THE EFFECT OF A PLANE BOUNDARY ON WAVE IN- DUCED FORCES ACTING ON A SUBMERGED CYLINDER Houston Keith Jones ^uulhY KNOX LIBRARY .v.A\/AL POSTGRADUATE SCHOOL 3STGR Monterey, California p II r i H RIB 1 1 in« ^Ba*-' H «aw^ •THE EFFECT OF A PLANE BOUNDARY ON WAVE IN- DUCED FORCES ACTING ON A SUBMERGED CYLINDER by Houston Keith Jones June 19 75 The? ;is Advisor: T. Sarpkaya Approved for public release; distribution unlimited. T167V79 SECURITY CLASSIFICATION OF THIS PAGE C*n»n 0«la Enfrmd) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM i. REPORT nuMeER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 3. TYPE OF REPORT ft PERIOD COVcRED Master's Thesis Thesis ' '. title (*r.d Suftmiaj The Effect of a Plane Boundary on Wave Induced Forces Acting on a Submerged Cylinder «. PERFORMING ORG. REPORT NUMBER 7. AjTmORi'v Houston Keith Jones 8 CONTRACT OR GRANT NUMBERf*.) S. PERFORMING ORGANIZATION NAME AND ADDRESS j Naval Postgraduate School r Monterey, California 93940 10. PROGRAM ELEMENT. PROJECT. TASK AREA a WORK UNIT NUMBERS III. CONTROLLING OFFICE NAME AND ADDRESS t j Naval Postgraduate School J Monterey, California 93940 12. REPORT DATE June 1975 13. NUMBER OF PAGES 14. MONITORING AGENCY NAME ft AOORESSf// dlllermnt from Controlling Olltc*) 40 ■ ■ ^> l«wr«i i v ^i ri u ajlh^; n r~, ■* i_ « AUunLjJiu U I Naval Postgraduate School i Monterey, California 9 39' 15. SECURITY CLASS, (ol thla rjportj Unclassified 13«. DECLASSIFI CATION/ DOWN GRADING SCHEDULE I 16. DISTRIBUTION STATEMENT (ol 5fa [Approved for public release; distribution unlimited Report; f : 17. DISTRIBUTION STATEMENT (ol th* mbtlrmct mnefd In Block 20, II dltlormnt from Rmpott) IS. SUPPLEMENTARY NOTES It. KEY v.'CKOS 'Connnut on rtrmram mid* II nac»fj-y and Identity by block nwnbtr) 20. ABSTRACT (Contlnuo on .-•v»/«» aid* II nacaaaary and Idantity by block numbar) The in-line and transverse forces acting on cylinders of 1.0 inch to 2.5 inches diameter placed near a plane wall in a harmonically oscillating flow have been measured. The drag and inertia coefficients Cd and C for the in-line force and the lift coefficients C.- and C, for the transverse forces away and DD i jan*7; 1473 EDITION OF 1 NOV C8 IS OBSOLETE (Page 1) S/N 0102-014-6601 | SECURITY CLASSIFICATION OF THIS PAGE (Whun Dmlm Knfr»d} ffllCUHITV CLASSIFICATION OP THIS P«GE'»',nn Di-la Enlmfd- toward the wall, respectively, have been determined as a func- tion of the relative gap e/D between the wall and the cylinder and the period parameter V T/D. The relative gap ranged from 0.014 3 to unity and the ' period parameter from zero to 35. In the subcritical range where these experiments have been performed, the effect of the Reynolds number, which ranged from 4,000 to 30,000, was found to be secondary and certainly ob- scured by the excellent correlation provided by the period para- meter. The frequency of the transverse force was found to range from one to five times the frequency of oscillation of the harmonic motion, but a dominant frequency of twice the oscillation fre- quency was seen over most of the values of V T/D. J DD Form 1473 , 1 Jan 73 S/N 0102-014-G601 SECURITY CLASSIFICATION OF THIS P»CEf«i*n Datm Enfrmd) The Effect of a Plane Boundary on Wave Induced Forces Acting on a Submerged Cylinder by Houston Keith Jones Ensign, United States Navy B.S., United States Naval Academy, 1974 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL June 1975 DuULlY KNOX LIBRARY i\AVAL POSTGRADUATE SCHOOL ABSTRACT The in-line and transverse forces acting on cylinders of 1.0 inch to 2.5 inches diameter placed near a plane wall in a harmonically oscillating flow have been measured. The drag and inertia coefficients C, and C for the in-line force and a m the lift coefficients C, , and C,m for the transverse forces 1A IT away and toward the wall, respectively, have been determined as a function of the relative gap e/D between the wall and the cylinder and the period parameter V T/D. The relative gap ranged from 0.014 3 to unity and the period parameter from zero to 35. In the subcritical range where these experiments have been performed, the effect of the Reynolds number, which ranged from 4,000 to 30,000, was found to be secondary and certainly obscured by the excellent correlation provided by the period parameter. The frequency of the transverse force was found to range from one to five times the frequency of oscillation of the har- monic motion, but a dominant frequency of twice the oscilla- tion frequency was seen over most of the values of V T/D. TABLE OF CONTENTS I. INTRODUCTION 9 A. EXPERIMENTAL JUSTIFICATION 9 B. BACKGROUND THEORY 9 C. RELATED RESEARCH 12 II. METHOD OF ANALYSIS ■ 17 III. EXPERIMENTAL EQUIPMENT 22 IV. DISCUSSION OF RESULTS 28 V. CONCLUSIONS 38 COMPUTER PROGRAM (IN LINE FORCES) 39 COMPUTER PROGRAM (TRANSVERSE FORCES) 41 LIST OF REFERENCES 42 INITIAL DISTRIBUTION LIST 44 LIST OF FIGURES 1. Schematic drawing of the U-channel and orientation of the test cylinder 23 2. Elevation and Acceleration traces 24 3. Force traces 25 4. Drag coefficients versus period parameter 29 5. Inertia coefficients versus period parameter 30 6. Lift coefficients versus period parameter (force towards wall) 31 7. Lift coefficients versus period parameter (force away from wall) 32 NOMENCLATURE A amplitude of the motion C, average drag coefficient a C,, maximum lift coefficient away from the boundary C,-, maximum lift coefficient toward the boundary C average inertia coefficient m D diameter of test cylinder e normal distance between the boundary and the cylinder surface F instantaneous total force acting on the test cylinder F, drag force acting on the test cylinder F, lift force acting on the test cylinder h water depth k wave number (2tt/1) L length of test cylinder 1 wave length T period of oscillation t time u horizontal fluid particle velocity V instantaneous velocity V maximum velocity m w vertical fluid particle velocity z vertical position of the test cylinder A percent error v fluid kinematic viscosity p fluid density o wave frequency (2tt/T) ACKNOWLEDGEMENTS This project has been carried out under the direction of Professor T. Sarpkaya. I would like to express my sincere gratitude for his guidance, but especially for the knowledge he imparted to me. Appreciation is also extended to Professor E. Thornton for his assistance in completing this thesis and Doctor D. F. Leiper for his time spent to read it. I. INTRODUCTION A. EXPERIMENTAL JUSTIFICATION With the current energy deficit becoming of vital impor- tance, offshore oil production assumed great priority, and pipe lines became the most feasible mode of liquid transporta- tion to the shore. This, in turn, necessitated the deter- mination of the wave and/or current induced forces on submerged pipes placed at or near the ocean bottom. The problem, how- ever, is not unique to the pipe lines. Underwater cables for communication and sensing purposes and all submerged structures, be they offshore oil platforms or oceanographic research instruments, are subjected to wave induced forces. The proximity of pipe to a solid boundary, such as the ocean bottom, further complicates the problem, and, unless the submerged pipe lines are anchored to a buoy system or buried, the bottom effects must also be considered. Even buried pipe lines, subject to scouring can become uncovered and have their support dug out from under, leaving them sen- sitive to wave induced forces. B. BACKGROUND THEORY The in-line force acting on a cylinder immersed in a time- invariant flow may be expressed, according to Morison and his co-workers [6] as: F=FJ+F = l/2LC,DpV2+C ttD2/4 LdV/dT (1) d m d m ' in which C, and C represent time-invariant drag and inertia d m ^ 3 coefficients, respectively. The first term on the right-hand side of this equation represents the velocity-squared depend- ent drag force and the second term the acceleration-dependent inertia force. A simple dimensional analysis shows that the time- averaged drag and inertia coefficients for a cylinder immersed in a harmonically oscillating uniform flow, represented by V=-V COS 27Tt/T m depend on the period parameter V T/D and the Reynolds number V D/v. In addition to the in-line force, the cylinder is m J subjected to an alternating transverse force due to separa- tion and vortex shedding. The proximity of a solid boundary can alter significantly the magnitudes of both the in-line and transverse forces and requires the determination of the wall proximity effect on the drag, inertia, and lift coefficients. In fact, it is the determination of this effect that prompted the present study. A harmonically oscillating uniform flow represents only approximately the oscillatory motion induced by waves about a submerged cylinder. Nevertheless, the wave-induced forces may be predicted with sufficient accuracy through the use of drag, inertia, and the lift coefficients obtained with har- monic flow, provided that the velocities and accelerations 10 about the cylinder are either measured or calculated through the use of an appropriate wave theory. For small amplitude waves, Airy's theory may be used to evaluate the velocities and accelerations of a given time and depth. The limitations of this theory are well-known and will not be represented here. The horizontal and vertical components of velocity may be written as [11] , u=Ao(ccsh k(h+z)/sinh kh) cos 9 (2) w=Aa(sinh k(h+z)/sinh kh) sin 6 (3) in which A represents the wave amplitude, k the wave number defined by 2tt/1, h the water depth, z the vertical position 2tt of the cylinder, a the wave frequency given by and 6=kx-at. The components of acceleration are given by 6u/5t=-Aa2(cosh k(h+z)/sinh kh) sin 9 (4) 2 6w/5t=Aa (sinh k(h+z)/sinh kh) cos 9 (5) The in-line force may be expressed through the combination of equations (1) , (2) , and (4) to yield, 2 2 F=Aa LC D cosh k(h+z) sin 9 4 sinh kh 2 2 -A C,LpD(cosh k(h+z) | cosat | cosat (6) (sinh kh) * The transverse force cannot, however, be calculated in a similar manner through the use of equations (1) , (3) , and (5) . 11 It is fundamentally the eddy behavior that determines the transverse force. Thus, the lift resulting from a nonlinear motion is not susceptible to estimation by superposition of individual responses to each harmonic acting in isolation. The lift coefficients in the present study are evaluated as follows: C, = (maximum transverse force away from the wall in a cycle) 0.5pDLV 2 (7) m C, = (maximum transverse force toward the wall in a cycle) 0.5pDLV 2 (8) m Force coefficients based on other time-invariant magnitudes such as the root-square values could have been easily obtained. It will suffice here to remark that a simple Fourier analysis for the in-line force and the peak value for the transverse force proved adequate for all purposes. The determination of the coefficients C,,C ,0,-/0,,- con- d m 1A IT stitute the essence of the present study for a cylinder placed within the proximity of a plane wall in a harmonically oscil- lating flow. C. RELATED RESEARCH Yamamoto, et al . , in 1974, studied wave forces on cylinders near a plane boundary in the range of Reynolds numbers from 2,000 to 30,000 and the surface period parameter from about 0.3 to 3. The total force was considered as the sum of components due to water particle velocity squared (lift and 12 drag forces) , and due to water particle acceleration (inertia force) . They have correlated the force coefficients C, and C, with the relative distance of the cylinder from the bound- ary. The vertical and horizontal hydrodynamic forces were obtained for unseparated flow over the submerged cylinder by using the method of double images and applying Blasius1 theorum, The results indicated that the proximity of a plane boundary modified both the lift and the inertia coefficients. However, these results are applicable only to unseparated flows where drag is negligible compared to inertial forces. Within a narrow range and relatively low values of the period parameter, it was shown that the maximum horizontal force oc- curred at zero crossings of the surface wave, indicating that wake dependent drag force is negligible and the horizon- tal force is composed mainly of the inertia force. The verti- cal force, for larger values of the relative distance was unsinusoidal with large downward forces (toward boundary) at the surface wave crests and small upward forces at surface troughs . In 1967, a theory for simple shear flow across a circular cylinder in proximity to a plane wall, was developed by Arie and Kiya [2] . Theoretically, they showed that a force acts downward (negative uplift) for uniform velocity, but a force acting upwards (away from the plane) exists for flow having a velocity gradient. Their experimental data failed to enforce 13 this theory, but did show that lift on the cylinder increased as the clearance between the cylinder and wall decreased. A wind tunnel was used for their experimentation. Studies on the effects of ocean currents on pipes anchored just above the ocean floor were conducted by Wilson and Cald- well [12] in 1970, and the lift and drag coefficients were determined. The frequency of the forces on the pipe was found to be effected by the eddy-shedding frequencies. In 1971, a study by Grace [3] on the effect of the clear- ance of pipe above a flume and the effect of orientation of the pipe to the wave crest was made. With clearance ranges from 1/32 inch to 1-1/2 inches and wave periods of 2 to 6 seconds in 3 foot deep water, he found that the horizontal force normal to the pipe was not effected by bottom clearance and that this force decreased as the pipe became perpendicular to the wave fronts. He also found that the "vertical force" decreased as the bottom clearance decreased. In much of the work conducted on wave forces, correlations between C , C,, and the Reynolds number V D/v, where v is m d m the kinematic viscosity, have been sought with little success. Wiegel, Beebe, and Moon (1957) [11] found no relation between these parameters in studies of ocean wave forces on cylindri- cal piles. Jen [4], in 1968, also failed to correlate C or C, with Reynolds number in a laboratory study of a 6 inch diameter pile. In 1958, Keulegan and Carpenter [5], investigating the inertia and drag coefficients on circular cylinders, found 14 a correlation between these force coefficients and a dimen- sionless parameter, V T/D. Working in a rectangular basin with standing waves, they covered V T/D values between 2.7 and 120 over a Reynolds number range from 4000 to 29,300. The lowest value of C and the highest value of C, were found to m 3 d occur at a V T/D of about 15. m Sarpkaya and Tuter [8], in 1974, with an oscillating U- shaped channel and various cylinders, reconfirmed Keulegan- Carpenter data except for a small spread in the upper regions of V T/D, where the Sarpkaya-Tuter data may be more accurate and reliable. An important finding of the Sarpkaya-Tuter work was that the lift or transverse force was as larage as the in- line force. The close correlation between C and C , and the period m d ^ parameter V T/D is evidenced by the close fitting curves re- lating these parameters. The C curve of Kuelegan-Carpenter forms a lower envelope for ocean data gathered by Wiegal, Beebe, and Moon, while the C, curve forms an upper limit for the same data [11]. The work on wave induced forces on cylinders has attracted much attention, but results seem quite scattered and some- times contradictory. Also, the effect of a plane boundary in proximity to the cylinder presents a new variable. It is the purpose of this thesis to analyze the forces on a cylinder in the proximity of a plane wall and to correlate the various coefficients with the period parameter V T/D, and the relative 15 distance between the cylinder and the wall, through the use of harmonic motion. 16 II. METHOD OF ANALYSIS To analyze the time dependent force in unsteady periodic flow, a Fourier analysis will be made, beginning with the Morison's equation given by F=l/2 CdpAV|v|+ CmpV dV/dt (9) 2 The enclosure of one of the components of V term in "absolute value" brackets accounts for the change in sign of the resist- ing force as velocity changes direction in an oscillating flow. For a circular cylinder, the equation reduces to: F=C 7r/4LD2pdV/dt + ^d_LDpV|v| (10) m 2 Representing the velocity of the harmonic motion by V=-V cosot (11) m and inserting into equation (10) , one has 2 C sinat - CJ I cosat I cosat (12) pLV ^D/2 V T m d m m Multiplying both sides by sin t and integrating, one has, 2^ 2tt 2 r F smct d(ot) = .ttD- . . . . j , .\ J —^ i^—L o; _c smatsinat d(at) pV^D/2 . L m m 2tt r d -9f Cj | cosat | cosat | sinat d(at) (13) 17 which gives 2\ F sinat d(gt) = tt2D C_(tt) .... f ~ m (14) ° pLV D/2 V T w m ' m and 2 V t 2lI Fsinot d(gt) ,. _, Cm= VT™~ °f — 2n (15) D pLV D Multiplying both sides of the force equation by cosot, and integrating, results in 2 it 2lT 2 ' F cosot d(ot) =0f tt D C sinat cosat d(at) - of 2 , V T m pLV zD/2 m (16) 2tt Qf C, | cosat | cosat cosat d(at) 2tt 0f F cosat d(at) = -8/3 Cd (17) DLV 2D/2 m Finally, 7 F cosat d(at) C = -3/4 d(t/T) (20) D LAp C, = -3T2 EFcos(^) d(t/T) (21) 8pD7TLA As an alternate method of computing the force coefficients, a least-squares method was also employed. The method of least squares consists of the minimization of the error between the measured and calculated forces. Letting F represent the in- stantaneous measured force and F the force calculated through the use of equation (10) , and writing E2= (F-F )2 (22) 2 2 and dE /dC =0 and dE /dC =0, one has m d 2tt i i c 23. r F | cosat | cosat d(at) (23) d 3tt pDLV 2 K m and 2tt _ . 2V T 0f F sinat d(at) (0A. D pV LD K m It should be noted that the Fourier analysis and the method of least-squares yield identical C values and that the C, values n J m d differ only slightly. The error between the measured and calculated forces, particularly in the neighborhood of the maximum forces, may 19 be further minimized by choosing the square of the measured force as the weighting factor in the least-square analysis. Thus writing 2 2 2 E =F^(F-F )* (25) 2 2 and dE /dC,=0 and dE /dC =0, one has / d m C =— Mh^h- (26) d pDLV 2f4fl"f3f3 m and 2 T f f -f f C =-4 0-i5jl -43-^2- (27) m 3T.r.2 f.fn-f_f_ 7T LAD 4 1 3 3 The functions f, are given by 27T 2 4 27T 3, f,=/ F cos at d(at) , f = / F |cosatj cosat d(at) * o « o 2tt ~ f3=0/ F sinat cosat| cosat|d(at) (28) 2tt _ 2ir _ f4=0/ F sin at d(at) , fg= 0/ F sinat d(at) Equations (27) and (28) may be shown to reduce to equations (23) and (24) by replacing F in equations (28) by F and carrying out the necessary integrations in which F does not appear . Each wave cycle was divided into 36 time intervals of At=2. 86/36 seconds, (T=2.86). The force for each interval was found and incorporated into the computer program to calculate 20 the appropriate coefficients through the use of the three methods of analysis cited above. 21 III. EXPERIMENTAL EQUIPMENT The basic oscillating flow system consisted of a U-shaped vertical water channel with a channel cross section of 18 by 20 inches shown in Figure 1. Oscillations were created by pneumatic pressure applied at a closed end of the channel. The closed end was opened by means of a slider crank mechanism uncovering a large exit and releasing the air pressure. The resulting oscillation was a near perfect harmonic as evidenced by the elevation and acceleration traces shown in Figure 2. The maximum amplitude of the oscillation was 11 inches and the natural damping of the oscillations was in the order of 1/8 inch per cycle. The water level at its minimum oscilla- tion height was twenty inches above the test cylinder. The cylinders were manufactured out of plexiglass tubes or rods at desired diameters and at lengths approximately 1/16 inch less than the channel width. Self-aligning bearings were imbedded at each end of the cylinders and caution was taken before each experiment to confirm the clearance of the cylinder end from the channel walls. The lateral and in-line (with respect to the fluid flow) force measuring devices consisted of various cantilever beams mounted to the ends of the test cylinder. Eight piezoresistive strain gauges were mounted on each cantilever beam and properly waterproofed . These force transducers were repeatedly calibrated by hanging known loads at the midsection of the cylinder in the 96 ■18" — \ e - / S ju: 58" S^T -^ ^ Fig.l Schematic drawing of the U~ channel and cylinder orientation 23 ! 1 \ _ 1 1 > /<. / V \ : r / l L i -; f 1 \ 1 \ / \ 1 / - / i 1 \ \ / ! \ r-l/i 1 J _ :\ / / L ! T 1 \ -i-/ I \ LEV ATI ON \ ■\A hvm " ! ,/ RACE _ _L \ / 1 !J fi " . z' ■ i . ' X \ JS i U-| i t I 1 i i : - • T. " i 1 ■ ! r ^* ""*"HC^ - ■ i lr*] *""' i X / \ / X ■ i i / ' ■ ! X I— ! ' / IX ; V / ■ i X' /ACC :elebatiok x^ i X^^^ ^s ' -viU X ^ TRACE N ' ■■ :- \ 1 1 i ' j . Fig. 2 Elevation and acceleration traces 24 — I | 1 f 1 1 1 II II j... 1 • [ LU L_ j 1 J_|_ -ft- 1 1 J J. 1 ■ ! 1 /x - i — I .1 - U-LIX- I U....L.. — ^*™^K. / l . ~1 1 Ill : i i i \ 1 ' 1 1 f 1 1 1 i I i ! ! ! ; — 1 1 IK i 1 j i j ■■■- |^V i • ■■• |- i t i / /_ V i „ i \ I - U^v- 1 ' :■ / * • _j_ 1_ / I \ r / \ 1 □ 1 i \ { i LI \ ■ II 1 ■ N : ; I / J , \ / \ v/V / V ! ; v J. \ >. / ' >^ ^~\ 1 N. s ^ 7 4. J 1 L_L^ X - j 1 ^"_ Y__ i 1 ! ' : L i :!-- A A j 1 \ i Ll3- :_ :: i 1 1 ' LIFT FORCE TBACES e/D= 0.3$ : : • '! ; • i i i - ' T T^ : 1 I: _[ l_i- • 1 - \ ' l-i 1 V! , :| L '.' . .:.. ■ 3 A . : :\ i* *mm : — : 1 mmi ; 1 i / V 1 - vs. /\ i i ! ■ . 1 t y \ i ' r_i ■ ^W 1 j^ ' \ \ ■ i ■> r .i ' V ! 1 y/ \ . \ / i I i i y/ ..- - l:- _1 *v / ! I:-" ^ I ^ 1 1 ' - '" i «^ L / 1 \- ■Fs / ' \ ; / ■J 11 ^L . i ■ i i 1 1 I ! i \ : \: ' ! •' i - i 1 /\ Mill ; 1 t / \ L 1 l i / 1 \ ■ / V i - UM4- "i i ; J- 1 ''l i \ 1 . - n i ,1 \S l \ • - J i V ■1 i 1 / \ 1 / ! 1 i / Y\ i \ 1 1 1 i, \ / i \ i / •■ J _L _;J .., — l-PsJ \ i ' r 1 |t*>v / L_ N- i ^i\ne/fc i TRACES e/D= 0.38 \ - iii.' E i L.. . 1 A i i I .=. n i tti rrn i ! ! 1 Fig 3. Force traces 25 horizontal and vertical directions. Some of the transducers were comprised of two pieces of thin cantilever beams cut orthogonal to each other and were able to measure both the in-line and lateral force simultaneously. Other transducers were capable of measuring only the in-line or lateral force. However, being rotatable -90°, this transducer could first measure the in-line force and with a 90° rotation, measure the lateral forces. In many cases the in-line and lateral forces were measured at each end of the cylinder and compared with each other. In no case did the force curves deviate from each other more than 5%, indicating a fairly uniform response along the cylinder with respect to the resultant forces. Throughout the investigation, the monitoring cf the char- acteristics of the oscillations in the U-channel was of prime importance. Most of the difficulties in past determinations of C ,C,, and C, resulted from the difficulty of producing a m d 1 purely harmonic motion or from having to determine oscilla- tion characteristics indirectly. The U-tube provides a per- fectly sinusoidal oscillation and the instantaneous displace- ment and acceleration are continuously recorded. The instan- taneous elevation in one leg of the channel was determined through the use of a capacitance wire connected to an amplifier- recorder system. Such wires have been used in the past to measure wave heights in open channels. The response of the wire was found through calibrations, to be perfectly linear within the range of oscillations encountered. 26 The instantaneous acceleration was measured by means of a differential-pressure transducer connected to two pressure taps at the mid-section of one of the channel walls. The acceleration was then calculated from Ap= psaz (29) where A P is the differential pressure, p the fluid density, s the distance between pressure taps, and a the instantane- ous acceleration. The effect of pressure drop due to the viscous forces was found to be negligible. The displacement and acceleration are, of course, in phase and may be used independently to calculate the velocity and displacement of the fluid, although displacement was actually used. The smoothness of the variation of acceleration and elevation traces shows the success in obtaining a purely harmonic motion as shown in Figure 2. 27 IV. DISCUSSION OF RESULTS The drag coefficients are shown in Figure 4 as a function of the period parameter V T/D, which also equals 2ttA/D, for III — various values of e/D. The actual data points are shown only for one value of e/D, partly to simplify the presentation of the data and partly to give some idea of the variation of C, with respect to the three methods of analysis used in its evaluation. Firstly, it is apparent that all three methods yield nearly identical results and that the data exhibit very little scatter. Secondly, the drag coefficient can reach values as high as 3.75. For e/D larger than unity, C, values approach those found for V T/D=°°. Evidently, it is not possible to explain the complex variations of C, with V T/D since it is largely determined by separation effects. The inertia coefficients are shown in Figure 5 in a manner similar to that for C,. The values of C corresponding to d m ^ V T/D=0 are obtained from the potential theory [10] . The ex- perimental values approach in all cases those predicted by the potential theory as V T/D->0. Once again, the inertia co- efficients approach those obtained for the limiting case of e/D=°° [8] for e/D values larger than unity. The lift coefficients C,A (force away from the wall) are shown in Figure 7 and the lift coefficients C,T (force towards the wall) are shown in Figure 6. Also shown in Figure 6 are 28 ' 0 0 «*■ in cd Q O Mr- r^ S.O ^>. >— no Men a* O CD CD ." OO H- - . Q O «» r// ; ' ! ! : CD UJ 2C UJ / *** ! 1 ! CC • f 1 • i • Q «a: ♦ '.lit / / tit ! ' It'* ! 1 1 • 1 O ID o- oo UJ C3 UJ »— / / ' 1 : / •/» / .' i : * / : I ' * wo / ; ! ; • ; / • ! •' / / / / < UJ oo LU a: UJ > < _J «t r> UJ cr CO a UJ ►— « »— H J— u_ cc oo »— < rs ct a o UJ o u. _l © 2: < I 1/ / f - ■ / 74 / .f • / • » • / - • { / l n 1 / / / / / rf // • 'x / • % \ \ / * • *\ < QD \\\ o 1 1 1 1 V CO 0 CO s- x Q> 0, CO 3 to lO 5— 1 •— > - *n GO c _0 ._ V) CO CN O <♦-< © as CJ ->* bo fa 29 4> s a *-> O O > o o to bo 30 CO o CO o> s «o Ctf CN «3 ex o O CN J-l 0) ft. CO > CO — -*-> C3 C £ o> o «-> -a t—° •— as o o •o o to -t-> o o ho fa 31 CO o CO lO 03 CN -a o O (_< CN a> ex CO 3 co ^^ tl __ to CO _. c3 > CO S: -«J s c o a> «-. o — (4-4 »— o >> <— < c3 «*-< £ ER/D I A ) C AMP=AMPL IUQE OF MOTION IN FEET C C 2I=FORCE COEFFICIENTS C C CM=INERTIA COEFFICIENT C C PFR=PEPIOD IN SECONDS (CHART PERIOD/CHART SPEED) C C CD=D^AG COEFFICIENT C C REMF=REMA INOER FUNCTION C C AI ,BI=FOURIER COEFFICIENTS C C N=NUM3ED Of DATA SETS C C CN=CONVERTION FACTOR C C . c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc DIMENSION FORCE( 36) N = 5 DO 100 1=1, N G = 32. 174 READ(5,55) (FD^CE(K), K=l,36) RHG=6?. 4/G PI=3. 14159 CNU=0. 0000105 C INITIATE DATA SETS C C READ IN PARAMETERS WHICH CHARACTERIZE THE DATA SET C READ(5,10)DIA,A*P ,PER,NCARD,UMX, CN»FMMM, NRUN C C COMPUTE FORCE COEFFICIENTS Zl = (2-PER*PrH/(PT*P!*-Pi*0IA*p IA*CL*RHO*AMP ) Z2=(-^-PER-:°ER) / ( S*PHO*DI A* PI -CI *AMP*AM° ) Z 2L S = < - 4V; P E^ * ? E * ) / ( 3 *P I *P I * R H2 * "J I A*C L * A ,-i P* A MP ) C COMPUTE BETA BETA = 2*PI*AMP/0I A REYNG=( UMX*1I4)/CNU WRITE (6,1 5)DIA,AMP,PER,UMX,CN,NRUN WRITE(6,20) TIME =0.0 SUM=0,0 DO 65 K=1,NCARD SUM=S'JM+FORCE(<) 65 CONTINUE FMEAN=SUM/NCARD DO 85 K=1»NCA*D FORCE (K ) = FORCE (,< J-FMEAN 85 CONTINUE FMMM=F^MM-FMEAN| CM=0.0 CD=0.0 CMLS=0 CDLS=0 FAA=0.0 FBP=0.0 FCC=0.0 FDD=0.0 FFE=0.0 DELTAT = 0. 027972028 DO 200 K=l, NCARD F=CN*FORCF(K ) ALPHA=2*PI*TIM= SINA=SIN( AL°HA ) COSA=CCS( ALPHA) FSINA=DELTAT*F*SINA FCOSA=DELTAT*F*COSA FLS=OE LTAT*F*C3S A* AB S ( COS A 1 FAAS=2*PI*F*DhLTAT*C0SA*C3SA*C0SA*C0SA*F FBBS=2*P I*F*F*C3 SA*AB S( CDS A ) *3E LT AT*F FCCS=2*PI*F*SINA*C0SA*AbSlC0SA)*DbLTAT*F 39 200 300 100 10 15 20. 30" 35 40 45 50 55 FDDS=2*PI FEES=?*PI F AA=F AAS* FBB=F33S+ FCC=FCCS«- FDD=FD0S+ FEE=FEES«- CM=FSINA+ CD=FCOSA+ CMLS=FS IN CDLS=FLS+ WRIT£(S ,3 TIME=TIME CONTINUE CM=Z1*CM CD=Z2*C0 CMLS=Z1*C CDLS=Z2LS CMFF=(Z1/ CDFF=(-4. ♦ FCCI WRITE(6 ,3 *F *F FA FB FC FO F£ CM CD A CD 0) + D *SI\'**SINA*DELTAT*F *F*SINA*DELTAT*F A B r 5 E ♦ CMLS LS FIMEf ALPHAfCOSfcfSIMA,FfFCOSAfFSINA ELTAT MLS *C3LS 2.0)*(FEE*FAA-FCC 0*Z2/( 3.0*PI ) )*(F 5) *FBB)/(FD3*FAA-FCC*FCC) EE*FCC-FD0*FB3 )/( FDD*FAA-FC WRITE(6,^0):.Mt:D,3ETA,REYVJ3,CMLS,CDLS,:MFF,CDFF ANGLE=0.0 WRITE(6,A5) ] 1 1 TIME=0.0 DO 300 K= F=CN-F3RC THETAl=( ( Cl=( ABS(C C2=RH0*( ( C3=( (PI** F1=(CM*C3 FLS=(C*LS FFOR=(CMF F=F/C2 REMF = AE?S( FMAX=FMMM REMF=RErtF RLS=(A3S( RFOR=( ABS WRITE(6,5 ANGLE=ANG TIME=TIME CONT I.MUE CGNT INUE FORMAT ( FORMAT* F8.4,2X FORMAT ( • ,2X,'F FORMAT ( F 0 RM A T ( t CMLS = ' FORM AT ( FORMAT ( •RLS' »1 FORMAT ( FORMAT ( STOP END 1 ,MCARD E(KJ 2 . O-3 I ) /360)*AMGL 0SCTHETA1 )) )*COS( UMX**2. 0) /2.0)*DI 2 )*Oia*SIN< THETA1 -CD*C1 ) *C3-CDLS*C1 ) F*C3-CDFF*C1 ) THETA1 ) A*CL ) ) /( UMX*PER) F J - A B S ( F 1 1 *CN/C2 /FMAX F)-ABS(FLS) )/FMAX ( F)-ABS( FFOR) )/FMAX OJTIMEi F,cl tREMF, FLStRLSfFFORtRFOR LE* 10.06993 +DELTAT 3F ■ 1 ' 0 •0 • 0 ,7 •0 " 0 2X J0 Fl 8.4, IS ' »2X,« UMX=« , 7X, I c 3F12 7X,« »CDL f F 12 7X,« FFOR 6F12 4) ,3F8. 3 I A = ■ F 8 . 4 t T I ME/ DSA' , .4,F1 CM=», S = « ,7 .4, 2F TI ME ' • , 12X .4,2F 4,18) i F3.4, 2X, 'CM PER' , 7 5X, «FS 9.4,3F 7X, 'CD X, «CMF 12.4) t9Xf •= , • RFOR 15.4) 2X, ' AMP = ' ,F9. 4,2X,'PE3=' , -l ,F3.4,2X,« \IRUN=«, 16) X, 'ALPHA' , 7X, » CCS A' , 15X, 'SIN INA» ) 12.4) =',7X,'8ETA=',7X,»REYNO=',7X F-«, 7X, 'CDFF=' ) ' ,9X, 'Fl' , 9X, 'REMF' ,9X, 'FLS' « ) 40 APPENDIX B (LIFT COEFFICIENT PROGRAM) CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C LIFT COEFFICIENTS C C H=GRAPHED WAVE HEIGHT (MM) C C E=ELEVATION CALIBRATION (FT/MM) C C A=AMPLITUDE (FT) C C D=CYLINDER DIAMETER (FT) C C X=CYLINDER LENGTH (FT) ■ C C B=BETA IDIMENSI3.NLESS PERIOD PARAMETER! C C REYNO=REYNOLD«S NUMBER C C UM=MAXIMUM WAVE VELOCITY (FT/SEC) C C T=WAVE PERIOD (SEC) C C FMU=GR^pHED FORCE AWAY FROM BOUNDARY MM) C C FMD=GRAPHED FORCE TOWARDS 3CUMDARY(MM) C C RHO=DENSITY ( SLUGS /CUB I C FT) C C CLU=LIFT COEFFICIENT AWAY FROM BOUNDARY C C CLD=LIFT COEFFICIENT TOWARDS BOUNDARY C C ATTF=FCRCE ATTENTUATICN COEFFICIENT C C ATTA=AMPLI TUDE ATTENUATION COEFFICIENT C C CN^FORCE CALIBRATION (L3/-IM) AT ATTF=133 C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC WRITE(6,200) M = ? DO 30 K = l ,M READ (5,100) EOiX|TfRHO,CN,N,J DO 20 1=1 ,N READ I5i400I H, ATTA,FMU,FMD,ATTF A=CH/2)*E*(ATTA/100J UM=2*3.1417*A/T B=UM*T/0 REYNO*UM*D/0. 0000105 CLU=( FMU*ATTF*CN )/( •5*RH0*UM*UM*0*X*1DD ) CLD=(FMD*ATTF*CN }/ i . 5*RHO*UM*JM*D*X*103 ) WRITE (6,300) J,D,B,REYNG,CLU,CLD J = J+1 20 CONTINUE 30 CONTINUE 100 FORMAT (6F8. A, 13,13) 200 FOFMATl //, 3X, » RUN U • , 4 X , • D I AM' , 5X , ■ BE T A ' , 6X , ■ RE YNO« , 5X , «CLJ« , 16X,'CLD, // ) 300 F0RMAT(//,4X,I2,6X,F6.4,3X,F5.2,hX,F8.1,3X,F6.A,2X,F6. 400 FORMAT ( F8.4,F8.^,F8.