f,, + £,,). Defining the inverse transform of a**(k,m;e€): a(x,z3;€) = $,(x,0,z5€), (2-11) and inverting the above series term-by-term, we obtain: ( z3€) ~ a ( se) to ely] ( zie) - oy €° (YP le ta) >, XsVsZys nh%sZs 2 O,\X 24s 21 Nyy Nes 1 3 3 Ty 2s3it : [| (ony ss Tnzz ) {2h ite aos + + Se Ee ; IY | ene olor 5 leas (2-12) This is the inner expansion of a typical term in the outer expansion. In order to combine the expansions of the separate terms into a single inner expansion of the outer expansion, let us assume that n : go, and a, are both O(c). (It is not necessary to assume this, it is merely convenient.) Then we have for the desired expansion: 683 Ogilvie g(x, y,z;€) ~ Ux O(1) + a (x,z5€) O(e) + a5(x,Z;€) +S ly |o, (x, z;€) O(e?) 1 1 2 + a3(x,z5€) a) ly |on(x,z3€) -5 ly| (a + a.) O(e?) + O(e*). (2-13) Note that we have reverted to far-field variables. We must here consider that y = O(e€) in order to recognize the orders of magnitude as indicated above. Next we must find the inner expansion of the exact solution. Substitute y = €Y in the formulation of the problem. The Laplace equation transforms as follows: 2 byy= - Udy + by,)- (2-14) The kinematic condition on the body is: + ph, - $y $h,=0 on y= + h(x,z), which transforms into: o,=+ e°(6H +H) on Y= + H(x,z). (2-15) We assume that there exists a near-field asymptotic expansion of the solution: N (x,y »Z3;€) ~ >, @,(x,Y,z;€), where py =0(@,) as en 0, n=0 for fixed (x,Y,z). (2-16) We could show carefully that: @)(x,Y,z;€) = Ux. (Perhaps it is obvious to most readers.) We then express the con- 684 Singular Perturbation Problems tin Ship Hydrodynamics ditions on the near-field expansion as follows: ~~ e7[ ie 7 oie +@o + e., iF paral (2-17) [H] 4 +@,, +3 +...~ + @[ UH, + @ Hy, + OH, +...] on Y =+ H(x,z) . (2-18) Solution of the @, problem. From the [L] condition above, it is clear that: Pi yy= 0 (2-19) in the fluid domain. Therefore ®; must be a linear function of Y. In view of the symmetry of the problem, we can set: ® (x, Y,z;€) = A, (x,25€) + B, (x,z3€) hy; for ly | > H(x,z). (2-20) The body condition reduces to: (x, +H(x,z),z3€) =e e° UH, (x,z) ee B, (x,z3€) = O(e?). (2-21) It appears that we have determined the value of B,(x,z;€) -- but this is wrong, as we shall see ina moment. The two-term inner expan- sion appears to be: x,y,z;€) ~ Ux + A,(x,z;€) + B, (x, z5€) ly |. Its outer expansion is obtained by setting Y = y/e: o(x,y,z;€) ~ Ux + <8, (x,z;€)|y| + A (x, z5€) . O(1) Ofe) O(e*) The order-of-magnitude estimates were obtained as follows: B, is O(e*), from (2-21). If our expansion is consistent (as we insist), then A, is also O(e®), by (2-20). Now, in the outer expansion of the inner expansion, the B, term is lower order than the A, term. 685 Ogilvie The two-term outer expansion of the two-term inner expansion is: o(x,y,z;€) ~ Ux +4 B (x,zse) ly| . O(1) Ofe) On the other hand, the two-term inner expansion of the two-term outer expansion is, from (2-13), o(x,y,z3€) ~ Ux + a, (x,z3€). There is no linear term here at all, and it seems that we cannot match the two expansions. It is a very comforting feature of the method of matched asymptotic expansions that things go wrong this way when we have made unjustified assumptions. Our mistake was this: When we found that apparently B, = €°UH, = O(e?) » we eliminated the possi- bility that there might be a term which is O(e) in the inner expan- sion’. Now we rectify this error. Once again, let ®, be given by (2-20), but suppose that both "constants" are, in fact, O(e). The body boundary condition immediately yields the condition that: B,(x,z3€) = 0, and so we have: ®, (x, Y,z;€) = Aj(x,z3€) . The inner expansion, to two terms, is now given by: $(x,y,z3€) ~ Ux t+ A; (x,z3€). When we match this to the inner expansion of the outer expansion, we find that: A, (x,23€) = a,(x,z5€) = $(x,0,z5€). (See 2-11.) Now we have matched the expansions satisfactorily, but *This trouble would have been avoided if I had started by assuming that the expansion is a power series in €, as many people do in such problems. However, that procedure can lead to even greater diffi- culties sometimes. 686 Singular Perturbation Problems in Shtp Hydrodynamics the result is not yet of much use, since we do not know either function, A; or a,. It is worth noting, however, that the inner expansion can be rewritten: (x,y »Z;€) ~ Ux + $,(x,0,z3€). Thus, to two terms the inner expansion is determined entirely by the far-field solution, the latter being evaluated on the centerplane. In other words, in the near-field view, the fluid velocity (to this degree of approximation) is caused entirely by remote effects. Solution of the @,; problem: This is much more straight- forward, and the results are more interesting. We may expect that @p = O(e*), since we still have the nonhomogeneous body condition to satisfy. In this case, then, @,(x,Y,z3€) = A,(x,z3e) + Bo(x,zse)|Y|, and the body condition requires that B,(x,z;€) = €°UH,(x,z) « The three-term inner expansion is: o(x,y,z;€) ~ Ux + a (x ,2;€) 2 A,(x,z5€) + UH, (x,z) |Y O(1) Oe) O(e*) Ole") The two-term outer expansion of this three-term inner expansion is: P(x,y,z3€) ~ Ux t+ a,(x,z;€) + Uh,(x,z3€) ly O(1) O(e) O(e) The three-term inner expansion of the two-term outer expansion is, from (2-13): 1 o(x,y,z5€) ~ Ux + a,(x,z5€) +5 ly |o,(x,z3e). (The a, in (2-13) is not carried over to the above expansion, since it originates in the third term of the outer expansion.) These two match if: o, (x,z3€) = 2Uh,(x,z;e€) = O(e). (2-22) Thus, finally, we have found o,(x,2;€), the source density in the first far-field approximation, as a function of the body geometry. It is the familiar result from thin-ship theory. In addition, we can 687 Ogilvte now also write down a,(x,z;€) by combining (2-7) and (2-11): tee ae Uh Cre) de de _ | ae aS Sy [(e-8) + (tye We have the two-term outer expansion -- with everything in it known -- and the three-term inner expansion -- with the "constant" A,(x,z;€) not yet determined. Solution of the higher-order problems: From the [L] con- dition, (2-17), it can be seen that &, (x, Y,z;€) is not linear in Y. However, the differential equation for @) is easily solved, the body boundary condition can be satisfied, and matching can be carried out with the outer expansion. The result is: $3(x,Y,z;€) = A,(x,z;€) + B,(x,z3€) ly | - 5 Ya, + a.) where B3(x,z3€) = €[ (a, H) + (aH), ], A,({x,23€) = a(x, Z5€). We also obtain o,, through the matching, o(x,z3€) = 2[ (ah), + (a, hj], and this information also gives us ap and Ap. Summary: Symmetrical Body. The results for both near- and far-fieId expansion are stated in terms of the far-field coordinates (the natural coordinates of the problem) in Table 2-1. Ina sense, the results are rather trivial. There could be difficulties near the edges of H, but, barring such possibilities, the inner expansion could be obtained from the outer expansion and then matched to the body boundary condition. This is actually the classical thin-ship approach. The outer expansion is uniformly valid near the thin body, except possibly near the edges. In the classical approach to the thin-body problem, there is usually a legitimate question concerning the analytic continuation of the potential function into the region of space occupied by the body. Sometimes one avoids the problem by restricting attention to bodies which can legitimately be generated by a sheet of sources, but this is not very satisfying. The method of matched asymptotic expansions avoids the question altogether by eliminating the need to ask it. What 688 Stngular Perturbation Problems in Ship Hydrodynamics TABLE 2-1 SYMMETRICAL THIN BODY Near-Field (Inner) Expansion y = O(e) (x, yszse)~™ Ux + o,(x,23e) + ,(x,z3e) + $a, (x, ze) ly| oo cuEEESESEEEEENES aumeenEmEeeet O( ) Ole) O(e*) i i 2 tas(x,z3€) + > o,(x,23€) ly] - > lyl (a + 1.) ER ee O(e°) + Ole’) Far-Field (Outer) Expansion y = O(1) N e ~ = Jt. Gnlesose) d& dG o(x,y,Z3€) ~ Ux = » Op Tleeb)e+ yy? + (zt) = ~ an e A Gary! 2 O(1) O(e") From Matching o,(x,z;€) = 2Uh,(x,z5€) ; o,(x,z3€) = 20 (qh), + (a, h),] 5 etc. s t n(S,G3€) do d @,(x,25€) = - Fe aie Pex-f)” + (n-tr] ate = : z 689 Ogilvie we are really saying is this: From very far away, the disturbance appears as if it could have been generated by a sheet of sources, but close-up we allow for the possibility that this observation from afar may be somewhat inaccurate. In fact, there is no analytic continuation presumed in the present method. One can show by the use of Green's theorem that the far- field picture is valid even if the analytic continuation is not possible. A particularly appealing (to me) version of such a proof has been provided by Maruo [1967] for the much more complicated problem of a heaving, pitching slender ship moving with finite forward speed on the surface of the ocean. I suppose that the uniformity of the thin-body solution is the result of the fact that a well-posed potential problem can be stated by giving a Neumann boundary condition over a surface. The situ- ation will be quite different when we consider slender-body theory: in the far field, it would be necessary to give boundary conditions on aline, and that does not lead to a well-posed potential problem in three dimensions. Similarly, we may expect trouble at the con- fluence of two boundary conditions, and this indeed occurs when we try to treat a ship problem by the method discussed above. The free-surface conditions cannot be satisfied, and the difficulty can be traced back to the behavior of the far-field potential near the intersection of the centerplane and the undisturbed free surface. 2.12. Unsymmetrical Body (Lifting Surface). For the sake of simplicity, Tet the body have zero thickness. Then it can be represented as follows: y = g(x,z3e) = €G(x,z) for (x,0,z) in H, (2-23) where H is now the projection of the body onto the y=0 plane. Again, there is a uniform incident flow in the positive x direction. The analysis is quite similar to the symmetrical-body case, at least in the near field, and so most ot the details will be omitted here. In the near field, let there be an expansion: N (x,y ,235€) ~ > @n(x,Y,z5€), n=O just as in (2-16). The first term is, again, @o(x,Y,z;€) = Ux. The terms again satisfy the transformed Laplace equation, (2-17): + i: + ead fel Porch Dean ier ae yy ‘4yy 2 T= Cf Oi) ) TOD ot Oost ia sti abel > 690 Singular Perturbation Problems tn Ship Hydrodynamics the body boundary condition is now: [H] &, +2 +3, +O, +... Y a ex { S [ UG, + (dG, + ie G;) aR (Bp Gy I: ®, G,) lj oe =| on Y =1G (x77). (2-24) The solution for ®, is generally an expression linear in Y, but, for the same reasons as in the symmetrical-body problem, only the "constant" term can ultimately be matched to the far-field solution, and so we take for 9®,: &,(x,¥,23€) = Aj(x,zse) = Ole). The superscript + has been attached to the solution to indicate that this quantity may be different on the two sides of the body. This was not necessary in the previous problem, because of the symmetry, but in the present near-field problem the body completely isolates the fluid on its two sides and there is no reason to assume that A; is the same on both sides of the body. (It turns out, in fact, that Aj = - Aj -) One next obtains: + + $,(x,Y,2;€) = A,(x,z3€) + Ba (x,zse) Y . From the body boundary condition, the following is true: + $2,(x,G,z3€) = Ba (x+z5€) = "UG, (x,z) : (2-25) Thus, we find that Ba(x,z3€) = Bg(x,z;€) = B,(x,z3e). Similarly, one can proceed: 1 2.2 $,(x,Y, ze) = A, (x,z3€) + B sboze)¥ -5e YA, + Ay), where + ‘ 2 se oe B,(x,z;€) = €[(GA,) + (GA, ),]. 691 Ogtlvie It is interesting to note the following about the symmetry: It turns out that ®, and @®» are odd with respect to Y, but 3 is neither even nor odd. The linear term in $3 , namely, Ba(x,2e)¥, is even, since it turns out that B; = - Bs. Careful study of the ®, problem shows that it actually implies that there is a generation of fluid in the body, but the rate of generation is higher order than the ®,; term. Physically, of course, there can be no fluid generated, and so a compensating source-like term appears in 5. The far field is again the entire space except for the plane y = 0. The relations (2-3) to (2-6) are again valid, as well as the discussion of them. But now it will not suffice to provide only source singularities on the centerplane; clearly we must also provide singularities which lead to antisymmetric potential functions. In fact, since the body has zero thickness, weshall expect the leading- order approximation to be strictly antisymmetric. These require- ments are all met by a distribution of dipoles which are oriented with the y axis. The potential of such a sheet of dipoles can be expressed: f(x,y,z) = ee eS rey ae aa (ees Pia sie The inner expansion of such an integral can be obtained by the same Fourier-transform technique that was used before. One finds that: Lyi(ke* mey2 **(k3y3m) = = (sgn y) pb” “(k,m) e : The exponential function can be expanded into a series, which is then inverted term-by-term. Define a new function (cf. (2-10)): 2 ye a 388 Oe eau ¥ mn (kc in)iee~ek Ee — ee The following relationships exist between the two functions ,(x,z) and y(x,z): béee) = ie oa (0) a de (2-28) “0 [(x-€) + (z-t)*] aos Lem + Hee) do oe E y(x,z) = a0 ie 7 ee re ae Vv ; (2-29) 692 Singular Perturbation Problems tn Shtp Hydrodynamics (Note the comparison between (2-28) and the relation between a, and on in Table 2-1. In fact, (2-29) gives the inversion of the formula in Table 2-1.) The inner expansion of f(x,y,z) can now be written in terms of these two functions: f(x,y ,z) = F{wlx,2(sgn y) - y(x,z)y - + y(sgn y) (jy, + Hg2) 1 1 7 31 y7(Yxx + Vz) + 41 y(sgn Y) (pexxxx + 2uxx22 + zzz) +.. “ ° (2-30) This may be compared with (2-12). Now let us assume that the two-term outer expansion is: 00 ° ° $(x,y,z;€) ~ Ux ES (€,0;€) d& d& -0 [ (x-§)° ty” + (x-t)*] 7 Furthermore, assume that p; and y, are both O(e). (If these assumptions are too restrictive, that fact will become clear in the subsequent steps of the method of matched asymptotic expansions.) Then the inner expansion of the two-term outer expansion is: 4 Ax,y ,2z3€) ~ Ux + p(x, z3€) - y¥,(x,23€) = 5 yh, +p) O(1) O(e) O(e?) O(e>) I have kept four terms, as indicated by the order-of-magnitude notes under the terms. (Recall that y = O(€) in the inner expan- sion.) Matching with the appropriate forms of the outer expansion of the inner expansion, we find that: = A, (x,25€) = + p,(x,z5€)3 (2-31) B5 (x,z3€) = - ey, (x,zZ5€). (2-32) From (2-25), we find that: ¥; (x,z3€) = - €UG,(x,z) =a= Ug,(x,z3€). (2-33) It appears now that we could use this knowledge of y, in (2-28) for determining p,. But this is wrong. Note from (2-30) that yj,(x,z) 693 Ogilvie is the normal velocity component on the y = 0 plane caused by the distribution of dipoles, pw,(x,z), over the same plane. Now we would presumably restrict the dipole distribution to the region H, and so (2-29) is valid if the range of integration is reduced to just H, since the integrand is identically zero outside H. But the same is not true in (2-28). There is a generally non-zero normal com- ponent of velocity, y,(x,z), over the entire plane, and the range of integration in (2-28) cannot be reduced to just H. Unfortunately, we know y;(x,z) only on H, from (2-33), and so we have solved nothing. This difficulty is hardly surprising, since we are really formulating here the classical lifting-surface problem, and its solution requires either the solution of a two-dimensional singular integral equation or the introduction of further simplifications -- which will be discussed presently. In the lifting-surface problem, we really should distribute dipoles over two regions, the centerplane H and the part of the plane y=0 which is directly downstream of H. Let the latter be called W. Pressure must be continuous across W, since there is no body there to support a pressure jump. In the usual aerodynamics manner, one can then show that dp,/8x must be zero on W. In this way, the integration range in (2-29) can be reduced to an integral over just H. Of course, lifting surface theory is usually worked out in terms of vorticity distributions. I happen to prefer using dipole distributions, mainly because then I do not have to worry about whether a vortex line might be ending in the fluid region. The con- nection is fairly simple between the two versions, of course. A single discrete horseshoe vortex extending spanwise between z=s and z=-s and downstream to x = oo corresponds to a sheet of dipoles of uniform density, spread over the plane region bounded by the vortex line. The potential function can be written, for unit vortex strength, 6 @ us -\ d ae = (sdb xh A eisebc Lc! aaa [ (x-£)° + y? + (2-271? Ss it LGN ere se Seis y? + (2-0)? [ ety roan | 2 ‘ Af a / 2 Leije =| tan! - tan Ce... [xt ty" tlassy] Z-s Zak x(zZ-s Singular Perturbation Problems in Ship Hydrodynamics The normal velocity component in the plane of the vortex is: 1/2 Meme at (ye tee) fi oe ee) Z-Ss ZTS A lifting line can be described in a similar way if we allow the dipole density to vary with the spanwise coordinate, z. For simplicity, let us assume that p(z) = p(-z), and that p(s)=0. The potential for a lifting line is: ul s 00 at (xy +2) x5 dt ute) f eC era ICT (2-34) (x-€)* + y? + (2-2)? ]¥? s s' rr) cae Ah ds* u's! d hoe VEeitole ness oa s' p(s f 4 \ [ (x-6)2 ty? + (2-EF]2” (2-35) and the normal velocity component is: s t ! $)(x,0,2) = - maf ds" p'(s!) E + Lx? + (z-s'P ee als | mi(easa} -$ Z-S Note that this reduces to the result for the single horseshoe vortex if a) we set p'(z) = 6(zts) - 6(z-s)™, and b) we integrate over a span from -s-f to s+®, where B is a very small positive number. This may lend some credibility to the procedure frequently advocated by aerodynamicists in wing problems, viz., when integrating by parts in the spanwise direction, extend the range of integration slightly beyond the wing tips so that quantities which become infinite at the tips do not yield infinite contributions that cannot be integrated. (This is terrible mathematics, but apparently the physics is sound, since the results seem to be correct.) Finally, we can use the above procedures to derive the cor- responding expressions for a lifting surface. The important quantity is the normal velocity component, given by: s(€) 2 21 1/2 $,(x,0,2) = J at ee: wal fy 4 LosnBik + (2-oF)*) (2-37) * 6(z) is the usual Dirac delta function. 695 Ogilvte where L is the range of x covered by the lifting surface (the length of L being generally the chord length), and s(x) is the half- span at cross section x. On H (the projection of the wing on the plane y = 0), the normal velocity component, dy, is known, either by direct application of the body boundary condition or by matching to a near-field solution, and we obtain the usual integral equation for a lifting surface. We shall not be concerned here with the various methods of attempting directly to solve this integral equation, either by analyti- cal or numerical methods. In fact, analytical methods do not exist, so far as I know, except for a few special geometries, such as elliptical planforms. The pair of equations (2-28) and (2-29) forms a remarkable analogy to a standard boundary-value problem in two dimensions which is analyzed thoroughly by Muskhelishvili [1953]. One three-dimensional case has been solved analytically by a method that has some similarity to the standard methods for the 2-D prob- lem; this was done by Kochin [1940]. Even his circular-planform wing led to so much difficulty, it seems unlikely that it will be generalized to other planforms. Analytical solutions have also been obtained for circular and then elliptic planforms by formulating the problem in terms of an acceleration potential in coordinate systems appropriate to such shapes of figures. This was all done long ago. See Kinner [ 1937] and Krienes [1940]. There are many numerical techniques for obtaining approxi- mate solutions of this problem. However, I ignore these and proceed to analyze a special configuration which can be treated approximately as a limiting case of the general lifting-surface problem. 2.2. High-Aspect-Ratio Wing It is an interesting historical fact that Prandtl's boundary- layer solution really contains the essence of the method of matched asymptotic expansions, but Prandtl failed to observe that the same technique would work in his lifting-line problem. In the boundary- layer problem, he really required the matching of two complementary, asymptotically valid, partial solutions. It was probably Friedrichs [1955] who first recognized that the high-aspect-ratio lifting-surface problem could be treated the same way. Van Dyke [1964] discusses the derivation of lifting-line theory in some detail from the point of view of matched asymptotic expansions. My presentation is not different from Van Dyke's in any startling ways. There are some differences, partly because I have in mind applications to planing problems eventually, partly because I am not an aeronautical (or aerospace) engineer at heart. The conventional approach to solving the problem of a wing of 696 Stngular Perturbation Problems in Ship Hydrodynamics high aspect ratio is to simplify (2-37) by arguments that relate the sizes of the terms involving (x-&)© and (z-t)*. (Quite comparable arguments are used in the conventional approach to the theory of slender wings.) If the radical in (2-37) can be simplified, then the € integration can be performed, and one is left with just the integral over ¢. In this way, the 2-D integral equation is reduced to a one- dimensional integral equation, which is of a standard form. Using the method of matched asymptotic expansions, we return to the original formulation of the problem and derive a sequence of simpler problems, rather than try to work out approxi- mate solutions of the integral equation. The large-aspect-ratio wing is "slender" in the spanwise direction. This means that cross sections parallel to the z=0 plane vary gradually in size and shape as z varies; in particular, the maximum dimension inthe z direction, say 2S (the span), is much greater than the maximum dimension in the cross sections. We shall make whatever further assumptions of this kind that we need in order to keep the solution well-behaved. The small parameter can be defined as the inverse of the aspect ratio, that is, e€ = 1/(AR) = (area of H)/4S*, where H_ is the projection of the wing onto the y=0 plane. As before, it is not necessary to be so specific about the definition of €, and in fact it may be misleading. A wing with aspect ratio equal to 100 might be slender in the required sense if, for example, there were discontinuities in chord length in the spanwise direction. In any case, the wing shrinks down to aline, part of the z axis, as € 7; 0.. Let the body be defined by the following relation: y= g(x,z) = h(x,z), (2-38) for (x,0,z) in H. See Figure (2-1). It is not necessary that the body be a thin one, in the sense of the previous section. I do, however, specify that it should be symmetric with respect to z, for the sake of simplicity in what follows. Both of the functions g(x,z) and h(x,z) really depend on €, of course*, but we shall generally omit explicit mention of the fact. There is an incident flow which, at infinity, is uniform in the x direction. Let the far-field solution be represented by the asymptotic expansion: * In fact, g and h are both O(e). 697 Ogtlvie Y= 9 (*,2) 2h(x,2) Fig. (2-1). Coordinates for the High-Aspect-Ratio Wing N o(x,y,z) ~ Ux + yS $,(x,y,Z), where $a = o(4,) as e~0O, n=l for fixed’ '(x,y;z) + (2-39) (Again, the dependence on € is suppressed in the notation.) Since the body shrinks to a line (x= 0, y=0, |z| 0, the terms denoted by $, all represent flow perturbations which arise in the neighborhood of this singular line. They can be expressed in terms of singularities on that line, and the strengths of such singularities should be o(1) as € ~ 0. In an ideal fluid, we could expect the occurrence of dipoles, quadripoles, etc., on the singular line. We also take the realistic point of view that viscosity cannot be completely neglected and that there may be some circula- tion as a result. In the usual aeronautical point of view, this implies that there may be a vortex line present, complete with a set of trail- ing vortices. Inthe point of view adopted in the previous section, I assume that there may be a sheet of dipoles behind the singular line. I also make the usual assumption that these wake dipoles (or vortices) lie in the plane y =0. This part of the y=0 plane (0 ty? + (z-£) N SO i An(6) at ; ht Set (2-40) ae i. [ x? ty? + (z- pie The first sum contains terms which are exactly of the form given in (2-34), that is, they represent a lifting line with a strength yn(z). The second and third sums represent lines of dipoles Oriented vertically and longitudinally, respectively. It is implied above that the sums are asymptotic expansions, in our usual far- field sense. We shall presently require the inner expansions of these ferme. We obtain the inner expansions by assuming that r= (x? + y 2) Ve = O(e€), which implies that both x and y are small. Inner expansion of the lifting-line potential: Each of the double integrals containing a y can be rewritten as a single integral: 4) y { i ( y(t) dé dg TJ Jo 2 2 2, 2 [(x-§) ty + (z-¢)] i) VE x2 + y2 + (z-0)*] == AY” at yen (ton! ty + tant! what ey BF] ) | (2-41) Now break this into two parts: 1) The first term in brackets on the right-hand side does not depend on x. As yO (i.e., for y = O(€) ), its contribution can be represented: Ps ' ezved- By SVE [1+ of), where the double sign is chosen according to whether y>O or y < 0, respectively, and the special integral sign indicates that the Cauchy principal value is intended. This representation is valid only for |z| tan x]. This can be true only if the following are separately true: 1 1 55 = 0; NO = Y,(z)3 Aoo =F y,(z)- (2-52) 700 Singular Perturbation Problems tin Ship Hydrodynamics N x,y .z)~ y @.(x,y»Z), yy = 01%) ‘as -Ee= 0, n=0 with (x/e, y/e, z) fixed. (2-48) The first term in this expansion satisfies the conditions: fo. 7 20, = 0 inthe fluid region, (2-49) Eto = 0 onthe body. (2-50) From (2-45a), it is clear that the one-term inner expansion, $(x,y,z) ~ &o(x,y,z), must match the one-term outer expansion, 6(x,y,z) ~ Ux. Thus @o(x,y,z) is the solution of a two-dimensional potential problem, and a rather conventional problem at that: Ina section through the body drawn perpendicular to the spanwise axis, the potential satisfies the Laplace equation in two-dimensions, a homogeneous Neumann condition on the body, and a uniform-flow condition at infinity. The direction of the uniform flow is the same as the direction of the actual incident stream as viewed in the far field. Since @) does satisfy the Laplace equation in two dimensions, the methods of complex-variable functions are available for deter- mining its properties. In particular, if we assume that V@p is bounded everywhere in the fluid region and single-valued too, then @9 can be expressed as the real part of an analytic function of a complex variable, the analytic function being such that its derivative can be expressed by a Laurent series. Thus, we can write for @o(x,y3z): a re) @)(x,y3z) = Ux + 6) log r + nm) tan £ 2 Ago + Aoi ces + Bo sin 0 + Az cos 20 + Boz oa 20 Itt oe Sah r r r where r = (x? +y?)'/2, The "constants" are all unknown functions of z, the spanwise coordinate. The first term represents a uniform stream at infinity, and I have already performed one matching to determine this term. The second and third terms represent a source and a vortex, respectively; the fourth term, a constant, is included for generality; the fifth and sixth terms represent a dipole; etc. Such an expansion as (2-51) is valid outside any circle about the origin which encompasses the body cross section. 701 Ogtlvie I have taken the trouble of writing out the inner expansion of the outer expansion in three ways just to point out how, in this problem, there is an additional term in the lowest-order expression each time we add another term of higher order in the outer expansion. Each of the three terms metoded In (2-44) contributes to the € term in (2-45c). This phenomenon occurs frequently, and its occurrence is the reason that one must proceed step-by-step in the matching. In the present problem, one would be in some difficulty if he tried to write down an arbitrary number of terms in each expansion and immediately start matching. Next we formulate the near-field problem. Instead of making the formal changes of variable, x= e€X and y = €Y, we shall simply understand now that, in the near field, x= O(e) and y=O(e); | also 0/ax= O(e!) and 8/8y = Ole"). Of course, differentiation with respect to z does not affect orders of magnitude. The Laplace equation can be written in the form: 9 TF Syy = - $27, (2-46) where the right-hand side is e* higher order than the left-hand side. The boundary condition on the body is: 0 = 6,(g,+h,) - 6+ $(g,#h,) on y=gth. (2-47) The last condition is equivalent to requiring that 8¢/én =0 onthe body, where 8/8n denotes differentiation in the direction normal to the body surface. An alternative statement is the following: ) S (+ gx + hy) by Fy _ _ (hz + gz)$z =gth. (2-47!) aR v1 + (gy + h,)*] v[ 1 + (gy + h,)] en P where 986/@N is the rate of change in a plane perpendicular to the z axis, measured in the direction normal to the body contour in that cross section plane. Note that the left-hand side is O($/e), since differentiation in the N direction has the same order-of- magnitude effect as differentiation with respect to x or y. The right-hand side, on the other hand, is O(ge), since g and h are both O(e). Now let there be an inner expansion: 702 Singular Perturbation Problems tn Shtp Hydrodynamics y,(z) = Ole); w(z), Aylz), y9(z) = O(€*); also, all other terms in (2-40) are o(e?). These statements can all be proven. The description of the problem is greatly simplified, however, by their being assumed now. We can write the three-term outer expansion now: (x,y,z) ~ Ux + $\(x,y,z) + (x,y,z), (2-44) where Gay. )=% a AG de ; (2-44a) OAXsy +z oh ae Me ve ae a i S z. Y2(6) dé dt Gea S, i [ (x-€)? +y? + (2-0F]¥? fa Lyp, (6) +xa,(0)] do . (2-44b) Se ey leet The inner expansion of the one-term outer expansion is, of course: $(x,y,z) ~ Ux, [ Ofe)] (2-45a) to any number of terms. (Recall that x = O(e) in the near field.) The inner expansion of the two-term outer expansion is: (x,y,z) ~ Ux “s y, (z) [1-= 2 4an ¥ | [ O(e)] (2-45b) £5 sx ob seo [ O(e?)] Finally, the inner expansion of the three-term outer expansion is: + (x,y,z) ~ Ux +5 y,(2)[ 1 - = tan -| x | a. (z) xr (z) [ O(e)]} 2n(x? + y%) (2-45c) Ss dé yilS) 4 1 Laren 2 -£5. ME) 4 Sy tey[1-Ltan' ZL]. [ove 703 Ogilvie this term represents a distribution of vorticity extending to infinity both upstream and downstream. Thus, it leads to a discontinuity across the y=0 plane, even upstream. The second term must compensate for this behavior, since there can be no discontinuities in the region x <0. 2) The second term in brackets on the right-hand side of (2-41) must be considered carefully with respect to the branches of the square-root function. With a bit of effort, one can show that, as r— 0, its contribution is: x2) (1 - 2 tan! XY) (1 + Ofe*)), 0 < tan’ L 14 Condition at infinity for ®, problem 3 {e) 2 Vorticity, y,(z), in far field 2 , + $,<— +4, + ®, 3 Downwash velocity (condi- tion at infinity for 9, problem) 4 Correction to vorticity in far field; densities of vertical, horizontal : Fo #13 Ge dipoles in far! field One point in particular should be noted: The near-field problem was not linearized. If one can predict the flow around the two-dimensional forms which appear in the near-field problem, one is not limited to consideration of, say, thin wings. All that is necessary is that the spanwise length be much greater than the dimensions in the two-dimensional problems and that there be gradual change in the body and flow geometry in the spanwise direction. Needless to say, the latter condition is usually violated at the wing tips, and so the analysis breaks down there. It may be hoped that the prediction of important physical quantities is not affected too seriously thereby, but higher and higher approximations certainly cannot be found until the extra singularities at the tips are removed somehow. 708 Singular Perturbation Problems tn Shtp Hydrodynamics 223+. Slender Body In the previous section, we considered the flow around a slender body which was oriented with its long dimension perpendicu- lar to the incident flow. Now we consider the flow around a slender body which is oriented with its long dimension approximately parallel to the incident flow. The same geometrical restrictions will be applied to the body in this problem, namely, that its transverse dimensions should be small compared with its long dimension and that cross-section shape, size, and orientation should vary gradually along the length. Although both this section and the previous section concern slender bodies in an incident flow, convention says that only this section really presents "slender-body theory." In ship hydrodynamics problems, slender-body theory has been applied mostly to nonlifting bodies, i.e. , bodies not generating trailing vortex systems. * T shall limit myself here to such prob- lems too. Specifically, I assume that there is no separation of the flow from the body; furthermore, there are no sharp edges at which a Kutta condition might be applied. The potential function should be continuous and single-valued throughout the fluid domain. This restriction is not generally desirable. Certainly an important aspect of aerodynamics is the calculation of lift ona slender body which does generate a vortex wake; modern high-speed delta-wing aircraft and many slender missiles are genuine slender lifting bodies. There are several important ship-hydrodynamics problems which may ultimately be best analyzed by a slender- wing! approach. Most important, perhaps, is the problem of a maneuvering ship. An attempt is made in this direction by Fedyayevskiy and Sobolev [ 1963], but it is not very successful because they use the conventional methods of slender-wing theory, and these break down in application to wings which are not more-or-less delta shaped. + A modern approach to slender-wing theory is given by Wang [ 1968]. *Obviously, a ship is a "lifting body," but I think it is commonly understood that the term implies a dynamic lift process, and that is the way I use it. Tusiender wing," "wing of very low aspect ratio," and "slender lifting surface" are all equivalent terms in my usage. * Conventional slender-wing theory can be used for wings in which the span increases monotonically downstream, ending in a squared-off trailing edge. If the incident stream is uniform and steady, the wing does not have to end at the location of the maximum span, but the part of the wing aft of this location must be uncambered. Not all of these conditions are satisfied in the interesting ship maneuver- ing problems. 709 Ogtlvte uae RADIAL COMPONENT | | | FLUID VELOCITY AXIAL COMPONENT To \ Fig. (2-2). Fluid Velocity Near a Slender Body in Steady Motion The physical ideas behind slender-body theory were developed fifty years ago, and the original way of looking at this problem is perhaps still the best way. Take a reference frame which is fixed with respect to the fluid at infinity. As a slender body moves past, one may imagine that its greatest effect on the fluid is to push it aside; the body also imparts to the fluid a velocity component in the axial direction, but this component should be quite small compared with the transverse component. Both components should be small compared with the forward speed of the body. In modern slender-body theory, we attempt to formalize this estimate of the relative velocity-component magnitudes. We devise a procedure that automatically arranges velocities in the anticipated order: 1) Forward speed 2) Transverse perturbation 3) Longitudinal perturbation When this pattern comes out of the boundary-value problem, we then investigate further to see what other patterns follow from the same assumptions. The whole body of assumptions, results, and intermediate mathematics constitute what we call "slender-body theory." In aerodynamics, the original intuitive approach of Munk was not completely displaced until the late 1940's. The newer, more systematic approach which developed then is described well by Ward 710 Singular Perturbation Problems in Ship Hydrodynamics [1955]. For the first time, it was possible to predict with some confidence how the flow around the various cross sections interacted. There were some difficulties in principle, even with the new ap- proach; what we now call the "outer expansion" of the problem was in effect forced to satisfy body boundary conditions. The difficulty is somewhat comparable to trying to force a Laurent-series solution to satisfy prescribed conditions which are stated on a contour inside the minimum circle of convergence. A readable, refreshing account Se ripe aoa theory in the 1950's has been provided by Lighthill 1960]. During the early 1960's, slender-body theory was applied to ship hydrodynamics problems by several investigators. Probably the earliest to try this on a major scale was Vossers [1962]; he attacked a variety of steady- and unsteady-motion problems by slender-body theory. He used a Green's function approach, which apparently avoids the fundamental difficulty in principle of the previous method. However, it is really too much to hope to obtain asymptotic estimates of five-fold integrals -- without making mis- takes. Apparently Vossers did hope for too much, but Joosen [ 1963] and [1964] corrected many of his mistakes. Newman [1964] also advocated the Green's-function approach and produced some inter- esting results. The modern (i.e., fashionable) alternative is to use the method of matched asymptotic expansions. In ship hydrodynamics, Tuck [ 1963a] first used this method in his doctoral thesis at Cambridge University. It avoids the difficulties in principle of Ward's approach, and it is easier to work with than the Green's- function method. Of course, the method of matched asymptotic expansions has its own set of difficulties of principle. However, it is the method that I shall pursue here. * In any case, the analysis can be no better than the assump- tions which are made at the beginning. Therefore I shall be (perhaps painfully) explicit about the assumptions. 2.31. Steady Forward Motion. Let the body surface be specified by the equation: *A very recent account of slender-body theory, particularly with respect to its applications in ship hydrodynamics, has been pub- lished by Newman [1970]. I think that his presentation and mine generally complement each other (and perhaps occasionally contra- dict too). Newman has provided a survey that seems comparable in intent to the one by Lighthill [1960], mentioned above, whereas I am trying to place slender-body theory into a hierarchy of singu- lar perturbation problems. My emphasis is on the development and application of the method of solution. ceo Ogilvie ra ro(x, 8) x in K, where r= (y? +t: z2)V2 , and @ is an angle variable measured about the x axis. It will be assumed that r = O(e). In this section, I take the most conventional definition of 8, namely, that it be measured in a right-handed sense from the y axis. (In ship prob- lems, it is more convenient to measure the angle from the negative vertical axis.) A is the part of the x axis which coincides with the longitudinal extent of the body; typically, one might take it to be the interval, - L/20, where e€ is the slenderness of the parameter. These order relations should be valid near the body. Far away, there will be the uniform stream, which is O(1), but there is no reason to assume that the perturbation velocity will have components with differing orders of magnitude. These order-of-magnitude relations all come about auto- matically if, in the near field, we define new variables: r= eR, y=eY, Z=€Z, and assume that differentiation with respect to x, Y, Z, R, and 08 all have no effect on the order of magnitude of a quantity. Thus, suppose that the potential in the near field can be written: (x,y,z) = Ux + @(x,Y,Z). Then the derivatives have the following orders of magnitude: Ob _ Od _ 0h _ Ob _ 1 OO _ ge Ut ges Ol) + OG); GEST = Ty = Olb/e); 882 0(6/e); 2= O(6/e). It will turn out that @ = O(e*). This means that the transverse velocity components, $y, $2, and 4, are all O(e), that is, they are proportional to the slenderness parameter. Note also that a circumferential velocity component would be given by (1/r)8o/80 = (1/eR)d8@/80 = O(@/e), when we interpret R= O(1) (that is, in the near field), and so circumferential and radial velocity components have the same order of magnitude. The perturbation of the longi- tudinal velocity component is O(®) = O(e*), which is, appropriately, a higher order of magnitude than that of the transverse velocity components. In the far field, we assume that differentiation with respect to any of the natural space variables has no order-of-magnitude effect. Thus, we use the Cartesian coordinates (x,y,z) and the (13 Ogilvte cylindrical coordinates (x,r,®) in a very conventional manner. As € ~ 0, the slender body becomes more and more slender, shrinking down to a line which coincides with part of the x axis. (This is the line segment that I defined as A previously.) In the limit, there is no body at all and thus no disturbance of the incident uniform flow. In the far field, the disturbance is always o(1). Therefore the far field consists of the entire space except the x axis, and the potential function must satisfy the Laplace equation everywhere except possibly on the x axis. At infinity, it is reasonable to require that the perturbation of the incident flow should vanish, which implies that the pertur- bation potential must be regular even at infinity. A velocity potential cannot be regular throughout space, including infinity, unless it is trivial. Therefore the velocity potential must be singular some- where, and the only place in the far field where such behavior is permitted is on the x axis. Our far-field slender-body problems all reduce to finding appropriate singularity distributions on the x axis. The Far-Field Singularity Distributions. In the far field, the first term in the asymptotic expansion for the potential function will be Ux. All of the following terms must represent flow fields for which the velocity approaches zero at infinity; they represent distributions of singularities on the x axis. The nature of the singularities can only be determined in the matching process, and so we must generally be prepared to handle all kinds of singularities. One of the easier ways of doing this is to apply a Fourier transform to the Laplace equation, replacing the x dependence by a wave-number dependence. The resulting partial differential equation in two dimensions can be solved by separation of variables in cylindrical coordinates. When we require that the potential functions be single valued, we find that the solutions must all be products of: K,(|k|r) or 1,(|k[r) and sin nO or cos n@, where Kn and I, denote modified Bessel functions. Since I, is poorly behaved when its argument is large, we reject it, so that the solution consists of terms: K,(|k|r)[@ cos nO + B sin né]. The quantities @ and B are constants with respect to r and 6, but they are both functions of k. They also depend on the index n, of course. The general solution is obtained by combining all such possible solutions. Any term in the far-field expansion of the potential function might be of the form: 714 Singular Perturbation Problems in Ship Hydrodynamics oo ik bq(X2¥ 12) = = » i dk e™ K,(|k|r)[az,(k) cos n@ +b, .(k) sin n6] , no 0 (2-59) where ama(k) and be Ak) are unknown functions. The most general far-field expansion comprises the incident-flow potential, Ux, and a sum of terms like the above, that is, M x,y,z) ~ Ux + b(x yz) - fortixed (x,y,z) as € => 0. y. m y m=l (2-60) It will be necessary to have the inner expansion of the outer expansion. This means that we must interpret r to be O(e) in the above expressions, instead of O(1) as heretofore, and re- arrange terms according to their dependence on €. The easiest procedure is to replace any of the Kp, functions in (2-59) by its series expansion for small argument. We obtain formulas such as the following: ra, FS (OE oo 1 ikx * = dk e™ K({k [nya o(k) -0 00 Co) _ logr ikx 1 ikx Clk]. —— i. dk e amolk) - aS. dk e a nolk) log Tse (2- 61a) n>0: 1 ied ikx * ! cos x. dk e K,(|k|r)a,,(k) (CP) 0 Coe aie _ikx ~ 2 (n= 1)) (£22) no { Se ama(k)- (2-61b) 2nr” a -00 [1k | Physically, the n=0 integral represents the potential for a line of sources. This can be seen directly from (2-61a): As r~ 0, the function is proportional to log r, which is the potential function for a source in two dimensions. However, the strength of the apparent 2-D source is a function of x. In fact, the integral defin- ing that strength is identical to the integral which gives the inverse of a Fourier transform. Let ao) be the function having a* (ik) 715 Ogilvie as its Fourier transform, and further define: o,(x) = - 27a, (x). Then the result in (2-61a) can be rewritten: or) iS. dk e| * Kol [k |r) aol k) oa Om(x) log r - qa f(x), (2-012') © f(x) -{ d€ of (§) log 2 |x-€ | sgn (x-&). (2- 61a") By manipulating the full integral containing Kp, one can also show that: i esate * 1 (° __ om(é) a€ ikx oa dk e Kol |k | r) anol) =- ae Fee ae » (2-62) which is easily recognized as the potential function for a line distri- bution of sources. Similarly, the other integrals can be interpreted in terms of dipoles, quadripoles, etc. In particular, we see that for n=l the inner expansion of the integral reduces to the potential in two dimensions for a dipole. We may consider the variable x asa parameter, and then we have a different 2-D dipole strength at each ora The Sequence of Near-Field Problems. In the near field, we can formalize our procedure by making the changes of variables already mentioned, r=eR, y = €Y, z = €Z, then assuming that differentiation with respect to R, Y, or Z does not affect orders of magnitude. Instead of doing this, I shall simply retain the ordi- nary variables, r, y, and z, and Iask the reader to recall that differentiation with respect to any of these three variables causes a change in order of magnitude. Thus, for example, 06/dr = O(¢/e) in the near field. In cylindrical coordinates, the Laplace equations and the body boundary condition can be written as follows: 716 Singular Perturbation Problems in Ship Hydrodynamics [L] 9er +2 , ty $09 = - Pxxi (2-57') [ H] 86 - Nv = 29,4 (1/20) 70949 = PkTOy Vv[it (r99 /r0)"] il dt (r099/roF] on r = ro(x,@). (2-58') The definition of N is analogous to that in (2-47'). It is a unit vector lying in the cross section plane at some x, perpendicular to the contour of the body in that cross section. It has the three com- ponents: (0,-1 »TO9/To) V[4 + (r99/r0)*] measured in the x, r, and 9 directions, respectively. Equation (2-58'), like (2-58), expresses the fact that 8¢/8n =0, where n is the unit vector normal to the body surface. Let the inner expansion be expressed as follows: N o(x,y,z) ~ » @,(x,y,z) as €—~0O for fixed (x,y/e,z/e). n=0 Substitute this expansion into the [L] and [H] conditions above: 2 (ee) | Vy,2(Po + P + a + 3 +...) = - (Go + Ot eee); ae@ To(Po, + O, Tec) 8Dp , AD - : [ H] Sh aon i aT re cere a [1 + (r96/t9)"] The operator Vy2 is the 2-D Laplacian inthe y-z plane, that is, It can be proven that the first term in the expansion, 9, represents just the uniform stream: (x,y,z) = Ux. 717 Ogilvte This appears so obvious that I pass on immediately to the 9, problem. Fromthe [L] and [H] conditions, we find: [Lui] Viz % =0 inthe fluid domain; [H,] ES ee on r=. (2-63) OR Ee att (r9,/20) Finding ®, is strictly a problem in two dimensions. In fact, it is just the problem that the early aerodynamicists put forth intuitively at the beginning of their slender-body analysis. (It was also the end of their analysis!) For an arbitrary body shape, we might have to solve this boundary-value problem numerically; that is not much of a problem today. However, we are not yet ready to work with num- bers, because the formulation of the problem is not quite complete: we have not specified the behavior of #, at infinity. To do so requires that we match the unknown solution of this problem to the far-field expansion. First, note what (2-63) tells us about the order of mag- nitude of ©. The right-hand member is O(e) and the left-hand member is O(@,/e) (because of the differentiation in the transverse direction), which together imply: ®, = Ole’). Actually, (2-63) says only that ®, cannot be higher order than e*; it could be lower order if the matching introduced some effect that required € to be o(®,), but this does not happen. This , problem is remarkably similar to the @) problem in Section 2.2. If we can assume that V4, is bounded at infinity, then we can express ©, ina series just like the one in (2-51). Whether V®, really is bounded at infinity can only be determined from the matching, of course, but we go ahead with the assumption, trusting that our method will show us if we have made unwarranted assumptions. It should be noted too that there are important differences between this problem and the problem of Section 2.2. The Neumann- type of condition on the body was homogeneous there, but it is not homogeneous here. Thus, one may expect that there may be a non- zero net source strength inside the body in the present problem. What happens at infinity is also different. In the earlier problem, the potential had to represent a uniform flow at infinity, and we supposed that there might be the proper circumstances that a circu- lation flow could occur. In the present problem, the uniform flow 718 Singular Perturbation Problems in Ship Hydrodynamics at infinity has been included in ), and so we might expect that ©, will represent a flow with velocity vanishing at infinity, and there appears to be no reason to expect a circulation in the 2-D problem. It would be tedious to go through the same arguments that were used previously, and so I shall only summarize the results that would be obtained after a careful matching process. In the near field, , does indeed yield a velocity field which is bounded in magnitude at infinity, and there is no circulation. Thus, it can be represented by the series: Ai cos @ + By sin 6 4 a wee » (2-64) A ® (x,y,z) = C, sis is log r t+ The "constants" are all functions of x. Inthe near field, all terms must be the same order of magnitude, by definition, and so Aj and Bit are O(€Ajg). (Iam, as usual, ignoring quantities which are O(log e).) In the matching, the 1/r terms are lost in the first round, and the log r and constant terms are forced to match the inner expansion of the outer expansion. In the outer expansion, (2-60), only a line of sources in the $, term of (2-60) can match the near-field expansion properly. That is, in (2-59) and (2-60), we have the following: * a," (Ik) = by, (k) =0 except for n=0. The two-term outer expansion and its two-term inner expansion are: [e6) ik * (x,y +z) ~ Ux + i. dk e Ko [k| r)a,o(k) (2-65a) UUs oe tiloge tote) (2-65b) on 9 8 an 1%! » where (2-61a') has been used to express the latter. Matching between the near-field and far-field then shows that: Aig = 0, (x) (2- 66a) Cy ge f(r). (2-66) In obtaining an actual solution, one proceeds through the following steps: 1) Matching shows that ® represents a flow with bounded velocity at infinity. 2) Thenthe #, problem is completely 719 Ogilvie formulated and can be solved. 3) From the solution of the ®, problem, the function Ajo(x) can be determined, which, through the matching, gives o,(x), and the far-field two-term expansion is known, 4) From the matching relation for C,(x), along with formula (2-61a"), the near-field potential is known completely to two terms, and the C,(x) term includes the most important effects of interaction among sections. This sequence of steps shows what an intimate relation- ship exists between near- and far-field expansions. The source strength, o, (x) = Aio(x) » can be computed without the necessity of solving the flow problem. In the near-field picture, draw a circle which encloses the body section. The net flux rate across this circle is just Ajg. From the body boundary condition, (2-63), one can show that there is a net flux rate across the body surface, and it is given by Us'(x), where: er 1 s(x) -3\ dé ré(x, 0) = cross section area at x. (2-67a) The two fluxes must be equal, and so we find that: o, (x) = A\o(*) = Us'(x). (2-67b) Thus, the source strength is proportional to the rate of change with x of the body cross sectional area. I shall not pursue the solution to higher order of magnitude, although there is no insuperable difficulty in doing so. Rather, I prefer to point out several interesting facts about the solution and then close this section. In the far field, the solution to two terms is axially sym- metric, although the body is not a body of revolution. The near- field two-term expansion is not symmetric in this way unless the body is circular and is aligned with the incident flow, However, the near-field solution can be represented by the series, Ai; cos 8+ By sin 0 x 4 o(x,y,z) ~ Ux + AO) jog x - aa f(x) + er rd os r and, at large r, the axially symmetric terms dominate this series. If the far-field expansion is carried to three terms, it will be found that the third term can be interpreted in terms of a line of dipoles, both vertically and horizontally oriented. Such terms will be of the form given in (2-61b), with n= 1; they contain unknown functions a3,(k) and b3,(k), which must be determined through matching. These unknown functions will depend entirely on the 720 Singular Perturbation Problems in Ship Hydrodynamics solution of the ®, problem discussed above. In fact, one finds explicitly that: ikx * atic): a Sele is = ike oo dk * Ay (x) = he. Teh e bo, (k) ° Thus, the two-term inner expansion contains enough information to determine the strength of the dipoles which appear in the third term of the far-field expansion. The same inner expansion would deter- mine the strengths of quadripoles in the fourth term of the far-field expansion, etc. , étc. On the other hand, the far-field expansion (even at the second term) contains much information about three-dimensional effects, information which is largely lacking in the near-field expansion. I have already pointed out that only the "constant" term contains im- portant information about 3-D effects in the two-term near-field expansion. The rest of the ©, solution depends on just the shape of the local section and the local rate of change of section shape and size. If higher-order near-field terms are found, it will be seen that they are influenced even by the two-term outer expansion. In fact, the "constant" term in ©, can be interpreted as a modification to the incident stream, caused by the presence of all the other cross sections of the body. The effects of this extra incident flow on the transverse velocity field are not perceived until one finds a higher order expansion of the solution in the near field. The briefest account of slender-body theory would be seriously lacking without mention of the possibly catastrophic effects of body ends. If a body has a blunt end, then s(x) increases linearly in some neighborhood of the end. Accordingly, s'(x) is discontinuous, jumping from a value of zero just beyond the end to a finite value at the end. This is an obvious violation of our assumptions about "slenderness." But trouble develops even without a blunt-ended body. For example, if the tip is pointed (but not cusped), there will still be a stagnation point right at the point. Thus this case violates the assumption that longitudinal perturbation of the incident flow velocity is a second-order quantity. Sometimes these end effects can be overlooked with impunity. There are major examples later in this paper. However, even when we have such luck, we must be prepared to have higher-order expan- sions go awry. 2.32. Small-Amplitude Oscillations at Forward Speed. In this section, we consider the same Kind of body as in Section 2.31, namely, a slender body which is aligned approximately with an incident stream. However, now we formulate a time-dependent problem in which the body performs small-amplitude oscillations while it moves through the fluid. G24 Ogilvie It would be entirely feasible to consider the general problem in which the body oscillates with the six degrees of freedom of a rigid body. (We could even include more degrees of freedom by allowing deformations of the body.) However, the major concepts should be clear if we allow only two degrees of freedom, a) a lateral translation, comparable to the heave or sway of a ship, and b) a rotation, like the pitch or yaw of a ship. In this section, I shall depart from my usual approach and first treat the problem for a perfectly general body, then introduce the slenderness property at the very end. This introduces a bit of variety, but more important is the fact that some general properties of the physical system can be pointed out, without any confusion over the effects of assuming slenderness of the body. We use two coordinate systems: Oxyz is fixed in the body with its origin at the center of gravity, and O'x'y'z' is an inertial system which moves with the mean motion of the body center of gravity. With respect to the stationary fluid at infinity, the mean motion is a translation at speed U inthe negative x!' direction; thus, inthe O'x'y'z' system, there appears to be a flow past the body in the positive x' direction. The two reference systems differ because the body oscillates inthe z direction, the instantaneous displacement being denoted by §,(t), and rotates about the y axis, the angular displacement being denoted by & (t). In a more general problem, we could let & (t), E,(t) and Es t) denote surge, sway, and heave (displacements along the x, y, and z axes, respectively) and &4(t), E5(t), and E,(t) denote roll, pitch, and yaw (rotations about the x, y, and z axes, respectively). It will be assumed explicitly that §j(t) is a small quantity, so that squares and products can be neglected in comparison with the quantity itself. Furthermore, it will be assumed that €; (t) varies sinusoidally in time and it will be represented by the real part of a complex function varying as e!t, We shall not usually bother to indicate that only the real part of a complex quantity is to be implied. Thus we can write: E(t) = iw (t). (2-68) The relationship between the two coordinate systems is as follows (see Figure (2-4) ): x = x' cos & - (z'-§,) sin€,@ x' - z', ; yes yes (2-69) z =x' sin &. + (z'-€,) cos &, = ee Ss . 122 Singular Perturbatton Problems in Ship Hydrodynamtes Fig. (2-4). Two Coordinate Systems for Oscillation Problem The absolute velocity of the center of gravity is: DR OE te: iw6,k! = (- Ucos && - iwé, sin €,)i + (- U sin &,+ iw€, cos ey) k (2-70) =- Uit (iw6., - UE.) k (2-70') where (i,j,k) are unit vectors in the Oxyz system, and (i', j',k') are unit vectors in the O'x'y'z' system. Let the body surface be defined by the equation: S(,y,z) = 0. (2-7 1) Denote the unit vector normal to the surface, inwardly directed, by n: n = ni + nj + n,k- (2=TZa) It is convenient to make a number of other definitions, as follows: 1) nj: Extend the above definition of ny to j =4,5,6 as follows: ACS Ogilvie rXn =n,i tn,j + nk (2-72b) where = Xj + yj + zk. In particular, note that: n,= n'k and n,= zn, - xn,. (2-72') 2) j: This is a normalized velocity potential. It satisfies: Ping | Pix, + oj,, =0 in fluid region; Dd; sel = nj on S(x,y,z) = 0; (2-73) |v,| > 0 at infinity. 3) v(x,y,z): This is a normalized fluid velocity, equal to the fluid velocity at (x,y,z) when an incident stream flows past the body, the stream having unit velocity, i , at infinity. It can be represented as follows: v(x,y,z) = V[x - $,(x,y,z)] : (2-74) 4) mj: This quantity is related to the rate of change of v(x,y,z) in the neighborhood of the body, as follows: mi +m,j tm3k= m= - (n'V )v; (2- 75a) magi + msj +t mgk= - (N-V)(r Xv). (2- 75b) In particular, note that: a, _ z = ° = ! sae on (2-75a') mM, = - ae r Xx v) = - on (2% - XV3) i Fi io [z(1-4, ) +x, =) ee +e (26, = xo). (2-75b') 724 Singular Perturbatton Problems tn Ship Hydrodynamics 5) Wj: This is another useful normalized velocity potential. It is related to mj; the way 9; is related to nj. It satisfies: Hiyy + iyy + hi,, = 0 in fluid region; It oi mj; on S(x,y,z) = 0; (2-76) Ivy; | > 0 at infinity. In particular, it can be seen that these conditions are satisfied for i=3, 5 if: a(x sy »Z) = (x,y 2Z)3 (2-76') b.(X,y +2) =e $(x,y»Z) = (29, I x9, )- (2-76") x z (The last term does satisfy the Laplace equation.) Now we can write down the velocity potential for the combined translation and oscillation in terms of the above-defined quantities. It is a well-known fact of classical hydrodynamics that the fluid motion can be expressed as a superposition of six separate motions, each of which would be caused by the motion of the body in one of the rigid-body degrees of freedom. However, it is essential for the use of this fact that the description be made in terms of a coordinate system fixed with respect to the body. Note that there is no lineari- zation implicit in this superposition, in the sense that there is no requirement that motions be small in any way. In the body-fixed reference frame, the velocity potential is: [- Ucos E. 2 iw, sin E.] , (x,y +2) +[-Usin & + iwé,cos .]ofx,y,z) + iw6, (xy »Z)- The nature of the superposition is obvious when we compare the first two coefficients here with (2-70). However, it must also be recalled that the velocity potential obtained in this way gives the absolute velocity of the fluid, that is, the gradient of this potential is the velocity in a reference frame fixed to the fluid at infinity. Thus, we must add to this potential an extra term to provide for the x This can be concluded also by recalling the definition of qj: its gradient vanishes at infinity. 725 Ogilvie apparent incident stream in the observation reference frame. The latter has the velocity potential Ux', and so the complete potential is: x', y',z',t) = Ux''='(U cos E.t iw6;sin €) (x,y »z) + (- U sin & + iwS; cos &,)bfx,y,z) + iwSsoe(x,y,z) (2-77) = Ux'- Uo (xsy>2) t [ iw (x,y »z)] 6 5(t) + [iwo(x,y,z) - Ud lx,y,z)] &,(t) . (2-77') The potential @ has been defined basically in terms of the inertial reference frame, although most of the right-hand side here is ex- pressed in terms of the body-fixed system. Note that not only the incident stream is defined in terms of primed coordinates, but also the body motion is really defined in those coordinates as well; in particular, heave motion is a translation of the body along an axis fixed with respect to the fluid at infinity. The Bernoulli equation must be used for computing the pres- sure: ind 2 2 2 - = >; + = ($y + $y: ar $,')« The linear approximations of the derivatives here are as follows: 2 6, = [(ia) 3 + (io) ,,] £5(t) + [ (iw) $, - (io) $, + (io) (24, - 26, )] 5 (5 $. = UL1 - $,] + Livdglés(t) + [ings - Uds,- Ud, ]Ep(t)s by = - UL 4] tind. ésit) + [log - Udy] Es(t)s $,, = - UL4,] +[inbs,] 83(t) + Linge, - Uds, + U4] E5(t). Some simplification has been done through the dropping of quadratic terms in §|. Substituting these expressions into the Bernoulli equation and simplifying somewhat, one finds that: 726 Singular Perturbation Problems in Ship Hydrodynamics 2 v? +[ (iw), + (iwU) (bs + V+ $9] S5(t) + [ (iu)?$5 + (ioU) (Ys + V+ 7 $5) - Ulds + V+ 7 4)] a(t) In the terms containing §j, one can use primed and unprimed co- ordinates interchangeably, since the difference leads to terms of higher order. The force (moment) corresponding to the j-th mode of oscillation is given by: Fj (t) = if ds njp(x,y»zZ,t) = Tji g (t) + Fio» s 7 where S is the surface of the body at any instant and Fjg is the steady force component. (For j =1,2,3, the latter is zero.) The "transfer functions" Tjj are: T33=- P i ds nf (ia) $5 + (iwU)(¥, t+ vi¥ ,)] 5 ae of dS nj (iu), + (iwU)(Y, + V9 4) - Ul, + ¥-V OJ]; Ty3 = - of ds nl (ia), + (iwU) (3 + v-.V $3) | ; Ty5 = - | ds n,[ (1c), + (iwU) (Hs + v:V $,) - urw,+ v-V $3]. These formulas can be simplified considerably, even before we introduce the slenderness approximation. We use two theorems: One is an extension of Stokes’ theorem, proven by Tuck (see Ogilvie and Tuck [ 1969]): f. dS nj(v-V $)) = J dS mj4j - The other theorem is Green's theorem; in applying it, we note that all of the functions decrease sufficiently rapidly far away that there is no need to account for effects at infinity.* Thus, in T33 and Tz, we have: * >, and $, appear to represent dipoles at infinity; thus, both are proportional to 1/ r@ as r—~ oo. $5 appears to represent a quad- ripole, thus is proportional to 1/r> at infinity. 727 Ogtlvie J, asagiigt vv) =) a5 (4,,45~ 5,6) =o Similarly, in T 55) f. dS n(p, + ve V $,) = 0)4 In T 35° we manipulate one integral as follows: dS + “V = dS - i nal, + v°V $5) \ (nsb; - m5$,) © ‘ a dS (ngs - 45,4, + ts%5, - 4305) a i ds n3$, ay ds z(n3,_ - n\$,,)- Similarly, in Ts, and T,,., we find: i ds ns (5 + v-V $,) = i ds n3$3 - f. dS 2(n,$, - n>, )- S The last integral in the last two expressions can be rewritten: i‘ dS z(n3$, - n,9, ) =j- \ dS znxV 9, j- { dS [n Xv (zo) -n X ($k)] ) \ dS n, 9; S the last equality following from application of Stokes' theorem to the first term. Combining all of these results, we find for the Tjj: 2 T33 = - p(iw) { dS n,$33 s T35= - p (ie)" {. dS nz, + p(iwU) i ds (n3$, - n, $,); 728 Sitngular Perturbation Problems in Ship Hydrodynamics Ts3 = - p(iw)* i ds n39, - p(iwU) { dS (nz - n,,); s Ty, = - pliw)” i dS ng, + ul dS (n3¢, - n,$,)- These results have been obtained with no assumptions made about the shape of the body. The only assumption was that the sinu- soidal oscillations had very small amplitude. Now, finally, let us assume that the body is slender. The only effect is that we lose the terms containing n,$,. For a slender body, n3 and n, are O(1) as the slenderness parameter, €, approaches zero, whereas n, is O(e€). From (2-73), we see that oly is therefore higher order than $, and 9, by a factor of €. E us: r7r lS. dS n, 4, /S. ds nsés| = O(e?). Seldom in practical problems do we ever retain terms with sucha great difference in orders of magnitude, and so we neglect the terms containing n,$, if the body is slender. In the ship-motion problem, the quantity corresponding to T33 will be (iw) [ ag, + é bgsl » where a3, and b,, are the heave added-mass and damping coeffi- cients, respective viet The other Tjj's have a similar interpreta- tion in terms of pitch added-moment-of-inertia and damping coef- ficients, cross-coupling coefficients, etc. We note that there are three kinds of terms here: a) Terms independent of U. These are all of the same form: Tt =- Gal f. dS nj $j. (2-78) b) Terms proportional to U. These occur only in the cross terms, Tij » with i#j. For a slender body, we have: * In the ship-motion problem, $j is complex. Here, of course, >; is purely real, and so there is no analog to bij. 729 Ogilvte 71) (0) Tg digs r (U/iwl bans (2-79) 7) (0) ee + (U/ia) Ty (2-80) c) A term proportional to U'. This occurs only in T 55 pO) T55 (0) sa + (Ufa)? TS). (2-81) Even at zero forward speed, there is coupling between the heave and pitch modes, unless the body is symmetric ) fore- an o aft. Ifthe body is symmetrical, one can show that T3, and T, are zero. But even in this case, the existence of forward meen. causes aloss of symmetry, and so a pure-heave motion causes a pitch moment, and a pure-pitch motion causes a heave force. The symmetry between T,, and T,, should be noted: The speed- independent parts are ‘equal, whereas the speed-dependent parts are exactly opposite. One remarkable fact is that there is no interaction between the oscillatory motion and the perturbation of the uniform stream by the steady forward motion. If the above formulas are derived from the kinetic-energy formula by use of the Lagrange equations, this fact is perhaps obvious. When we derive expressions for force and moment on an oscillating ship, it is anything but obvious. For the sake of completeness, I write out here the final formulas for the ti s for a slender body in an infinite fluid. We note first that, by the same procedures used in the steady-forward- motion problem, the following is true to a first approximation: iy, +j,,=9, in the near field. From (2-72'), it is rather obvious that, for a slender body, ees sl (| + O(e*)] 5 3 4 and thus, from (2-73): $,= - xpi + O(e*)]. Now let: m(x) = mat déin,$,= added mass per unit length, (2-82) C(x) 730 Singular Perturbation Problems in Ship Hydrodynamics where C(x) is the contour around the body in the cross section at x. Then clearly, (0) 2 T33 = T33 = - p(iv) i ax ( df nz, = - (ie) if dx m(x), L C(x) L where L is the domain of the length of the body. Similarly, we obtain: Ole 2 - 2 ‘ wee? Tags p (iw) f. dx «J df no, = (iw) \ dx x m(x) ; = (ih , dx x*m (x) . ae a uN Collecting these results, we have: ve I a3 = - (el i" dx m(x) ; ou w | ae a U (iw) A dx xm(x) - To T 3s ; (2-83) FI N 53 Gaye ih dx xm/(x) + ~ T33 $ ry 1 2 so (ay dx x*m(x) + (=) Tg-6 Ill. SLENDER SHIP Of all the problems discussed in this paper, the slender-ship problem has led to the most important practical consequences. Therefore it is not unreasonable to devote the longest chapter to the problem. Even so, some aspects will not be covered; perhaps the most important missing example is the case of sinkage and trim of a ship. In the four sections, two steady-motion and two unsteady- motion problems are discussed. The first steady-motion problem is the wave-resistance problem, that is, the problem of a ship in steady forward motion on the surface of an infinite ocean. In the second section, the problem treated is essentially the same, but the Froude number is assumed to be related to the slenderness param- eter in such a way that Froude number approaches infinity as slender- ness approaches zero; this rather unnatural relationship is discussed at some length. In the third section, I discuss in some detail the problem of heave and pitch motions of a ship at zero forward speed; the results are not at all surprising, but the method is quite clear in this case, which helps one in approaching the final section. It is concerned with the problem which is the combination of the first and third problems: heave and pitch motions of a ship with forward speed. W3i Ogtlvie 3.1. The Moderate-Speed, Steady- Motion Problem The theory presented here is due to Tuck [ 1963a] *. The analysis -- as far as I carry it here -- is not very much more diffi- cult than the analysis of the infinite-fluid problem, and so it will only be sketched here. The theory is attractive for its simplicity and its elegance, but unfortunately it has not been successful in predicting wave resistance. The reasons are not entirely clear, although they have been discussed for many years. See, for example, Kotik and Thomsen [1963]. The difficulty could very well be that real ships are just not slender enough for a one-term expansion (or perhaps any number of terms) to give an accurate prediction of wave resistance. This is the old question, "How small must the 'small' parameter be?" Another possibility is that the error arises because the lowest-order slender-body theory places the source of the disturbance precisely on the level of the undisturbed free surface, and so there are no attenuation effects due to finite submergence of parts of the hull. (These two possible causes of error are not entirely separate.) Still another possible cause is considered in Section 3.2. The hull surface will be specified by the equation: r = ro(x,@). (3-1) Now it will be convenient to measure 9 from the negative z axis, since most ships are symmetrical about the midplane. We assume that rg = O(e) and that 8°r,/8x" = O(€), as needed. There is a velocity potential satisfying the Laplace equation and the same kinematic body boundary condition, (2-58), as in the infinite-fluid problem. The incident stream is again taken in the positive x direction, that is, with velocity potential Ux. The two free-surface conditions are: ebtslertoytdd = 50%, on z= Ulxyls (3-2) C,o+ by Py- >, = 0, on z= C(x,y). (3-3) Finally, there is a radiation condition to be satisfied. * This reference is not readily available, but the material which is of interest here can also be found in Tuck [ 1963b] , Tuck [ 1964a], and Tuck [1964b] , all of which are gathered into Tuck [1965a] . 732 Singular Perturbation Problems in Ship Hydrodynamics As usual, we assume that there is a far-field expansion: N (x,y,z) ~ » $(x,y,z), where dng = (by) ase. == 0, “ee for fixed (x,y,z). (3-4) and a near-field expansion: N (x,y,z) ~ bt @(x,y,z), where @, =o0(@) as e-0, n=O for fixed (x,y/e,z/e). (3-5) These expansions are substituted into all of the exact conditions, from which we obtain two sequences of problems which must be solved simultaneously. In the far field, the first term in the expansion for ® must be just the incident uniform-stream potential, Ux, since the body vanishes as € —~ 0 andthe asymptotic representation $ ~ ¢9= Ux satisfies the free-surface conditions (trivially). The second term represents a line of singularities on the x axis. One really ought to allow the most general possible kind of singularities on this line, but it is no surprise to find that just sources are sufficient at first, and so we consider the special case of a line of sources on the free surface. One can show that higher-order singularities could not be matched to the near-field solution. Alternatively, one can construct a far-field solution using Green's theorem and show that it really represents just a line distribution of sources. See, for example, Maruo [1967]. One can use the classical Havelock source potential to ex- press the desired potential for a line of sources, but Tuck's pro- cedure is more convenient in the slender-body problem: Apply a double- Fourier transform operation to the Laplace equation, re- ducing it to an ordinary differential equation with z as independent variable: ~ (ke? + 07) 6 *(e,£5z) + bee (k,£3z) = 0, where k and &£ are the transform variables, and the asterisks denote the transforms. Assume for the moment that the line of sources is located at z =Z,)< 0. The above differential equation can be solved generally, with a different solution above and below Z=Zqg- The solution in the upper region is forced to satisfy the linearized free-surface condition, the solution in the lower region must vanish at great depths, and the two must have the discontinuity at Z= Zp appropriate to the source singularities. Finally, one may 433 Ogtlvte allow Z—~ 0. In physical variables, the result is: foe) 1 5 $(x.y +2) = - 5 dk e! 6 *(k) Ko (|k |r) @ 2 ro : 00 r ify +2-V(k2+L%) ime \ dle J c2a*(K) § aie 2 2492 eT, 2,’ rOeT “-m 00 f(x? +02)[ (2402) -(Uk- ip /2)"] (3-6) where yp denotes a fictitious Rayleigh viscosity, puaranteeing that the proper radiation condition is satisfied, and o (k) is the Fourier transform of o(x), the source density. The two-term outer expansion is: (x,y, 2) ~"Ux + $,(x,y»z), which has the two-term inner expansion: (x,y,z) ~ Ux +2 o(x) log x - 3 f(x) - glx), (3-7) where @ f(x) =a dé o'(&) log 2|x-&| sgn (x-&) (3-8a) ee) . = + a dice o diiloc ielLs ; (3- 8b) -@ 2 po od U ikx, 2 * dé g(x) = lim —— dk ek Go J SE pO 4m? vm : 00 (8 +8) [ eV (e487) - (Uk- ip /2)*] (3-9) The expansion should be compared with the corresponding expansion for a line of sources in an infinite fluid, as given in (2-65). We now have an extraterm, g(x), and the terms containing o(x) and f(x) differ by a factor of two from the earlier result. The latter variation is not important; it results from the fact that the line of sources was taken at z = zg <0, and those sources merged with their images when we let zo 0. * Define o(x) =0 for values of x ahead of and behind the ship. 734 Singular Perturbation Problems in Ship Hydrodynamics The most interesting feature of this inner expansion of the two-term outer expansion is that the wave effects are all contained in g(x) -- a function of just x. In the infinite-fluid problem, all 3-D effects in the near field were included (in the first approximation) in the single function of x, f(x) - We now have a generalization of this for the free-surface problem. In the near field, it is easy to show that the first term in the asymptotic expansion of the potential is again just the uniform-stream potential, Ux. The next term, %,, must satisfy the Laplace equation in two dimensions (in the cross-plane) and the same body boundary condition as before, (2-63): Ur, (x, 98) oar ae eae on r= To(x, 6). (3-10) a ty r Op 0 As in the infinite-fluid problem, this conditions suggests that ® = Ole*), since ro = O(e) and 9/9N = Ole”). Now consider the free-surface conditions. In the Bernoulli equation, note the orders of magnitude: 1 2 a 2 gf + UG, Feo NG Cy, i) Pees © on z= (x,y). O(t) O(e%) Ofe*%) Ofe?) Ofe?) The term containing or, can be dropped, but the others containing @, are all the same order of magnitude, and we have no reason to suppose that the € term is higher order. In the kinematic condition, note the orders of magnitude: Us, +O bx tO by - Ot... =0 on z= O(x,y). O(t) O(te?) O(f) Ole) Clearly, we can drop the term containing ),, but no others. Now we must relate the order of magnitude of ¢ with the order of magnitude of @,. From the kinematic condition, one might suppose that ¢ = O(e€). However, the dynamic condition then implies that ¢ ~ 0, which means only that ¢ is higher order than we assumed. In fact, the only assumption which is consistent with both conditions is that: 735 Ogilvie ae O(e?). The kinematic condition then reduces to: ®, = 0 on z= 0; (3-11) thus, ©, represents the flow which would occur in the presence of a rigid wall at z=0. From the dynamic free-surface condition, we can compute the first approximation to the wave shape: g(x,y) ~ - (UG, +5 é,) Ie 9° (3-12) It may appear to be a paradox that we have a flow without waves, from which we compute a wave shape! But, like all paradoxes, it is a matter of interpretation and understanding. We shall return to this point presently. Since ®, satisfies the Laplace equation in two dimensions and a rigid-wall condition on z = 0, it can be continued analytically into the upper half space as an even function of z. All of the arguments used in the infinite-fluid problem can then be carried over directly. In particular, at large distance from the origin, we can write, as in (2-64), o ~C, + Mog x + O(1/r), as ri 00. The two-term inner expansion can be matched to the two-term outer expansion. We obtain Aio 20(x); u C, - f(x) - g(x). Note that there is again a factor of 2 difference from the infinite- fluid results, (2-66). Of course, the term g(x) is new here. We can again determine Ajg andthus o in terms of body shape, without the necessity of solving the near-field hydrodynamic problem. By the simple flux argument, we find that: Aio = 2Us '(x) ? (3- 13a) where s(x) is the cross-sectional area of the submerged part of the hull. With this convention, we find that 736 Stngular Perturbation Problems in Shtp Hydrodynamics o(x) = Us'(x), (3-13b) just as in the infinite-fluid problem. Again, we have been able to determine the complete two-term outer expansion without explicitly solving the near-field problem. This occurs because the source- like behavior which dominates far away from the body (still in the near-field sense) can be found simply in terms of the rate of change of cross section, and it provides all the information needed for determining the two-term far-field expansion. Enough information is now available to determine a first approximation of the wave resistance. It can be computed in either of two ways: 1) integrate the near-field pressure over the hull surface, or 2) use the far-field expansion and the momentum theorem. In either case, one obtains: Dy = wave resistance © 0 4 ~ p do(x) da(&) Y, (k|x-& We ee where o(x) = source density, given in (3-13b), [a g/U*, dete ae ae YQ(z) = Bessel function of the second kind, of order zero, argument z. This is the slender-body wave-resistance formula which is so notoriously inaccurate. At speeds for which one would hope to use it, it gives values that are too high by a factor of 3 or more. Generally, one could not (and should not) expect to correct such errors by including higher-order terms, and so it is rather futile to pursue this analysis further. Streamlines, Waves, Pressure Distributions. I mentioned previously the apparent paradox of prescribing a rigid-wall free- surface condition, then using the solution of that problem to compute wave shapes, as in formula (3-12). Such a procedure really can be quite rational. Once a velocity potential is known everywhere, it is a fairly 137 Ogilvie simple task for a computer to figure out the velocity field and to pro- duce streamlines. Figure (3-1) shows the streamlines around a Series 60 hull, calculated from the near-field slender-body solution by Tuck and Von Kerczek [1968]. The upper boundary of the figure is the rigid-wall streamline. Figure (3-2) shows the same stream- lines in two other views. These drawings are accurate (in principle) to order €. This means, loosely speaking, that they show the streamlines on a scale which is appropriate for measuring beam and draft of the ship. Thus, we see that some of the streamlines start near mid-draft, pass under the bottom, then return to approxi- mately their original depth. These are variations which show ona scale intended for measuring quantities which are O(e). The wave height, on the contrary, is O(e*), as we found earlier. Therefore it should not show in these figures. Our assumptions have led to the conclusion that wave height is small compared with beam and draft. Thin-ship theory, on the other hand, predicts that wave height and beam are comparable -- without being very explicit about the ratio of wave height to draft. In the section of Fig. (3-2) showing hydrodynamic pressure along streamlines, only the waterplane curve (denoted byW) is really consistent. On any streamline, the pressure will vary mostly because of the changing hydrostatic head along the streamline. Such pressure variations are O(e). If we were to work out a second-order theory and plot the streamlines, the shift in streamline position from first- order theory to second-order theory would lead to a hydrostatic pres- sure change which is O(e*). This is the same as the order of magni- tude of the hydrodynamic pressure, but it is ignored in the figure. On the other hand, if we were inside the ship measuring pressure at a point on the hull, we would not care which streamline went past that point. We could use the Bernoulli equation to esti- mate the pressure at any point, and the estimate consistent to order €* would be found from the equation: . DP i 2 2 O= ry + gz fu + 2 (2), + @,)- 3.2. The High-Speed, Steady - Motion Problem In the preceding analysis, we have said nothing explicit about the speed other than assuming that it was finite. The first term in the velocity-potential expansions was Ux, and all other terms were assumed to be small in comparison. In principle, there is no reason to provide or allow a con- nection between Froude number and our slenderness expansion parameter. However, the practical manner in which a perturbation analysis is used may justify our making such an unnatural assumption. In practice, we work out an asymptotic expansion, which provides 738 Singular Perturbatton Problems tn Shtp Hydrodynamics Sia Se a ~ mx Ba ~ NX Se \Y et * \ \ \ ) } / eS : STREAMLINES -————-— SECTIONS Fig.(3-1). Steady-Motion Streamlines on Ship Hull According to First-Order Slender-Body Theory (Body-Plan View). From Tuck and Von Kerczek [ 1968]. 0.2 HYDRODRODYNAMIC PRESSURE ALONG STREAMLINES STREAMLINES IN SIDE VIEW Ww Fig. (3-2). Steady-Motion Streamlines and Hydrodynamic Pressure on Ship Hull According to First-Order Slender- Body Theory (Side and Plan Views). From Tuck and Von Kerczek [ 1968]. 139 Ogilvie a description that becomes approximately valid (in a certain sense) as the small parameter approaches zero. But we use the expansion under conditions in which the small parameter is quite finite, and we just hope that the resulting error is not too big. The size of that error may depend on other parameters of the problem, and we may possibly reduce the error by allowing such other parameters to vary simultaneously with the basic slenderness parameter. In the steady-motion problem that we have been considering, the small parameter e€ could be thought of as the beam/length ratio. There is a completely different length scale in the problem, namely, U 2/g = = F*L, where F is the Froude number and L is ship length. This length is proportional to the wavelength of a wave with propagation speed equal to ship speed. When we assume that F=O(1) as € ~ 0, we imply that the speed is such as to produce waves which can be measured on a scale appropriate for measuring ship length, and we imply that this speed is unrelated to slenderness. If we are interested in problems of very-low-speed ships or very-high-speed ships, in which the generated waves are, respectively, much shorter or much longer than ship length, it is entirely con- ceivable that our severely truncated asymptotic expansions may be made even more inaccurate by the extreme values of Froude number. We may increase the practical accuracy by assuming, say, that wavelength approaches zero or infinity, respectively as €~ 0. This is not to imply that there really is a connection between speed and slenderness. It is done only in the hope that wavelength and ship length may be more accurately represented when we use the theory with a finite value of e€. Formally, the low-speed problem may be treated simply as a special case of Tuck's analysis, as described in Section 3.1. One finds that the appropriate far-field problem contains a rigid-wall free- surface boundary condition (in the first approximation). Thus, both near- and far-field approximations are without real gravity-wave effects. However, this formal approach is quite improper. The diffi- culty is so serious that we devote a special section later to the low- speed problem. It is perhaps the most singular of all of our singular perturbation problems. The difficulty, in essence, is that we have treated all perturbation velocity components as being small compared with U, and this leads to nonsense if we allow U to approach zero. At high speed, a slender-body theory can be developed along lines paralleling Tuck's analysis. This has been done by Ogilvie [1967]. The resulting near-field and far-field boundary-value prob- lems are quite different from Tuck's however. No numerical results have been obtained yet from this analysis. Near-field and far-field regions are defined just as in the previous slender-body problem. In the far-field, the velocity- potential expansion starts with the uniform-stream term, Ux, 740 Stngular Perturbation Problems in Shtp Hydrodynamics followed by a term representing a line of singularities. The near- field expansion also starts with the uniform-stream term, followed by a term which satisfies the Laplace equation in two dimensions. The differences appear first in the boundary conditions satisfied by these expansions. The proper way of setting up these conditions is to nondimensionalize everything and then assume that Froude number, F, is related to the slenderness parameter, e, in such a way that _ F—-o as €~0O. Itis easier just to let the gravity constant, g, approach zero inthis limit. The only inter- esting new case, it turns out, is: g = O(e€). We now assume this to be’ the’ case. Since g appears only in the dynamic free-surface boundary condition, the body boundary condition will be the same as in the moderate-speed problem, Eq. (3-10), and in the infinite fluid prob- lem, Eq. (2-63). In the far field, the disturbance vanishes as € ~ 0. There- fore the free-surface disturbance is o(1). If we let the expansion of the velocity potential, (x,y,z), be expressed: N blxsy,z) ~ Ux + » (x,y,z), for fixed (x,y 2)4 n=l the dynamic and kinematic free-surface conditions are, approxi- mately: O= UL, - 4, on maa (3-14) Ue go + Ud), > We do not know the relative orders of magnitude of € and 9, a priori, but a study of the possibilities shows that only one combi- nation is possible, namely, that ¢ and @, are the same order of magnitude. Then, in the dynamic condition, the term containing g is higher order than the other term, and it can be neglected in the first approximation, that is, 9), =O on Ze Os (3-14) which implies also that $, = 0 on z=0Q, (3-15) Thus, the free surface acts like a pressure-relief surface, with no 741 Ogilvte restraining effect of gravity (to this order of magnitude). This condition points to a fundamentally different kind of solution from that of the previous problems. If we continue the function 4, analytically into the upper half-space, it must be odd with respect to the surface z=0. Thus 9, cannot represent a line of sources. The least singular solution represents a line of dipoles, oriented vertically. Assuming that 9, will consist only of such dipoles, we can write it: 6, (xy 2) f. sin 0 ." d& (6) | 1+, | «4 (Biz4 6) grb TE [ (x-8)) + 29] where y=rcos@and z=rsin®. The two-term outer expansion and the two-term inner expansion of the two-term outer expansion are, respectively: x,y,z) ~ Ux t+ $(x,y,z) ~ ux + 2sine (ae we). (3-17) 0 I am now assuming that the bow of the ship is located at x = 0; then, in matching to the near-field solution, we can show that the dipole density must be zero upstream of the ship bow. This expansion is unaffected by the downstream dipoles. In the near field, we assume the usual expansion: N Deaive zp Ux + ®, (x,y ,Z) for fixed (x,y/e,z/e). n=l The term @®, satisfies the 2-D Laplace equation: b +6 =0. yy zz The body boundary condition suggests that ®, = O(e*) , just as it did before in Eq. (3-10). From the dynamic free-surface condition, a wee 2 O= gh + US, +5 (G+ ,) on 2= (x,y), we see that { = O(e) (since g = Ole) ). This causes a new problem. We would like, as usual, to change this condition at z = C(x,y) toa 142 Singular Perturbatton Problems in Ship Hydrodynamics modified condition at z=0. But this is not possible. For example, the term oe would be transformed: © (xsysO(xsy)) = &, (x,y 10) + Slxsy)@,, (x+y0) +... - O(e") O(e?) O(e) Ole) Every term, in fact, will be the same order of magnitude, and so this ordinary kind of expansion fails. We must continue to apply the con- dition on the actual (unknown) location of the free surface. The kinematic free-surface condition is also nonlinear and must be satisfied on the unknown location of the free surface: O- UC, + 2, Cy- Pi, on z= C(x,y). Each term here is O(€), and so none can be ignored. We are left in the rather uncomfortable position of having to solve a nonlinear problem just to obtain a first approximation to the near-field potential function. However, that nonlinear problem is a two-dimensional problem, which is not an insignificant advantage, and, as we shall see, it is possible in principle to predict the loca- tion of the free surface, thus avoiding the necessity of searching for its We do not have a condition to apply at infinity in the 9, (near-field) problem. It is not so straightforward in this case to predict the form of the solution as r— oo, but Ogilvie [ 1967] showed that: [1 + O(1/r)] as roo, A 6 ®, (x;y ,z) = lle ah where Aj,; is a constant to be determined. There is no source-like behavior. This might have been expected, of course, since the inner expansion of the outer expansion, (3-17), showed the characteristics of a two-dimensional dipole. An intermediate expansion can be used to show that these statements are correct. A numerical procedure for solving this problem may be the following: Suppose that at some x we know the value of ®, onthe free surface, z = €(x,y), and that we also know (C(x,y) at that x. *T¢ we expand: ¢ ~ ae we could apply the condition on z=(,, then apply the usual kind of transformation, as above, so that conditions on higher-order terms would be applied on a priori known surfaces. 743 Ogtlvite Using Green's theorem, we can write: 1 ( 7 8@ fe) @, (xsy,z) = a) [=o log r' - ® 5p (log r') | alt, where r!*=[(y-y')? + (z-z')*], and the integration is carried out in the cross section, with (y',z') ranging over the body contour, the free-surface contour, and a closing contour at infinity. The last of these contours contributes nothing and can be ignored. We assumed that ®, is known on the free surface, and, from the body boundary condition, we know 0,/8N onthe hull. If we let the field point, (x;y,Z), approach the hull surface, we obtain an integral equation, with @, unknown on the hull and 984,/8N unknown on the free surface. This is not quite the usual form for an integral equation, but it should be possible to solve it approximately by essentially standard numerical methods. Then the Green's-theorem integral can be used to express ©, at all points in that cross section. Thus, the solution of an integral equation in one dimension allows the potential to be found. This procedure has not used the information contained in the free-surface conditions. Usually, we look on the free surface con- ditions as complications that cause tremendous difficulty in the finding of solutions. Now we take an opposite point of view: Supposing that we have solved the above problem at some x, we use the kinematic conditions to predict the value of ¢ just downstream: b(x + Axsy) = E(xsy) + Ax (x,y) toe. = C(x,y) +o [®,, (xyysGl — (2) if) go t+ o, on z=0; [ B] O=-o, tf, on Z = 04 These can be combined into the following: $,- vo=0, on Zz = 0, (3-22) where v = w°/g = O(e7'). In the far field, it is very difficult to guess how differentiation alters orders of magnitude. If the oscilla- tion frequency is very high, then the resulting waves are very short; it would be reasonable, perhaps, to try stretching the coordinates, and there would be no obvious basis for doing this anisotrophically. The approach which I take here is somewhat different: Solve the above- stated linear problem exactly, then observe the behavior of the solution for high frequency of oscillation. In other words, the problem is not stated in a consistant manner, but when we have the solution we rearrange it and make it consistent. The desired potential function can be written in the following form: o(x,y,z,t) = Re {$(x,y,z)e"}, (3-23) where: $(x,y,zZ) = sea dé ot — ekZ 3 p(k V llx-£) + 1) (3-23a) 7 gi tvezy (it?) (k VK 02) eee “ak elk Bae — (3-23b) 4n ee ores The form in (3-23a) can be obtained readily by superposing a distri- bution of free-surface sources: Jo is the ordinary Bessel function of order zero, and the wiggly arrow shows that the integral is to be interpreted as a contour integral, indented at the pole in the obvious sense indicated. Form (3--54b) is obtained by a transform method; o*(k) is the Fourier transtorm of o(x); details may be found in Ogilvie and Tuck [1969]. Again, the inner integral is to be inter- preted as a contour integral; there are two poles in this case. In both formulas, the path of the contour has been chosen so that the 750 Stngular Perturbation Problems in Shtp Hydrodynamics solution has a satisfactory behavior at infinity, viz., it represents outgoing waves. We need the inner expansion of this potential function, that is, we must find its behavior as r= (y2+ 2*)”2 +0. The basic idea here in finding the inner expansion is to use the second form of solution, convert the contour integral into an integral along a closed contour, and use the calculus of residues. The integrand of the inner integral has four singularities, located at £=+ 29 and at #=+tilk|, where fy = (v? - k?)!/”#, The first two are simple poles, but the second two are branch points. We "connect" the latter via the point at infinity; see Fig. (3-3). It is drawn for the case that k| v, all four singularities are purely imaginary. The contour is closed as shown if y>0O. (Otherwise, the contour is closed below.) The integrals along the large circular arcs approach zero as the radius of the arcs approaches infinity. Then the inner integral in (x,y,z) is equal to 2mi times the residue at £ = - £5, less the value of the contour integral down and back up the imaginary axis. The latter can be shown to be O(e€), and so the inner integral in (x,y,z) is: 00 i Ry+z+ ke+e P : 2.2 { dZe 2tTiv evz-iyvV-k + O(e) wo (Ke + H2y2_ y (v2 — «2/2 Next, we assume that the source distribution is smooth enough that o(x) does not vary rapidly on a length scale comparable with ship beam. This assumption implies that o'(k) decreases rapidly with increasing values of k, and so the value of the above inner integral -- a function of k -- does not really matter except when k is small in magnitude. Accordingly, we expand the above expression in a manner appropriate for small lk]. We obtain Fig. (3-3). Contour of Integration Defining the Velocity Potential of a Line of Pul- sating Sources: Zero-Speed Case 151 Ogilvie oO i i Be a (x,y s2) = a clic en ge (ie ae Pl e . @ . Eee ge iy dk Ag o*(k) {14 ieeeataay 27 og =iieee yg dicheh a oe (3-24) With the time dependence reintroduced, we have: d(x,yszst) @ Re {lo(xjeZell@t) p40 551. (3-25) This approximation represents a travelling wave; for y > 0, in particular, the wave is moving away from the line of sources. For y <0, we must start over, closing the contour for the £ integration on the lower side of the £ plane. It turns out that the result is the same if only we replace y by ly|. Thus, we have an outgoing wave for y <0 also. In both cases, the outgoing wave has the form appropriate for a gravity wave in two dimensions. In the approximations above, it is necessary to require that r be not extraordinarily large; if one assumes that r = O(i) and w= O(e-/2), then the above results follow logically. Thus the very simple approximation above is valid even in part of the far field. It is an example of the well-known physical principle that nearly unidirectional waves can be generated if the wave generator is much larger than a wave length. If we let r = O(€), no change occurs in this approximation. Since v = O(e7'), it is not permissible to expand the exponential functions even when y and z are O(e). The only effect of passing from far field to near field now is to change the scale of the observed wave motion. This far-field analysis has provided information that was probably quite obvious intuitively: In the near-field, the condition at infinity is that there should be outgoing, two-dimensional, gravity waves*. With this information in hand, we can move on to the for- mulation and solution of the near-field problem. In the near-field, we make the usual slender-body assump- tions: “7 cannot imagine that anyone would ever have doubted this fact, even without the above analysis to show it. But in the forward-speed problem, the condition at infinity in the near field is not at all ob- vious, and such an analysis seems necessary. 752 Stngular Perturbation Problems in Shtp Hydrodynamies ) ce) Cue -I By? Ba’ or = Ole )- To a first approximation, the potential function satisfies the Laplace equation in two dimensions: Pyy + $22 ~ 0, and the linear free-surface condition $,- vo~ 0 on Zz =. 04 (3-22) With the assumptions made above, the two terms here are of the same order of magnitude. (If we did not assume high frequency, we would obtain just the rigid-wall boundary condition, @,=0.) This condition implies that we shall be solving a gravity-wave problem in two dimensions. At infinity, we know from the far-field solution that the appropriate condition is an outgoing-wave requirement. All that remains is to put the body boundary condition, (3-21), into the appropriate form. Let 8/8N denote differentiation in the direction normal to the body contour in across section. Then, from (3-19) and (3-21), aN (ita? )¥/2 (ita2)”? = és - xts - Es, + dydy _ Es - xés : (3-26) (1+a5 yi”? (140% yV2 The last simplification involves an error which is O(e?) higher order than the retained terms. To the same approximation, we can write (see (2-72')): (14a2yv2' ge ees y Thus, the boundary condition is: o3 5o5 ? on Z= (x,y) » (3-27) oo _ DN. *.03 As in the infinite-fluid problem (cf. (2-73)), we can define normalized potential functions, 4;(x,y,z): 153 Ogtlvte ie + $j, = OR, in the fluid region; (3- 28a) te =n, on z = d(x,y); (3-28b) 6, - ¥% =O, on z=0, (3-28c) where v= w*/g. In the present case, the functions satisfy the 2-D Laplace equation and a 2-D body boundary condition, and they must satisfy the linearized free-surface condition. Instead of the previous simple condition at infinity, we must impose the 2-D outgoing-wave radiation condition and a condition of vanishing disturbance at great depths. Thus, the boundary-value problem is much more compli- cated than in the infinite-fluid case, but, thanks to the slenderness assumption, we have only 2-D problems to solve, and, thanks to the small-amplitude assumption, the problems are linear. The actual velocity potential function can now be expressed: (x,y ,Z st) “Re | » 1w6j (A Cx,y,2) | ° (3-29) j=35 It must be observed that each j is complex, because of the radi- ation condition. It is necessary to devise an appropriate numerical scheme for solving these problems. Both mapping techniques and integral-equation methods have been successfully applied. Note, incidentally, that the heave/pitch problem requires solution of just the 3 problem, since the slenderness assumption allows the approximation to be made that $,~ - XOz0 The result of this analysis is a pure strip theory, that is, the flow appears to take place in cross sections as if each cross section were independent of the others. It is consistent to follow the solution of this problem with a computation of the pressure field at each cross section, from which force-per-unit-length, then force and moment on the ship can be found after appropriate integrations. We obtain the following formulas for the force and moment on the ship resulting from the motion of the ship: m 2 F(t) = - | ds njl aby ~ xf) + al (Eg, +S 4)], 3-30) So where j =3 for heave force and j = 5 for pitch moment, and the symbol Sg denotes that the integration is to be taken over the hull surface in its mean or undisturbed position, which is specified by Eq. (3-18). The first term, involving g, is just a buoyancy effect. 154 Singular Perturbatton Problems tn Ship Hydrodynamics The following terms are purely hydrodynamic; they will be expressed in terms of added-mass and damping coefficients, as follows: Let: iw m(x) +=—n(x) = p i vat nyt, (3-31) C(x where C(x) is the contour of the immersed part of the cross section of x. Cf. (2-82). We call m(x) the "added mass per unit length" and n(x) the "damping coefficient per unit length." Using the slender-body approximations that $¢,™ - x, and ng@ - xn,, we find for Fj; (t): FF (t) = - val dS n(&5-x&) - iw)” ‘ dx (€-x6,)[ m(x) + n(x)/io] ; So (3-32) EM(e) = pg | dS xnglty-xf) + (iol | ax x(Eg-n6)[ mb) + (v0) /iel « So E Finally, we abbreviate these formulas: Fi(t) = - > [ (iw)*a,, + (lalby, +o; 18; (t) » (3-33) i=3,5 where as = : dx m(x); bss= J dx n(x); Ags = ag, - i dx xm(x); ba. = by, == { dx xn(x); E iE hye eee eceri(an) b -{ dx x?n(x) ; age i. x°m(x) 55 . x x°n(x c= ral dS n, = 208) dx b(x,0); 33 So "3 7 Crt “na 7 RE is ds Sons zoe dx xb(x,0); ce) Cog = al ds x, = ee | dx x“b(x, 0); So E b(x,z) is the hull offset at a point (x,z) on the centerplane. 95 Ogilvie The wave-excitation problem can be formulated as a singular perturbation problem, but such a problem has never been satis- factorily solved, even for the zero-speed case. Fortunately, another approach is available for obtaining the wave excitation; this is the very elegant theorem proven by Khaskind [1957]. It allows one to compute the wave excitation force, including the effects of the diffrac- tion wave, without solving the diffraction problem. Since we thus avoid the singular perturbation problem altogether, only the final results are presented here. (Reference may be made to Newman [1963] for details of the zero-speed case.) Let the incident wave have the velocity potential: yz+ilwt-vx) 5 ~ igh Oo(x,z,t) es € > the corresponding wave shape is given by: €o(x.t) = Hele e This is the head-seas case, For an arbitrary body, the heave force due to the incident waves is: F5(t) = pghe'™ ( ds ec" {(1 - vg)ng t ivdgn,} . So If the body is a slender ship, with axis parallel to the wave-propa- gation direction, this formula simplifies to the following: FY (t) = pghe's! ( dx “6 | dé ne” (1 - vo,). (3-34a) 3 (_ C(x) "3 we The corresponding expression for pitch moment on a slender body is: FS (t) = pghe@? ‘ dx Pile aS dl ne". (4 - vo). (3-34b) C(x) In the expression (I- v,) in the integrand, the first term leads to the force (moment) which would exist if the presence of the ship did not alter the pressure distribution in the wave; in other words, it gives the so-called "Froude-Krylov" excitation. This fact can be proven by applying Gauss' theorem to the integral, Dynamic effects in the wave ("Smith effect") are properly accounted for. The second term gives all effects of the diffraction wave. A final rewriting of the wave-force formula is worthwhile. The above approximate expression for FS (t) can be manipulated into the following: 756 Singular Perturbation Problems in Ship Hydrodynamics 20 M v + ipw | dx (xe) | df n_d,e””. L Fo, C(x) 33 The first term shows the Froude-Krylov force quite explicitly; the product of € (x,t) and the quantity in brackets is often called an "effective waveheight," the second factor being a quantitative repre- sentation of the Smith effect. The second integral term has been expressed in terms of the vertical speed of the wave surface, Co,(x,t). This term should be compared with the force expression for the calm-water problem, (3-30). For a slender body, the hydro- dynamic part of the latter can be written, for j = 3, soa) dS n,[&,(t)3;+6,(t) 5] = -tpo dx [ £,(t)-xé,(0)] Sn! N35. The last quantity in brackets is the vertical speed of the cross section at any particular x. Comparison with the second term of FY (t) shows that the latter is almost exactly the same as the hydrodynamic force that we would predict if each section of the ship had a vertical speed - Co,(x,t)- This analogy would be exact, in fact, if the expo- nential factor, e”*, were not present in the By (t) formula. Except for that factor, what we have found is that Korvin- Kroukovsky's well-known "relative-velocity hypothesis" is approxi- mately correct according to the analysis above. The hypothesis is particularly accurate for very long waves, in which case e”*= 1 over the depth of the ship, but it is less accurate for short waves. Again, it should be noted that we have no absolute basis for saying whether a particular wave is short or long in this respect. In com- puting the Froude-Krylov part of the force, it is well-known that the exponential-decay factor must be included in practically all cases of practical interest; this has been amply demonstrated experimentally. It suggests that one should be wary of dropping the exponential factor in the diffraction-wave force expression. Summary. In the far field, we assumed that the effects of the heaving/pitching ship could be represented by a line of pulsating singularities located at the intersection of the ship centerplane and the undisturbed free surface. For a first approximation, we tried using just sources, and these were sufficient to allow matching with the near-field solution. In particular, the inner expansion of the outer expansion showed that the near-field expansion would satisfy a two-dimensional outgoing-wave radiation condition, at least in the first approximation. With this fact established, we formulated the near-field problem; it reduced ultimately to the determination of a velocity potential in two dimensions, the potential satisfying a linear tot Ogilvie free-surface condition and an ordinary kinematic body boundary con- dition, as well as the outgoing-wave condition. This is a standard problem which must generally be solved numerically with the aid of a large computer; such programs exist. The force and moment were expressed as integrals of added-mass-per-unit-length and damping- per-unit-length, both of which could be found from the velocity potential for the 2-D problem. Finally, the determination of the wave excitation force and moment was carried out by application of the Khaskind formula, which permits us to avoid the singular perturbation problem involved in solving for the diffraction wave. 3.4. Oscillatory Motion with Forward Speed The problem of predicting the hydrodynamic force on an oscillating ship with forward speed is not fundamentally much differ- ent from the same problem in the zero-speed case. It is considerably more complex, to be sure, but no new assumptions are needed. The approach here is that of Ogilvie and Tuck [1969]. Alter- native approaches have been devised by numerous other authors; some of these were mentioned in the last section. The distinguishing characteristics of the Ogilvie-Tuck approach are: 1) application of the method of matched asymptotic expansions, and 2) assumption that frequency is high in the asymptotic sense that w= O(e"/@), while Froude number is O(1i). Also, the problem is broken down into a series of linear problems by the use of a "motion-amplitude" param- eter, 6, whichis a measure of the amplitude of motion relative to the size of ship beam and draft. The reference frame is assumed to move with the mean motion of the center of gravity of the ship. Thus it appears that there is a uniform stream at infinity, and we take this stream in the positive x direction. The z axis points upward from an origin located in the plane of the undisturbed free surface, andthe y axis completes the right-handed system. (Positive y is measured to starboard. ) Let the velocity potential be written: (x,y 5Z ,t) = Ux 7 Ux (x,y 32) + w(x, y»Z st) s (3-35a) where Ul x + x(x,y,z)] is the solution of the steady-motion problem discussed in Section 3.1. For the moment, we simply assume that W(x,y,z,t) includes everything that must be added to the steady- motion potential so that &(x,y,z,t) is the solution of the complete problem. We shall also divide the free-surface deformation function into two parts: C(x,y,t) = n(x,y) +t O(x,y,t), (3-35b) 758 Singular Perturbatton Problems in Ship Hydrodynamics where 1(x,y) is the free-surface shape in the steady-motion prob- lem (the {(x,y) of Section 3.1), and 6(x,y,t) is whatever must be added so that €(x,y,t) is the complete free-surface deformation. The body surface is defined mathematically just as in Section 3.3 for the zero-speed problem; see (3-18) and (3-19). The same assumptions are made about orders of magnitude: E(t} = O(ed); w= Oe"). From these assumptions and the subsequent analysis, it turns out that W(x,y,t) = Ofe%28), O(x,y,t) = O(es), as either € or 6—~ 0. Wecan look on the complete solution as a double expansion in € and 6. From this point of view, the expan- sion for the potential can be written: Gov e2st) = 1 Ux + UX, Gay 52) + «ee } Ol6°e) “Ol S-e-) tAWi(cry zt) + U(x, yee, t) tees } P'0(6).1 (3536) o(s'e*?) O(8' é*) The order of magnitude of the term Uy,(x,y,z) was found in Section 3.1. The order of magnitude of ~, may be somewhat surprisinge Physically, it implies that the effects of ship oscillations dominate the effects of steady forward motion -- in the first approximation. These orders of magnitude were derived by Ogilvie and Tuck. Here, I shall not prove them, but I hope to make them appear plausible. It should be noted that the high frequency assumption was made just so that the orders of magnitude would come out this way. (Cf. the discussion in Section 2.3, in which it was pointed out that the for- malism for the steady-motion slender-body problem is established to force certain expected results to come out of the analysis. We are doing the same here, forcing strip theory to come out as the first approximation. ) The linearity of the ~, problem permits us to assume that the time dependence of , and of the corresponding first term ina @ expansion can be represented by a factor e'™’. In order to find any effects of interaction between steady motion and oscillatory motion, it is necessary to solve for the term 159 Ogtlvie WAx,y,z,t). Thus, we must retain two terms in the time-dependent part of the potential function. (The problem is still linear, however, in terms of 6.) It is not convenient to be repeatedly attaching sub- scripts to the symbols, and so I shall simply write out equations and conditions which are asymptotically valid to the order of mag- nitude appropriate to keeping e€° terms in the expansion of W(x,y,Z,t). In the far field, the effect of the oscillating ship can be repre- sented in terms of line distributions of singularities. Again, we try to get along with just a distribution of sources, and we are successful if we allow for the existence of both steady and pulsating sources. The steady-source distribution is exactly the same as in the steady-motion problem. Let the density of the unsteady sources be given by o(x)e'“t; define o(x) =0 for the values of x beyond the bow or stern. The corresponding potential function must satisfy the Laplace equation in three dimensions, a radiation condition, and the usual linearized free-surface condition: (iw) + 2ioUY, + UY, tgu,=0 on z=0. (3-37) Then it can be shown that: eat fan ikx dé exp[ il + 2/k* +27] W(x, y »2,t)~ ~ 2 ) dk e o (k) 2 4m J« Cee 2 + Uk /w) (3-38) where o*(k) is the Fourier transform of o(x), and the contour C is taken as in Fig. (3-4), where k, and kg, are the real roots (k, < kp) of the equation: *There are two real roots if T= wU/g > 1/4; the other_two roots are a complex pair. Since we assume that w= Oltaler then also 1 Olive , and we are assured that T >> 1/4. However, if Tt | 1 7ay the complex pair come together, and our estimates are all very bad. Of course, it is well known that the ship-motion problem is singular at T = 1/4, For still smaller values of T, there are four real roots of the above equation, and the solution can again be interpreted physically and mathematically. From experimental evidence, it appears that our final formulas can be applied for any forward speed, at least in head seas, but the presence of a singularity at 7= 1/4 shows that this is accidental. Our theory is a high-frequency, finite-speed theory, and it really should not be possible to let U vary continuously down to zero. 750 Singular Perturbation Problems in Ship Hydrodynamics cia k < ky ® k> ko Fig. (3-4). Contour of Integration Defining the Velocity Potential of a Line of Pul- sating Sources: Forward-Speed Case [+]. Bf =o. and the contour is indented as shown at the poles on the real axis inthe £ plane. The contour C extends from - oo to too. The poles inthe £ plane all fall on the imaginary axis if k,; then the potential function would have to be tay UN) Ej (t), with Qj) (x,y,z) satisfying the conditions stated previously. This 764 Singular Perturbatton Problems tn Shtp Hydrodynamics "pressure distribution" is periodic in time, and it is also periodic in y as ly| — oo; the latter comes from the term containing 4)j,. Furthermore, the time and space periodicities are related to each other in just the way that one would expect for a plane gravity wave. This can be proven by studying the boundary-value problem for 9). Thus, there is an effective pressure distribution over an infinite area, and it excites waves at just the right combination of frequency and wave length so that we have a resonance response. In an ordi- nary two-dimensional problem, there would be no solution satisfying all of these conditions. However, our solution need not be regular at infinity; it must only match the far-field expansion. And the far- field expansion predicts an appropriate singular behavior at infinity. It is shown by Ogilvie and Tuck that the solution of this inner prob- lem does exactly match the above far-field solution. The way the pieces of the puzzle all fit together is rather typical of the method of matched asymptotic expansions, and it indicates at least that the manipulations of asymptotic relations were probably done correctly! (It still says nothing about the correctness of the assumptions.) There is no benefit to be derived by repeating here the solu- tion of the above detailed problems. Rather, we jump to the results for the heave force and the pitch moment, and we do little more than compare these results with the comparable formulas in two previous problems: CASE 1: The oscillating slender body, with forward speed, in an infinite fluid (Section 2. 32) CASE 2: The oscillating slender body (ship), at zero forward speed, on a free surface (Section 3. 3) In all cases, let the force (moment) be expressed in the form: m Ne : sO cia \ [ (iw) aj; + (io)by +), JE; (t) . i=3,5 We define cjj to be independent of frequency and of forward speed. (We must make some such arbitrary convention, or the separation into ajj and cj; components is not unique.) With this convention, cjj represents just the buoyancy restoring force (moment). Thus, cj, =O forall j,i in case 1; in cases 2 and 3, cj, is given by: [oj] = 208) dx {1-3 (4) bl, 0) . 765 Ogilvie Table 3-1 shows ajj and bjj for the three problems. InCases 1 and 2, the results have been obtained from Sections 2.32 and 3.3, re- spectively. ForCase 3, the present problem, the lengthy derivation will be found in Ogilvie and Tuck [1969]. Some points should be noted: 1. All of the terms’ in Case 3 include the corresponding Case 2 terms, i.e., the added mass and damping at forward speed can be computed in terms of the added mass and damping at zero speed, plus a speed-dependent component. Formally, we could also say that Case 1 includes all of the Case 2 terms, with n(x) set equal to zero. From this point of view, the only differences among the three cases are the forward-speed effects. 2. The coupling coefficients b35 and bsz include a forward- speed term +¥Ua3z3 in both Case 1 and Case 3. This means, first of all, that there can be some damping even in the infinite-fluid problem. Secondly, it means that this contribution to the damping coefficients is not altered by the presence of the free surface. Note that in neither case is it necessary to ignore the steady perturbation of the incident stream (the x terms in (3-41), for example) in order to obtain this result. 3. The other coupling coefficients, az, and a,., contain similar speed-dependent terms in Case 3; they arise at the same point in the analysis as the terms discyssed in 2 above. We could arbitrarily include such terms, +(U/w )b3z, in Case 1 too, without causing any errors since bg, is zero anyway in Case 1. 4, In Case 1, there is a speed-dependent term in age which is lacking in Case 3. The reason for the lack is that such a term is higher order in terms of € in the ship problem, because of the assumption that w= O(e"/2). There was no need for a high-frequency assumption in Case 1, and so the extra term could legitimately be retained. 5. If, in Case 3, one arbitrarily includes the forward-speed term, =(W/a)aans in the ag, coefficient, making it identical to the Case 1 coefficient, then it is consistent to modify b,, in a similar way, namely by changing it to: 2 b =f dx x*n(x) - (U/w) b,, 55 The relationship between these forward-speed effects is quite the same as that discussed above in paragraphs 2 and 3. Inthe bes coefficient of Case 1, we could also introduce an extra term, -(U/w)*b33, without causing any error, since b,, is zero anyway in this case. Thus we can maintain the symmetry between Case 1 and Case 3. 766 Stngular Perturbation Problems in Ship Hydrodynamics TABLE 3-1 ADDED-MASS AND DAMPING COEFFICIENTS IN THREE PROBLEMS CASE 1 - Body CASE 2 - Ship with CASE 3 - Body with Forward Speed Zero Forward Speed with Forward Speed in Infinite Fluid on Free Surface on Free Surface dx m(x) —7 dx n(x) ae) dx xm(x) + (U/w')b,, - (2pvU/u) Im {1} ' 7 dx xn(x) - Ua,, - (2pvU) Re {1} a dx xm(x) - (U/w)b,, + (2pvU/u) Im {1} : dx xn(x) + Ua,, + (2pvU) Re {1} ca iy dx x*m(x) - (U/w)"a5, dx x*m(x) L — dx x?n(x) oo NOTE 1) In all cases, m(x) and n(x) are defined: 1 m(x) + y> n(x) = ef df n,$,, Clix) where C(x) is the wetted part of the cross section contour at x, and n, and 4, have the same meaning as in Section 2.32 and 3.3. In CASE 1, $5 ia a real quantity, and so n(x) = 0. NOTE 2) The quantity I in Case 3 is defined as follows: Let $= $4, and let ¢q be a 2-D potential function which is sinusoidal in y, such that |¢- $,|—> 0 as y— oo. Then: co 2 2 1 2 I “J dx [So [¢ (x,y ,0) = $a(x»y»0)] s oy. oitxsb(x0),0) ’ where b(x,z) gives the hull offset corresponding to the point (x,0,z) on the centerplane. 767 Ogilvie 6. The only forward-speed terms not yet discussed are those in Case 3 which involve the integral I. They arise from the inclusion of the functions $j; in the potential function, as in (3-43), and the necessity for including those functions is a consequence of the fact that the right-hand side of (3-42), the free-surface condition, is not zero. Now, the right-hand side of (3-42) represents an interaction between the forward motion and the oscillation. One might try to simplify matters by assuming that one can neglect the effects of x, the perturbation of the incident stream by the body. But this reduces (3-42) to the following: Ves + gus = oa 2UYy,; on YT Oi. (3-47) O(c”? §) O(e 6) The right-hand side is still not zero, and we would still have the Qj functions to contend with. In fact, it may be recalled that this remaining term on the right-hand side was the one that caused the major trouble in interpreting the 92} problems. Neglect of the x terms leads to the condition on Q) (cf. (3-46)): Q) - vQj = - (2/g)4j,, on z=0, (3-48) jz and it is the one remaining right-hand term which causes the solution for &j to diverge at infinity. The usual procedure at this point is to set Q; = 0, turn the other way, and just ignore these problems. The results are in remarkably good agreement with experimental obser- vations, and one still wonders how this can be rationalized mathe- matically. Finally, we should at least mention the problem of predicting wave excitations in the forward-speed problem. The singular per- turbation problem involved in solving for the diffraction waves has not been satisfactorily worked out yet, at least, not in a manner compatible with the approach presented above. One might hope to avoid the diffraction problem by using the Khaskind relations, as in the zero-speed problem. (See Section 3.3.) In fact, Newman [1965] has derived what I call the Khaskind- Newman relations. These provide a generalization of Khaskind's formula, relating the wave excitation on a moving ship to the problem of forced oscillations of the ship when the ship is moving in the reverse direc- tion. Unfortunately for our purposes, Newman's derivation i based on an a priori linearization of the free-surface, in the sense that our terms involving x can be neglected. Therefore, the appropriate diffraction problem cannot really be avoided in this way. Also, it is necessary to have available the potential function for the forced- motion problem, and this includes at least a part of the 92} functions even if the x dependence is ignored. 768 Stngular Perturbatton Problems tn Ship Hydrodynamics In a not-yet published paper, Newman has applied the Khaskind-Newman relations in the forward-speed problem by arbi- trarily ignoring the 9j functions in the forced-motion potential function. He finds for the heave excitation force: ‘et zg : 1 (t) = pgh(1 + Uw)/g)e- dx e cag dfn ene L Uv Uv i E pes ay oe - al, where, as before, w is the frequency of oscillation (that is, the frequency of encounter) and v = w‘/g; the frequency measured in an earth-fixed reference frame is denoted by wo, and we define Vo = wo/g. The actual wave length of the incident waves js N= 2n/v. The two frequencies are related as follows: w= wot Uwy/g- These formulas are all valid for the head-seas case only. This formula should be compared with (3-34a), which was the corresponding result in the zero-speed problem. The first term in brackets yields the Froude-Krylov force, and the second term yields a pure-strip-theory prediction of the diffraction wave force, which can be interpreted approximately in terms of the relative-motion hypothesis. The remaining terms represent an interaction between forward speed and the incident waves. Again, it should be pointed out that more than just nonlinear effects have been neglected in setting 2j equal to zero. In fact, the usual linear free-surface condition for ship-motions problems can be written: by t BH, = - 2UW, - Udy, on 220, (Cf. (3-37) and (3-47).) Even the inclusion of the Qj) terms still omits some effects usually considered as linear, namely, the effects of the term - Ud,, inthis boundary condition. These effects are higher order in the theory presented here solely because of the high- frequency assumption. IV. THIN-SHIP THEORY AS AN OUTER EXPANSION It has already been shown how one can view a symmetrical thin-body problem in terms of inner and outer expans ions; the usual description of the flow around such a body is really just the first term of an outer or far-field expansion. It was not at all obvious that one had to use such a powerful method on such a problem, but it was clear that one could do this. Probably the only advantage of doing 769 Ogilvie so in the infinite-fluid case was that one could avoid possible ques- tions about the validity of analytically continuing the potential function inside the body surface. On the other hand, one had then to face all kinds of difficulties in principle in justifying use of matched asymptotic expansions. It was a rather academic exercise. The situation may be quite different in the thin-ship problem. The purpose of this chapter is to show one can obtain the first results of thin-ship theory in the same way as for the infinite-fluid problem but that a second-order solution leads to fundamental difficulty. The latter appears to suggest that a combination thin- body/slender-body approach may be appropriate. A limited amount of other evidence may be cited to support this idea. I wish to emphasize that there are no new results in this chapter. It is all a mater of interpretation. Perhaps someone will be able to show that the problem discussed here has a trivial expla- nation. On the other hand, perhaps someone will be stimulated to do further research on the subject. In either case, I shall be happy with the outcome. The problem may be partially stated just as the infinite- fluid, thin-body problem was stated. Let there be a velocity potential, (x,y,z), which satisfies the Laplace equation, [Li] xx + dyy + b22 = 0, everywhere in the fluid domain and the body boundary condition, [ H] O = dyhy F dy + $7h,, on y= h(x,z) = t€H(x pa): Now we add on the two free-surface conditions: [ A] 5 U'= gt +S Lont oy + 42), ony sz )=e6 (x53 [B] 0 = bySxt byby - $2 on 2 = £(x,y). Also, we must specify a radiation condition. In the far field, where y = O(1), we assume the existence of the expansions: N o(x,y,Z)~ >) b(x,y>Z)> n=O for fixed (x,y,z); C(x.) = os C,(x2y)> nsl 770 Stngular Perturbatton Problems in Ship Hydrodynamics N (x,y,z) a » Orcs vin 2) + n=0 . N for fixed (x,y/e,z). C(x,y) ~ > Zy(X,y) » n=l We assume right away that: bo(x,y,z) = @)(x,y 52) = |Upien In the far field, the ship vanishes as €—~ 0, and so we take the entire outside of the plane y = 0 (below the free surface) as the far field. It is easily seen that the second term in the outer expan- sion must be of the form: $(x,y.Z) = - #S. o(&,6)G(x,y,z35,0,0) d6 dt, (4-1) where His the portion of the centerplane of the ship below z=0, o,(x,z) is an unknown source density, and G(x,y,z35 »1,6) is the usual Green's function for a linearized problem of steady motion with a free surface. It has the important property: Gy tT KG) = 0), on Ze 0; (4-2) where K = g /U*. Of course, the potential @;, also has this property: Pie VOC, on z ="0. For later convenience, we define a,(x,z) = (x,0,z), (4-3) and so a,(x,z) has the property too: Te + Ka), = Or on Zia Ole (4-4) With $ (x,y,z) given by (4-1), the two-term outer expansion is: (x,y,z) ~ Ux + $ (x,y,z), Teh Ogtlvte and the inner expansion of the two-term outer expansion is: 1 Oi ny .z)i~ Ux.t.o,(x,2), + = ly | oy(x,2) een O(1) Ole) O(é) I have taken my usual liberty of indicating unproven orders of mag- nitude. I am not really assuming these orders of magnitude; I am saying that one can prove that these are correct, and I display them here now simply as an aid to the reader. Now consider the near field. Just as in the infinite-fluid problem, one may stretch coordinates, y =€Y, and follow through the consequences. This is effectively what I do, without writing the change of variable explicitly. Thus, the Laplace equation yields the condition: Plyy = 9; and so 4, must be a linear function of y. The same analysis as used in the infinite-fluid problem, Section 2.11, leads to the con- clusion that @; is even more restricted than this. It must bea constant with respect.to. .y.. Thus, let: ®, (x,y,z) = A, (x,z). The two-term inner expansion is then: x,y,z) ~ Ux + Aj(x,z). Matching gives the unsurprising result that: A,(x,z) = a,)(x,z). (4-5) In other words, once again the inner expansion starts out simply as the inner expansion of the outer expansion; it is not necessary to formulate a near-field problem to obtain this result. The same arguments lead to the prediction that: @o(x,y,z) = Ao(x ,z) + UbAx)Z) ly 3 (4-6) Thus the three-term inner expansion is formed, and it can be matched with the three-term inner expansion of the two-term outer-expansion, yielding the familiar result once again that: 772 Stngular Perturbation Problems in Shtp Hydrodynamics o Ge ,z) = 2Uh,(x,z). (See (2-22).) This obviously had to come out this way, since we have not yet introduced any effects of the presence of the free surface. It should be noted that only the function Ao(x,z) is not already deter- mined. (Knowledge of o, (x,z) allows us to express a(x,z) ex- plicitly, from (4-1) and (4-3).) A systematic treatment of the free surface-conditions leads to the following: [ A] 0 = gZ, + Ud, O(e) + gZp + Ud: + UZ, %,, +S (Gi, - 61.) +S (€3,) O( €%) GPiaracs, 5 Onez = 10; [ B] 0 = UZ, - O, O(e) + UZa, - @g, - 2,9, + ®,21, + &2 Ze, O(e*) ter isheverce on 2 = 0. The lowest-order conditions in [A] and [B] together require that: + KO. =70), on Zz =O We see that this is automatically satisfied by our ©, (x55) = A\(x,z) = a,(x,z).- (See (4-4).) The first term in the expansion for wave shape in the near field is also determined: Z Un Z, (x,y) =- ra @) (x, 0). This really says only that the free surface appears in the near field to be raised (or lowered) by just the limiting value (as y — 0) of C(x,y) in the far field. Again, a rather trivial result. When we consider the (a terms in the free-surface conditions, it is a different matter. The two conditions can be combined into the following: tts Ui 1 2 2 O= & + Kd2. - gl lee T Oy l@ly F yh (4-7) Wi7.2 i} + > (hy)x - FU aX, ‘lj TU Wy ly F +a h,Zay- 2 In condition [| A] , we note that differentiation of the € terms with respect to.y yields: 0 = Za, + Uhy. Therefore, in the complicated free-surface condition above, (4-7), only the first two terms involve y; all of the other terms are func- tions of just x. From (4-7) and (4-6), we can thus write the follow- ing: Ora 1h x, 0) Kh, (x,0)] Uly| + (a function of , x). XXX This must be true for any y, and so we obtain the condition: O7= hyyy ote Khyz; on Z= 0. If the ship is wall-sided at z= 0, the second term is separately zero, and so we would have to require that h,,,=0 at z=0. Now this is clearly unacceptable. Why should our theory work only for such a special case? (The waterline is made of circu- lar arcs inthis case.) As a result of our having stretched the coordinates, we came to the prediction that the fluid velocity near the thin body consists of a tangential component which is essentially independent of the local conditions plus a normal component which depends only on local conditions. Near the free surface, such results are simply untenable. I present here a formalism which apparently avoids this difficulty. Again, I point out that no new results are obtained. However, it does seem possible that the procedure might be fruitful if studied further,. The idea is to define a third region, complete with its own asymptotic expansions of @ and ¢. This region will be essentially the same as the near field in a slender-body analysis, that is, it is a region in which y = O(€) and z= O(e) as € ~ 0. It follows from this assumption that 8/8y and 8/8z both have the effect of changing orders of magnitude by a factor 1/e. What is most important is that this region is interposed between the thin-body near field and 114 Stngular Perturbatton Problems in Ship Hydrodyanamics the free surface. Thus, it is no longer necessary or even proper to try to make the previous inner expansion satisfy the free-surface conditions. We expect, as usual, that the first term in the expansion of @ in this new near field will be just Ux. Furthermore, we can expect the next term to be rather trivial, since the second term in the previous near-field expansion did actually satisfy the free-sur- face condition. Using the usual arguments of slender-body theory, we find in fact that the three-term expansion of $ in this new field is: (x,y,z) ~ Ux + a,(x,0) + Uh,(x,0) ly | - = ZQ,,(x,0). Olt) Ole) O(e*) O(e*) The corresponding wave shape is found to be: Gixry) ~ - Fay, (40) Of<) We - Z[ onyx, 0) ly | ecu ay, (x, 0) iggy (Xs 0) | g : O(e*) 2 - 35[, (x,0) + U hilx, 0) +(Z ai cx.0)| Pies It can be shown in straightforward fashion that these results match the far-field expansion as /(y2 + z2) ~ oo and they match the pre- vious (thin-body) near-field expansion as z—~ - oo. Furthermore, they satisfy the free-surface conditions without the necessity for imposing unacceptable restrictions on the body shape. There is just one aspect that requires special care: The free-surface condi- tions cannot be satisfied on the surface z=0 inthis near field. The reason is that the first term of the ¢ expansion is O(e), and differentiation with respect to z is assumed to change orders of magnitude by 1/e. Thus, suppose that we want to evaluate some function f(z) on z=6€ in terms of its value (and values of its derivatives) on z =0. The usual procedure is to write: HCh= 0) 4 CEO), Fai SAO) Pawn O(f) Ole)» O(f/e) O(e*)- O(£/e*) With our set of assumptions, this expansion is useless; we cannot C15 Ogtlvte terminate it. The one simplification which is admissible here is to evaluate f(z) and its derivatives on z=Z,, where € = Z, + o(e).* I have not worked out any more terms in any of these expan- sions, but I suppose that the next term in this near-field expansion will be much more interesting. In the far field, it is well-known that the third term in the expansion of the potential function will in- clude the effects of what appears to be a pressure distribution over the free surface. It was shown by Wehausen [ 1963] that at the inter- section of the undisturbed free surface and the hull surface the solu- tion is singular, and he represented the singular part by a line integral taken along this line of intersection. From the point of view of the method of matched asymptotic expansions, it should be possible to represent the far-field effects of that line integral in terms of an equivalent line of singularities onthe x axis. The strength of the singularities would be determined, as usual, by matching the solution to the near-field expansion. At this stage, thin-ship theory will have become a singular perturbation problem. V. STEADY MOTION IN TWO DIMENSIONS (2-D) Sometimes we study two-dimensional problems with the intent of incorporating the solutions into approximate three-dimensional solutions, as in the treatment of high-aspect-ratio wings and in slender-body theory. And sometimes we investigate two-dimensional problems simply because the corresponding three-dimensional prob- lems aré too difficult. The problems discussed in this section are in the second group. It is not likely that any of these problems and their solutions will have practical application before several more years have passed, even in the context of strip theories. Here are some of the most fundamental difficulties related to the presence of the free surface. The first two subsections concern a 2-D body which pierces the free surface. Such a problem is intrinsically nonlinear. We might try to formulate the problem as a perturbation problem, in this case involving a perturbation of a uniform stream. However, there must be a stagnation point somewhere on the body, and at that point the perturbation velocity is equal in magnitude to the incident stream velocity. It is not small! If the stagnation point is near the free surface, the free-surface conditions cannot be linearized. We must find methods which are adaptable to highly nonlinear problems. Such a method is the classical hodograph method, used since *The same difficulty arises also in Sections 3.2 and 5.42. 776 Stngular Perturbation Problems tn Shtp Hydrodynamics the nineteenth century for solving free-streamline problems. But it introduces a new difficulty: It cannot be used to treat free stream- lines which are affected by gravity, which means that only infinite- Froude-number problems can be treated directly. This leads to a further great difficulty, which is discussed in some detail in Section 5.1. In Section 5.3, a brief discussion is presented of the problem studied by Salvesen [1969]. It contains two aspects of interest: It is a case in which the free-surface conditions can be linearized because of the depth of the moving body, and I have already commented in the Introduction that there are very interesting fundamental ques- tions involved in such procedures. Also, it presents a clear example of the classical phenomenon discussed in the section on multiple scale expansions: The wave length obtained in the first approximation must be modified in subsequent approximations, or the solution becomes unbounded at infinity -- where we know perfectly well that the waves are bounded in amplitude. Finally, Section 5.4 describes two recent attempts to approach the problem of extremely low-speed motion. The difficulty is basically this: In the usual linearization, we assume that all velocity components (at least in the vicinity of the free surface) are much smaller than the forward speed -- which becomes nonsense if we subsequently decide to let U, the forward speed, approach zero. What is needed is a perturbation scheme in which somehow the small parameter is pro- portional to U. Then it is certainly permissible to allow U to approach zero. Section 5.41 shows a very straightforward procedure for doing this; however, it leads to a sequence of Newmann problems, and so the wave nature of the fluid motion is lost. In Section 5.42, an alternative method is discussed. It is an application of the multi- scale expansion procedure to which Section 1.3 was devoted. 5.1. Gravity Effects in Planing Before we try to treat this problem properly, let us consider briefly a well-known approach to the 2-D planing problem and deter- mine why it is not completely satisfactory. In the middle 1930's, A. E. Green wrote several papers on the subject, and the essence of his approach is well-presented by Milne-Thomson [1968]. A flat plate is located with its trailing edge at the origin of coordinates, as shown in Fig. (5-1). There is an incident stream with speed U coming from the left, and, at infinity upstream, there is a free sur- face at y = h. The effects of gravity are neglected. The fluid is assumed to leave the trailing edge smoothly (a Kutta condition), and a jet of fluid is deflected forward and upward by the plate. In the absence of gravity, the jet never comes down to trouble us again. In the figure, A marks the leading edge of the plate and C marks the stagnation point. The physical plane shown in Fig. (5-1) is also the complex ett Ogtlvie Fig. (5-1). Planing Problem in Fig. (5-2). Planing Problem in the Physical Plane the Plane of the Complex Potential. z=xtiy plane. Let F(z) = $(x,y) + id(x,y) be the complex velocity potential for this problem, Then F(z) effects a mapping of the z plane onto an F plane, as shown in Fig. (5-2), in which points are marked to correspond to Fig. (5-1). It is assumed that @=0 and w=0 atthe stagnation point. Furthermore, we have set w= Ua onthe upstream free-surface streamline, IJ, which implies that a is the thickness of the jet and that Ua is the rate at which fluid leaves in the jet. Of course, F(z) is not known yet. We can also consider that the z plane is mapped by the function w(z) = dF/dz. w(z) is the "complex velocity," that is, w=u-iv, where u and v are the velocity components inthe x and y directions, respectively. The entire fluid region is mapped by w(z) into the region bounded by a half-circle and its diameter, as shown in Fig. (5-3). Again, points are marked to correspond to Fig. (5-1). The diameter is the image of the planing surface, on which the direction of the velocity vector is known, and the circle is the image of the entire free surface, on which the magnitude of the velocity vector is known (from the Bernoulli eyactiani: Again, we note that the mapping function itself is not yet known. Fig. (5-3). Planing Problem in Fig. (5-4). Planing Problem in the Plane of the the Auxiliary (¢) Complex Velocity. Plane. 778 Stngular Perturbation Problems in Ship Hydrodynamics The functions w(z) and F(z) are, of course, very simply related, although neither is known explicitly yet. In order to obtain another relationship, one introduces the ¢ = € t+ in plane, in which the fluid domain is mapped into the lower half-space, as shown in Fig. (5-4). We can write out the explicit expressions for mapping the F and w planes into the € plane. The first is accomplished by means of the Schwarz-Christoffel transformation: deo Ua. tite. a) eateieic) Gat which can actually be integrated, yielding: ricien = B( £58 tog £22). The second mapping can be shown to take either of the equivalent forms: ia G=¢ Ue (1 - gc) ¢ivd - Avie? - 1) Jt Gee) ale - 2 v(t? - 1) - Cc w(z(f)) (5-1) The solution is then completed by using the relationship between F and w, along with these expressions, to obtain the relationship between z and ¢. Since: dz _ dz dF _ _Hi(s) C dF dt © wlz(t)) ’ (C) = H(6') dt! 20) = | wetery a = _— -c(G-1) + (1 tbc) log ea7 + if(1-cWi(2-1) - thy (1-c) log [¢ + vit®-1)] - iV(1-c?)y(br-1) log ee 1) T79 Ogilvie So now we have z as a function of €, as wellas F and w as functions of ¢. There are three parameters inthis solution, a, b, and c, none of which has been determined yet. By letting 4 | “> o.,7Green came to the conclusion that the flow far away is a uniform stream as required only if: ¢ = = cos @ and v(i - c*) = sina. (5-2) (Both statements are necessary to avoid an ambiguity in sign.) Also, one can use the z(€) formula to evaluate z at the leading edge of the plate: z(-1) = - be (Compare Figs. (5-1) and (5-4).) This provides a relationship among a,b, and c. Buc there are no more conditions to be found unless we introduce more information about the physical problem. For example, we could use the solution with unspecified values of a and b, and work out the formula for lift on the plate. (Milne- Thomson gives the formula.) If then we fix the value of lift, we have another condition on a and b. However, this is rather a back- wards way of going at the problem. We are most likely to want to solve the entire problem just to find the lift and other interesting physical quantities, and so we have not gained much if we must assume the value of the lift as a given datum. There is another anomaly in this result: The value of h (See Fig. (5-1)) has not been used in any way. In the formula for z(t), let G = &€, with |€| very large. Then every value of z com- puted in this way gives a point on the free surface far away from the planing surface. With a considerable amount of tedious algebra, one can eliminate § and express y asa function of x (at least asymptotically, as |x| oo). The first term is the most inter- esting: E ay (1 =e%) ‘er ors [log |x| + constant]. Thus, far away from the planing surface, the free surface apparently drops off logarithmically to - oo. The slope of the free surface approaches zero (« 1/|x|) and so there is no violation of our assump- tion that the flow at infinity is simply a streaming motion parallel to the x axis. But obviously the assumption that the trailing edge was located at a height h below the free-surface level at infinity was quite meaningless, and it cannot be enforced in the solution. 780 Singular Perturbation Problems in Ship Hydrodynamics There are thus two difficulties: 1) The above solution is not unique (a common difficulty in free-streamline problems); 2) It has unacceptable behavior at infinity. These difficulties were resolved by Rispin [1966] and Wu [1967] , who recognized that the solution of Green's problem is part of a near-field (inner) expansion of the complete solution. An inner expansion does not necessarily satisfy the obvious conditions at infinity; it must only match some outer expansion in a proper way. Rispin and Wu produced the appropriate outer expansions and showed that matching does occur. The effects of gravity appear first in the far field, which is hardly surprising, for two reasons: 1) Far away, one expects to find gravity waves as the only disturbance. 2) The divergence of the free-surface shape in Green's solution is so weak that one might expect the smallest amount of gravity effect to bring the free surface into the region where we expect to find it: thus, the small effect of gravity eventually would have a large consequence, but only far away from the planing surface. Rispin defines the small parameter: B= gf/U°=1/F*, where F is the usual Froude number. Inthe near field, the natural coordinates are used, which means effectively that £ is considered to be O(1). Smallness of B is achieved by allowing g—~0 or U— oo. Rispin treats his small parameter properly by nondimen- sionalizing everything, so that he then does not have to specify whether U-~oo or g~O. Rather than change all variables now, I shall treat g as a small parameter, as in Section 3.2; the results are the same as Rispin's, of course. In the far field, typical lengths are assumed to be O(1/) in magnitude, or O(1/g), in my loose notation. We could define new coordinates, say, z2=Pz; x= 6x; y= By, and consider that z = O(1) as g—> 0 inthe far field, while z= O(1) as gO inthe near field. Rather than do this, we shall just keep in mind that such orders of magnitude are to be assumed. Also, we note that d/dz = O(1) inthe near field and d/dz = O(8) in the far field. This problem is reversed from the most common kind of stretched-coordinate problem: The inner problem is solved by natural coordinates, and the outer coordinates are compressed. Note, however, that there is no distortion of coordinates between near- and far-fields. There is just a change of scale. 781 Ogilvie In the far field, the planing surface appears to vanish in the limit, and so the first term in a far-field expansion must represent just the incident uniform stream. That is, if the outer expansion is represented: N N F(z;B) ~ : F, (zB), w(z;B) ~ 3 W,(z38), for fixed Bz n=O n=O asB—O, then clearly we have: Fo(z;B) = Uz, and Wo(z;B) = U. This one-term outer expansion must match the one-term inner ex- pansion, the latter being just Green's solution. This much of the matching procedure is rather obvious, and Green already used this fact to determine the value of c, as given in (5-2). The next term in the outer expansion is not quite so obvious. In order to facilitate the matching process, Rispin solved the problem inthe ¢ plane, just as we did above for Green's problem. The free-surface boundary condition on W, is not much different from the familiar linearized condition. One can show fairly simply that: dW, , igA 3 A, Re[ Sr! + -B5 wi] =0 on = :0'; where A=a/n(b tc). (The factor A is just the value of dz/dt far away from the planing surface.) Note that the first term is O(BW,) because of the differentiation, and the second term is the same order because of the g factor. The solution for W, must be analytic in the lower half-space and satisfy this condition on 7 =0, |€ | > 0; note the exclusion of the origin, where singularities may Occur. As usual, we try to restrict the singularities to the simplest kind possible. In this case, we would find nothing in the near field to match with if we allowed all kinds of singularities in W,. A sufficiently general solution” is the following: -igat/uz°$ igAt/U*TC W, (638) = te aa { dt e" [St +S , [e.e) where C; and Cg are real constants yet to be determined (in the matching). SG Gales sae Rispin discusses more general solutions, which are needed in con- structing higher-order solutions. 782 Stngular Perturbatton Problems in Ship Hydrodynamics The two-term outer expansion is now: w(z3B) ~ U + W, (S38), with W, given as above. Its inner expansion to one term is easily found: w(Z3p)°o u- + ° We cannot really say positively that these two terms are the same order of magnitude, but it turns out that they must be if this expres- sion is to match the two-term outer expansion of the one-term inner expansion. The latter is obtained readily from Green's solution for w(z(€)) which was given in (5-1). It is: wipe + iU sina C Then, obviously, we find that: C, = - U sina. We cannot determine the other constants, C,, from the solu- tions so far obtained. It is necessary to solve for the second term in the inner expansion, and Rispin carries this through. Then, he matches the two-term outer expansion of the two-term inner expan- sion with the two-term inner expansion of the two-term outer expan- sion, finding that C, = - aU/n. Thus, C; is proportional to the rate at which fluid leaves in the jet; the C, term represents a sink, infact. (The C, term represents a vortex.) Rispin obtains estimates for h as well, but the results are rather complicated, and it would add no perspicuity to the present section to repeat them. The important point in principle is that it is possible now to specify the value of h and not come to a contradic- tion as a result. The far-field description has effectively provided a height reference, because of the effect of gravity. This effect does not change the first-order inner solution, but it does modify the second-order term. (The velocity magnitude is not constant on the free surface in the second approximation. ) In the second-order term of the inner expansion, there is another interesting phenomenon, namely, the apparent angle of attack changes. This means, physically, that the occurrence of gravity waves modifies the inflow to the planing surface. In the 783 Ogtlvte near field, it is still not possible to see the waves that exist far away, but the latter have the effect of making the incident stream appear to be rotated somewhat. It is like a downwash effect (although the physical origin is quite different). If one were given a planing problem such as we formulated early in this section, with the incident stream and all geometric parameters prescribed, it would be necessary to solve for the parameters a and b. One equation relating these parameters has already been mentioned, namely, the equation relating the length of the plate to these parameters. The other equation comes from the expression (which was not written out here) for has a function of a and b. Rispin avoided much tedious algebra by solving the inverse problem. He assumed that a, b, and c were given, then solved to find h. He also had to treat the angle of attack as an unknown quantity, and he found an asymptotic expansion for it. (Note that only two of the basic parameters can be prescribed arbitrarily, unless we are prepared also tolet £ be an unknown quantity.) One final comment on Rispin's work must be made. He finds terms of six orders of magnitude: O(1), O(f8 log B), O(f), 0(p? log? B) ; O(B* log 6), and O(8*). But he finds also that they cannot be deter- mined one at atime. Rather, they must be taken in groups: a) the O(1) terms, b) the terms linear in B (the logarithm being ignored), and c) the terms involving 8°. This is the same kind of matching procedure that would have been used if he had adopted the working rule that logarithms should be treated as if they were O(1). (See Section 1.2.) 5.2. Flow Around Bluff Body in Free Surface A problem related to that of Rispin [1966] and Wu [ 1967] has been studied by Dagan and Tulin [1969]. They have concerned themselves with the flow at the bow of a blunt ship, where any kind of linearization procedure must be completely wrong. In order to handle such a situation, they have adopted essentially the same pro- cedure that the previous authors used, namely, they set up inner- and outer-expansion problems in which the nonlinearity is confined initially to the near field and the effects of gravity are confined initially to the far field. Then, by limiting their study to a two dimensional problem, the nonlinear near-field problem can be solved by the hodograph method, and the far-field problem is a simple vari- ation of a well-studied problem in water-wave theory. The geometry of their problem is shown in Fig.(5-5), which is reproduced from their paper. They argue that at very low speed there will bea smooth flow up to and then down under the bow, with a stagnation point at the location of highest free-surface rise, but that that flow becomes unstable as speed increases, until finally a jet forms, as 784 Stngular Perturbatton Problems in Shtp Hydrodynamics Fig. (5-5). Bluff Body in the Free Surface sketched in Fig. (5-5). Regardless of whether their description of the flow at very low speed is correct , this jet model appears to be entirely reasonable physically; a barge-like body usually causes a region of froth just ahead of the bow, and this froth is probably caused by such a jet being thrown upward and forward, then dropping downward (which the theory overlooks). Thus it seems appropriate to study the formation of such a free-surface jet by the use of free- streamline theory, and one may expect that the details of the formation of the jet are not terribly sensitive to the effect of gravity. The body, as shown in Fig. (5-5), extends downstream to infinity. (In a sense, the whole problem is part of the inner expan- sion of a much larger problem, in which the stern of the body would be visible and in which waves would follow the body.) Thus, there is no Kutta condition or equivalent which can effectively cause a circulation type of flow in the fluid region. In Green's problem, for example, the flow at great distances appears to have been caused by a vortex. It is this property that causes the apparent logarithmic deflection of the free surface far away from the body, and it is this property that requires the far-field description (as in Rispin's problem) to contain a logarithmic singularity at the origin. Dagan and Tulin have no such logarithmic solutions. They find that the jet appears, from far away, to be caused by a singularity of algebraic type. Specifically, the outer expansion of their inner expansion shows the complex velocity behaving like Zaye: where Z is the complex variable defined in the physical plane, shown in Fig. (5-5). Thus, their far-field expansion must exhibit a singularity at the origin of this same type. This result, if correct, is most interesting, for, as Dagan and Tulin point out, it means that the far-field expression for * Their Section III. 2 has some questionable aspects. 785 Ogtlvte pressure is not integrable, and so one must use the near-field ex- pansion for any force calculation. Furthermore, it is a disturbing result, because it suggests that many previous attempts to incor- porate bow-wave nonlinearities into linear-theory singularities have been futile exercises. Personally, I am not yet willing to admit that the possibility of having the complex velocity behave like Z™ is really to be re- jected, as Dagan and Tulin claim. Wagner [ 1932] analyzed the region of the jet and the stagnation point for the flow against a flat plate of infinite extent downstream, and he showed that this flow, from far away, has the behavior of a flow around the leading edge of an airfoil, that is, the velocity varied with nee Physically it seems rather difficult to imagine that, by curving the body around just behind the stagnation point, one causes such a drastic change in the apparent singularity. Dagan and Tulin present a figure (their Fig. 2) in which they have placed many symbols showing beam/draft ratios of more than a hundred ships, and it is quite evident that most ships have values of this ratio considerably greater than unity. They then use this fact as an alleged justification for claiming that their 2-D model of the bow flow (as in Fig. (5-5)) will have some validity in describing the flow around the bow of an actual ship -- since most ships are pre- sumably of the "flat" variety. However, this claim is completely misleading. The theory might apply to a scow, but not to a ship. After all, beam/draft ratio is measured amidships, and even ships with the largest block coefficients have entrance angles less than 180°. Also, it is appropriate to mention again the warning against defining a small parameter precisely and then trying to interpret on some absolute basis whether a particular value of the parameter is "Small enough." For example, it is conceivable that a thin-ship analysis would be valid for a ship with beam/draft ratio of 10, whereas a flat-ship analysis might fail for the same ship. I am not saying that this is likely, but it is possible. In one problem, a value of 10 might be "small," whereas in another problem a value of 1/10 might be "not small." Notwithstanding these objections, the paper by Dagan and Tulin has provided a refreshing change in outlook on the bow-flow problem, and perhaps it will be more fruitful eventually than the usual attempts to place complicated singularities at the bow in the frame-work of linearized theory. 786 Singular Perturbatton Problems in Shtp Hydrodynamics 5.3. Submerged Body at Finite Speed Since the principal difficulty in solving free-surface prob- lems follows from the nonlinear conditions at the free surface, we are always seeking new arguments to justify linearizing the condi- tions. One possible basis for linearizing is that a body is deeply sub- merged. Then its effect on the free surface will presumably be small, even if it is not appropriate to linearize the problem in the immediate neighborhood of the body itself. Such problems were discussed by Wehausen and Laitone [1960], where the previous history may also be found. Tuck [1965b] introduced a more systematic treatment for the case of a circular cylinder. Salvesen | 1969] solved the problem for a hydrofoil (with Kutta condition and thus with circulation), and he compared his results with the data from experiments which he conducted. In the earlier studies of such problems, the approach was usually an itera- tive one in which the body boundary condition was first satisfied, then an additional term was added to the solution so that the free- surface condition would be satisfied; the latter would cause the body boundary condition to be violated, and so another term would have to be added to correct that error, but then there would again be an error in the free-surface condition. And soon. The free-surface con- dition that was satisfied once during each cycle was generally the conventional linearized condition. Thus, if the procedure converged, one obtained a solution which exactly satisfied the body boundary condition and the linearized free-surface condition. The contribution of Tuck seems to have been in systematizing the procedure in terms of a small parameter varying inversely with depth of the body and in pointing out that a consistent iteration scheme involves using the exact free-surface conditions as a starting point. Then, as the boundary condition on the body is corrected at each stage, so also is the free-surface condition made more and more nearly exact. Tuck concluded, in fact, that it was more important to include nonlinear, free-surface effects than to improve the satisfaction of the body boundary condition if one were most interested in certain free- surface phenomena, e.g., predicting wave resistance and near- surface lift. Salvesen agreed with this conclusion only on the con- dition that the body speed be not too large. At fairly high speed, his results indicated that precision in satisfying the body boundary condition was just as important as precision in satisfying the free- surface condition. Figure (5-6) is taken from Salvesen's paper; it shows the theoretical wave resistance of a particular body as a function of (depth) Froude number, the resistance being calculated by three different approximations: 1) linearized free-surface theory, 2) theory in which the free-surface condition is satisfied to second order, and 3) theory in which both the free-surface condition and the body boundary condition are satisfied to second order. The differences are quite apparent. 787 Ogtlvie 0.03 aw? 0.02 06 001 +——40.5 03 } = 04 = 04 03 J+. los RESISTANCE J 03 05 07 09 1 FROUDE NUMBER, U/Vgb) —___._._ ,_ first-order theory; ————_,_inconsistent second-order theory (neglecting body correction effects); ———-— ' consistent second order theory (From Salvesen (1969)) Fig. (5-6). Theoretical Wave- Resistance Gurves for € = t/b = 0.30. The figure is a very interesting one. The difference between the linear-theory curve and either of the other two curves is pre- sumably a second-order quantity, and yet that difference is -- in one case -- of the same order of magnitude numerically as the linear-theory curve itself. The problem is worth further discussion. Salvesen defines his small parameter as follows: = t/b;, where t is the thickness (or some other characteristic dimension) of the body, and b is the submergence of the body below the undis- turbed free-surface level. It is not assumed that the body is "thin" in any sense; it could be a circular cylinder (Tuck's problem), for example. Salvesen's calculations and experiments were carried out for a rather fat, wing-shaped body with a sharp trailing edge. The body was symmetrical about the horizontal plane at depth b. Ifthe free surface had not been present, there would have been no lift on the body. A complex velocity potential, F(z) = $(x,y) + iu(x,y), can be defined for the problem, with z =x tiy measured from an origin located in the body at a depth b below the undisturbed free surface. 788 Stngular Perturbation Problems in Shtp Hydrodynamics Salvesen expands the complex potential in a series which he groups in two alternate ways: F(z) il [ Uz + Fol pa be + Fp] eas (5-3) Uz +[ Fp, + Fe] Pho, + Ff] ae (5-4) These terms are defined in terms of the iteration scheme already mentioned. The grouping in (5-3) is to be used near the body, and the grouping in (5-4) applies far away from the body; in particular, the latter applies on and near the free surface. Salvesen points out that this distinction means that: a) near the body, we are consider- ing the zero-order flow to be that flow which would occur in the presence of the body and the absence of the free surface, and b) near the free-surface, the basic flow is just the uniform a stream, Thus, in (5-3), we must determine Fpo so that [Uz + F satisfies the kinematic boundary condition on the body and ae that Wore, => 0 as |z| — oo (in any direction). Next, Salvesen assumes that Fpo is O(e) far away from the body. The two terms so far obtained do not satisfy a free-surface condition, and so F¢, must be determined so that, when it is added to the first two terms, the sum satisfies the appropriate free-surface condition, which is: Re (Fy + Fe, tikFy) tixF,}=0 on y=b (5-5) where K= g/U*. Since Fpop is assumed to be O(e€) near the sur- face, then the same should be true for Fe, as Now the three terms in the series do not satisfy the body condition, and so Fp, is determined so that, when it is added to the first three terms, the sum satisfies the condition properly. Then Fp, is assumed to be O(e*) near the free surface, and a new function F¢, is found to provide a further correction needed near the free surface. It is in this last step that the Tuck-Salvesen approach differs from the previous treatments of such problems. If Fp, is really O(e?) , then the free-surface condition ought to be gafiatied to that order of magnitude. It can be shown that this implies the following condition on Fe Re {Fy f FY, +ikFp, + ikF4,} = 7, {Fp + FY, + iKFy + iKFy J : (1/2U)| Fy. - (5-6) 789 Ogtlvte The right-hand side of this equation takes account of the nonlinearity of the free-surface conditions , since obviously it involves just the potential function from the previous cycle of the iteration. 1, is the free-surface elevation from the previous approximation; it is given by: n(x) = - (U/g) Re {Fo + Fy}; with the right-hand side evaluated on y= b. One might try to cut corners in (5-6) in either of two ways, namely, 1) ignore the right- hand side by setting it equal to zero, 2) Drop the terms involving Fp, on the left-hand side. The first is equivalent to retaining just a linear free-surface condition. The second is equivalent to neglect- ing the effect of the second-order body correction at the free surface}; this is the "inconsistent" second-order theory to which Fig. (5-6) Ferrers. Apparently , Salvesen did not prove one important step in his development, namely, his claim that Fbpg is O(1) near the body and O(¢e) far away from the body. In fact, with his definition of e = t/b, it appears that the statement is wrong. The potential Fbo represents just a thickness effect, since it is the solution of the problem of a symmetrical body ina uniform stream. Although the body can be replaced by a distribution of sources, the disturbance will appear from far away to have been caused by a dipole, and so it must have the form: Fbo~ C/z. If the body were a circular cylinder, we could evaluate C: C= Ut’, where t is the radius of the cylinder. The complex fluid velocity on the free surface caused by the body is, in the first approximation, - C/z*= O(e*), since z =x +tib onthe level of the undisturbed free surface. This con- clusion contradicts Salvesen's assumption that the free-surface disturbance is O(e€), but perhaps it does not matter. At this point, the results would presumably be just the same if he had defined: €= (t/b)! e (The argument above for a circular cylinder agrees with Tuck's conclusions.) When the first free-surface correction is found, namely, Fe,» its effect in the neighborhood of the body is not diminished by an order of magnitude, since at least one part of F¢, involves an exponential decay with depth, the exponent being K(y - b). Near the body, y= 0, and so the exponential-decay factor is e“", and it has been assumed that Kb is O(1). (See Salvesen's paper.) Since Fy, is O(e*) near the body, the order of magnitude of the next correction term, Fp); must be the same. This time, however, the nature of the body disturbance is quite different from a dipole disturbance. The effective incident flow corresponding to Ff, is not a uniform stream, and so the presence of a sharp trailing edge on the body requires that a Kutta condition be imposed, and then a circulation flow occurs. From far away, it appears that Fp, 790 Singular Perturbation Problems in Shtp Hydrodynamics is caused by a combination of a vortex and a dipole. If the strengths of the two apparent singularities were comparable, the vortex behavior would dominate the dipole behavior far away, and the induced velocity would diminish in proportion to 1/z, rather than 1/z*, which was the case for the dipole. Thus, Fp, would be O(e>) near the free surface. In the absence of a sharp trailing edge which can cause the formation of a vortex flow, the corresponding Fp, would be O(e€4%). This matter remains to be resolved. There are other interesting aspects to this problem. One relates to the interpretation of the small parameter, € =t/b. In defining such a dimensionless perturbation parameter, one nor- mally assumes that the smallness of € can be realized physically either by letting t be extremely small or by letting b be very large. In the present problem, this choice is not really available tous. The reason is that there is another length scale in the prob- lem, namely, 1/xk = U‘/g, and this length scale appears generally in combination with the dimension b. It has been assumed that Kb = O(1) as € ~ 0. Therefore, if we want to consider the problem of a body which is more and more deeply submerged, (b —~ oo), then we must also restrict our attention to higher and higher speeds. This is awkward. Finally, one more important aspect must be mentioned. The relation between wave number, xk, and forward speed, U, namely, K = 2/U-, is based on linearized free-surface theory. In general, if one seeks to find the nature of nonlinear waves which can propa- gate without change of form, the wave length of those waves is not related to their speed in this simple fashion. To be sure, the relationship is approximately correct if the waves are not terribly big in amplitude, and so one might expect that the wave length or the wavenumber can be expressed as an asymptotic series in € K~ Kot K, + Ky toes F with Kg = g/U*. This can indeed be done, but it turns out to be much more convenient to assume that kK is precisely given and then to find the value of forward speed that corresponds to that wave number. Thus, one expands the forward speed, U, into an asymptotic expansion: Ut ug Fy Pech ss This procedure is discussed by Wehausen and Laitone [1960], and Salvesen uses it in his hydrofoil problem. I was able to omit mention of it in writing Eqs. (5-5) and (5-6) because it turns out that u, = 0, and so the effect of this speed shift (or period shift) does not enter the problem until the third approximation is being sought. However, this is a classic example of the kind of expansion described in ioe Ogilvie WAVE ELEVATION,FT ~~ BODY LOCATION ne, ae, he FIRST—ORDER THEORY a 4 SECOND— ORDER THEORY THIRD— ORDER THEORY es EXPERIMENT FROUDE NUMBER =0.79; &=t/ b=0.30 (FROM SALVESEN (1969)) wwe eee eee | Fig. (5-7). Third-Order Effect on Wave Length Section 1.3. If one did not allow for a variation in either K or U, the third approximation would not be valid at infinity, and so one would have great difficulty in predicting wave resistance, since that quantity depends explicitly on the wave height at infinity. Figure (5-7) is taken from Salvesen [1969]. It shows very clearly the change in wave length that arises in the third-order solution. In fact, it appears in this case that the change of wave length is practically the only third-order effect. This figure also speaks well for Salvesen’s experimental technique! 5.4. Submerged Body at Low Speed Salvesen [ 1969] computed the wave height behind a hydrofoil up to the third approximation, as already mentioned in Section 5.3. Although his third approximation is not really consistent, he gives what appear to be sufficient arguments to demonstrate that the con- sistent result would not be much different from the results presented in his paper. Figure (5-8), from Salvesen [ 1969], presents the wave- height computations in a way that shows the relative importance of the first-, second-, and third-order term. Let the wave amplitude be expressed by the series: He yh Bett ss where Has = 0(H,) as t = 310. 192 Singular Perturbatton Problems in Ship Hydrodynamtes F = U/y(gb) E=t/b fo) ° WAVE -HEIGHT RATI 2 3 gt its = €bk —-— H, / (Hy +Ho*Ha) —---- Hb/ Wane) ———_—= Ha/(H)+HotH 3) From Salvesen (1969) Fig. (5-8). First-, Second-, and Third- Order Wave Heights at Low Speeds. (t is the thickness of the foil, as in the last section.) Then the figure shows the three ratios, H,/(H, + H2+H;), for n=1,2,3; that is, each curve shows the relative contribution to the wave height of one of the first three terms in the wave-height expansion. As speed decreases (toward the right-hand side of the figure), the second-order part comes to dominate the linear-theory part, and then the third-order part dominates the first two. It seems quite likely that the fourth-order term would take over if the graph were extended, then the fifth-, sixth-, ... order terms. Salvesen's analysis is based on the condition that t (or, more properly, t/b, where b is the body depth) is very small; the Froude number is simply a parameter unrelated to t, which is equivalent to saying that Froude number = U/V(gb) is O(1) as t/b + 0. Perhaps it is not surprising if Salvesen's expansion is not uniformly valid with respect to Froude number. That is all that Fig. (5-8) really says. The reason for its nonuniformity has already been mentioned: In the expansion of the solution near the free surface, it has been assumed that the lowest-order approximation is just the uniform- stream term, Ux; all other terms in the expansion of the potential must be very small compared to this term. And this is nonsense if we consider the limit process U~0. Of course, we might have been lucky: It could have turned out that the velocity perturbation approached zero more rapidly than U. But it does not. And so we have here a genuine singular perturbation problem. Let us consider a sequence of steady-motion experiments, each lasting for an infinite length of time. We arrange the sequence of experiments according to decreasing values of body speed, U, T93 Ogilvie and we suppose that all conditions except forward speed are identical in all experiments. We shall discuss what happens when "UO," and we shall understand by the limit operation that we are passing through the sequence of experiments toward the limit case in which there is no forward speed at all. In each experiment, U isa constant. * As U~>0O, we certainly expect all fluid motion to vanish. But we would like to know to what extent the velocity field vanishes in proportion to U (that is, what partis O(U)), what part vanishes more rapidly than U (that is, what partis o(U)), and what part, if any, vanishes less rapidly than U. In an infinite fluid, the velocity everywhere is exactly pro- portional to U. Far away, the velocity approaches zero; it drops off like 1/r if there is a circulation around the body, and it drops off like tps if there is no circulation. But in both cases the constant of proportionality is O(U). No matter how distant our point of observation is from the body, the velocity is O(U) as Ui —= Oe At very low speed, one expects that gravity will force the free surface to remain plane. The constant-pressure condition will be violated to the extent that the magnitude of the fluid velocity on that plane is not quite constant, but the error in satisfying the dy- namic condition will be proportional to the square of the fluid velocity magnitude. The kinematic condition will be satisfied ina trivial manner. Accordingly, it seems quite reasonable to assume that the free-surface disturbance is O(U‘) as U0, and so the velocity potential in the first approximation is the same as if the free surface were replaced by a rigid wall. Let the rigid-wall velocity potential be denoted by (x,y). Clearly, it is true that: $o(x,y) = O(U). This follows by the same arguments as those used in the preceding paragraph. The more important problem is to determine the order of magnitude of [ $(x,y) - $o(x,y)], where (x,y) is the exact velocity potential for the case of the body moving at speed U under the free surface. In order to be specific now, let (x,y) be the velocity potential in two dimensions which satisfies the conditions: £90=0, on body; |$o- Ux| +0, as x— - a} Fe = 0 on y= 0. * This is the same point that I belabored in the last paragraph of Section 1.2. Again, I apologize to those to whom it is obvious. 794 Stngular Perturbatton Problems in Shtp Hydrodynamics The body is at rest in our reference frame. The rigid-wall solution satisfies all conditions of the free- surface problem except the dynamic condition on the free surface. The latter could be used to define the free-surface shape. Thus, if the free-surface disturbance is expressed by mix) nolx) tk) ce Gf the dynamic free-surface boundary condition says that: nlx) ~ nolx) = 35 LU" - 40,(x,0)] (5-7) Of course, the kinematic condition is now violated, but an additional velocity field which is O(U*) can correct that. And so it appears plausible that: (x,y) - $o(x,y) = 0(U). (5-8) One point should be noticed from this conclusion. The limit process "U-—-0O" implies that Froude number goes to zero. Nothing has been said about the length scale used in defining Froude number, but it does not matter so long as all dimensions are fixed. The submergence and the body dimensions may be quite comparable, for example. Thus, we are not considering t/b as small, in the sense that Salvesen did. However, both t and b are supposed to be large compared with the length U*7s; we imply this if we state that all dimensions must be fixed as U0. It would be wrong to take (x,y) as the potential for the flow around the body in an infinite fluid (without either free surface or a rigid-wall substitute). The body can be quite near to the free surface in Salvesen's sense, and so the effect of its image cannot be neglected. Furthermore, at least part of the effect of the image is O(U), even if the body is very far away from the free surface, and such an effect must be included in the first term of the approximation which is supposed to be validas U~>0O. The next problem is to find [ $(x,y) - $9(x,y)]. We consider two possible approaches in the following subsections. 5.41. A Sequence of Neumann Problems. As above, let there be a velocity potential, (x,y), which provides the solution of the exact problem: 195 Ogilvie 2 en) +5 ent ey) -5U=0, on y=nlxs (5-9) OxNx - y= 9, on y = (x); (5-10) oe = 0, on the body; (5-11) (2 5.y))-) Ux Fe 105 as xo, = 003 (5-12) The rigid-wall potential, (x,y), satisfies (5-11) and (5-12) too, but it does not satisfy the free-surface conditions, of course; instead, we have 8 od =05 on y =0. (5-13) Now we introduce one more potential function, the difference between the above two potentials: @(x,y) = (x,y) $ oo(x,y) ° (5-14) It must satisfy the body boundary condition, of course, and it vanishes far upstream. On the free surface, which we now define as: y = N(x) = No(x) + H(x) , (5-15) where Tox) is defined as in (5-7), the new potential satisfies the two conditions: 0 = gH(x) - 5 66,x,0) 2 2 2 +5140, + $0, + 240% + 2hoydy te + Fy lyeqoy § (5-16) 0 =[nolx) + H'(x)][¢0, + Pl enti - [Hoy # Py] Jenin” (5-17) These conditions are still exact. An obvious approach to solving for ®(x,y) and H(x) is to re-express these conditions on y = n(x) as conditions on, y = 0. Here I shall assume that this can be done in the usual way.” Then it follows from the exact conditions *This is the crucial point which distinguishes this section from the next section. 796 Singular Perturbatton Problems in Shtp Hydrodynamics that the following are appropriate simplifications: O= gH(x) + 0,2, » on y = 0; (5-18) d, = To) $0, - Nolx) Poy on y=0O. (5-19) The second condition is a Neumann condition; the right-hand side is known, and the condition is prescribed on a known, fixed surface. In fact, (5-19) is satisfied by the real part of: 00 *) ds p(s) s-z’ 00 where z=x tiy, P(X) = N(x) $9 (x, 0). (5-20) This follows from the Plemelj formula. (See, e.g., Muskhelishvili [1953].) The function p(x) can be interpreted in terms of the fluid velocity which is needed to correct the flow field because of the error incurred by taking the free surface at y = N(x) while using the potential function $ x,y) to prescribe the velocity field. This is the same correction which was discussed above in gonnection with (5-8). Now we may observe that, since No= O(U) and G(x,y) = O(U), it follows that p(x) = O(U"). Thus also: |Vve|=o(u) as UO. (5-21) This is certainly a much stronger conclusion than (5-8)! The integral expression given above is not the solution of the @ problem, even in the first approximation, since it does not satisfy the body boundary condition. However, since the existence of ® arises from a defect of #9 in meeting the free-surface conditions, it is difficult to imagine that the above estimate of the order of mag- nitude of @ is not correct. Numerical procedures could readily be worked out for solving problems of the above type. In fact, all that is needed is one algorithm which handles the problem of a given distribution of the normal velocity component on a surface in the presence of a plane rigid wall. The integral part of the solution given above would 197 Ogtlvie lead to a non-zero normal velocity component on the body, and this would have to be offset by a flow which does not change the condition at the plane y=0. Presumably, all higher-order approximations would be solutions of problems which are identical in form to this one, A variation on this approach has been discussed several times by Professor L. Landweber, although he has not published the work. He points out that the usual linearized free-surface condition, 2 a og 2 —_ + = = = Ae K 35 0, on z=0, where K= 2/Us, becomes the rigid-wall condition when U~— 0, and so one might try an iteration scheme in which @ is expanded ina series, $¢~ hy eo; and the terms are obtained as the solutions in an iteration scheme: 2 x In order to test the scheme, Professor Landweber proposed trying to obtain the potential function for a Havelock source in this way; this obviates the need to satisfy a body boundary condition, and the known potential for the source can be expanded in a series in terms of 1/kK. Neither of the above schemes appears very promising to me. Salvesen's findings about the singular low-speed behavior seem to condemn any approach which overlooks the peculiar nature of the free-surface problem at low speeds. The next section should make clear why I am pessimistic about these approaches. It should be obvious even now that the wave-like nature of the problems has been lost, but the difficulty is more serious than that. 5.42. A Dual-Scale Expansion. According to linearized wave theory, the wave-like nature of a free surface disturbance loses its identity exponentially with depth. A disturbance created at the free surface is attenuated rapidly with depth, and a disturbance created at some depth causes a free-surface disturbance which de- creases with the depth of the cause. The depth,effect is essentially proportional to e*“Y, where, as above, kK=g/U~ and y is measured as positive in the upward direction. As U approaches zero, this depth-attenuation factor ap- proaches zero for any fixed y. In other words, the free-surface effects are restricted to a thin layer which approaches zero thickness as U->0O. We might say that the free-surface is separated from the main body of the fluid by this "boundary layer" in which there is 798 Singular Perturbatton Problems in Ship Hydrodynamics a rapid transition from conditions at the surface to conditions inside the bulk of the fluid. From our experience with viscous boundary layers, we should expect the occurrence of large derivatives in this region and also some difficulty in satisfying boundary conditions on a face of the boundary layer. In a viscous boundary layer, of course, the derivatives are much greater in one direction than in another, and this fact allows us to stretch coordinates anisotropically and apply the limit pro- cesses of the method of matched asymptotic expansions. In the free- surface boundary layer, however, this does not appear to be a possible approach. From the linear theory, we expect that there will be a wave motion with wave lengths which are O(U?/g). Thus, derivatives will be large in at least two directions inside the boundary layer -- in the direction normal to the layer and in one direction parallel to the layer. When I tried to solve this problem two years ago (see Ogilvie [ 1968]), I did not apply very systematic procedures. Rather, I simply assumed that the first approximation to @, as defined in (5-14), would have certain properties, namely, = 5 3 B(x,y) = OU); (x,y), Fy(x,y) = O(U); also, the surface deflection function would be given by (5-15), with: Hix) =/O(U 1. H'b) =-0(0). The order of magnitude of, ® was chosen just so that the velocity components would be o(1y>), and I assumed that differentiation changes a quantity by 1/uU? in order of magnitude. The arguments leading up to (5-21) contributed heavily to the conjecture about veloc- ity components, and the 1/U® effect of differentiation was chosen just because the free-surface characteristic length is U‘/g. It is important to note that the rigid-wall potential, 9), is still part of the solution, and these statements about orders of magnitude and differentiation do not apply to it. In fact, lassume that ¢) is com- pletely known, and so it is not necessary to conjecture about the effects of differentiation. In terms of the general approach of the multiple-scale ex- pansion method, I have assumed that an approximation to the solution can be represented as the sum of two functions. The first depends only on the length scale appropriate to the body geometry. The second function depends primarily on lengths measured on a scale appropriate to U#/g, but it also depends on the first function and thus on lengths typical of the body. However, it seems to be possible to keep clear when differentiations are being carried out with respect to each of the length scales. 799 Ogilvie Physically, the situation may be described in the following way: If U is small enough, the body extends over a distance of many wave lengths of the surface disturbance. The initial dis- turbance is caused by the body, of course; this is the "rigid- wall" motion, and its dimensions are characteristic of the body. It causes a free-surface disturbance, with the result that waves are created. But these waves are very, very short, whereas the initial disturbance from the body appears to be just a slight nonuniformity in flow conditions when viewed on the scale comparable to the wave length. The method is, in fact, quite similar to classical methods such as the W-K-B method. When the assumptions listed above are actually applied, we find that the approximate free-surface conditions given in (5-18) and (5-19) must be replaced by the following: R BH(x) + bo, 0) (2, Mol) = 05 (5-22) G(x, No(x)) - 4,(x.0)H"(x) = p'(x); the function p(x) is the same that was given in (5-20). Note that @ in both conditions here is to be evaluated on y = N(x), rather than on y=0. The reason is the same that was given in Section 3.2 in the near-field problem: If we tried in the usual way to expand @(x,9), say as follows: 4.2 B(x, 1g) = B(x, 0) + nob (x,0) + 5 NoByy(x,0) +... we would find that every term on the right-hand side is the same order of, magnitude according to my assumptions. In particular, No= O(U‘), and, symbolically, we have: 8/dy = O(1 /U‘). So this expansion procedure is not useful. The two conditions above can be combined consistently into the following: Bylves Mole) +5 4 (2610) By (Mg) = PICA). (5-23) This is remarkably similar to the free-surface condition for another problem. In the ordinary linearized theory of gravity waves, sup- pose that a pressure distribution, p(x), is travelling at a speed U. The free-surface condition would be: 2 &, (x, 0) + Gyx(x,0) = p'(x) , 800 Stngular Perturbatton Problems in Ship Hydrodynamtes if (x,y) were the potential function for the problem. Replace U by 0,(x, 0), the "local stream speed," and evaluate the condition on y= T(x) 5 then this condition transforms into the condition found for (x,y) in the low-speed problem. Thus, ona "local" scale (in which a typical lengthis U 2/g), the free-surface condition is just a very ordinary condition; one cannot see that the stream velocity changes slightly along the free surface, because the change occurs on a scale in which a typical measurement would be a body dimension; the change is very gradual. Also, the level of the undisturbed free surface appears to change gradually, as given by (5-7); this change also cannot be detected on the "local" scale. It is now clear that the two length scales are quite distinct. We cannot separate the fluid-filled region into distinct parts in each of which only one length scale needs to be considered. Rather, the gradual changes which appear on the body-size scale appear to modify the short-length wave motion in the manner of a modulation. In trying to find a potential akon which satisfies (5-23), I made a nonconformal mapping: x! =x, y!=y - M(x). Then ©® soos a complicated partial differential equation in terms of x' and y', but the terms in the equation can be arranged according to their Hepentence on U, and it is found that the leading-order terms are simply the terms in the Laplacian, that is, Oyye + ®y'y! = 0; all other terms are higher order. In this new coordinate system, the free-surface condition, (5-23), is transformed too, but again the leading-order terms are just the same after the transformation (but expressed as functions of x' and y'). Furthermore, the boundary condition is then to be applied on y'=0. Let us now drop the primes on the new variables, for convenience. Then the problem is as follows: Find a velocity potential, @(x,y), which satisfies the Laplace equation in two dimensions and the free- surface condition: 2 Dy(xsy) += Gof%+ 0). (x0) = p(x), where p(x) = No(x) $o,(x,0) . In addition, the potential must satisfy a body boundary condition; this has not been carefully formulated yet, and, in any case, the only solution that has been produced so far is one that satisfies the free-surface condition but not a body condition. There may be 801 Ogilvte some good justification (or rationalization) for proceeding this way, but it is really an open question. With such restrictions and reservations expressed, we can write down a "solution" of the above problem. Define: @ = Re {fp(z)}; O(x,y) = Re {F(z}; -2 Note that: fifse) = 0,000); ke) = ef doe 00) Then the solution is given by: H 00 a2 z LT a = fa) ds p'(s) { ia exp E if du (u) | é -© -00 C The ¢ integral is a contour integral starting at x = - o, located entirely in the lower half-space. It should pass above the location of the singularity in k(z). This solution represents no disturbance at the upstream infinity, as one would expect. Far downstream, this solution can be approximated: 0 z F'(z) = 2ie i ds p"(s) exp [ixs = i du[k(u) - id] ; -00 s where K= g/U% Then, from (5-22), we obtain the wave shape far aft of the body: 00 H(x) = - aul ds p"(s) sin[ x(x - s) + K(s)], & J-o where K(s) -{ du [k(u) - kK]. Ss Calculation of the wave resistance is then very simple in principle. (In practice, it is a very tedious calculation.) Notethat the expres- sion for the wave shape downstream does not require knowledge of 802 Stngular Perturbatton Problems in Ship Hydrodynamics F'(z) (or &(x,y)), that is, the surface disturbance far away is a real wave, but its shape and size depend only on the solution of the rigid-wall problem. This is not true of the wave disturbance in the vicinity of the body. It would be very useful, I am sure, to formulate this problem carefully by the method of multi-scale expansions. The approach described by Ogilvie [1968] is very heuristic and leaves much to be desired. ACKNOW LEDGMENT The preparation of this paper was supported by a grant of the National Science Foundation (Grant GK 14375). REFERENCES Benjamin, T. P., and Feir, J. E., "The disintegration of wave trains on deep water. Part 1. Theory," Journal of Fluid Mechanics, 27, 417-430, 1967. Cole, Julian D., Perturbation Methods in Applied Mathematics, Blaisdell Pub. Co., Waltham, Mass., 1968. Dagan, G.,and Tulin, M. P., Bow Waves Before Blunt Ships, Technical Report 117-14, Hydronautics, Inc., 1969. Fedyayevskiy, K. K.,and Sobolev, G. V., Control and Stability in Ship Design, English translation: JPRS: 24,547; OTS: 64- 31239; Joint Publications Research Service, Clearinghouse for Federal Scientific and Technical Information, U.S. Dept. of Commerce, 1963. Friedrichs, K. O., "Asymptotic Phenomena in Mathematical Physics," Bull. American Mathematical Society, 61, 485- 504, 1955. Joosen, W. P. A., "The Velocity Potential and Wave Resistance Arising from the Motion of a Slender Ship," Proc. Inter- national Seminar on Theoretical Wave Resistance, pp. 713- 742, Ann Arbor, Michigan, 1963. Joosen, W. P. A., "Slender Body Theory for an Oscillating Ship at Forward Speed," Fifth Symposium on Naval Hydrodynamics, ACR-112, pp. 167-183, Office of Naval Research, Washington, 1964. 803 Ogilvte Khaskind, M. D., "The Exciting Forces and Wetting of Ships," (in Russian), Izvestia Akademii Nauk S.S.S.R., Otdelenie Tekhnicheskikh Nauk, 7, 65-79, 1957. (English translation: David Taylor Model Basin Translation No. 307 (1962)). Kinner, W., "Die kreisformige Tragflache auf potentialtheoretischer Grundlage," Ing.-Arch., 8, 47-80, 1937. Kochin, M. E., Theory of Wing of Circular Planform, NACA Tech. Memo 1324, (An English translation of the Russian paper), 1940. Kotik, J. and Thomsen, P., "Various Wave Resistance Theories for Slender Ships," Schiffstechnik, 10, 178-186, 1963. Krienes, K., "Die elliptische Tragflache auf potentialtheoretischer Grundlage," Zeit. angewandte Mathematik und Mechanik, 20, 65-88, 1940. Lee, C. M., "The Second-Order Theory of Heaving Cylinders in a Free Surface," Jour. Ship Research, 12, 313-327, 1968. Lighthill, M. J., ''Mathematics and Aeronautics,'' Jour. Royal Aeronautical Society, 64, 375-394, 1960. Maruo, H., "Application of the Slender Body Theory to the Longi- tudinal Motion of Ships among Waves," Bulletin of the Faculty of Engineering, Yokohama National University, 16; 29-61% 196%. ia Milne Thomson, L. M., Theoretical Hydrodynamics, 5th Ed., Macmillan Co., New York, 1968. Muskhelishvili, N. I., Singular Integral Equations, English transla- fion by J. R.. M. Radek, bP. Noordhott N. | V.,, 1953. Newman, J. N., "A Linearized Theory for the Motion of a Thin Ship in Regular Waves," Jour. Ship Research, 5:1, 34-55, 1961. Newman, J. N., "The Exciting Forces on Fixed Bodies in Waves," Jour. Ship Research, 6:3, 10-17, 1962 (reprinted as David Taylor Model Basin Report 1717). Newman, J. N., "A Slender Body Theory for Ship Oscillations in Waves," Jour. Fluid Mechanics, 18, 602-618, 1964. Newman, J. N., "The Exciting Forces on a Moving Body in Waves," Jour. Ship Research, 9, 190-199, 1965 (reprinted as David Taylor Model Basin Report 2159). 804 Singular Perturbatton Problems in Ship Hydrodynamics Newman, J. N., "Applications of Slender-Body Theory in Ship Hydrodynamics," Annual Review of Fluid Mechancs, Ed. by M. Van Dyke and W. G. Vincenti, Annual Reviews, Inc., 1970. Newman, J. N. and Tuck, E. O., "Current Progress in the Slender Body Theory for Ship Motions," Fifth Symposium on Naval Hydrodynamics, ACR-112, pp. 129-162, Office of Naval Research, Washington, 1964. Ogilvie, T. F., "Nonlinear High-Froude-Number Free-Surface Problems," Jour. Engineering Mechanics, 1, 215-235, 1967. Ogilvie, T. F., Wave Resistance: The Low-Speed Limit, Report No. 002, Dept. of Naval Arch. and Mar. Eng. » University of Michigan, Ann Arbor, Michigan, 1968. Ogilvie, T. F. and Tuck, E. O., A Rational Strip Theory of Ship Motions: Part 1, Report No. 013, Dept. of Naval Arch. and Mar. Eng., University of Michigan, Ann Arbor, Michigan, E67. Peters, A. S. and Stoker, J. J., The Motion of a Ship, as a Floating Rigid Body, in a Seaway, Report No. IMM-203, Institute of Mathematical Sciences, New York University, 1954. Peters, A. S. and Stoker, J. J. "The Motion of a Ship, as a Floating Rigid Body, in a Seaway," Comm. Pure and Applied Mathe- matics, 10, 399-490, 1957. Rispin, P. P., A Singular Perturbation Method for Nonlinear Water Waves Past an Obstacle, Ph.D. thesis, Calif. Inst. of Tech. , 1966. Salvesen, N., "On higher-order wave theory for submerged two- dimensional bodies," Jour. Fluid Mechanics, 38, 415-432, 1969. ae Stoker, J. J., Water Waves, Interscience Publishers, 1957. Tuck, E. O., The Steady Motion of Slender Ships, Ph.D. thesis, Cambridge University, 1963a. Tuck, E. O., "On Vossers' Integral," Proc. International Seminar on Theoretical Wave Resistance, pp. 699-710, Ann Arbor, Michigan, 1963b. Tuck, E. O., "A Systematic Asymptotic Expansion Procedure for Slender Ships," 8:1, 15-23, 1964a. Tuck, E. O., "On Line Distributions of Kelvin Sources," Jour. Ship Research, 8:2, 45-52, 1964b. 805 Ogilvte Tuck, E. O., The Application of Slender Body Theory to Steady Ship Motion, Report 2008, David Taylor Model Basin, Washington, 1965a. Tuck, E. O., "The effect of non-linearity at the free surface on flow past a submerged cylinder," Jour. Fluid Mechanics, 22, 401-414, 1965b. awe Tuck, E. O. and Von Kerczek, C., "Streamlines and Pressure Distribution on Arbitrary Ship Hulls at Zero Froude Number," Jour. Ship Research, 12, 231-236, 1968. Van Dyke, M., Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964. Vossers, G., Some Applications of the Slender Body Theory in Ship Hydrodynamics, Ph.D. thesis, Technical University of Delft, TOGz. Wagner, H., "Uber Stoss- und Gleitvorgange an der Oberflache von Flussigkeiten," Zeit. f. Ang. Math. u. Mech., 12, 193-215, 1932. Wang, K. C., "A New Approach to 'Not-so-Slender' Wing Theory," Jour. Mathematics and Physics, 47, 391-406, 1968. Ward, G. N., Linearized Theory of Steady High-Speed Flow, Cambridge University Press, 1955. Wehausen, J. V., "An Approach to Thin-Ship Theory," Proc. International Seminar on Theoretical Wave Resistance, pp. 821-852, Ann Arbor, Michigan, 1963. Wehausen, J. V. and Laitone, E. V., "Surface Waves," Encyclopedia of Physics, IX, pp. 446-778, Springer-Verlag, Berlin, 1960. Wu, T. Y., "A Singular Perturbation Theory for Nonlinear Free . Surface Flow Problems," International Shipbuilding Progress, 14, 88-97, 1967. 806 THEORY AND OBSERVATIONS ON THE USE OF A MATHEMATICAL MODEL FOR SHIP MANEUVERING IN DEEP AND CONFINED WATERS Nils H. Norrbin Statens Skeppsprovningsanstalt Sweden ABSTRACT This paper summarizes an experimental and analytical study of ship maneuvering, with special emphasis on the use of a research-purpose simulator for evaluating the behaviour of large tankers in deep water as well as in harbour entrances and canals. In an introductory Section some new results from full-scale measure- ments and simulator studies are given to illustrate the demands put on a mathematical model in the two ex- treme applications: course-keeping in deep water and manoeuvring in a canal bend. Well-known derivations of rigid body dynamics and homogeneous flow solutions for forces in the ideal case are included to form skeleton of the mathematical model, Separate equations handle helm and engine controls. Coefficients and parameters are made non- dimensional in a new system — here designated the "bis" system as different from the SNAME "prime" system generally used — in which the units for mass, length and time, respectively, are given by the mass of the ship, m, the length, L, and the time required for travelling one ship length at a speed corresponding to Via oe 1, L/e. Semi-empirical methods are suggested for estimates of the force and moment derivatives. Special consider- ation is given to added mass and rudder forces in view of their predominant importance to course-keeping behaviour; the rudder forces measured ona scale model are corrected for differences in wake and screw 807 Norrbtn loading before application to full-scale predictions. Non-linear contributions to hull forces are included in second order derivatives, relevant to the cross- flow concept. The extension of the mathematical model to the con- fined-water case is based upon the theoretical results by Newman and others, and upon relations found from special experiments. In the model the hydrodynamic interferences appearing in forces and moments due to the presence of port and starboard side wall re- strictions and bottom depth limitations are represented by additional terms containing higher order derivatives with respect to three suitable confinement parameters, N=. tp» N=Ns- Np» and €~ Ina canal the asym- metrical forces are considered as due to the aaded effects from port (p) and starboard (s) walls rather than as the effect of an off-centreline position; primarily n is a measure of this position, 7 a measure of the bank spacing. The mathematical model is here applied for evaluation of model test data obtained for a Swedish 98 000 tdw tanker in the VBD laboratories. Oblique towing and rotating arm tests were performed in "deep" and shallow water. Oblique towing tests were also run at various distances from a vertical wall in the deep tank, and in two Suez-type canal sections. The effect of shallow water was especially large in force non- linearities. Missing data for bottom and wall effects on added mass and inertia are taken from theory and from test results due to Fujino, respectively. The deep-water predictions for zig zag test and spiral loop prove to be in good agreement with full-scale trial results. Analogue computer diagrams are given to show the effects of shallow water upon definite manoeuvres and upon course-change transients follow- ing auto-pilot trim knob settings. Finally a few results are included to illustrate auto-pilot position control of the tanker in free water, in shallow water, between parallel walls and in a canal. 808 Shtp Maneuvering in Deep and Confined Waters I. INTRODUCTION On Course-Keeping in Deep Water The average depth of the oceans is some 3800 m. Small native crafts still steer their ways between nearby islands in these oceans. New ships are built to transport ever larger quantities of containers or bulk cargoes at a minimum of financial expense between the continents. It is not necessarily obvious that the helmsman shall be able to control a mammoth tanker on a straight course. A few years ago ship operators were stirred by the published results of an analytical study, interesting in itself, which in fact did indicate, that manual control of ships would be impossible beyond a certain size. Upon request by the shipbuilders a series of real-time simulator studies were initiated at SSPA in autumn 1967 to investigate manual as well as automatic control of large tankers then building, [1]. At an early stage of these tests the helmsman was found to constitute a remarkably adaptive control, which could not be simu- lated by a simple transfer function. As could be expected a rate dis - play proved to make course keeping more easy; the rate signal was even more essential to the auto pilot. The simulator findings were confirmed in subsequent proto- type trials. The diagrams of Fig. 1 compare simulator and proto- type rates of change of heading and yaw accelerations for a large tanker as steered by the author in a Force 6 following sea. (In the simulator case the sea disturbance was represented by a cut-off pseudo-random white noise of predetermined root mean square strength, that was fed into the yaw loop.) This particular tanker is dynamically unstable on a straight course, and the steady-state L(6)-diagram from a deep-water spiral test exhibits a hysteresis loop with a total height of 0.5 °/s and a total width of a little more than 3° of helm. If yaw rate is maintained within some 40 per cent of the loop height value it has been found possible to control the straight heading by use of small helm only. The use of the computer-type simulator for the prediction of ship behaviour implies the adoption of a suitable mathematical model and the knowledge of a number of coefficients in this model. An alternative technique that simulates full-scale steering by controlled free-sailing ship models is still in use. Mostly the steering has been exercised by manual operation of the controls, and it has been claimed that at least comparative results should then be valid. It is likely that the truth of this statement depends on the actual speed and size (and time constants) of the prototype ship as well as of the model scale ratio used. 809 Norrbtn 7 | 7 H ¥ é z : At | 3 fg 0,01 °/5 40,002 %st ner Full scale single-channel record (Helmsman: Author) BESNE 2 0 cel tment poe \ \ p 4 Ip | Af » VAY Simulator records (Helmsman: Author) Fig. 1. Manual steering of an unstable 230 000 tdw tanker in quartering sea. Prototype and simulator records. 810 Shtp Maneuvering in Deep and Confined Waters Time is scaled as square root of length. Human response time may be "scaled" within certain limits only. The w(6)-diagrams of Fig. 2 demonstrate results of simulated steering of the tanker prototype already referred to, as well as of her fictive models of four different sizes. (Note that curves run anti-clockwise with time.) The smallest "model" is in scale 1:100, i.e. it has a length of 3.1m, which should permit free-sailing tests in several in-door facilities. The two helmsmen, which each one seem to represent one kind of steering philosophy and who were allowed a short training period in each case, both failed to maintain the proper control of the two smaller "models." The control of a ship on a straight course is governed mainly by the effective inertia, by the yaw damping moment, by the rudder force available, and by the time this force is applied. A mathemati- cal model intended for studies of manual or automatic steering may therefore be quite simple; in contrast to the test basin model it may include proper corrections for the large scale effects often present in rudder force data. (Cf. Section VII.) Figure 3 repeats the original simulator b(5)-curves from real-time straight running, recorded by use of the "complete" mathematical model, but it also presents results from tests with a linear model as well as with a model, which contains no other hydro- dynamic contributions than those in lateral added inertias and rudder forces. No major differences were experienced in using these three models of increasing simplicity. On Manoeuvring in Confined Waters Manoeuvring, involving yaw rates and drift velocities, which are not small compared to the forward speed, demands a mathemati- cal model of considerable complexity. A useful presentation of non- linear characteristics has been given by Mandel, [ 2]. One particular non-linear model designed to include manoeuvres in confined waters will be more fully discussed in subsequent Sections of this paper. The average depth of the oceans is some 3800 m. But ocean voyages start and terminate at ports behind the shallow waters of the inner continental shelves. Additional confinements are presented by many of the important gateways of world trading, such as the Straits of Dover and Malacca, the Panama Canal, and the Suez Canal now closed. The maximum draughts of "large" ships have always been limited by bottom depths of docks and harbours, and of canals and canal locks. With few exceptions the requirements placed on under- keel clearances ~ by ship owners or by authorities — have been chosen solely with a view to prevent actual ship grounding or exces- sive canal bed erosions. Thus the Suez Canal Authorities accepted 811 Norrbtn Helmsman W Helmsman N Fig. 2. Simulator tests of manual steering of an unstable 230 000 tdw tanker in real and speeded up time. Yaw rate versus helm angle. (Numbers along curves indicate minutes in real time.) 812 Shtp Maneuvering tn Deep and Confined Waters “Complete” model: Linear model: Helmsman N Helmsman W Fig. 3. Simulator tests of manual steering of an unstable 230 000 tdw tanker using alternative mathematical models. Yaw rate versus helm angle. 813 Norrbtin a nominal blockage ratio of 1:4 for ships in northbound transit at a maximum speed of 13 kilometres per hour, corresponding to a mean back-flow velocity of some 1.5 m/s. Today new limits are imposed by the depths of ocean sills as well as by the depths and widths of open sea port approaches. The potential dangers of a large oil tanker navigating in such waters under, say, the influence of an unexpected change of cross current must not be denied. Whatever nautical experience the master or pilot may possess, he is still in need of actual data and of means to convert this information to helm and engine orders. Automatic systems on a predictor basis are likely to appear in a near future, [3]. In the planning for dredged entrance channels and harbour turning basins the maneuvering properties of the ships must no longer be overlooked. The upper drawing of Fig. 4, reproduced from Ref. [4], shows part of the plan view and a typical section of the buoyed channel for 200 000 tdw tankers unloading at a new oil ter- minal. Before entering the 90° starboard turn the speed is brought down to less than 2 knots, and the tanker then proceeds under slow acceleration by own power. Braking tugs are used on quarters, and forward tugs assist in the S-bend. The lower diagram of Fig. 4 is taken from SSPA records of yaw rates in the passage; the initial curvature corresponds to r' = 0.175, and the maximum rate of change of angular velocity is of the order of 0.0005 o/s? at a forward speed of 2.3 knots. In general the lateral forces on the ship will all increase as water depth turns smaller, and the dynamical stability is also likely to increase. From extensive measurements by Fujino it appears, however, that the picture is not so simple, and that for some ships there may be a "dangerous" range of depth-to-draught ratios, in which the dynamic stability gets lost, [5]. Recent model tests indicate that the large-value non-linearities, such as the lateral cross-flow drag at high values of drift, do increase even more than the linear contributions governing the inherent stability conditions. Whereas these non-linearities may be omitted in the mathematical model of the ship in a canal the bank effects here intro- duce destabilizing forces, that are again highly non-linear. The effects of well-known forces experienced by a ship sailing parallel to the bank of a canal are clearly apparent in the record from a Suez Canal transit here reproduced in Fig. 5, [6]. (The positions in the canal as well as the width between beach lines were derived from triangulation by use of two simple sighting instruments designed for the purpose.) Upon approach to the Km 57 bend the ship is slightly to port of the canal centre line. The pilot orders port helm for two minutes, by which the ship is pushed away from the near bank and the desired port turn is also initiated. Back on centre line the ship mainly turns with the canal. In spite of a starboard checking rudder Ship Maneuvering in Deep and Confined Waters Plan and typical section of dredged channel comer a pave Ser Gine ao SS Jee te eae a a Che H i Baad eee ! ee - c=] : Bape rears oa ta geen bibs ne Cebets se See Seep cean ae ea th i N ; : H N Le : Si ROG! SL MMe Le ek SO SER ON AE Mest OAT Ree cee oeR Oe” Oe Wm) GD LEG eee t an ae ee Er Oe a ee EE ie ae ee ee eee Gy i j : min ce Ag ‘0,01 %s ies os 4 Ship Speed 2,3 Knots Pease es. mS oh Q}. : . So : ae 1 “|: i sine 3 c 0 eee 2 x ee vb 5 6 7 8 9 0 1 3 4 Part of yaw rate record in transit Fig. 4. Example of yaw rates recorded on 210 000 tdw tanker in harbour approach, 815 : ot lirrited to mancewres initieted by pilot orciers. ugh Suez Canal on 36' draught. ds in KM 57 Bend. uthbound thro Fig. 5. 60 000 tdw tanker so Abstract of Recor Shtp Maneuvering tn Deep and Confined Waters she again moves closer to the port bank, and again port rudder has to be applied, etc. So far analytical studies of ships moving in canals have been dealing with straight running. It is believed that the mathematical model which is presented here may also be extended to the case of slowly widening and bending canals. II SYMBOLS AND UNITS, ETC. When applicable the symbols and abbreviations here used have been chosen in accordance with the ITTC recommendations, [7]. Some new symbols are introduced to define the position and orientation of a ship in confined waters. (See also Section X.) The system of axes fixed in space is 0)x9y9z,, that fixed in the body or ship is Oxyz. The point of reference O lies at distance Lpp/2 forward of A.P. of the ship. (Cf. Fig. 6 and Section IV.) Zo Fig. 6. Inertia frame and body axes, etc. 817 Norrbtin Dimensional numbers are given in metric units unless other- wise stated. Generally coefficients and relations are expressed in non-dimensional forms. In addition to the non-dimensionalizing "prime" system usually adopted use is here made of a new "bis" system, further presented in Section III. A dot above a variable stands for a derivation with respect to time. Partial derivatives of forces and moments are designated by the proper subscript attached to the force or moment symbol. TABLE I Symbol Definition ca deaesk Remarks A, Channel section area ie A Section area of hull isp Ajj Added mass.) Mt= 1.2.53 9 =1,2:3. M noon i= 4,5,6;)=4,5,6 ML* : 2 b= 1,52;33j = 495.6 ML Ay Added mass in horizontal oscill. inka free surface, neplecting gravity M Ay Added mass in horizontal oscill., unbounded fluid M A, Total proj. area of rudder i Bea Moveable proj. area of rudder ic B Beam of hull hb Cp Cross-flow drag coeff., 3-dim. - D Diameter of propeller ie F Force vector MLT Bae Froude number on depth - Pye v/Vgh Fy Froude number on length - Fat = V/VeL =v" I Moment of inertia Mi Ti Mass product of inertia ML* ee Ke J Propeller advance coefficient - J=u(1-w)MD K Rolling moment about x axis Mir? Kg Propeller torque coeff. - Kg = - Q?/pn*p° K, Propeller thrust coeff. - K,= T?/on*p* 818 Ship Maneuvering tn Deep and Confined Waters Symbol Definition Fy Soe Remarks SE Te a a a ee ere ee L Length of hull L L = Lpp M Pitching moment about y axis Mier M Moment vector is ag N Yawing moment about z axis MI?T* N" = N/mgL Q Torque about propeller shaft ML*T*® Qt= turbine torque R Turning radius L ri= 17R R Resistance MLT~ xX(R)=-R + Hull draught Te EG Propeller thrust MLT™ rs Kinetic energy of liquid ML T~ U Total flow velocity LT” V Velocity of origin of body axes LT y"= V/VeL ve Speed of water current LT"! V; Ship speed over ground LT” W Channel width in general io WwW Bank spacing, half of L 2W = We - Wp X,Y,Z Hydrodynamic forces along body -2 axes MLT y* Y-force due to rudder MLT™® ee Y-force on rudder proper MLT a Depth to top of rudder I a Water surface elevation L ay Slope of lift coefficient curve - b Height of rudder 1 c Flow velocity past rudder ot Cp Cross-flow drag coeff., 2-dim. - g Gap between rudder and hull 18 g Gravity vector LT h Depth of water ae h Vector in general Undef. kj; Coefficients of accession to inertia - rete eee: 819 Norrbtn Physical Symbol Definition Die nnion Remarks ki, Coefficients of accession to inertia - ic= 45.56 k, Corr. factor for rudder inflow - Cf. eq. (7.4) k, Corr. factor for rudder inflow - ee Non-dim. radius of gyration - m Mass of body M rm a n Number of revs. of prop. in unit time a -| - 2) Pressure in general ML FT. : -| _- q Stagnation pressure Mi... al : p.q,r Angular velocity components an Dang Max. radius of equivalent body of revolution L s Lateral thrust factor - Cf. Section VII s Sinkage L t Time T t"=t//L/g t Thrust deduction factor - u,V,;w Components of Vv along body EF axes Le w Wake fraction - X2V 9% Orthogonal coordinates of a right- handed system of body axes cL X92Vo2Z%, Orthogonal coordinates of a right- handed system of space axes (inertia frame) i A Weight displacement MLT* 4A=ppgV, =mg ny Volume displacement L3 Normal approx.! V=WV Vo Volume displacement at rest = A Aspect ratio - A, Aspect ratio of rudder x A, = b®/A, Re Do for rudder + plane wall image - A,= 2h, Bl is ® Velocity potential Ls wi PM y= ®/LV gb 820 Physical Shtp Maneuvering tn Deep and Confined Waters Symbol Definition Miimennioen Remarks ot Angular velocity of ship iT a Angle of attack - B Angle of drift - tanB = - v/u Y Frequency parameter - Y = Vw/g = u"w" Y Coeff. of heading error term in proportional rudder control - "Rudder ratio" 6 Rudder angle (deflection) - 6* Rudder angle ordered by auto pilot - 5e "Effective" rudder angle - Se = 6 for v=r=0 € Phase lead angle - ec Restricted water depth parameter - C70 (eee) n Ship-to-bank distance parameter - n=n, t Np n Bank spacing parameter = 1 =7s - Np m1, Port bank distance parameter - Np = L/(Wp - yo) "1. Starboard bank distance parameter - Ns = L/(W, - Yo) 's] Angle of pitch - y. Body mass density ratio - ik = m/p V0, or norm. surface ships p= l p Mass density of water ML”? o Coeff. of rate of change of heading term in proportional rudder control ae "Rate (time) constant" 9? Prismatic coefficient - > Angle of roll or heel - Wy Angle of yaw, or heading error - -| w Circular frequency T w" = wL/g w! Reduced frequency - wo = wl /V = w/a" 821 Norrbtn Ili, NON-DIMENSIONALIZING BY USE OF THE "BIS" SYSTEM The use of non-dimensional coefficients is accepted in all branches of ship theory, and when motion studies are considered even the variables of the equations are often normalized. Within the field of maneuvering a unit for time is usually the time taken by a body to cover the distance of its own length, and the unit for velocity then is most naturally given by the momentary speed V = (u* + v*)'*. Ifthe body does not move forward this defi- nition is less attractive. In the system just mentioned — which is recommended by ITTC and which in most cases is fully adequate — symbols for non-dimensional quantities usually are indicated by a prime. The unit for length almost always is chosen equal to the length L of the body, and for the common surface ship more specified L=L pp* The unit for mass is mostly taken as the mass of a certain volume ofthe liquid, defined in terms of the body or ship geometry. In the "prime" system already referred to reference volumes are, say, 3L° [8] or $1°T [9], the latter one used with the reference area LT suggested by the wing analogy. In case of bodies, which are supported mainly by buoyancy lift, the main hull contour displacement V, is perhaps the most natural reference volume: if body mass thenis m=p* p* Vo the non-dimensional mass is equal to wp. (When treating heavy aircraft dynamics Glauert chose ppV in place of pV for the mass unit, [10]-) In normal ship dynamics = 1, whereas for heavy torpedoes p=1.3 - 1.5, say; the symbol p will be rejected in certain appli- cations. Here a consistent normalization of motion modes and forces will be made in a new system, the "bis" system, where the unit for mass is m =ppV,, the unit for length is L and the unit for linear acceleration is equal the acceleration of gravity. From this the unit for time is ¥L/g, and it also follows the Table below: ‘Ship Maneuvering in Deep and Confined Waters TABLE II Unit for "bis" system "prime" system P73 Pye mass (M) LPV, 5 L 5 Pat length (L) L Ie t time (T) JL/g L/V L/V linear velocity Vel V Vv linear acceleration g v-/L V7/L angular velocity V¥g/L VE V/L angular acceleration g/L Maye vin force upgV> gv SwLt p «2.3 p 12.2 moment wpgVyL 5 VL Vota Reference area aM ie Ly It will be noted that, in the system suggested, a non-dimen- sional velocity is given by the corresponding Froude number, and that all forces are related to the displacement gravity load A =ppgV, of the body. (Cf. quotients suchas R/A, "resistance per tons of displacement," used in other fields of applied naval architecture. ) It is customary to form a non-dimensional force coefficient by dividing by the product of a stagnation pressure (q = (p/2)V*) and a reference area, and of course the new system will not demand any different rules. In place of the velocity V, however, here is chosen that particular velocity which corresponds to F,, = 1, i.e. the normalized stagnation pressure is q = (p/2)gl. The reference area then is seen to equal p(2V,/L). IV. KINEMATICS IN FIXED AND MOVING SYSTEMS The two orthogonal systems of axes here used, 0,xoyoZp fixed in space — the inertia frame — and Oxyz fixed in the body, are shown in Figs. 6 and 7. The orientation of the body axes may be derived, from an original identification with the inertia frame, by Norrbin ah, = bh + Sdech at oy Lae =F. SR ¥ @ Fig. 7. Graphical deduction of the absolute time derivative of a vector OF, =h defined in the moving body system the successive rotations through the angle of yaw, \w, the angle of pitch, 0, and the angle of roll, $, respectively, defined around the body axes z, y, and x in their progressively changed positions. In_a certain moment of time the,relation between the space vector 0,P = xp and radius vector OP = x,, invariant in the body system, is given by = Ax (4.4) where the orthogonal transformation matrix reads 824 Shtp Maneuvering tn Deep and Confined Waters coswcos® -sinwcos¢$ +cosWsin@sin®é sinysind tcosWsin®cosd> Az=|sinJcos® coswcos$+sinWsin®@ sing -cos sind tsinsin® cos > - sin® cos 8 sind cos 8 cos¢ (4. 2) When applied in opposite direction the transformation is — -l => — ~ — =_ xp=A (Xp - X49 = A(x, - X,9) (4. 3) where A is the transposed matrix, in which rows and columns appear in interchanged positions. In particular, note that the gravity vector ‘Bo = QZ, will be given by the column vector 0 - g-* sin 6 g = A} o} = g cos @ sin (4.4) g g cos 8 cos in the moving system. From Fig. 7 will be seen how the absolute (total) value of the time derivative of any vector h in the body system may be calcu- lated from the relation ap ae —>— — hy. =H tM xh (4.5) The angular velocity vector oy may now be expressed in terms of the Eulerian angles and their time derivatives: For the vector h there is h,=Ah and +AAh (4. 6) and so the column vector 2 is obtained from the carresponding anti- symmetric angular velocity matrix for the product AA, 825 Norrbtn p @- sin @ % = q| = cos @ sin +0 cos (4.7) r icon Gica-sheue ain The angular velocity components resolved in the inertia frame are $=p +q sin @6tan 0 trcos $tan 6 8 q cos $6- r sind (4.8) wW=rcos #sec 8 +q sin sec 8 In the special case of motion in a horizontal plane in absence of rolling and pitching itis Yp=r. In Section VIII an expression will be required for the absolute acceleration of amass element dm at station P(x,y,z) in a body moving through the water with velocity V. From (4.5) then u O -r q vd u -ry tqz ape = lv | +t] r O -p y| = |v trx -pz (4.9) w -q p 0 Zz w -qx tpy and by a repeated application of the transformation formula u - rv t+ qw - (q2 + r?)x + (pq - r)y + (rp + q)z 2 = : = + = 2 + 2 + th ce ae ry ° , (ap)bs v- pwtru- (r° + p*)y + (qr - p)z + (pq + r)x (4,10) w-qut+pyv - (p? +q2)z + (rp - q)x + (qr + py =F ~_,, Inthe presence of a homogeneous steady current Vg aterm Gig is to be added to the right-hand member of Eq. (4.9). In practical applications this current may be assumed to take place in planes parallel to the horizontal, so that Vis fully identified by u and v,°. It is easy to show that the column matrix for the acceleration in (4.10) will remain unchanged. To the surface ship 826 Shtp Maneuvering tn Deep and Confined Waters in horizontal maneuvers, this homogeneous current will only mean a steady shift of the path; alternatively, if a certain straight course is required heading shall compensate for the steady drift. The local finite current, on the other hand, generates varying outer disturbances and shall be handled by other means. V. FLOW PHENOMENA AND FORCES ON A SHIP IN FREE WATER Ideal-fluid Concepts As a source of reference for further discussions this Section recapitulates some of the characteristics of the flow past a ship in free or open water. When a double-body ship form — i.e., a body which is sym- metrical about the xy-plane — moves forward in a large volume of ideal-fluid water the streamlines adjust themselves according to the laws of continuity. The shape of those streamlines remain the same at all speeds. The increase of relative velocity past the wider part of the body corresponds to a back-flow or return flow of the water previously in rest. This disturbance in the potential flow pattern extends far into the fluid volume — a beam-width out from the side of the body the super-velocity still has a value, which is some 80 per cent of that just outside the body. From a resistance point of view the steady forward motion within this ideal homogeneous fluid may lack some realism. Accord- ing to the d'Alembert's Paradox the body will experience no resultant force. However, if the body is to be accelerated the kinetic energy of the fluid must be increased. This energy increase is manifested by a resistance, which for a given geometrical form is proportional to the mass of displaced fluid and the amount of acceleration, i.e. to the product of an "added mass" and the acceleration component in the direction considered. The resultant force is not necessarily orientated in the same direction. In the simple steady motion the total energy certainly will remain constant, but as the body moves forward through virgin fluid there takes place in each transverse section a repeated particle acceleration and transformation of energy. The impuls pressure distribution thus generated will normally be unsymmetric, and soa free moment results on the body. This moment may be expressed by a combination of total-body added mass coefficients. In the general case of a complex motion in the ideal homo- geneous fluid all the forces and moments will then be available in terms of added masses and inertias, according to the theories 827 Norrbin originated by Kirchhoff [11] and Lamb[12]. In spite of the fact that these forces will be modified by the presence of viscosity in the real fluid, and that new forces will also be generated by the viscous effects, these ideal results should be considered when formulating the mathematical model. If U is the velocity vector of the lqcal fluid element the total kinetic energy is given by T, = (p/2) U* dr, or in a potential flow generated by the impuls pressure p® eh £0Py Vi a T,=-£{o% ds (5.1) The integration is to be extended over the total boundary, i.e. over the wetted surface of the body. Let the potential be written in linearized form as $= Gut dv + o,w + &p + O&q + Or (5.2) with respect to the six component body velocities uj. The six coefficients @; then are functions of the body geometry and of the position in relation to the body. The condition for fluid velocity - 8@/8n at the body boundary to equal the body normal velocity may be formulated by use of the directional cosines for the normal inthe Oxyz-system, whereby or 2T =- X:u*- Y.v*- Z.w?- 2¥. vw - 2X.wu - 2X.uv i) Vv w Ww Ww Vv 2 2 2 2 Kp ae M,q - N,r - 2M. qr - 2K; rp - 2K,Pq (5.3) - 2(X,u + Y¥,v + Z.w)r Here there are 21 different added masses (Ajj) or "accelera- tion derivatives." Force derivatives with respect to a linear accelera- tion are of dimension M, and moment derjvatives with respect to an angular acceleration are of dimension MIL’, as are the mass moments of inertia. Cross coupling derivatives such as X,=- Ajq are of dimension ML. 828 Shtp Maneuvering in Deep and Confined Waters If the body has a plane of symmetry there remain 12 different acceleration derivatives , and for a body of revolution generated around the x axis there are only the three derivatives A,,, A,, and Ae The motion of the ideal liquid takes place in response to the force and moment expended by the moving solid. At any time this motion may be considered to have been generated instantaneously from rest by the application of a certain impuls wrench. The rate of change — cf.Eq. (4.5) — of the impulse wrench is equal to the force wrench searched for. Again, the work done by the impulse is equal to the increase of kinetic energy, and as shown by Milne- Thomson [13] the force and moment on the body may therefore be expressed in terms of the kinetic energy of the liquid, F =~ g (7it) -@x 23 oV OV (5.4) M=-$ (23) -dx-¥x4 a aQ oV (The partial derivations shall be considered as gradient operators.) The complete formal expressions for the inertia forces in the ideal fluid have been derived from Eqs. (5.3) and (5.4) by Imlay [14], and they are here given in Eq. (5.5). = ia = Xyu + X,(w + ug) + X.q + Zywq + Zq? + Xv + Xp + Xr 2 =~ Y,vr - Yorp - ¥,r° - Xyur - Yywr + Yyvq + Zspq - (Y; - Z,)qr Yi =Xu+Y,wt Y¥4q +Y¥,v + Yjp + ¥;r + Xwr - Yyvp + X;r? 2 + (X,- Z;)rp - Zp - X,(up- wr) + Xur - Ziwp - ZaPq + Xsqr X,,(u - wq) + Z yw t 24 - X,uq - Xx," ee Zp ge Ste tad Gag) 2 . +Y,rp + Y;p + Xyup + Yywp - Xyq - (X,- Yq)pq - Xqr 829 Norrbtin Kig = X;u + Z,w + Kya - Xywu + X;uq - Yyw? - (¥, - Z:)wq + Mzq? +Y,v + Kp + Kx - (¥, - Z,)vr + Zavp - M,r° - K.rp + X,uv = (Y, = -Z,)vw = (¥.°* Z,)wr - Y, WP Saree oi Z,)v4 + Kpq « (M, - N;)qr + Yyv" Mi4= X,(u + wq) + Z.(w - uq) + M,q - X,(u* =P WA) oss (Z, - X,)wu +Y,v +K,p +M,r + Y,vr - ¥;vp - K,(p® - r*) + (K; - N;)rp - Yyuv t Xww - (X; + Z.)(up - wr) + (x, - Z;)(wp + ur) - M;pq + K,aqr Ni = Xa + Z.w + M,q + Xu + Ywu » (X, - ¥,)uq - Z,wq - Kia" +¥,v + Kp + N.r - Xv" - X.vr - (X,- Y,)vp + M,rp + K,p? - (Xx, - Y,)uv - X,vw + (X, + Y,)up + Y.ur ct Z wp - (X, + ¥;)vq - (K, - M,)pq - K:qr (5.5) Forces in Horizontal Motions - General Especially, for a body which is symmetrical with respect to its xz-plane and which is moving in the extension of its xy-plane, there are Skin Yves Yer, + X(v - ur) + Xr Yig = Yyv + Xjur + ¥;r + X,(a + vr) + X;r* (5. 6) Nig = Nr +(¥, - Xuv + ¥,(v tur) [+ Xu? - v4) + X,(a - vr) By careful application of sound reasoning it is suggested that terms 830 Shtp Maneuvering in Deep and Confined Waters to the right of the bar may be dropped. Terms containing the coef- ficient Y; have been retained in view of the fore-and-aft unsymmetry present particularly in propelled bodies. The coefficients for u in xX, for v in Y, and for r in N — with signs reversed — are the most commonly well-known added masses and added moment of inertia respectively. These inertia coefficients also appear in some of the cross-coupling terms. Lamb's "coefficients of accession to inertia" relate added masses to the mass of the displaced volume V (kjj, i= 1, 2, 3) and added moments of inertia to the proper moments of inertia of the same aac volume (ki; > = 4, 5, 6). Lamb calculated k,,, and kg.= ky, for the sphereoid of any length-to-diameter seein. ais]. For ellipsoids with three unequal axes the six different coefficients were derived by Gurewitsch and Riemann; convenient graphs are included in Ref. [16]. For elongated bodies in general the total added inertias may be calculated from knowledge of two- dimensional section values by strip methods, applying the concept of an equivalent ellipsoid in correcting for three-dimensional end effects. (See further below.) Of special interest in Eq. (5.6) is the coefficient Yy - X, in the "Munk moment," [17]. (See also discussion in [18]. This free broaching moment in the stationary oblique translation within an ideal fluid defines the derivatives . 2v kee- ki patel “I ete NaS ’ B L3 (Lie Nuv= a L (kao - ky) (5.7) (Cf. Table II.) The factor kg 9- kj, may be looked upon as a three- dimensional correction factor. Due to energy losses in the viscous flow of a real fluid past a submerged body the potential flow picture breaks down in the afterbody. In oblique motion there appears a stabilizing viscous side force. So far no theory is available for the calculation of this force, but semi-empirical formulas give reasonable results for con- ventional bodies of revolution. Force measurements on a divided double-body model of a cargo ship form have demonstrated that some de-stabilizing force is still carried on the afterbody but that most of the moment is due to the side force on the forebody, predictable from low-aspect-ratio wing or slender body theories, [18]. Similar measurements on a divided body in a rotating arm shall be encouraged. Contrary to the case of stationary pure trans- lation the pure rotation in an ideal fluid involves non-zero axial and lateral forces. From Eq. (5.6) the side force is given by Xyur, whereas the moment here is Y;ur. For bodies of revolution the distribution of the lateral force may be calculated as shown by Munk 831 Norrbtin [17] whereas strip theory and two-dimensional added mass values may be used for other forms. The magnitude of ideal side force as well as moment are small, however, and ina real fluid the viscous effects are dominating. There are reasons to believe that the main results of the theories for the deeply submerged body will also apply to the case of a surface ship moving in response to control actions at low or moderate forward speeds. Potential flow contribution to damping as well as inertia forces depend on the added mass characteristics of the transverse sections of the hull, and as long as these character- istics are not seriously affected by the presence of the free surface the previous statement comes true. However, an elongated body performing lateral oscillations of finite frequencies will generate a standing wave system close to the body as well as progressive waves, by which energy is dissipated. The hydrodynamic character- istics then are no longer functions ofthe geometry only. At a higher speed or in a seaway displacement and wave interference effects will further violate the simple image conditions. VI. CALCULATIONS AND ESTIMATES OF HULL FORCES On Added Mass in Sway and Added Inertia in Yaw A brief review will here be given of the efforts made to calculate the added mass and inertia of surface ships in lateral motions. Four facts will be in support of this approach: The added masses are mainly free from viscous effects; the added masses appear together with rigid body masses in the equations of motions, and relative errors are reduced — this is especially true in the analytical expression for the dynamic stability lever, which involves only the small X,y the added masses are experimentally available only by use of non-stationary testing techniques, and in many places experimental data must therefore be supplemented with calculated values; the added masses are no unique functions of geometry only, and experiments must be designed to supply the values pertinent to the problems faced. The velocity potential for the two-dimensional flow past a section of a slender body must satisfy the normal velocity condition at the contour boundary as well as the kinematical condition for the relative depression velocity at the free constant-pressure surface, In case of horizontal as well as vertical oscillations this latter linearized conditions is w* + g(9@/8z) = 0 — cf. Lamb[ 12] — or, pe tape the non-dimensional potential @" = @/Lygl and w!"! = L/g, o® iy = n@ u" at | © ® (6.4) Shtp Maneuvering in Deep and Confined Waters For steady horizontal drift at moderate forward speeds one finds a similar condition BOM ona? 27) re e F ° oz" nL aty" (6.2) which shall govern the local accelerations of the flow in the trans- verse plane penetrated by the moving body, [18]. As is seen from the two equations above the vertical velocities at the water surface are zero in the limit of zero frequency or zero drift, and negligible for w << V¥g/L or 8 Fir << 1. The water sur- face may therefore be treated as a rigid wall, in which the underwater hull and streamlines are mirrored, i.e. the image moves in phase with the hull. For high frequencies, where w>>yg/L, the condition at the free surface is ®=0. The water particles move up and down normal to the surface, but no progressive waves are radiated. At the juncture of the horizontally oscillating submerged section contour and the free surface this condition may be realized by the added effect of an image contour, which moves in opposite phase. (Cf. Weinblum [19].) The value of added mass in this case, "neglecting gravity," is smaller than the deeply submerged value by an amount equal to twice the image effect. Added masses Aj for two-dimensional forms oscillating laterally with very low frequencies in a free surface have been cal- culated by Grim [ 20] and by Landweber and Macagno [ 21], using a LAURENT series with odd terms to transform the exterior of a symmetric contour into the exterior of a circle (TEODORSEN map- ping). By retaining the first three terms this transformation yields the well-known two-parameter LEWIS forms [ 22]; other combina- tions of three terms have been studied by Prohaska in connection with the vertical vibrations of ships [23]. Two terms (and one single selectable parameter for the excentricity) define the semi-elliptic contour as that special case with given draught, for which the added mass is a minimum. Landweber and Macagno also made calculations of the added masses Ay in the high-frequency case. For the semi- elliptic contour A, /A; = 4/n*, which result was first found by Lockwood-Taylor, [ 24] é A basic theory for the dependence of the hydrodynamic forces on finite frequencies was developed for the semi-submerged circular cylinder by Ursell, [ 25]. By use of a special set of non-orthogonal harmonic polynomials he found the velocity potential and stream function that satisfied the boundary conditions and represented a diverging wave train at infinity. Based upon similar principles Tasai extended the calculations of added masses (and damping forces) for two-dimensional LEWIS forms to include the total 833 Norrbtin practical range of swaying frequencies, [ 26]. His results are con- densed in a number of convenient tables and diagrams; the added mass values are seen to vary even outside the limit values cor- responding to zero and infinite frequencies. An application of a generalized mapping function technique to ship section forms of arbitrary shape was performed by Porter, who studied the pressure distribution and forces on heaving cylinders, [27]. A way of solving the two-dimensional problem without resort to conformal mapping was developed by Frank, who represented the velocity potential by a distribution of wave sources over the sub- merged part of the contour, now defined by a finite number of off- sets. The varying source strength was determined from an integral equation based on the kinematical boundary condition Vugts [ 29] contributed an extensive experimental and theo- retical study of the hydrodynamic coefficients for pure and coupled swaying, heaving and rolling cylinders, based on the previous works by Ursell, Porter and de Jong, [30]. The coefficients of the THEODORSEN mapping function were defined by a least square fit of the geometry of the cylinder contours to off-sets in 31 points. Of special interest is the good agreement obtained between experiments and the theoretical predictions for the added mass of a typical midship section; the oscillation experiments do not cover the very low fre- quencies, however. Although small the difference in the calculations for the actual section fit and for an approximate LEWIS form was mainly confirmed by the experiments. When used with the strip method the integrated section contri- butions to total added mass and inertia shall be reduced by the appropriate "longitudinal inertia factors" for three-dimensional effects. Following Lewis these factors are usually taken equal to those derived for the prolate sphereoid in a similar mode of motion. This is only an engineering artifice, and it is certainly not correct, say, in case of accelerations in yaw for normal hull forms; thus these correction factors are mostly omitted in hydrodynamic studies of sufficiently slender bodies. In a discussion of the strip theory Tuck [31] included the results of all the added mass and damping coefficients of a surface ship at zero forward speed, calculated by use of Frank's close-fit method with 15 off-sets for each of 23 stations. The total added mass (Aj) and moment of inertia (Az 69 oF of : Series 60 Block , 70 form are here represented by full lines in Fig. 8. Tuck also examined the forward speed corrections to be applieg to, ae a Nee values; thus, especially, he put Agg = Ace = (U 1 jes age or in present notation Ni(u" th .= NI( my es r x" my (6 3) ran oh ea ae u''=0 wile v5 tO i 834 Shtp Maneuvering in Deep and Confined Waters -Y¢" Ne 1.6 Speed corr. to Fry = 0.2 0.10 1.4 Theory ( Tuck ) 1z2 $Hi2.0 J g Theory ( Tuck ) 1.0 Exp. Fry = 0.2 (v Leeuwen ) 0.8 0.05 0.6 Exp. F,,=0.2 (v Leeuwen ) 0.4 0.2 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 WwW w" Fig. 8. Total added mass and added moment of inertia for a Series 60 Block .70 form according to theory and experiments. (Note that the strip theory is not valid for small "reduced frequencies" w' = w'"/u", where it shall be replaced by a slender body theory, [ 31] .) The dotted curves in the diagrams indicate predictions for Fa = ul = 0.20. The Series 60 Block .70 form was subjected to oscillator experiments in lateral modes at several frequencies and forward speeds by van Leeuwen, [32]. The results for the naked hull with rudder at F,, = 0.20 are compared with the predictions from strip theory in Fig. 8. The experimental values fall well below these predictions in the entire range of frequencies, especially in case of the moments in yaw. Although it is inherent in the testing tech- nique that very low frequencies could not be included van Leeuwens results do cover the critical range around w" > u" = 1/4, Consider a surface body in steady motion along the centre- line between two parallel walls width W apart; the diverging bow wave displays an angle to the centreline. If the motion is steady the reflected wave will pass aft of the body only if W/L > tg, regardless of the speed. For the simple travelling pressure point the cusp line angle is equal to 19947 according to the Kelvin theory, whereas-slightly different values may be observed for real ship forms. In case the body is oscillating (as in the simple example may 835 Norrbtn be illustrated by a pulsating source) additional waves will form, which move with speed g/w. At low frequencies these waves move faster than the body, so that the diverging wave front folds forward, and at a certain forward speed there is now a new requirement on basin width to avoid wall interference. For combinations of w and V (or u), in which y = w"u" = 0.272, the opening angle equals B = 90°, and with a further reduction in speed it rapidly reduces again to 55° as y approaches 1/4, This latter condition is associ- ated with a special phenomenon of critical wave damping, as has been shown from theory as well as experiments by Brard, [33]. In model tests with a ship form in lateral oscillations a narrow range of critical frequencies may be identified by a change of the distribution of the hydrodynamic forces, which was clearly demon- strated by van Leeuwen's analysis. Whereas there is a discrepancy in the absolute values of added masses compared in Fig. 8 this discrepancy could be reduced by the application of a three-dimensional corrector; more elaborate theories of forward speed effects for slender bodies at low fre- quencies may further improve the comparison. In the main, there- fore, it may be stated that the variation of added mass with frequency is well documented. Added Masses in Maneuvering Applications The performance problems set up in maneuvering studies usually involve a short-time prediction of a transient response to a control action, and it is therefore convenient to be in the position to use ordinary non-linear differential equations with constant coef- ficients. This, of course, is in contrast to the linearized spectrum approach to the statistical seakeeping problem, which will more readily accept frequency-dependent coefficients. (Frequency- or time-dependence as a result of viscous phenomena will be touched upon below.) Which values of added mass are now to be used in the equations for the manoeuvring ship? It shall be noted that it is hard to judge from the behaviour of a free-sailing ship or ship model which is the correct answer unless special motions are carefully examined. It was early suggested by Weinblum that the low added mass values of the high-frequency approximation should be adequate for use in dealing with problems of directional stability, where starting con- ditions should simulate impulsive motion, [19]. Weinblum also drew attention to Ref. [ 34] , in which Havelock proved that the high- frequency values appeared in horizontal translations with uniform acceleration, regardless of the initial velocity. The impulsive pressures experienced on the tapered bow and stern portions of a slender body in oblique translation may be calcu- 836 Ship Maneuvering tn Deep and Confined Waters lated from the sectional area curve slope and the added mass characteristics of the transverse sections, as shown by Munk [ 17] and experimentally verified for the submerged doublebody ship form in Ref. [18]. The good agreement obtained between total yawing moments measured on this form and its surface ship geosim suggests that the deeply submerged added mass values should apply in this case. It is observed, however, that the water particles in way of a certain section station here are not repeatedly accelerated from rest as is the case when considering the cylindrical part of the hull. Again, if the principle of superposition of damping and inertia com- ponents to the total hydrodynamic force shall be retained for general motions it shall be necessary to adopt the zero-frequency added mass values. An illustrative discussion of added masses with special application to the design and analysis of experiments is due to Motora in Ref. [35]. For the determination of the added mass in sway to be used in the aperiodic equations of a maneuvering ship he recorded the direction of the acceleration imparted to a model by a force suddenly applied in a certain direction. The added mass then could be found from a reasonable estimate of virtual mass in surge. To obtain the added moment of inertia in yaw he recorded the angular acceleration following the impact by a pendulum, the momentum loss of which was also known. He suggested that the inertia values so derived should correspond to the impact or high-frequency type, but the results included from tests with a series of ship models indi- cate sway mass values of the same order as those valid for the deeply submerged case, and moments of inertia in yaw of magnitudes cor- responding to finite frequency surface values. In a recent paper Motora and co-authors [36] compare the results of new experiments and calculations of an "equivalent" added mass for a ship model in a sway motion, which is initiated by a ramp- or step-form impact input of finite duration. The calcu- lations are based on Tasai's section values in the frequency domain [26] , and in agreement with the experiments they confirm that the value of the equivalent added mass defined is a function of impact duration. (Cf. Fig. 9.) If the duration is infinitely small only the equivalent added mass is equal to its high-frequency value, and it becomes larger the longer the duration. Thus these results help to explain the earlier findings for added masses as well as for added moments of inertia, for which latter the impact technique then used did generate rather short input impulses. For application to normal ship maneuvers it may now seem justified to use the low-frequency or deeply submerged values. In recent years it has been widely accepted that the accelera- tion derivatives for a surface ship model may be evaluated from a set of "planar-motion-mechanism" tests in pure sway or yaw. The acceleration amplitudes are varied by an adjustment of oscillator 837 Norrbtin Vv cm/s 10 8 Sway acceleration v(t ) 6 rl Calculations — -—O- -— _ Experiments 2 0 Fig. 9. Motora's equivalent added mass coefficient as defined by acceleration due to step input impact of duration T amplitudes, whereas the frequency is kept as low as running length permits, [32]. A typical reduced frequency w' =w-+ L/V willbe of order 0.5, corresponding to w"=y/u", wo" =o' + u" =0.4. in Fig. 8. The derivatives so obtained may be expected to be somewhat higher than the zero-frequency values. The theoretical zero-frequency added mass values for two- dimensional LEWIS forms as well as for semi-submerged ellipsoids of finite lengths indicate the main dependence on principal geo- metrical characteristics. Especially, for very large length-to- draught ratios the ellipsoid values tend to those of a semi-elliptic cylinder, (1/2)pT?, so that - Yj = T/B. Moreover, it will be seen from [ 24] that for LEWIS eee general - YJ! likewise is rather close to T/B for fullness coefficients corresponding to midship sections. The ellipsoid family has a constant prismatic coefficient g = 2/3. The correction for finite length involves a slight dependence on B/T, as may be seen from Fig. 10. In amore general case 838 Shtp Maneuvering in Deep and Confined Waters a) b) B wre) -Yy el Ye - Nr a il 0.4 < 1 0.3 oe 2 » SLCDAY 77 | ° * yf fos I ° / Zero freq. ofA y 0-2 F for B/Te a & & 31325 20 ., ots Se | § Weep tyes y, > A g | zs ess . oO 2 : A 4 Ellipsoid 2 2 —_ (Theory ) Fe 0.1 g O afellipsoid (Theory) H =| 3 4A Experiments : = Oo Experiments : 4 Motora (B =1.83) of a 4 Motora (¢ = 0.0319) Oe (2565) 5 igh reg OF a 054n Ei 9566) i £ ae O —"— ~ (0:067)) * HyA (Misc. ) * HyA ( Misc.) » vy Leeuwen 0.05 =» vy Leeuwen 0 0.5 1.0 0.1 0.2 0.3 0.4 0.5 Y - a -¥ Fig. 10. Non-dimensional added mass (a) and added moment of inertia (b) from theory and experiments T this correction will also depend on g and on lateral profile, etc. For the inclusion of ship form values in Fig. 10a the diagram is drawn to a base of g - (2T/L). The ordinates are given by the product - YJ (B/T)¢, by which the intercepts on the vertical g - (2T/L) = 1 then corresponds to the infinitely long cylinders. In addition to the ellipsoid and LEWIS cylinder values the diagram include the experimental results by Motora just referred to as well as a number of oscillator results, chiefly from tests run for SSPA inthe HyA PMM. The general character of the three- dimensional corrector is clearly seen, and it is suggested that the diagram may be used for approximate estimates. Non-dimensional added moments of inertia, in terms of 339 Norrbtn product - N}'+ B/T, are displayed in Fig. 10b, compiling experiment data from different sources. Here the two-dimensional LEWIS-form values for high as well as low frequencies are indicated by off-sets to the left in the diagram. Motora's 1960 impact test data, which appear on a level close to the high-frequency prediction, do not indicate any definite dependence on draught-to-length ratio. These data as well as low-frequency PMM data clearly indicate an increase of moment of inertia with reduced fullness. This trend may be expected in view of the deep and narrow bow and stern sections in fine forms — certainly the deeply-submerged ellipsoid is not repre- sentative for a ship form in yaw acceleration. Semi-Empirical Relations for the Four Basic Stability Derivatives Among the large number of first-order force and moment derivatives, that are used to describe the linearized hydrodynamics of the moving hull, only four appear in the analytical criterion for inherent dynamic stability with fixed controls. These are the stability derivatives proper, Yyy, Nyy, Yy, and Nyy» From simple analogy with the zero-aspect-ratio wing theory of Jones [37] they turn out as in Table III. TABLE III Non-dim. system: "Prime" tr Ref. area: Ld Symbol and analogy value: 1 ES ° Mm N RAE es es ee ee ~ ola AW WA NIA Sa NN 5 oa WA WA MA Ala Ny Ny Although this analogy has been verified in principle for a submerged double-body model as well as for the surface model at small Froude numbers [18], it shall not be expected to furnish an adequate nu- merical prediction. It suffices to point on the alternative relation for a closed body in a perfect fluid, given by Eq. (5.7), and to the fact that at least some negative lift is still carried on the run of normal ship-form hull. The bow lift or transverse force is not 840 Ship Maneuvering tn Deep and Confined Waters concentrated to the leading edge as in case of a rectangular wing but distributed over the forebody as an effect of fullness and section shape. Certain modifications to the hull form are known to affect the force derivatives, but do not appear in the simple form parameters of Table III. The fin effect of screw and rudder con- tributes to the derivatives even in the case of vanishing aspect ratio of the hull. From the analysis of a large number of derivatives it has been found that the scatter of data in a plot of, say, May versus the parameter LT /V is somewhat smaller than the scatter of VG on base of aspect ratio 2T/L. The diagrams Fig. 11-12 include stability derivative data for normal ship form models with normal-sized rudders propelled at medium Froude numbers on even keels. The dotted lines shown correspond to the simple wing analogy. The full lines are derived by linear regression and upon the tentative assumption of a - 1:2 relation of moment and force intercepts at zero aspect ratio. Their equations are given as -Yuv -Nuv 1.5 0.6 a 8G V 1.0 0.4 Number of Number of Prop. Rudder : Prop. Rudder 0.5 0.2 1 1 2 0 0.2 0.4 0.6 0.8 LT2/V Fig. 11. Stiffness force and moment derivative data with mean regression line. (Cubic fit to experimental results.) 841 Norrbtn Fig. 12. Rotary force and moment derivative data with mean regression line, (Cubic fit to experimental results.) 2 2 LT gh pes Yiy=- 2.6647 - 0.04= - 1.69- J+ ae - 0.04 nue - 1.01 £2°+0.022-1.28-7-S2+0.02 uv ° LY, e eo 4 VAR ° (6.4) 72, tors DS 3 = 7, iT = vies 1.02 V 0.18 _ 1.29 4 VW 0.18 LT? _ + LT — 1.88 ro aya +0.09 Ni,= - 0.74 + 0.09 and of the data 100, 86, 67 and 79 per cent respectively, appear within + 20 per cent of these mean values. It is obvious that these expressions should be regarded as guide values only, but they may also be used for comparative studies, especially when steering on a straight course is of main concern. In this latter case it is more important to have a proper knowledge of the control derivatives, whereas Eq. (6.4) may furnish adequate estimates for the hull forces; they again shall be corrected for alternative control arrangement alternatives, however. 842 Shtp Maneuvering in Deep and Confined Waters In the next Section an approximate method will be given for finding the control derivatives of a rudder of conventional design. In the hypothetical case of an isolated rudder experiencing the nomi- nal inflow at the stern of the ship it would be easy to calculate its contribution to the total "hull + rudder amidship" derivatives from a knowledge of its control effectiveness. In general the interference effects in behind condition are much more complicated, and in fact the contribution searched for mostly is quite small. Even more, then, the effect of a modification to rudder and control derivatives comes out as a very small change in the stability derivatives. The diagram in Fig. 13 is compiled to correlate the effects of such modi- fications as reported by Eda and Crane [38] and documented in test results available at SSPA. Obviously new experiments are required. Reference shall here also be given to the methods of estimating stability derivatives for surface ships as suggested and successfully tested by Jacobs, [ 39]. The aerodynamic wing analogy should only be valid for small Froude numbers as the limit solution of a general lifting surface integral equation. The effects of finite Froude numbers on the lateral stability derivatives of a thin ship of small draught-to-length ratio was studied by Hu, [40]. According to Hu the force and 0,15 Dav. Lab.Exp. with Series 60 Block.60 Form HyA Exp. with SSPA Twin-Screw/ Twin-Rudder Tanker 0 0,005 0,010 0,015 a(t] Fig. 13. Change of control force derivatives and total force derivatives in sway and yaw with change of relative size of rudder. 843 Norrbtn moment derivatives at F, = 0.1 are increased by some 20 per cent above their zero-speed values, an increase which is not fully realized in model tests. A comparison of the results of this theory with various experiments is presented by Newman, [41]. Newman also points out that the free surface may give rise to a steady side force as a thickness effect, and indicates a solution to that problem. From an inspection of the experimental results for the drift moment, which are the more consistent, a first approximation to the speed dependence is given by (NO) = (NE), + 2 Nie" (6.5) uv'u uuy where 3Ni,,* 1.3(NjV),- This suggests that the zero-speed values will be some 20 per cent lower than those indicated by the mean line of Fig. 11. Viscous Frequency Effects and Small- Value Non-Linearities in Lateral Forces In dealing with the free-surface effects on added masses it was concluded that so far the frequencies involved in manoeuvring motions were to be regarded as low, but that frequency (or memory) effects should be expected to appear in time histories were viscous phenomena were of more concern. The extreme exemplification is furnished by the pitching submarine, the stern planes of which are operating in the downwash behind the bow planes, but in case of submarines as well as normal surface ships also the very stern portion of the hull is exposed to velocities induced by vortices trailing from upstream hull and appendages. Moreover, local separation within the three-dimensional boundary layer flow over the stern directly affects the cross-flow momentum and the impulsive pressures. The forces and moments experienced by the hull in transient motions can then only be calcu- lated by use of convolution integrals over the entire time history, such as derived by Brard in case of a special descriptive model, [ 42]. For application to the mathematical model defined by ordinary differential equations it is again still possible to use frequency dependent coefficients, but unfortunately this frequency dependence is likely to be subjected to scale effects. It is therefore advisable to design experiments for Strouhal numbers or reduced frequencies, which are low enough to produce steady-state values. From a sum- mary of published data in Ref. [41] the limiting frequency will be expected to be somewhere in the region 1 C)) is the mean section drag coefficient the moment-due=to-yaw derivative is sNii =e (c,/32) + (LEST 72% except for a three-dimensional correction factor. (For rough esti- mates ZN], = 0.03 + 2Yiyy » which is verified from experiments.) The force-yaw velocity derivative now is zero to this approximation. Additional effects of skegs and screws contribute to non-zero values 1 of zNj\,, as well as Gare In the general case the local cross-flow resistance is pro- portional to lv + xr|(v + xr), and from symmetry relations the coupling terms are seen to include the derivatives Yjyj,, and Yvypr), etc. (In the cubic fits more often used these couplings are repre- sented by terms in Yyyr and Yy-,, etc. — cf. Abkowitz, [51].) The contribution to Y due to the combined sway and yaw may be written Yj, |v|v(r/v) + Yiote |r|r(v/r), iee., Yjyj, may be looked upon as the derivative of Yj,,, with respect to yaw velocity r per unit v, etc. Forward Speed and Resistance The principal effects of viscous and free-surface phenomena on the resistance to steady forward motion are well-known to naval architects. The correlations of wavemaking and separation with ship geometry are still less satisfactory. However, alternative methods are available for full scale powering predictions from standard series or project model data. As will be further discussed in next Section the adequate synthesis should supply information not only on shaft horse power and r.p.m. but also on hull resistance and wake fraction. Speed trial data therefore require an analysis suchas proposed and used by Lindgren; in case of very large and slow- running ships it may be necessary to include scale effects also in the open- water characteristics of the screw propeller, [52]. 848 Ship Maneuvering tn Deep and Confined Waters A simple guide to ship resistance values may be obtained from the mean line of Fig. 16, which summarizes the results ofa limited number of SSPA trial trip data in terms of the total specific resistance R/A = - >See on basis of Froude number Fy or u", (A similar plot of "total resistance in Ibs to displacement in long tons" versus Taylor speed-length quotient, based on model data, was published by Saunders, [| 53].) The mean line also reflects the general trend of the resistance-speed-dependence for the individual ships in the proximities of their design speeds. 8 ! Black circles relate to full speed, | open circles to lower speeds at same trial f H 0 0.10 0.20 0.30 Fry Fig. 16. Specific resistance figures as evaluated from ship trial data at SSPA. 849 Norrbin A close approximation to a resistance curve with typical humps and hollows requires a multi-term polynomial inu. Estab- lished practice in naval architecture makes use of a single exponen- tial term R, (u/u, )° to characterize the curve in the vicinity of Uy. For large slow-running tankers p* 2 over the entire speed range of interest, which is associated with an almost constant advance ratio for the screw. In confined waters it may be necessary to include a higher-order term; see Section IX. Forward Resistance Due to Lateral Motions When the ship deviates from the true forward motion addi- tional forces appear in axial direction. The main cause of speed loss in a turning motion is due to the axial component of the centri- petal mass force and the hydrodynamic contribution X,,° rv, of second importance is rudder drag and finally the axial force due to oblique-hull lift and wave-making shall be considered. Ideal-flow hydrodynamics identifies X,;, with - Yy, i.e. the mass effect is virtually almost doubled. (Cf. (5.6).) A recent analysis of turning trial data indicates much lower values of X,,. In a steady turn the ship proceeds with her bow pointing inwards, so that (m + X,,)rv = - (m t+ ed igs) -B indicates a force opposed to forward thrust. In running on a straight course the fre- quency of the yawing motion normally is so low that yaw rate and drift angle are in phase during most (but not all) of the time, and so an average parasite resistance results. : Let the response to a sinusoidal motion of the rudder be b= q° sin (wt te,) and B = By°* sin (wt + €g). Averaging over a number of complete periods gives =B = Fao cos (e, - &) (6.7) As the normal merchant ship will pivot round a point closely aft of the bow at low frequencies a rough, estimate of the average product is given by (rv)w.9* - (OP /2)uq. A plane wing in a uniform flow will experience an induced drag as given by Cpj = (1/mA)C,?. According to certain experiments this simple relation may still be used with a correction factor for the twisted flow over a rudder behind a screw. The calculation of rudder lift will be shortly discussed in the next Section; using a nominal aspect ratio equal to twice the geometrical one the correction factor just mentioned will be of the order of 1.2 - 1.4. 850 Ship Maneuvering in Deep and Confined Waters Typical estimates for tankers give as a guide value a relative increase in forward resistance due,to a rudder deflection of 6 radians AX(5)/X(u) = 3.5 or 5° S For small sinusoidal helm angles on a straight course the quasi-stationary application gives AX(6)/X(u) = 1.75 or 26,°, which may be compared with the relation given from propulsion tests with a Mariner ship model in Japan, AT(5)/T = 2+ 6°, [54]. At propeller advance conditions removed from the steady forward motion state the induced rudder drag will be given by + Xcess? |c|c6°, where c=c(u,n) is the effective flow velocity past the rudder and where the coefficient a tens is proportional to the control derivative +Y(,s and to the ratio a Vale In com- puter applications a soft-type limiter will be used fo simulate the conditions for a stalled flow. The viscous lift experienced by a slender ship hull in oblique translation is also accompanied by an induced drag, but the axial component of the resultant force still is expected to be positive. (According to the zero-aspect-ratio wing analogy the resultant force will bisect the angle between the normal to the hull and the normal to the flow. With increasing aspect ratios the resultants move towards the normal to the flow.) The break-down of the ideal flow over the stern causes a change of viscous pressure resistance, however, and wave-making effects will cause a further increase of forward resistance. These effects are here illustrated in Fig. 17 by results of axial force measurements on the surface ship model and the sub- merged double-body form otherwise described in Ref. [18]. From an inspection of these and other surface ship model experiments it is suggested to use a term lvf|ve (6.8) uvvv x(a, v) = 3 xX to represent the axial force due to lateral drift. An approximate value of the derivative is given by §X',, = - 200. 851 Norrbtn Submerged Model 0.06 Surface Model Y'= ? u 9 9 fe) > v2 LT | | | | ° 9 lee | Fry =0.21 Zero - Aspect - Ratio | Wing Theory l 79 ‘ | | | | -0.020 : 0 -0.010 -0.005 e fy2_rT ov . =3 vi Numbers at spots indicate I drift angle A in degrees -3dd-3 -0.02 -6 1g Pees lr (es Change of longitudinal force with hull lift in oblique towing of ship model and submerged double-body geosim 852 Shtp: Maneuvering tn Deep and Confined Waters VII. SPEED AND STEERING CONTROL In general the subject of steering and maneuvering may not be separated from that of propulsive control, and this is specially true in case of ship behaviour at slow speeds. Moreover, in model testing the interactions between hull, propeller, and rudder are likely to cause the main problems of model-to-ship-conversion, including scale effects of a hydrodynamic nature as well as other model effects due to the dynamics of the testing equipment. Large seagoing ships are usually propelled by a single centre line screw, or by wingward twin screws. In case of a tandem contra- rotating propeller arrangement most of the characteristics discussed below may be calculated for an equivalent single propeller. In case of close-shafted twin screws of overlapping or interlocking types the interaction with the rudder should be specially considered. It has been repeatedly proven by handling experience that twin screw ships should be fitted with twin rudders. Recent model tests indicate that with a suitable design of the rudders, including a certain neutral position toe-out, this arrangement may favourably compete with the centre line rudder alternative also from a propulsive performance point of view. In the application of the first-order steering theory, first introduced by Nomoto in 1956 and strictly valid only for inherently stable ships, there appear only two constants: a (desired high) "sain" K, which represents the ratio of rudder turning moment to yaw damping, and a (desired low) "time constant" T, which measures the sluggishness of the ship response, and which repre- sents the ratio of ship inertia to yaw damping. As was subsequently also shown by Nomoto [ 55] the non-dimensional quotient K'/T' turns out to be proportional to the parameter LA,/V for ships with similar stern arrangements. This quotient may therefore be looked upon as a rudder-on-ship effectiveness factor, proportional to the initial yaw acceleration imparted to the ship by a given helm. Some ten years ago maneuvering trials were run with three tankers of the Gotaverken 40 000 tdw series, all similar except for the stern arrangements, [56]. The SSPA analysis of zig-zag tests with respect to the rudder-on-ship effectiveness factor just mentioned offers a unique illustration of the merits of these arrangements, Fig. 18. In particular, note that the two alternatives with rudder behind screw (screws) prove to be equivalent in case of same total area of rudder, and that the use of the larger area of a twin alter- native therefore is especially favourable. A propeller or a rudder, or the combination of a propeller and a rudder, acts as a stabilizing fin as well as a manoeuvring device; the contributions to the fin effect from the propeller and from the rudder-behind-propeller are of equal order. It should be 853 Norrbtin 100 A,/LT Fig. 18. Results from first-order analysis of full-scale zig zag tests with three 40 000 tdw tankers, similar except for stern arrangements. realized that a minor modification to a rudder does not appreciably affect this fin effect or the size of a hysterisis loop in the yaw- velocity-versus-steady-helm diagram of an unstable ship. However, the higher control force per degree of helm then possibly achieved will help in actual directional control, where the history of yaw velocities and helm angles takes place well within the height of the steady-state loop. (See also Section I.) The general propulsion case will be represented by an arrange- ment including one centre line screw and two wing screws, develop- ing thrusts T,, Tg and Tp, respectively. Hull interference generates axial forces t,T,, tT, and tpI,, in the opposite direc- tions, as well as lateral or sideward forces s,° T and s,° T,- In order to adhere to the thrust deduction concept the factors t — which are not necessarily constants — will be taken as positive, so that’the force in postitive’ x’ direction is -t* T. The facter S, will be positive, and sp= - Sg. Roughly sg=tg* cot a, where a is the effective waterline angle in front of the propeller. Normally the lateral forces due to Tg and Tp are in balance, but if Tp# Ts there is a resultant force applied some 0.4L behind the C.G. of the ship. The turning moment thus obtained is much larger than that produced by the axial forces along the shaft lines, [ 57]9 854 Ship Maneuvering in Deep and Confined Waters Ya Bo Splp Ss Ts On > tpTp 3 ts Ts A I. Normal twin - Starboard screw screw propulsion idling Fig. 19. Force fields on twin-screw tanker on straight steady course. The diagrams in Fig. 19 illustrate the symmetric force field around a twin-screw tanker in normal straight course conditions, and the steady state situation when running with starboard propeller idling. The non-symmetric suction force on the port quarter is balanced by the forces due to drift and checking rudders. The drift angle is a fraction of a degree only, and some 90 per cent of the compensation force is due to the rudders, set at some 5 to 7 degrees, With the twin rudder arrangement it should be possible to maintain 75 per cent of the speed in this condition. The induced resistance due to rudder lift would be larger in case of a single rudder between the propellers, but the main cause of speed loss of a ship propelled by one of its screws only is the additional drag from the idling 855 Norrbtin propeller; again, that drag may well be increased by a factor of 3 if the propeller is locked. The characteristics of a propeller in axial open-water flow are usually given by tables or curves of well-known Ky and Kg coefficients versus advance ratio J. In yawed flow the propeller also experiences a lateral force and a (small) pitching moment, [ 58]. In behind conditions the effective angle of drift at the pro- peller still is roughly 2/3 of the nominal local angle, high enough to let the propeller contribute the fin effect already mentioned. (The sidewash behind the propeller then has a further straightening effect on the flow to the rudder.) The effective advance ratio is modified by the effective wake in the factor 1-w; here w will be chosen as for thrust identity. The effective wake, again, is modified by the drift of the ship, being higher for a starboard drift angle than for a port one and a right-handed propeller [59]; here that effect will be taken as of second order. Finally, the vertical asymmetry of the flow field is responsible for the appearance of a lateral force on the propeller of a ship even if drift or yaw are zero. In case of a single screw ship this latter force may be put equal to 3 to 5 per cent of the thrust, [60]. A right-handed screw tends to throw the stern of a loaded ship towards starboard, thus requiring a small starboard helm to be carried on straight course. Other free-running model tests prove that draught conditions may change this picture, and that the ship on light draught may have a tendency to turn to starboard, (oa). The hydrodynamic'thrust T (T,; T,, T,) and torque: © (Q¢, Qs, Qp) — which is negative in case ofa right-handed screw on a driving shaft — will be given as quasi-stationary functions of instantaneous values of forward ship speed, u, and screwr.p.s., n (ng, ng, np). The thrust is a major factor governing the flow velocity past the rudder, and this velocity likewise will be given in terms of u and n. Rudder control derivatives usually are deter- mined from model tests in one or two conditions of screw loading only. In order to find an adequate prediction of full scale control derivatives for the more general propulsion case it is necessary to combine model results with a simple procedure for calculating the total control force due to rudder deflection. From the hydrodynamical point of view the typical all- movable rudder in behind condition is equivalent to a twisted wing on a pointed afterbody. There are a number of additional complica- tions, however: The spanwise velocity distribution is highly non- uniform, the flow along the chord is accelerating or decelerating, the gap between wing and body is within a retarded boundary layer flow and it also varies with the angle of deflection, the boundary conditions at the free surface violate the vertical symmetry aspect even if there is no suction-down of air, the shape of the body stern 856 Shtp Maneuvertng in Deep and Confined Waters is far from say a simple axisymmetric cone. The modern half-spade rudder on a fixed horn (the Mariner-type) is a hybrid of the alle movable and the flapped types, and other common forms all have their special characteristics. The procedure here adopted is not a substitute for the detailed calculations necessary for a certain project design, but it will furnish a good estimate of control forces and make possible the extended use of model results referred to above. The Rudder or "Control" Derivatives It will be assumed that for each rudder configuration may be defined "equivalent" values of rudder area, rudder aspect ratio, rudder angle and rudder advance velocity. A detailed study of the velocity field in the slipstream ofa propelled tanker model and of the pressure distribution over the rectangular rudder fitted to this model was reported by Lotveit, [62]. The distorsion of the spanwise loading due to slip-stream rotation was clearly demonstrated, but the diagrams did not indicate any definite influence of the rudder image in the hull and free surface; the gap distance from top of rudder to stern profile was some 12 per cent of rudder height. Straightforward calculations of rudder lift from known relations of lift curve slope versus geometric aspect ratio and an average advance velocity based on the simple momentum theory proved to give good agreement with the rudder forces measured by a force balance or integrated from the pressure field. Unfortunately in this case no simultaneous measurements were made of the total hull-and-rudder forces, and there is stilla lack of such data for normal surface ship forms. However, already from the old experiments by Baker and Bottomley [ 63] it was seen that the total force due to rudder deflection was increased by some 40 per cent in presence of a deep cruiser stern close above the rudder, and that a third of the total force then was carried by the hull. Let b be the height of the rudder at the stock, or the higher value forward of it, and let a be the depth to top of rudder at the same station. With a projected area A, of the pudder the aspect ratio of rudder + plane image is equal to A = 2b*/A,. The lift curve slope aj is taken from the theoretical curve derived from the Weissinger theory [64], or from empirical curves available. The geometrical aspect ratio usually is of the order of 1.5, i.e. the rudder is not a low-aspect-ratio fin, but it seems still to be possible to make use of the results for wing-body interferences applicable to such fins. In particular, the ratio of the liffon a rigid combination of a wing and a cylindrical central body, Lae, to the lift of the abridged wing alone, Lg, is simply given by 857 Norrbtin (1 + a/atb)*, [65]. Next, for the calculation of the lift carried on the axially oriented body and on the wing deflected to the flow, it is observed that the exact t theory by Mirels [66] may be approximated by 1 = yee 7 LNS= EN. (1 +a/atb). Except for a correction factor the control derivative for the ship will be calculated as a2 oo A 1 Isgls W = eM 4S ' = py nacre Are where Y{ unlike Y. is defined also for zero forward speed. The Pedeon nolt spade or Mariner type rudder has a fixed horn, which divides the upper part of the rudder in ratio A,/(Ay - A,)- The right-hand member of (7.1) may then be multiplied by a factor i - (1/4) (ATR). The effective rudder advance velocity c (squared) is calcu- lated from the mean square velocity of the screw race and an esti- mated mean square velocity past the rudder outside the race. If w is the wake factor as integrated by the propeller (thrust identity) the effective square velocity above ae race in a normal single screw arrangement may be taken as u “(1 - 4 w)?, Inside the race, which in average conditions has a diameter some 10 per cent smaller than the propeller, the ultimate mean square velocity is given by u*(1 - w )*(1 + (8/m) * (K,/J*)), where, for u>0, J by Ky = Ky, * ey M Ky, s Ky J eae Ky, J (7.2) is to be approximated from the open water propeller diagram. Where- as the thrust may be analytically defined for all combinations of u and n — see below — the working conditions of the rudder are known only for a positive thrust, in which case =e 2 2 eve ites ee COT 2Cyy BF Cy, UN F 2Ciniq |n|n F2Cqy (7:3) From an analysis of a large number of control derivative measurements on models it appears that a correction factor of 0.7 - 0.8 shall be applied to Ae 1) when combined with (7.3) to give the force Y(u,n,6) =pV/L> aie - &&. This correction factor is understood to take care of gap ‘effects and non-ideal geometry of the hull + rudder arrangement, etc. The four constants in Eq. (7.3) depend on screw character- istics and wake factors, and they are therefore unique for the model scale. To facilitate a correction for this scale effect in the control derivatives the diagram in Fig. 20 has been compiled, chiefly from Ref. [67] and data available at SSPA. The slope of curves of wake factors against ship or model lengths increases with hull fullness; especially SSPA experience of full scale tanker trials rarely include 858 Ship Maneuvering in Deep and Confined Waters 0,4 .0;3 Fine Forms ee ae 0,1 Hull length L 1 2 5 10 20 50 100 200 m Fig. 20. Scale effects on wake factor w as integrated by propeller in model and full scale. effective wake factors above 0.38. In Fig. 21 the control moment derivative Ng for a 98 000 tdw tanker is presented as a function of forward speed u and shaft speed n, for a 1:70 scale model as well as for the prototype. (Extrapolation to slowly reversed propeller is shown dotted.) In particular it is seen from the diagram that the turning moment from the rudder at self propulsion point of ship is only some 60 per cent of the model test value. During a maneuver the effective change of angle of attack of the rudder is a function of nominal helm deflection 6, drift v, and yaw rate r, and change of screw loading. Again accepting this quasi-stationary model it is E quay 6e= S t+(ky> — +k, St )[8| (7.4) 859 Norrbtn Mm RPM on 20 150 10 15 100 10 5 50 Dae oes O here given as 861 Norrbtn eT" =! - dr"? + Tun +L 37") [nin tL + at eu D el -(/2 +1/2 EW (Q, - Qin=Lg-Q etl 'g - OF er ae me (7.6) ] -l tls 2Qhiu +L Qy,un + 2Qihin [n|n + 20a 3 sy FeV «8 (i-2 wir 1K etc. The steady-state hydrodynamic thrust and torque are given as functions of forward speed, wu, and rate ae revolutions, n, based on open water K. and K, characteristics; and K, are first approximated by square functions of J = apie -w) or 4/J. (Note that a linearization of these characteristics does not result in a linearization of the (u,n)-dependence.) The Nordstrom data [70] may be used when reversing or transient maneuvers are con- sidered. In general it is then necessary to confine the analytical functions to limited ranges of propeller advance coefficients, i.e. to use alternative coefficients as in Eq. (7.2). Harvald has presented useful information on the propulsive factors at arbitrary steady-state advance conditions, [71]. The effects of separating boundary layer flow along the stern of a retarding ship are still less predictable. The added mass and moment of inertia involved in unsteady maneuvering of the propeller are functions of the momentaneous advance coefficients as well as of the rate of change of r.p.m. In small changes from normal propulsive conditions the added inertia is small as blade angles of attack are small. Naval architects often use a value of 30 per cent of rigid screw inertia for the added inertia; although this figure originates from model tests with screws oscil- lating at zero advance coefficient it may still be used as an effective average value during the short reversing stage of an engine maneuver. In fact this stage is dominated by the large control torques and by the way they are used. When simulating maneuvers with diesel-powered ships it shall be observed that normal r.p.m. control is not possible for n less than some 35 - 40 per cent of design shaft speed ny. The torque delivered is here rapidly reduced, mainly due to loss of charge air pressure. (For high r.p.m. Q; is almost zero.) Slow speed maneuvering must be performed by intermittent use of the propeller, which requires repeated starting of the engine. Reversing maneuvers must await drop of speed to some 60 per cent of the full speed value, at which lower speed braking air may be applied. There is alsoa 862 Shtp Maneuvering tn Deep and Confined Waters certain astern r.p.m. which must be attained before fuel may be injected to start engine back. For a discussion of detailed features of diesel maneuvering the reader is referred to a paper by Ritterhoff, [ 72]% The energy-converting efficiency of a turbine wheel has a maximum of some 80 per cent at a certain ratio of blade velocity to nozzle steam velocity, attainable at the design point. Assuming this ratio equal to 0.5, and a parabolic curve of efficiency symmetric to the design point, the following simple formula is obtained for the torque output: Q’ = 2xQk (1 - 2° n/M ) (7.7) Here o- and n, refer to torque and shaft speed at design conditions for full steam inlet xk = 1. The formula furnishes a good approximation also for present multi-staged ship turbines. In practical applica- tions to studies of slow-speed port approach maneuvering it must be realized’ that steam production may then be limited to say kK=0.7. VIII. MODELLING THE DEEP-WATER HORIZONTAL MANEUVER The General Case The ship will be regarded,as a rigid body moving under the_, influence of the gravity force mg and the buoyancy force -p* Vo°g — where Vo is the volume displacement at rest — as well as under that of the external forces, including the control forces applied by use of rudders and thrusters. Before reducing the problem to the normal merchant ship case the more general form of the rigid body dynamics will be included. The centres of mass (G) and buoyancy (B) may be off-set from the origin of the moving system (0), and it is then practical to apply Newton's laws in a summation of the acceleration forces on the mass elements (cf. (4.10) and (4. 4)): dm 0 0 ay x m-pV, 0 0 0 0 dm 0 ay =a ly er 0 m-pV, 0 A JO Only Or-idm. | ba Z 0 0 m-pVo g Z jabs 863 Norrbtn es abs K : —VoraG Po Za) “YG. Plane y M+] mzg-pVoZe 0 -(mxg- pV, Xp) A 0 N -(my,-PV yg) m™mx,-PV>Xg ) g ey Upon summation the coefficient matrices of the acceleration terms, the mass and inertia tensors, expose as me 0 0 m = 0 Myy 0 = dm 0 0 my (8. 2) Ax “Ly “Ly Paepins tao aipeph ara % Sidm “1. “I, L, where the elements are defined by )x.dm s,m x, > ty? + 22 dm = Iyy ), xy dm =1,y Yydm=m+y, >) (2*+x)dm=ly Yyzdm=l, (8. 3) yy z2dm=m-z, >) (x? + y*)dm = 1,, )) 2x dm = 1, Many authors prefer to introduce the virtual masses and moments of inertia into the equations given above. Here the "added" masses will consistently be assigned to the hydrodynamic reaction forces in the right-hand members; in Section Vit was seen that these forces may include other inertia terms otherwise easily overlooked. 864 Ship Maneuvering in Deep and Confined Waters In most practical applications the xz-plane is a plane of symmetry, so that yg= yg=0 and I,yy=0. Except in a few special cases, such as when dealing with hydrofoil crafts, etc. — the dis- cussion of which is outside the scope of this paper — other terms may be safely ignored in view of the smallness of the products of inertia and the perturbation velocities involved. The Merchant Type Displacement Ship In what follows the discussion is restricted to displacement ships, for which m= pV, and V*\V,. Forward speed is always associated with a sinkage and change of trim, most obvious as "squatting" in waters of finite depth, but the manoeuvring dynamics will be sufficiently well described by the equations in four degrees of freedom, i.e. the surge, sway, roll and yaw. Then mfu - rv - ea + zorp{= x my + ru +xgr - zp b= ¥ re LP - lee . mz_(v + ru) = K - mg(Z, - Z5) sin > ep mx,(v + ru) =N Whereas the initial roll as well as the steady outward heel may be appreciable in case of say a highspeed destroyer these angles are also known to be quite insignificant in the tanker case. In steady turning a heel, proportional to - (L/Re) * ie , may produce an effective camber of the waterline flow around a fine hull, but this hardly applies to merchant ship forms. Leaving the roll equation the present deep-water model is given as in Eq. (8.5). It shall be pointed out that the derivative Yy, includes the potential-flow contribution Xj} and the derivative Nj, the potential-flow contribution Y;' . In the forward speed equation X\, is given a value that is smaller than its ideal value equal to - Yi Ones, cae o eee ee 4 (4 - Xiu ie = Xqyt ae ee jot 55 Siu 2 = joy a LL | - t) #(L+X™ yh L(x tox yb +L gt Ext ulylv? vr G jams A Y & 6 uw =| 4 (2 +L | Xess [el cd, 865 Norrbtn F me ; “V2, <2. . 4 24 (i VAS Ey Styy st tes tap Ae Z Yuu tb lon 3/2 -1/2 Ae Ss hee? =| 8S Seon 1 XN ay fide af a aN el = Ye eoltany 1 " rite [ey " ‘ W ‘ Ce rir Wie + YN, lv[ety vy -| a 1 "l W +L + Zig lelede + ket i a : : 5 ‘ -3/2 -I2 : (ko, FN )p = LU (NS - xt)v +L (NE - x8up+L og ee fe rat -2 “8/2 -V2... A nn 12 ee | PDN ay? ie 5g Nu 2S ay Ivlv 1 " Foulleg = A tt ‘ -I S u ‘ #5 Nie IEP PEO Nie tele EEO Ne fa Fy 2 Nt. leleoue is tom" 2 lelcS e ys (8.5) Eq. (8.5) is to be combined with Eqs. (7.3), (7.4) and (7.6). In case of twin-screw ships (7.6) is to be properly modified and terms corresponding to sp°* Tp and s,° Ts are to be introduced in (32:5). Some Elementary Concepts So far as small motions are considered forward speed and r.p.m. remain almost constant and the rudder force and moment may be regarded as functions of nominal helm 6. The yaw-rate/helm relation is given by the transfer function eee | 1+ T3s ——— (8.6) v5 et es | eee and the open loop heading response by Y, = (1/s) - Yys which may be used with Y, from Eq. (7.5) to study the closed-loop system with transfer function F = Y,/(1 + Y,Y,). The static gain and the three time constants in (8.6) are built up from the coefficient of Eq. (8.5). T3 is always positive. The two constants T, and T, are given by the roots of the characteristic equation. If s, = - YT) 5 the root to the right on the real axis, turns positive the ship is inherently unstable. The analytical criterion for dynamic stability suggests the dynamic stability lever 866 Shtp Maneuvertng tn Deep and Confined Waters qe sqws= xg - ue ee Nuv (8 7) ai as ea cio mgs oh uv to be a suitable measure for the degree of stability. In particular it provides a good illustration when studying the effects onto the stability characteristics of changes in the stability derivatives. Most modern large tankers are slightly unstable,or marginal stable, i.e. 1, =1,. For such ships the pivoting point position is given by the simple relation OP 1-y" ss ees. 7 ola eee) uv which may be approximated by OP/L = 0.45 + (1/3)(6,,(B/T) - 2). For a typical tanker this corresponds to OP/L =0.5. (The formula in fact indicates an acceptable value also for the destroyer, about 0.3.) Again, the pivoting point position — or the drift angle B — is a critical parameter to study when entering shallow waters. IX. CONFINED WATER FLOW PHENOMENA AND SOME RESULTS FROM THEORY Mostly on Resistance In his notes for a third volume of "Hydrodynamics in Ship Design" Saunders collected a number of citations, ranging from Scott- Russel to Moody, which all illustrate the classical picture of ship behaviour in confined waters as it has been derived from obser- vations in full scale and in model tests, [| 73]. He also concluded that, by 1960, the ventures and progresses made in analytical studies of ship manoeuvring in shallow waters remained scarce. One exception was offered by the papers by Brard, [74]. The problems of inter- action between meeting or passing ships, or between ships travelling abreast — closely related to the bank effect problem of the single ship — had been dealt with by Weinblum [ 75], Havelock [76], and Silverstein [ 77]. Undoubtedly much more effort had by then been devoted to the changes of frictional and wave resistance of ships in axial motion in confined waters, and an important survey and contribution had been given by Schuster [78]. Ocean-going ships generally move at low speeds in shallow or narrow waterways, and hence the deformation of the wave system is small. According to Schuster the wave resistance is not notably affected by a limited depth for speeds below Ean = 0.7, at which speed the excentricity of the orbital ellipse corresponds to a diameter 867 Norrbin difference of about 5 per cent. In case of a bottom depth of 15m this again corresponds to a ship speed of 16 knots. In Ref. [79] Weinblum demonstrated that the wavemaking in a canal is a complicated function of speed, depth and width. In general it is therefore not possible to define a single effective length to characterize the canal dimensions in a speed number. However, effective canal speed includes the back-flow, and just as a critical speed in shallow water is defined by the speed of the solitary wave, Vgh, experimental evidence advocates a critical speed in a certain canal corresponding to a certain Boussinesque number B= Fy,y(h/W) +1. (Here W is equal to half the mean width of the section.) For a rectangular section Muller proved that the maximum wave resistance occurred at Fp, = (2(h/W) + 1)-V2 [80]. In a canal 15 m deep and 120 m wide this corresponds to Fy, = 0.81. Again, let it be assumed that a significant change of the wave resistance due to the confinements will be found only at a speed equal to or higher than 70 per cent of this critical speed: this now gives a speed of about 13 knots, much to high to be experienced in canal transits involving normal blockage ratios. It may be concluded that the additional resistance terms to appear in the speed equation normally need not to account for the oscillatory wave-making com- ponents. Reference shall here be given to recent studies of the un- steady flow conditions existing within a critical speed range fora ship in a canal; this range tends to zero when the width of the canal tends to infinity, [81, 82]. At sub-critical speeds the wave-making itself may influence the lateral force and moment on a ship moving along a bank, as shown by Silverstein, [77]. In case of the low Froude numbers met with in practice also these effects may probably be ignored, and the water surface may thus be treated as a solid wall. At F,, = 0.078 or F,, = 0.32, realized for a 98 000 tdw tanker proceeding ata speed of 14 km/h through the Suez canal, the longitudinal waves will have. a,length.of some:,10.m,, i,.e.; only,4 per cent of the length ofthe ship. The back-flow producing an increase of frictional resistance will also produce an increase of sinkage, and in case of small bed clearances this will of course indirectly affect the lateral forces sensitive to the clearance. These secondary effects must be born in mind when comparing predictions from theory with results from force measurements on models, which are free to heave and trim. In the normal evaluation and presentation of such measurements, however, it will be considered more practical always to use the nominal under-keel clearance. The viscous resistance, including frictional as well as viscous pressure resistance, may be calculated accepting a plate 868 Ship Maneuvering tin Deep and Confined Waters friction line and a form factor, characteristic for the super- velocities arene the hull. This resistance now may be written [ X"lwehza= 2 Xyyu*, where u is the forward speed of the ship. In confined waters there are additional supervelocities, the effect of which is equivalent to a back-flow along the hull and waterway bottom, where another boundary layer is generated. The two bound- ary layers will reduce the effective under-keel clearance, which tend to increase the trim by stern. Separation and unsymmetrical eddy-making within the boundary layers may initiate yawing ten- dencies in straight running, or change the behaviour of the ship in manoeuvres. Graff has suggested to consider part of the mean back flow, AUp, to be due to the lateral restriction, and the other part, AU,; to be due to the finite depth, [83]. In normal applications AU is small compared to u, so that XY = Exyur(1 + SOUP ys + SOU) = xt + KY + Ky) (9.4) The effects of a plane bottom at distance h below the ship waterline and a pair of parallel vertical walls, each one at distance W from the ship centreline, are those produced by an infinite array of image bodies with spacings equal to 2h and 2W respectively. At the double-body ship centreline the lateral perturbation velocities cancel whereas the axial components add together. (This simple concept is not valid for W or h small comparedto B or T, in which case additional doublet distributions are required to prevent a deformation of the body contour.) Graff choose to calculate an approximate value of K, for an elliptic cylinder, extending from the surface down to the bottom and having a beam given by the three- dimensional form displacement. (Thus K, is dependent on canal depth, although the final calculation is manely two-dimensional.) For the calculation of K, he used an equivalent spheroid and results for supervelocities Sanies published by Kirch, [84]. His final results are given in graphs and compared with model measurements, which confirm that this method offers acceptable values of resistance allowances for moderate confinements. It is thereby also possible to define a suitable form of resistance derivatives to be evaluated from model experiments from case to case. In particular, a limited re-analysis of some of the data given by Graff indicates that the resistance increase in shallow water will be proportional to the increase of an under-keel clearance parameter € = T/(h-T). Further analysis of the results for sinkage in shallow water according to Tuck's theory are likewise in favour of the use of this parameter. (See below.) In waterways severely restricted in width as well as in depth the increase of resistance is a complex function of blockage conditions. 869 Norrbtin From model tests with a Rhine vessel [85] it appears that the added resistance at a given forward speed may be approximated by an expression of the form AR=a-+ (BT/Wh) tb- (BT/Wh- ¢, or, roughly, AX(u;& n) = 5 Xyuts OF +. Xwurtar 67 (9.2) where 1/2 = L/W is a bank spacing parameter defined from the mean width 2W of the canal cross section. (See Section X.) The higher resistance in confined waterways is associated with a lower propeller efficiency, and the total propulsive efficiency is further reduced by an increase of the thrust deduction. The influence of flow restrictions on thrust deduction and wake factors has also been considered in a paper by Graff, [| 86].In most simu- lator applications this letter influence may be ignored. However, the computed values of r.p.m. and speed attained at a given engine setting should be compared with, say, diagrams compiled by Sjostrom, [87]. Sinkage and Lateral Forces Within the last decade the application of slender-body theory has furnished new understanding and quantitative estimates to the old experience on sinkage and lateral motions in confined waters. Further developments of the theories and more accurate measure- ments are required to bridge a gap still remaining in force pre- dictions. In an essentially forward motion of the ship in shallow water the back-flow is increased all round the frame sections, and according to the first-order theory of Tuck the dynamic pressure is largely constant in the water around a cross section of the hull and over the bottom bed close below it, [88]. Upon assumption of a water depth of same order as the draught, the draught and beam being small compared to the length of ship and waves, and by use of the new technique of "matched asymptotic expansions" Tuck derived formulae for the vertical forces and so also for the sinkage and trim at sub- and super-critical speeds. In case of ships with fore-and-aft symmetry the theory pre- dicts zero trim for subcritical speeds, and zero sinkage for super- critical speeds. For small to moderate Froude numbers based on depth the sinkage varies as speed squared,and, using the under-keel clearance parameter defined here, according to the upper curve of Big 22). 870 Shtp Maneuvering in Deep and Confined Waters Infinite width 0.05 04 0.2 0.5 1.0 2-0 5.0 2-0 (flow af 1.5 A o2.4 6231 12 1+ 78 e-(1-F Ee) 1 Finite width 1-05 1-02 0.2 05 1-0 2-0 wee Af 2b pqs? 12 aw RAT = UF) Fig. 22. Sinkage in shallow water of infinite and finite width, recalculated from Tuck's results. 871 Norrbtin In Ref. [89] Tuck has extended the theory to canals of finite width, in which the ratio of sinkage into the water (or trim) in the canal to the sinkage (or trim) in shallow water is given by a unique curve on basis of a simple width-and-speed parameter. Replotting this curve as in the lower diagram of Fig. 22 Tuck's results are shown to yield a square dependence on the bank-spacing parameter 7 when FA, << 1. In canals presenting higher blockage the total sinkage or "squat" is dominated by the contribution from water level lowering as a consequence of flow continuity. From the Bernoullie and conti- nuity equations an approximate relation for the hydrostatic ship sinkage in terms of ship lengths is given by A 2 Tey x° Fa Q A L Wp ; 2 Here g and aw are the prismatic and waterline-area coefficients of the ship. Other methods of the practical calculation of squat are discussed in Ref. [ 90]. At low speeds wave making is concentrated to bow and stern of the ship, where changes of the local velocities do not influence the blockage conditions, and it shall be possible to calculate the forces on the ship without regard to wave making. The absolute speed still is a parameter, as it is seen to affect the hydraulic as well as the dynamic squat in a canal. Kan and Hanaoka first presented low-aspect-ratio wing results for the calculation of transverse forces and moments ona ship in oblique or turning motions in shallow water , [91]. As the theory predicts the same correction factor to be applied to all deep- water values it seems to be essentially a two-dimensional theory as it is in deep water. Newman studied the same problem by use of the method of matched asymptotic expansions and by the assumption of a three-dimensional flow, differently orientated close to the body and close to the bottom (and upper image wall), [92]. His results bear out the effects of finite length, most obvious in case of moments due to yaw acceleration. Newman considers the inner flow to be a two-dimensional cross-flow of reduced velocity, at each section depending ona blockage parameter in the velocity potential. The outer solution assumes flow to take place in planes parallel to the bottom wall at nominal transverse velocity as the body is reduced to a cut normal to the flow, this being physically similar to the flow past a porous plate. The results as applied to forces on a wing of low aspect ratio (or to a ship) are given in a simple diagram in [92], and here 872 Shtp Maneuvering in Deep and Confined Waters they are used for comparisons with ship model values in Figs. 33 and 34. (A limited comparison of sway force and moment derivatives derived for the SSPA tanker model was included in [92]. A small adjustment of model force derivative appears in the present compari- son, due to modified assumptions for non-linear viscous cross-flow contribution; cf. Section X.) The lateral forces acting on a body of revolution in axial motion in close presence of a vertical wall have also been studied by Newman, [93]. The source distribution inside the body is mirrored in the wall, and in addition the calculations require the original distribution to be off-set towards the wall. This three- dimensional source distribution defines the velocity potential and so the forces may be found by use of the Lagally theorem. As expected from experience and approximate image theories for bodies not close to the wall there is an attraction towards the wall, increasing monotonically up to a finite value of body-and-wall con- tact. It is concluded that for geometrically related bodies with same sectional-area distribution the suction force will be inversely proportional to the length, whereas the yawing moment will be inde- pendent of length variation. The results also indicate that there will be a bow-away-from-wall moment for bodies with a stern, which is blunt compared to the bow, and vice versa. In Fig. 23 calculations by Newman's method are compared with the results of force measurements on a tanker model towed along the vertical wall of a ship model basin. (Cf. Section X,) Basin depth was equal to 0.29 Lp», total basin width equal to 2.7+* Lj. The diagram is plotted on ratio of wall distance to maximum radius of equivalent body of revolution, defined by length and displacement of model hull t image. The better agreement is obtained for that equivalent body, which also has the same sectional area curve, but even then the experimental results are some 25 per cent in excess of the prediction. At larger separations the differ - ence is still larger. Comparative calculations using Silversteins "not-too-near-wall" results for an equivalent ovoid [77], are included in the diagram; in this case the prediction is better for larger separations, but in all much too high. As long as the body is not too close to wall contact the Newman theory gives a linear dependence for the lateral force on ratio of body radius to centre-line wall distance, i.e. it is propor- tional to nN, or 1p, defined for starboard or port wall distances in next Section. This linear dependence suggests that the lateral force on the ship between two parallel vertical walls may be obtained by adding the effects from each one, which idea may also be supported by the new presentation of old DTMB data [94, 95] given in Fig. 24. The diagram includes force and moment measurements on a twin- screw tanker model in several canal sections of simple rectangular form. 873 Norrbin 0.5 = u"2 "' Equivalent '' ovoid ( Silverstein theory ) 0.2 0.1 0.05 Tanker model (Spots from tests : at Fri= 0.078 ) Equivalent spheroid (Newman theory ) 0.02 e Body of revolution with tanker hull + image section area curve (Newman theory ) 0.01 FF 1 2 5 10 15 -Yp/ Tome Fig. 23. Lateral force on a body moving parallel to a vertical wall. Measurements on propelled tanker model and theoretical results for bodies of revolution. 874 Shtp Maneuvering tn Deep and Confined Waters 6 uN \ < ‘ 1.0 se ~~ SS S ‘ xX ~~ . L=720.6 feet ate & L=720-6 feet \ a © 2W= 268 feet © 2W= 268 feet rN 500 ‘SY 4 500 ae N ia) 770 0 Fa eat. 10 0 5 nn, 10 Fig. 24. Asymmetric forces and-moments on a twin-screw tanker model moving parallel to the vertical walls of canals of differing widths and depths (From DTMB test data) The theoretical results for bow-away-from-wall moments are somewhat modified in practice, where bow wave and screw action contribute to make the tendency felt in ships of all types. Thus, in general a ship that moves off the centre-line of a canal must use helm towards the near wall and it takes up a small bow- from-wall equilibrium angle. Typical values derived from [ 94] for the 721' tanker off-set 50' from centre-line of the 500' X 45' section are 15° helm and 1° drift. The motion in shallow port approaches may involve much larger drift angles, and the behaviour of the ship is markedly affected by the increase of lateral cross flow resistance due to under-keel blockage. The diagram in Fig. 25 is compiled from shallow water test data in Ref. [ 49], and from Japanese data in Ref. [ 96], which also include measurements in presence of a wall. Again the parameters ¢ and n are used for the presentation. For moderate € cross-flow drag increases in proportion to €, just as the linear force derivatives, butthe dependence on 1 is of higher order, The cross-coupling between § and ) may probably be ignored in practical applications. 875 Norrbtin AC, (Co)jan20 —O— Tujizal. ,/96/ ——-=—= US Navy Tests, /49/ nle-j Fig. 25. Ship model cross flow drag coefficients as influenced by change of depth and presence of vertical wall X. FORMAL REPRESENTATION OF CONFINEMENT EFFECTS Waterway Description The uniform straight canal with a rectangular section is the most simple case of a waterway confined in depth and width, but even there several parameters are required to characterize the flow phenomena taking place. It was seen in the last Section that the wall distance parameter n and the under-keel clearance param- eter € both were useful tools for the description of certain effects. Their first merit, of course, is due to the zero values defined in unrestricted deep water. Figs. 26 and 27 show a more general section of a canal. Such a canal is usually described by its mean depth between the bed lines, its widths at bed and beach lines, and its cross section area, related to the midship section of a transiting ship by the blockage ratio. The position of the ship in the canal is mostly given by the off-centre distance, and by the angle to the canal centre-line. Here approximate expressions involving the new parameters only will be given for the main geometric characteristics. 876 Shtp Maneuvering in Deep and Confined Waters / Vap w I Esa 5 on & tx 4 Wp.—yo Ws-yo Yo Fig. 26. Ship moving parallel to walls in a straight canal The depth h is considered constant between the bed lines. The mean width 2W is defined as the quotient between cross section area A, and depth h. The ratio 2W/B is a better param- eter for width-to-beam relations the more shallow is the canal. For use with theoretical results for thin ships the width parameter will here not be related to beam but to ship length L. As seen from Fig. 27 the bank and ship positions may be given by coordinates normal to a datum line essentially parallel to the main direction of the canal. The orientation of the ship is given by the heading angle J, measured from the same datum line. The basic geometric parameters are defined as Under-keel clearance parameter 6 = T/(h - T) Port bank distance parameter Np = L/(Wp - Yo) (10.1) St'bd bank distance parameter Ns = L/(W, - Yo) 877 Norrbin — Datum Line Fig. 27. Ship moving in a canal of slowly changing form Note that W. > : Pa Wes so that 7,>0 and TN) < 0. Itis also con- venient to introduce 1 = Ns + Np dom n= Ms - Np 878 Shtp Maneuvering tin Deep and Confined Waters The mean width of the canal at the station considered is = _ e n : ZW = Wy - We = aap i (10.3) where the parameter ratio in the right member is constant for all lateral positions of the ship in the cross section. Expecially, when ship on centre-line so that 1, = - Np = "/2, there is L/2W = 7/4. In fact 7/4 alone is an acceptable approximation to the "ship length-to-canal width" parameter ratio also at ship positions slightly off-set from the centre-line: with yo = W/4 7/4 over-estimates the ratio L/2W =7,/4 with less than 7 per cent. As a consequence } and € may be used to define an approximate blockage ratio BT 1 B 7 10 =—_- ant —_— _ °? L 4 2Wh Zan ae ae oa 4 ( For small ¢ the blockage ratio is proportional to 7, for large € to nm alone. Force Representation The asymmetric forces appearing in presence of a single wall or in a canal are highly increased by an increase of the under-keel clearance parameter, and the general model will include complex couplings. If a single canal depth is studied on basis of special model tests it is of course possible to express the wall effect forces in terms of Np and 1, only. Although the geometry of the inflow to the propeller may be modified in confined water it is assumed that the control derivatives remain unchanged and that changes of rudder forces are due to changes of screw loading only. When suitable theoretical and experimental information becomes available it shall be possible to include the effects of ship motions towards the wall and of the angular orientation along it. At present solutions to the problem of motions oblique to a wall seem to be known only for elementary singularities such as circular cylinders and spheres, [97]. In particular, these results give a re- pulsion by the wall on the body moving toward or away from it, but, again of course, an attraction on the body moving parallel to it. For the present investigation it shall be assumed that the effects of the walls on a ship moving not to close to them will be approximated by the quasi-steady asymmetry, and that the added masses may be taken as those derived for low-frequency oscillations in the centre of the confinement. In the previous Section was shown that the attraction force 879 Norrbtin on the ship in motion parallel to a wall is essentially inversely pro- portional to the separating distance, i.e. Y(u,n,) = Zz uinee u"N"> and that the effects of two walls may be approximated by super- position. Thus for Y(u,Ns»Np) 2 2 tuapel tT t 2 Yuunp Np = 2 Yyugh (10. 5) where Yuyun= Yuuns= Yuunp* As the ship moves closer to one of the walls, or as the walls are closer, this expression shall be completed by terms in ng and Tio | Tipe or alternatively, in nn. The effect of a limited bottom depth is included by additional terms in n& and nn&. The forces due to steady sway and yaw are assumed to be increased in proportion to uvn and urn, andto uvnt and urnG respectively. The dependence of, added inertias on the, confinements are represented by terms in vt and vn rt and mt, all so far evaluated from the results published by Fujino, [5]. XI. MODEL TESTS Test Program and Model Five years ago an experiment program was designed for a tanker model with a view to put to test the analytical model set up as well as to obtain basic simulator data for a first canal transit study. Full scale measurements should subsequently be made with the 98 000 tdw prototype in the Suez Canal, but these plans could not be fullfilled, of course. In November 1965 a first series of three component force measurements were ordered to be run witha 1:70 scale model at the VBD Laboratories in Duisburg. The test program included straight- line oblique towing of the propelled model in "deep" and shallow water in the large Shallow Water Tank and rotating arm tests at same depths in the Manoeuvring Tank. It also included straight-line oblique towing of the same propelled model in two Suez-Canal-type sections witha water depth equal to that of the shallow water tests. Most of these tests were run at self-propulsion point of model, determined from straight course speed runs in the waterways studied. All tests were performed at maximum "Suez draught." Resistance and propulsion tests had earlier been completed at SSPA with a 1:35 scale model on several draughts, and ship speed trials were analyzed to support the prediction of full scale screw loadings and control derivatives on model test draught. (Cf. Section VII.) 880 Ship Maneuvering tn Deep and Confined Waters Additional tests in the VBD Shallow Water Tank were ordered in April 1966 to establish near-to-wall stiffness derivatives from straight-line motion close to one of the vertical basin walls in "deep" water. After a series of repeated tests with a modified recording system the full captive model test program was completed in April 1967. The test data are included in reports [98] and tables from VED. The test program is condensed in Table IV. It shall be ob- served that no acceleration derivatives could be obtained from these tests. Most of the force measurements were made at a model speed of 0.465 m/s corresponding to a ship speed of 7.6 knots or 14 km/h. The ship prototype was a single-screw/single- rudder turbin tanker of the Kockums 90 000 - 100 000 tdw series, delivered to the owners in October 1965 for use in the crude oil trade through the Suez Canal on a reduced draught. The main dimensions of ship and 1:70 scale model are given in Table V, and the body plan is shown in Fig. 28. The prototype has a Mariner-type rudder, normal bow and no bilge keels; a few tests were run to investigate the effects of a bulbous bow and of bilge keels of common design. Wl Fig. 28. Model of 98 000 tdw tanker -- body plan and profiles. Model tested on "Suez draught" 881 Norrbtn DIZ Il peues II-SD ofc 00 oS¢F o9F 00 o9F G6°L s 0 : 7S-0F-9S 0- : 8t°¢ = Ee 30 z 62°0 5 Le Ge BESe LE SE 099} i 0086 O0EZ2 Ta 0086 02 OT = 0086 zi 00S = FITZ PTC PZ I I3}JeM IojeM Teue) MOTTEYUS MAOTTIEUS ie Si Sv MS (WI G9T°O = L ‘Ut FI9°E = 44-7) [ePOW Teyxue], LOZ wrerzso01rg isa OSOt Ire“ 0} Te3N MN 00 oS¢F 00 o9F : 0 ¥S “07-9S. 0- = = Le 0 62 °0 64 °O 67°O = 0086 = 0086 5 0086 00St2 = OSOt OSsOT IayeM IoyeM oo1q ay | AVY MA Q fo osuey mh 20 ¢g jo osuey |°u zo Su oxey U/T = ,t fo oduey SUOTJETISA IOJOWUIeIeYg M2/T = $/%U “/ (Z— O)A= MZ ‘UIPTA Yue y WPed An SUIPT Yue, POU M ‘IPT Yue L, Is ‘loJoWwe{Ip utseg y ‘yydap rt903eM :SUOTSUSTUIIP [Opopy uolyeustsep AVM IZIEM SsoTias “AI ATAVL 882 Shtp Maneuvering tn Deep and Confined Waters TABLE V Ship 1:70 Model Length, Lop =e m 253,00 3.614 Beam, B m 38.94 0.556 Design draught, Tow. m he) Oatyc Suez draught (38'), T m 14.558 0.165 Displacement, Suez draught, V m°> 91 933 0.2680 Slenderness ratio, Suez draught, - 5.606 "8 Midship coefficient, Suez draught, - 0.991 p CB forward of L,,/2, x¢@/Lop +0.0185 Long. radius of gyration/Lpp - 0.23 Propeller diameter, D m fk Ge Oe HOal Pitch ratio, P/D = 0.74 Area ratio, Ae ~ O65 Number of blades, z - 5 Rudder area, total, A, m* 64.8 0.01382 Horn area/A, - 0.182 Relative rudder area, A, json t Z 0.0221 Height at stock, b m ete 19) 0.140 883 Norrbin Results for Force and Moment Coefficients Figs. 29 - 32 show plot of force and moment coefficients from tests in deep (free) and shallow water, and the analytical approximation obtained by stepwise regression analysis. Results from the near-to-wall experiments have been given in Fig. 23. In the evaluation it was consistently assumed that the changes of first and second order derivatives due to finite depth could be approximated by terms proportional to ¢. As the tests did include two values of & only, one of which very small, this does not effect the derivatives derived for these two depth conditions, nor the "true- deep-water" values. Further, because of the scatter of experimental data it proved suitable to perform the analysis with an assumed value for the deep- water cross-flow resistance corresponding to Cp = 0.7; cf. Section VI. In agreement with earlier findings the test results indicate a very marked influence of shallow water on the non-linear force contri- butions, and on the lateral force due to yaw in particular. It shall be observed that the analysis involves a change of sign in the first-order rotary force derivative as water depth is reduced. The force and moment derivatives derived from shallow water and canal tests will be presented in next Section. Fig. 29. 98 000 tdw tanker — Force coefficient ¥"(B) Ja"? in deep and shallow water. 884 Shtp Maneuvering tn Deep and Confined Waters 12 Fig. 30. 98 000 tdw tanker — Moment coefficient N"(B)/u in deep and shallow water. “ Note: Rigid mass included Tests at Fry = 0-078 - 0.108 Fig. 31. 98 000 tdw tanker— Total force coefficient Y"(r') /u"® in deep and shallow water. 885 Norrbtn 02 Tests at F nL = 9:078 - 0.108 Wigs 32. ye 000 tdw tanker — Moment coefficient N(x!) fa? in deep and shallow water. XII. RESULTS FOR CONFINEMENT DERIVATIVES Figs. 33 - 38 present available empirical or semi-empirical results for the main lateral hydrodynamic derivatives appearing in the confinement terms of the completed mathematical model. The derivatives Yyyy and Nyy, for the 98 000 tdw tanker have been derived from the near-to-wall tests in deep water, and are not shown here. (Cf. Fig. 23.) In Fig. 33 the shallow water results obtained for the SSPA tanker are compared with the experimental results published by Fujino [5], and with calculations from Newman's theory,[93]. The SSPA analysis is based on a linear dependence of the derivatives on C and the results are given by plots on straight lines, also sug- gested by the theory. Fujino's derivatives are evaluated separately for each depth. In general the theory seems to underestimate the influence of finite depth, especially for the stiffness moment. Increase of added mass and added moment of inertia as obtained from Fujino's experiments and Newman's theory is shown in Fig. 34, again in poor agreement. 886 Ship Maneuvering in Deep and Confined Waters 5 § , 5 —O— SSPA/VBD Tanker BY / OY ii uv | uf as T TYavij=0 | / Yor )fa0 ujino Tanker 7 & Fujino “Mariner” (Black spots for higher speeds) Fig. 33. Stability derivatives as influenced by finite depth -- results for SSPA tanker compared with Fujino tests and Newman theory. 887 Norrbin AYo { Yu)j=0 O . Fujino Tanker & Fujino ‘Mariner’ ( Black spots for higher speeds) <4 _ -— =_—_ _ _ =—_—_— c _Newman_T heory Fig. 34. Increase of linear and rotary acceleration derivatives with increase of parameter € according to Fujino tests and Newman theory. Je) 7 8 vA (Yur), urs} (urls $20 yy iat / 7 7 7 red Fa = 0-0675 7 7, ae A pore a a Z p= 2-00 ye ie PA ! Pig oO Le Say x ee " ~ : ae ee ei Be ai Beer bain ‘ al Fig. 35. Rotary force derivative for tanker as a function of waterway depth and width, replotted from Fujino PMM data 888 Ship Maneuvering in Deep and Confined Waters Fig. 36. Rotary moment derivative for tanker as a function of waterway depth and width, replotted from Fujino PMM data. Fig. 37. Rel. change of lateral acceleration force derivative fora tanker as a function of waterway depth and width, replotted from Fujino PMM data. 889 Norrbin y, Fr = 0-0675 Fig. 38. Rel. change of rotary acceleration moment derivative for a tanker as a function of waterway depth and width, replotted from Fujino PMM data. The diagrams in Fig. 35 - 38 are compiled from Fujino's measurements of rotary and acceleration derivatives in shallow waters and in canals. The dotted curves suggest a linear increase of all these derivatives with ¢€ in unrestricted water, and a more complex dependence of ¢ and y inacanal. (Cf. end of Section X.) XIII. SOME ASPECTS OF SHIP BEHAVIOUR IN CONFINED WATERS Here a few comments will be given on some of the results obtained in a computer and simulator study performed for the 98 000 tdw tanker. The diagrams in Figs. 39 - 45 all include results directly drawn on the analogue computer recorder. The only full scale maneuvering trials with the prototype ship so far available are a 20°/20° zig-zag test and a Dieudonné spiral, both run at full speed on full draught. These results are compared with the computer predictions — or hindcasts — for the ship on Suez draught in Figs. 39 and 40. As the difference in draught is not likely to have a significant influence the agreement is quite good. It shall be observed that the derivatives with respect to 890 Shtp Maneuvering tn Deep and Confined Waters Full scale / (7 =13,45m) / (T =11,58m) 0 1S min 10 Fig. 39. 98 000 tdw tanker zig-zag test in deep water. Comparison of full scale trials and computer prediction. v|r| and |v|r are not derived from measurements with this model but taken from an analysis of rotating arm tests with another tanker form, and the almost exact prediction of overswing angles might be somewhat accidental. The good correlation of speed loss in the zig-zag maneuver is satisfying. The phase difference is likely to be due to the stern position of the ship's pressure-type speed log. The ship (and simulator model) is slightly unstable on straight course in deep water; the total loop width is about 3.5° at slow speed as well as at high. In Fig. 41 is also shown the spiral prediction for shallow water (¢ = 3.37 or h/T =1.3). Here the initial stability is further impaired, whereas the stability ina turn is increased. A major factor governing the dependence of initial stability on water depth is the change of Yy,-- From Fig. 33 was seen that Yure is negative for this model, so that the value of lf = (xg - Ni - Nut c)/i - Yi, - ure 6) may diminish much faster with increasing © than does 1} = (Ny + Nuvt Ae Yuve aa 891 Norrbin Fig. 40. 98 000 tdw tanker — ,(5,)-diagram from spiral tests in deep water, Comparison of full scale trials (x,o) and computer prediction ( ). 892 Shtp Maneuvering in Deep and Confined Waters Fig. 41. 98 000 tdw tanker — Low speed spiral diagram from computer predictions for deep and shallow water A similar trend is not unique, but it shall be observed that it may be necessary to include higher order derivatives in ¢ to account for a finite range of "dangerous depth" as defined by Fujino, [5]. Figure 42 shows predictions for 20° rudder step responses in change of heading, yaw rate, and drift angle. The small drift angle obtained in the shallow water case is associated with the large increase of cross-flow drag. Similar results have earlier been reported by Schmidt-Stiebitz, [99]. From simulator and full scale experience is known that the helmsman may have some difficulties of controlling the ship in maneuvers that involve a change of course in shallow waters. Maneuvers by use of auto pilots are repeatable and well suited for 893 Norrbtin Y ¥% ¥° ~* Q3 3pi2 Y 0,2 20} 8 Pp 0) 104 qo Fig. 42. 98 000 tdw tanker — Computer predictions of 20" rudder step response in deep and shallow water. Approach speed 7.6 knots. Approach Speed 16Knots -5 Auto Pilot Constants:Rudder Ratio 3 Rate Constant135 s Approach Speed 7,6 Knots Fig. 43. 98 009 tdw tanker — Computer predictions of 10° course change manoeuvres by use of auto pilot knob setting. Two speeds in deep water. 894 Ship Maneuvering in Deep and Confined Waters v% Approach Speed 7,6 Knots f 23,367 05 10 0,10 gost * 0 0 14min -G@os -s5 6 Auto Pilot Constants: Rudder Ratio _g Rate Constant 135s Yi-8° 1S Y $=0 Wy. 0 Approach Speed 7,6 Knots IS 0,10 0,05 . 0 0 14 min -0, Fig. 44. 98 000 tdw tanker — Computer predictions of 10° course change maneuvers by use of auto pilot knob setting. Shallow and deep water. es Re aati Vas In Deep Water (} =0) ye between Parallel Walls (7,=8,71) Goin en ee In Shallow Water ( | =3,37,7)=0) In Deep and Wide Water (}=0, 7] =0) 6%: 10¥+135¥* Q0175ye Datum Line S*210¥ + 135Y+0,0175ye 8°P10'¥ +135 40,0175 yo Datum Line (€) In Canal (} =3 37, 4=8,71) Fig. 45. 98 000 tdw tanker — Computer predictions of on-track control by auto pilots in deep and confined waters. Approach speed 7.6 knots. Initial off-set 20 m to starboard. 895 Norrbin comparative studies. The diagrams in Figs. 43 and 44 refer to 10° course change maneuvers predicted for the tanker at two speeds in deep water and at the lower speed in shallow water, all executed using the same normal setting of auto pilot controls. There are several overswings in shallow water, and checking helm is large. The final diagrams in Fig. 45 furnish a condensed illustra- tion to the changing problems of course control in shallow waters and in canals. These problems are also dealt with by Eda and Savitsky [100], and in considerable detail by Fujino, [101]. It is assumed that the ship is moving at low speed on a straight course parallel to the required track (in a buoyed channel, say) but off-set 20 m to the starboard side. A signal proportional to this lateral error, calling for 1 degree rudder per m off track, is fed into the auto pilot. The upper curves for the free water con- ditions demonstrate that the rudder ratio setting must be increased (from normal 3 to say 10) in order to stabilize the ship on the re- quired track. This control works reasonably satisfactory also in shallow water, but it tends to make the ship over-shoot the centre- line in the alternative case between parallel walls in deep water. Obviously the presence of the near wall accelerates the first swing towards and beyond the centre-line. The two lower curves of Fig. 45 relate to the ship ina typical part of the Suez canal. In the shallow water the effect of the near wall is even more pronounced, and the stern of the ship is in danger of hitting the bank. However, by turning down the lateral error knob to zero the auto pilot is made to behave like the ex- perienced helmsman, already referred to in the introduction. Thus, the ship first sheers bow-off the wall before the auto pilot applies a counter-rudder in order to slowly press the ship laterally away from the wall. The ship is seen to be almost steady on to the centre- line within two ship lengths. REFERENCES 1. Norrbin, N. H., and Goransson, S., "The SSPA Steering and Maneuvering Simulator," (in Swedish), SSPA Allm. Rapport No. 28, April 1969. 2. Mandel, Ph., "Ship Maneuvering and Control," in "Principles of Naval Architecture" (revised), New York 1967. 3. van Berlekom, W., "Summary of a Simulator Study of Maneuver- ing Large Tankers in the Approach to a Port at Brofjorden," SSPA PM No. 1643-10, Sept. 1969. (Contract Report) 896 fi. 12. 13. 14. £5. 16, a7. Shtp Maneuvering tin Deep and Confined Waters Bohlin, A., "Problems Arising from the Use of Very Large Ships in Connection with the Alignment and Depth of Approach Channels and of Maneuvering Areas," Paper S. II-3, Proc. XXII©® Congrés International de Navigation, Paris 1969. 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F., "Screw Propeller Characteristics," SSPA Publ. No. 9, Goteborg 1948. Harvald, S. A., "Wake and Thrust Deduction at Extreme Pro- peller Loadings," SSPA Publ. No. 61, Goteborg 1967. Ritterhoff, J., "Beitrag zur Erhohung der Sicherheit von Schiffsantriebsanlagen durch Untersuchung ihres Manoververhaltens," Schiff und Hafen, 22. Jahrg., Heft 3, March 1970. 901 73. (ee hoe Tcye Tales 78. 19. 80. 81. 82. 83. 84. 85% Norrbtn Saunders, H., "Hydrodynamics in Ship Design," Vol. III, Chapter 6, publ. by SNAME, New York 1965. Brard, R., "Maneuvering of Ships in Deep Water, in Shallow Water, and in Canals," Trans. SNAME Vol. 59, 1954. Weinblum, G., "Theoretische Untersuchungen der Stromungsbee- influssung zweier Schiffe aufeinander beim Begegnen und Ueberholen auf tiefem und beschranktem Wasser," Schiffbau 34. Jahrg., 1933% Havelock, T. H., "Wave Resistance — The Mutual Action of Two Bodies," Proc.: Roy. Soce,; Series" A,-Vol. 1553 London 1936. Silverstein, B. L., "Linearized Theory of the Interaction of Ships," Inst. Engng. Research, Report Series 82, Issue 3, Univ. of California, Berkeley 1957. Schuster, S., "Untersuchungen uber Stromungs- und Widerstandsverhaltnis se bei der Fahrt von Schiffen in beschranktem Wasser," Jahrb. STG, Bd. 46, 1952. Weinblum, G., "Wellenwiderstand auf beschranktem Wasser," Jahrbs STGY Bdo39);- 1938. Muller, O., "Aus dem Grenzgebiet des Wasser- und schiffbaulichen Modellversuchswesens," Schiffbau, 36. Jahre.§. LISS, Constantine, T., "On the Movement of Ships in Restricted Waterways," J. Fluid Mechanics, Vol. 9, Part 2, Oct. 1960. Hooft, J. P., "On the Critical Speed Range of Ships in Restricted Waterways," Intern. Shipb. Progress, Vol. 16, No. 177, May 1969. Sturtzel,,W., and Graff, W., "Untersuchungen uber die Zunahme des Zahigkeitswiderstandes auf flachem Wasser," 85. Mitteilung der VBD, Forschungsberichte des Landes Nordrhein- Westfalen Nr. 1777, 1967. Kirsch, M., "Die Erzeugung von Rotationskorpern aus gegebenen Singularitaten-Verteilungen," Schiff und Hafen, Heft 11, 1959) Sturzel, W., and Heuser, H. H., "Widerstands- und Propulsionsmessungen fur den Normalselbstfahrer Typ Gustav Koenigs," Forschungsberichte des Landes Nordrhein- Westfalen, Heft 868, 1960. 902 86. 37. 88. 89. 90. 91. 92. 93% 94. 95, 96. Ue 98. Shtp Maneuvertng tn Deep and Confined Waters Graff, W., "Untersuchungen uber Anderungen von Sog und Nachstrom auf beschrankter Wassertiefe in stehendem und stromenden Wasser," Schiffstechnik, 8. Bd, Heft 44, Nov. 1961. Sjostrom, C. H., "Effect of Shallow Water on Speed and Trim," Naval Engineers Journal, April 1967. Tuck, E. O., "Shallow Water Flows Past Slender Bodies," Journ. Fluid Mechanics, Vol. 26, Part 1, Sept. 1967. Tuck, E. O., "Sinkage and Trim in Shallow Water of Finite Width," Schiffstechnik, 14. Bd, Heft 73, Sept. 1967. Hay, D., "Harbour Entrances, Channels and Turning Basins," The Dock and Harbour Authority, Jan. 1968. Kan, M., and Hanaoka, T., “Analysis for the Effect of Shallow Water upon Turning," (in Japanese), Journ. SNAJ, Vol. 115, 1964, Newman, N. J., "Lateral Motion of a Slender Body Between Two Parallel Walls," Journ. Fluid Mechanics, Vol. 39, Part 1, Oct. 1969. Newman, N. J., "The Force and Moment on a Slender Body of Revolution Moving Near a Wall," DTMB Report 2127, Dec. 1965. Garthune, R. S., Rosenberg, B., Cafiero, D., and Olson, G.uR.,; "The Performance of Model Ships in Restricted Channels in Relation to the Design of a Ship Canal," DTMB Report 601, Aug. 1948. Schoenherr, K. E., "Data for Estimating Bank Suction Effects in Restricted Water and on Merchant Ship Hulls," Proc. First Symposium on Ship Maneuverability, Washington D.C., DTMB Report 1461, Oct. 1960. Tuji, T., Mori, N., and Yamanouchi, Y., "On the Water Force Acting on a Ship in Oblique Flow (Restricted Water Effect) ," Contr. to the 12th ITTC, Rome 1969. Kennard, E. H., "Irrotational Flow of Frictionless Fluids, Mostly of Invariable Density," DTMB Report 2299, Feb. 1967. Nussbaum, W., "Messung von Quer- und Langskraften am Modell eines Grosstankers,' VBD Berichten Nr. 396 I-II and 406, June-Sept. 1966. (Contract Reports.) 903 Norrbtn 99. Schmidt-Stiebitz, H., "Die Manovriereigenshaften der Schiffe in Abhangigkeit von Schiffsform und Fahrwasser," Schiff und Hafen, 16. Jahrg., Heft 2, Feb. 1964. 100. Eda, H., and Savitsky, D., "Experimental and Analytical Studies of Ship Controllability in Canals," Dav. Lab. Techn. Note 809, Sept. 1969. 101. Fujino, M., “Experimental Studies on Ship Maneuverability in Restricted Waters -- Part II," Intern. Shipb. Progr., Vol. 17; No. 186 Feb. 1970. 904 THE SECOND-ORDER THEORY FOR NONSINUSOIDAL OSCILLATIONS OF A CYLINDER IN A FREE SURFACE Choung Mook Lee Naval Ship Research and Development Center Washington,: D.C: ABSTRACT A nonlinear hydrodynamic response resulting from vertical oscillation of a horizontal cylinder in a free surface at the sum of two monochromatic frequencies is investigated. The fluid surrounding the cylinder is assumed incompressible, its motion irrotational and its depth infinite. It is shown for the case of a semi-submerged circular cylinder that when the two frequencies are close to each other the hydrodynamic force associated with the difference of the two frequencies is greater than the steady force. Inthe limit as the two frequencies become equal the above two forces also become equal. It therefore appears reasonable to include the differ- ence-frequency force in the calculation of the maximum steady force when the excitation of a body consists of narrow-band frequencies. I. INTRODUCTION The hydrodynamic problem dealing with a horizontal cylinder undergoing a vertical simple harmonic motion in a free surface has been investigated by many authors. Ursell [ 1949] treated a semi- circular cylinder using the method of multipole expansion and ob- tained the pressure distribution, added mass, and damping of the cylinder. Later Tasai [1959] and Porter [1960] extended Ursell's work to cylinders of ship-like sections using conformal mapping. Frank [ 1967] dealt with the foregoing problem by the Green's func- tion which resulted in a distribution of singularities. Lee [ 1968] , following Porter's work, extended the potential solution to second- 905 Lee order in a perturbation series in the ratio of motion amplitude to half-beam. In the present work this cylinder-oscillation problem is extended to the case where the cylinder is oscillated at the sum of two monochromatic frequencies. In this case the second-order forces acting upon the cylinder include the effects of interactions between two frequencies, in particular, the sum and difference fre- quencies of the basic spectrum. The magnitudes of these second- order hydrodynamic quantities provide a measure of the non- linearity of the frequency response of an inviscid incompressible fluid to a periodic disturbance generated by an oscillating body in a free surface. Hydrodynamic quantities such as added mass and damping obtained from the theory of oscillating cylinders in a free surface with a monochromatic frequency are extensively used in the studies of ship motions. Most of these studies are based on the assumption of linear frequency response of ships to waves. Recently Tasai [1969] and Grim [1969] emphasized the necessity of further investi- gation on nonlinear ship responses to waves. The present investiga- tion is an attempt to provide information on the nonlinear relation between the motions of a body and surrounding fluid. This informa- tion might lead to the study of nonlinear ship motions in waves, perhaps by using the scheme suggested by Hasselmann [ 1966]. The problem to be investigated in this work is the following. An infinitely long horizontal cylinder which is symmetric about its vertical axis is semi-submerged and forced to oscillate vertically at the sum of two monochromatic frequencies. The maximum dis- placement of the cylinder from its mean position is assumed to be small compared to the half-beam of the cylinder. The fluid in which the cylinder oscillates is assumed inviscid, incompressible, and infinitely deep. The motion is assumed to have existed for a period significantly long that the initial transient phenomenon of the response of the fluid has completely decayed. This problem can be formulated as a boundary-value problem for a velocity potential. The kinematic and dynamic conditions to be satisfied on the free surface are non- linear and the position of the free surface is a priori unknown. An exact solution of this problem in a closed form cannot be attained, so an approximate solution based on a linearization of the problem is pursued in this work. The linearization of the problem is carried out by a perturbation expansion of the velocity potential in terms of a small parameter formed by the ratio of the half-beam to a typical displacement amplitude of the cylinder motion. The first-order perturbation potential consists of two potentials, $,(x,y,t) and ,» each of which involves only one of the two fundamental frequencies. The second-order perturbation potential consists of five potentials. Two of them are $, and 4, which are associated respectively with frequencies of twice the fundamental frequencies. Two more are @, and $, which are associated respectively with the sum and the difference of the fundamental frequencies, and the last one, (x,y) 5 is independent of the frequencies and is a steady potential. The 906 Nonstnusotdal Oscillations of a Cylinder in a Free Surface solutions for the first-order potentials were given by Ursell [1949], Tassai [1959], Porter [1960], and Frank [1967] among others. The solutions for three of the second-order potentials, $3, $,, and $,, were given by Parassis [1966] and Lee [1966]. Inthe present investigation, the solutions for the remaining second-order potentials, $5 and ¢¢, will be given. These solutions are based on the method of multipole expan- sions similar to that employed by Lee [1966]. An interesting prob- lem arising from the present work is a surface-wave problem con- cerning a non-decaying pressure distribution on the free surface. The solution of this pressure-distribution problem is shown in detail in Appendix C. The potential $,, associated with the difference fre- quency, is of particular significance in practical problems. $¢¢ is a potential which is slowly-varying in time if two fundamental fre- quencies are close. In the case of bodies with insignificant restoring forces, such as submersibles and floating platforms, any hydrody- namic force, which is constant in time or varies slowly with time, could cause large excursions from the mean positions of such bodies if it acts for alongtime. 4, must be calculated in order to deter- mine this slowly-varying hydrodynamic force. In the present work numerical results obtained from the solu- tion of ¢, are shown. These include the pressure-distribution about a semi-submerged circular cylinder, the hydrodynamic force acting on it, and the outgoing waves. These results are shown with other first- and second-order quantities for comparison purposes. II FORMULATION OF THE PROBLEM A Cartesian coordinate system is used with origin at the interaction of the undisturbed free surface and the vertical line of symmetry of the cylinder. The x-axis is in the undisturbed free surface and the y-axis is directed upward. Any point in the space is described in complex notation by z=xtiy= rel9, (1) The region outside the cylinder and the cylinder boundary is mapped from the region outside a circle inthe €-plane and its circumference by the conformal transformation 00 Z, _ -(2n+1) ae ae: (2) n=O t=Etin=re™, 21, (3) 907 Lee where a and Bone are real constants. Points on the surface of the cylinder at its mean position are given by = Ben Skene A) x a} ro cos gee 3 y) Xene , (4) és a sin (2n+1)@ Yor a}hgsin a - s eT Sa ? n=O 8 where i, is the radius of the reference circle. When the cylinder is at the rest position its half breadth and draft are given respectively by b = x,(A,, 0); (5) aeelyiikly=n/2)| + (6) The forced motion of the body is assumed to be the sum of two vertical simple harmonic motions with different frequencies. The motion of a point fixed in the body is expressed by y(t) = h,(sin o,t + sin ot) (7) where o, is greater than o, and h, represents the amplitude of the individual simple harmonic motions. The fluid is assumed to be incompressible and its motion to be irrotational so the continuity of mass in terms of velocity potential, @(x,y,t) is expressed by 2 92 (soe + gon) ® = V°B = 0. (8) The boundaries of the fluid are the free surface which extends to infinity along both the positive and negative x-axes, the fluid bottom which is at infinity, and the immersed surface of the cylinder. If we let the equation of the free surface be expressed by y= Vixsth, |x| > b, (9) 908 Nonstnusotdal Oscillattons of a Cylinder in a Free Surface the kinematic and dynamic boundary conditions on the free surface can be given respectively by (x, Y(x,t),t) ¥, (x,t) - a! + ¥Y, = 0, (10) and &, (x, ¥(x,t),t) + g¥ +3(O +B) = 0, (11) where a constant atmospheric pressure and an absence of surface tension on the free surface have been assumed. Taking the substantial derivative of Eq. (11) and eliminating Y(x,t) by using Eq. (10), we obtain + O25, + 20,6,6,, + O56,=0. (12) Let the equation of the cylinder surface at its rest position be given by S(x9,Yo) = f(x) - yo = 0 (13) where f(x,) represents an implicit functional relation between x, and y, through the parameters X, and a. Then the equation of the oscillating surface can be written as S(x,,y, + y(t)) = £(x,) f h(sin ot + sin ot) - y= 0. (14) The kinematic condition to be satisfied on the cylinder surface is VO(x,,y, + y(t).t) > n= Vn=- st (15) where n is the unit normal vector on the cylinder surface and points into the fluid and Vn is the normal component of the cylinder- surface velocity. Since eae VS = CE ,-1) 2 Nei: - del S : Eq. (15) becomes 909 ® (xo,y, + y(t), t)f'(x,) - o, mir ho, cos ot + a, cos o,t). (16) In terms of the stream function which is the harmonic conjugate of @ the boundary condition on the body is ay _ 8 _ dy(t) dx op EL Ee where s_ is the arc length of the cylinder contour in the counter- clockwise direction. Thus we have Weagsy, + y(t) t) = - x, SO | To complete the specification of the boundary-value problem, following conditions should be also given; the symmetry of the flow about the y-axis implies that @(x,y,t) = O(-x,y,t), the zero normal component of the fluid velocity on a rigid surface at the infinitely deep horizontal bottom is described by @,(x,-0, t) = 0, and the solution should represent outgoing plane waves as | x | 7160. Ill. PERTURBATION EXPANSION Assume a frequency-response system in which the relation between the input X andthe output Y is given by YaiAx+ Bx? where the input X is given by ae ee and A and B are constants; it can then be shown that the frequency components involved in the output Y are o,, 05, 20,, 205, 0, +O; vg, - 0, anda '"d.c." shift. Therefore, we make a perturbation expansion of the complex velocity potential 910 Wonstnusotdal Osetllations of a Cylinder in a Free Surface H(z st) = @(x,y,t) i i(x,y,t) in terms of a perturbation parameter € = h,/b in the following fashion: = eH! zh <*H'?) rs eH?) fice = ¢(o'" ig” y+ (Bl?) + jy + eH) +) 400, (17) Each velocity potential and stream function given above is further expanded as a" = $(x,y,t) + o(x,y,t) t ot -jo + P(xsy)e 2 ’ =) 9 (x,y)e g!?) -j2o,t -j2o4t = 9,(x,y)e + ye +¢@ -jlo,-o,)t ae ie t % + g(x,y) y= Ulxry.t) + bplx,y,t) -jo,t -jo5t W(xye | + W(x,y)e iu, Y= Ys(x,y.t) +y(x,yot) tU(x,y.t) + olx,yst) + H(x,y) -\lo,+o5)t 5° -j2o,t -j2o,t -j( )t = W,(x,y)e ee We ieee Vie de -j(o,- o)t + Ye mete + W(x, y) etc., where P= Gegt Jeg? v k Ves + jw $(x,y.t) + o(x,y.t) + o(x,. yt) + o(x,y.t) + d(x, y,t) ks’ (17a) (1 7b) (1 7c) (1 7d) for k=1,2,..-,6, and j=v-1. In these expressions, only the real 911 Lee parts are needed, so whenever there appears an expression of the product of two complex functions one of which is a time harmonic involving with j = v-1, it should be understood that the real part of the expression is to be taken. The convergence of these perturbation expansions will not be discussed. As usual it is hoped that the first few terms of the ex- pression would yield an adequate approximation to the exact solution of the complex potential H(x,y,t). The expansion given in Eqs.(17) and the Bernoulli equation (11) suggest that we also assume the expansion e€ ] -j2o\t -j@o5t + e[ ¥,(x)e + Y,e Y(x,t) = €[ ¥, Eyer)" a -jlo,+o,)t -j(o,;-o.5)9 rye | at Yee ee . + Y(x)] + O(e%) (18) where Tp Soe oR) Oke for fe 1G 23 Se oe Substituting these expansions into the Laplace equation and the boun- dary conditions and equating the terms of the same order in € as well as of the same harmonic time dependence, we obtain a set of linear boundary-value problems. In this linearization process the instantaneous boundary of the fluid is expanded in Taylor's series about the undisturbed position of the fluid boundary. The linearized boundary conditions for the functions 9j (j = 1,2,3,4,7) are shown in Appendix A where it is shown that in the limiting case of o, =o, the relations 9, = 92, 93= 94 = p,/2 and Yg= %7 can be established. These identities mean that when o, =o the perturbation expansion given in Eqs.(17) reduces to that for the case of a simple harmonic oscillation which was investigated by Lee [ 1968]. The linearized boundary conditions for the functions ?. and Yg are given next. 3.1 The Boundary-Value Problem for Pa(x,y) gy, is harmonic in y <0 except in the portion occupied by the cylinder at its mean position. On the free surface Oey +0) - Kyo, = h(x) (19) 912 Nonstnusotdal Oscillations of a Cylinder in a Free Surface whe re K,= (co, +o,)"/g: h(x) = - jo, /(28)19,(*, 09, ~ jo5/(28) {99 x.0)(% yy in which 2 jel On Bh ae On the cylinder surface, - K,9))) - 2(9), P25 + Pry Pay)3 = Ki%y) a 2(9,, Pox + PiyPay)s Pia, ! =- 5 4(XoVo)E(X,) - M5) = Mp(X,rY,) where rele. m,(X9, Yo) == Jey {(Pixy (Xo Yo) + Poxy)£ (XQ) ~Piyy - Poyy) ’ or, in terms of stream functions, Pals, We 5s Vo) ea J 2 (Wy (xy, Yo) a Wy) (20) (21) (22) (23) In the far field 9,,—~ 0 as y~-o and 9g, should represent out- going plane waves as flow condition which is expressed by $5 (%»y) = %5(-X,yY)+ 3.2 The Boundary-Value Problem for 9,g(x;y) yg, is harmonic in y< 0 except in the portion occupied by the cylinder at its mean position. On the free surface where K, = (o, - o,) /g; 913 |x| ~ oo. Furthermore there is a symmetric- (24) Lee h(x) = - jo, /(2g){9,(x,0)(9,,, - K9,) - 219%, + 9) Py} 3 5%/(28) {Pl Pyy r Ky) y 2(P1, Pox + P1yPoy) (25) and the bar signs mean the complex conjugates, i.e. 1 =%ic-ji?is for i=1,2. On the cylinder surface Pex(Xo2 Vo) f (Xp) = Pey = M,¢(Xo+ Vo) (26) where og ID ee = , M65 mae 2 {Poy y(Xo, Yo) ms Piyy ay (Poxy a Pixy)f (x,)h (27) or, in terms of the stream functions, We(Xo, Yo) = j 2 (Why = Wey). (28) In the far field ggy~ 0 as y~-o and 9 should represent out- going waves as |x| —~ oo. The symmetric-flow condition implies that 9_(x+y) = 9,(-x,y)- IV. SOLUTIONS FOR os AND 96 It will be assumed that solutions for the first-order potentials gy, and g, are known. The method of multipole expansions for finding g, and 9, is described in Appendix B. The main difference between the first- and second-order problems is in the free-surface conditions. A first-order problem has a homogeneous differential equation for the free-surface condi- tion (see e.g. (A-1) of Appendix A) whereas a second-order problem has an inhomogeneous one (see e.g. (19))- When there exists a non- constant pressure distribution on a free surface of negligible surface tension the first-order free-surface: condition for an incompressible irrotational flow is represented by an inhomogeneous differential 914 Nonstnuostdal Osctllattons of a Cylinder in a Free Surface equation such as Eq. (19) or (24). The term on the right-hand side of the equation of the first-order free-surface condition represents the pressure distribution on the free surface. Thus the problems for the second-order potentials Ps, and 9, presented in Sections 3.1 and 3.2 are the same type of boundary-value problems as those for the first-order potentials except for the "non-constant pressure" on the free surface. If we assume there is no body in the fluid, then these problems can be treated as problems for surface waves arising from variable free-surface pressure distributions. Solutions for these problems are given in Wehausen and Laitone [1960]. If we denote the velocity potential associated with the problem of variable pressure distribution on the free surface by W and if we assume that it is known,we can use it to find the potentials gs and gg. This is done by introducing a new function G=g- W, where 9 could be either g, or 9.,s0 that the free-surface condition for G is given by a homogeneous equation such as Gy(x,0) - KG=0 where K is either Ks or Kg. The boundary-value problem for G is then identical to those for the first-order potentials and the solutions to these are well known. Once G is known the solution for @9 is readily obtained from g=Gt+twW. This scheme was used by Lee [ 1968] to find the second-order potentials gy, and g,. However there are certain requirements on the "free-surface pressure functions," h, and hg given by Eqs. (20) and (25) respectively, to be satisfied before the known methods can be used to find the potential W. These requirements are that the functions hs(x) and h,(x) should be absolutely integrable in (-00,00) and should satisfy the Hdélder condition. Although the proof of these statements may not seem obvious from Eqs. (20) and (25), it can be ee that both h, and hg satisfy the Holder condition and as x] —* hy = O(1/24 (29) and i€(K,-Kp)Ixt - B} e yea + O(1 /x?) (30) where a, and 6 are given by Z ay = — Q,9,K,K (0, - 02) (30a) and oo di - do- 1/2. (30b) Here the quantities Q, and q, for k=1,2 are associated with the 915 Lee first-order potentials and can be best described by the expressions of asymptotic behavior of the first-order potentials such as Kyy j(K,Ixl-q, ) data xe “e K kK for Keak 2 as |x| — o. It is apparent from Eq. (30) that hg is not absolutely integrable. This implies the necessity of further consideration in deriving the solution of W which is associated with hy. 4.1 Solution for a Case Where Free-Surface Pressure Distribution is Specified In this section we consider a potential-flow problem with a given pressure distribution on the free surface. We restrict our attention to the pressure distributions which have harmonic time dependence and are even in x. Furthermore we consider the two special cases: the one where the pressure distribution decays in the manner of 1/x? as |x| — o and the one where the pressure distribution behaves like that for outgoing plane waves as |x| ~ o. Let w(x,y,t) be a harmonic function defined in y <0 and with its time dependence of the form -iwt w(x,y,t) = W(x,y)e where W=W,+jW, and w is an angular frequency. The free- surface boundary condition is Wy(x,0) - KW = h(x) (31) where K = w*/g and h is a known pressure distribution and is even in x. We expect that the solution of W should represent outgoing plane waves as |x| — oo and furthermore that Wy (x, -00) = 02 "Wie seek solutions to this problem in two cases. Case 1: h(x) = O(1 /x?) as |x| — o. (32) The solution for this case is given in Wehausen and Laitone [ 1960] in the form of a complex potential te F(z) W(x, y) + iW (x, y) oe) : Hs ; 1/1 Ae n(é)e'Kl2-8) & (-iK(z-€)) dé + 2i ( hleve iK(z-€) dé 00 : + (j-i) ‘a Helens 5 dé. (33) 916 Nonstnusotdal Oscillations of a Cylinder in a Free Surface Here E, is the exponential integral defined by 00 et E,(z) - =~ at for larg (z)| <7. Case 2: h(x) = Pe ase O(4 /x?) as |x| — c (34) where K' (# K) is a wave number and A is a complex constant with A = A, + jAg. Before trying to solve for W, we introduce a harmonic function W,(x,y) which is even in x and satisfies W), (x0) - KW, = AelN™! W(x» -00) = and W, ~ Be as |x| OS where B is a complex constant with B=B,+jB,. The solution for W;, is found in Appendix C as Ky Yo) (42a) and Gel(x,+y,) = is (W, (x52 V9) 35) We = B,(x,y,)> (42b) Furthermore Gy, is evenin x, Gyy 0 as y~- oo, and G, 919 Lee should represent outgoing waves as |x| — Os These boundary-value problems are almost identical to those for the first-order velocity potentials. Thus we find them by the same method used to find the first-order potentials which is described in Appendix B. It is often called the "multipole-expansion method" since the potential is expanded in an infinite series of poles, located at the origin, of increasing order with unknown strengths. Each pole satisfies Laplacian equation everywhere except at the origin, the linear free-surface condition of the type ® (x, 0) -\KO@ = 0 and the infinite-depth condition, and is evenin x. However, since each pole vanishes as | x | — o the radiation condition of outgoing plane waves is not satisfied. To circumvent this a source singularity which has all these properties plus the property of outgoing waves at | x = oo is added to the multipole-expansion series. The unknown source and multipole strengths are found by satisfying the remaining condition which is the boundary condition on the body. Specifically we assume the solution for G, to be ee) G (x,y) = S (Dem * 5S pan) My m(X(X52) sy(A,a))e cK, (43) m=O Here b,, = Q, = unknown strength of a source at the origin, c,,=0, vee -J cre OsUpee Area sels conte (44) 0 = a source of unit strength at the origin, ioe) where f indicates that a Cauchy principal value is to be used, M cos 2ma K sin (2m - i)ea km em k (2m - 1)xem (2m+2n+1)a (2n ti) a,- 27 sin ~ 2m +2n ag yemeenel for m= i1 (45) = multipoles of unit strength at the origin, by, and c,, form=i1 are unknown multipole strengths, and q represents unknown phase relations between the forced motion of the body and the pulsating singularities at the origin. The expression for the harmonic conjugate of G, is co G* (x,y) = » (by mn + 5C pan) Man (0 9 @) ry(\,@)) (46) m=O 920 Nonsinusotdal Oscillations of a Cylinder in a Free Surface where 00 DYae- * K M. = f -j2o,t -j2o,t -jlo,+o,)t we {Pie + Piye + Pye +P -j(o, -o,)t +P fs i} + ole3) vie vir'*07% | Y : 922 Nonsinusotdal Oscillattons of a Cylinder in a Free Surface We expand P, (i=I,II,...,VII) in Taylor series about the mean position of the cylinder (x,,y,) and substitute y(t) = h,(sin ot + sin ot) €b(sin ot + sin oot) in this expansion. We rearrange the terms in powers of € andin time harmonics to obtain -jo,t -jo,t P= € {Pi (xorye)e + pe ot -j2o,t e 4 -j2 -j(o,+0,)t + e%{pe +p geuhil = tp -j(o, -o,)t oper |e Peale) (54) From the Bernoulli equation, 2 2 P = -~ pOj(xo, yo + y(t),t) - Palys + y(t) - $ (& + y), we eliminate the static pressure pgy,, expand the right-hand side in accordance with Eq. (17) and equate terms which are of the same power of € and of the same time harmonics. We then find the ex- pressions for pj (i=1,2,...,7) in terms of velocity potentials 9, and their derivatives. The expressions for these P;'s are Pj jp (o) 9) (X95 Vo) - gb) for i=1,2 (55) ee er PER Gt tee for (1220 (56) Pigg PF Ve dOU Mie Yo vive aie a iy Os aa? - \- J(o, + o)e Axo ¥,) +5 (Px Pox t P1yPry) ue) a N +2 (0, 9,, a5 5% oy )\ ? (57) Pe =~ PySoy ~ oppelx ) +5 (Po, 4 Oye 6 JAG, 2PeXorVol 7 F N\PixPoy PiyPoy) z 2 (7% + T2Pey yt i in 2 b 1 — = F 923 Lee where the bar sign means the complex conjugate e.g. Pox Prex~ IPogy° If we let 6; =tan (Re, pj/Im, pj) for i=1,2,...,6, we can express these pressures in the form p, = jlp ler?! for L=ilads ek 9.40 or, in association with the time harmonics, -jwjt pje = Ip; | sin (wt + 6.) for i7s4l 2 ,..<57,0 where iT] —_ w ies) oj for i Wj = (61) 5.2.) Vertical Hydrodynamic Force on the Body The vertical hydrodynamic force acting upon the body is given by L(t) Erste i Pi cos (njy) dl. £ 0 Here £,(t) is the instantaneous position of the point of intersection between the bottom of the body and the y-axis, 2(t) the instantaneous position of the point of contact of the body with the free surface, and cos (n,y) the direction cosine of the unit normal vector on the body surface inthe y-direction. The positive direction of the unit normal vector is into the fluid, and the integral is taken along the cylinder contour. Eq. (54) enables us to show that for the family of cylinders 924 Nonsinusoidal Oscillations of a Cylinder in a Free Surface mapped according to Eq. (2) b A : -jo ft -jJo.t F = 2! dxo[e{P\(x-y0)¢ ad + p,e 52 \ O -j2o,t -j2o,t -j(o,+o5)t + lp. + pye + pee -j(o,-o,)t £ Pee J o, os + p7t| + O(e*) c (62) If we let -jo,t -jo5t F=e(fe"' +f, °°? ) 2 -j2o,t - j2ont -jlo,+o,)t te (f,¢ 2g f,e a: f,€ + fg +i) +O), (63) then we find that b a) = 2 | P(X (A522) s¥Q(hy-%))T(2) dw = for f= 142,20%,7 ~ (64) -17/2 where the expressions for p,; are given in Eqs. (55) through (58) and 2 en¢ Xo 00 T(@) = - a4}, sina + » ase eriic sin (2n +1) a}. n=0 If we let y, = tan” (Re; £, /tm, £) for irs Dale ness we can show that fi. Il jlfife eM" lf, | sin (wt + y;) for =) de seg 0 (66) 925 Lee where the w,'s are defined in Eq. (61). 5.3 Outgoing Waves at |x| = 0 Equation (11) shows that (x,t) = - - {B xb, ¥(x,t) +t) +5 (&; + a5). If we substitute the expansions given in Eqs. (17) and (18) into this equation and equate the terms of the same order and time harmonic, we find that ¥j(x) = j + i (x,0) for i= 1,2, (67) 1. Kj Lp 2 2 : Vis = 5 1520) ¢1,2%-0) - > PiPiy - | (Vix +o) for i=1,2, (68) ; Tt Ys(x) = j A? og(x,0) - ZIP(o,o2y + P21y) 1 =e Pitan” Tage (69) 0, -Cv 0 | 0. = = Y,(x) = j 1 le,(x.0) ur re (PP oy + PP iy) 1 _ ie a 2g (91, Pox + PiyPay! » (70) 2 ¥ Ax) = - a (9;,(% 10) Pj, + PiyPiy - 2Ki9;Piy)- (74) If we let |x|— o (or X}—~o for @=0 or - 1) only the pulsating sources contribute non-vanishing values (see the expression for the first-order potentials in Appendix B, and for gz and g, in Lee [1968]). Thus we find that 00 -jqj e?¥ cos px Kiy (x,0) ~-Q.e lim ( dp + jre cos K;x}}]__ g (x, 0) sl y. ip Sa ee i Jly=o j(Kjlx!-qj) Sb MaGher ie Uhitemik inlet ha x3 ice (72) J i where the Q;'s are the source strengths, the qj's the phase re- 9126 Nonstnusotdal Oscillations of a Cylinder in a Free Surface lationship between the forced motion and the pulsating singularities at the origin, and os Bie 2 Kig2 = ot F for es er Ae We can also show from Eqs. (43) through (45) by letting |x| — oo (or Xo for @=0 or -rn) that 00 -)q; py é G,(x,0) ~ - Qie ' lim (5 ee dp ane cos K;x}| =0 |x |—oo re) P - Kj va j(KiIxl-q;) = jmQje for 1=55,.0 (73) where 2 K, = Aumar)s ua > g 2 K. = (21> 52) 6 & ? and then Eqs. (38) and (39) can be used to show that 00 ; st) We(x.0)~ 5) gee er® dé, (74) i{(K,-K)IxI-B} oo -é) Wales 0)~ S85 eas ti made at, (05) where ao and B are defined in Eqs. (30). The far-field behavior of the derivatives of the functions g; (i=1,2,3,4), Gj (i= 5,6), Ws, and Weg can be shown to differ from those exhibited in Eqs. (72) through (75) by factors of the appropriate wave numbers Kj. If these results are applied to Eqs. (67) through (71) and some mani- pulations are carried out,we can show that as A > 0 BLE, i ¥i (x) ~ F Qie for i= 1,2, (76) 927 Lee 2 ol l4kjlxl- aj, 2) 4. (7Q;K)) j(2KjIxl- 2q;) eon eee as 00 rau F 4K i Ixl-@; peut h, (6) cos 4K,& dé | aad g b +2 I Vier ¥; , {*) z Q for i= 27 where h (x) and h (x) are defined in Eq. (A-2) of Appendix A and s/ - Im, (Sf, digalé) cos 4K,& dé) 9i,2 = tan - Re; (Sy risel6) cos 4K;§& dé) Bae j(Kgx!-q5) Y,(x) Z Q.(c, a v,)e 2 i{(K,+K )Ix!-q,-a.} pe Q,Q,0,0,(K, + Ke | athe 00 2. ~ Std | h,(6) cos KE at| ears (78) b where 6, = tan" Ps es Im j (J, gl) cos K,& dé) - Re; pass cos K,& dé) BIL! j(K dx! - gg) ¥,(x) : Q, fo, - o,)e 2 i{(K)-K>)IxI-(q,- a2} 00 ae Si ie, = OP) ie Bey caster ag | qilKelx! +86) (79) where 0. = tan” aaa) Cm h(E) cos Kg dé) 6- tan jayteNAD pod ow. tuo Doiaw -Re, (f, he(é) cos Keé dé) ¥5(x) aug OP 928 Nonstnusotdal Oscillations of a Cylinder in a Free Surface VII. NUMERICAL RESULTS A semi-circular cylinder of unit radius (b= 1) is chosen for sample calculations of the pressure distributions, hydrodynamic forces, and out-going waves. The inputs for the calculations are the values of the fundamental frequencies o, and o,. Three values of g, are chosen such that the corresponding length of gravity waves in deep water, , = 2mg/c,*, are equal to 2b, 10b, and 20b. For each value of o, the values of the eo cne sy ge o> are chosen so that the wave lengths obtained by Ao omen lie in the interval of 4, X—/, as afunction of ,/X, is shown in Fig. 1. This figure shows the wave length corresponding to the difference-frequency o, - 7, compared to the fundamental wave lengths \; and ho. In the rest of the figures the abscissas are \ which is defined as X= o/dy- For the values of X,= 2b, 10b, and 20b, the corre- 929 Lee Fig. 1 Difference-frequency wave length vs. fundamental-frequency wave length sponding frequencies are respectively ov, (= V2mg/i,)=Vag/b , mg/5b, and Vng/10b. o, is obtained from X= d/d, = Cre hay as co = ¥ 298/008). - Thus we have ur ° Le J Pa iT] ia) ion q ine) tH a = lon a 930 Nonstnusotdal Oscillations of a Cylinder in a Free Surface for ,=20b, o,=¥mg/(10b)). In Table 1, the values of 6) = osb/g and 6,= (0, - o,) b/g are given for = 1.0 (0.1) 2.0 at each given 2X, = 2b, 10b, and Z0b. Thus the results in the subsequent figures can be referred to appropriate individual dimensionless frequencies suchas 62, 54= 40%b/g = 465, and 6¢= 2mg/d,(1 +1/X- 2V1/X once X, and X are given. TABLE 1 5, and 5¢ versus i, at three different values at h, and 6, h, = 2b and 6, = wm |,=10b and 6,=m/5 |\,=20b and 6, = 7/10 0.365 x 10° 0.136 x 10°? 0.477 x 10°? 0.950 x 1072 15t X40" Lethe alo: .276.X 107 341 x 10°! .406 X 107! .474 x 107! .540 x 10°! 0.680 x 1073 0.238 X 10% 0,475-x 10 0-753. x 10:4 0.106 x 10°! 0.138 x 10°! O70 X10" 0.203 x 10°! 0.237 X 107 0x270 < 107! 238 x 107 .475 X 10°! 153 6-10"! . 106 . 138 .170 . 203 eit . 270 1 BA 3 4 5 6 it 8 ) 0 eee Ee FE NY DY DN Dd DPD Soo ©. O Go oO 6 Oo © 6 So © ,O..0-.0170 SC 2 CO © 16 © 1 1 i. i. si i i; 1 nla Es ea © —& & © Ce, oS © In Figs. 2, 3, and 4 the maximum hydrodynamic pressures at three points on the cylinder, 0= - 900, - 450, and - 5°, are shown as functions of \ for \,=10b. The maximum pressures Ip. and lp. are obtained from Eqs. (58) and (59). The pressures are non-dimensionalized by pgb and are denoted with bar signs e.g. De= |p,|/pgb. The values of p, and_p3 which are not shown in these figures are respectively equal to pp and pg at A=1.0. The maximum hydrodynamic forces fp (= lt] /2pgb?) ; hgh? epcend £7 which are obtained from Eq. (64) are shown as functions of X in Figs. 5, 6, and 7 for \, = 2b, 10b, and 20b, respectively. Figure 8 shows the phase angles y,, y,, and y, which are defined by Eq. (65) 931 Lee 18 20 1.0 1.2 1.4 1.6 First- and second-order pressures vs. \ = h,/h, Rig. 2 at 0 =- 90° for dy, = 10b 1.8 20 Vn 1.0 1.6 Fig. 3 First- and second-order pressures vs. Ge No/ dy at 0 = - 45° for \,= 10b 1.2 932 Nonsinusotdal Oscillations of a Cylinder in a Free Surface d, = 10b : 6 =-5 DEG Fig. 4 First- and second-order pressures vs. X= Xo/hy at @=- 5° for X,= 10b for 4, = 10b. The radiating-wave amplitudes at |x| = oo are shown in Fig. 9 for 4; = 10b. In this figure Y, is defined by Y, = |Y,|/b = 7™Q,0,/(bg) and Ne and Y,, are obtained in the following way. We can show from Eq. (79) that -j(o,-o.)¢ Ye(x)e ASieee) oa A, cos 1K |x| - (o, * o,)t a de | +A, cos} K,|x| - (0, - o,)t + o,f - Yepcos }(K, - Ky |x| - (0, - o,)t - (a, - at where A, ar: (0, = T)Q, ’ 933 Lee LY are if pdf fe i fi AWN E ANY Fig. 5 First- and second-order forces vs. \ for X= 2b Oh) tte So oa = |) ae) aur > x 1.0 12 1.4 1.6 1.8 2.0 X Fig. 6 First- and second-order forces vs. X for , = 10b 934 Nonsinusotdal Oscillations of a Cylinder tn a Free Surface >| Fig. 7 First- and second-order forces vs. \ for A, = 20b 2 ad Ap= e (o, - o,) | \ h (5) cos K,§ dét , 2 T Furthermore we can reduce the above expression to -i(o,-o,)t = ! Y,(x)e Ye cos 1K, |x| - (o, ~ o,)t + cm = Lan cos }(K, - K,) |x| = (oc o,)t - (q, - q,) where (a 2 1/2 Yeo= LA, tA; +2A,A, cos (a,+,)] Q' = tan’! Az sin 0¢- A, sin gg | $ A, cos 0,+ A, cos qd, 935 Lee Fig. 8 Phase angles of first- and second-order forces vs. X for Y= 10b Fig. 9 First- and second-order wave amplitudes at lx| = co ve. \% for. X= 10b 936 Nonstnusotdal Oscillations of a Cylinder in a Free Surface 7) w rT) oc 9 wi a) Fig. 10 Phase angles of first- and second-order waves at |x| =00 vs. X for d, = 10b We then define Y Y In Fig. 10 the phase angles gq, (see Eq. (76)), 6,', and q, - q> are shown as functions of X for i, = 10b. VII. DISCUSSION If the forcing motion on a floating body has a very narrow frequency band,most of the second-order hydrodynamic responses occur in a frequency range of about twice the forcing frequencies. However, two components of the second-order force are exceptions to this case. One is the steady-state force and the other is the force with frequency equal to the difference-frequency between a pair of the frequencies in the narrow-band spectrum. If the value of the difference-frequency, 0, - oj, is very small, the force which 937 Lee is associated with the difference-frequency changes very slowly in time and often may be treated as a pseudo-steady force. In fact we can show that in the limiting case of o, = og that the difference- frequency force reduces to the steady-state force or in our notation, f,=f, when o, =o 9. It is shown in Appendix A that for o, =o5 P= Po» P3=%4=%/2, and 96=9- Equations (55) through (57) show immediately that p, = pp, P3= Py = p,/2, and p,=p-7- Substitution of these relations into Eq. (64) leads to f,=f;, for o, = aie 4p eae in Figs. 5, 6, and 7 that as Mar 4.0, ie, ups oe £1 and in Fig. 8 that yg -0/2 as Na 1.0. Since ee 265 eee sin yg from Eq. (66) and f7 is nega- tive in this case, we'see ‘that Y6lo,2 05 = 7 m/2 in order to maintain the relation f, = f. for 0, = >- For sufficiently small values of o, - 63, we can show that the expression of the forcing motion becomes _y(t) ho sin o,t + sin Tot R 2 sin opt cos sto t (80) and the corresponding expression for the hydrodynamic force can be derived from Eqs. (63) and (66) as F = e2|f,| cos ee t sin (opt if Y>) a efalt,| cos (co, - o,)t sin (20,t + y4) + lf, sin y,cos (o, - o4)t Dy £,{+ O(e*). (81) This is a beat oscillation for small values of o, - o5. The response of hydrodynamic forces to this beating motion is made of two kinds of beat oscillations: a slowly-varying sinusoidal oscillation, anda steady component. For comparison purposes the relative magnitudes of the different components of the hydrodynamic force given in Eq. (81) are shown in Table 2 for \,;=10b and do= 11b i.e. XSdeg4e The values in Table 2 are obtained from Fig. 6. 938 Nonstnusotdal Oscillations of a Cylinder in a Free Surface TABLE. 2 The magnitudes of hydrodynamic forces for \, = 10b and = 1.1 2f, 1.58 4f, 1.00 f,| sin Yel 0.014 f, 0.04 ¢, - 1éiL = for i= 2,;436,7) 2pgb It is clear from Table 2 that the first-order force dominates the second-order forces. For instance, if we assume € = 0.1 the ratio of the first-order force to the largest second-order force is 2e|f,| /4e*|f,| + 16. It is also clear that the magnitudes of the difference-frequency force and the steady force are much smaller than the first-order force, so they appear unimportant. However, when such forces act upon a body which has very small restoring force for a sufficiently long period of time a considerable excursion from its mean position can occur. One can see from Figs. 5 through 7 that the |f,| is largerthanthe |f,| in 1.0¥ 4) for i= 3 and 4, (A-5) and ' b ' P2(XQr¥Q)E (x) - 92, = - x Im, [xy (X69 Vq) + Poy Jt (XQ) = Piyy - Poy | = m{x),Y,)- (A-6) In the far field P jy(x»- 00) = 0 for La lj2y 35 45 and 7 and at | 3x | = oo the potentials g,; for i=1, 2, 3, and 4 should represent outgoing plane waves. For the steady potential g_ the condition at |x| = oo should be determined by the law of mass con- servation (see Lee [ 1968]). Symmetric flow condition: yi(x,y) = 9 (-x,y) for le=) 2 ae ands In the limiting case of o, = 0, the forcing motion given by Eq. (7) reduced to y(t) = 2hgsin ojt and if we let € = 2h,/b, the perturbation expansion given by Eq. (17) reduces to -jot - j2o,t 2 B(x,y,t) = €9,(x,y)e + €g,(x,y)e + €°g_(x,y) 942 Nonstnusotdal Oscillations of a Cylinder in a Free Surface which is the same expansion as that assumed by Lee [ 1966]. We can easily establish the identities gy, =, and g3= 9. It will now be shown that for T= o, we also have 9, = 29, and g,=9,. Equation : 2 5 3 6 G (20) gives e . Go hg(x) = - § 5b} o,(*,0)(Pay - Kp,) . F = 201, ox + Pry Poy) t - 52] PalPiy ~ Kiry) - 210), Pox + Py Poy) f > LS ne 2 2\) hex) = - jt }o,(x,0(g,, - Kye) - 2, +e, )f. Comparison of this with Eq. (A-2) shows that h,(x) — 2h{x). Equation (22) gives a m(x,,y¥,) = - Jz | (Piny Coa Yg) 7 Pony)? Oy) Sig > Peyy t. so for 6, =o, Ng Ta F jb } P ixy (qr Yo)f'(X,) ~ Piyy t . Comparison of this with Eq. (A-5) shows that m, = 2m, The far field conditions and the symmetric-flow condition for both ?, and 9g, are essentially identical. The above results lead to the conclusion that 9, = 29;- Q,_ can be shown to be equal to g_ by a similar proof if h(x} (Eq. (25) for o, = ¢2) is compared with’ h{x) (Eq. (A-3)) and m,(x) (Eq. (27) for ov, = o,) is compared with m_(x) (Eq. (A-6)). 6 943 APPENDIX B Evaluation of the First-Order Velocity Potential There are two first-order potentials, g, and 9,, involved in this work. Since their solutions are es sentially identical (they differ only in the frequencies), g, will be chosen as the representa- tive first-order potential. As shown in Appendix A, the boundary- value problem for g, is V9, = 0, (B-1) (x0) - Kg, = 0, (B-2) P(X or ¥)f(x,) - 9 = - boy, (B-3) P(X» -00) = 0, (B-4) 9) (x,y) = 9 (-x,y), (B-5) and the radiation condition can be explicitly written as lim Rej(9), = jK,9,) = 0. (B-6) x->+00 There are two methods for the solution of the above problem. One of them is the method of multipole expansions (see Ursell [ 1949]) which is essentially an eigenfunction expansion of the unknown function. The other is the method of source distribution (see Frank [ 1967] ) i.e. the method of Green's function. A brief description of the method of multipole expansions will be given. First we consider the problem without the boundary condition on the body given in Eq. (B-3). If we transform the problem into the {-plane! we find that 7°M(X,e) = 0, (B-7) (2n + i)a aM Ka} - 2, Se Sarina | Min.0} - SE = 0, (B-8) "Here, it should be recalled that the transformation given by Eq. (2) maps the €- and y-axes into the x- and y-axes and maps the contour of the semi-circle inthe (-plane onto the contour of the cylinder in the z-plane. 944 Nonsinusotdal Oscillations of a Cylinder in a Free Surface M(X,@) = M(\,7 - @), (B-9) and in place of Eq. (B-4) and (B-6) we require M— 0 as X — oO in -7w=as0. (B-10) The solution of this problem is a) = £28 2ma@ , jae (2m - i)a@ ide ea hem VU (2m - 1) 2m! oo - », (2n+1)agne, sin (2m+2n+1)a \ (B-11) n=O where m is a positive integer. M, is often called the multipole of order m. Although this expression for Ma trivially satisfies Eq. (B-6) in the C-plane,the expression above still does not repre- sent the outgoing plane waves. To satisfy this radiation condition we introduce a source function M,(x,y) which satisfies all the required conditions except the boundary condition on the body. The expression for the function M, is M,(x,y) = ‘ a ea dk - jre” cos K,x (B-12) .e) 0 where f means that a Cauchy principal value is to be used. There- fore we represent our solution as oo 9, = y (Dm + 5Cm) My (x(X,2) sy(X,@))e!4 (B-13) m=0 where b,, and cm are the unknown strengths of the singularities, q is the phase difference between the motions of the body and the fluid, by = Q= source strength, and c,= 0. The unknown constants bm» Cm, and q are to be determined from the boundary condition on the body given by Eq. (B-3). We introduce the stream function W, which is the harmonic conjugate of the velocity potential g,. The Cauchy-Riemann relation gives 89,/dn 2 OW /8s along the contour of the cylinder where s is the arc length of the contour in the counter-clockwise direction. The boundary condition for W, on the cylinder can be shown to be Wi (x,,y,) =- bo,x,- (B-14) The expression for W, in terms of the harmonic conjugates of My, 945 Lee denoted by N, is easily found to be 00 <= » (bm + 5em) Nm(x(X@) sy (X,a)Jer*. (B-15) m=O where N. = - Sin 2me + Kya {£28 (2m - 1)a m 2m | (2 1)y2m-t cy (2n+i)a cos (2m+2nti)a 7 2 2m 1824 2 ment \ for m= 1, (B-16) n=O and eX sin kx Kiy No=- 4) apap dk + jme ' sin Kx. (B-17) On alvala, Substituting Eq. (B-15) into (B-14), we get 00 >. (bm zw jcm)Nm(x(5 2) ,y(h,a))e/4 i bo, x,. (B-18) m=O We choose any point on the contour of the cylinder between 6=0 and == 7/2, say (x!,y5) in z-plane and (\,,@') in the ¢-plane, to show that 00 Pe Ne ’ e 14 ie box! /{ > (5, r 3S on). Nin (Not 2) + QNo( x5, ye) } . (B-19) m=! Equation (B-19) can be substituted into (B-18) to give 00 ) Am Ne(hor) = Nig(qo $4 = Nolxdsye) - Nol%orYe) (B-20) Deantal€ ‘ Ag = —a 5m | In principle we can choose an infinite number of points on the cylinder (-1/2< @< 0) to set up an infinite number of simultaneous equations from Eq. (B-20) for the unknown coefficients A,. However the 946 Wonsinusoidal Osetllattons of a Cylinder in a Free Surface infinite series in Eq. (B-20) is truncated to a finite series to obtain an approximate solution by a matrix inversion. After finding some finite number® of b, and c, and using these coefficients in Eq. (B-19) we find the values of Q and q. APPENDIX C Solution for the Problem of Sinusoidal Pressure Distribution on a Free Surface We seek a solution for the following boundary-value problem: VW (x,y) = 0 in v <0, W, (x,0) - KW, = Ae (Gan) where K =w*/g, A is areal constant, and K'=w'*/g# K Further- more we require that Wiy (x , - 00) = 0; Ww, ~ Bek Yeik'Ix! as |x| — oo where B is a complex constant, and W, (x,y) = W, (-x,y)- If we let Wy > Wig Aig we can easily show that Wi cy(*> 9) - KW, = A cos K'x,; (C-2) W),y(x.0) - KW,, = A sin aa a (C-3) ?The exact number is determined in the sense of "an approximate in the mean" for the function on the right-hand side of Eq. (B-20) by the series on the left-hand side. 947 Lee The value for the function W,, which has all the required properties can be shown to be AeKy ie qe cos K'x. (C-4) It takes little more effort to solve for Wj),. We find it by using a transform. Let 00 e 3 Wis -{ e PAWis (x,y) dx. - 00 The Laplace equation requires that 2u,* * = - p Wis OW fey, 0 or * Ipl W,,(psy) = c(p)e’ (C-5) If this is substituted into the Fourier transform of Eq. (C- 3),the left- hand side yields Wry = KW, = (|p| - K)c(p) and the right-hand side yields 00 : co al e”'P* sin K'|x| dx = 2A \_ e7!* sin K'x dx on fe) oo lie i(k" 2AK' = (eilk'-Px _ Gi(K'ePe) Gy 5 1 fe) p--K"= (C-6) where the apparent improper integral above is interpreted as a generalized function.> Thus we find that - 2AK' - 2AK' (|p| - K)c(p) a PLiae or c(p) = Ge K'2)(]p|] - K) 3 Another way of interpreting this is that of Lighthill [1967] who let jK' jKolx! = w'=wot+ je, € = 0 so that eK ixt LQ Mom! oo where K5 = (we - €*)/g approaches K' when yp (= €2w,/g) — 0. 948 Nonstnusotdal Oscillations of a Cylinder in a Free Surface If this expression is substituted into (C-5) and the transform is converted we find that AKC ably (p2 - K'2)(|p| - K) | W,, (x+y) P* dp . _ 2AK' © eP%cos px aa ™ J, (p?- K%)(p - K) A py { 1 -= e"’ cos px { ———_——>--—___ T Jo (K +K')(p +K') Ges tl 2K! | y dp. C-7 (K'- K)(p - K') (K'- aaa Pp ( ) Apparently there are poles at p=K' and p=K. However if the inverse transform of the right-hand side of Eq. (C-6) is taken, it is readily seen that the integral must be integrated as a principal-value integral in order to recover the original function A sin K'|x|. This means that the integral in Eq. (C-7) associated with the second and third terms in the square bracket should be taken as P.V. integrals. If we let ward cos px en Re 124 S808 PX ap = Re, { —— dp, 0 pt+K 'Jo ptK and make the change of variable t = i(p+K')x, we can show that ‘ 00 sy rage I, = Re, eX? \ — dt = Re; | e'*7E, (iK'z) | : iKz Again the change of variable t = i(p-K')z enables us to show that oo py 00 _-ipz -5 e__COs8 px 4, Re; 5 & - dp 2 p- K' Y7O P- K -iK'z Re; [e E, (-iK'z) = ime“ 4 where + signs correspond to the case of x2 0. Similarly the change 00 of variable t = i(p-K)z in fs (e”! cos px) /(p- K) dp leads to -iKz -iK I,= Re, [e""E,(-iKz) ¢ ine*7} 949 Lee for x 20. This integral is the last term in Eq. (C-7), and we observe that ae Iz= - meXY sin K|x|. This implies the existence of a sinu- Xl» 00 soidal wave with wave number K in the far field. It obviously vio- lates the radiation condition that the outgoing waves have wave number K'. However a careful examination of the integral I, shows that it is just one of the homogeneous solutions of the problém which can be discarded, if desired, because of the radiation condition. Thus sub- stituting the expressions obtained above for I, and [, into Eq. (C-7) and discarding the last integral in that equation, we find that A I if Wty) = - 2 [eaber tebe | 7 LK+K' iM tie pear t “1K Zi page K'y --4Re([* EV (iK'z) ,e E | ( Re) +e sin eulaole T K+K' K' 7K K"-K (C-9) We combine Eqs. (C-4) and (C-9) to finally obtain : Aek Ye ik Ix! Wilsall RRS RT A eRe Kay , eke (-iK'2) : jAre,| £ =k) + ———L | . (C-10) T K + K' K-K tiz Since lim e E (+iz) = 0, we see that |x — 00 as is required. 950 Nonstnusotdal Oscillations of a Cylinder in a Free Surface DISCUSSION Edwin C. James Californta Institute of Technology Pasadena, California I would like to direct a question to Dr. Lee concerning the pure steady force. Apparently this type of force can arise in free surface problems and is attributed to a mean drift of mass in the direction of wave propagation. The action of such a force applied to an unrestrained body results in a sinkage or alift. The question is then, how does one physically explain the steady force when the symmetry of the problem dictates that the mass transport at the station x = 0 should be zero? REPLY TO DISCUSSION Choung Mook Lee Naval Shtp Research and Development Center Washington, D.C. A mass transport phenomenon arises in the higher-order theory of surface waves (see, e.g. Wehausen and Laitone [1960, pp. 660-661]). Since the present work deals with a second-order problem of free-surface waves, it may be expected that mass- transport will occur in the present problem also. Although I have not touched upon this subject in the text, I discussed it in some detail in my previous work (Lee [1968, pp. 317-318]). As the discusser pointed out, there is no mass flux across the y-axis. Then, the question arises as to the origin of the mass to supply mass transport. I answered this question in this previous work by showing that the role of the steady potential gy7(x,y) is to counteract the mass transport phenomenon. This means that 9, should behave like a steady sink whose strength is equal to the total mass drift through two vertical control planes encompassing the cylinder, divided by 217. The lowest-order contribution from ¢, to the steady force is fourth order, as is proved by Bernoulli's equation. Thus, the second-order steady force still exists while the mass transport phenomenon is nullified by the pure steady potential 97. 951 . a Ho vA » od), ae” «th sy: Des ey ¢ ™ » oe Pee ee +ROw . Tr v i Oa wien te ' Rasa ss Se 1 ee "4 Prec athe ol rhes, | 4 ; cove Pee \ s fr57 ry . ; 3 “ 4 ‘ aha in A. C ¢ * 1 { est ni 6 saxo 3 baddtyisan ! ASS oo de Corey s 4 (iiss ‘ ie * J 7 ) y fel < tite @ he aie 4 “ pate > | rut BS r ? E y a ‘ i ’ La. ¢ a? r “i wy vig bY es ; +e 4, Oy 4 H a % i \ ; * * ane | Pt) \ ¥ e na hy i o i: : “ ‘ > a * % 1="2 90.4 é pa eT he dn » et ae i ’ ba | A ge { iO F! & eR RS sé ; ; ‘ } ie hy do Qabiosbroske @ ddiw easeb tick in edt © see ery 038 ne ee Wes iad? Dosaoex mw yaar 3 ta } -o=4) to onpotd SeBs i Rs ae. midos rae ‘ cian ia PrOges Pret as Tf GOSBHOGLD | .IRB? Sc hd ish? get sed HOS iMiE-TEE .ga@ ,% A.) diow avolyeny wit ad fe asoros xii: a heey oa al Stas? . b mis eoely say sh => | He Le aby | rf. Ve : 1! toe | ' mj as % Kane S Pom ea 7 or ~medt ‘ate BOLE sol att SOs ROMP- B17 POP Bie 4 ALGOQVENE TT PARES = OF E Ot YORSIS tut) 7 lot ov? tectd amie ode a ax " Ase Mutenreooy T86qeagt) 66 sett oo) Fae 1 SSM Bl syeris Geodw Sola Vbsole = 2411 aod bles or? gccia SCRA UMS BP y 10% tia5 [andivey owl tng sie On, oe} mort noWbe iiees sehro-deawod auth eh he sbi ‘tbh tebe e ‘ille@as iti WE PeYOIG & LS «eee ee Oho Oe, eT Cs. vhuate a ofisw eseing Lite sarot ybastr 7abyo-ba 0243 ,enlT Oe ot; vbeede stia oad yd be fit! i = WULF qj Gone ss BPKIT,, a Li texeetat HYDRODYNAMICS IN THE OCEAN ENVIRONMENT Friday, August 28, 1970 Afternoon Session Chairman: T. Y. Wu California Institute of Technology The Drifting Force on a Floating Body in Irregular Waves J. H. G. Verhagen, Netherlands Ship Model Basin, The Netherlands Dynamics of Submerged Towed Cylinders M. P. Paidoussis, McGill University Hydrodynamic Analyses Applied to a Mooring and Positioning of Vehicles and Systems in a Seaway P. Kaplan, Oceanics Wave Induced Forces and Motions of Tubular Structures J. R. Paulling, University of California, Berkeley Simulation of the Environment and of the Vehicle Dynamics Associated with Submarine Rescue H. G. Schreiber, Jr., J. Bentkowsky, and K. P. Kerr, Lockheed Missiles and Space Company, Sunnyvale 953 Page 955 981 1017 1083 4141 vy S46 wy kitt peve WW yy eek £ lees oF . YROAOMUIIS £ 6O GINTIBO BittoOuUlmG ined loboM git® ubantvecdio ,negatiasV oo JER nak frases at arteboilyvs powoT besusmdue to aoke vilazevint! Loi9loM i te ¥ vpendy? OF i | mri hi ‘ome a Deke Mas tlre Fa RA higak an ay Leta, olecenybork t WB B fee 46 mi erceote y wna esl shia V to yninobimosbi, aolvssot) .selaeal en easpioutia talodeT 29 ano! sal ms 2eoto0t beoubal oval rolonesd .ainxotiie> to yitexvevianl ,gotiives ot a meat, eisideV odd te bone Josrmureriynd edd Yo eee salxsardve dilw batsincesA eok bas ,yiaw odgnett ot th pgneyn? d32 wi. olavy USE . eae THE DRIFTING FORCE ON A FLOATING BODY IN IRREGULAR WAVES J. Hs Gs Verhagen Netherlands Shtp Model Basin The Netherlands I. INTRODUCTION A floating body in waves experiences a hydrodynamic pressure force which is exerted by the surrounding fluid. Several factors contribute to this wave pressure. One of them is undoubtedly the conventional unsteady exciting force, which makes the body oscillate at frequencies in the region comprising the bulk of the energy con- taining waves. Another factor originates from higher order forces due to various non-linear effects. In general these non-linear effects are too small to influence the high-frequency motions of the body. They can, however, be of importance in that part of the frequency domain in which the wave energy is very small, i.e. in the low-frequency range, in particular if one of the natural frequencies of oscillation of the body lies in that range. In the limiting case -- zero frequency -- one arrives at the well-known drifting force. It will be obvious, to assume that the force on a floating body in irregular waves comprises not only a steady part but also a slowly varying part, slow in comparison with the mean period of the wave spectrum. The steady as well as the slowly varying part of the wave force, both of which are proportional to the square of the wave height, are denoted by "drifting force." The present paper is con- cerned with the slow drift oscillation of a moored vessel in irregular waves. It is based on general observations revealed by an extensive test program on the behavior of moored bodies in a seaway. 955 Verhagen Il GENERAL OBSERVATIONS OF TEST RESULTS A study of the test results on the behavior of moored floating bodies in irregular seas revealed the following general observations: 14. The horizontal modes of motion -- surge, sway and yaw -- show two separate frequency regions. A low frequency region corresponding to the low natural frequencies of the moored system, and a frequency region corresponding to those of the energy containing waves. 2. The long periodic motion is excited by waves or by a wave group with amplitudes high compared to the mean wave height. In the considered cases, where a linear stiffness of the mooring system is employed, it appeared that the amplitude of the long periodic motion for a given vessel and mooring system is proportional to the square of the significant wave height divided by the mean wave period Coe for various long crested seaways coming from a given direction. 3. For a given body and mooring system tested in various seaways no clear relation could be discovered between the time averaged excursion from the equilibrium position in still water and the amplitude of the long periodic motion. These observations are obtained from extensive model tests conducted in the Seakeeping Laboratory of the N.S.M.B. at Wageningen. The behavior is not unique to moored vessels. Also the towing force of a vessel towed in irregular seas show the same tendency as well as for instance, the slow oscillations in torque and thrust of the propeller of a self-propelled model as observed in the seakeeping model tests. III. DISCUSSION OF THE RESULTS The third point of the above mentioned observations deserves particular attention. The drifting force on a floating body in regular waves -- the time averaged position of the body is fixed in space -- is dependent on the joint action of waves and body motions. The force is proportional to the square of the wave height and dependent on the phase between wave and vessel motion. If we consider an irregular wave as build up of a regular wave whose amplitude is a slowly varying function of time (slow as com- pared to the wave period) and a stochastic variable phase, the cor- responding energy spectrum will be narrow. The drifting force on the floating body in that case will show the same dependency of the time as the square of the wave amplitude. The amplitude of modulation will be the same order of magnitude as the time averaged drift force. 956 Floating Body in Irregular Waves For a given moored system with approximately linear spring stiffness and damping coefficient the same linear relationship must be found between mean excursion and low frequency amplitude of motion for various seaways. This appears not true. Especially in heading waves large discrepancies can occur. From this observation I am led to suppose that it will be not allowed to describe a practical wave spectrum by a slowly modulated regular wave in order to explain the obtained test results. Based on the mentioned observations I am led to suggest the following hypothesis. Hypothests: The wave forces on a moored body in irregular waves which are responsible for the excitation of the mass-spring system in its resonance frequency are the second order low-frequency wave forces on the body in fixed condition, i.e. the drift force due to the reflection of waves. The influence of the ship motions can be neglected. One of the conclusions of this hypothesis is that the exciting force for the long-periodic motion is a function of wave character- istics and shape of the body alone, and not dependent on the mooring system or on the weight distribution of the moored body.. The hypothesis is supported by numerical motion calculation for comparison with experiments, which will be shown later on. It is needed however to extend the number of comparisons in order to obtain the restrictions of the proposition. The proposition can be made acceptable in the following way: Suppose the irregularity of the wave could be described by a more or less regular wave pattern in which a few discrete steep waves are present. Intuitively it can be stated that the occurrence of a few high waves in an otherwise nearly regular wave pattern gives rise to some violent ship motions. Through inertia effects these motions occur mostly after the corresponding high waves have passed the vessel. Hence, interaction between the high waves and the resulting motion on the pressure distribution around the ship's hull is drasti- cally reduced, by the mentioned retardation between exciting force and resulting motion. Hence, the effect of such a single high wave on the floating body is consequently restricted to the instantaneous effect, i.e. the effect of the wave reflection, on a fixed body. The corresponding exciting force due to reflection is only dependent on body form and wave characteristics. Conclusion: The mean drifting force on a floating body in irregular waves is dependent on the joint action between waves and body motion. 957 Verhagen The slow drifting oscillations of a moored vessel are caused by forces due to the reflection of the waves against the fixed obstacle, The remarkable observation that the amplitude of the long- periodic motion for a given body and mooring system_in various seaways of given direction is proportional to C¥,,, /T can now be explained as follows: The force due to reflection of irregular waves against a fixed obstacle is proportional to Cy ,3 if the body dimensions (L) in the wave direction is not too small compared to the mean wave length \ (L/\ >37). A "high" wave can be defined more formally as a wave with amplitude a fixed number times the significant wave height. The change of occurrence per unit time of our socalled "high" wave is then proportional to 1/T provided all of the considered random waves are Gaussian distributed, are at least distributed in the same way. If the combination of floating body and mooring system is considered as a mass-spring system with linear stiffness and damping coefficients the resulting long- periodic motions will be proportional to Cwy 3 /T. In consequence of the many assumptions made in the above reasoning, one should be careful to adapt the explanation without reservation. Cwiyse 1.25m 729.0 sec. 0.3 SU (W) In msec. eee ss Bi dal FE 0 dV Ba Fh FP SB DS Fa ee a | i i PG Ra a a nia A NSS 1.5 W in alot 108 Fig. 1. Wave distribution and spectrum 960 Floating Bodies in Irregular Waves WAVE ELEVATION IN m percent occurrence Measured Sway 2.00 m, f=96 sec. ------ Theoretical : Gupae 2.00 m, T= 9.0 sec. TTT TLL pol SEO Se a Paeee eee W in ect Fig. 2. Wave distribution and spectrum 961 Sc(w) in m? sec. Verhagen WAVE ELEVATION IN m 20 CTT A CCCCaTe CC -1 Oo 1 2 wave trough wave crest percent occurrence Measured Cwyys" 236m, Ts 6.6 sec. Theoretical : Cw" 250m, T= 6.0 sec. a Se Aerie Sa A ae a PTALN Agi eee teee Eee W in ae Fig. 3. Wave distribution and spectrum 962 Floating Bodtes tn Irregular Waves Wave elevation in m. § 40 : - WwW i 4 [4 =) oO 3 Se 20 - — | o hi y ee ai " ia — ° = nm 2 1 ° 1 2 WAVE TROUGH WAVE CREST MEASURED , fi1/3= 2.52m., #s 95 sec. ———— —— THEORETICAL (Pierson- Moskowitz), Ai 2.501m:,, qt s 9.0 sec. a ed se a pa thf aa Pe Ed aaah 1.5 cay smal W in rad.sec. | Fig. 4. Wave distribution and spectrum 963 Verhagen Wave elevation in m. G max.es 46 m. max.- s 44 m. = zs MEASURED, Hy32 4.80 m., Tz 10.1 sec. THEORETICAL (Pierson - Moskowitz), Fits: 5.00 m., ¥. 9.0sec. fnh (W) in m® sec. Rl nn es eS SS W in rad. sec”! Fig. 5. Wave distribution and spectrum 964 Floating Body in Irregular Waves SWAY WAVE mean ie] Fig. 6. Irregular wave and sway motion 965 TIME Verhagen SURGE DEVIATION IN m © G = 0.87 m e 49 max.+= 2.4 m t max.-= 3.0 m =) U Yy ° 20 Ee p © a2 0 aii | be -4 -2 ie) 2 4 —=> ship forward Head sea : =2.30m, fT. 6 6sec. wr Gwi/3 W inrad.sec” Fig. 7. Distribution and spectrum of surge 966 Floating Body in Irregular Waves SURGE DEVIATION IN m percent occurrence Sy (W) in msec. W inrad.sec> Fig. 8, Distribution and spectrum of surge 967 3 3) Sy (W) in m? sec. Verhagen SURGE DEVIATION IN m by of c¢ 40-—max.+ 7) L max.- 8.9 m 5 Vv 13) ° 20 E © 5 ii i rT) a oO ape fi Sia -8 -4 oO 4 8 —=~ship forward SURGE SPECTRUM 7] Head sea : =4.883m, T=101 sec. w Gw13 W inrad.sec” Fig. 9. Distribution and spectrum of surge 968 Sy (w) in m?sec. Floating Body in Irregular Waves c-o7m > i. mMax.«2z 3.2 m maxcz_ 23m [Nmeen | | a eee Saez -4 -2 re) 4 ship to starboard ship ta port SWAY SPECTRUM = 120m, #098 sec. 40 percent occurrence ~ re) oO Beam sea : Goris a ea ee LaRES ZS SERRA LESERPSESASe See BEReeeo NS2ae BE GR Z aa aise 2 cies A 1. W inrad. sec-' Fig. 10. Distribution and spectrum of sway 969 Sy (W) in m? sec. Verhagen SWAY DEVIATION IN m percent occurrence -10 0 5 10 W inrad.sec~' Fig. 11. Distribution and spectrum of sway 970 Sy (w) in m?sec. Floating Body tn Irregular Waves SWAY DEVIATION IN m “ a Beam sea Cwiy/3" 2.36 m , T=66 sec. W in rad.sec.~' Fig. 12. Distribution and spectrum of sway Ar ge Verhagen V. COMPARISON BETWEEN EXPERIMENTAL RESULTS AND CALCULATIONS _ Some tentative calculations have been carried out amplifying and illustrating the aforementioned suppositions. Mean Drifting Force The mean drifting force on the moored vessel in the long- created irregular waves has been determined by a linear superposi- tion of mean drifting forces in the regular wave components of the known wave spectrum. As has been shown by Maruo [1] a.o. the principle of superposition can also be applied to determine the non- linear mean drifting force. For head seas the mean longitudinal drifting force is 2nr® F. = he Fl) _ S.(w) dw (4) © petiB/L § In beam seas the mean lateral drifting force is ) peie @ F = 2pet | mre S(u) do (2) O 2pgo L Formulas for the mean drifting force in regular waves on freely floating, completely restrained or elastic moored bodies are obtained by Maruo[2] and Newman[3]. Numerical calculations are usually carried out on specific formulas for the drifting force on a slender body. [3] As is well known, the agreement of these calculations with experiment is on the whole not very satisfactory. The slender body approximation results f.i. in a vanishing mean lateral drifting force. Therefore an engineering approach is pre- ferred using available experimental data on a similar ship form. The estimated curves of the longitudinal drifting force in regular head waves at zero forward speed and the transverse force in beam waves are given in Figs. 13 and 14. Using these data the expressions (1) and (2) for the mean drifting force in irregular waves can be solved. The results compared with the experimental results are shown in Table I. The agreement is reasonable. 972 Floating Body in Irregular Waves 06 Oo 0.5 1.0 W in rad.sec~! Longitudinal drift force for head seas Pig. 13. 20 15 > | Xin 640 re ee fo) a 0.5 oO Oo 0.5 1.0 1.5 W inradsec>! Fig. 14. Lateral drift force for beam seas 973 Verhagen TABEE: 1 Wave Mean drifting forces characteristics in tons *® in sec force measured force calculated force measured force calculated height Cy,, in m Average period Wave direction Longitudinal Longitudinal Transverse Transverse » G @?2 0 ot MS ot G ter) ran YW) The low frequency drifting force This part of the drifting force has been estimated as follows: The measured wave height record {€,(t) is squared c(t). This squared wave height function can again be analys ed by a spectral analysis. The spectral density function of 7¢°(t) is determined. For an example see Fig. 15. As can be seen from this figure it contains the sum and difference frequencies of the original wave height spectrum. The difference frequencies are now of special interest. They are related to the envelope of the original wave height record. The r.m.s. value of the low-frequency energy variation is: V2 Cedi | f Si p2(w) ao diff. freq. Te att is proportional to the mean square value of the fluctuating part of the wave height envelope. The energy in the fluctuating part of the envelope curve depends largely on the occurrence of "high" waves. As discussed earlier the occurrence of "high" waves is pro- portional to 1/T per unit time. So the relation between the r.m.s. 974 Floating Body in Irregular Waves 0.3 Vv © rr ae Eoce £ ra) ww) x WY 0.1 oO 05 1.0 1.5 2.0 2.5 W inrad.sec.~ Fig. 15. Spectral density function of a half times the square of the wave height value of the low-frequency energy variation and the characteristics of the waves becomes =n ows = constant ° TE ditt Aa Figure 17 shows that the produced wave spectra fit this relation quite well. Now the exciting force must be determined keeping the vessel in a fixed position. In that case the exciting force is due to wave reflection alone. In regular head waves this force is F, = $pga*B sin* a x 975 Verhagen ord SE ate in m: w2 2 3 Swiys /# in m2. sec.-1 Fig. 16. Relation between the low-frequency energy variation and wave characteristics when a is the wave amplitude and sin*@ is the mean square of the angle between the tangent at the ship's waterline and the longi- tudinal axis. Now the re-m.s. value of the low-frequency exciting force becomes 2 Op, = PEF aitt B sin” a and the r.m.s. value of the low-frequency motion is a x By * Woy The surge damping B, is obtained from an extinction curve. The numerical values are B,* w,,= 1.50 ton/m, sin*a@=0.24, The results compared with the experimental results are shown in Table II. The agreement is good. 976 Floating Body in Irregular Waves TABLE II o, inm o, inm calculated measured 0.96 0.87 0.86 0.77 2.87 2.54 where y is a coefficient depending on the beam-wave length ratio. For the wave lengths under consideration (\/B = 5) the mean value of y is obtained from experimental data on Series 60 models is about 0.5. This value increases up to one for shorter waves and decreases for longer wave lengths. Now the r.m.s. value of the low-frequency exciting force in irregular beam seas becomes Oy = YPB% aieg £ and the r.m.s. value of the low-frequency sway motion is The sway damping is again obtained from an extinction curve. The numerical value is Byw., = 4 ton/m,. The results compared with experimental results are shown in Table III. TABLE III ~ x o, inm co, inm sec 1.19 9.8 0.72 0.67 1.96 9.6 2012 2.05 2.30 6.6 Se? 4.56 a Verhagen Taking in mind that in the last case the value of y must be higher than 0.5, due to the shorter wave length, the agreement is very good again, © Gy SURGE uw2 Swis /# in m2 sec.-' Fig. 17. Relation between surge, sway and wave characteristics ACKNOW LEDGEMENT The author would like to thank the staff of the Seakeeping Laboratory of the N.S.M.B. for the many fruitful discussions. He is indebted to Mr. Tan Seng Gie who carried out the experimental program. 978 2. 3 4, Floating Body in Irregular Waves REFERENCES Maruo, H., "The excess resistance of a ship in rough seas," Int. Shipbuilding Progress, 1957. Maruo, H., "The drift of a body floating on waves," Journal of Ship Research, 1960. Newman, J. M., "The drift force and moment on ships in waves ," Journal of Ship Research, 1967. Verhagen, J. H. G. and van Sluijs, M. F., "The low-frequency drifting force on a floating body in waves," Int. Shipbuilding Progress, 1970. 979 : eee CY. a ote > _ ¥iee TT oY, a go few BEN! eet ufot eer on Bee ik mn $4 | bo op Oye Shel Ghar laa vel Ul) yee Dg nak meet ‘erry tLe oP wy Gee: ie ip ee st pe a4 Op? 7 ®ageadgs0% ri olde 6 te sankielsad eaeoxe odT" , Tee 4 : BEA a aenthar el galb lt aga le 7 a farane! * ~ovaw fe gaitach ypod 6 io yh vot 4 cil fe ty td C07 Se | { aly boxes de ne " ry a4 Arye vaew al bqlde nm doerrets bos Satoh Mish say a ae a oy) l steel dozaoeoH que 40 cas ab Weutrnus.-wolets” g.% aoa ,a(icte cav bos .O. oH ic a ives ...rat -" dovaw! Ab ybod ys Pools ao so7ok yale - | OTe! ,sastgon | ; 7 ' . - ‘ i, ‘ ; al i i ae 1a 7 a Bs i ' - , uv = { ry s a rs . J =e -e hy oAaenty ‘ - ' a } s | . cas . 7a 4 ew i vyav¢ t 7 >) ACN IS Lal Ss . » > «¢ , = F 28 @ "ine Y= 1% a H > Covi Hs A ced ire amie ap acc paewry e1 Ahe N, £) it. Be. for thy mary Leu fal Gis rea Brin. te, Letitod) & if Tun ig Cale ws sPYisd oft the levi” af : Biya Tair. DYNAMICS OF SUBMERGED TOWED CYLINDERS M. P. Paidoussis MeGtll University Montreal, P.Q.; Canada I. INTRODUCTION Interest in the dynamic stability of towed ships dates back to the halcyon-days when solutions to engineering problems could still be obtained by experience, without the aid of sophisticated analysis. Certainly, operators of horse-drawn barges in canals must have been aware of possible instabilities and remedial actions. Never- theless, to the author's knowledge, the first substantive paper on the subject, by Strandhagen, Schoenherr and Kobayashi [ 1], did not appear until 1950. This is also surprising, if one considers that both the analytical techniques and physical concepts were understood long before that; indeed much earlier work does exist on the closely related topic of stability of airships moored to a mast and kite bal- loons, starting with the work of Bairstow, Relf and Jones [ 2] in 1915, and followed by the work of Munk [ 3], Glauert [4], and Bryant, Brown and Sweeting [5], for instance. Strandhagen et al., and the discussors of their paper, firmly established the following important criteria for stability of a towed ship: (i) the point of attachment of the tow-rope should be ahead of both the center of mass and the center of pressure of the (static) lateral hydrodynamic forces acting on the ship; (ii):the ship should be stable when moving untowed; (iii) in cases where (ii) is not satisfied, then the system could be rendered stable by either short enough or long enough tow-ropes. It is noteworthy that the criteria for stability, at least for the linearized theory of small departures from course, apply to all towing speeds, so that for a given configuration a (rigid) towed ship is either stable or unstable irrespective of how fast it is being towed. Instabilities were found to be of two distinct types: (a) yawing, i.e. azero-frequency, amplified motion which in aero- elasticity would be referred to as 'divergence', and (b) oscillatory instability, where the system, when disturbed, oscillates about its position of rest with increasing amplitude. More recently, interest in the instability of submerged towed bodies has arisen mainly in connection with sonar applications. Here 981 Patdoussis the body housing the sonar device is towed deeply submerged by sur- face craft, for the purpose of hydrographic survey, submarine detection, or location of schools of fish. We must refer to the work of Strandhagen and Thomas [6], Richardson[7], Laitinen [8], Patton and Schram [9], Jeffrey [10], Schram and Reyle [11], and Whicker [12]. The stability problem for the sonar-type towed bodies is of course quite similar to that of a towed glider [5]. The work referred to here deals with the dynamics of the towed body system as a whole; the geometry of the towed body for these applications tends to be fairly complex, and the analysis quite elaborate. A considerable amount of work also exists on the equilibrium configuration and thedynamics of towing cables, starting with McLeod's [13] and Relf and Powell's [14] work, to more recent work by Landweber and Protter [15], Pode [16], [17], O'Hara[18], Kochin [19], Eames [20], and Albasiny and Day [ 21]; this represents a by no means exhaustive list of references. The author's interest in this field comes from work associ- ated with yet another application: that of the Dracone flexible barge, which is a flexible sausage-like container towed behind a small craft, and used for the transportation of oil and other lighter-than-water cargoes, including the sea transport of fresh water to arid lands (e.g. to some of the Aegean islands from the mainland). The new element that enters the problem in this case is that of elastic forces, making this a problem in the general area of fluidelasticity (cf. 2a) \e The first analysis of stability of the Dracone was by Hawthorne [ 2a » Later, the author studied systematically the dynamics of flexible slender cylindrical bodies immersed in axial flow, for various con- ditions of end-constraint [ 24] , [ 25], including the case of a towed slender cylinder [26]. In the latter case, both rigid-body type in- stabilities and flexural instabilities were shown to exist; stability was highly dependent on the towing speed. It was suggested [ 27] that cylindrical or quasi-cylindrical containers towed underwater by a small submarine could be used to transport liquid cargoes to and from arctic ports, avoiding the hazards of surface transportation in ice-covered seas. The con- tainers could be either flexible or, more likely, rigid; there could of course be a string of such containers towed by the same submarine. This idea has taken added poignancy since the oil discoveries in the Arctic. In this paper we shall re-examine the problem of stability of a submerged cylindrical body, both flexible and rigid, towed by a submarine craft. 982 Dynamtes of Submerged Towed Cylinders Il THE EQUATION OF SMALL LATERAL MOTIONS OF A FLEXIBLE SLENDER BODY IN AXIAL FLOW We shall derive the equation of small lateral motions of a slender body of revolution of the type shown in Fig. 1(a); the body is supposed to be supported somehow so that it is not washed away downstream. The fluid is incompressible and of density p; it is flowing with velocity U parallel to the x-axis, which coincides with the undisturbed longitudinal axis of symmetry of the body. The body is of mass pér unit length m(x), cross-sectional area S(x), and flexural rigidity ElI(x). Fig. 1(a) Diagram of a flexible, slender body of revolution in axial flow We consider small motions y(x,t) and assume that y, dy/ax, 8*y/ax* to be all small, so that no separation occurs in cross- flow. Moreover, we assume that dS/dx is small everywhere, except perhaps at the ends of the body, so that no separation occurs in the axial flow (except perhaps at the rear end), and so that slender-body theory may be used. Also d(EI)/dx is assumed to be small, which, together with the restrictions on the displacement function, allows us to use the simple Euler-beam approximation to describe the flexural forces. The body is further assumed to be of null buoyancy and uniform density, so that no constraining force in the y-direction nor a moment is necessary to keep it lying along the x-axis, at least at zero flow velocity. Furthermore, the motions are considered to take place within the (x,y)-plane, which for the sake of simplicity is assumed to be horizontal, Finally, we neglect internal dissipation in the material of the body. We now consider an element 6x of the body. The forces and moments acting on it are shown in Fig. 1i(b). Q is the transverse shear force, ™ is the bending moment, T is the axial tension, F, and F, are the normal and longitudinal components of frictional forces per unit length, and Fy, is the lateral inviscid force per unit length. 983 Patdoussis — we ce ais [ove nev, ~*~ |e ao tad x Fig. 1(b) Forces and moments acting on an element a(x) of the body Taking force balances in the x- and y-directions and a moment balance we obtain aT dy _ Sx0 Fu En Bam (1) aQ 8 in by dy a*y _ da 7 Fut q(T By) + FL GR - Fa mogee = 0 ® am Q= a3. 9 (3) where the inertia forces in the x-direction have been neglected. We next consider the functional form of the forces. The lateral inviscid force F,5x represents the reaction on the body of the force required to accelerate the fluid around it, and may be written as F, = [(2/at) + U(8/dx)] (Mv), (4) as discussed by Lighthill [ 28], [29], where v is the lateral relative velocity between the body and the fluid flowing past it, and M is the virtual mass of the fluid. Here the effects of sideslip have been neglected, effectively assuming that each cross section of the body is 984 Dynamites of Submerged Towed Cylinders part of an infinite cylinder; boundary layer effects have also been neglected. The virtual mass M(x) = pS(x), and v(x,t) = [ (8/8t) + U(8/8x)][ y(x,t)] , which substituted into (4) yield F, = pS[(8/at) + U(8/ax)]*y + pU[(dy/at) + U(dy/8x)] (dS/dx). (5) The frictional forces, as proposed by Taylor [ 30], and elabo- rated by Paidoussis [ 24], [25] are taken to be tr} " 2¢,(PS/D)U* sini and F, = 3¢,(pS/D)U* cos i, where i is the instantaneous angle of incidence on the cross-section and is given by i= sin! (v/U), and D= D(x) is the diameter. Accordingly, Fy, and F, are given by Fy = 2¢y(PS/D)U[(ay/at) + U(ay/ax)] and F, = 2¢,(PS/D)U?. (6) Finally, we note that the bending moment is related to the flexural rigidity by M = El(0*y /ax?). (7) Now, substituting (6) into (1), neglecting terms of second order of magnitude, and integrating from x to L, we obtain L T(x) = T(L) + e,pu*{ [ S(x) /D(x)] dx, x where T(L) is the value of T at the downstream end. We consider that T(L) is non-zero and that it arises from possible form drag at the end. We accordingly write e 2 T(x) = 2copS(L)U nF 2c, purl [ S(x) /D(x)] dx, (8) x where cy, is the form-drag coefficient. Substituting now (3), (5), (6), (7) and (8) into (2), making use of (1), and neglecting terms of second order of magnitude, we obtain 985 Patdoussts 2 2 os &, (er 24) + 05(2 tuhyy +eu(R+u¥) ss + 5 uC) + Us i 2 -5 ous [¢,S(L) + J oo ax |33 tm2y =0, (9) xX which is the equation of small lateral motions. For a uniform cylin- der this equation becomes eroy +u(2 ru zyy +5 onl (ge + Use 2 2 1 2 L-x] 9 8 4 -3 MU [cg +c D |< +m Be = 0, (10) where the diameter, D, and M=pS are now constant. We note that in the absence of frictional forces, (10) becomes the governing equation for small motions of a cylindrical beam con- taining flowing fluid [31], where we interpret M as the mass of the contained fluid per unit length. The physical similarity between the internal and external flow cases is striking, albeit that in the former case fluid friction does not enter the problem. We shall refer to this later. We finally note that Eqs. (9) and (10) also hold to describe the motions of a towed flexible body, if we identify U as the towing speed, provided the tow-rope forces are taken into account as part of the boundary conditions. III. BOUNDARY CONDITIONS Clearly the boundary conditions will depend on the mode of end constraint. Let us consider the case of a towed flexible cylindrical body shown in Fig. 2. The body consists of a uniform cylinder ter- minated by a rounded 'nose' and a streamlined, tapering 'tail', incor- porated to provide reasonable axial flow conditions over the body. We assume that the towing craft moves horizontally in a straight course with uniform velocity U, so that the tow-rope in its undis- turbed state lies along the x-axis; we also consider the assumptions made at the beginning of §2 to hold. We may use Eq. (9) to analyze the system, together with boundary conditions stating that (a) at the downstream end, x=L, 986 Dynamtes of Submerged Towed Cylinders Fig. 2 Diagram of a towed flexible, slender cylinder with streamlined "nose" and "tail" the bending moment and shear force are zero, and (b) at the upstream end, x= 0, the bending moment is zero, but the shear force is equal to the normal component of the tow-rope pull. It is obvious, however, that the very form of Eq. (9) will depend on the shape of the nose and the tail. As we are only looking for the general characteristics of the dynamical problem, this is not convenient. We shall instead proceed as follows: (i) we shall use Eq. (10) which satisfactorily applies over the uniform, cylindrical part of the body; (ii) the forces acting on the non-cylindrical ends will be lumped and incorporated in the boundary conditions. For this process to be meaningful we must have 4, < IV. EQUATION OF MOTION AND BOUNDARY CONDITIONS OF [ 26] The equation of motion given by Eq. (10) is not identical to that previously derived by Paidoussis [ 26]. The difference is in the frictional terms, because of the different manner in which frictional forces were resolved in [26]. The boundary conditions are identical. As we shall make use of the results obtained in [ 26] , we give the equation of motion below, for reference. 4 erge+m(2+uZy)y+3e(MZ)(¥ bc, ( MU ates = 0. (10a) The equation of motion and boundary conditions used in the ‘new theory' presented in this paper are believed to be more self- :onsistent than those of the 'old theory' of [ 26]. 989 Patdoussis Ve. DYNAMICS OF TOWED FLEXIBLE CYLINDERS 5.1 Method of Analysis Upon expressing the equation of motion and the boundary con- ditions in dimensionless form, the dynamics of the system may be found to depend on the following dimensionless parameters: (i) £, and f,, which were defined in §3; (ii) €cy, €c,, c, and c,, where € = L/D; (iii) A= s/L; the ratio of tow-rope length to body length; (iv) x, =x,/L and x, =x,/L, where x, and x, were defined in §3; (v) u=(M/El)'”UL, the dimensionless towing speed. It is noted that according to the assumptions made in the theory, m=M. We shall not present the analysis here, as it is adequately documented elsewhere [26], [27]. Suffice it to say that solutions were obtained of the type where Y is a function of x/L, T is a dimensionless time and w is the dimensionless frequency given by Q being the circular frequency of motion. In general, w will be complex. Clearly, we have an infinite set of frequencies, W;, as the system has an infinite number of degrees of freedom. If the imaginary components of the frequencies, Im/(wj;), are all positive, then the system will be stable. If, on the other hand, for the jth mode we have Im (w;) < 0, then the system will be unstable in that mode; now if the corresponding real component of the frequency, Re (w, ), is zero this will represent a divergent motion without oscil- jailons: which we shall call yawing; if Re (wj) # 0, then the insta- bility will be oscillatory. 990 Dynamics of Submerged Towed Cylinders The calculation procedure was as follows: (a) a set of values GE tp fp» EC.» ECL, Cy» Co» Xj» X_ and A were selected; (b) the complex frequencies of a few of the lowest modes of the system were traced as functions of u, starting with u = 0. MERGES WITH FIRST MODE 3.6 3.5 ZEROETH Fig. 3 WwW a a % 2 > ) MODE (A x Re (w)=0 Re(w) The dimensionless complex frequencies of the zeroth and first modes of a flexible cylinder with ecy=ec,= 1, f,=f,=1,¢, = co = 0, A= 1 Ne = 2 = 0.01, as a function of the dimensionless towing speed u. (Theory of [ 26]). 991 Patdousstis SECOND MODE 40 50 60 Re (w) WwW a re ” z 5 -0.2 fe) 20 Fig. 4 The dimensionless complex frequencies of the second and third modes of a flexible cylinder with ec,=ec,=1, f,=f,=1, c,=c,=0, Aas Ge Xj =e = 0.01. (Theory 5.2 Results Based on the Theory of | 26 Typical results are shown in Figs. 3, 4 and 5, obtained by using Eqs. (10a), (11), (12) and (13). We first consider Figs. 3 and 4 applying to bodies with well streamlined nose and tail, and A=1, Figure 3 shows the behavior (with increasing towing speed) of the two modes which at zero towing speed have frequencies w)=w,=0; these are the so-called zeroth and first modes and, at low towing speeds, are associated with quas!- rigid body motions -- a matter to be further discussed in §6. Figure 4 shows the loci of the so-called second and third modes of the system as functions of towing speed. The frequencies of these modes at zero towing speed correspond to the second- and third-mode frequencies of the flexible body treated as a free-free beam; accordingly, these (and all higher modes) are flexural in character. 992 Dynamics of Submerged Towed Cylinders We observe in Fig. 3 that both the zeroth and first modes lead to instabilities for small, finite u. The instability associated with the zeroth mode is a yawing one, while that associated with the first mode is oscillatory. We see that for u> 3.05 the oscillatory instability ceases in the first mode, re-appearing at u* 3.65. However, at much lower towing speed (u* 2.3) the system loses stability in its second (flexural) mode, as shown in Fig. 4, and at u* 4in its third mode. In short, this particular system is subject to several types of instabilities; at low towing speeds it is subject to quasi-rigid body instabilities, and at higher towing speeds to flexural oscillatory instabilities as well. Figure 5 shows the zeroth and first mode of a system with a well streamlined nose and a very blunt tail. We see that it is not subject to yawing instability, and the first mode is only unstable in the range 0 5.29. Accordingly, a blunt tail stabilizes the system considerably. Also shown in Fig. 5 is the first mode of a system with a less than perfectly streamlined nose; we see that the range of first-mode oscillatory instability in this case is larger, i.ewg O 4d, = 0, Ol. Also the first mode with f, = 0.8. (Theory e [ 261). 993 Patdoussts Based on such complex-frequency calculations it was possible to construct stability daigrams illustrating the effect of various parameters on the stability of the system. Examples are given in Figs. 6 and 7 showing the effect of stability of f, and A, respectively, Other similar stability diagrams may be found in[ 26]. The following general conclusions may be drawn: (a) for optimal stability the tail should be blunt (f, small, cy large), the nose should be well-streamlined Oe ae) and the tow- rope length should be short (A small); (b) asystem that is.unstable by yawing, within a range of towing speeds, can be stabilized by blunting the tail, but not by manipulating the length of the tow-rope; (c) in some cases it is possible to stabilize a system which is unstable at low towing speeds, by towing it faster, within a specified range of towing speeds. Conclusions (a) above are not contrary to reported experience with rigid bodies. On the other hand, (b) may sound surprising. The fact is that the onset of yawing is not a function of A, nor is its cessation (§6.2). This is also true with f,, Finally, conclusion (c) is characteristic of the dynamical behavior of towed flexible cylinders. 6 SECOND MODE OSCILLATORY INSTABILITY FIRST MODE 5 OSCILLATORY INSTABILITY fy, ‘ Oy Ne SS 4 ae u STABLE REGION ZEROETH MODES aus FIRST MODE Za YAWING YAWING AND OSCILLATORY INSTABILITY FIRST MODE OSCILLATORY INSTABILITY 0 0.1 0.2 0.3 0.4 O15 0.6 0.7 0.8 0.9 1.0 BEUNT TAIL ———— f, ——-» ELONGATED STREAMLINED TAIL Fig. 6 Stability map showing the effect of the tail shape fora flexible cylinder with ecy= ec;=1, f,=1, c= 0; A= X, =X> =0001 and c,=1-£, (Theory of 126i fe 994 Dynamies of Submerged Towed Cylinders / / / THIRD - MODE OSCl LLATORY INSTABILITY FIRST- MODE OSCILLATORY INSTABILITY SECOND - MODE OSCILLATORY INSTABILITY SECOND MODE STABLE REGION ~ wor YAWING A/S YAWING AND FIRST- MODE FT OSCILLATORY INSTABILITY | va fo) 0.1 0.2 0:5 o:5 | 2 5 5 10 A Fig. 7 Stability map showing the effect of A for a flexible cylinder with €cy=ecy=1, f,= 1, c,=0, f,=0.6, cy = 0.4, X, =X, = 0.01, (Theory of [ 26]). Some experiments were performed designed to test the theory [26]. Rubber cylinders of neutral buoyancy were held in vertical water flow by a nylon 'tow-rope'. Provided the tail was streamlined and the tow-rope not too short, 'criss-crossing', essentially non- flexural oscillations developed at very low flow; these were inter- preted as corresponding to first-mode oscillatory instability. At higher flow velocities, flexural oscillations developed with a modal shape corresponding to that of the second mode; sometimes, at yet higher flow velocities, oscillations with a third-mode modal shape developed. These flexural oscillations were interpreted to cor- respond to second- and third-mode oscillatory instabilities, Se- quences of ciné-film frames depicting these oscillations are shown in Figs. 8 and 9. Finally, it was observed that for sufficiently blunt tail and short tow-rope, the system was completely stable. Thus the experimental results were in generally good qualitative agree- ment with theory. 995 Patdoussts (a) (b) Fig. 8 Photographs in consecutive frames showing a cylinde. 11.1 in.long and 0.54 in.diameter with streamlined nose and tail executing (a) criss-crossing, essentially rigid- body, oscillation (8 frames/sec), and (b) second-mode flexural oscillation (24 frames /sec) Quantitative agreement in the various instability thresholds and stable zones, based on estimated values of some of the theoreti- cal parameters, was also fairly good. 5.3 Results Based on the New Theory Typical results based on the new theory and obtained by using Eqs. (10) - (13) are shown in Figs. 10 to 13. 996 Dynamtes of Submerged Towed Cylinders (a) (>) (c) Fig. 9 Photographs in consecutive frames showing a cylinder 15.8 in. long and 0.68 in. diameter executing (a) criss- crossing, essentially rigid-body, oscillation (8 frames/sec), (b) second-mode flexural oscillation (24 frames/sec), and (c) third-mode flexural oscillation (24 frames/sec) Figures 10 and 11 show the dynamical behavior, with in- creasing towing speed, of the zeroth, first, second and third modes of a system with well streamlined nose and tail and A = 1; this is the identical system, the dynamical behavior of which, according to the 'old' theory, is shown in Figs. 3 and 4. We observe that, accord- ing to the new theory, the system is considerably more stable than predicted by the old theory. Thus, the first mode is unstable only for u<0.74 (not discernible in the scale of Fig. 10); moreover, the unstable locus originating from merging of branches of the zeroth and first modes regains stability at u = 6.3. Similarly, the system loses stability in its second and third modes at respectively higher towing speeds than predicted by the old theory. Further calculations were conducted for the same system as above but with other values of f,, always taking c,:= 1\- foe It was found that the first mode is not uniformly stabilized with decreasing f5 as was the case with the old theory (cf. Fig. 6). The ranges of instability of the first mode, for various values of fo, were found to be as follows: 0 MULE CT ty. L 2 fxg MU Srey erf.-f,, 1. L7° + MUL [p< +saayp]¢ 11 f, +f, @MUL 3 erp 5 Je=o, aL where Crp= roe By (1B +c, tCo. The same equations could have been obtained from first principles. It is noted that here L is the length of the cylindrical portion of the body which is smaller than the over- all length, as used in §5, by L, +45, the difference never exceeding a few per cent. We non-dimensionalize these equations by introducing n= y,/L, A=s/L, €=L/D, x, = x*,/L, Xp = x,/L and 7 = Ut/L, and consider solutions of the form 1=He”* and d= @e where w is a dimensionless frequency defined as w= QL/U, being the complex circular frequency of oscillation. Substituting 7 and $ into the non-dimensionalized equations, and noting that our assumptions require m = M, we obtain {[2 +x, (1 +4) +xe(t t£)] (-04) +[ 5 ecy +4, -£,] wi) +[ 5c, /A]{ H + {- Fix, +8) - x, H6)] (-04 +L 2-5 (E, +6] (oi) +[5 Ecytf,- f,-4 (1/Mcql | ® = 0, (17) and 1 + Xi (a +2) - 4 Xp(4 + £,)(-o*) -[ Sf, 7£,) | (wi) - [ 5 Cr p/4\) 1) + \[Z +4 x) (te, Sxl! + £,)] (-w*) +S, - f,) +3 €cy] (wi) + [5 (1/Mcyp- 5 (ie sito] i = is (18) 1004 Dynamics of Submerged Towed Cylinders Similar equations were obtained when using the theory of [ 26], i.e. Eqs. (10a), (11), (12) and (13), namely }[2+x, +4) +xo(1 +£,)] (- ) +L Ecytf, - fal (wi) +15 cy)/A]{ H at }- 4 [x,( +f) - X2(4 + f5)] (-w*) 7 [ 2 - Fig, +£,)] (wi) +5 ele, he) +£,-£,-4cqp/A] 1 = 0, (17a) }- [5 x14 +) - Sxolt +e] (-04 -[ 4 +2)] (od) - [4 c,,/Al tH +} taxi (+8) +4 xo Hey) (-04) +14 (e- 6) + A ecy] (wi) tlgcp/A-F(, +4)]1 6 = 0, (18a) For non-trivial solution, the determinant of the coefficients of H and ® in (17) and (18), or in (17a) and (18a), must vanish, yielding a quartic in w, i PAG Bao eco =o, (19) 6.2 Calculations Based on the Theory of | 26] The aim here was to compare the dynamical behavior of the rigid body to that of a flexible body; as the rigid body may be regarded as a flexible one of very large flexural rigidity, it would be reasonable to expect correspondence of the dynamical behavior of the rigid body to the 'rigid-body' modes of the flexural one, i.e. the zeroth and first modes. Recalling that the dimensionless flow velocity in the case of a flexible body was defined as u = (M/EI)'/* UL, the dynamical behavior of the rigid body should approach that of the flexible one as uO. Two sets of calculations were conducted, as described below. The four rigid-body frequencies, given by (19), were computed for a number of cases and the values compared with the existing com- plex frequencies of the flexible body. As an example, let us compare the case corresponding to Fig. 3. The four frequencies are w, = 1.956, we= -0.761, w,,= + 0.582 - 0.3571. These compare well with the four frequencies associated with the flexible body for u=0.7, namely w, = 1.934, O5.=9 0. 734, = + 0.580 - 0.350i; the first ek 1005 Patdoussits two are associated with the zeroth mode, and the other two with the first mode and its mirror image about the [ Im (w)] -axis. Surprisingly, the correspondence of the rigid-body frequencies to those of the flexible body, for the apparently arbitrary value of u = 0.7 persists for other values of f,, as shown in Table 2. This value of u= 0.7 can be explained as follows. We have defined the dimensionless frequency of the rigid body by w,p= QL/U. On the other hand, the dimensionless frequency of the flexible body was defined as w,,= [ (M+m)/EI]’“QL*, which may be rewritten as o.,=[(M +m)/M]/2u2L/U, where u is the dimensionless flow velocity (§5.1). The assumptions made in the theory require that m-=M, sothat wy,= V2uQL/U. Now, if the dimensional frequency, 22, of the rigid body and of the flexible body are identical, we may re- write this as Wey = ANE and we can see that identity of the dimen- sionless frequencies will occur when u = 1//2* 0.707, Calculations were also conducted to pin-point the thresholds of yawing and oscillatory instability in terms of f,, A etc., and to compare with the existing stability diagrams, e.g. Figs. 6 and 7. TABLE 2 RIGID-BODY AND FLEXIBLE-BODY FREQUENCIES COMPARED Other parameters: A=1;€cyaé€cps1, f) = t-c, = 1, 1¢, = 1h X; =X2 = 0.01 Rigid body Flexible body (u = 0.7) 0. 58-0. 36i 0.58-0.35i 0. 84-0. 39i if 0. 83-0. 38i 4.28-0.04i 1. 28-0.004i In the case of a rigid body with parameters corresponding to those of Fig. 6, it was found that oscillatory instability exists for 0=f,=1 and that yawing occurs for f,>0.5. Correspondingly, for the case of Fig. 7 it was found that yawing persists throughout, and "Upon examination, the st: »le branch of the zeroth mode as given in[26] was found to be in error; the locus moves away from the [Re (w)] - axis much faster than shown in Fig. 3, | 26]. The corrected value for u= 027 is given here. 1006 Dynamtes of Submerged Towed Cylinders that oscillatory instability occurs for A>0.20. Once again agree- ment in behavior of the rigid body and the flexible body (for u = 1/2) is good. Similar calculations confirmed agreement with the other stability maps of [ 26]. Two stability diagrams were constructed (Figs. 14 and 15) showing the effect of ECy, €Cr, fp and A on stability, for comparison with those to be obtained using the new theory. a YAWING a Pg Fig. 14. The effect of €cy, €c; and f, on stability of a rigid cylinder with A= 1, f, = 1-c,=1, Co= 1-f2 and x, oa 01, —— ec, = 0.1; =-— €¢,= 0,5; --- ec,=1, (Theory of [26]). In Fig. 14 we observe that unless ec, is considerably less than €cy, the region of oscillations practically covers the whole plane; moreover, oscillations persist to lower values of f, than yawing does. In Fig. 15 we see that a sufficiently short tow-rope has a very definite stabilizing effect on the system, as far as oscillatory insta- bility is concerned. Very long tow-ropes, on the other hand, evi- dently have a very weak stabilizing effect. The foregoing clearly establish that the dynamical behavior of the rigid body is represented by the behavior of the zeroth and first modes of the flexible body at small u. One noteworthy aspect of the analysis is that the existence of yawing instability cannot be affected by varying A, i.e. by altering 1007 Patdoussts OSCILLATIONS Fig. 15. The effect of A on stability of a rigid cylinder with f,=1-c,=1, ecy=ec,=0.5, Cy = 1-f 2 and xX, =X, = 0.01. (Theory of [26]). the tow-rope length. In the case of the rigid body this becomes obvious upon considering equation (19). Since the threshold for yawing instability implies w= 0, this threshold is established by the equation E=0. Now E is found to be E = (c4p/2A)[5 €(cy + Cy) - 2£,] é Clearly we see that the threshold is not dependent on A. This seems to be in contradiction with Strandhagen's et al. [1] criterion (iii) for the stability of towed ships (as given in §1); on closer examination of their own work, however, we see that the equivalent of term E, in their case also, contains A as a common factor. Accordingly, we must conclude that the only form of instability the existence or non-existence of which may be controlled by the tow-rope length is oscillatory. 6.3 Calculations Based on the New Theo Calculations were also conducted with the new theory. It was found that, in this case also, the dynamical behavior of the rigid body corresponds to that of the zeroth and first modes of the flexible one at low towing speeds -- quantitative correspondence of frequencies occurring at u=1/72 as before. Stability plots were also constructed (Figs. 16and17). These are markedly different to those given by the old theory (Figs. 14 and 15), the main difference being in that oscillatory instability according 1008 Dynamtes of Submerged Towed Cylinders OSCILLATIONS (ec =1) fe OSCILLATIONS (€c,=0-5) OSCILLATIONS (€ c;= eae se OSCILLATIONS(€c,=1)_ — ees OSCILLATIONS = (€c,=1) {-- <= Fig. 16 The effect of. €c,, €c, and f, on stability of rigid cylinders with A=1, c,=,1-f,, c¢) =0 and xX, = Xo= Oe Oty a ee f, = 0.8; --- f, = 0.7. (New theory). OSCILLATIONS Fig. 17 The effect of A on stability of a rigid cylinder; Co= 1-fp, ECy= €Cp7= 0-5, X,; = Xo = 0,01; — f, = 1-c, = 1; ---f,=0.8,c,=0. (New theory). 1009 Patdoussts to the new theory occurs over a much more limited range of system parameters, while yawing is more prevalent. Comparing Fig. 16to Fig. 14 we note the following essential differences: (i) yawing, being independent of c, according to the new theory, is represented by a single line; (ii) according to the new theory oscillations persist to progressively lower values of fp, as €cy, is reduced, while the opposite trend was predicted by the old theory; (iii) according to the new theory, for f, = 1, there are large regions in the (€cy, f,) parameter space where yawing occurs alone, but not where oscillations occur alone; on the other hand, according to the old theory the opposite is true. However, this last point applies only for f, = 1. It may be seen that for f,= 0.8 and 0.7, the results of the new theory become much more like those of Fig. 14 in this respect. We note that the onset of yawing is independent of f, as well as Cy, so that the line shown in Fig. 16 applies to all cases examined therein. Once again considering term E of Eq. (19), which in this case is given by E = (CAN €c,- 2f,), we see that £,, Cy, Cos Cy and A are all parameters that cannot affect the onset of yawing. We next compare Fig. 17 to Fig. 15. The results are quite similar, except that (when f, = 1) oscillatory instability occurs over a more limited range according to the new theory than predicted by the old theory. However, the results of the new theory for f, = 0.8 when compared with those of the old one for f, = 1 are quite similar. The results for f, = 0.7. not shown in Fig. 17, are of interest in that oscillatory instability, in that case, occurs practically over the whole plane, i.e. for fg>0.013 for A=0.1 and for f,> 0.008 for N= 02. VII. CONCLUSION In this paper we have reviewed an existing theory for the dy- namics of flexible cylindrical bodies towed underwater, and developed a parallel theory for rigid cylinders. It was shown that, whereas the dynamical problem in the case of rigid cylinders is independent of towing speed, in the case of flexible cylinders the dynamical behavior (and stability) of the system is highly dependent upon towing speed. It was found that, in general, flexible towed cylinders are subject to both flexural and 'rigid-body' instabilities, the latter occurring at relatively low towing speeds. It was also established that at low towing speeds, the dynamical behavior of the flexible cylinders in their two lowest modes (the so-called zeroth and first) correspond to that of rigid cylinders, which of course have but two degrees of free- dom. Thus the study of the dynamics of towed flexible cylinders yields sufficient information to establish the dynamical behavior of the corresponding rigid bodies. 1010 Dynamics of Submerged Towed Cylinders A new theory was also presented (for both flexible and rigid cylinders) which, it is believed, represents the physical system more closely. The main difference in the results obtained by the old and new theories are associated with the behavior of the rigid- body modes of the system; specifically, the new theory predicts the system to be more stable in its first (oscillatory) mode and less stable in its zeroth (yawing) mode than does the old theory. The new theory is in general qualitative agreement with ex- periment. Quantitative agreement cannot be assessed definitively until a means is found for accurately determining the values of some of the dimensionless system parameters, particularly f, and fp. Nevertheless, it is possible to make intelligent estimates of these parameters based on experience from other experiments [25]. On that basis quantitative agreement between theory and experiment, for one particular experiment (Table 1), is seen to be fair, although clearly leaving a good deal to be desired. In all the above discussion, as in [ 26] » the observed criss- crossing instability was identified with the theoretically predicted first-mode oscillatory instability, despite the fact that in most cases theory predicts that the system is also subject to yawing instability over the same range of towing speeds. This is supported by the observed frequency characteristics of the oscillation and the ob- served effect of varying A, for instance, being essentially as theoretically indicated for the behavior of the first mode. It has thus been presumed that oscillatory instability is the prevalent form of instability. There is, however, an alternative interpretation of the observed behavior, namely that criss-crossing oscillation is a nonlinear manifestation of yawing. This may be postulated, but can- not be proven by the present linear theory. In fact a number of questions remain. More careful and ex- tensive experiments, including experiments with rigid cylinders, and more extensive theoretical calculations are necessary to resolve these questions. We next consider briefly the mechanism underlying the onset of instabilities, to the extent of identifying the physical forces at work. We first consider the mechanism involved in yawing. The first thing to recognize is that yawing must involve angular motion as opposed to pure translation. This is evident upon considering the cylinder momentarily displaced parallel to the x-axis; in this case the forces acting on the cylinder are exactly as in the equilibrium configuration, except that the tow-rope exerts a restoring force on the body. We next imagine the cylinder momentarily displaced such that the y-displacement of the nose is positive and that of the tail negative. Then, considering boundary conditions (11) and (12), we note that the inviscid hydrodynamic force at the nose is £ MU(dy/dx) 1011 Patdoussts while at the tailitis - f MU (ay /8x) » producing a moment tending to exaggerate the original inclination. However, there are Coriolis forces proportional to MU(@y/8x 8t) which always oppose rotation [ cf. Eqs a10)].. ; To understand this action of the Coriolis forces we consider the related physical system of a hinged-free tube containing flowing fluid (as mentioned in §2), depicted in Fig. 18(a), which was first considered by Benjamin [32]. We see that if the system rotates about A without bending, the fluid suffers a Coriolis acceleration which has a reaction on the tube always opposing the motion. This is clearly a stabilizing effect, as energy has to be expended by the tube to keep the motion going; as further elaborated by Benjamin, this represents the action of a pump from the energy-transfer point of view. Fig. 18 Rudimentary representation of a pump and a radial-flow turbine We next consider flexural instabilities. Clearly everything mentioned so far applies here also. But we also have another force coming into play. Once again we consider the hinged-free tube con- taining flowing fluid, as shown in Fig. 18(b), where the tube is momentarily 'frozen' in the bent shape shown. The centrifugal force of the fluid acts to increase the curvature further. This is clearly a destabilizing force, energy flowing from the fluid to the tube; it is the action of a radial-flow turbine. In flexural oscillations we have a play between these 'centrifugal' forces and the Coriolis forces; «hen the former prevail, then instabilities may develop. 1012 Dynamites of Submerged Towed Cylinders More formally, we may consider the work done, AW, on the cylinder over one period of oscillation, ti» in much the same way as was done in[ 24]. We find that t, \ e ° AW = (1- ans by + Uyy'] veg ade etd fgmul (ees Uyy'],., dt 0 Seg iG: MU) (5? + Uyy') dx dt. (20) If AW< 0, oscillations will be damped, while if AW >0O oscillations will be amplified, i.e. the system will be unstable. We first note that if f, = f,= 1, then instability can only arise from viscous effects (cf. [24]). We next consider the first two terms of (20). We note that for ice ae sien i stability will be governed by whether (1-f 3 yé dt - (1-f OI i yi dt is positive or negative; it is clear, therefore, that a well streamlined nose (f, ~ 1) and a blunt tail (f2< 1), both tending to make AW <0, will promote stability. For higher U, however, the situation becomes more complex, as Ulyy' >y may now obtain, and yy! may be either positive or negative, the bar representing the mean value over one period of oscillation. (It is noted that from Figs. 8 and 9 it may be found that for oscillatory instabilities we generally have (yy')o being strongly negative, and (yy vy"), also negative but with smaller absolute value.) Stability will depend on the magnitude of f,, f,, Vos val etc., and no simple general rules can be formulated beyond the statement of Eq. (20). It was found that the most effective way of stabilizing a towed system is by making it blunt at the tail, which has the disadvantage of increasing the towing drag. Clearly, what is needed is a blunt tail without separated flow! The present work and that of [ 26] indi- cate that small f, and large cy» (both associated with a blunt tail) have individually stabilizing effects on the system. Clearly then what we need is a sufficiently small f2 for stability, and a small Cg for moderate form drag. From the boundary conditions we note that a small f, has the effect of reducing the lateral shear exerted by the tail on the cylinder. Accordingly, if the tail is made very flexible with the rest of the body essentially rigid, the full shear force might not be transmitted to the cylinder, simulating the effect of a small f,; yet insofar as axial flow conditions are concerned, they would be fairly good. Of course, this particular solution might give rise to other problems, e.g. whiplash-type behavior of the tail may be envisaged. Another point of possible practical interest hinges on the fact that a towed flexible body, which is unstable at low towing speeds, may be stable at an intermediate range of towing speeds. (On the other hand, a rigid towed body of the same shape would be unstable 1013 Patdoussts at all towing speeds.) Accordingly, in the case of a flexible towed system this suggests the possibility of removable stabilizers; these would be operative only at low towing speeds, and would be removed at the operating speed to reduce drag. Incidentally, the above would generally also apply to articulated towed systems, made up of a number of rigid tubular sections flexibly connected [ 32], [33]. ACKNOWLEDGMENTS The author is grateful to his student, Mr. Jean J. Baribeau for assistance in the preparation of this paper during the summer of 1970. The author wishes to thank the National Research Council (Grant No. A4366) and the Defense Research Board (Grant No. 9550-47) for financial support making this research program possible. REFERENCES [1] A. G. Strandhagen, K. E. Schoenherr, P. M. Fobayashi, "The Dynamic Stability on Course of Towed Ships," Trans. Soc. Naval Arch. and Marine Eng., Vol. 58, pp» 32-66,°1950. [2] L. "Bairstow, E. F.-Relf, R. Jones, "The Stability of Kite? Balloons: Mathematical Investigation," A.R.C. R& M ZOOS), 2o158 [3] M. M. Munk, "The Aerodynamic Forces on Airship Hulls," NACA Rept. No. 184, 1924, [4] H.Glauert, "The Stability of a Body Towed by a Light Wire," ASRZG SOR Gin £st2 74930). 5 L. W. Bryant, W. S. Brown, N. E. Sweeting, "Collected y g Research on Stability of Kites and Towed Gliders," A.R.G. R°& M 2303; 1942. [6] A. G. Strandhagen, C. F. Thomas, "Dynamics of Towed Underwater Vehicles," Rept. No. 219, U.S. Navy Mine Defense Lab, Panama City, Florida, 1963. [7] J. R. Richardson, "The Dynamics of Towed Underwater Systems," Eng. Res. Associates Rept. No. 56-1, Toronto, Canada, 1965. [8] P.O. Laitinen, "Cable-towed Underwater Body Design," Rept. No. 1452, U. S. Navy Electronics Lab. , San Diego, California, 1967. 1014 [12] [13] [14] [16] [17] [18] [19] [ 20] Dynamites of Submerged Towed Cylinders T. Patton, J. W. Schram, "Equations of Motion of a Towed Body Moving in a Vertical Plane," ReptsJNo. 750, U.S. Navy Underwater Sound Lab., Fort Turnbull, Conn., 1966. E. Jeffrey, "Influence of Design Features on Underwater Towed System Stability," J. Hydronautics, Vol. 2, pp. 205-13, 1968. W. Schram, S. P. Reyle, "A Three-dimensional Dynamic Analysis of a Towed System," J. Hydronautics, Vol. 2, pp. 213-20, 1968. F. Whicker, "Oscillatory Motion of Cable-Towed Bodies," Ph.D. Thesis, Univ. of Calif., Berkeley, California, 1957. R. McLeod, "On the Action of Wind on Flexible Cables with Application to Cables below Aeroplanes and Balloon Cables," Wake G. R& M554, 1918. F. Relf, C. H. Powell, "Tests on Smooth and Stranded Wires Inclined to the Wind Direction and a Comparison of Results on Stranded Wires in Air and Water," Adv. Comm. Aero., R & M 307, 1917. Landweber, M. H. Protter, "The Shape and Tension of a Light Flexible Cable in a Uniform Current," Rept. No. 533, David Taylor Model Basin, Navy Department, Washington, D. C., 1944, Pode, "An Experimental Investigation of the Hydrodynamic Forces on Stranded Cables," Rept. No. 713, David Taylor Model Basin, Navy Department, Washington, D. C., 1950. Pode, "Tables for Computing the Equilibrium Configuration of a Flexible Cable in a Uniform Stream," Rept. No. 687, David Taylor Model Basin, Navy Department, Washington, DeGeg 19515 O'Hara, "Extension of Cylinder Tow Cable Theory to Elastic Cables Subject to Air Forces of a Generalized Form," A.R.C. R& M 2334, 1945, E. Kochin, "Form Taken by the Cable of a Fixed Barrage Balloon under the Action of Wind," Appl. Math. Mech, Vol. 10, pp. 152-64, 1946. C. Eames, "Steady-state Theory of Towing Cables," Quart. Trans. Roy. Instn. Naval Arch., Vol, 110, pp. 185-206, 1968. 1015 [ 21] [ 22] [ 23] [ 24] [ 25] [ 26] [ 27] [ 28] [ 29] [ 30] [31] [ 32] [ 33] 5 s Patdoussts L. Albasiny, W. A. Day, "The Forced Motion of an Ex- tensible Mooring Cable," J. Inst. Maths. Applics, Vol. 5, pp. o>-71; 19609. H. Toebes, "Flow-induced Structural Vibrations," J. Eng. Mech. Div., Proc. ASCE, Vol. 91, No. EM6,-pp. 39-66, 1965. R. Hawthorne, "The Early Development of the Dracone Flexible Barge," Proc. Instn. Mech. Engrs., Vol. 175, pp. 52-83, 1961. P. Paidoussis, "Dynamics of Flexible Slender Cylinders in Axial Flow -- Part 1. Theory," J. Fluid Mech., Vol. 26, pps (17-30; 1966; P., Paidoussis, "Dynamics of Flexible Slender Cylinders in Axial Flow -- Part 2. Experiments," J. Fluid Mech., Vol. 26,5 PPe 731-51, 1966. P. Paidoussis, "Stability of Towed, Totally Submerged Flexible Cylinders," J. Fluid Mech., Vol, 34, pp. 273-97; 1968. P,. Paidoussis, "Stability of Towed, Totally Submerged Flexible Cylinders," Rept. Eng. R-5, Atomic Energy of Canada, Chalk River, Ontario, 1967. J. Lighthill, "Mathematics and Aeronautics," J. Roy. Aero. SOG si> Vol. 64, PPpe 375-94, 1960, J. Lighthill, "Note on the Swimming of Slender Fish," Tie Fluid Mech., Vol. 9; PPe 305-17, 1960. I. Taylor, "Analysis of the Swimming of Long and Narrow Animals," Proc. Roy. Soc. (A), Vol. ‘214, pp. 158-83, 19522 W. Gregory, M. P. Paidoussis, "Unstable Oscillation of Tubular Cantilevers Conveying Fluid -- I. Theory,' Proc.” Roy. Soc. (A); Vol. 293, pp. 51 2-27 P1966? B. Benjamin, "Dynamics of a System of Articulated Pipes Conveying Fluid -- I. denies " Proc. Roy. Soc. (A); P. Paidoussis and E. B. Deksnis, "Articulated Models of Cantilevers Conveying Fluid: The Study of a Paradox," J. Mech. Eng. Sc., Vol. 12, pp. 288-300, 1970. 1016 HYDRODYNAMIC ANALYSES APPLIED TO A MOORING AND POSITIONING OF VEHICLES AND SYSTEMS IN A SEAWAY Paul Kaplan Oceantes, Ine. Platnvtew, New York I. INTRODUCTION At present, increasing interest is being devoted to the prob- lems of deep sea operations of vessels that must remain on station for an extended period of time in order to accomplish their intended mission. This concern was given its initial impetus by the success- fully conducted preliminary operation of drilling through the ocean bottom from a surface ship in the operation known as the "Mohole Project," as well as the increase in oil exploration in deeper water depths. On the other hand, from the point of view of military operations, there is need for placing instrumentation packages and other military systems on the ocean floor for various purposes of National Defense. These operations require a definite degree of precision, safety during the course of the operation, and the capa bility of returning to a particular locale and retrieving information and/or the equipment itself for further study of data or for emplace- ment in another location. As a result of this emphasis on deep-sea operations, it is necessary to determine the response of representative moored ships in the open sea, and also to determine the characteristics of the important parameters associated with lowering loads from sucha vessel to the ocean floor and returning them to the ship. The parameters that are of interest to the personnel aboard the ship are the forces in the mooring cables, the displacements and tensions in the lowering lines, the degree of precision in placing the loads, the accelerations acting on the loads, and the magnitudes of impact on the ocean bottom. In order to arrive at some appropriate engineering estimates of the capabilities of carrying out such operations, appli- cation of available theoretical hydrodynamic studies can be made to deal with problems of this nature. The study of motions of ships at sea is a general problem of 1017 Kaplan naval concern, and has received increasing emphasis during the last fifteen years or so by virtue of the advance of statistical methods which describe the effects with greater realism than in previous studies based on simplified wave representations. Major concern has been devoted primarily to the problems of an advancing ship in head seas, with the prime variables of concern being the heave and pitch motions. Recent studies, however, have been concerned with motions in oblique waves, wherein lateral motions (sway, yaw, and roll) are also important. All of these studies involved large ships advancing in waves, and only limited theoretical studies have been developed to predict adequately the motions in all six degrees of freedom under these operating conditions. A treatment of the motion of a free ship with six degrees of freedom in waves is a formidable problem that has not achieved a complete solution at the present time, and when the influences of moorings are also included, the problem is further compounded. Nevertheless, there exists a need for some means of preliminary estimation of the expected motions of a moored vessel, and there is sufficient hydrodynamic information available to allow a study that will indicate the expected range of amplitudes of motion so that the results obtained can be used as guide-lines for operating personnel. Another related problem that is assuming more significance recently is that of a moored buoy system. These smaller payloads are planned for use over large ocean regions to provide a network of environmental reporting stations that will yield continuous data on the important properties of the ocean and atmosphere for use in weather forecasting and other technologies dependent on air-ocean interaction. The effective design and engineering development of such systems requires an ability to predict the buoy (and hence the transmitting antenna) oscillatory motions and structural acceleration loadings in various seaways; the determination of the tensions along the cable under various operating conditions; etc. Knowledge of such results will greatly enhance the design of handling equipment for both launching and retrieving of buoys at sea, and will also provide basic information on system survivability under extreme environ- mental conditions. A tool that can provide engineering estimates of such informa- tion is a mathematical model that describes the essential mechanical- dynamic characteristics of a moored buoy system. This mathemati- cal model will be a system of equations and relationships that allows the calculation of the spatial configuration, dynamic motion and internal tensions of a specified moored buoy in a given excitation environment. The hydrodynamic force acting on the buoy hull and the forces acting on thecable system (hydrodynamic, inertial and elastic) are coupled so that each affects the other, especially when considering dynamic effects and rapidly varying motions. Certain similarities exist between this problem and that of a moored ship, together with definite differences as well. The applicability of basic techniques of analysis from one problem to another provides useful 1018 Mooring and Postttoning of Vehicles tn a Seaway insight and extends the utility of basic "tools" used in hydrodynamic and dynamic investigations. When considering the problem of maintaining a ship on station for a long time period, various concepts for achieving a minimum deviation from a derired operating point are possible, with the two main methods being that of fixed mooring or by use of a dynamic positioning system. In certain situations where mobility is required, as well as due to the high capital cost of a mooring system for very deep water operations, the associated high cost of emplacement and the dangers of damage due to large storm conditions, a mooring system does not appear to be attractive. Dynamic positioning is a more recent development, which has only received limited applica- bility to date. In order to provide the information ncessary to determine the possibility of an application of dynamic positioning, it is neces- sary to carry out particular analyses to determine the environmental conditions appropriate to possible operating areas; the resulting forces and moments acting on the ship; the arrangement and type of control effectors; the possible signal systems that provide the error and command signals for actuation of controls; possible control system concept designs; etc. The important quantities that must be determined for proper design of the positioning system are the disturbing forces that act on the ship. The major forces and moments that affect the ship stationkeeping ability in this case are the more-or-less steady type of "drifting" forces imposed by the environment, and these quantities are amenable to computation by means of hydrodynamic analyses using available theory. In all of the foregoing situations the importance of hydrodynamic force evaluation and its applicability to obtain desired engineering performance data is paramount. Many publications are available in the literature on ship motion theoretical studies that can be applied to the above problem areas, with reasonable expectation of validity for the results. The central theme of this Symposium, "Hydrody- namics in the Ocean Environment," is certainly appropriate to the present International Decade of Ocean Exploration which will em- phasize the technology that will yield benefits to Mankind. The application of the basic developments in hydrodynamics of ship motion to the applied engineering problems associate with maintaining vessel operations at fixed positions in the ocean, which will be required as part of this extensive international effort, is a vital element in achieving improved system performance. It is also a good illustration of the direct application of many years of basic research toward the solution of problemsthat are anticipated as further and deeper ventures into the sea are made. The present paper is aimed at providing a limited description of the use of hydrodynamic analysis when applied to some of these problem areas. 1019 Kaplan II. SCOPE OF INVESTIGATION It is easily seen that there are a host of problems associated with the subjects considered in this paper. As a result, some limi- tations are imposed so that only certain aspects are considered in detail. The region of application of the results in this paper is in deep water, so that no shallow water effects are considered. This limitation thereby excludes problems of ship oscillation when moored at docks in harbors, which is an important problem that can be treated in a similar fashion to those herein by proper inclusion of shallow water effects. The main emphasis within this paper is on the seaway and its effects, and in some cases the influence ofa current will not be considered. However it is known that currents are often present together with sea waves, and their combined effect is often very important. In addition the presence of a current is often necessary to establish certain static equilibrium conditions for a vehicle about which the seaway disturbances are imposed, and in that case certain assumptions are made as to the existence of such initial conditions for purposes of simplifying the analysis. Similarly, the presence of any wind effects is also not considered in detail within this paper. When considering the problem of the motions of moored systems, it is known that the effects of drift forces are also present and that they produce an important influence on the resulting motions and cable forces. However, in an effort to obtain tractable solutions and to provide information on the characteristics due to different force mechanisms, these effects will be considered separately. Illustrations of the different influences that act on vehicles and systems in a seaway will be presented separately, with some dis- cussion given to the expectations with combined effects in a realistic situation when more than one mechanism is acting on a system. The discussions of results are devoted to the more important phenomena influencing performance of a system in the sea, and they will be given throughout the paper for each case treated. III TECHNIQUES USED FOR MOORED SHIP ANALYSIS In order to determine the motions of a moored ship in irregular waves, it is necessary to determine the response in regular sinu- soidal waves. The aim is to predict these motions, and the technique to be utilized is that of spectral analysis [1] wherein the statistical definition of the seaway in the form of its energy spectrum is used as the initial data. The energy spectrum of the time history of each motion of the vessel in response to irregular waves is evaluated for the corresponding degrees of freedom to the energy spectrum of the seaway. These operators are obtained from the solutions for the motions in sinusoidal waves, and in accordance with the basic premise of this technique of analysis, a linear theory of ship motions is a prerequisite. 1020 Mooring and Postitioning of Vehicles tn a Seaway The equations of motion in regular waves, for six degrees of freedom, are formulated according to linear theory by the balance of inertial, damping, restoring, exciting, and coupling forces and moments. Both hydrodynamic and hydrostatic effects due to the body- fluid interaction are included in the analysis, together with the influences of the mooring system. The longitudinal motions (heave, pitch, and surge) are coupled to each other, and similarly, the lateral motions (sway, yaw, and roll) are also coupled. There is no coupling between the two planes of motions, in accordance with linear theory. The hydrodynamic forces and moments such as damping, exciting effects due to waves, etc., are determined by application of the methods of slender-body theory. Essentially, this theory makes the assumption that, for an elongated body where a transverse dimension is small compared to its length, the flow at any cross section is independent of the flow at any other section; therefore, the flow problem is reduced to a two-dimensional problem in the transverse plane. The forces at each section are found by this method, and the total force is found by integrating over the length of the body. A description of the application of slender-body theory to calculate the forces acting on submerged bodies and surface ships in waves is presented in [2], where simplified interpretations of force evaluation in terms of fluid momentum are also given. The hydrostatic and mooring forces and moments are combined with the hydrodynamic terms, resulting in linear combinations of terms that are proportional to acceleration, velocity and displacement in the various degrees of freedom. All of these expressions, when related to the appropriate ship inertial reactions by Newton's law, lead to the set of six linear coupled differential equations of motion. Solutions of the equations are found for regular sinusoidal seas with varying wave length and heading relative to the barge. The response amplitude operators are found from these solutions together with the phases of the motions relative to the system of regular waves. Assuming a knowledge of the oncoming irregular sea conditions (e.g. in terms of sea state, as specified by an associated surface-elevation energy spectrom from information in[3]), the set of energy spectra for the ship motions are determined. Information on average values and probabilities of relatively high values of the amplitudes of oscil- lations in the ship-motion time histories for the different degrees of freedom are found from the ship-motion energy spectra in accordance with the methods of [1]. Cross-spectra are also used to determine the energy spectra and hence the various average values and the pro- babilities for the remaining quantities of interest, such as load- displacement time histories and other quantities which are linear combinations of the ship motions and their time rates of change (the presence of lowering lines for placing loads on the ocean floor is considered inthis analysis). These energy spectra may also be obtained from the solutions of the differential equations by linear superposition, and explicit use of cross-spectra here is necessary 1021 Kaplan only for obtaining phase information. The ship is assumed to be placed in a currentless seaway, with no wind effects being considered. This may be somewhat un- realistic from the practical point of view, but since concern here is devoted only to the motions induced by the seaway, this neglect is reasonable (as discussed previously). The ship is assumed to be moored with bow and stern moorings of conventional line and anchor type. The line and anchor mooring system utilized for this study is a particular system especially suited to deep-sea operations [4], and utilizes a taut line. Other types of mooring lines can be con- sidered as well, but separate analyses to determine the static orientation, restoring force variations, etc. must be carried out. The extent of linearity for these different mooring arrangements must be determined for use in the present type of analysis. The effects of the moorings will be to provide restoring effects in the particular displacements of surge, sway and yaw, thereby providing "spring-like" terms in the equations for these degrees of freedom. As a result, there are certain natural frequencies associated with these motions, which do not ordinarily occur in case of free (un- moored) ships. The moorings are assumed to have a negligible influence on the motions of heave, pitch, and roll, which have large hydrostatic restoring effects. Following the evaluation of the various motions of the moored ship, equations are formulated to determine the forces in the moor- ing cables, and the displacement of and tension in the lowering line, as a function of the different degrees of freedom of the oscillating platform moored inthe seaway. The lowering line displacement and tension, which are functions of the ship motions are then related to the seaway and all of the resulting spectra determined. Operations on these quantities provide information on expected amplitudes for particular sea states, and in addition the vertical accelerations of the loads are determined and similarly expressed, where this infor- mation is useful for study of impact ofthe loads on the ocean bottom. IV. EQUATIONS OF SHIP MOTION The equations of motion of the moored ship are derived on the basis of linear theory, with the body allowed to have six degrees of freedom. A right-hand cartesian coordinate system is chosen with the axes fixed in the body, and with the origin at the center of gravity of the body. The x-axis is chosen positive toward the bow, the y-axis is positive to port, and the z-axis is positive upward. These axes are defined to have a fixed orientation, i.e. they do not rotate with the body, but they can translate with the body. The body angular motions can be considered to be small oscillations about a mean position given by the axes. The dynamic variables are the linear displacements x, y, and z along the respective axes, and the angular displacements $, 8 and yw which are defined as positive in 1022 Mooring and Postttoning of Vehteles tn a Seaway a direction of positive rotation about the x, y, and z axes, respectively, (i.e. port upward, bow downward and bow portward). The positive directions of the forces and moments acting on the body are similarly defined. The force (or moment) acting on the body is composed of the inertial force due to dynamic body motions (denoted as Fj), the force due to damping (denoted as Fy), the force due to hydrostatic restoring action (denoted as F)), the force due to the moorings (denoted as Fm), and the force due to waves (denoted as Fy). The equations of motion are then established as me = F au Dept (1) for rectilinear motions (with s representing any rectilinear dis- placement, and m the mass ofthe ship), with similar representa- tions for the angular motions. A discussion of these different types of forces is given below, together with some results obta ned, for purposes of illustration. The hydrodynamic forces and moments due to dynamic body motions are of inertial nature, and do not contain any terms of dissi- pative nature. The effect of the free surface is accounted for by different frequency-dependent factors that modify the added masses of each section. All couplings of inertial nature are exhibited in the results of the analysis. In the case of dynamic body motions, the simplified results of slender-body theory states that the local force on any section is equal to the negative time rate of change of fluid momentum [ 2]. For the vertical force (z-force), this is expressed by D ae = at | A535 > (2) where A is the added mass of the cross-section and Wy is the body vertical velocity, given by w, = yy (2 - £8) = % - 60. (3) In the above equations, the coordinate € is a "dummy”™ variable along the longitudinal coordinate x (and coincident with it), and the time derivative D/Dt is just the partial derivative 0/8t, since there is no forward speed. The quantity A33 is the added mass of the cross section, including free-surface effects, which is obtained from the work of Grim [5] for the class of sections known as Lewis forms. The total vertical inertial force is then found to be 1023 Kaplan £p ! eo &b 1 ee zi =~ | aggadt- 2 +) Pape at - 6 (4) & bs where § and & are the bow and stern €-coordinates respectively. In a similar manner, the lateral force (along y-direction) may also be expressed by use of this same procedure, but certain addi- tional factors enter in that case. These factors are the necessity of including roll effects which influence the lateral velocity, and also the fact that the representation of the lateral force is based upon added mass terms that are evaluated for motions relative to the free surface level, rather than the body center of gravity position. Cor- rections to refer the final forces to the center of gravity position are made after finding the forces referred to the free-surface position. The detailed procedures for determining these inertial force (and moment) results, as well as all other forces of hydrodynamic, hydrostatic, etc. nature are described in [ 6] , which is the basic report on which the present section of this paper is based. In view of this, only limited discussion of the remaining forces and moments will be presented. The damping forces and moments are dissipative in nature, and are primarily due to the generation of waves by the ship motions on the surface, which continually transfer energy by propagating outward to infinity. In accordance with the two-dimensional treatment used for the analysis of inertial forces due to body motions, the same concept is used in evaluating the local forces at a section of the ship due to wave generation. With the ratio of the amplitude of the heave- generated two-dimensional waves to the amplitude of heaving motion of the ship section denoted by A,, the vertical damping force per unit vertical velocity of the ship section is expressed as o—2 1 _ pg Az _ rR =2 N= nae po (>) A,2 (5) where A, for Lewis-form sections are available as a function of w°B* /2g = mB '/i, for different beam-draft ratios and section coef- ficients, where B- is the local beam and 2 the wave length. The vertical damping force at each section is dz 1s . “ze == N,,(z - E80), (6) and this is integrated over the ship length to determine the total vertical damping force, given by 1024 Mooring and Postttoning of Vehtecles in a Seaway Zaz - Nz +NiO, (7) where 2 ee N, = po(%) VA? at, (8) fs and é 28> Nzg = po(>) \. Aré dé. (9) Ss Similar treatments yield the lateral damping force, pitch damping moment, etc. In the initial discussion of damping, emphasis was placed upon energy dissipation due to wave generation. Actually, viscous effects also manifest themselves and contribute to damping. The contribu- tion of the viscous damping term is quite negligible’for most motions, with the possible exception of roll. Roll damping due to wave genera- tion is often small for most normal ships and viscous effects (or other drag mechanisms, such as eddy-making) assume greater importance, especially if the ship is fitted with bilge keels. In that case, the roll damping is often of nonlinear form, and an approxi- mation is used to determine some equivalent linear representation. Knowledge obtained from model experiments [7] was used to deter- mine the value of roll damping used in treating the illustrative ship case in this paper. The hydrostatic restoring forces and moments are, as the name implies, due to buoyancy effects arising from static displace- ments. The only displacements that will result in hydrostatic restoring effects are heave, pitch and roll. On the basis of linear theory, the local hydrostatic vertical force change due to vertical displacements is ae =- pgB*(z = 50), (10) where the ship is assumed to be almost wall-sided near the inter- section with the free surface, and the effective buoyancy change comes from the total immersion. Similarly, the hydrostatic restoring pitch moment is dMp _ dZp “xe fe ome 1025 Kaplan leading to total hydrostatic restoring vertical force and pitch moment given by b x bx Z, = - PS B d&«z + pe B & de = 65 (42) s és and fb fb M, = ee) BE ab 2- pe) B*t’ dé +e. (13) Ss Ss In the case of roll motion, the hydrostatic restoring effect is given by K, = - pgV|GM|¢= - W/GM|4, (14) where VY is the displaced volume, |GM| is the metacentric height, and W = pgV is the ship displacement. The exciting forces and moments due to waves are obtained as the sum of terms due to buoyancy alterations as the waves progress past the ship hull, together with hydrodynamic terms of inertial and damping. The buoyancy effect for the vertical force is repre- sented by pgBn(é xt) (15) at each section, and these contributions are combined to determine the total forces and moments due to waves. The analysis includes an allowance for the waves to be propagating at an oblique heading with respect to the ship, and a further allowance for the influence of the non-slenderness of the ship is also included. A correction factor, relating the beam to the wave length and the heading, is included for this purpose since the shipforms considered for mooring application are often not very slender. Details of the evaluation of wave forces and moments by these methods are presented in [6]. Before discussing the mooring forces and moments, informa- tion on the characteristics of the vessel studied in this investigation is given below. The particular vessel for which the equations are formulated and solutions carried out is the CUSS I, which was the vessel used in the preliminary Mohole drilling operation. This ship is considered representative of the class of construction type barges which will be utilized for deep-sea construction operations. A diagram of the barge, together with its mooring and load-lowering lines, is shown in Fig. 1. A summary of the numerical values of the parameters characterizing the moored-barge system is presented in Table 1. 1026 Mooring and Postttoning of Vehicles in a Seaway Draft: 10 Center of gravity 12,000- foot mooring cable Fig. 1. Schematic diagram of moored barge (Profile and plan views) 1027 Kaplan 260 ft 48 ft LO ft 9.8 £ ot Nae 8 = 15.2 2 8.16 ft 2823.2 long tons 6.324x10® lbs Table 1 Numerical Values of Moored-Barge System Length = Beam = Draft Vertical distance from CB to CG = Vertical distance from free surface to CG = Vertical distance from CG to keel = Metacentric height = Displacement Weight = Mass = | Pitch moment of inertia = Yaw moment of inertia: = Roll moment of inertia” = Total roll moment of inertia (including added inertia due to fluid) = Surge period® = Sway period = Heave period = Pitch period = Roll period = Effective spring constant for mooring cable = Effective mooring system spring constants: Surge = Sway = Yaw = Depth of barge ' Assuming longitudinal gyradius = 0.25 L. T surge T sway Theave T pitch Trott 197.624x10° slugs 706. 7x10°® slug-ft* 706. 7X10° slug-ft? 49x10° slug-ft® 6 78.69X10 slug-ft- 79 seconds = 64.5 seconds 4.6 seconds 4 seconds 7.75 seconds 4250 lbs/ft 1250 lbs /ft 3750 lbs /ft 633. 75X10° lb-ft /rad 15 ft 2Without added fluid inertia; it is assumed that transverse gyradius = Bys. 3For all motions these are uncoupled periods determined in terms of effective spring constants and values of total masses or inertias. The effects of coupling will change these somewhat, but for first app roxi- mations andinterpretation of critical conditions, this. will suffice. 4F rom model tests [ 7]. SBridge strand wire rope, of cross section 0.595 in, 1028 Moortng and Postttioning of Vehicles in a Seaway In analyzing the mooring forces and moments, the barge is assumed to be moored by a conventional line and anchor system, with both bow and stern moorings. However, for application to deep-sea conditions with depths of the order of 1000 fathoms, a certain particular mooring scheme is utilized. This scheme utilized a long-wire rope for each mooring leg assembly (12,000 ft in length), which is supported in the water by a series of submerged spherical buoys. The buoyancy of these buoys keeps the rope taut along its entire length, thereby not allowing it to assume the usual catenary shape. Withthis arrangement, an initial tension is applied along each mooring let, and any changes in mooring forces on the ship (and therefore also in the cables) occur as a result of elastic forces resulting from ship displacements. A layout drawing of such a system is shown in [4], which has direct applicability to ships of the same general displacement as the construction barge presently studied. The displacements having greatest influence on the moorings are in the horizontal plane, and these are surge, sway and yaw. Since the mooring lines are fairly taut and are under an initial tension, the elastic restoring effects may be taken to be fairly linear, i.e. the restoring force is proportional to the displacement. The proportionality factor for an effective displacement along a single mooring cable is found from a knowledge of the modulus of elasticity of the cable material. For the present case of 1-inch diameter bridge strand wire rope, which is 12,000 ft long, has a cross section area of 0.595 in®, and an assumed modulus of 25 X 106lb/in®, the effective spring constant for a single wire rope is found to be C = 1250 1b/ft. This linear result only holds below the yield point of 60,000 lb of static force (in a single cable), but it is anticipated that the maximum deflection necessary for attaining this force (viz. 48 ft) will not be experienced in the present case. For the purposes of analysis, the barge is assumed to be moored in an arrangement similar to that shown in the following sketch of the mooring plan. A longitudinal displacement of the barge along x, denoted as Ax, leads to an effective displacement along a single cable given by Axcos @, where @ is defined in the sketch above. The force in a single cable is then C Ax cos a, The longi- 1029 Kaplan tudinal force component at one end of the ship is represented by (CAx\cos @)cos @ + (CAx cos*e)cos,o = .2CAx cos* Q, and since an extension of the cable at one end of the ship requires a contraction at the other end, a similar force occurs. These forces are restoring forces and the net result is a longitudinal force in the barge due to the moorings, given by X, = 4C cos‘ a@*x=- k,x, (16) where x is the surge displacement variable. In the case of sway displacement, the effective displacement along the cable is y sine, and combining components for net Y- force on the barge, accounting for all the cables, leads to a net mooring lateral force given by Xm etc gin oy = — bys (17) For yaw displacements, Y* L/2 where L is the ship length. The lateral force at one end of the ship is then 2C sin®a- Sy = Cl sin?a- y, (18) and the contribution to the yaw moment is 2 1 4 2 we CL sin’ @* ¥ (5) =5 CL sin a-w (19) at each end. Since the forces at each end are equal and opposite (approximately, since the origin is not exactly at the ship center), the net yawing moment acting on the barge is given by Nm=- CL’ sin?a* w= - kyl (20) The variations in the force in the mooring cables due to the motions of the barge can easily be found, since they are related kinematically to the motions. It is seen that the longitudinal dis- placement, x, andthe net lateral displacements, y +(L/2)W at the bow and y - (L/2)\ at the stern, can be combined to determine the net variation in elongation of each mooring cable. The cable displacements due to surging motion on the barge are x cosa, while the cable displacement due to the motions of sway and yaw are‘ [y+(L/2)W] sin a, according as the cable is at the bow or the stern. 1030 Mooring and Posittonting of Vehicles in a Seaway Different effects as to the cable displacement directions occur for the cables, at either the bow or the stern, for the influence of the lateral motions, while the same direction of displacement (at either bow or stern) occurs for the surge motion. The general expression for the fluctuating cable force may be written as Fe = c[x cos a+ (y+ 5v) sin a| where C is the effective spring constant for a single wire rope, and particular values for each of the four cables are given in the following, where a positive cable force is defined as that which pulls on the restraining anchor support on the ocean floor. The expressions for the individual cable forces (c.f. sketch of mooring-line system) are listed below: Bow F, =- c[x cos @ +(y +>) sin a | port (21) _ zi ‘ b Fa — - Ci cos a-- (y >) sin a | starboard Stern F, = c [x cos @ - (y - 1) sin a| port (22) F, = c [x cos @ + ( - >) sin a | starboard 4> y "2 For the present case where the barge is moored with a = 60°, L = 260 ft, the mooring system restoring constants are k, = 1250 lb/ft ky = 3750 lb /ft (23) ky = 633.75 X 10° 1b-ft/rad These values are the effective spring constants for surge, sway and yaw, and as a result there also exist natural periods for these motions in the case of moored ships. There still exist natural periods of heave, pitch and roll, as in the case of free ships, and these natural periods are relatively unaffected in the present case. The introduction of the existence of natural periods in surge, sway and yaw (with possible large motions associated with resonances in 1031 Kaplan these degrees of freedom) is the main characteristic of moorings applied to ships that distinguishes the resulting motions from those of free ships in waves. V. SOLUTION OF EQUATIONS The equations of motion result from combining all of the constituent terms discussed above, and solutions can be obtained by converting them to a simpler form for sinusoidal waves. Since the exciting forces and moments are sinusoidal functions, the motions will also be sinusoidal with the same frequency. Defining — iwt — iwt — iwt = iwt = _lwt c= xe ; y=ye ; VA AS he Xy = Xe , Yw.= YC rq s, sete. the equations of motion are then converted to (complex) algebraic linear equations. In matrix form the equations may be represented by a I 0 as ]f x x 0 Aon aos Z = 4. (Zi (24) a ass as 6 M for the longitudinal motions, where the coefficient matrix is sym- metric, i.e. aj3 = a3,, a,,= a3,- The matrix elements are defined by: a, = (- mo* + ioN, + k,) (25) aj3 = a3, = m|BG|o (26) aon= = (m + ( As, at) ot ioc No + ral Bae (27) és €s 2 bot 4 b * Ags = ago = w (; A336 d& - iwNzg - af B Gude (28) Ss Ss {p i ee 2 fb 2 an =| I +( A.W dé )'@ + iwCgNo + pe | BtE dé (20) 33 GF f 33 ) es The lateral equations are represented by 1032 Mooring and Postitioning of Vehicles in a Seaway by be bsry ng bo, bop = Bag ff P| =| N be Bean bs i) K + (OG)Y (30) where the matrix here is also symmetric, i.e. bia = bo, by3 = bas by, = b3, The elements are defined by €b by =-(m¢t \ Ago d&) w + iwCyNy + ky és 2 Sb ; b,, = b,, = - w ( Ae dé + iwNyy s bit ss be bs, Jf (Ago + (OG) Age) dé + iwCyNy | BG | s E b= - (1, + ( ” Ast dé) w° + iwCyNy + ky és €b by, = Dap = - wf (Ago + (OG) Ax.) & dé + ioNyy| BG | s bene ol, + iwNg + W|GM| The presence of symmetric matrices helps in effecting an (31) (32) (33) (34) (35) (36) easier solution of the equations, obtained by matrix inversion on a large digital computer. The solutions are then available for each degree of freedom and also for any linear combination of degrees of freedom. The real form of the final solutions is obtained by taking the real part of the complex function, which was the original defini- tion implied in the complex representation of the solution variables. The lowering line displacements are related kinematically to the body motions, and hence they are relatively simple to determine once the different methods of lowering loads are specified in this study, viz. center-lowered loads and boom-lowered loads. Center- lowered loads, as the name indicates, are lowered through some sort of opening through the ship's keel, and it is assumed that this is done at just about amidships. The instantaneous displacement vector components of the load and lowering line are s,, s and are then given by simple geometry as ; 1033 and s, Kaplan S, =x sy=y + |KG|¢ (37) S$, = 2 where x, y, z and @¢ are the instantaneous ship motions of surge, sway, heave, and roll, respectively, and KQ@ is the vertical distance between the center of gravity and the keel. The tension, T, in the lowering line is given by the relation T-W =—s = ; (38) Wigs FW pat —- Zz g where Wg is the weight of the load, and only vertical effects are considered to affect the tension. At rest, so that upon representing the tension as - = ! Lots t.t aM, gee where T' is the tension change due to dynamic effects, one obtains Thus the tension variation due to the dynamics of the ship motion is directly related to the vertical acceleration of the load, and it is also proportional to the weight of the load. In the derivation of the formulas given above, it is assumed that the trajectory of the load attached to the line is such that at each instant it is on the vertical line through the point of attachment of the lowering line to the barge. It is also assumed that the elastic effects of the lowering lines may be neglected; the only dynamic influences considered being those due to the ship motions. The neglect of elastic effects in the lowering line appears to be a fairly safe assumption, since the major influence would occur only if the wave frequencies excited the natural frequency of wave propagation in the lowering line. In view of the lack of specification of the line's physical characteristics, as well as the expectation of wave-propa- 1034 Mooring and Posittontng of Vehicles in a Seaway gation frequencies out of the range of interest in the present problem, the tensions and accelerations are considered adequately represented by Eq. (39). For boom-lowered loads, the situation may be visualized by reference to the accompanying sketch, where the boom length £ is stern elevated at an angle a'. An appropriate value for the relevant horizontal projection of the boom length (£ cos @') is considered, for computational purposes in the present case of a 260 ft length barge, to be 150 ft. As shown below, the boom is also oriented horizontally at an azimuth angle y, measured from the bow, positive in the counterclockwise sense, viewed from above. The load and stern bow lowering-line displacements about their respective equilibrium positions are given by s? =x - (f cos @') sin y> w slay + (2 cos a') cos y> w (40) az = it cos a') cos y° 6 + (£ cos @a') siny > and the line tension (fluctuating part), Tt!” and vertical acceleration are represented by 1035 Kaplan rT” 2 it os ee oo Moe Eh S cos a'(siny > 6 - cos y= @)] (41) where 4, 8, and W™ are the rotational barge motions, roll, pitch and yaw, respectively, and the superscript y denotes the boom azimuth angle. These quantites are derived on the same basis as those for center-lowered loads, it being assumed that the boom pivots about the ship CG. The instantaneous magnitudes of these quantities thus appear as linear combinations of the instantaneous ship-motion solutions. Each motion of the barge in response to a regular sinusoidal wave having a given frequency and propagating in a given direction will also be sinusoidal, of the same frequency, but will, in general, possess a different phase. In addition, the amplitude of each motion will, in general, differ from that of the wave, the ratio of the former to the latter being a function of the wave frequency and the heading of the wave relative to the heading of the barge, and this amplitude- ratio function is known as the response amplitude operator for the particular motion of interest. In order to arrive at an effective characterization of the barge motions in a random sea, in which case these motions themselves have a random nature, the function known as the spectral energy density, or the energy spectrum, of each motion must be found. This spectrum is a measure of the vari- ation of the squares of the amplitudes of the sinusoidal components of the motion, as a function of frequency and wave direction. The total area under the spectral-energy density curve contains much of the statistical information on average amplitudes, near-maximum amplitudes, etc., for the particular motion considered. For an arbitrary motion, represented by the i-subscript, the energy spectrum of that motion, due to the effects of irregular waves, is given by (i,i) @"' (w) = | Tig(w) | A (o) (42) for a particular fixed barge heading in a unidirectional irregular sea, where A (w) is the wave spectrum and (Tia is the response amplitude operator for that heading. For computational purposes in the present study, the Neumann Pierson spectral-energy description of the seaway has been adopted, and calculations made for these particular sea states, corresponding to three particular wind speeds. The following table illustrates the conditions. 1036 Moortng and Posittoning of Vehicles in a Seaway Table 2 Sea Wind Speed Sig. Wave Surface Elevation (Time State Vw (knots) Ht. Hyy3 History and Energy Spectrum) (ft) rin... Value,, Lt. energy, o , (ft) o%, (ft)? 20% = E 3 14 S05 0.81 0.66 pW Ye = 9 6.9 iit 3.05 6.10 5 Ze 10.0 2.50 6.25 12.46 The Newmann wave spectrum for a unidirectional fully- developed sea represented by 2 6 -29°/(wVy) iw tS A*(w) = C (43) where C is an empirical constant having the value 51.5 ft*/sec, vw is the wind speed in units of ft/sec, and A*(w) has the units ft?-sec. The wave spectrum for a non-unidirectional sea, allowing for angular variation (a two-dimensional spectrum), is represented by 2 -6 -29°/(wVy)* T T — Cw e “ cos® By, for -353 0 + So 60 90 120 150 180 Z (feet) 1.0 180 150 120 90 60 30 - O + 30 60 90 120 150 180 @ (radians) 025 Eee 180 150 120 90 60 30 - O + 30 60 90 120 150 180 A (degrees) Fig. 2. Amplitude of response for unit-amplitude wave as a function of direction of wave relative to barge. Longitudinal motion; X= 300. 1039 Kaplan Y (feet) 10 0.8 180 150 120 90 40 30 - 0 + 30 60 90 120 180 180 ¢ (radians) .08 .06 180 180 120 90 60 30 - 0 + 30 60 90 120 150 180 y (radians) .010 .008 .006 180 180 120 90 6Q 30 - 0 + 30 60 90 120 150 1860 43 (degrees) Fig. 3. Amplitude of response for unit-amplitude wave as a function of direction of wave relative to barge. Lateral motion; X= 300°. 1040 Mooring and Posittoning of Vehicles in a Seaway Taq |? (dimensioniess) w (rad/sec) 2 (Response amplitude operator)~ for surge, Tis I [Tz 9 (dmenssoniess) w (rad/sec) 2 (Response amplitude pparataris for heave, IT2 9 = = .0004 ~ i] a ° a me + 0002 § = w (rad/sec) (Response amplitude operator) for pitch, Teel” Fig. 4. Response amplitude operators for longitudinal motions. 1041 Kaplan Fy 2 10 0.8 90° v_ 0.6 = - 04 45° 0.2 °e 0 (e} 10) 05 10 Le j w (rad /sec) (Response amplitude operator)@ for sway, [Tval” 006 N = .004 x 902 nN wv ec «002 AS 45° 19) 0.5 10 1S w (rad/sec) 2 (Response amplitude operator) for roll, IT gal .00006 45° .00004 .00002 o°, 90° te) 0 0.5 1.0 us w (rad/sec) IT py I?, (rad? ft?) (Response amplitude operator)* for yaw, [Toel* Fig. 5. Response amplitude operators for lateral motions. 1042 Mooring and Postttoning of Vehicles in a Seaway 3 SEA STATE 5 WwW (rad/sec) - SEA STATE 5 ENERGY DENSITY (ft*sec) {@) 0.5 10 Ww (rod /sec) Fig. 6. Spectral energy density for translational barge motions for indicated barge heading Bg 1043 Kaplan 8 SEA STATE 5 Ww (rad/ sec) .005 coe SEA STATE 5 .003 @ =0° 002 Py =O 001 ENERGY DENSITY ( rad@-sec) 0 l ee) eee ee ee ee ia] 05 1.0 15 W (rad/sec) nOn09 SEA STATE 5 .0003 38 = 135° 0002 0001 My 15 w (rad/sec) Fig. 7. Spectral energy density for rotational barge motions for indicated barge heading Bp 1044 Moortng and Posittoning of Vehicles in a Seaway In the present study the angle for the predominant wind direc- tion was taken to be By = 0, and a variable barge heading angle, Bp, introduced to allow for the relative heading of barge to wind. The relationships of the wind direction, the wave heading, and the barge heading are shown in Fig. 8, together with the difference angle Bw- Bg representing the wave heading relative to the barge heading. Also shown in this figure are the conventions made use of later for the designation of the forces in the mooring cables and the azimuth angle for the boom used to lower loads from the barge. VECTOR WAVE PROPAGATION DIRECTION PREDOMINANT WIND DIRECTION L, MOORING LINES ? DIRECTION OF BARGE HEADING La (RELATIVE To PREDOMINANT WIND) Fig. 8. Orientation and relations between barge, wind and waves 1045 Kaplan Figures 9 and 10 show, for each of the six barge motions, the variation of total spectral energy with barge heading, for each of the three sea states considered. The ordinate plotted for each of the curves is the r.m.s. value, oj, for the time history of the barge otion represented, and it will be convenient to refer to the variance gj of the function as the total energy. The r.m.s. value of any time-history function is therefore the square root of its total energy (assuming, here, as always, the mean value of any time-history function to be zero. Representative examples of calculated spectral energy density functions for the case of the center-lowered load are presented for two sea states and two barge headings relative to the predominant wind direction in Fig. 11. The spectral energy density functions shown for the load were calculated from those of the fundamental set of cross-spectral energy density functions, i.e. those of the six barge motions. Since the time histories of the load and amplitude operators for the former may be obtained by forming appropriate linear combinations of the complex response operators, Tjn, for the barge motions, and calculating their squared absolute values. The r.m.s. values (as defined here) were obtained for all quantities of interest for the load lowering operation such as dis- placements, accelerations, tensions, etc. as well as the forces in the mooring cables, for each sea state and barge heading relative to the waves. Similarly variations of these quantities as a function of the boom azimuth angle were found, from which an optimum boom angle (which minimizes the r.m.s. values of any one of the time histories of interest) may be determined. As an example of results obtained for a 200 ton load lowered in a State 5 sea with a crosswind barge heading and with the optimum boom azimuth angle (here 180°, i.e. boom over the stern), the r.m.s. value of the added-dynamic line tension given by Fig. 12 is (2.38)(200)/32.2 = 14.8 tons. From data on the normal probability curve for this r.m.s. value, it can be shown that the downward force of impact on the bottom would exceed 25 tons approximately 2.3% of the time, if the instant of im- pact were allowed to occur at random. For a center-lowered load under the same conditions, the r.m.s. value of its acceleration is 1.17(200)/32.2 = 7.27 tons, and the downward impact force on the bottom would exceed 14.3 tons approximately 2.3% of the time. The r.m.s. value of the fluctuating component of the force in each of the four mooring cables is shown in Fig. 13 as a function of barge heading for each sea state. The four are seen to have nearly the same r.m.s. value for any particular barge heading in a Sea State 3, with the actual values varying between 300 and 600 lb. For a Sea State 4 the range is from 1100 to 1750 lb, with the differences between r.m.s. values for the four fluctuating cable forces being as much as 150 1b. For Sea State 5, the range is from 1800 to 2700 lbs with differences in r.m.s. values between cables of 250 1b. In all 1046 Mooring and Postttoning of Vehicles itn a Seaway SEA STATE 3 RMS VALUE Mee 180 150 120 90 60 30 - O + 30 60 90 120 180 180 Ag. BARGE HEADING (degrees) SEA STATE 4 RMS VALUE (feet) 1.2 1.0 Z 0.8 Y¥ 06 x 0.2 180 150 120 90 60 30 - O + 30 60 90 120 150 180 $y, BARGE HEADING (degrees) SEA STATE 5 RMS VALUE eae 1.6 0.6 0.4 0.2 180 150 120 oe 60 307° — 3 O}) + 50 60 90 120 150 180 Ag. BARGE HEADING (degrees) Fig. 9. RMS values of the translational barge motions as a function of barge heading at indicated sea state. 1047 Kaplan SEA STATE 3 RMS VALUE (radians) .018 016 O14 p O12 002 180 180 120 90 60 30) =" “0? + - 30 60 90 120 180 180 A, , BARGE HEADING (degrees) SEA STATE 4 RMS VALUE eet ba 180 150 120 90 60 30° =~ '0)'+°'30 60 90 120 150 180 4,, BARGE HEADING (degrees) SEA STATE 5 RMS VALUE (radians) .09 .08 01 180 150 120 90 60 207 — Ors 30 60 30 120 150 180 4, . BARGE HEADING (degrees) Fig. 10. RMS values of the rotational barge motions as a function of barge heading at indicated sea state. 1048 Mooring and Postttoning of Vehicles in a Seaway (TT) $(w) FoR Bg = 90° —e "e T' = Added dynamic tension in lowering line I f load Bi wering line per slug o SEA STATE 5 aan SEA STATE 3 Oo (¢) 0.5 1.0 1.5 Ww (rad/sec) 16 (Sy,8y) @(w) FOR SEA STATE 5 Sy = Port-starboard displacement of center-lowered load SPECTRAL ENERGY DENSITY (ft2-sec) fo) 0.5 1.0 1.5 w (rad/sec) Fig. 11. Spectral energy density functions for added-dynamic tension in lowering line and lateral displacement for center-lowered load, for indicated barge heading and sea state. 1049 Kaplan MINIMUM RMS VALUE SEA STATE 5 3.0 iE i* 2.4 7 =+180° Y=+165 345 7 =-165~ 7 =-180° 7=180% 722 SEA STATE 4 1.8 ° ° 1. 6 7 =+180 7= +165 7=-165° 7 =-180° 7 =180Y + 1.4 1.2 1.0 SEA STATE 3 wine 7 =+180 0.4 0.2 180 150 120 90 60 P 30 - O + 30 60 90 120 150 180 Bs, BARGE HEADING (degrees) Fig. 12. Minimum r.m.s. values of added-dynamic line tension (pounds /slug) and vertical load acceleration (feet/second ) for boom-lowered load, as a function of barge heading at indicated sea state cases the cable force r.m.s. values are greatest near crosswind, and least for upwind and downwind barge headings. All of the results obtained in this study provide useful infor- mation for application to many operations that can be performed at sea, using a moored ship as the base. The major questions con- cerning these results are their degree of validity, as well as the capability of extending the results of related situations such as shallow water operation, different mooring systems, the effects of nonlinearity, etc. Some extensions and/or applications of the present theory have 1050 Mooring and Postttoning of Vehicles in a Seaway RMS VALUE (pounds) 3000 SEA STATE 5 aie ae p crams ( Fa Fe SEA STATE 3 iw ~ 500 mat Sap Re RI 2055. idea hn RRs ay 180 150 120 #90 60 30 s— 10) 14,30 60 90 120. 150 +180 Ag, BARGE HEADING (degrees) Fig. 13. RMS values of mooring cable forces as a function of barge heading at indicated sea state. 1051 Kaplan been carried out, where comparisons between theory, model experi- ment, and (in some cases) prototype behavior were made (see [8], [9]). The conclusions of those studies were that linear theory produced good predictions of motion response operators; shallow water effects may be easily incorporated; the effects of other mooring arrangements (such as the usual catenary form of weighted chain cables) can be represented in linear form and produce results agree- ing with theory, within the range of lower sea states. While agreement between theory and experiment was generally obtained for almost all conditions, some degrees of freedom of the vessels considered in [8] and [9] were not properly predicted for irregular sea conditions. The motions of surge, sway, and yaw exhibited large spectral response characteristics at very low fre- quencies where little (if any) wave energy was present, but close to the natural frequencies of those motions (due to the mooring "spring™ forces). Since these motions are very lightly damped, a very small amount of input excitation can still produce relatively large motions at these low frequencies. There are a number of possible explana- tions for this behavior, but the most plausible one is related to the influence of the nonlinear "drift" forces and moments, which will be discussed in a later section when considering dynamic positioning. The theory described here supplements what may be known qualitatively for moored vessel behavior by furnishing quantitative estimates for the motions and their inter-relationships. While the validity of these analytical results for any particular vessel is sub- ject to test, the results for other moored vessels using this same analytical procedure give support to the reliability of the predictions. Thus, it is feasible to treat all six degrees of freedom of a moored ship in a realistic seaway and obtain results for response character- istics of various motions of the ship and any associated load. Considering the results and examples concerning the load- lowering operation, there are two main conclusions. First, motions having amplitudes of oscillation or giving rise to forces and accelera- tions sufficiently high to influence construction operations may occur under certain of the environmental conditions considered in this study, particularly when loads are lowered by means of a boom ina high sea state. Secondly, the violence of these motions, forces, or accelerations may be significantly reduced by the proper choice of vessel heading relative to the wind, and boom azimuth angle. The latter factor regardless of the sea state, has by far the greater effect in minimizing the energy of the fluctuating tension in the lowering line, the vertical acceleration of the load, and its three displacement components. These results provide useful information for conducting operations from a moored ship platform, and hence the capability of obtaining guidelines for operating vessels and performing engineering work at sea is available with the tools of theoretical hydrodynamic analysis presented here. 1052 Mooring and Postittoning of Vehicles itn a Seaway VIL. MOORED BUOY ANALYSIS A moored buoy system is similar in many respects to the moored ship case, and simplifications are made in order to treat a representative problem. The buoy system is assumed to be a single point mooring, with a surface floating buoy hull connected by a flexible line to the ocean bottom. Both slack and taut types of moorings are included in the analysis, and the surface buoy form can be either a ship-like form, a spar shape, or an axisymmetric discus shape. The analysis is restricted to motion in a single plane and the current direction and wave direction thereby lie in this plane, making a two-dimensional problem. Allowance for current magni- tude variation with depth is considered, with its main influence being in the static equilibrium problem (which will not be treated in detail here). Considering the static equilibrium problem, a free-body diagram of a differential element of the cable in the plane of interest is shown in Fig. 14. The cable bends and the tension varies along its length so as to keep all the indicated forces in equilibrium. The cable weight acts vertically and the tension forces are directed along the cable axis. The hydrodynamic forces due to the current are resolved into components normal and tangential to the cable direction. These unit forces are represented as follows: F($) = Cys 5 pc(Ve sin 4)° (48) G($) = C_+ 5 pc(V, cos 4)” (49) 1 where p = mass density of fluid cable chord length (in current direction) ie) Ul and C,, C, are appropriate drag coefficients. These coefficients depend on the cable cross section and surface geometry. The summation of forces along the direction of the cable axis yields: T + G(o)(1 + €) ds - We, ds sin > - (T - dT)cos (dd) = 0 For a differential element, d¢—~ 0, so that cos (d¢) ~ 1.0. This gives the differential equation for cable tension in terms of the inde- pendent variable s, as follows: £053 Kaplan (p)(I+e) ds F(¢) (1+ «)ds Ve = current velocity JT = cable tension g = distance along relaxed cable « = cable strain @ = angle of cable from horizontal W,= unit submerged weight of cable G¢) = tangential unit force componet due to current a = Hl normal unit force component due to current Fig. 14. Cable Free-Body Diagram 1054 Moortng and Posittoning of Vehicles in a Seaway dT =[- G(¢)(1 + €) + W, sin ¢] ds. (50) The summation of forces normal to the cable axis yields: F(¢)(1 te) ds + W, ds cos $ - (T - dT) sin (dé) = 0 and with d@é—~ 0, we can approximate sin (dd) by dq. Neglecting higher order terms involving products of differentials, results in an equation for the differential angle: do = = [F(t +e) + We. cos 4] ds. (51) The strain, or cable elongation, is obtained from the following simple relationship for an elastic cable material: €= a (52) where cable (load-bearing) cross section area > uN E. = effective static elastic modulus of cable material. Associated with these equations is the representation of the forces and moments acting on the buoy due to the wind and the cur- rent (not considered here, but discussed in[10] from which the present analysis is abstracted). All of these effects are considered to be in equilibrium with the weight, buoyancy, and cable forces. All the forces and moments acting on the buoy due to current and wind are considered to be in equilibrium with the weight, buoyancy and cable forces. At the surface buoy we then have: D, + Dp = T cos 6 L, + B(6,h) = W+T sin $ (53) 2 2,1/2 lize M, +M,= T(4, +24) sin (@ + © - tan 5 me + B(6,h)GZ(0,h) c where 1055 Kaplan D, = drag due to wind acting on a buoy Do» Les Me = current-induced drag, lift, and moment acting on buoy T = cable tension (at buoy attachment point) @ = cable angle from horizontal (at same point) B(@,h) = buoyancy force W = total weight of buoy 4,, Z, = horizontal and vertical distances respectively from cable attachment to CG Z(8,h) = hydrostatic righting arm Thus, for a given buoy'configuration in a particular condition of sub- mergence in a given current, Eq. (53) can be solved for 0, T and $; that is, the buoy trim equilibrium and the cable tension and angle at the buoy. This result then becomes the initial condition for the static equilibrium cable geometry calculation. When considering the problem of the dynamics of the complete moored buoy system, separate considerations in the analysis are given initially to the buoy and to the mooring system, with ultimate combination (i.e. coupling) exhibited later. The motion of a buoy in waves considers the buoy to be equivalent to some type of hull form, and the restriction to planar motion results in analyzing only three degrees of freedom which may be considered to be surge, heave and pitch. The equations of motion of the buoy are formulated in the same general way as for a surface ship, described previously. The only possible additional influence in the present case of the buoy is to allow for the effect of a uniform surface current, which can be included in the equations by interpreting the current as an equivalent forward speed of the buoy hull through the water. However, for simplicity here, this effect is deleted when analyzing the buoy wave responses. For the case of a ship-form buoy hull the hydrodynamic force derivation is similar to that shown previously for the moored ship. The general equations of motion in the vertical plane for the coupled motions of surge, heave and pitch can be represented in a more specific form as ax tasx tia 6 = et x, (54) 1056 Moortng and Postttontng of Vehteles in a Seaway ee oe Zz a, me zt ang a a579 + a5,0 + A909 = Lan F Ly (55) a 5 a,x tayz ta,z ta,z ta,0 ta,0 ta,0 = Mn t My (56) 6 where the mooring forces are represented in general form and the wave forces can be represented as sinusoidal functions of time for different wave frequencies. The mooring forces will depend upon _ the mooring arrangement (i.e. number of cables, attachment point, etc.), whereas the functional form and degrees of freedom in the force representation depend upon the geometric arrangement. For the surge degree of freedom, the coupling with the pitch equation, and vice versa, occurs as a result of hydrodynamic inertial coupling (potential flow theory) and hence the symmetry relation a,7= a3, is attained. This result is due to the equivalence of the off-diagonal terms of the added mass tensor representation of inertial forces. With the longitudinal force mx assumed to act through the center of buoyancy (CB) of the hull, a pitch moment m|BG|x occurs, i.e. a,,=m|BG| = a,7z where LBG| is the vertical distance between the CB andthe CG (center of gravity). The remaining terms in the surge equation are a,, = m and some estimate for surge damping Ajos The surge damping can be represented in a number of ways, either linearly with allowance for the current by means of perturba- tion theory, or in a nonlinear form as a drag coefficient representa- tion, etc. (see [10] for more details). Ordinarily, this surge damping term is not very important in its influence on the resulting ship or buoy motions since there is no natural resonant response in surge. However, in the present case of a moored buoy there isa restraining surge force from the mooring cable and there may be some resonant surge motion. Thus, the proper inclusion of the surge damping force on the buoy hull can be important for dynamic behavior calculations. The mooring cable forces acting on the buoy hull are con- sidered separately further ahead in this study. They are important since such forces affect the buoy motions, and the buoy motions in turn determine the boundary conditions as well as the input excitation for the cable dynamics. The techniques for inclusion of these effects in the overall mathematical model are considered later in this inves- tigation. A spar buoy hull form is axisymmetric about the vertical axis and hence motion analysis can be carried out for the three degrees of freedom with slender body theory techniques used in the analysis of the hydrodynamic action on a long slender spar form. mi = = pgs, aihiy Pa ee (57) 1057 Kaplan where S, is the spar cross section area at the waterline intersection. The mooring force will depend upon the mooring arrangement and geometry, and is deleted temporarily from consideration. The wave exciting force can be evaluated and the wave generation damping is determined from work in[11]. The velocity potential and the pressure on a slender axisym- metric body in waves are found using the results of [12] for the case of a vertically rising body in waves, at zero forward speed and evaluated for the condition where only the submerged portion from the waterline down is of interest. The fluid pressure on the body in regular sinusoidal waves is (from [ 12]) P,=.pea ES a sR cos 8' cos wt - sin wt) (58) where € and @! are the longitudinal and angular coordinates of the spar hull and R is the local hull radius. The local vertical force on a section of the spar buoy is then wv a =-2R tana p dae’ (59) e 0 with tan a = dR/d€ (the slope of the body contour), the local vertical force is 2nrt/ ds! SEs = pga sinwts+ e m™ ae (60) where S' is the local hull cross section area, leading to 0 ké ds! Zw = pga sin ut | Oar dé (61) where k= 2n/N= w/z, which can be simplified further by integration oy parts (with S'(L) =0). The surge equation for the spar hull form is represented as 0 eo mx = - of. st(é)[x + (€ - E,)8] d& + Xqt Xy t+ Xm (62) where &, is the C.G. location along the €-axis, and the wave force is obtained as 1058 Mooring and Postttoning of Vehicles tn a Seaway O Xy -f i p cos 6'R dé dé = pau | Bete Nae ocoaai Co) The surge damping force expression due to wave generation (pro- portional to x) in [11] and to this should be added the surge damping due to real fluid drag effects, which is nonlinear. This drag term is represented in the form 0 ESP ay) [e+e - eg ag (64) where Ap is the lateral projected submerged area of the buoy and Cy is a drag coefficient whose value is = 1.2, the value for long slender bodies and sections in an oncoming normal flow. The pitch equation for the spar hull form is represented by @e 0 Ho =-) suenlx +6 - 9616 - &) a Bt fe) =: oa (€ - &)S'(&) dé > 0 +Mg+tMy+Mm (65) and the wave induced exciting moment is given by fe) dX oe he - &,) Se ag 2paw aa - Eg)eMs! (E) d& * cos ut (66) M The pitch damping moment coefficient due to wave generation is given by [11], and as with the surge damping above, the pitch moment equation has additional nonlinear damping given by O S04, ) Pea DOl(E-&) 48 (67) which must also be induced. By considerations of symmetry, a cross-coupling damping term due to pitch angular velocity, 0, will appear in the surge equation, which is given by 6) 0 ° x,=- 289 Moye ag Je - toielsuey asd (6 1059 Kaplan and a similar term will occur in the pitch equation, proportional to x, given by nee ° ; Me = - oan este) ag - Mn ( - Eg)e“*s'(é) d&- x. (69) All of the above expressions can be combined to produce the coupled surge and pitch motion equations of the spar buoy, together with the heave motion equation. The effects of the mooring are included in terms of the appropriate degrees of freedom to allow computation of the complete system response, which will be the end product of the program. The disc-shaped buoy hull is analyzed as a case of a shallow draft vessel. The section in the water is a circular cylinder with a small draft compared to the cylinder diameter, and the form is axisymmetric. The hydrodynamic and hydrostatic forces are found using the shallow draft approximation, as in [13], together with other simplified representations for the wave-induced forces. Because of symmetry relations, where the disc-shaped buoy is assumed to be circular shape, some of the coefficients in the basic equations of motion, Eqs. (54) - (56), are immediately evident: (70) aeg9 = az6 = 0. Specific values of certain other coefficients are readily evaluated for a circular discus shape, and they are given below. Assuming that the discus buoy is a cylinder of radius R, and draft d', the following heave restoration coefficient value is found: 2 aog= pgmR. (71) For the case of the pitch restoring moment, the basic term (cor- responding to the hydrostatic portion) of the coefficient az, is obtained from the expression M, = - W|GM|é (72) where = Mr weight of buoy |GM| = metacentric height 1060 Mooring and Posittoning of Vehtcles in a Seaway |GM | is the difference between the distance between the CG and the CB of the submerged portion of the buoy, and the metacentric radius between the CB and the intersection of the displaced buoyancy vector with the vertical axis. The metacentric radius is determined from the value of the lateral displacement of the CB of the submerged section of the disc cylinder. This is determined as the ratio of the moment of inertia of the waterplane area about the buoy vertical centerline plane, to the displaced volume. With this moment of inertia found to be mR*/4, and with the valug of the buoy volume given by mR*d', the metacentric radius is R /4d', leading to a metacentric height |GM|: given’ by ne |GM | Sore |BG| (73) where |BG| is the vertical distance between the CB and CG of the buoy. Similarly, values for added mass and added inertia for the disc-shaped buoy can be found from the work of [13], based on con- sidering this hull as a shallow draft vessel. In that case the total added mass in the vertical direction is given by ‘ Al, dx = pR°My (74) where the value of My is given in[13]. Similarly, the added pitch inertia term is represented by { Al.x’ dx = pR°I! (75) where the value of I} is given in[13]. These values are weakly frequency dependent for the range of significant wave lengths of concern in the buoy problem, and an appropriate approximate constant value can be used. The damping coefficients for heave and pitch are represented, respectively, by a es Ni dx = pR*wN, (76) and ; 2 5 asg= | Nyx" dx = pR'wH, (77) 1061 Kaplan where the quantities N, and H, are indicated as frequency-dependent parameters in (77). For the surge degree of freedom the same expressions as for the ship hull form for the coupling terms with pitch, and vice versa for pitch with surge, are valid for the case of the disc-shaped buoy. The damping due to surge also has the same expression, and can be carried over to the present case of a disc-shaped buoy, with appro- priate values of drag coefficient and reference area for the disc. The wave forces acting on the disc-shaped buoy are primarily due to hydrostatic action and are evaluated on that basis. The verti- cal wave force is expressed as R Zy = 2pg io £(x)n(x,t) dx (78) where f(x) = /R* - x? (79) is the lateral offset of the buoy circular section and (x,t) is the wave height representation, as follows: n(x,t) =a sin (x cosB - cyt) a sin(=™ - wt) (80) where the dependence on the heading angle B is deleted due to sym- metry. The resulting expression for the vertical wave force is Zy = - ApgRa sin at | (1 - o* cos Yo) do (81) 10) where o =x/R and y = 2mR/\, leading to Zw=- 2mpgRea , 240) sin wt (82) where J,( ) is a Bessel function. The pitch moment term due to waves is given by R M,,= - 20g | xf(x)n(x,t) dt (83) 1062 Mooring and Posittoning of Vehicles in a Seaway leading to _ 7 My => pgR-alJ,(y) - Jgly)| cos ut . (84) For the surge force due to waves the pressure component in the axial direction is required, and this is found in terms of the axial gradient of the wave amplitude record along the disc. Fora hull of draft d', the surge force due to waves is given by R X, = - 2pgd' \ f(x) a n(x,t) dx (85) which leads to X, = - 2egmd'RaJ\(y) cos at. (86) Thus the above expressions complete the representation of the terms required for treating the motion of a disc-shaped buoy in regular waves. VIII. MOORING DYNAMICS The initial treatment of mooring cable dynamics will be based upon a complete formulation of equations of motion for a continuous line that is assumed to be completely flexible and extensible. The analysis is restricted to two-dimensional motion in a single plane, which is coplanar with the oncoming current, and the velocities, etc. are converted from directions along x- and y-axes (fixed in space) to those along the normal to the cable, which leads to con- sideration of the velocities U (normal) and V (tangential) relative to the cable as basic variables. The basic equations of motion inthe x- and y-directions are du _ 9 Ox Kae =Ry (r dette) + G(ite) cos $+ F(ite) sin > (87) av 8 3 fegy = s(t sais) + G(ite) sin @- F(ite) cos ¢- We (88) where p is the sum of the cable mass and added fluid mass per unit length, when considering an elongated element of the cable, of length (1{+e) ds. From geometric considerations 1063 Kaplan ox = (ite) cos 4, oy = (ite) sin $ (89) and with the definitions ney veer (90) U=usin @-vcos ¢ (91) V=ucos ¢+vsin $ (92) it can be shown that U, - Vo, = - (1+ 94, (93) and V, +t Ud, = & (94) where the s and t subscripts represent partial derivative operations. By considering effects normal and tangential to the cable, the basic dynamic equations can be expressed in the form pl U, - Vo,] = - To, + F(ite) + W, cos # (95) pl V, + Ud] = T, + Glite) - W, sin 4. (96) In addition the relation E yA (97) is also necessary, so that the basic equations governing the cable dynamics are Eqs. (93) - (97), where the first two relations are basically kinematic. For the steady state case, i.e. neglecting time derivatives, Eqs. (95) and (96) reduce to the same expressions as given in the static equilibrium case, i.e. Eqs. (50) and (51). To solve the quasilinear partial differential equations given in Eqs. (93) and (97), a linearization procedure can be applied. Defining the expressions 1064 Mooring and Posittonting ‘of Vehicles in a Seaway U = Us) + U'(s,t) (98) V = Vols) + V'(s,t) (99) T = T,(s) + T(s,t) (100) = $(s) + $'(s,t) (101) € = €,(s) + €'(s,t) (102) where the '-symbol quantities are perturbations about the equilibrium positions (o-subscript terms), and expanding various constituent terms up to first order terms alone, leads to sin @= sin ($9t+ $') = sin 6, + $' cos $5 (103) cos $= cos ($,t+9') = cos $, - $' sin $9 (104) The hydrodynamic loading terms are expanded in the form Fak, + Fu" + FyV't+ Fy?" (105) G=G,+GyuU' + Gw' + Gyo! (106) where the partial derivatives of the loading functions with respect to particular velocities are indicated. This can be accomplished when considering the steady state velocity solutions (from Eqs. (93) and (94) when time derivatives equal zero) which, when combined with Eqs. (114), are Up, = 0 (107) Vior= 0 (108) 0 The linear perturbation equations are then given by -pU' = Tod; +790 (ite, )[F,U'+FV'+ Fo] +F,e'+ Ww, ¢'sin d, (109) -pV' = Tj +(1te,)[ GU'+GWV' +Gyo'] +G,e'- Wed! cos $o (110) 1065 Kaplan (1 + €,)o, = V'bo, + Vos - Us (141) €} =e vert U" do, (11:2) ut oe oer (113) which are a set of first order linear partial differential equations. The boundary conditions for this set of equations is the next task of importance. For a moored buoy in a combined current and seaway, the current is felt acting on the buoy and also on the cable. How- ever, the wave effects attenuate rapidly with depth, and hence the wave forces act on the buoy along with no influence assumed on the cable. In that case the buoy motions due to a regular sinusoidal seaway (assumed for analytical simplification) are transmitted to the cable at its attachment point, and the cable motions are then sinusoidal in time at that point. The boundary conditions at the anchor point at the sea bottom are given by U=V=0, s=0 (414) and at the buoy attachment point for the cable, s =£, the boundary conditions are much more complicated. The velocities at the buoy attachment point are given by U= x = 2°6 (115) c v=z- 4,0 (116) where x; z and @ are the wave-induced surge velocity, heave velocity, and pitch angular velocity, respectively, and Ze and f¢ are the vertical and horizontal distances from the buoy CG to the cable attachment point. The normal and tangential velocities are defined by Eqs. (91) and (92), and considering the wave-induced motions to be of the same order as linearized perturbation terms, the boundary condition relations for the perturbations are U! = (x - z,0)sin $, - (2 - 4,8) cos 9%, (417) V' = (x - z,0)cos &+ (z - £,0) sin $, (118) at s=2, where x; z and @ canbe represented in the et form for a sinusoidal wave input. The boundary conditions at s=0 are 1066 Mooring and Positioning of Vehicles in a Seaway U'=v'=0 (119) where only four boundary conditions are necessary since e' can be eliminated as a variable by use of Eq. (113). The representation of the boundary conditions at the upper end of the cable (s = 2), at the attachment with the buoy, shows how the cable motions are influenced by the buoy motions. However, the buoy motion is also influenced by the cable system dynamics since a mooring force acts on the buoy as well. The mooring force that affects the buoy motion is due to the component of tension at the attachment point, which leads to Xm=- T'(L) cos off) + To(L) sin o(£)'(£) (120) Zm=- T'(L) sin $,(£) - TolZ) cos (2) $'(£) (124) MS Se ee eZ (122) where these expressions are component terms on the right-hand sides of the respective equations, e.g. Eqs. (54) - (56). With T'(£,t) and #'(£,t) represented as f(s)e®t forms the total system of buoy and cable can be solved using linear equations of motion for sinusoidal wave inputs at different frequencies (assuming the non- linear damping terms in the buoy motion equations are linearized). The "feedback" nature of the equations governing the buoy motion and the cable motion is illustrated by the above discussion, where the cable tension force influences the buoy motion directly and the buoy motions determine the cable upper point boundary condition. As mentioned earlier, the study of a moored buoy system is closely related to other mechanical cable system problem areas. The case of a moored ship is, of course, very similar to a moored buoy but the distinguishing difference is the relative masses that are involved. For a moored ship case, the ship is so large (rela- tively) that it can be realistically assumed that only the quasi-static forces applied to it by the mooring cable are significant, and that the cable dynamics do not influence the ship's response; that is, mooring cable dynamic forces can be assumed small with respect to other excitation forces. Thus, the dynamic problem of the ship and the cable can be treated separately. Similar reasoning applies to the surface condition of a cable-towed body system. However, the analysis of the component forces involved in such systems is appli- cable to the present case of a moored buoy system, keeping in mind the required coupling in the mathematical model, as shown above. The equations developed here for a moored buoy system have to be solved in order to determine the necessary information on 1067 Kaplan system performance. The methods to be applied should recognize the problem as involving complicated two-point boundary value prob- lems, or alternatively another technique that replaces the equations by a set of difference equations or differential-difference equations similar to the case of a beam vibration problem can be applied (see [10] for a discussion of different computational techniques for this problem). The solution of the class of partial differential equations given above is a specialized simulation problem that is the subject of presently on-going research so no further detailed discussions can be given. The development of these equations is another illustration of the application of knowledge of hydrodynamics of ship motion toward other related problems of engineering significance. IX. DRIFT FORCES DUE TO WAVES When a floating vessel is acted upon by waves it experiences forces and moments that are predominantly oscillatory-like in nature, with the frequency characteristics similar to that in the spectrum of the oncoming wave system. These forces are also linear with regard to wave amplitude. In addition there are also nonlinear force contributions that arise from the presence of the vessel hull modifying the incident waves by virtue of its function as an obstruction, as well as the effect of interaction between the vessel motion and the incident waves. These nonlinear forces are much smaller than the linear wave forces, but nevertheless exert a significant effect on certain degrees of freedom of the vessel. The major nonlinear drift forces of importance to the problem of maintaining a desired position in a seaway are in the longitudinal and lateral directions relative to the vessel, as well as the yawing moment that tends to rotate the vessel in heading. Some theoretical studies of these quantities have been made, but only for determining average values in regular waves, with the work of Havelock [14], Maruo [15], Hu and Eng [16] and Newman [17] serving as typical examples. Havelock [14] treats only the drift force in head seas. His formula is based on the heave and pitch motions and their relative hase to the incident wave. The theoretical approach of Hu and Eng Pt 6], which follows that of Maruo [15], yields expressions only for the lateral drift force and draft yaw moment in waves. (Maruo's results only considered the lateral drift force.) While their results are quite general, they have only been reduced to workable formulas under the restrictive assumptions of a thin ship, with small draft, in long waves. These results indicate infinite (practically unrealis- tic) forces and moment as the wave length goes to zero. The maxi-~ mum lateral drift force occurs in beam seas (varying as sin°§), and the maximum moment occurs at an angle of 45° (varying as sin B). 1068 Mooring and Postitioning of Vehteles in a Seaway Newman's method [17] , based on slender body theory, does show some comparison with very limited experimental work for lateral drift force and yaw moment which indicates rough agreement. His results for longitudinal force, for which no experimental com- parison is given, indicate that this force in head seas generally exceeds the lateral forces for a given wave length condition. Further- more, these results indicate ‘that the maximum lateral force occurs in bow waves (B = 45°), with the force going to zero in both head and beam waves. The results of Hu and Eng [16] include the effects of sway, yaw and roll motions, with no influence of heave and pitch included (as to be expected for thin ship analysis) while Newman [17] only accounts for heave and pitch motion effects without any influence of the three lateral degrees of freedom. Thus there is a question as to the proper representation of the drift forces that would reflect the influence of the important dynamic motions that produce these forces. The analysis by Maruo [15] presents a final expression for the average lateral drift force in beam seas that depends upon the reflected wave amplitude, which in turn is defined in terms of the relative motion between the incident wave and the resulting heave motion. The presence of sway motion has no effect on the lateral drift force since the body acts like a wave particle in beam seas and no relative motion occurs (to that order). A similar result is indicated for submerged cylinders in the work of Ogilvie [18], where the average lateral force identically vanishes. In all of these hydrodynamic studies, the force is found to be proportional to the square of the incident wave amplitude, since the nonlinear pressures are represented in terms of squares of generated wave amplitudes, squares of fluid velocities, and products of first order oscillatory displacements with derivatives of fluid velocities +> While the previous hydrodynamic analyses have been concerned with the average drift force for a regular sinusoidal wave, it is important also to determine the actual time histories of these forces, especially for the case of an irregular incident wave system. In that case it is expected that the drift force will be a slowly varying function, in terms of the frequencies contained in the defining incident wave band- width, and it is of interest to determine the basic representation of the forces, the response of floating vessels to such forces, the statistical properties, etc. In order to illustrate the basic characteristics of these forces, particular attention will be given to the case of the lateral drift force acting on a vessel in beam seas. Using the results of Maruo (15); the later drift force acting on a cylinder (in the two-dimensional cas e) is given by FE 1 —2 2 1 2——2 2 > = 5 pgA,|2-71| = 5 Pga A,|=-1| (123) 1069 Kaplan where A, is the ratio of the amplitude of the heave-generated two- dimensional wave to the amplitude of heaving motion of the ship section, and |z- n| is the absolute value of the relative heave mo- tion. It is seen that this expression is thus proportional to the square of the incident wave amplitude, with the force only occurring due to the relative motion between the heave and the incident wave. It can be shown that the mean value of this force in an irregular sea characterized by a wave spectrum such as the Neumann spectrum is given by Qo —— : ~— mee SG) See (124) o a where A*(w) is the Neumann spectrum representation given in Eq. (43). The results obtained in the basic derivations for determina- tion of the drift force have been carried out for the case of a single sine wave at a fixed frequency. Inthe real case, the waves are composed of many frequencies in a band, and for the purposes of simplification of the following analysis it will be assumed that this bandwidth is relatively narrow. If a combination of two different frequencies is present in a wave, and hence in the relative heave motion represented by Z-N= b, sin wt +b, sin (wet + ¢) (125) the square of this term is given by (z - n)°= be sin’ wt + bs sin® (wot + 9) +2b,b, sin w,t sin (wot + >) 1.) 2 2 = she +b. + 2b, b, cos [(w, - w, )t + 6} 2 2 - 54) cos 2w,t + b, cos 2(wt +) - 2b,b, cos [ (wy + o,)t ral}. (126) It can be seen that this expression is made up of a group of terms that are essentially constants and slowly varying terms (due to the narrow band assumption), and another group of terms representing higher frequency oscillations, i.e. at higher frequencies than the wave terms. If a time average of this quantity is made, it will be seen that the combination of the constants and the slowly varying term remains and the higher frequency terms drop out, and that this first grouping of terms can be represented as the square of the envelope of the combined signal given in Eq. (125) (see Er9}). 1070 Mooring and Postttoning of Vehicles in a Seaway If the wave system, and the resulting relative heave motion, are represented by the sum of a larger number of terms of different frequencies, with these frequencies having only small increments relative to a single reference frequency (as a result of the narrow band assumption), the contribution to the drift force value in that case can be shown to be given by an expression that is also identified as the square of the envelope of the total signal. This identification and interpretation of this type of expression for the drift force can generally be extended to the case of an arbitrary input, including a random input which is assumed to be made up of a combination of different frequencies within a narrow band. Thus a simulation technique in the time domain for this term requires determination of the envelope of the input signal (relative heave motion), squaring this quantity, and applying the appropriate constants to produce the required time history signal. The drift force has been shown to be a nonlinear function of the wave amplitude, and in a random sea it is a slowly varying function of time, where this slow variation is considered relative to the wave frequencies and the linear wave-induced forces. Since the wave surface elevation and all linear terms derived from it are assumed to be Gaussian random processes, the drift force is known to be non-Gaussian in regard to its probability density. In order to obtain further characterization of the properties of the suction force, it would be useful to determine the probability distribution and spectral properties of this force. The accomplishment of this task will be aided by the simplified interpretation of the drift force that was presented above. Considering the drift force as the square of the envelope of a Gaussian random process, certain information is available concern- ing the probability density of this type of function. A square-law detector produces an output proportional to the square of the envelope of the input, if the input is a narrow band Gaussian random process [ 20] which is the assumption used in the present analysis. Fora particular input into such a square-law detector, the probability density function of the output (denoted as w) is given by 1 -w/ E(w) s p(w) - E(w) © ’ w= 0 (27) where E(w) is the mean value of the square-law detector output. On that basis, the probability density for the drift force in a random sea can be represented by -F//F p(Fy) = se de seke nabie ee (128) F y where Fy, is the mean drift force, and hence the probability distri- 1074 Kaplan bution is given by P(= 1.0 will be eliminated. Thus it can be seen that the drift force itself is concentrated at low frequencies, and that the response of a dynamic system with a very low natural frequency (i.e. the ship sway motion) will result ina response spectrum concentrated at an even smaller low frequency band. This is what is usually found in the results of model tests and full-scale experience, and thus an explanation is provided by the preceding analysis. X. APPLICATION TO DYNAMIC POSITIONING When considering the case of dynamic positioning, various forces act on a free vessel in the open sea that cause it to move from its required position. These forces are the relatively steady forces due to wind and due to current, the oscillatory-type forces due to waves and the drift forces. The wind generates forces and moments because of its impingement upon the abovewater surfaces of the hull and superstructure, while the current forces act on the underwater hull (and any submerged drilling equipment, if that is the purpose for the vessel). These steady forces can be overcome by the generation of steady forces by some type of thruster mechanism that will act to maintain the ship more-or-less in its desired location. The oscillatory-type forces due to waves are very large, and no force-generating system installed on a vessel is expected to be able to overcome such effects. The ship will therefore oscillate "back-and-forth" in response to these large wave forces with essentially no net deviation of significance from its average position. 1076 Mooring and Postttoning of Vehicles in a Seaway Sy (w) drift 0.04 2(pauk) ft.*-sec. 0.02 0 0.5 0 1.5 230 2.5 3.0 w, rad/sec. Fig. 17. Representative power spectrum of lateral drift force. It is the drift forces and moments whose mean values tend to move and/or rotate the ship off position, with their level of fluctuation causing dynamic responses that produce ship motion. Thus some means of control must be applied to "modulate" the forces developed by the thruster system in order to minimize the ship's average motion relative to its desired position. The ship will experience drift-like forces in the lateral and longitudinal directions, as well as a yaw moment, and the philosophy of applying control forces to the ship will be aimed at countering these forces by orienting the ship in the proper direction so that the resultant force is acting along the ship's longitudinal axis and there will be no significant moment. It would then be possible to use the main propulsive thrust of the ship, assuming controllable pitch propellers, to counter this resultant force. Lateral forces that are developed by particular thrusters, or other force systems, will be used to overcome any tendency of the ship to rotate out of its pre- ferred direction due to any resulting yaw moments. The force magnitudes in regard to average values can be estimated from the results of some of the cited references given in this paper, and some idea of the time history variations and maximum magnitudes expected relative to the average values can also be inferred from the work presented here. 1077 Kaplan In a complete simulation study of the resultant motion of a ship in which a dynamic positioning system is to be installed, all three degrees of freedom (surge, sway and yaw) will be coupled and the forces and moments will be dependent upon the relative orienta- ticn with respect to the incident wave system. This will be a some- what complicated analysis, but the tools are generally available for determining the various hydrodynamic parameters entering inte such a study. It will also be necessary to consider the type of signal system that would inciate the position errors of the ship, together with a signal processing operation (i.e. control system design) that will be necessary in order to achieve the desired type of operation. Similarly, some estimate of the response time of the thruster force development must be included in determining ship response so that a measure of positioning accuracy can be obtained as a result of the analysis. In view of the complexity of this problem, a discussion of a simple application will be given for the case of sway motion alone in beam seas. In that case the equation of motion will be similar to that given in Eq. (137), without the presence of the linear spring term (that was due to the mooring in the previous case). The response due to the linear wave forces will be generally the same for this case as in the case of the moored ship, as shown in Fig. 15. However, the effect of the drift forces will cause the ship to continu- ally deviate in position within a very short time. The deviation will be almost a quadratic growth with time since the response is similar to that of a constant force acting on a system primarily represented as a pure second derivative dynamic response. Thus a control force is necessary, and the control rule should include terms proportional to sway displacement and velocity, i.e. the control force will be of the form Y, = pr hl vio. > soimaia5M a io , ‘2D, sti Gis dP 8 ok Ast re bse zh ,@eegere oT ee i. i 4 4 ai" | a sipoF Ski so, Sal Sods soqueveGd os Yonyxoeut3 1 ot colbpub e OT. coat, .o) Root Te ov _ , ¥ e Mov lem al ie Sagi u Be +e he : ¢ 1% AS ge Bt a ' ; y i] Biee Apts ( ‘ ' -uiar 4 2 t rao . WAVE INDUCED FORCES AND MOTIONS OF TUBULAR STRUCTURES J. R. Paulling University of Caltfornta Berkeley, Caltfornta ABSTRACT Many types of stable ocean platforms consist of space- frame assemblages of tubular structural and buoyancy members. An approximate method of predicting the hydrodynamic forces and resulting motions of such structures is described. In this procedure, the force on each member is’computed by assuming that the member is long and slender and all other members are absent. Such forces for all members are summed and introduced into the linear equations of motion of the entire structure, which may then be solved for the re- sulting platform motions. Reasonably good agreement is obtained between the results of such analysis and model experiments with several different platform configurations. I. INTRODUCTION Many of the stable floating platforms which have been pro- posed or constructed for deep water drilling, mining, or emplace- ment and recovery of heavy objects can be described as space- frame assemblages of tubular members. R. H. Macy [1969] describes and illustrates several oil-drilling platforms of this type. One of them, BLUE WATER II, consists of a square base configura- tion approximately 200 feet square, made up of cylindrical members 14,5 feet in diameter, with four vertical corner caissons 24.7 feet in diameter supporting the main deck. This platform normally operates at a draft of about 40 feet. A second platform, the SEDCO 135, consists of three main vertical caissons located approximately at the vertices of an equilateral triangle, several diagonal tubular truss members, and, at the bottom of the main caissons, elongated pontoons of oval planform. These platforms are moored by a spread array of anchors, and operate in water depths of up to 600-1,000 feet. 1083 Paullting McClure [ 1965] described a platform which was designed for the MOHOLE deep sea drilling project. This platform was to consist of two submerged main horizontal pontoons 35 feet in diameter, and 390 feet long, with a centerline separation of 215 feet. Three vertical caissons extended from each horizontal pontoon through the free water surface to support the main working deck. This platform was intended to operate in a water depth of 14,000 feet, and was to be dy- namically positioned by means of trainable propulsion units controlled through a central computer system. The foregoing types of platforms are referred to as "column stabilized," which implies that pitch and roll static stability are obtained primarily from the waterplane ma- ment of inertia of the surface piercing vertical column. A third type of platform for which a tubular space frame con- figuration has been proposed is the tension leg platform, an example of which is shown by Macy [1969] and described by Paulling and Horton [1970]. This is a moored stable platform for which the buoyancy exceeds the platform weight, and the net equilibrating vertical force is supplied by vertical tension mooring cables secured by deadweight or drilled-in anchors. As a final example of tubular stable platform structure, we mention the spar-type platforms, the prime example of which is FLIP, described by Fisher and Spiess [1963]. The platform consists of a single cylindrical member of tapering cross section arranged to float vertically, with a small portion of its length projecting above the surface of the sea. All of these platforms share a common characteristic in that their configuration consists of a space frame assemblage of relatively long, slender, cylindrical members, with the addition in some cases of small buoyancy chambers or pontoons. All share a common ob- jective of producing a working platform having minimum wave- induced motions, even under relatively severe sea conditions, i.e., a platform which is "transparent" to the waves. The positioning methods used differ greatly in each case, ranging from essentially no positioning, in the case of FLIP, dynamic positioning with no physical connection to the sea bottom, in the case of the MOHOLE platform, to various types of anchoring systems exemplified by the tension leg and the column stabilized platforms. The sea environ- ment and resultant platform responses are similar in each case, i.e., all are intended to operate in relatively deep water under severe environmental conditions, with platform motions which are small compared to the overall dimensions of the platform and to the length of waves involved. Our objective here is to describe a pro- cedure for analyzing the forces and motions which can be applied equally to all of these platforms if suitable account is taken of the type of anchoring or positioning restraint involved. Such an analysis of wave-induced forces and motions forms an essential part of the process of designing a platform to perform a specific mission. At least three such functions are envisioned. First, for a platform of given geometry, a range of sea conditions may be investigated to determine what limitations may be imposed 1084 Wave Induced Forces and Mottons of Tubular Structures on the platform's performance by the resulting wave motions. Second, given a set of sea conditions and platform requirements, we may investigate a family of platform configurations to determine those members of the family which will be able to perform the specified mission under the stated sea conditions. This kind of analysis, in turn, might form a part of a more extensive system study aimed at determining the most cost effective platform system. As a third function, the force distributions on structural members which are obtained during the force and motion analysis may be used in connection with the detailed structural design of the platform. The general procedure followed in analyzing the dynamic behavior of such a platform is to assume that it behaves as a rigid body having six degrees of freedom. The external forces which excite the motion of the structure are associated with the fluid motion relative to the structure, and with the structure's mooring or position- ing system. Two alternative methods are available for the computa- tion of the fluid forces. In the first, the fluid is assumed inviscid and its motion irrotational, and we proceed on the basis of classical hydrodynamic theory to seek a solution to Laplace's equation in the fluid region subject to certain boundary conditions. These include kinematic boundary conditions on the free water surface and on the wetted surface of the structure itself, a constant pressure dynamic boundary condition on the free surface, a dynamic boundary condition on the wetted surface of the body, which is derived from the rigid body equations of motion, and other conditions far from the body which are necessary for uniqueness of the solution. This approach yields great insight into the fundamental nature of the fluid phenomena, and is exact within the limits of the necessary fluid idealization and motion linearization. Its implementation, however, is beset with almost insurmountable difficulties unless the geometry of the body is extremely simple. The second method is less exact in principle, but provides approximate means of including real fluid effects and of dealing with geometrically complex, realistic configurations. This procedure, which is employed in the present analysis, is termed "hydrodynamic synthesis." Here we consider the complex structure to be assembled from a group of simpler bodies whose individual hydrodynamic pro- perties are known, perhaps as a result of an analysis of the first type above. A fundamental assumption is then made that the hydrodynamic force on the assembled structure may be computed by taking the sum of the forces of all of the component members. In the simplest:case, these forces are computed as though.each member were completely remote and independent of the rest of the structure, but subject to the same pattern of body and fluid motions. The forces computed in this way might be refined by introducing modifications to the fluid flow to account for the hydrodynamic interaction between adjacent members. The result of this hydrodynamic synthesis is a system of hydrodynamic forces acting upon the assembled structure, containing terms dependent upon the incident wave system and upon the motion 1085 Paullting of the structure itself. Additional restraint forces are introduced to account for the effects of the mooring or dynamic position system. This total system of external forces is then equated to the mass times acceleration of the body by Newton's second law, yielding a system of coupled differential equations of motion. These equations are then solved to obtain the time-dependent motion of the structure. In the present analysis, a linear relationship will be assumed to exist between all forces and the appropriate motion parameters. Two important consequences follow as a result of such an assumption: (1) The hydrodynamic forces acting on the structure may be divided into two independent parts, one depending only on the incident wave motion, and the second depending only on the platform motion. (2) A prediction of the platform response to a realistic random sea- way may be obtained by superimposing the responses to the seaway's regular wave components. The validity of such a linearization may be tested either empirically or by comparing its results with results of an "exact" analysis. An exact analysis is normally possible only for sucha simplified class of geometries that the validity of the comparison for the realistic case is subject to question. We are, therefore, forced to an experimental test. For the present, we have considerable evidence on the usefulness of linear techniques in predicting ship motions as in Gerritsma [1960], and motions of platforms of the present type, Burke [1969], Paulling and Horton [1970]. Some further experimental comparisons are given in the present paper. II. THE EQUATIONS OF MOTION The motion of the platform will be expressed as a small deviation from a mean position, and for this purpose it is convenient to define two coordinate systems. The first, OXYZ, is fixed rela- tive to the structure suchthat O is located at the structur's center of gravity, Y is directed vertically upward, and OXZ is parallel to the mean waterplane. In many cases, we may take advantage of symmetry to arrange these axes so that one or more of them is a principal axis of inertia. Also, in some cases, a designer's co- ordinate system may be used for drafting or other design purposes, which is parallel to OXYZ but whose origin is located elsewhere. Quantities defined in this latter system may always be transformed to the OXYZ system by simple coordinate transforms, and it is assumed that this is done. A second coordinate system, oxyz, is fixed in space such that it occupies the mean position of OXYZ as the platform moves in waves. In general, it is found most convenient to express the inertial properties and the forces acting on the structure in OXYZ since the geometry of the structure is fixed in this system. On the other hand, it is more convenient to express the equations of linear motion in the space system, oxyz, since this is an inertial system, and because we ultimately wish to obtain the motion of the platform 1086 Wave Induced Forces and Mottons of Tubular Structures in terms of time dependent deviations from this mean position. The linear displacements of the center of gravity of the platform from its mean position may then be expressed by the small quantities x(t), y(t), z(t), measured in the oxyz system. We next express the rotational motion of the platform in terms of the Eulerian angles e(t), B(t), y(t). These angles are so defined that the angular displacement between the two coordinate systems, oxyz and OXYZ, may be created by imagin- ing the platform as first oriented such that the two coordinate systems coincide. It is then rotated about OX through the angle a, then about the new position of OY through the angle 6, and finally about the new position of OZ through the angle y to bring the platform to the final position of angular displacement. For small values of a, B, y, the two coordinate systems will now be related by: x 4 -Y B Xx y = y 1 -a@ . Y e (1) Zz -B a 1 Z The equations of motion may now be written. It is convenient to first write the equation for translatory motion in oxyz, thus it is assumed that all forces acting on the body have been expressed in this system, giving f, = mx, , = 4i,. Zed s (2) where the x; = x(t), y(t), and z(t), respectively. The equations of angular motion may be most easily written in the body coordinates, OXYZ, since the moments and products of inertia of the structure are constant in this system. If the exter- nal moments are also expressed in OXYZ, the Euler equations for rotational motion in rotating coordinates are obtained: se ial 8. = components of the angular velocity vector in OXYZ, moment of inertia about i-axis if i= j, i (-) product of inertia if i# j. * Note that the transformation expressed by Eq. (1) can be applied to forces and velocities as well as to coordinates. 1087 Paulltng Now, M; may be transformed by (1) into components, m, i? expressed in the space ue system. We note, further, that if the angular velocities, , are small quantities, the moments, m,, causing the motions eee small of the same order. The transfor- mation of moments, therefore, after dropping products of small quantities, yields mj = M;.- (4) Similarly, the components of angular velocity, Q,, in OXYZ may be transformed into the space coordinate system. wae small angular velocities, this results, approximately, in or oe (5) Here, the @, are the components in oxyz of a small rotation of the structure about an instantaneously fixed axis in space. In other words, for the small angular motions to which we limit the present analysis, the Euler angles are approximately equal to the components of the body rotation during a small time interval about the fixed space axes. We may now write the translational and rotational equations of motion, in the fixed coordinate system, in the combined form 6 y mj; X; — 1, as a ae 6\. (6) jzl Here f. »>x.; i=i1, 2, 3, are the forces and displacements in the X,Y, Z- directions ; fi» x.3 i= 4, 5, 6, are the moments and rota- tions about the x, y, z-axes, m,,* M,5» m,,= m= mass of structure, 33 m,,° M,., M,, = moments of inertia about XYZ-axes, respectively. M4. = ™M,4= ih XY dm Mgg= Meg= - we YZ dm | Products of inertia about | the OXY Z-axes. Myg = Mgq= - seat XZ dm 1088 Wave Induced Forees and Mottons of Tubular Structures The force system acting on the structure comprises hydrodynamic forces resulting from wave and platform motion, hydrostatic forces from the changes in displaced volume associated with static displace- ments of the platform and the restraining forces exerted by the positioning or anchoring system. Tl. COMPUTATION OF THE FORCES The total hydrodynamic force exerted on the body is assumed to be composed of three parts: (1) The force resultant of the pressure exerted by the un- disturbed incident wave train on the stationary body in its mean position. (2) The force resulting from the disturbance of the incident waves by the body occupying its mean position. (3) The force resulting from the motion of the body computed as though it undergoes the same motion in calm water. In a formalized linear analysis, these terms would be associ- ated with velocity potentials representing the incident wave train, a diffracted wave train, and the waves generated by the motion of the body. The force represented by (1) above, i.e., that part ob- tained by neglecting the effect of diffracted waves and body motions, is termed the Froude-Krylov force. Procedures based on assumptions similar to these have, as previously noted, proven successful in predicting the motions of ships in waves. The present situation is somewhat simpler than the ship case because the structure has no forward speed. As noted in the Introduction, we shall obtain the total force on the structure by computing the force separately on each member of which the structure is composed. Two principal assumptions will be made in computing these forces. First, each member is assumed ‘to be either a cylinder whose cross sectional dimensions are small compared to the lengths of both the cylinder and the incident waves, or the member is a small pontoon (point volume) all of whose di- mensions are small compared to the incident wave lengths. Second, all hydrodynamic interaction effects between adjacent members will be neglected, thus the force on an individual member will be com- puted as though the member occupies its mean position in the field with all other members absent. In Fig. 1, we show a cylindrical member located in the flow field corresponding to a train of waves and undergoing motions cor- responding to the rigid-body motions of the entire structure. The force per unit length at a given point along the length of the member 1089 Paulling ,Y — NOTE: €€-PLANE IS MEAN WATER SURFACE LINE IN €C-PLANE PARALLEL TO X-AXIS 8 WAVE DIRECTION ORIGIN OF ent Fig. 1. Coordinate systems and nomenclature for member is assumed to be expressed as Foe are pn ds + C,\ we = Uap) af: Cuil ara anp)- (7) The first term is the resultant force obtained by integrating over the surface of the body the pressure, p, which would exist at that location if the body were not present and is seen to be the Froude-Krylov force. Note that n is the unit normal vector directed out of the body into the fluid. The second term is called the drag force and is assumed proportional to the relative normal velocity between the member and the fluid. The third term is called the added mass force and is assumed proportional to the relative normal acceleration between the member and the fluid. The first term and the first parts of the second and third terms are seen to be dependent on the wave motion while the remain- ing parts of the last two terms depend on the structure's rigid-body motions. In order to evaluate these forces for the cylinder, we must define two additional coordinate systems. The first, ox yz, is associated with the member such that the x-axis lies along the centerline of the member, the y-axis is directed generally upwards, the z is defined to form a right-handed system. The second co- ordinate system, o§nf, is used to express the waves, and is so defined that the €-axis lies in the mean water surface and is positive 1090 Wave Indueed Forces and Mottons of Tubular Stretures in the direction of propagation of the waves, 7 is directed upward, and € is defined so as to form a right-handed coordinate system. Wave Forces In order to be consistent with our assumption of small plat- form motions, an assumption of small incident waves must also be made. The velocity potential for regular infinitesimal gravity waves moving in the direction of the positive §€-axis is = h k(n+d) 9(&.,0,t) = $2 SOS AT sin (k§ - ut). (8) The pressure is given by C) P=- pen- par, (9) and the linearized velocities and accelerations in the &- and n- directions by _ 99 . 99 Se v= 1" (10) . . OU See BS U= 3p = ° We shall first compute the Froude-Krylov force which is given by the first term in (7). This integral, which is to be evaluated over the entire immersed surface of the body in order to obtain the total force, may be replaced by the following volume integral through the application of Gauss' theorem cee ae se The corresponding integral expressed for the moment referred to the Ent-axes is ah p(r Xn) ds SSS (cB ng. SSE 3 oe - & ge) av. (1 1b) Note that in defining these integrals for a member which projects ti — Th iN 1091 Paulling through the free water surface, the pressure and its derivatives vanish for that part of the member's surface or volume above the free suriace. The evaluation of both of these integrals is a straightforward but rather tedious process, yielding the components of force in the Eng-directions, and the moments about these axes. We first per- form the indicated differentiation of the two terms in the pressure equation with respect to €, n, and {, then substitute for €, n, and {, their values transformed to the oxyz-coordinates. The element of volume is given in oxyz by Adx where A is the constant cross sectional area of the cylinder. Since the cross sectional dimensions are assumed small compared to the wave length, the integrand may be assumed constant over A and, conse- quently, the volume integrals are reduced to one-dimensional integrals in x to be evaluated over the length of the cylindrical member. If the member is completely immersed, the evaluation of this integral yields two terms corresponding to the two terms in Eq. (9) for the pressure. The first term is the static buoyancy force or moment and the second is a time-dependent "variable buoyancy" corresponding to the variation in effective weight density of the fluid as a result of the wave motion. If the member projects through the free surface, the region of integration must be dealt with in two parts. The first part is the constant volume of the member below the mean water surface, and the second is the time varying part of the volume of the member lying between this mean waterline and the instantaneous water surface. Evaluation of the integrals over the first part of the volume, i.e., that part below the mean waterline, yields a result identical with that for a completely submerged member. In evaluating the integral over the second part of the volume, we note that this volume is of the same small order of magnitude as the wave amplitude. Since the velocity potential, (8), and therefore the second term in the pressure, (9), is of this same small order of magnitude, only the first term in the pressure, i.e., the hydrostatic part, has a linear contribution to this part of the integral. In evaluating these integrals, (11), for the pontoon or point volume, we note that the assumed small dimensions of the pontoon imply that the integrands are constant over its volume. Therefore, the integration reduces to taking the product of this volume with the appropriate derivatives of the pressure expression. The first parts of the last two terms in (7), i.e., the drag and added mass forces associated with wave induced water motion, are each computed in a similar manner. Let u; and uj; represent the components of fluid velocity and acceleration in off, with u; and u, the corresponding components in oxyz. A coordinate 1092 Wave Induced Forces and Mottons of Tubular Structures rotation may be expressed by ai such that The components of force on an element of length, dx, inthe oxyz- directions are given by dF, = @,u, +%,,u,) &, j=1,2,3 (13) where p; and }; are added mass & and linear drag coefficients. For the cylindrical member, only p55 Xoo, and X33 are non- zero, while for the point volanie FAT hee the diagonal terms in p» and X are nonzero. The forces may now be expressed in o€n{ by the inverse of the above transformation dF = aj, dF; it R rom hy = (piu, + u,) dx. (14) Here, the added mass and drag coefficients in of G are seen to be related to those in oxyz by the following expressions. al amenicl a as dij =a Rigi (15) _Corresponding expressions for the elementary moments about oxyz and o§nG may be written. The total forces and moment may be obtained by integrating these expressions over the length of the member, noting that uj and uj, contain terms with the same trigonometric functions of time which appear in the Froude- Krylov buoyancy integrals, (11), and may, in fact, be combined with them in carrying out the integration over the member length. Evaluation of these integrals yields a set of forces and moments in o§nf. A coordinate translation and rotation may now be applied to express these forces and moments in the space coordinate system oxyz. 1093 Paulling Motion-Dependent Hydrodynamic Forces We now consider a velocity vector, us whose components are the velocities of the center of gravity of the structure in oxyz and are given by x a‘ ip 1 x, in Eq. (6). We may similarly define an angular velocity. vectors QR, whose components are the three rotations about oxyz, and were denoted X4» X53» Xe in (6). The resultant velocity of a point P(x,y,z) is given by ee —> —> — us UTM Xr, where r is the radius vector drawn from the origin of oxyz to the point P. Similarly, the resultant acceleration is given by ea —_ =u + a rs) +Yxr. (Note that this neglects the radial or "centripetal" acceleration which is of the order of the square of the (small) angular velocity.) Point P may be thought of as lying on the x-axis of a member of the structure, We therefore may obtain the velocity and acceleration vectors u,p and app, appearing in Eq. (7) by a transformation similar to (12)... THe a, however, relate the member coordinates OxyzZ to the space coordinates oxyz inthis case. Also note that the velocities and accelerations are now unknown quantities. Applying these transformations, we obtain expressions similar to (12) where the Uj» u; are replaced by the components of up and ape The elementary forces are given by an expres: sion which is the negative of (13) but containing the same p's and X's. These are then transformed back to the oxyz-directions by the inverse of the above transformation. Integration of these elementary forces and their moments over the length of the member is somewhat simpler now since the velocities and accelerations vary in a somewhat simpler manner than in the case of the wave-induced velocities and accelera- tions. The Added Mass and Drag Coefficients The idealization made in arriving at the subdivision of forces illustrated in Eq. (7) is a computational expedient at best. In reality, the total force experienced by a member of the structure will be the resultant of distributed normal pressure forces and tangential shear- ing forces arising from viscosity. The subdivision into a Froude- Krylov force, a force proportional to relative fluid velocity, anda force proportional to relative fluid acceleration, is done partly on an intuitive basis and partly on the basis of our knowledge of simpler problems. Such a subdivision has the great advantage of leading to solutions in which there is a linear relationship between platform motion and the exciting wave motion. Let us review some of the 1094 Wave Induced Forces and Mottons of Tubular Structures justification for this simplification. Maruo [ 1954] and Havelock [| 1954] have discussed the forces on submerged bodies which are caused by waves of small amplitude in an ideal fluid. Maruo deals with the two-dimensional problem of a horizontal cylinder completely submerged below the surface of an inviscid fluid, and gives some results which can be compared directly with Eq. (7). In particular, he shows that, for the case of a deeply submerged cylinder, the "exact" total force is equal to twice the Froude-Krylov force, Eq. (7), and that this corresponds to a value of the added mass coefficient equal to that of the cylinder in an infinite fluid combined with the wave motion at the centerline of the cylinder. If the cylinder is near the free surface, the force differs from this deeply submerged value by an amount dependent upon the depth of submergence and the wave length. The error, however, is small if both the depth of submergence and the wave length are greater than several cylinder diameters. Similarly, C. M. Lee [1970] has analyzed the problem of an oscillating cylinder submerged beneath the free surface, and has shown that the added mass coefficient for forced motion approaches the infinite fluid value within a small error if the depth of submergence is more than two times the cylinder diameter and the length of generated waves is more than about five times the cylinder diameter. For our present purposes, these results imply that we may assume a constant value of the added mass coefficient in Eq. (7), since in the majority of practical situ- ations the cylindrical members will be sufficiently deeply submerged and of sufficiently small diameter compared to wave lengths of interest to fulfill the above conditions. Thus the first and last terms in Eq. (7) can be expected to give a good approximation to the non- dissipative parts of the force on the individual member considered here. The drag or velocity dependent force acting on an oscillating body under a train of waves is associated with two phenomena: (1) the dissipation of energy in surface waves which are generated as a result of motion of the body, and (2) the viscous effects which are felt both as tangential forces on the surface of the body, and as a deviation of the pressure distribution from its ideal fluid value. This latter effect, which is associated with the formation of a wake and vortices downstream of the body, will cause the added mass coefficient to differ from its ideal fluid value as well. The drag force associated with free surface wave effects decays to zero with increasing depth of submergence at the same rate that the added mass coefficient approaches the infinite fluid value. Therefore wave damping is of little significance to the configurations being considered here. The drag forces associated with viscosity are generally of much greater importance, and also less clearly defined. The usual method of approximating these forces, Wiegel [1964], is to assume that they behave in a manner similar to the drag on a body immersed in a flow of constant velocity. In such case, the drag force is expressed asa quadratic function of velocity and the drag coefficient is found to be a 1095 Paulltng function of Reynolds number. In applying this concept to the present situation in which we have a periodic fluid motion resulting from the superposition of wave and body motions, we might compute a Reynolds number using the mean absolute velocity, and choose the drag coef- ficient accordingly. The drag term in Eq. (7) should then be a quadratic function of velocity. This, however, destroys the linearity of our analysis, and in view of the crude approximation involved, it is not worth the added complication. In order to preserve linearity, an equivalent linear drag coefficient is therefore defined, as described in Blagoveshchensky [1962], such that the linear drag force dissipates the same energy per cycle of periodic motion as the nonlinear drag force which is being approximated. The derivation of the equivalent linear drag coefficient is as follows. Assume a sinusoidal variation of the relative fluid velocity given by Vv =v, sin wt. (16) The linear drag force is given by Dy FEY and the nonlinear drag by - n D, = Cyav - The energy dissipated per quarter cycle of motion is given in the linear case by ~W/2w Cc v? dt; OL 0 and in the nonlinear case by T/2w nl Gi vo pdt. ie) Equating the two energies enables us to solve for the equiva- lent linear drag coefficient in terms of the assumed nonlinear coef- ficient. In the case of n= 2 (quadratic drag) the result is +, Bo neass (17) ot ae oe 1096 Wave Induced Forees and Mottons of Tubular Structures Thus, it is seen that the use of such equivalent linearization requires a prior knowledge of the amplitude of the motion. No difficulty is introduced by this in the case of the wave force ona stationary member. However, the amplitude is unknown for the absolute motion of the member. This leads to the necessity for an iterative solution in which we first assume an amplitude of motion, compute the equivalent linear coefficient, and then solve the equa- tions of motion using this value. This solution then is used to com- pute a refined value of the linear drag coefficient, which is used for the second solution of the equations of motion, and so on. It is questionable whether the approximations involved warrant more than two iterations, as noted by Burke [1969]. Hydro static Forces A floating body which is displaced in heave, pitch, or roll from its equilibrium position experiences hydrostatic forces pro- portional to these displacements as a result of the changes induced in the immersed volume. There will be no forces in surge, sway, or yaw since these displacements, which are parallel to the free surface, cause no change in the immersed volume, These forces, including coupling terms, are computed by standard naval architectural formulas. Thus, the vertical force resulting from a small heave displacement, x,, is given by F = - pgA,x,, (18) where A, is the waterplane area. Similarly, the moments of this force about the x- and z- axes (static coupling terms) are given by M = pgAy zx x wow 2 (19) M ae PEAY XX, » The roll and pitch moments resulting from small angular dis- placements are given by = - pgVGMa, (20) where GM is the appropriate metacentric height, V is the volume displaced by the structure, and a is the small roll or pitch angle, either x, or x, in the notation of Eq. (6). Finally, the force in the y-direction resulting from a small 1097 Paulling roll displacement, X4, is Fy = PR AWZyX4\s (21) and for a small pitch displacement, Xe Fy = - pgAwXw%¢ » (22) Xw, Zw are the coordinates of the center of gravity of A,. These forces are included in the equations of motion as static restoring or coupling force terms. The Restoring Forces Three types of restraints have been described in the Intro- duction, dynamic positioning, spread array mooring, and vertical tension leg mooring. A dynamic positioning system incorporates two principal components: sensors for detecting deviations from the desired position, and thrustors which may be activated automatically or manually to exert a force tending to restore the structure to the desired position. In the simplest system, the thrustors are actuated without time lag to exert a force proportional to the displacement, This would be termed a pure proportional controller. Real systems seldom operate this simply but incorporate time lags, back lash, and other non-ideal characteristics. Increased sensitivity and response may be built into the system by having it sense velocity (rate control) and acceleration. If the system can be approximated by linear features, i.e. , if the applied thrust can be linearly related to displacements, velocity, and acceleration of the structure, then the control system constants may merely be introduced in the force terms of the equations of motion, (6), as additions to the already defined hydrodynamic and hydrostatic terms. In a spread mooring system, several pretensioned anchor lines are arrayed around the structure to hold it in the desired location. If the structure moves from its mean position, the tensions in the anchor lines change and these changes may be related to the geometry (catenary), elasticity, and hydrodynamic properties of the anchor lines. It is usually permissible to neglect the hydrodynamic forces on the anchor lines and to approximate the force by a linear relationship between force and displacements in the plane of the anchor line. The displacements at the point of attachment of the anchor line may be determined in terms of the coordinates of this point for given displacements of the structure. These are resolved into horizontal and vertical displacements, xy, yg, in the plane of the anchor line by a transformation similar to (12). The horizontal and vertical forces exerted by the anchor line may then be expressed 1098 Wave Induced Forces and Motions of Tubular Structures as Bay “a Its - Kye (23) hy NW Ly Ks + KO be These forces may then be transformed back to the oxyz coordinates for inclusion in the equations of motion. The anchor spring constants, ky eee ky are computed from a knowledge of the aforementioned elasticity and weight-shape characteristics of the anchor line. In a tension leg mooring, the mooring lines are vertical and provide essentially total restraint against vertical movement of their upper ends. Figure 2 illustrates the horizontal force which results when the upper end of such a mooring line is displaced horizontally as a result of surge, sway, and yaw. The restoring force in the direction opposite the displacement, x,, of the end of mooring line n is given by (24) The displacement, xX,» may be expressed in terms of its components in the x- and z-directions, in which case the corresponding com- ponents of F, in these directions will be given by (24). Xr“ HORIZONTAL COMPONENT _ _XnTn ft La Th MOORING LEG "n" Fig. 2. Restoring force in tension mooring legs 1099 Paullting IV. SOLUTION OF THE EQUATIONS OF MOTION The total system of forces described in the preceeding sections are now introduced into the equations of motion, Eq. (6). Our linearization of the problem has resulted in the subdivision of these forces into two categories: those forces resulting from the wave motion in the presence of the stationary structure, and those resulting from the motion of the structure in a stationary fluid. The former category contains the first term and the first parts of the second and third term on the RHS of Eq. (7). The platform motion dependent forces are contained in the second parts of terms two and three of the RHS of Eq. (7), plus the hydrostatic and restraint forces. The wave motion depending forces are seen, as a result of the velocity potential assumed to represent the wave motion, Eq. (8), to be sinu- soidal functions of time. If we rearrange the equations of motion into the standard form, placing the motion-dependent terms on the left-hand side and the time dependent forcing terms on the right-hand side the result is a set of six simultaneous second-order differential equations of the form 6 Ds [ (mj; a aij) x; + bij Xj 28 ci, x;] = F,j sin (wt + Ej), (25) j=l where the exciting force amplitude, F,;, is proportional to the wave amplitude, a. The solution of these equations may be expressed in the form x; =X; sin (wt + 6), where x,,; is proportional to Fj, therefore to the wave amplitude. The quantity ws or the amplitude of response to unit waves varies with wave frequency, since the exciting force Fo is a function of frequency and because the coefficients a, b, c the LHS of (25) may also vary with frequency. The square of this unit response is then the response amplitude operator, which may be combined with the wave spectral density function to obtain the platform response to a random seaway. V. MODEL EXPERIMENTS A number of model experiments have been conducted in the University of California Towing Tank in order to test several parts of the procedures described in the previous sections. The initial objective of the study was to evaluate the tension leg platform, and all experiments deal with this configuration. Initial experiments were made on single cylinder members to test some of the hydro- dynamic force predictions and the linearity of the resultant motions in regular waves. Next, experiments were conducted in regular 1100 Wave Induced Forces and Mottons of Tubular Structures waves using a platform model of triangular plan form to investigate the predictability and linearity of motions and mooring tensions of a composite structure consisting of a number of cylindrical members. Finally, experiments were made in random seas to test the applica- bility of linear superposition. The arrangement of the model and experimental apparatus is shown in Fig. 3. The model configuration shown is typical of a number of those tested, consisting of three base cylinders arranged to form an equilateral triangle with three or more vertical legs supporting the deck. An important geometrical parameter studied in these tests was the relative proportion of buoyant volume contained in the vertical and horizontal legs. PULLEY “ 0000 LEADS TO INTEGRATOR WAVE SURFACE X- AXIS PLATFORM MODEL INSTRUMENT TENSION LEADS TO RECORDER ad METERS DEAD-WEIGHT ANCHORS V TANK BOTTOM Fig. 3. Arrangement of experimental apparatus 11014 Paulling From Fig. 3 it is seen that instrumentation was provided for measuring model motions, tension variations in the mooring legs, and incident wave amplitude. The surge motion was sensed and con- verted to an electrical signal by a miniature low torque potentiometer driven by the model through a string and pulley arrangement. Yaw was sensed by a rate gyroscope mounted on the model, the output of which was integrated electronically to give the yaw displacement. Tension meters were installed in each mooring leg. These consisted of small proving rings fabricated from seamless stainless steel tubing and mounted with etched foil strain gages. Four gages on each ring were connected to form a four-arm Weatstone bridge, the output of which is proportional to the applied force. The bridge was balanced initially to bias out the initial static tension. Therefore only the time dependent variations are recorded. The outputs of these force and motion transducers, as well as the output of a resistance wire wave meter, were recorded, using a multichannel oscillograph. During experiments in random waves, a simultaneous recording was made of the same quantities in digital form on magnetic tape for processing by electronic computer. Single Cylinder Experiments The first group of experiments were conducted using a single circular cylindrical model having hemispherical ends and moored by two legs, one at each end. Only the incident regular waves and tension variations were recorded. The model dimensions and test conditions were: Length 3.44 ft Diameter 0.282 ft Depth of model 0.792 fit = 2.8 X dia. Weight 6.5 1bs Water depth 4.17 ft For this configuration, the computed tension variations in the mooring legs will be equal to the hydrodynamic forces expressed in Eq. (7), i-e., there will be no linear coupling between the tension variations and the motions of the model. These results, for regular waves of 1.45 second period, and several different amplitudes striking the model at 0, 45, and 90 degrees, are shown in Fig. 4. Experimen- tal points show the amplitudes of force variations which were measured in the two mooring legs. The theoretical lines have been determined by the method described here and by Havelock [1954]. Havelock's procedure gives the wave force and moment on a spheroid having its long axis horizontal, moving beneath a train of regular waves. For the present computation the approximating spheroid was assumed to have the same length and diameter as the cylinder model. The dashed curve labelled "present work" was computed by Eq. (7) assuming the infinite fluid value of unity for the added mass coefficient 1102 Wave Induced Forees and Mottons of Tubular Structures , LBS FORCE AMPLITUDE °6 Ol G2" 03804. 05). 06 OF; WAVE HEIGHT , FT Fig. 4. Tension teg single cylinder of the circular cylinder and a quadratic drag coefficient of unity. Within the limits of experimental accuracy, the forces are seen to vary linearly with wave height for a range of heights tested. The two theoretical procedures are seen to give results of about equal degree of conformity with experiments. Triangular Platform 1 in Regular Waves Experiments similar to those with a single cylinder were next conducted with a complete platform model. This model was similar to the one shown schematically in Fig. 3 except that the main horizontal pontoons were of oval cross section and the above-water deck was supported by a space frame arrangement of very small vertical and inclined tubes. The dimensions of this model, referred to as Model 1, are given below: 1103 Paulling Weight 40.13 lbs Buoyancy 56.40 lbs Length of side of equilateral triangle 3292 ft Main pontoon cross section, vertical and horizontal semi-axes 0.175 X0.109 ft Vertical member - radius 0.031 ft Draft to centerline of main pontoon 0.77 ft Water depth 4.18 ft This model was subjected to a large number of tests in regular and irregular waves. Only a few of the regular wave tests are reported here, - [NS WAVE PERIOD = 1.15 SEC WAVE DIRECTION = 0° Beal EXP THEORY LEGS IAND3 © LEG 2 WAVE PERIOD = 1.44 SEC AN Nai NE NE NLL, ERASE WAVE PERIOD = 1.74 SEC \ mt | LAAT PEAK-TO-PEAK TENSION VARIATION, LBS Co © ©, © Ss | ~ © i; © cs ¢* - © N I : Paz] er ise 3 isi] Ps wie HEIGHT, FT 4 0.5 Fig. 5. Tension variations in legs -- Model 1 1104 Wave Induced Forees and Mottons of Tubular Structures 0.5 WAVE DIRECTION = 0° THEORY: DRAG COEF = 1.0 DRAG COEF = 1.5 ° rs SURGE MOTION PEAK-TO- PEAK, FT ) 0.1 0.2 0.3 04 0.5 WAVE HEIGHT, FT Fig. 6. Surge motion -- Model 1 Figure 5 contains a comparison of measured and computed tension variations in the mooring legs for three different wave periods and Fig. 6 contains the platform surge motion. Again, the agreement between computed and measured values is about as good as in the case of the single cylinder. Two features should be noted here. First, a mean curve drawn through the experimental tension variations appears to curve slightly concave downward with increas- ing wave height, thus indicating a measurable nonlinearity in this quantity. Second, better agreement between the computed and measured values is obtained for motions than for forces. Note that the motions are shown for two different values of the assumed quadratic drag coefficient. The effect of a substantial change in this quantity is seen to be slight. Triangular Platform 2 in Random Waves A second triangular platform was tested in both regular and irregular waves. This platform, designated Model 2; was similar in arrangement to the platform depicted in Fig. 3 and had the following characteristics: Weight 28.49 lbs Buoyancy 32.30 lbs Length of side 3212 ft Main horizontal pontoon dia. 0.187 ft Vertical cylinder dia. 0.381 ft 1105 Paulling Draft to centerline, horizontal pontoon 0.78 ft Water depth 4.70 ft This model was tested in six different random sea conditions, representing two families of wave spectra. The first family of spectra, designated "A," have their peak ordinate at a wave period of about 0.9 second. The second family, or "B" spectra, have their peak at about 1.55 seconds. Both families for several different significant wave heights are shown in Fig. 7. These two spectra were used in order to adequately excite the model over the range of wave periods of interest. "B " SPECTRA SEA CENCE mn CORINA peel MET AL BPS aa oncgh SiN 0 0.4 08 12 16 20 oe PERIOD IN SECONDS Bakre A SPECTRA x 104 B SPECTRA x !0° WAVE AMPLITUDE HALF - SPECTRUM Fig. 7. Experimental tank wave spectra 1106 Wave Induced Forees and Mottons of Tubular Structures The model response is shown in Figs. 8 - 10 for waves mov- ing parallel to the X-axis. The first two of these figures show the mooring tension variations, and the last shows the surge motion versus wave period. In each case the ordinate is the double ampli- tude of the force or motion in question divided by wave double ampli- tude, thus the amplitude of the transfer function obtained from a time series analysis of the random wave tests. Points on the figures dis- play these random wave results for three significant wave heights within the applicable range of periods for the "A" and "B" spectra. Also shown on these figures are results from experiments in regular waves, and theoretical predictions. It is interesting to note that the random sea tension variation results display an apparent amplitude dependence in the range of longer periods. This is the range in which drag forces would be most strongly felt and no doubt points to a possible deficiency in the process of linearizing the drag force. As before, the surge motion shows very good agreement between experiment and theory. 0.2 0.6 1.0 1.4 1.8 2.2 26 WAVE PERIOD IN SECONDS Fig. 8. Tension variations in anchor leg 2 P2LO-7 Paulling 5) DIRECTION OF WAVE ADVANCE sre B-SPECTRA a EXPERIMENTS IN REGULAR WAVES THEORY 5a Ne 0.2 0.6 1.0 1.4 1.8 22 T IN SECONDS Fig. 9. Tension variations in anchor legs 1, 3 1.6 B 2 12 A-SPECTRA| B-SPECTRA : y > O08 P ae A (2) BSS a be oft 0.2 06 10 1.4 1.8 22 2.6 T IN SECONDS Fig. 10. Surge motion 1108 Wave Induced Forcees and Mottons of Tubular Structures VI. CONCLUSIONS In the previous section, a comparison is shown of experi- mental measurements and theoretical predictions of platform motions and mooring leg tensions, using the procedure developed here. The theoretical results were obtained by the simplest form of the pro- cedure in which constant infinite fluid values were assumed for added mass coefficients. Constant linear drag coefficients were used, and no attempt was made to account for the hydrodynamic interference between members of the structure. The first two groups of experi- mental results show that the observed performance of single members and assemblages does, indeed, follow a nearly linear pattern in regular waves, and this pattern is well predicted by the present pro- cedure. The last group of experimental results show nearly equally good results in both regular and irregular waves. There is, however, some consistent nonlinear amplitude variation in longer waves, as may be seen in Fig. 9. It is probable that the good agreement is obtained because the structures tested consisted of assemblages which satisfied the initial assumption reasonably well, i.e., (1) All members were long, slender cylinders relatively sparsely distributed throughout the structure. (2) The bulk of the members were submerged sufficiently deeply below the free surface. (3) The cross sectional dimensions of all members were small compared to the waves used in the experiments. (4) The motions of the models were small compared to the model dimensions and to the wave lengths. The aforementioned nonlinear behavior probably illustrates the failure of a single value of the linear drag coefficient to ade- quately represent this component of the hydrodynamic force over the entire range of frequencies. ACKNOWLEDGMENTS This work was conducted under the sponsorship of Deep Oil Technology, Inc. and the author expresses his appreciation for their permission to publish the foregoing results. The assistance of a number of individuals in conducting experiments and performing calculations is also acknowledged. Special thanks in this respect are due to Mr. O. J. Sibul, and graduate students Kwang June Bai and Nabil Daoud of the University of California, Department of Archi- tecture, and Mr. Paul Gillon of Deep Oil Technology. 1109 Paulling REFERENCES Blagoveshchensky, S. N., Theory of Ship Motions, Dover, 1962, p. 142. Burke, Ben G., "The Analysis of Motions of Semisubmersible Drilling Vessels in Waves," Paper No. OTC 1024, Offshore Technology Conference, Houston, 1969. Fisher, F. D. and Spiess, F. N., "FLIP -- Floating Instrumental Platform," J. Acoust. Soc. Am., v. 35, no. 10, 1963, pp. 1633-44, Gerritsma, J., "Ship Motions in Longitudinal Waves," Netherlands Research Center, TNO Report 35S, Feb. 1960. Havelock, T. H., "The Forces on a Submerged Body Moving Under Waves, Trans., RINA, 1954, pp. 1-7. Also "Collected Papers of ...," pub. by ONR, Dept. of the Navy, ONR/ACR- L036 Lee, C. M., private communication, 1970. Macy, R. H., "Drilling Rigs,"Ch. XVI of "Ship Design and Con- struction," A. M. D'Archangelo, Ed., SNAME, 1969. Maruo, H., "Force of Waves on an Obstacle," J. Soc. Naval Arch. Japan, v. 95, 1954, pp. 11-16. McClure, Alan C., "Development of the Project Mohole Drilling Platform," Trans. SNAME, v. 73, 1965, pp. 50-99. McClure, Alan C., "Delos: An Application of Oil Field Marine Technology to Space Programs," Marine Technology, v. 6, no. 2, 1969, pp. 156-170. McDermott, J. Ray, Inc., "Feasibility Study of a Floating Ocean Research and Development Station (FORDS)," Final Report to Dept. of the Navy, Bureau of Yards and Docks, under Contract NBy-37640, April 1966. Paulling, J. R. and Horton, Edward E., "Analysis of the Tension Leg Stable Platform," Paper No. OTC 1263, Offshore Technology Conference, Houston, 1970. Wiegel, R. L., Oceanographical Engineering, Prentice-Hall, 1964, p. 248ff, 1240 SIMULATION OF THE ENVIRONMENT AND OF THE VEHICLE DYNAMICS ASSOCIATED WITH SUBMARINE RESCUE H. G. Schreiber, Jr., J. Bentkowsky, and K. P. Kerr Lockheed Misstles and Space Company Sunnyvale, Caltforntia I. INTRODUCTION The U.S. Navy's first Deep Submergence Rescue Vehicle (DSRV) was launched at San Diego, California on January 24, 1970. This vehicle was designed and built by Lockheed Missiles & Space Company (LMSC) under contract to the U.S. Navy's Deep Submergence System Program Office (DSSPO) to provide the capability to rescue the crew of a submarine immobilized on the ocean floor. The DSRV is 50 feet long, 8 feet in diameter, has a fiberglas external hull and an inner (pressure) hull made of three interconnected HY140 steel spheres, Propulsion and control of the vehicle are provided by a stern propeller in a movable shroud, horizontal and vertical ducted thrusters located in pairs fore and aft, and a mercury trim and list system. An Integrated Control And Display (ICAD) system developed at the Massachusetts Institute of Technology Instrumentation Labora- tory enables the DSRV operators to correlate information from sonars, Closed circuit television, and advanced navigation devices, in order to perform this intricate rescue mission. The mission scenario of the DSRV is as follows. Word and position of a dis- tressed submarine is received and the DSRV and its support equip- ment are flown by three C141 aircraft to a nearby port. The DSRV is then loaded on to a mother submarine, by being attached to the after escape trunk, and transported to the area of the downed sub- marine. The DSRV then detaches itself from the mother submarine and descends to the disabled submarine, and mates to one of the escape trunks of the distressed vessel as shown in Fig. 1. The rescuees are then transferred into the aft two spheres of the DSRV and returned to the mother submarine, 24 at atime. Because of the possibility that the distressed submarine may be at an unusual atti- tude, and there may be bottom currents, the DSRV must be able to perform this hovering and mating maneuver in a one knot current and at attitudes up to 45 degrees in pitch and roll. se Fk Bentkowsky and Kerr Sehretber, Iojsuerly, sonosoy “ty "81a 1Vt2 Vehtele Dynamics Assoctated with Submarine Rescue This hovering and mating operation puts the DSRV in a new and growing class of submersibles which because of their missions, are required to hover, work, search, and otherwise maneuver at low speeds. This requirement for low speed, high angle of attack maneuverability is far outside the range of operation of the conven- tional fleet type submarine and consequently analysis designed to predict the dynamic behavior of conventional submarines is not com- pletely applicable to the prediction of motions of the DSRV and other submersibles of the same class. The adequate prediction of the DSRV dynamics requires six degrees of freedom and a simulation capable of predicting the forces and moments at high angles of attack wherein the vehicle will experience lateral forces equal in magnitude to the axial forces. To be useful, the simulation must be precise enough for use in the design of the automatic control system. The operational environment is also quite different from that normally simulated in that the vehicle must hover and maneuver in currents at near zero forward speed and in the presence of the disabled sub- marine which causes considerable disturbances to the flow field. This paper, which is divided into three general parts, presents one approach to the simulation of the dynamics of a highly maneuverable submersible. The first part describes the simulation of the free- stream vehicle dynamics or thé dynamics outside the influence of the distressed submarine. The second section deals with the inter- action forces and moments caused by the presence of the distressed submarine and includes a discussion of a test program conducted to measure these forces and moments. The third section describes the application of the resulting equations of motion in conducting a man-in-the-loop simulation of the DSRV motions during the mating maneuver. The equations of motion were developed at LMSC and pro- grammed on a Remington Rand 1103A computer. They were used to determine the preformance characteristics of the vehicle to be used in design studies and to provide equations of motion for use in the control system development. The interaction forces were measured in the 12-foot variable pressure wind tunnel at the Ames Research Center in Mountain View, California. Tests of this nature were necessary due to the lack of data on interaction forces and the possibility that these forces would provide a significant influence on the vehicle and control system design. The manned simulation was performed at the Marine Systems Division of the Sperry Rand Corporation to provide demonstration of the ability to manually con- trol the DSRV within the limits necessary for mating and to deter- mine operational limits for this mode of operation. 1113 Schretber, Bentkowsky and Kerr Il. FREE STREAM DSRV DYNAMICS SIMULATION EQUATIONS OF MOTION The development of a dynamic simulation of the Deep Sub- mergence Rescue Vehicle (DSRV) follows a different approach than the methods used in most submarine studies. This deviation from the standard approach is necessary because of the basic differences in the mode of operation of the DSRV compared to that of conventional submarines. While the analysis of a submarine is generally con- fined to prediction of the vehicle dynamics at speed in an infinite fluid, the DSRV dynamics must also be simulated while hovering and docking in the presence of a downed submarine. The conventional method used to simulate the dynamics of a submarine is to calculate the position of the center of gravity of the vehicle using linear force and moment coefficients for the complete vehicle which are refer- enced to its center of gravity. The basic equations of motion for the DSRV differ in two ways from this conventional method, first in the choice of an axis system and secondly in the manner of handling the forces on the vehicles and appendages. Axis System Since the DSRV is required to assume angles of 45° to the horizontal in pitch and roll (very unrealistic for a conventional sub- marine) a mercury trim and list system is incorporated which moves the vehicle's center of gravity (c.g.) to accomplish these attitudes. The fact that the vehicle's c.g. moves with respect to the vehicle during maneuvers makes it a poor choice as a reference point for describing force and moment coefficients since they would have to be changed as a function of c.g. position. Using the c.g. as a refer- ence axis system would also lead to complications in describing the vehicle's motion with respect to the distressed submarine since the motion of the axis system with respect to the vehicle would be in- cluded in the velocity of the axis system. Therefore, an axis system fixed to the body was used as a reference point. Since the axis sys- tem is not always at the center of gravity and terms to account for this shift must be included in the equations of motion there is no advantage in choosing the nominal vehicle c.g. as the center of the axis system. There are, however, advantages to having the x-axis lie along the vehicle centerline since the basic DSRV shape is a body of revolution. This axial symmetry provided by having one axis of the system lie along the vehicle centerline greatly reduces the number of cross coupling coefficients required to describe the forces and moments on the body. The positive direction of this axis is forward so that positive vehicle velocities are associated with vehicle forward motion. Similarly with the z-axis through the centerline of the transfer skirt (280.8 inches aft of the forward perpendicular) the number of cross coupling coefficients are reduced and the direct reference to the centerline of the transfer skirt simplifies the description of relationships between transfer skirt and the hatch during mating 1114 Vehtele Dynamics Assoctated with Submarine Rescue maneuvers. The positive direction of this axis is downward com-: mensurate with standard submarine dynamics analysis. The y-axis is through the x-axis z-axis intersection with positive direction to the starboard to provide a right handed orthogonal system. This xX, y, 2 body axis frame is related to an inertial axis system, X, Y, Z, through the ordered rotations WV, @, and @ about the Zz, y and x axis. The origin of inertial axis system is located at the origin of the vehicle axis system at the start of a computation and the X Y plane is parallel to the water surface with the vehicle axis in the X Z plane. The Euler angles are formed in the following manner. With the two systems initially coincident, a first rotation, (ZW), is performed giving the system (x,, y,, Z). Next a rotation, (y,®), is performed about the y, axis resulting in the system (x, y,, Zo). A third rotation, (x4), about the x axis brings the body axis system, (x, y, z) to the final position. The transformation matrix relating the (X, Y, Z) system to the (x, y, z) system through the above ordered rotations is then x cos@ cosw cos®6 sin sind x y |=| sin® sin@cos-cos@ cosy sin@ sindsinbtcosdcosi sinécosé fl y Zz sin8 cos@coswtsiny sing sind cosdsiny-sinécosW cosb cos@}} Z It remains to relate the Euler angle rates to the roll, pitch and yaw rates, (p,q,r) respectively. The relation is 11/15 Schretber, Bentkowsky and Kerr 4 p t+ tan 0(q sin #6+r cos @) @ |= qcos $- r sin $ w (r cos 6 + q sin $)/cos 0 High Angle of Attack Considerations The second major difference between the DSRV simulation and the conventional method is required because of the high angles of attack experienced during the hovering and docking maneuvers. This high angle of attack problem becomes accentuated by the vehicle which, by using its thrusters, is capable of turning without forward way, a maneuver entirely outside the scope of those covered by conventional analysis. These shortcomings of the conventional analys is were overcome by a technique used during development of LMSC's DEEP QUEST research submersible where the hydrodynamic forces on the body and appendages (in this case the shroud ring) are considered separately. This allows for adequate representation of stall characteristics of the shroud ring as a function of the local angle of attack at the shroud ring which is essentially impossible to account for when a total coefficient for the body-ring combination is used in the simulation. The coefficients of the body itself are handled through addition of a normal drag components to the standard small angle of attack representation of forces. These high angle of attack considerations will be discussed further in the following sections. The equations of linear motion are derived from the funda- mental equation e = 2 (mY) (1) external stating that the sum of the external forces acting on a rigid body of mass m, equals the time rate of change of the momentum of the body. The momentum, mVg,, is a vector quantity and V, is the inertial velocity of the center of mass. Expressing V. in terms of the velocities and rates about the vehicle fixed axis system described earlier u+aqZg- rY¢ +t Xe wtpY,- 4X_tZ 1116 Vehtele Dynamics Associated with Submarine Rescue where Xg, Yg, Z,g are the coordinates of the vehicle center in the body axis system. Neglecting the velocity and acceleration of the ceg.e with respect to the body (Xg, Yg, Ze¢, X_q Yo, Zg= 0) because they are small and performing the operations = d pee ne = Fexternal = ar (™Ve) =mV, body pmo aN we obtain u tqw - rv - X(q° + 1°) + X(pq - r) + Z,(pr + q) Pexternol =myfu tru - pw - Yg(r" + p*) + Z,(ar - p) 7 X(ap + r) w+ pv - qu- Z¢(p? +g?) + X,(rp - q) + ¥,(rq +) The equations of angular motion are derived from Mexternal = £- (le) (2) which states that the sum of the external moments acting on a body equals the time rate of change of the angular momentum of the body with both the moments and angular momentum expressed about the same point. Since the hydrodynamic moments are described about an off c.g. axis system the development of the right-hand side of the equations consists of expressing the time rate of change of the angular momentum of the body about the center of the axis system with the rotational vector expressed in the directions of the body axis, The results of this operation [1] yield: Tp + (i - i )qr +m ¥,(w + pv - qu) - Z,(v tru - pw)| Mexternal = fea AU ESE) yeep es! | Z,(u + qw- rv) - X,(w + pv - qu)} lLrt+(I, -I,pqtm X.(v + ru - pw) - Y,(u + qw - rv) z y x G G The moments of inertia I,, I, and I, are the moments of inertia about the center of the body axis system and not about the vehicle cénter of gravity. Since there is near symmetry in weight distribution about this body axis system, cross products of inertia have been dropped from the equations of motion. The external forces and moments, Foyterngi ANd Meyternai » on the vehicle during free stream operations come from three 1217 Schreiber, Bentkowsky and Kerr sources; the body, the shroud ring, and the thrusters. Fexternal = F body + Fghroud * Fthruster - The following sections will discuss the simulation of these three classes of forces. BODY FORCES The forces on the body can be divided into several types: the static forces, Fhoqy, static » associated with buoyancy and weight and the dynamic forces, Fhoqgy dynamic? which depend on vehicle motions with respect to the water. Static Forces The weight or gravitational force (W) acts inthe Z direc- tion and the buoyancy, A, acts inthe -Z direction. The total vehicle weight, W, can be best expressed as the vehicle weight when in neutral trim W, =A plus the change in variable ballast from initial conditions. The resulting static forces on the body are: - sin 6 F pody, static = (Wo +My, - S)| sin @cos 0 cos $cos 8 The DSRV contains the following ballast and trim systems which will affect the static forces on the vehicle: Main Ballast Variable Ballast Transfer Ballast Rescuee Ballast Mercury Trim Mercury List and BG Control In operation, the main ballast tanks are full when submerged. The Transfer Ballast tanks are empty except when they are being used in the dewatering process, and rescuee ballast is exchanged for rescuees providing a constant value during submerged operations. These systems do not vary during normal submerged operations and therefore are not included in the simulation. The tanks whose contents vary during submerged, unmated operation are the variable ballast (T, and T 3); trim (T, and Ts), 1118 a ee oe Vehtele Dynamics Associated with Submarine Rescue and list (T,, T,, and T;) system tanks, Fig. 2. The variable ballast tanks (6, 7) are hard tanks each witha capacity of 500 pounds of seawater. Water can be transferred to and from the sea from either or both tanks at a rate of 2 gallons per minute. The mercury trim system is a set of two tanks (4,5) con- taining 225 pounds of mercury and 15 pounds of oil. A pumping rate of 3 gallons per minute provides a net weight change of 5.3 pounds per second between the two tanks. With the list system reservoir full (Z, maximum) a 23 degree trim angle is attainable. T4815 16817 12 13 TI Tl LIST SYSTEM RESERVOIR T2 STARBOARD LIST TANK ene 13 PORT LIST TANK x 4g + X5Ws T4 FORWARD TRIM TANK CG = —W=Ww T5 AFT TRIM TANK . T6 FORWARD VARIABLE BALLAST TANK 17 AFT VARIABLE BALLAST TANK Y5W> «YW Y ~ = G + Z ti z(w__w_) 7. =< Z nf n n no G = “co* —Wl+8W, Fig. 2. DSRV Trim and List System Tanks The list system consists of three spherical tanks each witha capacity of 2780 pounds of mercury. The two list tanks (2, 3) are located 2.2 feet above the vehicle centerline and are separated by 4.4 feet. The reservoir (1) is located 3 feet below the centerline. The configuration of the list system piping and valving is shown in Fig. 3. It is noted that transfer can be effected between list tanks, between each list tank and reservoir and between the two list tanks tied together and the reservoir. Pumping rate can be varied between 1119 Schreiber, Bentkowsky and Kerr 1/2" TUBE N = NEUTRAL (AS SHOWN) E = ENERGIZED Fig. 3. List System Schematic 0 and 28 GPM (51.3 pounds per second net change). During roll damping operation the rate is controlled proportionally. In all other modes of operation, maximum rate is used. Variation of BG is accomplished by transferring from the reservoir to the two list tanks. Roll damping is accomplished by transferring between list tanks for list angles less than 22.5 degrees, and between reservoir and the appropriate list tank for list angles greater than 22.5 degrees. n = net weight in tank n - pounds rate of change of weight in tank n - pounds /second commanded rate of change 228% iT] net weight in tank n for vertical buoyancy, neutral trim and maximum z,- DSRV on surface D = Depth, feet Baseline operation is represented by 200 pounds plus a cor- rection for depth in each of the variable ballast tanks, 93 pounds in each list tank, 2404 pounds in the list system reservoir, and equal 1120 Vehicle Dynamics Assoctated with Submarine Rescue distribution of mercury between the two trim tanks. Note that all mercury system weights listed represent the difference in weight between mercury and an equal volume of oil. Table 1 summarizes the location of all tanks and the value of Wg: TABLE 1 Physical Parameters of Ballast and Trim Systems W max = i 140,369 pounds ZG = 0.1335 feet The effects of variations in the weight of water in the variable ballast tank on. zg have been included in the basic vehicle equations and the z_, variations computed in this section are due only to vari- ations in the list and trim systems. Changes in depth effect both the density of the water and the compressibility of the hull. The net effect on buoyancy, using tem- peratures and salinities corresponding to sub-tropical waters cor- responds to a gradient of 0.1 pounds per foot of depth. Under normal conditions the required ballast change is divided equally between the two variable ballast tanks. Thus, for neutral conditions W, = Wy + 0.05D All operations are written with t = 0 corresponding to neutral 1121 Sehretber, Bentkowsky and Kerr buoyancy at the initial operating depth of the problem. Thus Wn = War, * Wr dt aap Maree eS Pe arc 0 t Wy = Wyo +9-05D +f) W, dt n= 6,7 The term Wy which is used in the vehicle equations matrix is given by Wii = We t Wr - Weg - Wao> Location of the center of gravity is given by wee xqWa te x5W5 G Wo + ie Yo= W> + y3W gz ee Wo + DW D2 Wr aq] Wao) Zg= Cs) Wo " DW; Dynamic Forces The hydrodynamic forces arise because of the motion of the body with respect to the water and are defined in terms of hydro- dynamic force coefficients. The hydrodynamic force and moment coefficients used in this report are not the standard non-dimensional coefficients used in most studies. It has been found that a set of dimensional coefficients provides a much easier nomenclature. The force and moment coefficients are represented by subscripted capital letters of the form Xgpce._. The letter denotes the direction of the force (X for forces along the x-body axis, Y along the y-body axis, and Z along the z-body axis) or the axis about which the moment results (L, Mand N for moment coefficients describing the mo- ments about the x, y, and z body axes respectively). The subscripts vary in number and form and denote the variable quantities that the coefficient must be multiplied by to obtain a force or moment on the body. For example, Xyjy is the dimensional axial drag coefficient since when multiplied by u |u| the square of the axial velocity, u, it results in an axial force Pi22 Vehicle Dynamites Associated wtth Submarine Rescue Xywulul = Fx drage Similarly, Muwwlu| is the pitching moment used by the normal velocity, w. The use of absolute values in these coefficients provide for the proper signs on the force and moments, and because of most of the near fore-aft symmetry of the DSRV less the shroud, the coefficients are independent of the direction of various velocity components. The direction of the normal force Zwiui W}u} is dependent only on the direction of the normal velocity regardless of whether the vehicle is going forward (u > 0) or back- ward (u<0). Since the sign of Zwiy is negative the normal force due to normal velocity is always in the opposite direction of the normal velocity. A brief description of the development of the representation of hydrodynamic forces on the body follows. First consider the forces on the axisymmetric bare body of the DSRV and then add forces resulting from asymmetries, such as the transfer skirt and splitter plate. The representation of lift, acceleration and axial drag forces on an axisymmetric bare body is relatively well known and can be obtained from slender body theory, Ref. 2, other potential flow analysis, Ref. 3, or test data and is of the form Xyu t Xyy ula F Yyv + ¥pr + Yow rlul + Yuu v1 ul EXT lift, acceleration, axial drag Zyw + Zaq + Za a/ul + Zwiyiwlu| These lift, accelerating and axial drag forces are those normally used to simulate the dynamics of submarines and provide a very adequate representation of the forces and moments at low angles-of- attack (a < 15°), At high angles-of-attack, however, they become inadequate. For example Zwyjw|u], the only force resulting from normal velocity, w, goes to0as u goes to 0, while a vehicle normal to the flow experiences a significant normal force. This normal force is due primarily to flow separation and is, excluding Reynolds number effects proportional to the normal velocity squared w*, Wind tunnel and water tunnel tests on the Polaris, Poseidon, DEEP QUEST, and other vehicles have shown that a reasonably good representation of the forces and moments due to normal velocity can be obtained by using the two terms Zwyy wlu| i Zwiww | w | where Zww is measured force at u=0 (g = 909) divided by the normal velocity squared and Zyjw is the slope of the force versus angle of attack curve. Pitching (yawing) of the vessel will cause a variation in normal velocity along the vessel and therefore a variation in this normal drag over the body. It remains then to develop a method to account for the distribution of this local normal drag caused by pitching and yawing. The use of a strip theory method provides that the normal force due to normal drag can be expressed as Znormal drag= Ziawiw'|w'| dx where Zi‘, is the local value of the normal force coefficient and w' is the local normal velocity and can be expressed as w'=w+tqxX where X is the distance between 1123 Sehretber, Bentkowsky and Kerr the point in question and the center of the axis (- forward). This integral can be evaluated at each step in the integration of a simula- tion when the distribution of Z\,y is known but it proves both cumbersome and time consuming. On the other hand, for a nearly cylindrical body such as a missile or the DSRV, test data has shown that a fair representation of both the force and moment are obtained when a constant value is used for Zw, from the nose of the vehicle to the forward edge of the shroud, a distance L, from the nose. This then allows the value Z'wiw to be removed from the integral and sets the equality Zwiw= Zwiwi/Ls. Replacing the local normal velocity w' by its equivalent w + qX, the integration Z ata ‘Ioody (w + qX)+ |(w-qX)| dX still poses some problems because of the absolute value signs. To accommodate these two integrals are formed depending whether the center of rotation, the point where w' =0, is on or off of the body. Expressing the ratio of the distance the center of the axis system is off of the nose, Lj), and the length L, as K,= L/L, the center of rotation is forward of the nose when w/qLs < Ks and aft of the body when w/aLs >K, - 1 and the value w'|w'| can be replaced by (|w|/w)(w')"=(|w|/w)(w? + 2wxq + x?q?) and the integration ae LeLy Ziel w' |w'i|*dse= Z! ll { (w* + 2wxq + xq?) dx E w -L wlwl results in three terms wi 2 Cywlw] + Ga lw] + Cyl g where 2 mt C, = Lg, Zwiwi 3 ! 4 2 ' When the center of rotation is on the body K,= w/ql, = K,- 1 the integral must be divided into two parts to account for the sign change in w'|w'| at the center of rotation and Lg ly -w/q Lg ly 2 Zit ww? |SdX*="= Zs Jal] ¢ w' ax sh w! ax| -Ly -w/q where the lal/a is used to denote the direction of force since the local normal velocity forward of the center of rotation depends only 1124 Vehicle Dynamites Associated wtth Submarine Rescue on q. This when expanded to - Zwiwi lal hie (w? + 2qxw + (qx)*) dX ey) q Ly Le L, (w? + 2qxw + (qx)*) ax| Ww and integrated yields a four term expression for the normal drag with the center of rotation on the body C4a/la| w°/a + Cya/lal w? + wlal +c alal where Caw = 21/3 Ziel Cs Baele (1 - 2K )z' wiwil Cogn (i) Ke Ze wiwl Grit Kk, + kG - 2K) Zz Tw wiwl In a similar manner the lateral drag terms for the sway or equation are developed and result in C, abs us ie at lv | + GC, e/a for the center of rotation off oe the bod “vy /-rL or v/-rL, < K,-1 and Cqar/|r|v3/r + Cove /|t - my A # eta for the center of rotation on the body K, = ia = - 1 with 2 Ciy = Ls Y vivi 3 Co = fs (2K, - 1)Y'! vivl EN at a 3K et 3K. )yt Us < i} viv Cay = 23/3 Yvivi 2 C.. =, (2Ko- 2) iui > 2 Ce, = Le Lt -~K y ee 3 mg be 4 2 Cae, /3 = Kort tee aK, 4) y au This then completes the simulation of forces on the axisymmetric bare body of the DSRV. This representation has been developed 1225 Sehretber, Bentkowsky and Kerr keeping in mind the fact that it will be programmed on a digital and/or analog computer and several simplifying assumptions were made to allow for mechanization of the resultant equations. As in any simulation a trade-off must be made between the accuracy of representation of forces and moments, the ability to mechanize certain types of expressions, and the availability of data. The presences of the transfer skirt and splitter plate add additional coefficients to the body because of the asymmetries they provide. A complete set of these terms due to asymmetries were developed, their numerical values determined, and their relative importance established, with the reduction in the number of terms in the simulation as an end goal. It was doin that additional terms of the form Xquw > Xryrv, X, r’, Koad’: Xrprp» YpPp,» YpwPw, Yap QP » Yuiwiv] ws z ppP* » Zprpr; aa » Zryrv, Zeer , and Zyjyju u| would be required foe adequate simulation of the dynamics of the DSRV during the hovering maneuvers. qs, The addition of forces due to the shroud, Fyproug,s propeller, Forop» thrusters, Frthr» and the interaction forces during mating, Tae , complete the force equations. A similar line of development was used for determination of a set of moment equations. Values for most of the coefficients were obtained from model tests conducted at NSRDC (Ref. 4) and Hydro- nautics (Ref. 5), and theoretical values were computed for the re- maining coefficients. A complete set of equations of motion used for the DSRV Model for Analysis (Ref. 6) follow along with numerical values of the hydrodynamic coefficients, Table 2. Surge: m[u + qw - rv - xg(q? + r*) + yelpq - r) + zg(pr + q)] = Xu + Xyaw + Xyrvit+ Xywulul + Xr + Xa + Xprp - (Wo +) We - 4) sin © + Xghroua + Xprop + Xthr + Xdist Sway: m[v + ru - pw - ye(r* + p*) + agar - p) + x,(qp + r) | SY 7 Ypert Ypp + Yriu r|u| + YpwPw + Yqpqp | wi ch Monomer LY yi, V Pa act 1126 Vehiele Dynamtes Assoctated wtth Submarine Rescue * PhGeviy| + Conv + oul r*] # Cyst — + Caer y +Gevir| + Car|r(|] + (Wo + > wy, - A)cos 8 sin @ + Ygnroud + Yprop + Ythr + Yaist * Vv Vv Terms are cancelled when “1, >Kg, or ie K, or w/ql,< kK, - 1 ** Terms are cancelled when K, 2 w/ql, = K, - 1 Cw = pl, Z wlwil 3 Cow= > PLS (1 - 2K)Zyiwy 4 2 1 Caw pl, /3 (4 ~ 3K, + 3K, )Z iwi p2L,/3 Zalwi ie) a = 1 2 el, (i 5 2K)2 wliwl OQ o = N 3 2 2 pls [« - K,) + a) a 2) N Q N By = = I Ne Ne ne aK NK Ne NK Ne 4 2 pL, /a(i - K+ K, 1 - 2K)Z 4. Yaw: Lr + (I, - I,Jap + ml xg(v + ru - pv) - ye(t. + qw - rv)] = Nex B Niv + Npged + Nuvi lulv + Netut r{u| + NwpwP + Nyqvd + Nopwip |u| + Niwiv | wlv + (WoxX, a > Wy x4) cos 8 sin ¢ + Woycesin 8 + Yehroud*shrd * Nope + Npropt Naist 4 * + Cher] el + Copy A Corie Te? + Ca,|r|v] 1128 Vehtele Dynamtes Assoctated with Submarine Rescue ** Cs,|vir + Cy viv + epi r?] * Vv Vi Terms are cancelled when - FL. >Ks or - ai < K, - 1 A Vv Terms are cancelled when K,= - ns K,- 1 s 4 5 4 eee = pL, /4[K, + (1 - Ky Yui { 2 2 Cor = > pies /2lK, + (1 - Ke) Win 4 C. ae 5 pL, /6 Yyivi en 4 3 4 3 y1 ‘ates wire 2L,/3{ Kg - (1 = K,) Tein 4 4 ) 3 Cz, = o p 2125/3 Ket (i= Ks) aii iM 3 2 2 Ce, = 2 pl, [2 [K, ia (4 ~ Ks) 1 Yviw 4 5 4 4 xy: Cr, = > pl, /4[Kg - (1 - K,) ger Roll: Lp PL. = qr + ml Yg(w + pv - qu) = Zg(Vv + ru - pw)] = Kp + Ky + Kar + Ky, v|w| +K, rw + K, va + K,pw +Kyy viel +Kyy viv] +Kyyplul +K,jy rlul + Kyi P P| + Woyg cos 8 cos $ - (Woz, +) Wy, Z,)cos 8 sin > + Korop + K gist 1129 Schretber, Bentkowsky and Kerr Pitch: Iq + (I, - L)pr + m|[ zg(u + qw - rv) - X_(W + pv - qu)] = Mga + Myw + M,pr + Mutwi | ul w * Maui a| | + Mypvp +M,,,;u]ul + Myw? + Myyrv + Myr - (Woz¢ >) W;, 24,) in @ = (W,Ge +) Wy eos ®@ cos $ - ZshroudXshrd + Mprop + Mthr - * + Mais +1 Cigalal * Cogrdyw! + Cogpdy < + Cagl a ** #[Cgglwla + Cggwiw] +6, AL o] x Terms are cancelled when w/qlL, >K, or w/ql, + (4 - Rep Z re Cog= 5 ple /2[-Ky + (1 - Ke] Zhiq Ch, = 5 ple /Al - Ky + (1 - Kyl") Ziaw 1130 Vehicle Dynamics Assoctated with Submarine Rescue TABLE 2 DSRV (ML-493-03) Parameters and Coefficients we, (436s ates I, 3678X 10° slug-#t 26 0.1335 feet Iy 4.52% 10° slug-ft* i 49.33 feet Te 4550 40 alupor L, 23.4 feet L. 46.95 feet FORWARD MODE -3 2 | -6.87X 10 p72. = -1.67X 10 | i yd 2a ee tons 2.433 X 10° -1.24x 10° | am -4.6 X10° ai i2 x 10" L Zhu x10" 2.43 pa x 10° -1.95 x 10° | LAT 208, AO -5,59X 10: | Ye W422 Xa" ao = 13.36 X 107 | -9.41 X10 2.198 X 10° -~2.07 X 10° NOTE * Indicates coefficient value is zero when w is negative, ** Indicates coefficient value is zero when |= | > 0.173. (1) Under following conditions: 200# in each variable ballast tank, 93# in each mercury list tank, remainder in reservoir. 1131 Sehretber, Bentkowsky and Kerr TABLE 2 (Cont'd.) Coefficient Non- Dimensional Dimensional -1.2 x10° PY = -4. 44X10" x 10° 1.20X10° 3.72X10° -2 3 x 10 -3.60 X10 x 107 “4,32 x 10" x 10> 1.04% 10° x 10° 2.64 X 10° -2 3 x 10 3.60 X 10 -2 3 x 10 -3.60 X 10 -3 2 x 10 -4.44 X 10 -2 3 -3.4 X40 -3.72 X10 -3 3 2 5.32X10 af2Li= 6.32 10 -4 5.4 5G10 1.20 X 10° -6.6 R19 2.3 X107 2.76 X10° -2 3 -2.8 X10 =3,36 10 - 3 oO KAO" -3.6 X10 => 2 4.98 X 10 5.98 X 10 é 2 Heo tO. pel -8.3 X10 -4 6 2 -1.3 x10 5.92 xX 10 = fel x 10 4. 2 -1.34X 10 -7.93 X 10 1.4 x10 8.3 40 2 2 1.34xX10° 7.93 X 10 2 i.3 X10° 7.7 X10 =1.3 7207" =7.7 M40" 1232 b Vehtele Dynamics Assoctated with Submarine Rescue TABLE 2 (Cont'd.) Coefficient Non- Dimensional Dimensional 5.92 x 10° ok Seto) 8.3 x 10° -4 2 1 34.<40 7.93 X10 -4 2 =1e,10.. e110) -5.92 x 10 OL Ore 5.92 x 10. 0) 0) -3 2 2210 2.64 X 10 -3 2 1.419. 40 4,42 <* 10 -4 4 3 -3.16 X 10 py Zine = -1.87 X 10 -4 6 2 1.5510 5.92 x 10 9.17 X10 -4 3 =2.354 ~ 10 -1.39xX10 -4.5 x10° 2266 40" -4 2 1.3410 7.93 X10 -4 2 = 23) 3x 10 (ey eae 8) x 2 ia tO" -8.3 X10 -4 2 1.4 X10 8.3 X10 -4 2 ={.58 X10 -9.35 X10 n5le X40 322 10° “i G4 io -7.93 X 10° ee oe -7.7 X10" -4 2 -1.34X10 -7.93 X10 -5 5 3 -1),55 X 10 p/2L = -4,53 X 10 0 2.92 X 10° 0 L133 Sehretber, Bentkowsky and Kerr TABLE 2 (Cont'd. ) -1.13 X10 -3 -1.4 x10 -3 1.39 X10 0 -3 -1.4 x10 -3 -1.39 X10 -3 -8.05 X40 -2 -9.9 x10 5 -1.0 X10 -615 “TO 7.7 X107 3 5535 X 10 -2 -8.7 x 10 -3 5.35 X10 -4 x 10 -2 x 10 2.4 X10 -4 2652 Xf0 aa -7.25 X40 -3 5.4 X40 -4 -7.25 X10 -3 8.4 X10 2 p/2L = 2.433 X10° p/2L° = 5 1.20 x10 p/2L = 6 592 x 10 1134 Coefficient Non-Dimensional Dimensional 5 : 2.92 X10° 4 -3.3 X10 5 -4.1 X10 5 4.06 X 10 0 5 -4.1 X10 5 -4.06 X 10 | -1.96X 10 -2.41 X 10° -2.43 -1.58 xX 10° 9.24 10° 2 6.42 X 10 4 -1.04X 10 6.42 10° 566 SUG: 13.0 ° to" 2.88X10° -3.67X 10° -4,29x 10° 4 =e x 10 3 -4,29 X 10 4 -4.97X 10 Vehicle Dynamics Assoctated with Submarine Rescue SHROUD RING FORCES The control shroud is a circular movable wing located at the aft end of the vehicle, supported by four struts space 90° apart, Fig. 4. The force and moments on the shroud ring are obtained from curves of lift and drag on the shroud as a function of total angle of attack of the shroud, Figs. 4 and 5. Resolving the resultant force, F, into the directions of.the vehicle's axis system provides the components used in the equations of motion, Xgbroug» Yshrouqs 2nd Z shroud’ The relative velocity of the shroud with respect to the water in the directions of body axis system is se u Vop = vg f=|v- Fe Ww, w + X,q The ordered deflection of the shroud, 6p, a deflection in the vehicle pitch plane followed by 6y a deflection about the pitched shroud yaw axis results in the relationship SAE SHROUD ANGLE OF ATTACK, ag, DEGREES 1.4 Uer4 on ae 0.6 IS sles a Nia a oe 0.2 Pax BS REN SES Pa Nay roe al a ee eS ee ee FSF ae a poi aa ea Pm Viniaes ee ae 0 120 140 160 180 iets Gok i atti. a, DEGREES Fig. 4. Shroud Lift Coefficient, C, vs. Shroud Angle-of-Attack, aS 1135 Schretber, Bentkowsky and Kerr DRAG = C, 1/2pVv? Fibgc&s Ue Vg Ws 8 FT 1,83 FT 0 20 40 60 80 100 120 140 160 180 SHROUD ANGLE OF ATTACK, Ac IN DEGS, Fig. 5. Shroud Drag Coefficient vs. Shroud Angle of Attack Us Us ss = | Yes | = Teaae| Vs | = TsoeeYse Wes Wg between the velocity of the shroud relative to the water expressed in the direction of the body axis system Vcsg and the same velocity expressed in the direction of the deflected shroud axis system V,,. Us We cos 65 O- - sin 6p]f ug, are EE 0 4 0 Vs } Wy sin 6) 0 cos 6, |L_w. Wis 1136 Vehtele Dynamites Associated with Submarine Rescue Sy U2s “Is —_ es) 7 Chie Y2s ‘2s 28 Is cos 6, sin by 0 cos 6, QO -sin 5p Ug = |-sin 6, cos 6, 0 0 4 0 Vs = 20 0 1Jtisin6, 0 cos 6pJLw, Us cos dycos 6p sin by -cos éysin d5][ us = Ts22s| Vs |= |-Sin 6ycos 6) cos dy sin dysin 6p |] vg | = Ts22s VsB Ws sin 5p 0 cos 6p Ws The total angle of attack at the shroud can then be expressed as 1/2 -lf (vag tw -I = 2s = shroud = tan [! a 2s) ] = tan | vns/‘tps| 2s or performing the indicated operation 1/2 7 ( (-sinSycos dpus tcos Syvs tsindysin Spws )’ Hs inSpus tos SpweF| \ shroud = (cos § cos dpu, tsind,v, -cos 6ysind,wg) are then looked up on Figs. 4 and 5 and values of lift, L,, and drag, D,, on the shroud are calculated from 2 z L, = p/2S,6,.V and D, = p/25;CpV with S, = b,C, 2 Veutvtwe b, = 8 feet C,= 1.83 feet The lift and drag on the shroud are then resolved into forces on the shroud F,, and transformed into the body axis system, Fae ° 1 Scehretber, Bentkowsky and Kerr Np = Cc. cos @, + Cy sin a, on sin C.e- Cy cos @, — Fog se V25/Vns Ne -Was/VagNe and F Shroud = T2208 Fg The moments on the shroud are the product of the shroud force and the distance of the shroud center from the center of the axis system as shown in the moment equations. THRUSTER FORCES The propulsion system of the DSRV consists of a single con- ventional screw propeller for axial thrust and four ducted screw propellers arranged in forward and aft pairs for lateral and normal thrust. This system provides the vehicle with five degrees of maneuvering freedom (heave, sway, surge, pitch and yaw) and pro- vides forces and moments sufficient to meet hovering requirements in currents of the order of one knot. The control of the sixth degree of freedom, roll, is provided by the trim and list system. A com- plete treatment of the development of the maneuvering system is contained in Ref. 7. The following treatment will present the data used in the simulation with a little explanation of its development. Main Propeller The main propeller is a 6-foot diameter, wake adapted, three-bladed propeller with a blade area ratio of 0.24 and a maxi- mum speed of 1.64 revolutions per second. For estimates of the vehicle maneuvering performance the propeller thrust and torque characteristics are required for ahead and astern motion of the vehicle and for positive as well as negative propeller rpm. "Behind-the-ship" tests of the DSRV propeller were performed for all four operating modes at the Naval Ship Research and Development Center (Ref. 8). The curves of the thrust and torque coefficients, Fig. 6, are typical for ahead and astern operation of a propeller and can be expressed in the form eee her 1138 Vehtele Dynamics Assoctated wtth Submarine Rescue NOTE: DATA FROM MODEL TESTS CONDUCTED AT NSRDC THRUST COEFFICIENT, Kr, AND TORQUE COEFFICIENT, 10Kq APPARENT SPEED COEFFICIENT, Ja = 745 Fig. 6. Characteristics Curves for DSRV Propeller (Note: Data from model tests conducted at NSRDC (Ref. 8) ) The coefficients a, b, c have been evaluated separately for each quadrant of the propeller curves, and the resulting thrust and torque coefficients are used as an expression of the steady and transient characteristics of the propeller forces and moments. During maneuvering the propeller may also experience velocities normal to its axis. The resulting effect on the propeller thrust in the axial direction and torque about the roll axis have been estimated and the form of the coefficients can be rewritten including this effect as follows: 2 2 2 1/2 KeatbJ, te] +a(se +3.) re) where 1139 Schretber, Bentkowsky and Kerr vie ge Jy= nd ed nd u,v,w = X;y,Z components of vehicle velocity = components of vehicle angular velocity relative to yaw and pitch axis RK .Q J £ = distance of propeller from coordinate system origin The coefficients of the y and z components of the propeller force and the corresponding moment coefficients can be similarly estimated and are proportional to J, and Jy respectively. For the computations of the vehicle responses, all six force and moment components have been considered. The resulting propeller force and moment equations are: Xpop = 755 n(n| - 58 un - 3.8 ae +26 mn (ye twee” 7 U=) 05 <= -0.21 = - 365 n?-172un- 45u>+26nlvatwe)’ ;u2z0,>8-0.21 = 155 7 + 60 un + 22 n” + 26n(ve + wa)!” 3u< 0, n=0 = - 365 n@ - 13 un + 22 n® + 26 n(ve + wo) 5 u<.0% mes Yprop = - 30 nvp 5. ee 0 = - 12nvp ;n<0O Zprop = - 30 wp nO = - 12 nwp nex 0 2 2 /2 e Pars a0 mln|- 6) Sai AOS ure eentve + we F231 A n = > -0.21 2 2 /2 ° Wag Fee SS wal 38a + 22(v, + wp)” +130 a 4% >) We =,0% = -0.21 e|p 1140 Vehicle Dynamics Assoctated wtth Submarine Rescue 2 @e Ba0n #80 un + 35.200 + 22(vp + we)? Tn eee it. <= 0, oy = 20 i - 468 n - 8unt15,2u° + 22(ve two)! +131 a 5 u<0,n<0 Mprop= - 765 nwp - n= 0 = - 306 nWp ; n<0O Norop = 705 NVp > n=O = 306 NVp Di 0 where Vp=v- 25.54 RS " Ww ot 25% 5 '¢q Ducted Thrusters There are two pairs of ducted thrusters used for maneuvering in the pitch and yaw planes, as shown in Fig. 2. The four-bladed propellers are 18 inches in diameter and have a maximum speed of 9.8 revolutions per second and produce side force through a combi- nation of impeller thrust and a change in the pressure distribution on the hull (Fig. 7 and Ref. 7). The thrust due to variation in pressure distribution is very dependent on forward speed and the total thrust coefficient for the steady state n= 0 condition was measured at NSRDC as a function of forward speed (Ref. 9). Force coefficients derived from this test are shown in Figs. 5 and 6 as a function of u/|n| where n is the propeller RPS. The steady state force is obtained from the relationships 2 * Xthr mf = Nmfl 5 X thr ma = mat Lehr yf = an [nv T Yonr ya So Scehretber, Bentkowsky and Kerr Fig. 7... Ducted Thruster Schematic * Z the zt = ny¢|M2¢/T| k Zthr za = Hy, @se\s * Mebr 2¢ = 1-06 nz¢|nz¢ | M, * Méhr za = 1-13 Nzq|Nz9| M2 - 0.686 Ayatiyn Mehr yn * N the yf = 0.98 ny¢ | ny] My * N = 1.04n,,|n,,|Mz thr ya N thr zn - 0. 686 De, len where Kone mana X force due to thruster mn Yenr mn= * force due to thruster mn 1142 Vehicle Dynamics Assoctated wtth Submarine Rescue Zehr mn= Z force due to thruster mn M¢bhr mn = Pitch moment due to thruster mn Neher mn= Y2W Moment due to thruster mn M = Direction in which force acts f for forward-thruster, a for aft-thruster Dp It with the force eye ; Te : Ts Pay > and moment coefficients (M,*, Mo*, My, M,*) shown in Figs. 8 and 9. FORCE COEFFICIENT (LBS/RPS“) FORWARD SPEED/IMPELLER SPEED, TnT (FT/SEC/RPS) Fig. 8. Effect of Forward Speed on Duct Forces 1143 Sechretber, Bentkowsky and Kerr MOMENT COEFFICIENT (FT LBS/RPS2) FORWARD SPEED/IMPELLER SPEED inl (FT./SEC ./KPS) Fig. 9. Effect of Forward Speed on Duct Movements When the propeller in the duct is accelerating, the thrust component due to the propeller is a function of the ratio of jet velocity, Vj, to propeller speed, n. The thrust coefficient for a propeller of this type was estimated from data taken from Ref. 10 and fit with a second order curve resulting in V; Tpmn= 10.63 nmn|Mmn|- 5-04 Vijmon| Vjmnn| ; ina a = 10.63 nap|nmnal + 6-0 Mmal Vjmnl - 5+04 Vjmnl Vimal 3 Vimn < 9 mn 1144 Vehicle Dynamtes Assoctated wtth Submarine Rescue where the jet velocity is obtained by integration of the expression aVimn _ 9035 (T at ~ 1.91 Vimnl Vjmnl) pmn The forces on the body due to pressure distribution changes are a function of the jet velocity alone and are expressed as Top sOeG55 Viet | Vime| (Ty - 2.93) Tyma= 9+ 655 Vine | V. ett 2003) The resulting forces and moments on the body are then expressed as 2 * X thr mf — 0; 655 V jmt T3 2 * Xehemareeeze Vim 14 Y the yn Tpyn i T hyn Zthe zn = Tpz nt ie ? * ok = Mig 2M; M, Tio Ty Tio M _ 2M r M ‘ 22 2 2 29 ae T2 Teo Mig s0%e88 Uiya-t 0573 nye * = Mio M20 Nene yt = 9-98 Tr +1.31 “T.* Tox bt Riise * * N = 1,04 M207 44,34 2Me2 oie zo thr ya — T 5 pya T* a * bya 20 2 20 Nishi > 06238 Lin. 73 nee * * with Tio» Too » Mio» Mgo being the values of T,*, T,", M,” and M, for 0 forward way (u= 0). 1145 Sehretber, Bentkowsky and Kerr III SIMULATION OF DSRV/SUBMARINE INTERACTION FORCES There are two forms of DSRV/submarine interaction forces. The first is a mechanical type of force produced when the shock miti- gation system touches on the deck and transmits a force to the DSRV. The second type of force is caused by changes in the flow field cuased by the bottom and the downed submarine and will be called flow inter- action forces. SHOCK MITIGATION SYSTEM The shock mitigation system is primarily designed to protect the transfer skirt and absorb shocks in the event of obstacle collision. A secondary purpose is to act as a retractable base from which the transfer skirt may be slowly lowered to contact the hatch mating surface. ; The shock system consists of a bumper ring concentric about the transfer skirt (see Fig. 10). The ring is attached to eight struts extending from four points on the outer hull. Each strut has a hy- draulic piston/cylinder arrangement designed to attenuate impacts, as well as extend and retract upon command. A simplified model of the shock mitigation system is pre- sented for purposes of simulating near normal impacts during the controlled docking event. Figure 11 illustrates the DSRV with four vertical legs extending down from the four hardpoints on the outer FULLY eae FULLY a EXTENDED Fig. 10. Shock Mitigation System 1146 Vehtele Dynamics Assoctated wtth Submarine Rescue Fig. 11. Simplified Shock Mitigation System for Simulation hull. Each leg acts independently of the other three and is limited to axial deflections only. The force elements in each leg consist of a spring in series with either a damper or a constant force element depending on both deflection magnitude and rate. Other forces applied to the leg ends are due to lateral friction at the con- tact surface. The equations that follow, approximate the force effects on the vehicle due to near normal impact during docking. The approximation is good if the deviation of the transfer skirt mating flange plane from the plane of impact at instant of contact is less than 10 degrees. Also, the vehicle velocity parallel to the impact plane should be less than 0.5 fps at time of contact. The resulting equations will give a disturbing force and moment expression, X pist> Y pist> Z ois? Kost? Moist » and Noist for application to the vehicle model equations of motion. Using the direction cosine matrix coswcos@ coswsinOsind-sinbcosd sinpsindtcossinOcos¢ [D] = | sinbcos®@ cosycos$tsinbsinOsind sinwsinOcos ¢-coswsind sin 0 cosOsind cosOcos 1147 Schretber, Bentkowsky and Kerr Lp Por The forces on the i'” leg can be calculated: a: F pFerces i -1 Yy =[D] Yo Zi Zi Subscript V refers to vehicle axis systems. Xi BoXy. + Yvi * Yy sue Zyi = Zy + Zi Also, 1148 Vehtele Dynamics Associated wtth Submarine Rescue In all cases, { Xvi (eo) X; Miye LO wie=-< Yor Tl Dix Y Zj Zvi fe) Zi, Xyj F,; = Fy = Fy = 0 if Z - Aj =O Lg iT So Zi = A> 6 Ai < 0.58 ft = {K/c(z}- A,) “*} Aj< 0.5 ft/sec 22 Fj = -CAj F5j = - K(zj - 4;) Zi AO A,= 0.58 K(Z, - A,;) >11,000 lbs Aji = Zi - 14,000/K if A, = dA; /dt Z; - A, >0 Aj < 0.58 ft A,> 0.5 ft/sec F,j = - 11,000 lbs Ai iseZ ioe 000/K j; - Aj> 0 if A;= 0.58 ft K(Z; - A;)< 11,000 Ibs F,j = - 11,000 lbs | t Aj -\ Aj dt 0 °2 1/2 ERAT Pe (X,/(X, +¥, Wie 1149 Schretber, Bentkowsky and Kerr Fyi = bE /{¥,/(X, i ¥ 7} Once Fy, Fyj and F, have been calculated, they are transferred back into the vehicle frame of reference and summed: Pyyj Fxi } aM Py = [D] Fy; Poyi Fy 4 Xpist = > Pxyj i= | 4 Ypist -) By i=l 4 Zpist -) Fai i=l 4 Koist -) ( By ln eh i) i=l = ( ly fs Ste) Moist = (Fy, Zi - Fy, Xi) Npist = » (- Ry Vit Fy Xi) The physical and geometric properties of the system are shown below. 1150 Vehicle Dynamics Associated with Submarine Rescue K = 310,000 lbs /ft C = 44,000 Ibs-sec’/ft” y= 0.4 lbs/lb for wet rubber on steel TESTS TO MEASURE FLOW INTERACTION FORCES Because of the complex flow phenomena, tests were required to obtain measurements of the forces applied to the DSRV during the mating sequence with probable currents of 0 to 1.5 knots, wherein the flow forces are dependent on the approach attitude of the rescue vehicle, the orientation of the bottomed submarine, and the proximity of the two bodies. In order to obtain meaningful experimental data, the following test requirements had to be satisfied: (1) The tests had to be performed at full scale Reynolds number, and (2) The environmental conditions had to be known. Test Facility The operating characteristics of existing hydromechanic test facilities were investigated to determine the facility most suited to conduct the test program. Because of the stringent combination of test conditions , i,e., (1) test operating Reynolds number range of 2 to 6X 10° per foot, (2) rescue vehicle angles-of-attack up to 45°, and (3) varying proximity of two bodies in the test channel, it was determined that the test requirements extended beyond the operating characteristics of all hydromechanics laboratories. However, the National Aeronautics and Space Administration (Ames) 12-foot vari- able pressure low turbulence wind tunnel was capable of meeting all the test requirements. The Ames Research Center is located at the Moffett Field Naval Air Station at Mountain View, California. The DSRV test program was conducted at this facility, which operates at subsonic speeds up to approximately Mach 1.0. The facility was £151 Schretber, Bentkowsky and Kerr operated by the Arnold Research Organization under contract with NASA. The operating Reynolds number per foot versus Mach-num- ber range of the tunnel is presented in Fig. 12. With a 1/30 scale model of the bottomed submarine, Reynolds numbers up to 6 X 10& based on the model diameter could be achieved at a Mach number of 0.2. This corresponded to the full-scale Reynolds number ina 4.5 knot current. Compressibility effects at a Mach number of 0.2 are known to be insignificant. Because of the large size of the test channel and the available equipment and instrumentation this facility afforded a unique capability for the DSRV program. MACH NUMBER STAGNATION PRESSURE, LB/SQ IN abs a 70i 75 ec ie DYNAMIC PRESSURE — LB/SQ FT F 50] 0 l 2 3 4 5 6 7 8 9 LO} casei REYNOLDS NUMBER PER FOOT (x10~°) Fig. 12. Operating Characteristics of the Ames 12-Foot Pressure Wing Tunnel Test Section and Model Support System The test section is circular in cross section except for flat fairings. Figure 13 presents a schematic sketch of the general arrangement of the test section and the DSRV model support system. As illustrated, the sting-type model support consists of a fixed strut mounted vertically in the wind tunnel to which is attached a movable body of revolution carrying the sting and, in turn, the DSRV model. The strut functions as a support and guide for the body of revolution which can be pitched in the vertical plane by means of motor-driven lead screws. The range of pitch angles is 10 to 20 degrees; however, 1152 Vehicle Dynamics Associated with Submarine Rescue 12-FT DIAMETER SUNKEN 2 a SUB 10 DEG MODEL ~ as SEA BOTTOM PLANE | | FIXED SUPPORT STRUT DOWNSTREAM ELEVATION SIDE ELEVATION Fig. 13. Arrangement ofthe Test Section and Support System pitch angles of 45 degrees were obtained using a bent sting. The 1/30th scale submarine model was situated on a ground plane installed to simulate the ocean floor, and was oriented both into and normal to the current at 0, + 22.5, and + 45 degree roll positions. For these mating positions, the DSRV model was set at various attitudes when approaching the submarine model along its longitudinal axis and from the side (athwartships). Photographs of the actual model installation for various mating conditions on the forward and aft hatches are shown in Figs. 14 and 15. The displacement between the two bodies was regulated by vertical movement of the sting mount which supports the DSRV. Models The 1/30-scale model of the bottomed submarine, shown in Fig. 16, was constructed of poplar wood. The diameter of the model was 1.07 feet and its overall length was 8.2 feet. The model was constructed with a removable aft section in order to position the DSRV sting-support, which is located vertically along the centerline of the tunnel, over the area of the forward hatch. The model was also provided with a small and a large sail in order to simulate Permit (594) and Skipjack (588) class submarines. The forward hatch locations with the small and large sails was as shown in Fig. 5. Only one location of the aft hatch was considered. £153 Bentkowsky and Kerr Sehretber, sooid0q 0 JO Suypeoay ouyzeurqns e YIM YEH WV oY} UO SuyJep[-uopyeTLe su] Tepoyy ‘Dy °Sta 1154 Vehtele Dynamites Associated with Submarine Rescue sooid0q 06 FO Zupeopy euyTAeUIGNS e& YIM YOIe_] PIeMAOT oY} uO Suyzep -uoqeTpeysu] Tepoywy 3 it55 Sehretber, Bentkowsky and Kerr AFT HATCH LOCATION FWD. HATCH LOCATION WITH SKIPJACK SAIL FED, HATCH LOCATION WITH REMOVABLE SECTION PERMIT SAIL PERMIT SAIL NOTE: THE MODEL SAIL ASSEMBLY COULD BE ROTATED +22.5 AND 45 DEGREES ALL DIMENSIONS IN INCHES Fig. 16. 1/30 Scale Submarine Model The submarine model was mounted on a flat rectangular plate, 1/2" X 40" x 45", to distribute the load on the ground plane. The roll positions of the planes and sail were adjustable to simulate sub- marine roll angles of 0, + 22.5, and 45 degrees. The DSRV model (1/30 scale), shown in Fig. 17, was con- structed principally of aluminum. The diameter of the model was 3.3 inches and its overall length, 19.73 inches corresponding to a full-scale length of 49.33 feet. The transfer bell extended 1.3 inches below the baseline of the DSRV hull. Instrumentation The overall steady-state forces and moments acting on the DSRV model were measured by a strain-gauge balance mounted on the end of the supporting sting within the model. A type T-0.75 (i,e., 3.4 inch diameter) six-component internal strain gauge was used for the test program. A seven-track FM tape recorder was used to record the balance outputs and to provide a time code in order to locate specific data for later analysis. Additionally, a switching network was pro- 1156 Vehicle Dynamies Assoctated wtth Submarine Rescue DSRV MATING BELL ALL DIMENSIONS IN INCHES Fig. 17. 1/30-Scale Model of the DSRV vided for each data input to provide direct data entry to an oscillo- graph as well as the conventional "Record onto tape/Play-back from tape to graph" and for quick-look at model oscillation frequencies where they occurred. Test Conditions The test program (Ref. 11) consisted of 258 runs, at Reynolds numbers up to 6.0 X 10® per foot, and the DSRV positioned at 6 or more locations (distances from the submarine model). The data were recorded for about five minutes for each set of test conditions. The vertical distance from the mating surface of the DSRV rescue bell to the mating surface (hatch) of the submarine, Zg, was varied from 0 to at least 12 inches, which corresponds to 30 feet, or about one submarine diameter, in full scale. As shown in Fig. 13, an angle adapter was used to obtain different pitch angles, Orv, of the DSRV. For these test conditions, the angle of attach, @ry, of the DSRV was the same as Ory. Angle adapters of 0, 15, 30 and 45 degrees were used. When the sub- marine model was removed from the tunnel to obtain DSRV free stream conditions, the vehicle's angles of attack (pitch angles) were obtained by a combination of adapter arrangements and pitch of the strut mechanism. 1157 Scehretber, Bentkowsky and Kerr REDUCTION AND PRESENTATION OF DATA The proximity effects are described by a time independent term and a time varying term in each of the six equations of motion. These components are functionally dependent on the proximity to the distressed submarine, Zq, the DSRV attitude angle, Orv; and the orientation of the distressed submarine relative to the current. By means of the tests conducted in the Ames facility, these effects were determined primarily for two orientations of the submarine to the current: head-on and athwartships. The tests were conducted with the DSRV mating at both the forward and aft hatches with various attitude angles of the DSRV and roll angles of the distressed submarine. The DSRV yaw angle was zero for all test conditions. Time Independent Interaction Forces At the Ames facility the balance data were recorded by printing devices, punched onto paper tape by a Beckman 210 com- puter, and carried to the laboratory's computing center. The resultant steady-state force and moment coefficients were computed at the Ames center in the body-axis system. The force coefficients were non-dimensionalized by the DSRV's maximum cross sectional area; the moment reference arm for moment coefficients was the maximum diameter of the DSRV. In order to determine the DSRV characteristics in free stream, the submarine model was removed from the tunnel, and the forces on the DSRV were determined through an angle of attack (pitch angles) range from - 12.5 to + 35.0 degrees. The resulting normal force, pitching moment, and axial force coefficients versus angle of attack are shown in Fig. 18. These results are shown to correlate well with previous free stream tests conducted at Hydro- nautic'’s ane o(Reft 5). Interaction coefficients were determined by plotting the measured data and extrapolating the curves to the free stream con- ditions. The interaction coefficients are: Cy (-G)s Cy.» Cz (- Cy,) Force Coefficients he Cy, A Cy Moment Coefficients i i where wl Ory - Om Cy; = CX Om TK {5 weLGe Cz, = C26 + K Seve 2 Pp CLGs Vehicle Dynamics Assoctated wtth Submarine Rescue O DENOTES AMES DATA © DENOTES DATA OF REF, 2 AXIAL FORCE COEFFICIENT, Ca Sat! DSRV ANGLE-OF-ATTACK IN DEGS, PITCHING MOMENT COEFFICIENT, C -f NORMAL FORCE COEFFICIENT, Cy, Fig. 18. DSRV Free Stream Characteristics where C and K are functions of the vertical distance between the bottom of the transfer bell and the hatch, Zg. Ory is the pitch angle of the DSRV (in degrees). 6,, denotes the mean DSRV attitude angle (the angle to which the data is refer- enced) in degrees and can be converted to disturbing forces and moments as follows: wd max 1 4 2 X gist (lbs) = Cx; 2 pVc 57 CLC. wd, 1 2 List (ft-lbs) = CL =e dinax > pV, JCC. It was determined early in the experimental program that the flow forces encountered during mating on the forward hatch with the Skipjack sail configuration were more severe than those with the Permit sail geometry, and flow forces encountered during mating on the aft hatch are less severe than those encountered on the for- ward hatch. Hence, almost the entire test program was conducted with the Skipjack sail, and the data presented herein are repre- sentative of the most severe flow forces that the DSRV will experi- ence during mating with a downed submarine. 11F59 Sehretber, Bentkowsky and Kerr When the DSRV is approaching the distressed submarine along its centerline and headed into the current, a suction force is applied to the DSRV when it is within a one submarine diameter of the hatch, Fig. 19, because of the accelerated flow and the associ- ated reduced pressures between the DSRV andthe hull. This suction force increases as the displacement between the bodies, Zg, is decreased. The maximum value of the suction force, Fyyction » for ai knot current is 45.5 1bs. This result correlates well with the theory presented in Ref. 12. TUNNEL '€ = FWD. HATCH LOCATION B GROUND PLANE gou8 ee SUB @ METHOD OF REF. 3 Gace Zz,’ ni! vy, -0.40 -0.30 = E UO -0.20 - -0.10 = 0 oo 5 10 15 20 25 30 0 5 10 15 20 25 30 Zp IN FEET Zp IN FEET 2 IN FEET Fig. 19. Time Independent Flow Interaction Forces -- DSRV Mating Parallel to Submarine Centerline In contrast, when the DSRV is approaching the distressed submarine athwartships and headed into the current, interaction forces are applied to the DSRV when it is within 2-1/2 submarine diameters (75 feet) of the hatch. Referring to the solid Cyn .5 curve of Fig. 20, it is apparent that the maximum suction force is also 45.5 1bs, for this orientation in a 1 knot current. Note: The subscript 6, denotes that the attitude angle of the DSRV is zero. 1160 Vehicle Dynamics Assoctated with Submarine Rescue 40 a %o =— Kj -— Zp IN FEET 30 40 2p IN FEET 0) Zp IN FEET Zp IN FEET Fig. 20. Time Independent Flow Interaction Forces -- DSRV Mating Normal to Submarine Centerline Although the DSRV was heading athwartships directly into the current for this series, it is shown in Fig. 20 that lateral forces were applied to the vehicle, i.e., Cys Cy,» and Cj; were not equal to zero. This result is due to the fact that a cross flow results when the current is deflected off the sail and the DSRV is therefore not heading directly into the resultant flow. The magnitude of the side force, Fejgg, in ai knot current is 300 lbs. Time Dependent Forces During the test program, the outputs of the six-component balance (i.e., normal and lateral forward and aft gauges and the roll and axial-force channels) were recorded over a five minute interval. The long recording time was established to provide a high confidence level during analysis of the unsteady effects (Ref. 13). The information was digitized by data conversion and input to an IBM 7094 force and moment conversion program. The output of the 7094 program was then used as input for an existing LMSC Power- Spectral Density Computer Program. 1161 Sehretber, Bentkowsky and Kerr Plots of typical power-spectral energy versus frequency in model scale are shown in Figs. 21 and 22 for the unsteady normal force and pitching moments acting onthe DSRV. These conditions were for the DSRV mated (Zy = 0) athwartships on the forward hatch of a downed submarine with no roll. The full-scale natural pitch period of the DSRV can range from 41.5 to 72 seconds, corresponding to a BG value of 1 to 3 inches and a weight of 75,000 lbs. This information shown in terms of model data for a 1 knot current in Fig. 22 indicates that there is no significant concentration of energy near the DSRV natural fre- quency; therefore, motion excitation at resonant conditions will not be significant. The standard deviation of forces and moments in model scale were determined as the square root of the spectral energy, which POWER SPECTRAL DENSITY LBS/CPS FREQUENCY (CPS) Fig. 21. Normal Force Power Spectral Density vs. Frequency (Model Scale Values Shown) £162 Vehtele Dynamtes Assoctated wtth Submarine Rescue SUBMARINE POWER SPECTRAL DENSITY LBS/CPS FREQUENCY (CPS) Fig. 22. Pitching Moment Power Spectral Density vs. Frequency (Model Scale Values Shown) was obtained from the integrated power density spectrum. Then using Strouhal number and Reynolds number scaling, the full-scale unsteady (a.c.) component standard deviation of forces and moments were computed. For the corresponding test conditions, the inter- action data were used to compute the full-scale magnitude of the steady components. In the simulation the unsteady forces were approximated by a white noise perturbation on the current used to generate the steady forces. Based on the standard deviation of the test data a ratio of unsteady (a.c.) to steady (d.c.) forces of 0.15 was selected in the frequency range from 0 to 0.082 V, Hertz. The ducted thrusters of the DSRV 4re presently designed to provide 830 lbs of normal and lateral force and are required to maintain vehicle heading in a 1 knot current. The worst side force condition, obtained during tests with the sail rolled 22.5 degrees 1163 Sehretber, Bentkowsky and Kerr into a 1 knot current and mating on the forward hatch, was 690 lbs well within the thruster forces available, Furthermore, during mating, the operational procedure will be to head the DSRV, within practical limits, into the actual flow (the resultant of the current and the cross-flow due to deflected flow off the sail). This operation will result in a reduction to the side force to a much lower force level. INCLUSION IN THE MATHEMATICAL MODEL Attempts to mathematically simulate the interaction forces by using the potential solution of flow around a cylinder on a plane to generate the flow field were unsuccessful in the time available. In addition, not enough submarine-current configurations were tested to verify superposition techniques. The parameters Cxg and K of the previously mentioned interaction terms were apporixmated in the simulation by two-slope nonlinearities. Figure 23 shows the two slope approximations of the >< 10 ee 15 20'~°25' "30° “35 ons TO" 1S “20 * 25, “Sema <0.5 Spee ee) sy Sy d= 1B) 320 FORWARD HATCH 0.8 DISSUB -C 2.5 ROLL ANGLE: O 0.6 z8, DISSUB HEADING: 90° Or 5 oO 15: 20 25° 40,035 ae 2 ee Fig. 23. Interaction Force Parameters 1164 Vehicle Dynamites Associated wtth Submarine Rescue experimental data equivalent to Fig. 20 with the initial conditions starting with the DSRV 35 feet above the submarine 1.atch. The method of simulating the steady and unsteady interaction effects is shown in Fig. 24. GENERATE C AND K PARAMETERS AS XO APPROXIMATED BY TWO-SEGMENT FUNCTIONS OF DSRV ALTITUDE ABOVE HATCH Za GENERATE UNSTEADY OCEAN CURRENT (ALTITUDE INPUT FROM VEHICLE) VARIABLE BAND PASS FILTER PERTUBATION INPUT (UNSTEADY COMPONENT OF CURRENT) COMPUTE Cy,'s FORCE AND MOMENT COEFFICIENTS: GENERATE DISTURBING FORCES AND MOMENTS: ; < F | Xeaist), ¥ (cist), a, ee WHERE: : LeMans Now : (PITCH ANGLE Gre C. ce (dist), © (dist) ©“ (dist) FROM VEHICLE) Ko ea 2 Qa 0, “AL (@ 8) X (dist) i Cxe ( rigs 5 Mery) Ziaist) | OUTPUTS fo} * (aist) VEHICLE M dist) N(aist) Fig. 24. Simulation of Interaction IV. MANNED SIMULATION Early in the DSRV program it was decided to initiate a manned simulation study, whose primary objective would be the investigation of the operation of DSRV under manual control con- ditions, using minimum backup displays. The control system aboard DSRV is relatively sophisticated, providing substantial pilot assist- ance in the form of augmented stabilization, decoupling of degrees of freedom, and automated control loops. Although the primary operating modes of the DSRV were not to be manual, it was believed that a manual control capability was essential for backup in the event of failure or damage of the primary control system. The simulation program was confined to the most severe segment of the rescue mission, the mating of the DSRV to the hatch of the distressed submarine (DISSUB). This segment starts when the 1465 Sehretber, Bentkowsky and Kerr DSRV is approximately 20 feet above the deck of the DISSUB, and ends when the DSRV is sitting on the deck and has positioned itself to the accuracy required to assure a satisfactory seal. This simu- lation would be used to determine the limits of current magnitude and direction and distressed submarine attitude for which manual control is feasible. Mating aids would be used as required. The manned simulation program was undertaken sufficiently early in the design program to permit some design investigation of the parameters of various control elements. Several significant design changes were made as a result of these investigations. This computerized simulation program complemented the simulations undertaken with the LASS (Lighter than Air Submarine Simulator) vehicle. LASS operations (Ref. 14) had established the feasibility of manual control. However, they did not permit con- trolled variation of the environment. The facility used for the manned simulation program was located at the Sperry Marine Systems Division in Charlottesville, Virginia. This facility had previously been used for simulation studies of the NR-1 research submersible, and many of the programs developed for the NR-1i were available for the DSRV studies. Ron Rau, one of Lockheed's DSRV test pilots, served as test pilot for the simulation study. FACILITY DESCRIPTION The computer facility utilized in the simulation combined an Ambilog 200 hybrid computer with an EAI-231R analog computer. The Ambilog 200 has a basic 4,096-30 bit word memory witha memory cycle of 2ysec. The analog portion, used for multiplication and division, has a 50 psec cycle. The Ambilog 200 was used to simulate the vehicle, coordinate transformations, actuators and effectors, current interaction effects and the mating aids. The EAI-231R was used for display generation and for simulating the ballast and trim systems. In addition to the computer, the simulation facility included a cab driven in two degrees of freedom (roll and pitch). The cab contained a control station consisting of control sticks and other system inputs, various meter type displays, and a TV display. Figures 25 and 26 show the cab and its interior display arrangement, respectively. 1166 Vehtele Dynamtes Assoctated wtth Submarine Rescue Fig. 25. Simulation Cab 1167 Sehretber, Bentkowsky and Kerr VE VELOCITY= | RMR ATS PELL (CURRENT MAGNITUDE = TROLL RATE _— SWAY VELOCITY Fig. 26. Instrument Panel ELEMENTS OF SIMULATION Vehicle Because the mating operation is confined to low current velocities, vehicle control is obtained by means of the main propul- sion and thrusters; the shroud remaining locked amidships. The relatively limited capability of the Ambilog computer necessitated some approximations of the equations of motion to assure fitting the entire problem to the computer. The major approximation used ignored the stalling effect of the shroud. The effect of this approximation is shown in the comparison of the heave versus angle of attack curves of the analysis and simu- lation models of Fig. 27. At high angles of attack the simulation model provides higher heave forces than the analysis model, an effect which tends to make the results of the simulation conservative. The iteration interval used in the computation was 125 milli- seconds. Since some of the degrees of freedom, particularly the mating aids, require a higher frequency response, selected portions of the simulation utilized a 62.5 millisecond interval. A relatively simple integration routine was employed, as shown in Fig. 28. 1168 Vehtele Dynamtes Assoctated with Submarine Rescue HEAVE VS ANGLE OF ATTACK F/V2 +/ oe) 0 10 20 30 40 50 60 70 80 90 a, DEGREES -———>>— Fig. 27. Comparison of Simulation and Analysis Models TIME Noes | ical n EQUATIONS OF MOTION b= fF (u,v, W, Py r) SOLUTION Da Un-y2! Vinee"? ETG) AND vs Via + ut ITERATION INTERVAL 62.5 OR 125 MILLISECONOS Fig. 28. Computation of Vehicle Motions 1169 Sehretber, Bentkowsky and Kerr Throughout the course of the program, vehicle responses using the simulation model were compared to responses computed from the more complete analysis model, which had been programmed with a more sophisticated integration routine. Vehicle Control System The control effectors available are main propulsion for surge, a pair of horizontal thrusters for yaw and sway, a pair of vertical thrusters for pitch and heave, and mercury ballast control for list and trim. Experience with DEEP QUEST and other submer- sibles had shown that independent control of the thrusters was not effective, since each horizontal thruster, for example, affected both yaw and sway. To achieve effective control it is desirable to sepa- rate the commands to each degree of freedom. Thus, a sway com- mand would be applied equally to both fore and aft thrusters, while a yaw command would be applied differentially to the two thrusters. Vehicle control (except for trim and list) is obtained from two hand controllers. The block diagram of the system, including the actuators and effectors, is shown in Fig. 29. The output of DSRV STICK SUMMATION AND THRUSTER SIMULATION PROP RPM EFFECTORS | ACTUATORS : STICK SUMMATION Y x | x|x|-—* MAIN PROP eo (nes) are LEFT STICK y | T iene Fe Bel fa RIGHT STICK T AV | | A THRUSTER F x a l Ms l NIN REFINEMENT[ 7 OY | | | THRUSTER DESIGNATION ML — MAGNITUDE LIMIT v= SQUARE ROOT FH = FWD HORIZ THRUSTER D.Z. — DEAD ZONE FY — FWD VERT THRUSTER R.L. — RATE LIMIT AH — AFT HORIZ THRUSTER PALG — MAJOR LOOP GAIN AV — AFT VERT THRUSTER Fig. 29. DSRV Stick Summation and Thruster Simulation 1170 Vehtele Dynamtes Associated wtth Submarine Rescue each command axis is fed into a "signed square" circuit, so that the commands represent forces rather than propeller RPM. After mixing the signals appropriately, they are fed into square root cir- cuits so that the commands to the actuators represent RPM. Each actuator is represented by a delay network which includes a maximum motor acceleration limit. The effectors are represented by an (RPM)? network (to convert RPM to thrust) and a "thrust refinement" circuit. The latter accounts for the variations of thrust with the forward speed of the vehicle. The approximations used in the thrust refinement are shown in Fig. 30. The actuator/effector simulation shown in Fig. 29 ignores the lead-lag thrust effect of the thrusters which results from the delay in accelerating the water in the thruster duct. Exploratory experiments showed that, in manual control, the thrusters are used in a bang-bang fashion, and the effects of ignoring the lead-lag response of the thrusters are minimal, In the more sophisticated control modes used aboard the DSRV, the mercury list system control is integrated into the right- hand stick, and the pitch control motion of the right-hand stick affects both thrusters and mercury trim control. NORMAL FORCE FORWARD FORCE COEFFICIENT (LB/RPS“) AXIAL FORCE MOMENT COEFFICIENT (LB FT/RPS*) 0 0.2 0.4 0.6 0.8 1.0 0! 0.2" OFA 0760058. 11.0 FORWARD VELOCITY FT/SEC FORWARD VELOCITY Epes IMPELLER VELOCITY ’ IMPELLER VELOCITY ” Fig. 30. Effect of Forward Speed on Thruster Forces and Moments se BY a | Schretber, Bentkowsky and Kerr Displays Two sensors are available aboard the DSRV for assistance in performing the final mating maneuvers. These are a TV camera which looks through a viewport in the midsphere lower hatch, anda high resolution short range sonar (SRS) mounted to a retractable boom in the transfer skirt. Only the TV was simulated. Details of the performance of the SRS were not available at the time the simulation study was performed. Also, with good visibility, the TV is a much more informative sensor than the SRS. The midsphere TV display was simulated by photographing a model of a submarine. The photograph was then scanned by a CRT, using a flying spot scanner. The size of the area scanned is a function of the distance from the camera to the hatch. The dis- placement of the center of the hatch from the center of the screen is proportional to the distance between the hatch center and the inter- section of the camera axis with the hatch plane. Because of the relatively small angles between the DSRV and DISSUB planes, no attempt was made to provide foreshortening effects. Reproductions of the TV display are shown in Fig. 31. The four radial line seg- ments at the extremity of the picture represent the staples on the submarine deck surface to which a McCann Rescue Chamber can be attached. These staples provide precise centering information for the pilot. Fig. 31. TV Display 1172 Vehtele Dynamtes Assoctated with Submarine Rescue The most significant meter displays are those of doppler velocity, attitude rates. The doppler sonar, located 8.9 feet aft of the C.G. and 3.3 feet below the centerline, provides 3 axis ground velocity data. Since the doppler sonar is offset from the center of gravity, angular motions couple into the doppler signals, and in some situations were interpreted by the pilot as translation veloc- ities. Displays were also provided for roll, pitch and heading angles, and for sonar altitude above the DISSUB. Current magnitude and direction indicators were available, but were not used in the simu- lation, since the corresponding sensors were not installed in the vehicle. The additional displays shown in Fig. 26 are associated with the anchors and haul down winch control systems. Shock Mitigat ion System The shock mitigation system serves a dual purpose in the mating operation. The primary one is that of dissipating the kinetic energy of the DSRV when it lands on the DISSUB. The second function was realized only after the simulation study was started. Prior to dewatering, the DSRV is connected to the DISSUB primarily by verti- cal thrust forces from the DSRV and coulomb friction. Because of the existence of the shock mitigation system it is not necessary for the DSRV to land precisely on target. As long as the shock mitigation ring encloses. all the staples, the DSRV can slide on the DISSUB deck until precise alignment is reached. As described previously the shock mitigation system has been simulated as four independent damped springs. The natural frequency of the DSRV-shock mitigation system is approximately 11 radians per second, which is too high to simulate with a 62.5 millisecond iteration interval. Accordingly, the spring constant was reduced, with a re- duction in the natural frequency to 1.7 radians per second. The damping constant was also reduced to maintain essentially the same percentage damping. Anchor and Hauldown Exploratory runs were made using both the anchors and haul- down as mating aids. No help was obtained with the anchors, and very limited assistance was obtained from the hauldown. Schedule and budget limitations did not permit an intensive evaluation of this problem at the time. Some digital simulation was performed at a later date with the hauldown system, which indicated that it should provide substantial assistance, particularly when the DSRV is re- quired to mate bow up to the current. These results were confirmed on simulated tests made with LASS. 1173 Sechretber, Bentkowsky and Kerr CURRENT Fig. 32. Mating Geometry A TYPICAL SIMULATION RUN The mating situation to be described is depicted schemati- cally in Fig. 32. The DISSUB is rolled 225 degrees in an athwart- ship current, so that the DSRV is required to mate bow up. With the DSRV heading into the free stream, the interaction forces and moments are as depicted in Fig. 33. Deflection of current off the sail of the DISSUB causes a starboard sway force on the DSRV. The corresponding yaw moment is counterclockwise at large separa- tions, but becomes clockwise as the DSRV approaches the DISSUB. The normal force provides a suction effect at relatively large dis- placements, but becomes destabilizing as the DSRV approaches the DISSUB. Thus the normal force, due to interaction, adds to the force due to the free stream and tends to push the DSRV away from the hatch. Pitch moments remain rather constant over the distance Included in the run. In performing the mating operation the pilot attempts to head into the local stream rather than into the free stream. He sees and "feels" the DSRV sway and adjusts his heading to minimize the sway motions. Thus, the relative heading is not into the free stream but 14.74 Vehicle Dynamics Assoctated wtth Submarine Rescue 1 KNOT ATHWARTSHIP CURRENT, FORWARD HATCH, DISSUB ROLLED 22.5 DEG. DSRV BOW UP (6 = DSRV PITCH ANGLE) 6 = 30° 6 = 22.5° 500 = Be ui} 400 a 4 a O 300 & > uw S 200 Z wn 0 ~ OM 5 10-. 45920. 95° 30 re) DISTANCE DSRV TO DISSUB (FT) 4 © 5 7410 18 -20°- 25- 30 DISTANCE (FT) 2 s = os a Z Lau Li = = re re) = = uf = ~4000 = 0 a 0 5 10 5 20 25 30 x 0 5 10 15 20 25 20 DISTANCE (FT) DISTANCE (FT) Fig. 33. Interaction Forces and Moments rather is in a relatively arbitrary direction. The effects of heading changes on the interaction problem are illustrated in Fig. 34. First of all, as the DSRV heading is changed, Euler angle variations occur (Fig. 34a). Where initially the DSRV was not re- quired to list at all, it must now both list and trim, and can no longer make precise list and trim adjustments prior to landing. Concurrent with these changes, the free stream current components change with heading (Fig. 34b) which must be compensated appropri- ately. Assuming that the ideal heading corresponds to zero sway force, the variation of optimum heading with distance to hatch is shown in Fig. 34c. The optimum relative heading is approximately 30 degrees, with a significant heading change required as the separa- tion is decreased. It will be noted from Fig. 34d that the zero head- ing yaw angle does not correspond to the zero yaw moment angle, This is due to the horizontal gradients in fluid velocity along the length of the DSRV. Thus, there is no yaw plane equilibrium con- dition, and yaw plane control becomes a more severe problem than 1175 Schretber, Bentkowsky and Kerr DISSUB ROLLED 22.5 DEG A PITCH & ROLL ANGLES REQUIRED TO PARALLEL HATCH PLANE 8 8 C YAW ANGLE REQUIRED TO MAINTAIN ZERO SWAY FORCE ANGLE (DEG) i) ° YAW ANGLE (DEG) 0 5 10 15 20) 25) 30 DISTANCE TO HATCH (FT) D YAW MOMENT UNDER CONDITIONS F = fe) Z < = = = a OF (C =i) (ee OF MOMENT 3 = = TO INCREASE ANGLE Zz : : — é = 30 (OO 30 60 90 0 5 10 15 20° 25 Mao YAW ANGLE (DEG) DISTANCE TO HATCH (FT) Fig. 34. Counteracting Interaction Forces by Heading Changes pitch plane control. A six channel recording of the simulation test run under these conditions is shown in Fig. 35. The variables plotted are the displacements x, y and z of the camera of the DSRV from the center of the hatch in the coordinate frame of the DSRV as shown in Fig. 32. Also plotted are the roll, pitch and heading angles of the DSRV, zero heading being into the free stream. For a one minute interval during the run the roll, pitch and yaw angle traces are replaced with RPM traces from the forward vertical, forward horizontal and main propeller, respectively. The run begins with the DSRV aligned with the free stream, in a level attitude, and at a camera elevation of 20 feet (correspond- ing to 15 foot distance between the DSRV seal and the DISSUB hatch). This initial condition results in a more severe transient than would occur on a mission, so that the advantages of the relative proximity in distance is overcome by the necessity to restrore dynamic equili- brium. At the start the pilot turned to port about 20 degrees to try to maintain equilibrium. Simultaneously, the roll and pitch angles are adjusted to try to maintain the DSRV sealing surface parallel to the hatch plane. After about two minutes, the pilot descends about 1176 Vehicle Dynamies Associated with Submarine Rescue et dr an r+ Zeszs- BESSaay SCALE CHANGE =. fopert ee ae) Tate =3 ee 'BS2e 6 el : 4 Mt a i 4 1 i mee ei le Hd will eee a at wa a nga erage xt i eth te aii i eer ees q Hl A a HL TON ma Me ull oe espa 2 ttl z net 2 ca it! r i 3 im il 5 tram Cut Rt ty J =2s== = S253 penrisie aa 2SS= 22 PRET EPP EEEES EES a2 sa2= 1 ? PATTI N Ty Gaaeedacee Pee A geaea: ue LEPEyT ; Per PE ET EP EEE PEEPS PPG == = SS Se eS Et PEPE EPL EE EPEPP ERTS EE ES SS 2uS2 es BEECHER FEE PEER START FORE AFT PPE PEPE Bez zezee: =. gag aS PECVEFEEP EEL PEREFT CL gerr {Rees Rene Sen a Seo Oe we eae PEEEELA 2a2=2 = === iN in Hl S225 =| Epi trea zt = FEEEES EPP Ess Fe | == = = PHS == SSS LTT AA MA ‘OH i ma re f ATH A rH IMI HAE i AI. Fi $883.22 ce Y (CARHERA) BReSSeszezazz= ss Sz SS= ed : EF ez. Li BBS. zez= = =: HE FE = FF Pee] Pte =S = eee zzz. ZS BSzez. BaSS SSSSz== =3SSeS ee fe ===: 'FBBSelZ zeesene eeSe= zz BeEEz: = Fs = ——— 22a: 5SS> 4 == 2332: SEILPELLEEL DELP EEEUEL ESE EELLLL Pe rere stb Libbi SEL TEL ETS =a tom FwOD VERT THRUST RPM ezzE> =5 =: 11 #2: PEEP EE ER Po BS gSzesz=szSze SS =: Saee: wSSaaas2: Pps Fins B35 EA EF EEE Zz, == ==2: zz=SSe. ge] 2588s: 322223 = == 7==2 =sn5 Sie EE EF e2= === == ==25n= a2 SeSs=s is ORT TT STM IHN e222 SSes2 = INH yy SU mu HWM IW i HIM tH reuse Hn un I Mi Fa aaa Ni WH TMG Mig E 77 eT if hy! { in ag Ca tH 000» uevon reaen vot ete OR TRON PI LEDER TT AUTOR ‘We anni ‘Ill; ‘Ni ni Maat i Me Hv RRA Wil AR ga Lt bald ih Sh Ee TNT iia | ] nineet ae 2S et Eres tt fe | =3= m= > =’ —— MeL ul HT pa a IVR a ee tT] can ag HNN s a | b AT Hm van li “ania "I } Magy pa. Mata tan ya Simulation Run -- Forward Hatch, Athwartship 1 Knot Current DISSUB Rolled 20 Degrees Big, 3 Schretber, Bentkowsky and Kerr half the distance to the hatch, then attempts to maintain altitude while adjusting his x and y coordinates. In the process the yaw angle has been increased to about 35 degrees. The final descent is made, the DSRV touching the deck somewhat in excess of 3.5 minutes after the start of the problem. At touchdown the DSRV was almost perfectly aligned inthe x direction but was nearly 2.5 feet off center inthe y direction. Onthe deck, vertical thrust was applied (trace not shown) in an attempt to keep from being lifted off by the current. Roll and pitch angles were adjusted to try to position all four legs in contact with the deck. Note that during this time the DSRV continues to roll and pitch. The transients at impact caused the DSRV to slip aft about 2.5 feet. This was corrected, as was the misalignment in the y axis. After nearly 5 minutes the pilot had the DSRV under control and could commence the final retraction of the shock mitigation ring and start up the dewatering pump. At this time the computer was placed into HOLD. The final misalignments were 0,10 feet inthe x direction and 0.16 feet inthe y direction, within the required tolerances. The final yaw pitch and roll angles were 36, 6 and 16 degrees, respectively. Unfortunately, this would not have been a completely success- ful landing. At the time the pilot terminated the run, the DSRV seal plane and the deck hatch plane were misaligned about 7 degrees in roll and 2 degrees in pitch so that all four shock mitigation legs were not in contact with the deck and a satisfactory seal could not be made. No instrumentation exists on the DSRV to provide this relative attitude data which could result in a serious operational limitation. The problem can be alleviated in part by use of the hauldown system which provides a larger stabilizing moment, reducing the offset angles. A major assistance was obtained by installing, in the simu- lation, a set of 4 indicators which measured the stroke of the shock mitigation hydraulic cylinders. Roll and pitch alignment could be achieved with this system, while maintaining the horizontal plane alignment. A review of the thruster activity leads to two interesting observations. First, the thrusters are operated in a bang-bang fashion, that is, either maximum thrust or zero thurst is commanded. This, despite the fact that an accurate proportional control system is available. Second, the activity of the horizontal thrusters is much greater than either the vertical thruster or the main propeller. This was somewhat predictable from the environmental curves of Fig. 9. The frequency of the horizontal thruster activity has implications in the thermal design of the thruster motors. 1178 Vehtele Dynamics Assoctated wtth Submarine Rescue DSRV AFT_HATCH-FORE-AFT CURRENT FORWARD _HATCH-ATHWARTSHIP CURRENT DISSUB ROLL ANGLE = 0 DISSUB PITCH ANGLE = 0 E — — EXPLORATORY RUNS FINAL RUN wn — oO Z = - Se, ee 7 ail ———_ [°s) entiie ae : eel =e | U 45 22.5 (2) 22.5 45 BOW BOW BOW BOW DOWN UP DOWN UP DISSUB PITCH ANGLE DISSUB ROLL ANGLE Fig. 36. Experimental Results RESULTS OF MANNED SIMULATION PROGRAM The experimental results of the simulation study are sum- marized in Fig. 36. Two sets of curves are shown, those for the final runs, of which Fig. 35 was an example, and those of earlier, exploratory runs which were made prior to the inclusion of the shock mitigation system in the simulation. Performance of the final runs did not meet those of the exploratory runs. Schedule constraints did not permit much training with the use of the shock mitigation system. The pilot believes that with such training the results of the two sets of runs would match more closely. On the basis of these results, we have tentatively arrived at the following performance predictions. a) Mating on the aft hatch is feasible for currents in excess of one knot at all attitudes of the DISSUB (up to 45 degrees) and headings of the DISSUB with respect to the local cur- rent. b) On the forward hatch, we have to distinguish between longitudinal and athwartship currents. In athwartships currents the goal of mating in a one knot current can be achieved except possibly for very high DISSUB roll angles which require the DSRV to mate bow up to the current. With respect to longitudinal currents, on those submarine £179 Schretber, Bentkowsky and Kerr classes which have sufficient clearance between the sail and the forward hatch to permit the alignment of the axes of the two vehicles, mating is possible in currents well in excess of one knot. For those submarine classes which do not have sufficient clearance, mating is limited to about 3/4 of a knot. The manned simulation program yielded some important by- product results which impacted on the vehicle design. The most significant ones are as follows: a) The splitter plate behind the transfer skirt was originally incorporated to reduce flow separation behind the skirt and minimize axial drag. Model testing unfortunately did not verify this drag reduction. However, the splitter plate was found in the simulation to provide sufficient roll damping to permit mating without automatic roll stabilization (automatic roll stabilization is, however, provided even in the manual mode). b) The shock mitigation system had originally been designed to dissipate energy only for impact velocities in excess of 0.25 feet per second. Below 0.25 ft/sec, the system acted as a spring. However, because the DSRV is neutrally buoyant, it will bounce off any spring unless the impact energy is absorbed. As a result of observing this phenomenon in the manned simulation study, the shock mitigation system was redesigned to provide damping for all impact velocities. c) The relative attitude indicators required to assure angular alignment have not yet been incorporated in the design. The results of the Ames tunnel tests have been invaluable in gaining an understanding of the problems involved in submarine mating. In the early phases of the simulation program, before the Ames results were available, mating runs were made under free stream conditions. Although there had been apprehension about the ability of the pilot to perform the 6 degree of freedom control function manually, we found that experienced aircraft pilots, with nominal DSRV simulator training, could control the DSRV with ease. Success- ful mating to currents up to two knots were anticipated for virtually all orientations and in excess of two knots for the most favorable conditions. The inclusion of the interaction effects, particularly on the forward hatch, dampened our optimism. The performance goals could be met, but at considerably reduced current magnitudes, and requiring considerably more pilot training. 1180 Vehitele Dynamics Assoctated wtth Submarine Rescue Prior to the start of the Ames test program, it was believed that the most serious interaction effects would be in the pitch plane, due to Bernoulli or "suction" effects. The experimental program was organized primarily to determine those forces. As we have seen, yaw plane interactions are more critical than those in the pitch plane. It would be desirable to have additional data, particu- larly with respect to interactions as a function of yaw angle and (for the longitudinal current) as a function of lateral separation of the longitudinal axes of the two vehicles. Recognizing the above limitations in the test conditions, no attempt has been made to use the Ames data quantitatively in the control system design. The data has been useful in the following areas: a) It has provided an appreciation of the yaw plane problems associated with mating. b) The force and moment gradients observed have been used to select and verify the static gain requirements of the automatic control system. c) The non-steady interactions have provided an input which could be used to establish the dynamic requirement of the actuators and effectors, in particular the pumping rate requirements of the list system. The DSRV is completing preliminary sea trials and will soon be conducting mating trials. Before too long we will have some full scale verification of the usefulness of the Ames test results. IV. CONCLUDING REMARKS This paper has presented an approach to the problems of simulation of the dynamics of highly maneuverable submersibles. All elements of the simulation are covered in considerable detail to provide an adequate base to build on for others with similar problems. No such comprehensive reference was available for our use. Although the model test data are not presented in their entirety, a reasonably complete description of the test procedure and results should allow determination of the usefulness of the data. Several references are given for more complete test results. It is hoped that this paper illustrates where Naval Hydro- dynamics is a continually expanding field and must take into con- sideration aspects of control system design, man-in-the-loop analysis, and numerous other fields not normally considered as relevant to the theoretician. 1181 Sehretber, Bentkowsky and Kerr Finally a note about the methods used during the development of the simulation. The work was not done in a theoretically rigorous manner but rather the simulation was built upon the background of the personnel involved in its development and their ability to apply existing theories to the problem. A number of the elements in the complete simulation of the vehicle dynamics reflect engineering judgment and experience. The major output of the study was re- quired on a tight schedule and relatively little new theoretical analysis were initiated. Since the DSRV is presently preparing for sea trials the validity of the simulation should soon be checked and correlation between the sea trials data and the results of the simulation should provide valuable insight for futher simulations. REFERENCES 1. Johnstone, R. S., "A Mathematical Model for a Three Degree- of-Freedom Simulation of the Underwater Launch of a Rigid Missile," Lockheed Missiles and Space Company, TM5774- 69-21, May 1969. 2. Kerr, K. P., "Determining Hydrodynamic Coefficients by Means of Slender Body Theory," Lockheed Missiles and Space Company, IAD/790, 21 July 1959. 3. Hess, John L., "Calculation of Potential Flow about Bodies of Revolution Having Axes Perpendicular to the Free Stream Direction," Douglas Aircraft Company, Report No. ES29812, October i, 1960. 4, Feldman, Jerome P., "Model Investigation of Stability and Control Characteristics of a Preliminary Design for the DSRV," David Taylor Model Basin Report 2249, June 1966. 5. Goodman, H. and Ettis, P., "Experimental Determination of the Stability and Control Characteristics of a Proposed Rescue Submarine (DSRV) Using the Hydronautics High Speed Channel," Hydronautics Inc. Technical Report 511-4, November 1966. 6. Bentkowsky, J., et ale, "DSRV Model for Analysis ML 493-03 Vehicle," Lockheed Missiles and Space Company Report No. RV-R-0037A, May 1968. 7. Reichart, G., "A Propulsion and Maneuvering System for Deep Submergence Vehicles," presentation at AIAA meeting in Seattle, Washington, June 1969. 1182 Vehicle Dynamics Associated with Submarine Rescue 8. Beveridge, J. L. and Paryear, F. W., "Performance of a DSRV Propeller on Four Modes of Vehicle Operation (NSRDC Model 5128)," Naval Ship Research and Development Center T & E Report 099-H-05, August 1967. 9. Beveridge, J. L., "Static Performance of a DSRV Ducted Pro- peller Thruster at Discrete Pitch Ratios," David Taylor Model Basin Hydromechanics Test Lab Report 099-H-03, July 1966. 140. Chislett, M. S. and Bjorheden, O., "Influence of Ship Speed on the Effectiveness of a Lateral Thrust Unit," April 1966. 11. Kerr, K. P., "Experimental Determination of DSRV/Submarine Mating Forces Using the Ames Variable Pressure Wind Tunnel," Lockheed Missiles and Space Company, TMOS-H- 67-62, June 1967. 12. Ogilvie, A., "Force on an Ellipsoid Moving Near a Wall," David Taylor Model Basin Report, 1967. 13, Moody, R. C., "Statistical Considerations in Power Spectral Density Analysis," Technical Products Company, 1966. 14, Turpen, F. J. and Goodman, A., "Experimental Determination of the Performance Characteristics of the DSRV Based ona 1,25-Scale Lighter-Than-Air Submarine Simulator (LASSI)," Hydronautics, Inc., Technical Report No. 705-3, August 1968, 1183 es AUTHORS INDEX Bentkowsky, J., 1111 Newman, J. N., 519 Bessho, M., 547 Norrbin, N. H., 807 Brennen, C., 117 Ogilvie, T. F.5 663 Carrier, G. F., 3 Paidoussis, M. P., 981 Chan, R. K. C., 149 Paulling; J. RR. 1083 Coantic, M., 37 Savitsky, D., 389, 447 Dagan, G., 607, 625 Sawatzki, O., 275 actors, L.J:, 601 Schooley, A. H., 311 Havre, A., 37¢ Schieler, M., 3614 Fromm, J. E., 149 Schreiber, H. G. Jr., 1111 Hasselman, K., 361 Sharma, S. D., 601 Hogben, N., 446, 473, 540 Shwartz, J., 321 Holmquist, C. O., xi Street, R. L., 149 Hsu, B. Y., 11 Baylor, bs Wey OL 1g O59 James, E. C., 951 Tuck, E. O., 627, 659 Kaplan, P., 1017 Tulin, M. P., 321, 607, 626 Kerr, K. P., 1111 Van Mater, P. R. Jr., 239 Krishnamurti, R., 289 van Wijngaarden, L. ,235,287,622 Lackenby, H., 474 Verhagen, J. H. G., 955 Landweber, L., 449, 475 Wang, D. P., 189 Lee, C. M.,; 905, 951 Waters, O. D. Jr., xiv Le Méhauté, B., 71 Weinblum, G. B., 599 Linden, T. L. J., 477 Whitney, A. K., 117 Maestrello, L., 477 Yih; €.7S.,° 219, 236 Maruo, H., 624, 658 Yim, B., 573, 604 Miles, J. W., 95 Yu> Hs Y., 22 Munk, W. 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