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Numerical Solutions for Larg - Amplitude Ship Motions In the Time Domain W.-M. Lin (Science Applications International Corporation, USA) D. Yue (Massachusetts Institute of Technology, USA) ABSTRACT A three-dimensional time domain approach is used to study the large-amplitude motions and loads of a ship in a seaway. In this approach, the ex- act body boundary condition is satisfied on the instantaneous wetted surface of the moving body while the free-surface boundary conditions are lin- earized. The problem is solved using a transient free-surface Green function source distribution on the submerged hull. Extensive results are presented which validate and demonstrate the efficacy of the method. These re- sults include linear and larg~amplitude motion co- efficients and diffraction forces with and without forward speed, calm-water resistance and added- resistance with waves and motions, the large- amplitude motion history of a ship advancing in an irregular seaway, as well as load distributions on the changing submerged hull. Most of the large- amplitude results we obtained are new and illus- trate the importance of nonlinear effects associated with the changing wetted hull. Of special signifi- cance are the dramatic changes of the added mass, the steady resistance, and sinkage and trim forces as the motion amplitudes increase. The present method is a major step forward in the development of design and prediction tools for ship motions and loads, and represents a signifi- cant milestone towards a full~nonlinear capability in the forseeable future. 1 INTRODUCTION The accurate prediction of wave-induced motions and hydrodynamic loads is of crucial importance in ship design. In addition to concerns such as 4 efficiency and comfort, severe motions can limit operability and affect safety, while extreme loads may lead to structural failure. Thus the general problem of a moving body interacting with waves has been pursued actively since at least the time of Froude (1868) and Michell (1898~. Traditionally, the problem is linearized and formu- lated in the frequency domain, by assuming the motions to be small and time harmonic, and the resulting boundary-value problem is solved using a singularity distribution on the mean body bound- ary. For zero speed problems, this approach is quite successful and has become a standard tool for the design of large offshore structures (e.g., Ko- rsmeyer, et al, 1988~. In the presence of forward speed, the so-called Neumann-Kelvin problem is significantly more difficult due primarily to the complexity of the corresponding Green function. Thus, despite several earlier attempts (e.g., Chang, 1977; Inglis & Price, 1981; Guevel & Bougis, 1982), a truly satisfactory numerical solution is as yet unavailable. A promising variation due to Gadd (1976) and Dawson (1977) is the use of Rankine sources on the body surface as well as a portion of the free surface on which more general quasi- linearized free-surface conditions can be specified. Such approaches have been developed actively in the past 5 or 6 years (e.g., Chang & Dean, 1986; Xia, 1986; Larsson, 1987; Boppe, et al, 1987; Jensen, et al, 1988; Letcher, et al, 1989; Bertram, 1990; Nakos & Sclavounos, 1990), with increasingly encouraging results. In all of these methods, how- ever, the free surface and body geometry remain fixed in the undisturbed positions, and geometric nonlinearities are not included. An alternative to the frequency-domain approach is to formulate the time-domain initial-value prob- lem (cf., Finkelstein, 1957; Cummins, 1962~. The

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requisite time-dependent Green function which satisfies the linearized free-surface boundary condi- tion is simpler than the corresponding ones in the frequency domain, yet is capable of describing ar- bitrary (large-amplitude) motions when the proper free-surface memory effects are included. While linearized and even fully nonlinear time-domain re- sults have been available for problems in two di- mensions (or with vertical axisymmetry) for some time, developments for three-dimensional problems have been relatively recent. Such work include Ko- rsmeyer (1988) for the linearized radiation problem without forward speed, and Liapis (1986), Beck & Liapis (1987), King (1987), King et al. (1988) for the general linearized problem with constant for- ward speed. For submerged bodies, results for lin- earized free surface but large body motions have been obtained by Ferrant (1988) and recently by Beck & Magee (1990~. We remark that for lin- earized (small-amplitude) motions with zero or constant forward speed, these time-domain solu- tions are formally related to the frequency-domain results via Fourier transforms. In this paper, we extend the time-domain approach to arbitrary large-amplitude motions of a surface- piercing body in a seaway. The exact body bound- ary condition is applied on the instantaneous sub- merged hull surface while a linearized free-surface condition is used. This approximation can be jus- tified in principle upon the assumptions of small incident wave slopes and slenderness of the body geometry in the directions of the (large-amplitude) motions. The practical utility of this approach must, in the final analysis, be demonstrated by the validity and accuracy of its predictions. This is the focus of much of the present work. In a boundary-element approach, the submerged body surface at each time step is divided into a number of panels over which linearized transient free-surface sources are distributed. In contrast to earlier work, the problem is formulated in a coordinate system fixed in space. This is clearly necessary for the case of arbitrary large-amplitude motions and excursions which is the primary ob- jective of the present code. Under this formula- tion, a general and concise waterline integral term can be derived to account for arbitrary transla- tions and distortions of the body waterplane, and the diffraction problem can be included straight- forwardly by adding the incident wave contribu- tion to the body boundary condition. For gen eral nonlinear calculations, the position and ori- entation of the body is updated (by solving the equations of motion or as prescribed) and the un- derwater body surface is repanelized at each time step. Since the body boundary condition is satis- fied on the exact instantaneous hull, the so called "m-term" effects associated with forward speed (Ogilvie & Tuck, 1969) are automatically and ez- actly included. For the special case of constant forward speed and small oscillatory motions (the linearized seakeeping problem), the traditional lin- earization (and decoupling) of the latter is, how- ever, less explicit in the earth-fixed (time-domain) formulation. For these linear forward-speed cal- culations, the quadratic terms are included in the force calculations to account for the forward-speed couplings but the m-terms are otherwise neglected in the body boundary conditions. Linear and large-amplitude computational results are presented for a floating sphere, two Wigley hulls, and the Series 60 (CB = 0.7) hull under- going free or captive motions and with or with- out forward speed or incident seas. The program is applicable for general six-degree-of-freedom mo- tions (without lift) but we restrict ourselves to vertical plane motions in head seas in this paper. For the linear problems without forward speed, (time) impulse response functions are computed from which the requisite motion coefficients are obtained via Fourier transforms. For the nonlin- ear cases and for problems with forward speed, the bodies are started from rest and computations typically continued until steady states (limit cy- cles) are achieved. Where available, the results are compared with other time- and frequency-domain calculations and experimental data. In all cases, the importance of the nonlinear effects due to ge- ometry variation is identified. 2 MATHEMATICAL FORMULATION We consider a general three-dimensional body floating on a tree surface and undergoing arbitrary six-degree-of-freedom motion in the presence of in- cident waves. An earth-fixed Cartesian coordinate system is chosen with the x-y plane coincident with the quiescent free surface, and z is positive upward. The fluid is assumed to be homogeneous, incom- pressible, inviscid and its motion irrotational. Sur- face tension is not included and the water depth is infinite. 42

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The fluid motions can be described by a velocity Note that G = 0, so that potential 4iT(X,t) = ~(x,t) + 4~(x,t), (1) where ~! is the incident wave potential, 4} = AT- 4~r the total disturbance potential, t is time, and x is the position vector. In the fluid domain V(t), satisfies Laplace's equation V2~ = 0 . (2) On the mean free surface Butt, we impose the lin- earized condition ~` +g4iz = 0 on 7(t), t ~ 0, (3) where 9 is the acceleration due to gravity. On the instantaneous body boundary B(t), we require no normal flux: ~ = Vn - ~, ~ on B(t), t ~ 0, (4) where the unit normal vector to the body n is pos- itive out of the fluid and Vn is the instantaneous body velocity in the normal direction. For finite time, the conditions at infinity, SOO, are A, 4~' ~ O onSOO, t > 0, (5) and the initial conditions at t = 0 are 4} = 4~` = 0 on {(t), t = 0. (6) We introduce the transient free-surface Green func- tion for a step-function source below the free sur- face (see, e.g., Stoker, 1957~: G(P,t;Q,T)=G+Gf=~_ :1+ 2 / [1 - cos(~(t-Tall ek(2+~) Jo~kR) dk o for P 76 Q. t > T. (7) where P = (a, y, z) and Q = (6, 71, <) are the source and held points, r = IP- Ql, r' = IP- Q'l' Q' = ((, 9, -a), R2 = (x _ ()2 + (y - 71)2, G = 1/r - 1/r' is the Rankine part of the Green function, Gf = G-G is the free-surface memory part, and JO Bessel function of order zero. Eq. (7) satisfies the following initial-lo oundary-value problem: V2G = 0 G`` + gGz = 0 G. G' ~ 0 G = G' in V(t), t > T on{(t), t > T for t > ~ = 0 onT(t), t = ~ . GO = Gi = -2 / ~sin(~(t-T))ek(Z+~) Jo~kR) dk . (8) o To obtain a boundary integral formulation for A, we apply Green's identity to IRAQ, T) and Gr(P,t;Q,T) in a fluid domain V(T) bounded by ~F(T), B(\T), SOO, and a small surface SP excluding point P: ///V(~)~.VQG, - G7VQ.) dV = // ~ Agent-G~nQ`) dS, (9) sco +~(~+~(~)+SP where V(~T), ~F(T), I3(~T) are respectively V(~), jF(T), I3(T) with the exclusion of possible point P. The left-hand side of (9) and the integrals over SOO and SP on the right-hand side are all zero. In- tegrating the resulting equation with respect to T from 0 to t, we obtain / //:~(T)+~3(r)(~?G7nQ-G~r~nQ~) dS = 0. (10) To eliminate the integral over F(\T), it is neces- sary to exchange the time and surface integrals involving ~F(T). It should be noted that (To iS time-dependent in the earth-fixed coordinate sys- tem. Applying the linearized free-surface condition (3) and the transport theorem we obtain finally // (.G,nQ GT4in~ ~ dS 9 { Br //3F(T) T T) dS - / (.G??-G7.T\)VN dL} , (11) where (To iS the instantaneous intersection of the body with the free surface (excluding possi- ble point P), N the unit normal to P(T) on the free surface, positive out of the fluid domain, and VN the normal velocity of [(T) in the N direction. Combining (10) and (11) and applying the free sur- face condition on G and the initial conditions on 4i, we obtain //() ~ S + /0 dT //~(T)(.G'nQ-G,tnQ ~ dS 43

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+ -| do| (.G,, - G?~^,)VN dL = 0. (12) The integral over the free surface /(T) can now be eliminated by applying Green's identity again to 4~(Q,t) and G in F(t) and combining the result with (12). Finally, we have for P on B(t): 2~r4~(P,t)+/t (EGG-4inQG ) dS /o {//~`T'(qiG7nQ-Snags) dS + 9 /rt ,(.G77-4~'G])VN dL}. (13) The overhead bars on the integration surfaces are dropped since the integration over SO on these sur- faces gives no contribution. The waterline memory integral over [(T) in (13) is the general form for ar- bitrary large motions in the earth-fixed coordinate system. For a submerged body, this term vanishes. For the special case of horizontal planar motions only, this integral reduces to a similar term formu- lated in a body-fixed coordinate system given by Liapis (1986~. For large-amplitude problems, the evaluation of the tangential velocities on the body is of critical importance, and, in the absence of lift, a source formulation is preferred. To obtain the equation for the source distribution, we formulate the inte- rior problem governed by an equation similar to (13), which upon combining with (13) yields 'T {/Jrs(~( / [// ~ S 9 /r(T) aGf VNVndL] } (14) where a(Q,T) is the source strength at point Q at time T. and VN and Vn are related by VN = V /N Finally, we apply the body boundary condition for P on B(t) to obtain: = Vn(P, t)-VAMP, t) no 47r {//~(t) aGnp dS + / do + [// dS-9 /r~ ~aGnp`VNVn dL] ~ (15) Eq.~15) can be solved for the unknown UP, t) given B(t), Vntt), Arty, and a(P,~) and 8(T) for O < T < t. Once the source strength is found, ~ is evaluated by (14), and the velocity on the body, Vat, is obtained using a vector form of (15~. The total pressure is given by Bernoulli's equation, P = -Pi {~! + 2~V.T~2 + go), (16) and the force on the body is obtained by inte- grating (16) over the instantaneous submerged hull B(t). The formulation for the general arbitrary motions problem is thus complete. For linear (small mo- tions) problems without forward speed, the body boundary condition is linearized by applying (15) on the mean body position BO (the waterline term is absent) and the quadratic term is neglected in integrating (16) on BO for the forces. For the lin- earized problem with forward speed (the linear sea- keeping problem) in the present earth-fixed coor- dinate system, the quadratic contributions are in- cluded in the pressure integration (Eqs. 16 and 19 ~ to account for forward-speed effects, but the body boundary condition (15) on BO is otherwise not further expanded. The formulation is thus nei- ther strictly linearized (in terms of the motion am- plitudes) nor consistent (in that the associated m- terms are not included). This is a consequence of the fact that a formal decomposition of the prob- lem in terms of the steady forward-speed distur- bance and the (small) oscillatory disturbance.is no longer straightforward in the earth-fixed time- domain system. (In this system, the forward speed is in fact a large-amplitude motion.) A more con- venient formulation for the linearized constant for- ward speed problem is, of course, to adopt a coor- dinate system translating with the ship (c/., Liapis & Beck, 1985~. 3 NUMERICAL METHOD 3.1 Solution of the Integral Equation A panel method is used for the solution of the in- tegral equation (15~. The body surface B(t) is di- vided into No) quadrilateral elements over which the source strength is assumed constant. Non- planar quadrilaterals are mapped to planar ele- ments by fitting to the corner points in a least- squares sense (Hess & Smith, 1964). Similarly, the 44

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body waterline I`(t) is divided into NW(t) straight line segments on which the source strength c,* is assumed to be the same as that of the adjacent body panel. The equation is satisfied at collocation points corresponding to null points of the panels. A forward Euler scheme with constant time step, At, is used to integrate forward in time and the con- volution integral is computed using a trapezoidal rule. The discretized form of (15) is given by: NM ~ aM 1l Go p (Pi, Q j; O) d S i=l s =-Or [V(Pi, t)-V.I(Pi, t)] np M-1 _N,m -At ~ Em ~ aJ || G/pt(Pi, pi; t-T) dS m=0 i=1 S Nw -~ aim | G/pt(Pi, Qk; t-T)VNkVnkdL k=1 r. for i = 1,2,...,NM. (17) In the above, M and m are the indices for t and T respectively where t = MAt and T = mat; i, j are respectively the panel indices for collocation points P and field points Q (k is the index for the waterline field points); and so = 1/2 and em = 1 for m > 0. Note that the convolution summation ends at m = M - 1 due to a property of Gf. Eq. (17) is in the form of a system of linear equa- tions: NM ~ AijaM = Bi i = 1,2,...,NM, (18) j=1 which can be solved for the unknown panel source strengths, aM, by standard means. 3.2 Evaluation of the Free Surface Tran- sient Green Function The integrals involving the Rankine part of the Green function G are evaluated using a method similar to Hess & Smith (1964~. Special efforts have been made to completely vectorize these eval- uations, resulting typically in several factors of sav- ings on vector processors. The numerically more time consuming task is the evaluation of the memory term Gf and its deriva- tives which, because of the convolution integrals, must be evaluated a large number of times. There has been much effort in recent years to develop efficient and accurate numerical methods for cal- culating Gf and its derivatives, including Newman (1985, 1990), Liapis & Beck (1985), and Magee & Beck (1989~. The present approach is an improve- ment upon the method given in Newman (1985~. The domain for Gf is divided into a number of regions wherein, depending on the arguments, as- cending series, asymptotic series or a combination of these and two-dimensional economized (Cheby- shev) polynomial approximations are utilized. The final results are maintained to a minimal accuracy of 6D for Gf and 5D for its first and second deriva- tives. The entire procedure is completely vector- ized and the average computing time required for the evaluation of Gf and its two first and two sec- ond derivatives is 0~1-3) microseconds on a Cray- Y/MP depending on the relative frequency of eval- uations in the different regions. Details can be found in Lin & Yue (1990~. The numerical integration of Gf and its deriva- tives over the quadrilateral panels is performed using two-dimensional Gauss-Legendre quadrature after mapping the general quadrilaterals into unit squares. A similar one-dimensional quadrature is used for the waterline elements. 3.3 Implementation & Force Evaluation The present method is general for arbitrary large- amplitude six-degree-of-freedom motions which in general requires the modelling of a changing un- derwater body geometry. Depending on the type and amplitude of the motions, different numerical treatments are required: 1. Linear motions with zero or constant forward speed In this case, the underwater geome- try does not change with time and the matrix Aij in (18), which represents the Rankine in- fluences and depends only on the relative dis- tances between Pi and Qj, is constant. Thus, A'j needs to be evaluated and inverted only once at the beginning of the computations. Furthermore, since Gf and its derivatives de- pend only on Pi - Qj~ and t - T. for zero or constant horizontal velocity, the previous val- ues of Gf can be reused at each time step. 4s

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Thus only one new set of evaluations for Gf and its derivatives is necessary at a new time step, and the rest can be obtained from pre- vious calculations. The main differences be- tween zero and constant forward speed prob- lems are the absence of the waterline contri- bution in the former, and the need to include quadratic terms in the force evaluation to ac- count for forward-speed cross-coupling contri- butions in the latter. " Linear motions with arbitrary horizontal ex- cursions The body moves with variable speed on a straight course or perform arbi- trary excursions in the horizontal plane. In this case, the underwater body geometry still does not change in time and Aij needs to be evaluated and inverted only once. How- ever, the fixed relationships of Pi-Qj~ and t-T between time steps no longer exist and evaluations of Gf and its derivatives have to be done for different T,S at every time step. This increases the computational burden sig- nificantly. " Arbitrary large-amplitude motions-In this case, the underwater body geometry changes with time, and both Aij and all values of Gf and its derivatives must be reevaluated at each time step. For large-amplitude motions, a robust geometry processing capability is essential and an automatic repanelizer was developed for this task. The orig- inal input geometry must now include the above- mean-waterline portion. At each time step, the un- derwater geometry is represented by a number of vertical strips with each strip having a fixed num- ber of panels along its girth. As the body moves, its new location and orientation is updated in the global coordinate system and the new waterline is found from the intersection with the mean free sur- face. The underwater portion of each strip is then repanelized using a spline curve fitting. For sim- plicity, the number of panels in each strip is kept the same but any particular vertical strip may be eliminated completely if it is out of the water. The force evaluation follows from integration of (16) over the instantaneous or mean submerged hull for the nonlinear and linearized problems re- spectively. The quadratic terms are included in the nonlinear and forward-speed calculations and are evaluated directly given the normal and tangential velocities on the body. The calculation of It is separated into two parts: B.~/~1t is given from the incident waves, and B~/0t is evaluated by / By! Hi m Am _ Am-~ ~ fit J i At -Vim V.m , ( 19) where Vim, the 'grid' velocity at point P., is equal to ~ Am _ pSm~! )/At for the nonlinear problem, and is simply the forward speed, U. in the linearized forward-speed problem. We remark that because of repanelization for the nonlinear problems, the control points Pi in general do not represent the same global points or the same body points. When the number of panels changes between time steps, (19) can not be used in a straightforward manner and special care must be taken. This is not con- sidered in the present code. 4 RESULTS Numerical computations were performed for lin- earized radiation and diffraction problems; large- amplitude forced motions and free motions of a floating body with and without forward speed and the presence of ambient waves; as well as wave re- sistance and added resistance problems. For sim- plicity we limit ourselves to heave/pitch motions in head seas although the present code is capable of general six degree-of-freedom motions (without lift). A sphere, two Wigley hull forms, and the Series 60 (CB=0.7) hull were used for this study The first Wigley form was used by Gerritsma ( 1988) in his seakeeping experiments and has a beam-to-length ratio 2b/L=O.1, and draft-to- length ratio H/L=0.0625. This is designated as the ``WSK', hull hereafter. The half beam y is given by: Yb = (1 - X)~1 - Z)~1 + 0.2X) + Z(1 - Z4~1 - X)4 where X-(2:~/L)2 and Z-(z/H)2. The other Wigley hull is commonly used for wave resistance studies which we designate the "WRT" hull. This hull has 2b/L=O.1, and 11/L=0.0625 with half beam given by y/b=~1- X)~1- Z). For convenience, unless otherwise noted, all quan- tities in the following are nondimensionalized by fluid density p, gravitational acceleration 9, and body length L (or radius a for the sphere). The panel numbers, N. indicated are always for half 46

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Hulme (1982) SAMP N=18 SAMP N=150 . . . 0.0 5.0 10.0 15.0 20.0 told Fig. 1: Impulse response function for the heave force on a hemisphere, V _ 2~ra3/3. Of the (submerged) body. The present compu- tational results are referred to as SAMP (Small- Amplitude Motion Program) when the linear op- tion (fixed underwater geometry) is used, and as LAMP (karge-Amplitude Motion Program) when the large-amplitude capability with changing sub- merged geometry is employed. Where available, numerical calculations from the linearized time- domain method developed at the University of Michigan (Magee & Beck, 1988) are included and denoted as "Michigan". Strip theory results are based on Salvesen, Tuck & Faltinsen (1970~. Those used in Secs. 4.1 and 4.2 are taken from Magee & Beck (1988~. 4.1 Linear Radiation Problem Fig. (1) shows the impulse-response function, L(t), for the heave force on a hemisphere, radius a, as a function of time, obtained using SAMP with At=0.05. Here L(~) is defined as ~ ~ P.//l30 (20) for an impulse (delta function) acceleration at t=0. The curve for Hulme (1982) is obtained from Fourier transform of his analytic frequency-domain results. The time-domain results show the charac- teristic oscillations at larger times which are caused by the presence of irregular frequencies of the in- terior problem (Adachi & Ohmatsu, 1982~. Note that the amplitudes of these spurious oscillations decrease as the number of panels is increased. 0 r I D ~ ~ o jo Do - - ~. o A 1 Hulme (1982) SAMP N=18 SAMP N=150 \~_ ~ ___ (~ 1 it,, 1 Tier a/g | 1 A33/PV 0.0 2.0 4.0 6.0 8.0 w2a/9 Fig. 2: Heave added-mass and damping coefficients for a hemisphere. The added mass and damping coefficients are re- lated to the cosine and sine transforms respectively of the impulse-response function. These are shown in Fig. (2) as compared to the frequency-domain result of Hulme (1982~. For illustration the loca- tion of the lowest irregular frequency (w,2r~a/g 2.56, Hulme, 1983) is also indicated. It is seen that the oscillations in Fig. (1) are directed related to the irregular behavior near Micra although the singular nature is not fully captured because of the finite-time truncation of L(t) (at t=20 in this case). We note again the smaller and more confined os- cillations for larger N. For radiation problems with forward speed, a forced oscillatory motion is superimposed on a step-function jump of the forward velocity to the prescribed value. The force coefficients are then obtained from Fourier transforms of the SAMP force time histories after steady state is reached. In the presence of forward speed, quadratic con- tributions must be retained in the present space- fixed formulation and the impulse-response func- tion is not generally useful. This computational inefficiency for the special case of linearized sea- keeping is the main (in fact only) disadvantage of the spaced-fixed formulation. Figs. (3,4) show the heave and pitch diagonal and off-diagonal added-mass and damping coeffi- cients of the WSK hull at Froude number En -- U/~/~=0.2. The SAMP calculations use N=120 and At=0.1. These results are considered to have converged in that selected calculations (not shown) 47

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,1 o ~ Il. o /: /~ o \ 10 00 10- ZO- 0- t0 7^d/~Sa . U~, _ ~o o - ? _ ~ o o _~ o _~ o o o. _ o C~ . _ I,`+N + o'-)U '\ ~_o \ ~ '\ ~ \6 o '\\ tt _ '\\ o O + . ~,. o /,''+ O , + ', - n + ,' ~ O D n o n o 90 0 10 10 7^d/4Sa $~L,1~ ,'~ '+C ~o I Ct\. lo G- ~ o | ~ .-~ ~o CZ 0 910 900 900 _ 000 900- C10- S?0 l~d/sV ]^d/S~ o ~G 49 ~C~ _ C ~o o 11 o _ O -O o U] o a' 0 o C~ bo 11 ._ ,. ~ ',. C ~ 11 ~C~ O ~ ~ .. _` ~ ~ O .~ ~ _4 _' ,V 11 a) _ ~ o ~o . . . . U) ~.o . _

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using N=200 and At=0.05 show changes of less than 1% in the force coefficients. As pointed out earlier, the SAMP calculations do not contain the contributions of the m-terms while the general LAMP computations do. To gauge the importance of these effects, we include in the figures the small- amplitude limits of the large-amplitude LAMP re- sults (see Sec.4.5~. These values are obtained from extrapolations of the small finite-amplitude LAMP results to zero amplitude and are denoted hereafter as LAMPo. From Figs.~18), it is seen that LAMP approaches the LAMPo limit very smoothly so that the estimates are reliable. For the diagonal coefficients (Fig.3), the LAMPo and SAMP results are both reasonably satisfactory although the m-term effects appear to play an ap- preciable role as the relative frequency, To--U~/9, decreases and especially for the pitch coefficients. On the other hand, the Michigan results (which include m-terms) using the same N show unex- plained deviations for intermediate values of To Not surprisingly, strip theory performs better for larger To but appears to be affected significantly only for Bss at lower frequencies. The results are much more interesting for the off- diagonal coefficients. For the added-mass terms A3s and As3, the SAMP calculations compare well to measured data. When effects of the m-term are included, the deviations of LAMPo from SAMP are primarily at lower frequencies but the com- parison to experiments is otherwise not improved. For the damping terms B3s and Bs3, the SAMP results are inadequate, showing a large somewhat constant shift, and indicate that the m-term effects may play an important role for these coefficients. Indeed, when the m-term effects are included, the LAMPo and Michigan results are both reasonably close to experimental measurements. In summary, we see that the SAMP results are satisfactory for the off-diagonal added mass coef- ficients, and generally acceptable for the diagonal coefficients. Taken together, the LAMPo predic- tions provide overall good agreement with exper- iments and are clearly superior. By comparing SAMP and LAMPo results, the effect of the m- terms is quantified and shown to be important for pitch-related coefficients at lower frequencies and especially for the off-diagonal damping terms. In is noteworthy that since LAMP uses a linearized free-surface condition, the present results suggest that nonlinear free-surface terms such as those that may be included in Dawson-type codes (e.g., Nakos & Sclavounos, 1990) may otherwise not be as im- portant in the prediction of the motion coefficients. A well-known theoretical difficulty of the linear seakeeping problem is the singularity at the critical relative frequency To =1/4 (e.g., Dagan & Miloh, 1980~. Physically, this may be explained by the simple fact that at To = 1/4 the group velocity of the waves generated by the oscillatory motion matches the ship speed U and resonance occurs. In the frequency domain, the linearized result is un- bounded while in the time-domain the resonance is manifest as a slow unbounded growth of the so- lution with time. While actual predictions at the critical frequency may not be contemplated in the context of lin- earized theory, it is useful to be able to avoid the undesirable growing solution in geneml time- dependent simulations. An obvious idea (also sug- gested by Beck & Magee, 1990) is to simply trun- cate the duration of the memory effect convolution terms in (15) by requiring t - T < TCU`, say. This is illustrated in Fig. (5) which shows the time history of the vertical force on a WSK hull starting from rest with a heaving frequency of w=1.25 and a for- ward speed corresponding to To = 1/4 (Fn=0~2~. When the memory effect is not limited, the force grows slowly without reaching steady-state. When we set TCUI=1O' however, steady-state is quickly reached resembling the results at non-critical To,S. We point out that because of the relatively slow growth and the effective truncation of TCU' by the total duration of the simulation, the problem of the critical frequency may not pose a severe limi- tation in practice for general (not critically forced) time-domain applications. The motion coefficients obtained from SAMP calculations using different values of TCU! are presented as separate symbols on the To = 1/4 vertical in Figs. (3,4~. We cau- tion that these results should not be considered as 'predictions' at the critical frequency despite the fact that they are now finite and appear to give a roughly continuous trend. 4.2 Linear Diffraction Problem The presence of incident waves, ~I(x,t), can be included into the governing equation (15) in a straightforward manner. In practice, this is intro- duced instantaneously at t=0 and the diffraction ejects evaluated after the transients vanish typi so

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o- o o to - o 0 to to o- | SAMP IV= 120 unlimited memory effect memory effect truncated for t-T ~ TO 0.0 20.0 40.0 60.0 t~l~ Fig. 5: Peaks of the leave force history for a lose hull at Fn=0.2, TO=1/4. cally in a very short time. In the case of no forward speed, the linear diffraction problem is validated for a floating hemisphere. The SAMP results are indistinguishable from the earlier computations of Cohen (1986) and King (1987) and the compar- isons are not shown here. Figs. (6) shows the amplitude and phase of the wave exciting forces on a WSK hull moving with forward speed, Fn=0.2, in head seas. For the SAMP results, N=120 and 30 time steps per inci- dent wave period is used. The comparisons to the time-domain results of Michigan, strip theory and experiments are overall satisfactory. The Michi- gan calculations underpredict the secondary hump at high frequency although both of the 3D time- domain calculations show better correlations to the experiments than strip theory. For a more realistic and complicated geometry, we consider the Series 60 (CB=0.7) hull form for which extensive experimental data are available. Figs. (7) show a complete set of results for Fn=0.2 in head seas. The SAMP results with N=180 and 250 have clearly converged (30 time steps per incident wave period are found to be more than adequate) and compare very well with experimental data. Not surprisingly, strip theory results are generally ac- ceptable compared to SAMP predictions at higher frequencies. On the other hand, the Michigan re- sults do not indicate convergence and appear to un- derpredict the amplitudes especially at the higher frequency second hump. 4.3 Large-Amplitude Motions at Zero Speed As a first example, we study the large-amplitude heaving of a (complete) sphere. The sphere is ini- tially semi-submerged with its center at the ori- gin and a forced heaving motion with zest) = -Ahsin~t is imposed at t=0. We set w=1 and use lV=150 and wAt = 2,r/30. As pointed out earlier, at each time step, the submerged portion of the sphere is repanelized, the Rankine influence matrix reevaluated and solved, and the memory terms recalculated from T=0 to ~ = t. Fig. (8) shows the different components of the ver- tical force on the sphere as a function of time for the case of Ah/a=0.5. Steady-state (limit cycle) is rapidly reached (within one period) for all the com- ponents. For this geometry, the hydrostatic force is a large part of the total. The inertia (-~/0t) term shows distinct higher-harmonic components while the quadratic (-~V.~2/2) component is pri- marily at the second harmonic, as expected. Sur- prisingly, these higher harmonic contributions ap- pear to cancel closely so that the total force oscil- lates primarily at forcing frequency together with a negative (suction) steady component. To validate the large-amplitude capability of LAMP, systematic convergence tests were per- formed for this case. A typical example is Fig. (9) which shows the hydrodynamic force his- tory (FHD) for varying N and w/`t for Ah/a=0.5. The results are indistinguishable indicating that IV=150 and w/`t = 2,r/30 are more than adequate. Fig. (10) shows the time history of the vertical hy- drodynamic forces on the sphere for different Ah/a. As expected, the nonlinear curves approach the linear one as Ah/a decreases. For larger heaving amplitudes, the peaks become sharper and higher, while the troughs become shallower. The nonlin- ear effect, however, is relatively small even when the heaving amplitude is 50% of the radius. This was found also in earlier full~nonlinear simulations for heaving axisymmetric bodies (Lin, et al, 1984; Dommermuth & Yue, 1986~. A detailed analysis of the frequency components of the force can be obtained from Fourier transform of the steady-state (limit-cycle) time history. For convenience, we define the frequency components of the vertical force: F (`t) = OR (Lfo + feint+ f2ei2W'+ 51

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o en ~ to * ~ o ly -~ m 1 lo 11 ~ . 1 . to . to - . 1 o o 1 SAMP N=168 SLAW (Letcher et al, 1989) xs~ Experiment (ITTC, 1985) 0.0 t.0 2.0 3.0 to 5.0 t/To Fig. 12: Horizontal force time history for a WRT hull moving at Fn=0~3. to o o- 1 cut to * (a) 0 \ ~ 1 - 1 ~1 [q up _i 11 1 ~4 ~ 0 C`2- SAMP N=168 SLAW (Letcher et al, 1989) i 1 0.0 1.0 2.0 3.0 4.0 5.0 t/To Fig. 13: Vertical force time history for a WRT hull moving at Fn=0~3. to * ,~ 0 ~ ~- ~ 1 :a ~ 0 -I'q C`2- I lo 11 o_ 1 0 l 4, SAMP N=168 SLAW (Letcher at al 10~\ . . . . . 0.0 1.0 2.0 3.0 4.0 5.0 t/To Fig. 14: Pitch moment time history for a WRT hull moving at Fn=0~3. the pitch moment is very good while the two re- sults differ by about 15% for the steady vertical force. To evaluate the dependence of SAMP predictions on panel size and time steps, systematic conver- gence tests were performed for different numbers of panels and time steps for Fn=0~3 and 0.4. These results are summarized in Tables 2 and 3. Three different grid distributions-uniform, cosine and 'geometric' spacing-are considered in the lon- gitudinal direction while the girth-wise grid sizes are kept constant. The so-called 'geometric' spac- ing is based on the criterion that the projection of each panel on the y-z plane be constant. For the parabolic WRT hull, this corresponds to a square root grid distribution in the longitudinal direction. From the tables, some indication of convergence is observed, although a definite trend is still difficult to obtain. This suggests that possibly larger num- bers of panels may be required for more accurate predictions at these Froude numbers. Comparing the three grid distributions, the geometric grids seem to give the best and most consistent results. For the calculations in this and later sections, a geometric grid with N=168 (28x6) and ~t=2'r/40 was used. To obtain a complete resistance curve, the calcu- lation of Fig. (12) was repeated for Froude num- bers ranging from 0.2 to 0.45. The final results are shown in Fig. (15) together with the ITTC ex- perimental data. The overall comparison is quite satisfactory. Finally, we evaluate the possible effect of the start- up velocity profile on the steady-state force predic- tions. For illustration, three velocity profiles were considered with V(t)=0 for t < 0, V(t) = U for t > To, and for O < t < To: (i) impulsive start, V(t) = U; (ii) ramp start, V(t) = Ut/T~; and (iii) cosine-function start, V(t) = Uf1-cos(,rt/2T')~/2. Fig. (16) shows the horizontal force time histories for these three cases with T~/To=0.6. The initial transients and the phases of the later oscillations differ appreciably between (i) and (ii),tiii) but the same steady-state value is reached. As expected, the smoother ramp and cosine-function profiles produce smaller transient and To oscillations (asso- ciated with milder start-up disturbances) and are preferred for practical resistance predictions using SAMP. 56

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Table 2: Wave resistance coefflaents Cat x 103 of the WRT hull at Fn=0~3 an a function of grid size anu distribution. | At/2,r 1 uniform grid I cosine grid I geometric grid ~ | 1 1 20x6 1 28x6 1 36x6 1l 20x6 1 28x6 1 36x6 11 20x6 1 28x6 ~ 36x6 . l .. , 1/20 2.53 2.84 1.70 l 2.82 1.99 1.48 1.20 1.48 2.50 1/30 2.50 1.37 1.58 l 1.82 1.87 0.94 1.57 1.62 1.60 -1/40 1.68 2.79 1.26 1.22 1.84 1.78 . 1.29 1.64 1.75 Table 3: Wave resistance coefficients Cc x 103 of the WRT hull at Fn=0~4 as a function of grid size and distribution. | l~tj2,r 1 uniform grid I cosine grid I geomeh 36x6 2.07 1.94 2.09 lo ._ * 20-x6 1 28x6 1 36x6 1l 20x6 128x6 1 36x6 1l 20x6 1 28x6 1/20 1.32 1.56 1.53 ~ 1.43 1.62 1.71 1.42 1.68 lt30 0.33 2.50 1.96 ~ 1.68 1.98 2.12 2.24 2.07 1/40 4.22 1.24 2.24 1.61 2.33 2.10 2.05 2.15 0 0 cr)- . ~O SAMP N=168 Experiment(ITTC, 1985) lo N 0.15 0.20 0.25 0.30 0.35 0.40 0.45 050 Fn Fig. 15: Wave resistance coefficient for the WRT hull as a function of Froude number. 4.5 Large-Amplitude Motions with For- ward Speed A main objective of this work is to evaluate and quantify the importance of large-amplitude mo- tions on seakeeping characteristics. While the change of such relevant quantities as added mass (and hence natural frequencies and response ampli Impulsivestart Ramp start Cosine function start. rem I I r - -l 00 10 20 3.0 40 5.0 t /Lo Fig. 16: Horizontal force time history for a TORT hull moving at Fn=0~3 using different start-up ve- locity profiles (T1 /To=0.6 ). tudes) due to large-amplitude motions can be ex- pected on physical grounds, and has been reported in experiments (e.g., O'Dea & Troesch, 1987), the precise magnitudes or even dependencies are as yet not well known. To address this problem, the LAMP program was applied to the large-amplitude forced heav 57

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A 1 . A A l * t-:, A 1 to o- 1 00 0.1 02 0.3 04 ()5 Ah/0 Figs. 17: Steady surge, heave and pitch forces and moment on a heaving (frequency w=1.0, amplitude Ah) WRT hull moving with constant forward speed (Fn=0.2~. .: LAMP (N= 160~; - - - : 'calm-water' (Ah=O) value obtained with SAMP. ing of a WRT hull moving with constant forward speed. Specifically we chose a Froude number Fn=0.2, a heaving frequency w=1.0, and consid- ered a range of heaving amplitudes correspond- ing to Ah/B ~7.5 - 45% where H is the draft. Figs.~17) show the added resistance, added sinkage and added trim forces as a function of the heave amplitude. For contrast, the 'calm-water' values (cf. Sec.4.4) are also indicated. For small Ah/H, the results are close to the calm-water values as expected and the approach to this limit is very smooth. As the amplitude increases, however, the A c ~ . co . o . D ~o ~ , D - ~"added mass" . 0 0 0 "damping" added m ~ ~, . 0.0 0.1 0.2 0.3 0 4 0 Ah/0 Fig. 18: Excitation frequency components of the limit-cycle forces and moment on a heaving ~ w=1.0, amplitude Ah ~ WRT hull moving with con- stant forward speed (Fn=0.2~. ~ in phase with acceleration ~ "added-mass" ); O out-of-phase with acceleration ~ "damping" ~ coefficients using LAMP (N=160). Also indicated are I, \7 : small- amplitude limits (LAMPo); and , As: SAMP added mass and damping coefficients. added resistance increases rapidly and is as much as 0~5) or more times the calm-water value for the larger amplitudes considered. The picture is sim- ilar for the mean vertical force and moments but somewhat less dramatic. By analysing the limit-cycle force and moment his- tories for the frequency components at the forc- ing frequency that are in phase and out of phase respectively with the acceleration, the so-called large-amplitude "added-mass" and "damping" co- efficients with forward speed can again be identi- fied (cf. Sec.4.3~. These diagonal and off-diagonal s8

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seakeeping coefficients are shown in Figs.~18) as a function of Ah/H. For small amplitudes, the approaches to the linear (zero amplitude) limits are very smooth allowing accurate extrapolation for the linearized LAMPo estimates. As noted in Figs.~3,4), this small-amplitude limit of LAMP differs from the linearized SAMP prediction (also A, shown in Figs.18) because of the absence of the effect of the m-terms in the latter. As in the case for the heaving sphere with no for- ward speed (Figs.ll), the large-amplitude added mass shows a clear dependence on the amplitude and decreases appreciably as Ah increases. As an illustration, when the heave amplitude is ~40% of the draft, A33 has decreased to about 60~o of the small-amplitude value. For the present WRT hull, the waterplane area changes very little with heave amplitude and consequently the relevant heave natural frequency would increase by ~20% com- pared to the linear value in this case. This may explain some of the "unexpected" dependencies of the response amplitude functions on forcing ampli- tude observed in experiments. As with the sphere with no forward speed, the LAMP damping coef- ficients here are less sensitive to heave amplitude probably due to the fact that nonlinear radiation mechanisms are absent in the present approxima- tion. 4.6 Motions of a Floating Body and Un- steady Loads With the ability to determine the relevant hydro- dynamic forces at any given instant, the complete six degree-of-freedom motion history of the body can be obtained by a direct integration of the dy- namical equations. Thus LAMP can be employed to study general time-dependent motions such as in the case of a ship advancing in an irregular sea- way. Indeed, one of the important applications is the study of episodic events involving large loads and motions in the time domain. To validate the motions solver, we consider first the decaying motions of a freely-floating sphere re- leased from rest with a given initial vertical dis- placement. As a test, we integrate the problem di- rectly in time, although for the linearized problem the solution can be obtained by a Laplace trans- form involving the force impulse-response function of Fig. (1) (Liapis, 1986~. Consistent with the open time quadrature formula of (17), we employ an ex 59 SAMP LAMP Ah/a=0.50 0.0 8.0 16.0 24.0 32 0 told Fig. 19: Vertical displacement history of a float- ing hemisphere released from rest with an initial height Ah. plicit multi-step scheme for the motion time inte- gration. First, SAMP computations were performed and compared to the linearized calculations and small- amplitude experiments of Liapis (1986~. The re- sults are indistinguishable and very close to the measurements and are not shown. Fig. ( 19) shows the SAMP and LAMP (for initial ampli- tude Ah/a=0.5) results with N=150 and At=0.1. The difference in the (normalized) amplitudes is small but it is of interest to note that the LAMP oscillation periods are longer than the linearized values by about 5%. This is caused by the com- peting effects of added-mass decrease due to large- amplitude motions (cf. Fig. (11~) which tends to decrease the period, and the reduction in the av- erage waterplane area (and hence the hydrostatic force) which tends to increase the period of oscil- lation. In this case, the latter effect is greater and consequently an increase in natural period is ob- served. This again underscores the importance of nonlinear geometry effects. An interesting application of the motion program is to reexamine the calm-water resistance problem of Sec.4.4 but this time allowing the body to freely sink and trim until steady-state values are reached. Starting from rest, the ship is held to a constant forward speed but is otherwise free to heave and pitch. In the large-amplitude simulation, the un- derwater geometry is allowed to change as steady

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o ,(- o lo o GO * o C\2 lo lo - Pree Body: LAMP N = 196 :~ Experiments (ITTC, 1985; Bai, 1979, Fixed Body: O SAMP N=168 Experiments (ITTC, 1985) , . . . . . . 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Fn Fig. 20: Wave resistance coefficient for the WRT hull as a function of Froude number. - _` ~ + f _' CO lo. .~ C~2 lo lo ~ S;nkage LAMP JV=196 ;~ Experiment (ITTC, 1985) 4~1 trim 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Fr' Fig. 21: Sinkage (a) and trim (b) of the WRT hull as a function of Froude number. [A--draft at AP, [F-draft at FP. Positive trim means bow up. state is attained by an equilibrium between hydro- static forces of the final geometry and the steady vertical force and pitch moment. Fig. (20) shows the free sinkage and trim wave resistance of the WRT hull as compared to cor- responding experimental measurements as well as the fixed body resistance results of Fig. (15~. The present LAMP calculations use At=21r/40 and N-196 (28x 7) where an extra horizontal row of panels is added to the grid of Fig. (15) to model the extra draft due to linkage. From the figure, the increase in wave resistance due to sinkage and trim is clearly predicted. Comparisons between the LAMP results and experiments are excellent. The comparison for sinkage and trim is shown in Figs. (21~. The LAMP predictions are again sat- isfactory both in the magnitude and the forward speed dependence. The results are somewhat bet- ter for the sinkage and show a slight underpredic- tion of the trim at higher speeds. Since sinkage is the main reason for the resistance increase, the under-prediction of trim at high En does not affect the resistance prediction significantly. Finally, we applied LAMP to study free vertical- plane motions of a ship advancing in irregular head seas. A time record of the free surface was con- structed using a two-parameter Pierson-Moskowitz spectrum with a significant wave height H~/3=10 feet and a modal period Tm=12 seconds with low and high frequency cutoffs at 0.1 and 4.25 rad/sec respectively. A 400 foot long WRT hull with constant forward speed corresponding to Fn=0.2 was first chosen. Figure (22) shows the time history of the inci- dent wave elevation at the ship center of gravity, along with the pitch and heave displacements from LAMP calculations (N=160, At=0.3526 seconds). For contrast, strip theory predictions obtained by superposition of the linear frequency-domain re- sponses at the component incident frequencies are also included. To be consistent with the strip the- ory program used here, the LAMP calculations in- clude the hydrostatic (and Froude-Krylov) forces only up to the mean waterline. To remove the ef- fects of starting transients, the LAMP simulation has been started from rest at t=-10 seconds. As can be seen in Fig. (22), the differences between LAMP and strip theory are small for the pitch dis- placement, with the exception of apparent mem- ory (transient) effects around 10 c t < 60 and 60

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to cO T o co LAMP Strin theory Do 0 _ to 1 o CD 1 0 0 0_ 1 \ ~ 1/ I O ~BY' (J ~ INJI J ~I\ .' ~l:~` I' ~ 1 1~\ " :t ~7 :1: ~ ,1 v V V \`,1 V1 o 1 _ 1 30.0 60.0 ~ 90.0 t (see) Fig. 22: Incident wave elevation (~) at the ship's center of gravity, pitch angle All, and heave displacement (z) of a WRT hull at Fn=0.2 in irregular head seas. 6

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lo ~ o ~- 1 o r\' - ~o ~ o ~ - . 1 o - ll 600 90.0 Fig. 23: Incident wave elevation (~) at the ship's center of gravity, pitch angle All, and heave displacement (z) of a Series 60 (CB=0.7O) hull at Fn=0.2 in irregular head seas. 62

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| ~ t=0.9305 second v 1 0.2 ~ _ _ 1 4 _ - - - 1 V -1 Z~. t=4.4564 second vim Fig. 24: Non-dimensional dynamic pressure distribution on a WRT hull at two time instances. 80 < ~ < 90. The same is true for the heave dis- placement except for a nearly constant downward shift of the nonlinear result. This is caused by the steady sinkage force associated with forward speed which is present in LAMP but absent in the strip theory (from Fig. 21, this is estimated at about 0.53 feet for a 400 foot hull). Overall, the rela- tive good performance of the strip theory is not unexpected for the mathematical WRT hull (with relatively small geometry changes near the water line) and for head seas. For a more realistic geometry, we show in Fig. (23) the corresponding case for a 400 foot long Series 60 (CB=O.7) hull at Fn=0~2. In this case, strip the- ory overpredicts the pitch motion significantly, in some instances by over 50% (e.g., near t=75 see). For the heave displacement, there are appreciable underpredictions of the troughs in addition to the absence of the steady downward linkage. These de- ficiencies of strip theory for the more complicated Series 60 hull are consistent with the observations of Frank & Salvesen (1970~. As a final illustration, we display the pressure distribution on the hull as output from LAMP. Fig. (24) shows the dynamic pressure distribution on the WRT hull at t=0.9305 and 4.4564 seconds respectively corresponding to the case in Fig. (22~. Note that the pressures are given on the actual sub- merged surface. The complete unsteady loads on the ship hull can now be obtained from the pres- sure distribution and motion histories. 5 CONCLUSIONS A time-domain method, LAMP (karge-Amplitude Motion Program), was developed for the general large-amplitude motions of a three-dimensional surface-piercing body in a seaway. The body boundary condition is satisfied exactly on the in- stantaneous underwater body surface while a lin- earized free-surface condition is used. To validate the approach and evaluate its accu- racy, the method was applied extensively to obtain linearized motion coefficients for a number of dif- ferent geometries with or without forward speed. The results include added-mass and damping coef- ficients, wave exciting forces and steady wave resis- tance, sinkage and trim forces and moments. These are compared to experimental measurements and existing linear time- and frequency-domain calcu- lations. The comparisons are overall satisfactory for all the results and show that LAMP is equal or superior to any of the existing computational methods in terms of accuracy. The main feature and purpose of LAMP, however, is for general nonlinear large-amplitude motions. To illustrate its effectiveness, we apply LAMP to study the large-amplitude forced heaving of a (complete) floating sphere; the large vertical-plane motions and the free sinkage and trim of a Wigley hull moving with constant forward speed; and the general time-dependent large-amplitude motions of a Series-60 ship advancing in an irregular seaway. 63

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Some of the main findings are the importance of (added) steady (and higher-harmonic) compo- nents, and the modifications of the first-harmonic (excitation frequency) motion coefficients. In the first case, the presence of large-amplitude heaving motions is shown to result in significant increases of the wave resistance and steady sinkage and trim forces. In the latter, the nonlinear "added-mass" is found to decrease markedly with increasing heave amplitude. The consequent reduction of inertia and increase in natural frequency may have impor- tant implications to the motion dynamics of the ship. This may also explain some of the exper- imentally observed dependencies on amplitude of normalized motion response functions. When ap- plied to general time-dependent motions in irregu- lar waves, LAMP demonstrates the importance of transient (memory) and nonlinear geometry effects especially for realistic ship geometries where strip theory is found to be inadequate. The road is now laid for nonlinear simulations of extreme episodic events and complete load and motion predictions. The current version of LAMP is fully vectorized for high-speed vector processors. For a nonlinear (large-amplitude) simulation using 0~150-200) un- knowns on the body and a similar number of time steps, the typical CPU time on a single Cray Y-MP processor is 0~1-2) hours. Further code optimiza- tion may reduce this requirement by a small factor. For applications involving significantly larger num- ber of unknowns and time steps, the time domain formulation may be particularly suited for parallel algorithms on multiple processors. Acknowledgement This research was sponsored by the Office of Naval Research, the U.S. Coast Guard, and the De- fense Advanced Research Projects Agency. We are grateful to Cray Research, Inc., for the use of their Cray Y-MP/832 supercomputer. Some computa- tions were also performed on the NSF Pittsburgh Supercomputer Center Cray Y-MP. We thank M. Meinhold and K. Weems for valuable technical and graphical help. REFERENCES Adachi, H. & Ohmatsu, S. (1979), "On the influence of irregular frequencies in the integral equation solutions of the time dependent free surface problems" J. Eng. Math., 16: 97-119. Bai, K.J. (1979), "Overview of results," Proc. of the Workshop on Ship Wave-Resistance Computation, DTNSRDC, USA. Beck, R.F. & Liapis, S.J. (1987), "Transient motion of floating bodies at zero forward speed" J. Ship Res., 31: 164-176. Beck, R.F. & Magee, A. ( 1990), "Time-domain analy- sis for predicting ship motions," Proc. IUTAM Symp., Dynamics of Marine Vehicles & Structures in Waves, London. Bertram, V. (1990), "A Rankine source approach to forward speed diffraction problems," Proc. 5th Intl. Workshop on Water Waves and Floating Bodies, Manchester. Boppe, C. W., Rosen, B. S., & Laiosa, J. P. (1987), "Stars and Stripes 87: Computational flow simula- tions for hydrodynamic design,", 8th Chesapeake Sailing Yacht Symposium (SNAMEJ, pp. 123-146. Chang, M.S. (1977), "Computation of three-dimension- al ship motions with forward speed," Proc. 2nd Intl. Conf. Num. Ship Hydro., UC Berkeley, California. Chang, B. & Dean, J.S. (1986), "User's manual for the XYZ Free Surface Program," Report No. DTNSRDC- 86/029. Cohen, S.B. (1986), "A time-domain approach to three- dimensional free-surface hydrodynamic interaction in narrow basins," Ph.D. Thesis, U. Michigan, Ann Ar- bor, Michigan. Cummins, W.E. (1962), "The impulsive response func- tion and ship motions" Schiffstechnik, 9: 124-135. Dagan, G. & Miloh, T. (1980), "Flow past oscillating bodies at resonant frequency," Proc. 18th Symp. Naval Hydro., Tokyo, Japan, 355-373. Dawson, C.W. (1977), "A practical computer method for solving ship-wave problems," Symp. 2nd Intl. Conf. Num. Sh* Hydro., UC Berkeley, California. Dawson, C.W. (1979), "Calculations with the XYZ Free Surface Program for five ship models," Proc. Workshop Ship Wave-Resistance Computations, DTRC. Dommermuth, D.G. & Yue, D.K.P. (1988), "The non- linear three-dimensional waves generated by a moving surface disturbance," Proc. 17th Symp. Naval Hydro., The Hague, The Netherlands. Dommermuth, D.G. & Yue, D.K.P. (1986), "Study of nonlinear axisymmetric body-wave interactions," Proc. 16th Symp. Naval Hydro., UC Berkeley, California. Ferrant, P. (1988), "Radiation d'ondes de gravite par les deplacements de grande amplitude d'un corps immerge: comparaison des approaches frequentielle et instation- naire," these de Doctorat, Universite de Nantes. Finkelstein, A. (1957), "The initial value problem for transient water waves" Comm. Pure App. Math., 10. Frank, W. & Salvesen, N. (1970), " The Frank close 64

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fit ship-motion computer program," DTNSRDC Report No. 3289. Froude, W. (1868), "Observations and suggestions on the subject of determining by experiment the resistance of ships," The Papers of William Froude: 120-127. Gadd, G.E. (1976), "A method for computing the flow and surface wave pattern around full forms," Trans. Roy. Inst. Naval Arch., 113: 207. Gerritsma, J. (1960), "Ship Motions in Longitudinal Waves," Intl. Shipbuilding Progress, 66: 42-95. Gerritsma, J. (1988), "Motions, wave loads and added resistance in waves of two Wigley hull forms," Rep. No. 804, Technical University of Delft, Netherlands. Guevel, P. & Bougis, J. (1982), "Ship motions with forward speed in infinite depth," Intl. Shipbuilding Progress, 29: 103-117. Hess, J.L. & Smith, A.M.0 (1964), "Calculation of non- lifting potential flow about arbitrary three-dimensional bodies," J. Ship Res., 8: 22-24. Hulme, A. (1982), "The wave forces acting on a floating hemisphere undergoing forced periodic oscillations," J. Fluid Mech., 121: 443-463. Hulme, A. (1983), "A ring-source/integral equation method for the calculation of hydrodynamic forces exe- cuted on floating bodies of revolution," J. Fluid Mech., 128: 387-412. Inglis, R.B. & Price, W.G. (1981), "A three-dimensional ship motion theory comparison between theoretical prediction and experimental data of the hydrodynamic coefficients with forward speed," Trans. Roy. Inst. Naval Arch., 124: 141-157. Jensen, G., Soeding, H. & Mi, Z.X. (1988), "Rankine source methods for numerical solutions of the steady wave resistance problem," Proc. 17th Symp. Naval Hy- dro., The Hague, The Netherlands. King, B.W. (1987), "Time-domain analysis of wave ex- citing forces on ships and bodies," Report No. 306, Dept. Naval Arch. Marine Eng., U. Michigan, Ann Arbor, Michigan. King, B.W., Beck, R.F. & Magee, A.R. (1988), "Sea- keeping calculations with forward speed using time don main analysis," Proc. 17th Symp. Naval Hydro., The Hague, Netherlands. Korsmeyer, F.T. (1988), "The first and second order transient free-surface wave radiation problems," Ph.D Thesis, Dept. Ocean Eng., MIT, Cambridge, Mas- sachusetts. Korsmeyer, F.T., Lee, C.-H., Newman, J.N. & Sclavounos, P.D. (1988), "The analysis of wave effects on tension leg platforms," Proc. Intl. Conf. Offshore Mech. ~ Arctic Eng., Houston, Texas. Larsson, L. (1987), "Numerical predictions of the flow and resistance components of sailing yachts," Proc. Conf. Yachting Tech., U. W. Australia, Australia. Letcher, J.S., Weems, K.M., Oliver, J.C., Shook, D.M. & Salvesen, N. (1989), "SLAW: Ship Lift and Wave Code, theory, implementation, and numerical results," SAIC Report 89/1196. Liapis, S.J. & Beck, R.F. (1985), "Seakeeping compu- tations using time-domain analysis," Proc. 4th Intl. Conf. Num. Ship Hydro., National Academy of Sci- ences, Washirlgton, D.C., 34-54. Liapis, S.J. (1986), "Time-domain analysis of ship mm tions,", Report No. 302, Dept. Naval Arch. Marine Eng., U. Michigan, Ann Arbor, Michigan. Lin, W.M., Newman, J.N. & Yue, D.K.P. (1984), "Non- linear forced motions of floating bodies," Proc. 15th Symp. Naval Hyd rot, Hamburg, West Germany Lin, W.M. & Yue, D.K.P. (1990), "LAMP Large- Amplitude Motion Program)," SAIC report (in prepa- ration). Magee, A.R. & Beck, R.F. (1988), "Compendium of ship motion calculations using linear time-domain anal- ysis," Report No. 310, Dept. Naval Arch. Marine Eng., U. Michigan, Ann Arbor, Michigan. Magee, A.R. & Beck, R.F. (1989), "Vectorized compu- tation of the time-domain Green function," Proc. 4th Intl. Workshop on Water Waves &' Floating Bodies, Oystese, Norway, 139-144. McCarthy, J.H. (1985), "Collected experimental resis- tance component and flow data for three surface ship model hulls," Report No. DTNSRDC-85/011. Michell, J.H. (1898), "The wave resistance of a ship," Phil. Mag., 45: 106-123. Nakos, D.E. & Sclavounos, P.D. (1990), "Ship motions by a three-dimensional rankine panel method," 18th Symp. 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DISCUSSION Pierre Ferrant Sirehna S.A., France I would like to make a comment on your results about the heaving sphere. I do not completely agree with your analysis of the behaviour of the damping coefficient which you show to remain constant when amplitude is varied. My own results show that at least for a submerged body, the nonlinear phenomena associated to the body boundary condition are very frequency-sensitive. In fact, when dealing with oscillatory motions, even in the time domain, one cannot ignore the importance of frequency and this parameter must be varied before drawing conclusions. I would therefore be very interested if you could give result for the surface-piercing heaving sphere at lower frequencies, say about wet R/g=0.4. AUTHORS' REPLY We have preliminary results for the heaving (surface-piercing) sphere for normalized frequencies ranging from w ~ 0.4 to ~ 3 and amplitudes rangin, from Ah/a ~ O to ~ 0.5 or higher. In contrast to your results for the submerged sphere, the damping coefficient remains relatively independent of amplitude for lower frequencies (w < ~ 1) and shows some sensitivity only for intermediate frequencies. The precise mechanisms for these dependencies (an amplitude and frequency) are as yet not completely understood. The following table lists the results for the frequency ~ = 0.4 you su~gest~l. For comparison, the data for Fig. (1) at w = 1.0, as well as w = 1.5 and 3.0 are also included. Again, we note that these results are only preliminary. Normalized damping coefficients for a (surface-piercinz) heaving ~here. Ah/a 0.125 0.250 0.375 0.500 w=0.4 w= 1.0 w= 1.5 w=3.0 0.036 0.195 0.151 0.314 0.036 0. 195 0.154 0.3 12 0.035 0.195 0.170 0.312 0.035 0.194 0.185 0.316 66