Te she De! hale eT DAVID W. TAYLOR NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER Bethesda, Maryland 20084 DTNSRDC-82/036 wHOI| DOCUMENT COLLECTION DICTED MOTIONS OF HIGH-SPEED SWATH SHIPS IN HEAD AND FOLLOWING SEAS by Young S. Hong APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED SHIP PERFORMANCE DEPARTMENT RESEARCH AND DEVELOPMENT REPORT w —{ Wi ” ~) = = (o} - —_ 2) Le 2) Fd < a) < uu <= es ” = ake ” aa - = n a) tw Lu a Sl L ae x= ioe oO 2) 2 2 = Oo = July 1982 DTNSRDC-82/036 MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS DTNSRDC COMMANDER TECHNICAL DI ia fe OFFICER-IN-CHARGE CARDEROCK OFFICER-IN-CHARGE ANNAPOLIS SYSTEMS DEVELOPMENT DEPARTMENT AVIATION AND SURFACE EFFECTS DEPARTMENT SHIP PERFORMANCE DEPARTMENT 15 COMPUTATION, MATHEMATICS AND LOGISTICS Selmer STRUCTURES DEPARTMENT PROPULSION AND AUXILIARY SYSTEMS DEPARTMENT SHIP ACOUSTICS DEPARTMENT SHIP MATERIALS ENGINEERING DEPARTMENT CENTRAL INSTRUMENTATION DEPARTMENT RelA eeeae NDW-DTNSRDC 3960/43 (Rev. 2-80)/| UNCLASSIFIED ee AUG 31 1982 SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS 1. REPORT NUMBER 2. GOVT ACCESSION NO.| 3. RECIPIENT'S CATALOG NUMBER i DINSRDC-82/036 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED PREDICTED MOTIONS OF HIGH-SPEED SWATH SHIPS Einal 8. CONTRACT OR GRANT NUMBER(s) 7. AUTHOR(s) Young S. Hong 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS - PERFORMING ORGANIZATION NAME AND ADDRESS David W. Taylor Naval Ship Research and Development Center Bethesda, Maryland 20084 11. CONTROLLING OFFICE NAME AND ADDRESS Naval Sea Systems Command Washington, D.C. 20362 (See reverse side) } 12. REPORT DATE | July 1982 13. NUMBER OF PAGES | 52 H - MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 1S. SECURITY CLASS. (of thie report) UNCLASSIFIED - DISTRIBUTION STATEMENT (of this Report) APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED - DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES - KEY WORDS (Continue on reverse aside if necessary and identify by block number) Small-Waterplane-Area Twin-Hull (SWATH) Ship Unified Slender-Body Theory Heave Motion, Pitch Motion viaristtl ealibas of Strip Theory wae Following Seas Head Seas High Speed . ABSTRACT (Continue on reverse side if necessary and identify by block number) Unified slender-body theory has been applied to predicting the motion of small-waterplane-area twin-hull high-speed (SWATH) ships in head seas. Using this theory, numerical results indicate an improvement in accurately predicting the effects of added-mass and damping coefficients when com- pared with strip theory. However, the same improvement is not achieved in predicting the effects of exciting forces and motions. Corrections in (Continued on reverse side) DD , etal 1473 EDITION OF 1! NOV 65 IS OBSOLETE 3 sa S/ACOTDD Leta eeal UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) (Block 10) Program Element 61153N Task Area SR 0230101 Work Unit 1572-031 (Block 20 continued) the numerical computation of previous work have been implemented; previous calculations for following seas have also been repeated. The pitch motions of SWATH 6D in following seas are now in good agreement with the results of scale model experiments. However, the motion results of SWATH 6A in following seas show large un- explained peak values when the encounter frequency becomes very small. UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) IGMESWE OLY IIWEWINES G96 og 56 610 6 OF ILI It OY WAILERS =p 9 56 6 6 oo 6 8 MATHEMATICAL NOTATION AND CONSTANT TABLE OF CONTENTS SO OWL Oot ee OO OTT Carr Cn OMG scy aan ANI VAVIU NEON ZN) (UINICIES OLY MONSUIRIMIBNEE 56 5 56 6 6/656 6666 66455 6 6 IMISTINNGIE 9 6 0 0 0 6 6500 0 6 0 ADMINISTRATIVE INFORMATION ... TNAUROWUIGIENION 5 G6 6:65 60 6 50 0 6 o EQUATIONS OF MOTTON... ... . COORDINATE SYSTEMS ...... POTENTIAL OF STEADY FORWARD MOT TWO-DIMENSIONAL POTENTIAL OF OS TWO-DIMENSIONAL DIFFRACTION POT UNIFIED SLENDER-BODY THEORY . SINGULARITY OF THE KERNEL FUNCTION RESULTS AND DISCUSSION ..... SUMMARY AND CONCLUSIONS ..... ACKNOWLEDGMENT 275 3 5 5 a IRIN 5 ¢ 6 606 000000 1 - Coordinate System Fixed with with a Speed U Relative to a 2 - Sectional Coordinate System 3 - Heave Added-Mass and Damping in Head Seas at Zero Speed . 4 - Pitch Added—-Mass and Damping in Head Seas at Zero Speed . HLOIN 6: $0) 86 Von GO sd WOURON CG a glo 6.46) ects (GINMWAIHOM 56 6 0 00060000000 0 IINIEIVN ys 9 5 0 0 oO e e e e e e e e e e e e ° e e e e ° ° e e e e e e e e e ° e e e ° ° e e ° e e LIST OF FIGURES Respect to the Ship. This System Moves System Fixed in Space ....... ° ° ° e ° e e ° . . . e e e . ° ° ° ° . Coefficients of Twin Ellipsoid Coefficients of Twin Ellipsoid iii 40 41 16 17 10 jLaL LZ U3} 14 US) 16 LY 18 19 20 Heave Exciting Forces and Pitch Exciting Moments of Twin His psoudsaunsHeadmSeas wate Zeromspeeds turmeric en men ienneinte Heave and Pitch Amplitudes of Twin Ellipsoid in Head SEES US ic SoeQelig gud gisiio oo lo 6 6 60 6 6.6 Heave Added-Mass and Damping Coefficients of SWATH 6A in Head (Seaspait, 26) KNOSI (2 viele to) cect henner meena Pitch Added-Mass and Damping Coefficients of SWATH 6A in SKSEVGlWSSEIS CVE wc ISMONESI Go Vo va auo.ovalle ‘616 oavolo oo lo 6-6 Heave Exciting Forces and Pitch Exciting Moments of SAI OA alm each SES Ge ZI KNOES 6 65666566 0 0 6 oO Oo Heave and Pitch Amplitudes of SWATH 6A in Head Seas AER ZS ORMOES! 5: "sistas, ial Ciometere tat cre reek oe rel toy aiemeaaat GMa AO erm am ee Heave Added-Mass and Damping Coefficients of SWATH 6D in Headt Seas! ait 26 (Knotish "5 Sry sn” otic at Lecter ohmicin at Niclas Net tenet Pitch Added-Mass and Damping Coefficients of SWATH 6D in WSEIGl SSRN ENE NAS) MNOIES) = G /o0g+d 606) oO. ovo 6 G40 oro ohooue Heave Exciting Forces and Pitch Exciting Moments of SWATH 6D sim eaGl SEAS at 2 KOOES o)56 60000000066 60 6 Heave and Pitch Amplitudes of SWATH 6D in Head Seas Aer 2 OMKMOWS yey oaney Gomioitet ier seh vies ener kev ou et os Heave Added-Mass and Damping Coefficients of SWATH 6A 37), ROlLemaliae Seas aie ZO iWnOES 6 6 6 6 6 0 Pitch Added-Mass and Damping Coefficients of SWATH 6A abn Polllowaing S€aS Ge 20 KN@ES 56 6560050000000 Heave Exciting Forces and Pitch Exciting Moments of SWATH 6A in Following Seas at 20 Knots ...... Heave and Pitch Amplitudes of SWATH 6A in Following S08 Gi ZO KES o 6 oo 6 6 6 0 oOo Oooo Heave Added-Mass and Damping Coefficients of SWATH 6D iin WollioirAline Sees aie ZO uNMOES 5 6/5 515.6 0/016 Pitch Added-Mass and Damping Coefficients of SWATH 6D im P@iLLowaineS See aie ZO KAES 5 5616 6 056 6 05600 6 6 iv Page 18 IL) 21 22 23 24 26 2b) 28 29 30 Sal 32 33) 35 36 21 22 Heave Exciting Forces and Pitch Exciting Moments of SWATH 6D in Foltlowingyseassaty ZOBKNOESBAmanete tats) seein laibeiie «5 © Heave and Pitch Amplitudes of SWATH 6D in Following Seas ENE 7A) EAVES 5. oi Wastin oe tome MO! 6.0." dia! iG nGulicw SSING CHC ams so. oa LIST OF TABLES Values of Principal Parameters for Three Hull Models ... Comparison of Numerical Values of Pitch Determinant D for SWATH 6A and 6D Ships Using Slender-Body and Site WNEDEHES 5 56566000000 o oOo DOO Page 38 15 39 MATHEMATICAL NOTATION AND CONSTANTS A Amplitude of incoming wave AG Added-mass coefficients (j, k=3 for heave; j, k=5 for pitch) Bix Damping coefficients (j, k=3 for heave; j, k=5 for pitch) c. Coefficient for longitudinal interaction Ooi Hydrostatic coefficients D Displacement of SWATH ship D Pitch determinant F Heave exciting force (i=3) or pitch exciting moment (i=5) G Green function g Gravitational acceleration H, Heave (i=3) and pitch (i=5) amplitudes I Mass moment of inertia i= Eine Imaginary unit K = Galle Wave number of frequency of encounter (also, K, = w/e is the incoming wave number) L Characteristic length of SWATH ship M Mass of SWATH ship m(m, ,m m3) Normal vector due to the steady forward potential 2 m = w/e A constant may nn) Unit vector directed normal into the fluid Ol F537 52) 8) OCS 5S) Coordinate systems with origin at 0 P Pressure y = 0.57721... ae (Gitte Anas an) Three-dimensional source strength Area of immersed cross section Steady forward velocity vector Spatial coordinates Multipole strength Heading angle of incoming wave with respect to the x axis (8 0° is the following wave, and 8 = 180° is the head wave) Euler's constant Length of incoming wave Spatial coordinates Complex amplitude of ship motion Two-dimensional source strength Total velocity potential Velocity potential due to steady forward motion (i=l) or to unsteady oscillations (i=2) Two-dimensional velocity potential for heave (j=3) or pitch (j=5) due to harmonic motion Two-dimensional velocity potential for heave (j=3) or pitch (j=5) due to steady forward motion Encounter frequency Frequency of incoming wave Incoming wave potential Velocity potential due to motion of the ship with unit amplitude in each of six degrees of freedom Diffraction potential vii GM SWATH ABBREVIATIONS AND UNITS OF MEASUREMENT Metacentric height Kilogram Meter Principal value Second Small-waterplane-area twin-hull Waleiete ABSTRACT Unified slender-body theory has been applied to pre- dicting the motion of small-waterplane-area twin-hull high- speed (SWATH) ships in head seas. Using this theory, numerical results indicate an improvement in accurately predicting the effects of added-mass and damping coeffi- cients when compared with strip theory. However, the same improvement is not achieved in predicting the effects of exciting forces and motions. Corrections in the numerical computation of previous work have been implemented; previous calculations for following seas have also been repeated. The pitch motions of SWATH 6D in following seas are now in good agreement with the results of scale model experiments. However, the motion results of SWATH 6A in following seas show large unexplained peak values when the encounter frequency becomes very small. ADMINISTRATIVE INFORMATION This work was performed under the General Hydromechanics Research Program administered by the DINSRDC Ship Performance Department and was authorized by the Naval Sea Systems Command, Hull Research and Technology Office. Funding was pro- vided under Program Element 61153N, Task Area SR 0230101, and Work Unit 1572-031. INTRODUCTION An analytical method for predicting the motions of small-waterplane-area twin- hull (SWATH) ships has been developed using the strip theory developed earlier by ieee The method has been further improved for the computation of heave and pitch motions in head seas by adding the surge effect on the pitch exciting moment and by correcting the viscous-damping terms developed by the present emidinore., The numerical results correlate well with experimental results for a moderate speed range. The original strip theory has several limitations in application. First, when the encounter frequency is very small, strip theory cannot be properly applied. The small encounter frequency occurs when a ship proceeds with a fairly high speed in following seas. Second, the strip theory does not predict the motion correctly when a ship moves at high speed in head seas. The fundamental assumption of the strip theory is that the encounter frequency should be far larger than the longitudinal gradient of the body surface multiplied by the forward peed.” *A complete listing of references is given on page 41. Neonat recently applied slender—body theory to the problem of predicting ship motion. He has developed a unified slender-body theory and applied it to the compu- tation of added-mass and damping coefficients. Newman and Solemn: have computed added-mass and damping coefficients for surface ships, and their numerical results agree well with the three-dimensional results for a surface ship when the forward speed is zero. For the case of nonzero forward speed, the agreement is not as good as for zero speed. The author has previously applied the unified slender-body theory to the compu- tation of motions of SWATH ships in following seas” This resulted in an improve- ment in predicting the effects of added-mass and damping coefficients, but a rather inconsistent improvement in motion seatecione. Furthermore, the numerical handling of the hydrodynamic singularity was deficient; this singularity problem has been corrected. Numerical computations in following seas are repeated in this report. The unified slender-body theory is here applied to improve the prediction of motion of high-speed SWATH ships in head seas. Two-dimensional theory is applied in the inner region and three-dimensional theory is applied in the outer region of the hydrodynamic flow domain. The matching process is taken in an intermediate region, and the correction terms are added to the results of the strip theory. The numerical results are compared with those of strip theory and experiment. An improvement in predicting hydrodynamic coefficients has been achieved, but a similar improvement has not been obtained in the motion results. EQUATIONS OF MOTION COORDINATE SYSTEMS Two coordinate systems are defined: the first 0 559g Zo) is fixed in space, and the second O(x,y,z) is fixed with respect to the ship which moves with a forward speed U along the positive 00% axis. The Oz axis is directed vertically upward, and the 0x axis is positive in the direction of the ship's forward velocity (Figure 1). The Oxy-plane is in the plane of the undisturbed free surface. The two coordinate systems coincide when the ship is at rest at time zero. Using the slender-body theory, the length of the ship is assumed to be far larger than the beam or the draft. We separate the fluid domain (y,z) into two regions: the outer region where (y,z) is of the order of the length of the ship, and the inner region where (y,z) is of the order of the beam or draft. In the outer Figure 1 - Coordinate System Fixed with Respect to the Ship. This System Moves with a Speed U Relative to a System Fixed in Space region, the three-dimensional Laplace equation is solved under a three-dimensional free-surface condition and an imposed radiation condition. The inner solution is governed by the two-dimensional Laplace equation with two-dimensional boundary conditions. These two solutions are matched in the overlapping domain or inter- mediate region to determine the solution of the unified slender-body theory. The governing equations of velocity potentials, boundary conditions, and their solutions will now be given. The details of the derivations are given in References 4 and 6. POTENTIAL OF STEADY FORWARD MOTION The velocity potential of steady forward motion can be expressed as >, (x,y, 2) So W so (S992) (1) where >, (&y»2) satisfies Laplace's equation and the free surface and body boundary conditions given by ) + o + > = z <0 (Dy Il (=) N Il S) (*/e) o., + 4 (3) OZ (Vo +0.) -n=0 on the body surface (4) = Here We (-U,0,0) is the steady forward velocity vector, and n is the unit normal vector of the body directed into the fluid domain. The solution of Equations (2)-(4) is given by atteveike as - 2 Pen a d) 5 log r + 5 log a 0 2 (5) where a == S' = the strength of the source S = area of the immersed cross section S' = dS/dx ale /p2. = y) 2 r = [(y-n) +(z-Z) J and 1/2 = 2 2 E) = [(y-n)“#(2+2)°] Here, (y,z) is the point where the potential is solved and (n,Z) is the source point. TWO-DIMENSIONAL POTENTIAL OF OSCILLATION The two-dimensional velocity potential due to pure heave oscillation is given by (S)) in The first potential satisfies the following conditions 0°, 9°, SOREL when z < 0 @) oy Oz 205 ne Pega evel © on z = 0 (8) and 36, Bae cae iw n, on the body surface (9) > n, is the heave component of the unit vector n. The second potential of Equation (6) 3 satisfies the following conditions: 3b, 9°, + = 0 when z < 0 (10) 2 2 oy OZ 3b Dats 3 Wy) = ¥. ila ta 5 = (0) on z = 0 (11) and 36. a 3 on the body surface (12) Here Mm. is given by ap 3 p) iL mz = - (m) 35 My oS) Es CS There are two ways to solve for the potential 3: one is the multipole ex- pansion method given by eee.” and the other is the close-fit method by eel” The solution of > using the former method, is given by Be (0) cos(2m0) K cos[(2m-1)0] 3 = 93 Gop 2 on ( i pre 2m-1 oe) where 04 is the two-dimensional source strength located at the coordinate origin. The terms under the summation are the higher order multipoles which form the wave- (0) free potentials with multipole strengths on Gop function due to a source at the origin and is given by Wehausen and adizome 2 fi kz G Oy Lp BOSS) ay -i ae cos (Ky) K-k 0 where K = aie and PV in Equation (15) denotes a principal value integral. solution of >, by the Frank close-fit method is is the two-dimensional Green as (15) The (16) where o is the source strength distributed on the ship's contour and G is a two- 2D dimensional Green function due to a unit source on the ship's contour. Cone case, is slightly different from Equation (15) and is given by iL : : : A ODE Re oF log (y+iz-n-ic)—-log (y+iz-nt+iZ) Qa i co -ik(yt+iz-n+iZ) ‘ A 5 epy |p Sa ae Ree eae) K-k 0) + Re a log (y+izt+m-iT)-log (ytiz+nt+iz) eo) ik ( Ji ate +iT) -ik(y+iztnt+i iy é ap ee ev | oe en ak |) > Sh Rete ik(ytizmtit) ) K-k 0 in this (17) Here (y,z) is the point where the potential is sought and (n,t) is a source point (Figure 2). (7,6) @ (y,z) Figure 2 — Sectional Coordinate System 2D If (m,f) goes to the origin, the value of G in Equation (17) becomes twice as large as that of Equation (15). By similar analogy, the solution of 3 is given by (oe) a. ak (0) : cos (2m6) K cos[(2m-1) 6] 3 = 3 Gop D>, 2, ( 2m Yea 2nd Gs) m=1 2D is given by Equation (15) and Oo is the multipole strength. Both the or in Equation (14) and the 0 in Equation (16) can be solved with application of the body boundary condition [Equation (9)]; o (12). 3 in Equation (18) is solved using Equation The two-dimensional velocity potential due to pitch oscillation can be obtained by multiplying Equation (6) by -x to give TWO-DIMENSIONAL DIFFRACTION POTENTIAL The two-dimensional diffraction potential satisfies the conditions ————_- + ———_ = 0 when z < 0 (20) 2 2 dy Oz . = Ore =0 on 2 =. 0 (21) and 2m Saya ae ia ar on the body surface (22) Here Q, is the potential of the incoming wave given by Let : a : Q, a, exp (K zt+ik x cos B ik y sin 8) (23) where A = the amplitude of the incoming wave 8 = the heading angle of the incoming wave (8 = 180° is for a head sea, and 8 = 0° is for a following sea) o 2 mS Tae /g The solution of OS satisfying Equations (20)-(22) is (Ss) is €, = || G5 Gp dg (24) Cc where 0, is the complex source strength at the ship's contour c, and Gon is given by d Equation (17). UNIFIED SLENDER-BODY THEORY The velocity potential due to oscillation is given by Newman as eS) =o bin G, =, + ,(x) (6,48,) (3=3,5) (25) where 9, is the conjugate of Oh The coefficient Oe (es) is expressed by the integral { q, (6) £(x-€) d& L where q is the three-dimensional source strength distributed along the Ox axis. (26) This source strength is the solution of the following Fredholm integral equation of the second kind: q,(x) - 4 Jd | q,(6) £(x-£) db = 0, +9, 1 I The kernel function f(x) is given by £(x) = £n(2K) § (x) + + G(x) - in 8 (x) 2|x| The Green function G4 in this equation is 1 aL G3 (x,y,256,n, 0) ma CCS 575 25 Solo) == 1g al where 2S [Gye ee and 1/2 ry = [Gx-8)7#(y-n) 74 (240) 71 and G is the three-dimensional Green eumention-- given by (27) (28) (29) ° (z+t)u-i wu EXO S WHR sila) | au | a (30) il es (Vm u cos 6+VK) Here, w= (x-&) cos 6 + (y-n) sin 8, and m = eo /a. By substituting Equation (30) into Equation (29), G,(x,0,0,0,0, 0) = G, (x) is given by oo T 2 2 G4 (x) nd ke auf (/K+/m u cos 0) wexptet x ; cos @) de (31) Lee 0 u — (/K+/m u cos 6) The diffraction potential of the unified slender-body theory is given by e, =) + ci (x) (b49_) (32) 7 7 7 Simas) where oe is the symmetric function of ger ¢, is the conjugate of >5> and C_ (x) is 70) 2 4 | a, (E) £(x-&) dé (33) 27105 L Here, qd, is the solution of q7(x) - i i q,(E) £8) dé = - 0, (34) De in Equation (32) can be expressed as eG) (35) 10 in d Equation (24) is distributed on the ship's contour (Figure 2), and O05 is a concen- is given by Equation (15). The two-dimensional source strength oO (0) where Gon trated source strength at the origin. In other words, O4 is the solution of the Frank close-fit method and oO, is the solution of the multipole expansion method. 7 The relation between 0, and O7 can be expressed by d i 2 | Og as cos Kn d& (36) c The same relation between 0, and oO is given by (37) Q Il i) Q w) Nei Q ° n 3 au = The total unsteady potential 5 for heave and pitch motion is given by Equations (28047 (25) ee andi@2)% o, =P) +O, + 3 3 + Ee O, (38) where b3 is the heave amplitude and Es is the pitch amplitude. In the frame of the unified slender-—body theory, only heave and pitch motions are considered. The pressure p in the fluid is given by Bernoulli's equation p = p(iwp,-Vo,-Vo,) @ 0* - pgs (39) By substituting Equation (38) into Equation (39) and integrating on the body sur- face, the hydrodynamic forces are expressed by —iwt (j=3,5) (40) E EMBs [ i (ion, +m, ) (PtHP +E +E.) dS e S where ne and m, are given by 11 (41) and Me) Sa (42) The first two potential terms in parentheses in Equation (40) represent the exciting forces, and the other two potential terms are due to the added-mass and damping forces. Complete lists of added-mass and damping forces are given in Table 1 of Reference 6. Further, the detailed derivations of all equations not derived in this report are given in References 4 and 6. SINGULARITY OF THE KERNEL FUNCTION The procedure employed here to compute the hydrodynamic forces and the motion begins by solving for 0, and o, in Equations (14) and (18) through the use of strip theory. The kernel function f(x) of Equation (28) is computed and retained for later use. Next, the three-dimensional source strength q, Ox) is computed by solving the Fredholm Integral Equation (27). Once q, Ox) is obtained, the sectional inter- action coefficient 2s) may be computed through Equation (26). The most difficult and time-consuming part of numerical computation is the evaluation of the kernel function Equation (28) and the three-dimensional source strength Equation (27). In a previous pabielente tena the present author presented a numerical method for the evaluation of the kernel function, Equation (28), through a small-order procedure in order to avoid the 1/|x|-type singularity. However, this method is not generally applicable to the SWATH ship motion problem since the so- lution is not uniform, i.e., the predicted response depends upon the size of the small-order parameter. In order to avoid this singularity, Equation (27) may be integrated by parts to yield 6. Lyf 2 nn q. (x) - if A 4) }-q. (E)F@-é) +fat @)EGeE)ae = 6. +6, G=355)) Gs) J 2n6, J =H J J where lg (C9) is the derivative of q, Ox) with respect to x; the function F(x) is the integral of Equation (28) with respect to x: IL x F(x) = $ Sgn x(2n(2K|x|)-inty) + vf G,() dé (44) 0 Here, Sgn x is defined as 1 when x < 0, -1 when x > 0, and undefined at x = 0, and y, Euler's constant, is 0.57721.... The integrated term in Equation (43) can be expressed as E=x-€ E=L/2 =4; (EG) (45) Lim a (€) F (x-€) E=x+e €>0 E=-L/2 Since a5 (Sti2) and q,; @/2) are assumed to be zero, Equation (45) is further reduced to im {-q. (x-€)F(e)+q. (xte)F(-e) } (46) €>0 J J As q, () is a continuous function along the ship's length, Equation (46) can be written as - Qaim {q, (x) [F(e)-F(-e) ]} (47) e>0 J F(€) has a singularity as € + 0. This singularity occurs because of the limits of the integral and can be omitted with the interpretation that the "finite part" defined by Hadamard obeys many of the ordinary rules of mieSeeatsfign, Po By dis- carding the integrated term, the integral equation for q, Ox) becomes oto, Q qo) [aw F(x-E) dE = 0, +9, (j=3,5) (48), J 2n0 J J L The numerical solution of Equation (48) is accomplished by iterating. One dis- advantage of this method is that the numerical method does not converge when the 3) forward speed is high. To overcome this convergence problem, it is instructive to compare Equations (27) and (48). Equation (27) can be solved by the matrix in- version or Gauss-Jordan method as long as the singularity problem in f(x) is solved. By defining a new function, h(x-€) = £(x-&) EF x (49) and h(x-€) = £(«-€) + [F(e)-F(-e) ] E=x the integral equation o.t0, HY q.(x) - if -Lt lao h(x-€) dE =o, +0, (j=3,5) (50) al 2no J J} J j Ib, has the same solution as Equation (48). The advantage of solving Equation (50) is that the matrix inversion method can be applied to it and has been found to yield solutions for all speed ranges. In the low-speed range, the iteration method works satisfactorily and, in this case, the solution of Equation (50) compares well with that of Equation (48). RESULTS AND DISCUSSION In order to validate the numerical results of the present unified slender-—body theory, three hull forms have been selected: a twin ellipsoid hull, the SWATH 6A hull, and the SWATH 6D hull form. The principal parameter values for these three hull forms are given in Table 1. The twin ellipsoid is a mathematically exact hull, and its performance at zero speed may be compared with the predictions of the strip theory and the three-dimensional theory. The computations for SWATH 6A and SWATH 6D have been carried out for speeds of 28 knots in head seas and 20 knots in following seas. These results are compared with those of strip theory and experiments. Numerical computations for the twin ellipsoid have been carried out at zero speed in head seas and are shown in Figures 3-6. In this case the three-dimensional 14 TABLE 1 —- VALUES OF PRINCIPAL PARAMETERS FOR THREE HULL MODELS* Hull Form SEEEMSESE (gine UAE, SWATH 6A | SWATH 6D Twin Ellipsoid Displacement (long ton) 2815 Characteristic length L (m) U3 ok Length of waterline (m) 68.0 Length of main hull (m) V352 Beam of each hull at waterline (m) 2D Hull spacing between centerline (m) 22.9 Draft at midship (m) Jo dL Maximum diameter of main hull (m) 4.6 Longitudinal center of gravity aft of main hull nose (m) Bo il 3.048 Vertical center of gravity (m) Hod 0.381 Longitudinal GM (m) 26.4 6.096 Radius of gyration for pitch (m) 19)0 iL, 2Ue2 Waterplane area (m) DiI, 2 7.297 Length of strut (m) 25.8/strut Strut gap (m) 16.4 Maximum strut thickness (m) 35 iL *Dimensions are full-scale. theory is exact. The added-mass coefficients Ass> i = 3, 5, computed with the slender-body theory compare better with the three-dimensional results than with those of the strip theory. This is especially true at low frequencies. At high frequencies the results of the strip theory and the slender-body theory are almost the same, as in Reference 4 in which the kernel functions of Equation (27) vanish. The abrupt changes in the curves of added-mass and damping coefficients are due to the hydrodynamic interaction of the two hulls when the (nondimensional) frequency is approximately 2.5. The exciting forces F, and moments F are shown in Figure 5. The slender—body theory predictions for pitch moments Fe are found to be nearly the same as those of the strip theory, but higher than those of the three-dimensional theory. The 15) q f SNMP Ure has SLENDER-BODY THEORY Wales Oils = wales 8ix%¢ a2 {0) i ot A33/M B33*xLxxO.5/MxGxxO0. 5 ie 258 iwi wRSee 4.0 NON-DIM. FREQUENCY (3) i@) Figure 3 - Heave Added-Mass and Damping Coefficients of Twin Ellipsoid in Head Seas at Zero Speed 16 b= Foie in BOR o = SLENDER-BODY THEORY + = Tee Olt in BO key ASS/MxL»*x*2 BS55/MxLx (GxL) xO. 5 i=) (>) IEC). 2.0 Sue 4.0 NCN-DIM. FREQUENCY Figure 4 - Pitch Added-Mass and Damping Coefficients of Twin Ellipsoid in Head Seas at Zero Speed / e= SUP Waka y O= SLENDER-BODY THEORY ca AMAING eS Oie eae lOlK 1=) (>) ~ 20.5 P3xL/DxA (5,0) O58 wt FS/D0*A 20 cD choke 2.0 3.9 4.0 S.C WAVE-LENGTH RATIO Figure 5 - Heave Exciting Forces and Pitch Exciting Moments of Twin Ellipsoid in Head Seas at Zero Speed 18 Solin ORY O~=- SLENDER-BODY THEORY eS WINE GM. Wale Gin wd h3/A 3.0 0.8 2.0 HS*L/ (2*A) fa) ss i ee 2.0 3.0 4.0 WAVE-LENGTH RATIO uo oO Figure 6 —- Heave and Pitch Amplitudes of Twin Ellipsoid in Head Seas at Zero Speed 19 slender-body heave amplitudes shown in Figure 6 are greater than those of both the strip theory and the three-dimensional theory. The pitch amplitudes agree quite well even though the pitch moments are computed differently. For the twin ellipsoi- dal hulls, the damping coefficients are almost the same, the added-mass coefficients are computed as slightly less, and the exciting moments are higher than those pre- dicted by the three-dimensional theory. In the case of pitch amplitude, it appears that the underprediction of the added-mass coefficients and the overprediction for the exciting forces cancel out in the motion results. In a previous weno: this author presented an explanation for the rather large discrepancies in the added-mass coefficients computed through Equation (37) employing the singularity distribution o Since that publication, the author has computed the singularity distribution z, directly through the multipole expansion method for the semicircular section and has compared this result with Equation (37); the two computations agree quite well. Unfortunately, for the twin-hull section, the direct computation or through Equation (14) is apparently impossible. Therefore, 04 has been computed for all models through Equation (37). The large discrepancies mentioned above have been identified as the result of a different numerical handling of the singularities of the kernel function. As shown later, the prediction of the added-mass coefficients in following seas has the same tendency--lower at low fre- quencies and higher at high frequencies than those of the strip theory. The results for SWATH 6A at 28 knots in head seas are shown in Figures 7-10p The added-mass and damping coefficients do not show any difference when compared with those of the strip theory. While our interest range is in wavelength ratios between 1 and 6, the corresponding nondimensional frequency number varies from 2 to 6. These frequency numbers are fairly high, and, therefore, the slender-body theory does not improve the prediction of the added-mass and damping coefficients. However, the exciting forces and moments show large discrepancies when compared with those computed through theory. This is due to the fact that whereas the two-dimensional source strength OF for the diffraction potential has the factor exp (iK)x cos 8) in Equation (23), O5 in Equation (36) varies harmonically along the ship's length. ‘This sinusoidal change of OF affects the solution of q7 in Equation (34) and also the so- lution change of C_ in Equation (33). The oscillatory behavior of the exciting 7 forces and moments is caused by this solution for Cl. The heave and pitch ampli- tudes show some discrepancies between experiment and the strip theory. The mixed 20 y= SURG eine ORM oO = SLENDER-BODY THEORY Cl AS3/M So(8 OS ] B335*L»**xC.5/Mx6xx0.5 ie) ons 2.0 4.0 6.C 8.0 NGN-DIM. FREQUENCY Figure 7 - Heave Added-Mass and Damping Coefficients of SWATH 6A in Head Seas at 28 Knots 21 By SURE MIE ON OF SEE NOB = SORNe ina ONY. A55/MxL¥*2 B55/MxL» (GxL) **O.5 2. a5) 6.0 8.C NGN-DIM. FREQUENCY (2) (@p) Figure 8 — Pitch Added-Mass and Damping Coefficients of SWATH 6A in Head Seas at 28 Knots 22 BS SUM Wisin. O=" SEENDER SOU Mh TheORY F3xxL/DxA FS/DxA 3.5 eS Bale) dis 6.0 WAVE-LENGTH RATIO. Figure 9 - Heave Exciting Forces and Pitch Exciting Moments of SWATH 6A in Head Seas at 28 Knots 3! SUE Minies Chins SEENDERS BODY THEORY EXPERIMENT MIXED METHOD .) f x + O io H3/A HSxL/ (2%A) 3.95 aR 3.6 a8) 6.0 WAVE-LENGTH RATIO Figure 10 - Heave and Pitch Amplitudes of SWATH 6A in Head Seas at 28 Knots 24 method is the result of the combination of the strip theory and slender—body theory. In the mixed method the added-mass and damping coefficients are computed through slender-body theory, while the exciting forces and moments are computed through the strip theory. The results obtained with the mixed method are generally the same as those of the strip theory. This means that the effect on motion amplitude of the added-mass and damping coefficients computed with the slender-body theory is negligible in this example and that the oscillatory results in the pitch motion are caused by the pitch moments. The results of SWATH 6D at 28 knots in head seas are shown in Figures 11-14. As previously noted for the SWATH 6A, the added-mass and damping coefficients of the slender-body theory do not show any difference from those of the strip theory, except in the low-frequency range where the slender-body theory values for the hydrodynamic coefficients are lower than those of the strip theory. In the (nondimensional) fre- quency range between 1.5 and 4.0, the slender—body heave forces F, show the same 3} tendency as those of the strip theory, and the pitch moments F. display oscillatory behavior at low wavelength ratios (Figure 13). This en is also observed in the results of the SWATH 6A. In the previous wenoee,” the results for SWATH 6A and 6D at 20 knots in follow- ing seas have been compared with those of the strip theory and experiment. Since the numerical procedure has been corrected in the computation of the kernel function, the computations of the following seas have been repeated in this report. These results for SWATH 6A are plotted in Figures 15-18. When the nondimensionalized frequency is 0.56, there exists a discontinuity in the curves of the hydrodynamic coefficients. As mentioned in the previous nemo,” when wU/g is 0.25, G3 of Equation (31) is singular and the numerical results become unstable. The damping coefficients are nearly the same as those obtained through strip theory, except where this dis- continuity occurs. Compared with the previous results, the damping coefficients do not show any change; however, the added-mass coefficients are quite different. While the previous results are uniformly greater than those of the strip theory, the present ones are less at low frequency and greater at high frequency. The results of the exciting forces and moments computed by the slender-body theory and by the strip theory are similar, except there is a peak value in the pitch moment computed by the slender-body theory for a wavelength ratio of approximately 2.0. The motion results are also similar to those computed by strip theory, except at the peak value for a wavelength ratio of approximately 1.5. When the wavelength ratio is 1.3, the 25 Peo SURUP inland Qe SUENDER ASIN Url ini A33/M B33xL**0.5/MxGxxO.5 0.0 1.0 2.0 §.0 4.0 NON-DIM. FREQUENCY Figure 11 - Heave Added-Mass and Damping Coefficients of SWATH 6D in Head Seas at 28 Knots 26 eS e nlel Ohne OFS SEENDERS BOD THEORY ASS/MxL»*x2 0. 100 0.125 0.075 Oa5 05050 ian) i) q BSS/MxL» (6*L) xx0.5 oO (ee) (>) me 2.0 5a 4.0 GON-DIM. FREQUENCY Figure 12 - Pitch Added-Mass and Damping Coefficients of SWATH 6D in Head Seas at 28 Knots 27 Bonini: mime ein o = SLENDER-BODY THEORY F3»xL/DxA FS/D0*A (@) oO NaS) S40) | 4 6.0 WAVE-LENGTH RATIO Figure 13 - Heave Exciting Forces and Pitch Exciting Moments of SWATH 6D in Head Seas at 28 Knots 28 BES SINE alee Oo = SLENDER-BODY THEORY Hae iain iene Seo ile watOi ae Yet 3.6 4.5 6.C WAVE-LENGTH RATIO Figure 14 —- Heave and Pitch Amplitudes of SWATH 6D in Head Seas at 28 Knots Ze, o=- STRIP THEORY o = SLENDER-BODY THEORY A33/M 1 2.00 0.50 whe) 1 B33*xL*x0O.5/MxGx0O. 5 O25 0.50 ON7S 1.00 NGN-DIM. FREQUENCY Nn Nn ye. IU Figure 15 - Heave Added-Mass and Damping Coefficients of SWATH 6A in Following Seas at 20 Knots 30 B= SIRIEY MieORY o = SLENDER-BODY THEORY 2.0 a0 8} O il, ASS/MxL»*%*2 6 0.6 0.0 O BSS/MxL» (GL) **0.5 0.4 (@) Oo Oo O 0.2 0.00 0. 25 0.50 0.75 1.00 NON-DIM. FREQUENCY Figure 16 - Pitch Added-Mass and Damping Coefficients of SWATH 6A in Following Seas at 20 Knots o = STRIP THEORY o = SLENDER-BODY THEORY 3.0 2.0 F3xL/D»A 1.5 0.0 1.0 FS/D*A 0.5 0.09 1.5 | 3.0 4.5 6.0 WAVE-LENGTH RATIC (qs) (a>) n Figure 17 - Heave Exciting Forces and Pitch Exciting Moments of SWATH 6A in Following Seas at 20 Knots 32 Ge Sik iPaslheORY 2 -SSEENDERS SOD IhEORY. PS (Pele MleiNl x = MIXED METHOD a0) 2.0 H3/A H5x*L/ (2A) 6.0 9.0 0.9 SAO 0.0 0.0 1S 3.0 4.5 WAVE-LENGTH RATIO Figure 18 - Heave and Pitch Amplitudes of SWATH 6A in Following Seas at 20 Knots 3)3) 6.0 nondimensionalized frequency number becomes zero and the computed hydrodynamic coefficients are far smaller than those predicted by strip theory. Therefore, the motion amplitudes are magnified at this wavelength ratio. If we compare the results of the strip theory and the mixed method, this magnification effect is easily seen in the motion results. In the mixed method, the hydrodynamic coefficients are computed by the slender-body theory, and the exciting forces and moments are computed by strip theory. The results for SWATH 6D at 20 knots in following seas are plotted in Figures 19-22. When the nondimensionalized frequency number is 0.65 (wU/g=0.25), there is a discontinuity in the curve of the hydrodynamic coefficients similar to that observed in the calculations for SWATH 6A. These coefficients are found to be smaller at low frequencies and larger at high frequencies than those computed through strip theory. While the heave exciting forces are overpredicted when compared with those of the strip theory and the experiments, the pitch moments are underpredicted for both results. All computed heave motions are in close agreement at low wavelength ratios, but for high wavelength ratios, discrepancies between experiment and computations are quite large. Pitch motions are in good agreement with the experimental results, except near a wavelength ratio of 1.0 where the encounter frequency becomes zero. The pitch moments computed by the strip theory are less than those obtained experi- mentally, but the pitch motions are larger. Generally, the motion results are affected by the inertia, hydrodynamic, and hydrostatic forces. Therefore, it is very difficult to determine the exact cause of these discrepancies. Although there is a peak response shown in the motion of SWATH 6A in Figure 18, SWATH 6D (Figure 22) does not show such a peak value. This difference can be analyzed through the motion equations. If the coupling effect between heave and pitch motion is neglected, the solution of the pitch motion becomes Coal I 2 — - {F[C..- (tA, a ]-F 4B, .}/D i 2 = ee - {F,[C,.-(1+A,5)@ ]+F B55) /D where 2 D) 2 eo" I) se (@B.«) 34 R=vouiwgk Minavny o = SLENDER-BODY THEORY A33/M B33»*xL»*xO.5/MxGxx0.5 6.068 0.25 0.50 0.75 1.090 NON-DIM. FREQUENCY Figure 19 - Heave Added-Mass and Damping Coefficients of SWATH 6D in Following Seas at 20 Knots 35 B= SURUE ThieORY O-'SeENOEk-SODN Theory ASS/MxL»¥x*2 /MxL»* (GL) x*xO.5 B55 () (@) (@) C. 25 6.50 0.75 wee NON-DIM. FREQUENCY Figure 20 - Pitch Added-Mass and Damping Coefficients of SWATH 6D in Following Seas at 20 Knots 36 Ge SURE eine he O= SEBNDERS BOD ThEOkhY + = EXPERIMENT F3xL/D»A FS/D*A 3.9 Ns 8 55 6.0 WAVE-LENGTH RATIO Figure 21 — Heave Exciting Forces and Pitch Exciting Moments of SWATH 6D in Following Seas at 20 Knots 37 | k ON a ewOhy SLENDER-BODY THEORY EXPERIMENT MIXED METHOD x se © on} H3/A HS*L/ (2xA) 3.0 1S 3.0 4.5 6.0 WAVE-LENGTH RATIO Figure 22 - Heave and Pitch Amplitudes of SWATH 6D in Following Seas at 20 Knots 38 The subscript R defines the real part and I the imaginary part of the complex func- tions. The peak value of SWATH 6A occurs near the point where the frequency becomes zero. Near zero, the pitch exciting moments for both SWATH ships are almost the same for both the strip theory and the slender-body theory. In this case, the major factors which determine the pitch amplitudes are C » and B The peak value a 55° 455 55° occurs when the pitch determinant D becomes small. Numerical values for these co- efficients are presented in Table 2 where it is also shown that the value of D differs by a factor of six for SWATH 6A between the strip theory and slender—body theory but less than a factor of two for SWATH 6D. The peak value of SWATH 6A is therefore caused by this change in the factor D. TABLE 2 - COMPARISON OF NUMERICAL VALUES OF PITCH DETERMINANT D FOR SWATH 6A AND 6D SHIPS USING SLENDER-BODY AND STRIP THEORIES Quantities Needed to Calculate D Slender- Strip Slender- Strip (and Units) Body Theory Theory Body Theory Theory L (m) 54.3 73.1 6 6 Me Cgesscee ny) 0.2673.10 0.2917.10 Co. (kg-m) 0.3483.10° 0.9664.10°" i) (Misee) 0.0212 0.0183 A/L 11925 0.96 0.7825.10 1.0521.10° 9 IL Gpomecen) 9 ee Ge—mesae") 66.11.10 92.67.10 573.23.10 644.33.10 9 9 Bee (kg-m-sec) 0.1599.10 0.5994.10 0.8222.10 1.58.6.10 -0.0685.10 -0.9537.10 -1.1918.10° 8 0.1504-10° 0.2894.10 0.9322.107° WS OuneOn 0.0246.10° 0), WA7AL IL) 0.3186.10"" 2.0847.10"" i) (kg-m) 5 D (eecem) 59 SUMMARY AND CONCLUSIONS This report presents the application of the unified slender-body theory to predict the motion of SWATH ships in head seas. At zero speed, the computed results for a twin ellipsoid show overall improvement in the hydrodynamic forces and motions compared with three-dimensional theory. For SWATH ships with high forward speed in head seas and in following seas, there is some improvement in the added-mass and damping coefficients compared with two-dimensional results, but for the exciting forces and motions the same improvement is not expected. In following seas, the motions of SWATH 6A show unexpected large peak values at small encounter frequencies. The motion results of SWATH 6D agree very well with the experiments. From the present study, the following conclusions have been drawn: 1. The slender-body theory improves the two-dimensional predictions of added- mass and damping coefficients. The same improvement has been shown for surface ships in Reference 5. 2. The application of the slender—body theory for the prediction of exciting forces has not been entirely successful. Further investigation is necessary in this area. 3. Except for the pitch motion of SWATH 6D in following seas, the computed motions have not been significantly improved by the application of the slender-body theory. When the sectional interaction terms are solved with the sectional source strength of the slender-body theory, they are strongly influenced by strip theory in head seas. Further investigation of these interactions is therefore warranted and is necessary to the final development of the slender-body theory. ACKNOWLEDGMENT The author gratefully acknowledges the support of Ms. M.D. Ochi, Dr. D.D. Moran, and Mr. V. Monacella. 40 REFERENCES 1. Lee, C.M., "Theoretical Prediction of Motion of Small-Waterplane-Area, Twin-Hull (SWATH) Ships in Waves,'' DTNSRDC Report 76-0046 (Dec 1976). 2. Hong, Y.S., “Improvements in the Prediction of Heave and Pitch Motion for SWATH Ships," DTNSRDC Dept. Rept. SPD-0928-02 (Apr 1980). 3. Salvesen, N., E.O. Tuck and 0. Faltinsen, "Ship Motion and Seas Loads," Trans. SNAME, Vol. 78, pp. 250-287 (1970). 4. Newman, J.N., "The Theory of Ship Motion," Adv. in Appl. Mech., Vol. 18, pp. 221-283 (1978). 5. Newman, J.N. and P. Sclavounos, "The Unified Theory of Ship Motion," Proc. Thirteenth Symposium on Naval Hydrodynamics, Tokyo, Japan, p. 373 (Oct 1980). 6. Hong, Y.S., "Prediction of Motions of SWATH Ships in Following Seas," DINSRDC Report 81/039 (Nov 1981). 7. Tuck, E.0., "The Steady Motion of a Slender Ship," Ph.D. thesis, University of Cambridge, Cambridge, England (1963). Se Unrselile er. On the Heaving Motion of a Circular Cylinder on the Surface of a Fluid," J. Mech. Appl. Math., Vol. 2, pp. 213-218 (1949). 9. Frank, W., “Oscillation of Cylinder in or below the Free Surface of Deep Fluids,'' DINSRDC Report 2375 (1967); avail. DINSRDC. ' 10. Wehausen, J.V. and E.V. Laitone, "Surface Waves," in Handbuch der Phys., Vol. 9, pp. 446-778 (1960). 11. Lighthill, M.J., "Fourier Analysis and Generalized Functions," The University Press, Cambridge, England (1970). 12. Hadamard, J., "Lectures on Cauchy's Problem in Linear Partial Differential Equations," Yale University Press (1932). q 41 Cou Nena gk sgh athens NAC emma T a a ‘ide spnitegthinandaiane’ ay aa ont ta predict hive HHO LALON (ME CTA) GR, iendeuher init “ hia if Day 4 rit yy Wd valde bv’ eaneue How, MOR ie send hang te ye hapa MAY, in, how haan sdomp wea) wit tht idaaa tate OER) wk ah cones it had wean add te tide seh ana Rime i’ o wie! ai i oo) | if este a2 eens, Oe ee damp " epee # rage PE *, ee : cae t, kl or ra i vals mm 1 wteay ASG wok bina 4 by ii asa ie peed Vina ae sill Fy: tibia Wea a uh tion coenlta ae ocehP™ BRS serena tt ae | a £448 isi tas : Ober aN beg vr ek ond Janping coafilemenig,) The: wae) la . atta wt egine ATAWC tn ans "esata a i) heen 2G basin RR ip. 4 Oy: The Oper chon: ue Che, Bere eee i ge F itdiores Wiese tin € a“ bas get “aah fa ahee a ee ri ea pede - ; es is) % i 2. i 7 oh nf 4 . 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