Cover Image

PAPERBACK
$203.25



View/Hide Left Panel
Click for next page ( 174


The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement



Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 173
Nonlinear Free Surface Waves Due to a Ship Moving Near the Critical Speed In a ShaBow Water H.-S. Choi, K.J. Bai, I.-W. Kim, I.-H. Cho (Seoul National University, Korea) ABSTRACT This paper describes two methods of solution to the nonlinear free-surface waves generated by a ship moving steadily with a transcritical speed in a shallow water. As a mathematical model, a non- linear initial/boundary-value problem is formulated within the scope of potential theory. One method is based on matched asymptotic expansion techniques and the Kadomtsev - Petviashvili equation is ob- tained as the leading-order solution for a slender ship. The other one is based on classical Hamil- ton's principle and the finite element method is im- plemented for numerical calculations. In order to examine the effect of the tank width on the wave field and resulting hydrodynamic forces, computa- tions are made systematically for the Series 60 ship model with Cb = 0.8 by these two different methods. For wider tanks, the pressure distribution on the free surface, equivalent to the ship model, is treated. The results obtained by two different methods are com- pared each other and with experimental measure- ments available. Also discussed are the appearance of stem waves at the tank wall and the evolution of the crestline of diverging waves in a wide tank. NOMENCLATURE A cb D Fh 9 J L N NO Ni : typical wave amplitude : block coefficient of ship : fluid domain : waterdepth-based Froude number : gravitational acceleration : water depth : functional : Lagrangian or ship's half length : total number of nodes : total number of free surface nodes : trial function basis non, nil, no) : outward unit normal vector p q S (x) SO SF SF sm SO t t* T. g U W X,y~z xc CY p INTRODUCTION . pressure : source strength : longitudinal distribution of cross sectional area of ship : blockage coefficient : free surface : projection of SF : maximum cross sectional area of ship : ship surface : time : final time : generation period between first two solitons : ship's speed : tank's half width : rectangular coordinates : x-location of the crestline at y = 0 : speed parameter or unwinding parameter : blockage index : slenderness parameter or variational operator : nonlinear parameter : surface elevation : tank width parameter : dispersion parameter : water density A free-surface flow of an ideal fluid caused by a ship translating with a constant speed near the shal- low water celerity is described by an initial/ bound- ary value problem governed by the Laplace equation with the free surface as a part of solution. In the past, problems of this type were nor- mally treated after the boundary conditions on the unknown free-surface had been linearized. Recently, however, there are growing interests in solving the Hang S. Choi, Kwang J. Bai, Jang W. Kim, Il H. Cho Department of Naval Architecture, Seoul National University, Kwanak-Ku, Seoul 151-742, Korea 173

OCR for page 173
nonlinear Resurface problems more exactly. In some cases, it is of vital importance since linearized solutions fail to predict experimentally-identified phe- nomena. One example is the generation of upstream- advancing solitons by moving disturbances in shal- low water. A comprehensive explanation on the physics involved is given by Wu A. There is a line of investigations on this non- linear free-surface problem based on shallow water approximations which result in a variety of theories such as the Korteweg- de Vries (KdV), Kadomtsev - Petviashvili (KP), Boussinesq equations and the Green - Nagdhi formulation (GN); Many references in this area can be found in Ertekin and Qian [2~. To name few, Mel and Choi Al, Katsis and Akylas A, Wu and Wu [5] and Ertekin, Webster & We- hausen t6] considered thre~dimensional problems. There is another line of approach based on a nu- merical method as finite difference or finite element methods. Bai, Kim & Kim [7] were the first who ap- plied the finite element method to a 3-dimensional nonlinear shallow water wave problem. In the present paper, we concern with theo- retical and numerical methods for solving a nonlin- ear three-dimensional Resurface flow problem in a shallow water. Specifically, a ship moving near the critical speed is treated to numerically simulate the experimental condition in the towing tank. It is formulated as an initial/boundary value problem within the scope of potential theory. As the solu- tion procedure for the nonlinear problem, two dif- ferent methods are described herein. In the first method, the given problem is reduced to a homo- geneous KP equation with flux conditions on the boundaries. The ship is simplified to an equivalent slender body. Then the KP equation is numerically solved in the two-dimensional horizontal Resurface plane by an explicit finite difference scheme. In the second method, the original problem is replaced by an equivalent variational problem based on Hamil- ton's principle applied to water waves derived by Miles [8~. Then the variational functional defined as an integral in the unknown three-dimensional fluid domain is solved numerically by the finite element method. The variational functional used here is basically the same as the well-known Luke's vari- ational principle A. However, the present func- tional is more advantageous in numerical computa- tions compared to Luke's principle. Recently, these two methods have been suc- cessfully applied to the generation and emission of solitons in the upstream and complicated waves in the downstream due to a moving ship in shallow wa- ter [7,10~. In these papers, however, no systematic investigations on the effect of the side walls have been undertaken. In the present study, it is our intention to clarify the effect of the width of the side walls on the wave response and hydrodynamic forces. Thus the numerical results of the free sur- face elevations, hydrodynamic forces (i.e. wave resis- tance, lift and trimming moment) acting on a ship obtained from the both methods are presented and compared partially with the experimental findings of Ertekin [11~. The formation and development of stem waves is illustrated, when the generated waves are reflected at the tank wall. Also discussed are the evolutions of the crestline of diverging waves in a quite wide tank. I N I T I A L / B O U N D A R Y - V A L U E FORMULATION We consider a ship advancing steadily with a transcritical speed U along the centerline of a shal- low tank. A rectangular Cartesian coordinate sys- tem moving with the ship's speed U is used, in which the x-axis coincides with the longitudinal axis of the ship and the z=0 plane is the undisturbed free sur- face. The ship directs toward the negative x-axis and the positive z-axis points upward (see Fig.1~. Under the usual assumptions in potential theory, fluid motions are expressed in terms of a velocity potential, (x,y,z,t), which is the solution of the Laplace equation V2 = 0 -h < z ~ ~ (1) in the fluid domain D, where h and ~ are the wa- ter depth and the free-surface elevation, respectively. The kinematic condition is imposed on the ship sur- face So in--Un2 ~ (2) where n = (no, no, no) denotes the outward unit nor- mal vector on So. No net flux condition also holds at the tank bottom and side walls (z=0 z=-h, +~=0 y=+W. (3) (4) The kinematic and dynamic boundary condi- tions on the free surface SF must be satisfied id = (t+(U+z)(z+~(v , (5) 9~+t+Uz+ 2 1 Vat 12 +P =0 , (6) where 9 is referred to the gravity constant, p to the fluid density, and p = p(x,y, t) to the pressure, which is taken zero when the pressure distribution on SF is absent. By assuming that the fluid is initially at rest, the initial condition may be given as 74

OCR for page 173
==0 at t=0, (7) and the radiation condition yields to ) ~ O as X2 + y2 ~ oo (8) It is to note that a modified radiation condition is utilized in computations for the downstream bound- ary. METHODS OF SOLUTION Since the concept of the two methods employed here is quite different by their nature, a brief descrip- tion on each method is necessary. We begin with the theoretical part,then the numerical part follows. Matched Asymptotic Expansion Technique In order to analyze the above nonlinear prob- lem, further assumptions and limitations are required The first step is to introduce appropriate smallness parameters, with respect to which the expressions given in the previous section are to be perturbed. Hereby we define two small parameters ~ = A/h' ~-h/L (~9~) and assume ~ = p2, where A and 2L mean the typi- cal wave amplitude and the ship length, respectively. It corresponds to the Ursell number of order of unity, which implies that the nonlinearity and the disper- sion are both important to the leading-order solu- tion, i.e. we are dealing with a weakly nonlinear dispersive wave system. However, it may not be a serious restriction because the above assumption seems to be valid in a wide range of Ursell numbers as shown by Lee, Yates & Wu t12~. Since we are in- terested in the ship's speed in the neighborhood of the critical Froude number, it is expanded as follows: Fh = 1 - 2ap2 with ~ = 0~1) . (10) In order to include the lateral dispersion as well as the longitudinal dispersion, we have to choose a wide tank in comparison with the ship length W/L = 1/~n with ~ = 0~1) . (11) It is in general recognized that the governing parameter of the problem is the blockage coefficient, which is simply the area ratio of the midship to the tank cross-section. As pointed out by Mel t13], the order of magnitude of the blockage coefficient must be O(p ~ SB = Sm/2Wh = 0~4), (12) where Sm is the maximum cross-sectional area of a ship. If we assume the ship to be slender, of which the characteristic transverse dimension is denoted by RO, then the slenderness parameter becomes ~ = Ro/L = o(~2) . (13) It indicates that the nonlinearity arises directly from the disturbance caused by a slender ship. As a result, we have four characteristic lengths in this problem; water depth (h), tank width (2W), ship's length (2L) and transverse length (RO), which have vastly different scales each other. h/L = 0~), W/L = 0~~~), RO/L = 0~2~. (14) To accommodate these in our analysis in a consis- tent manner, it is adequate to divide the fluid do- main into three regions; near the ship, far from the ship and an intermediate region therebetween. The procedure of the derivation has been reported in de- tail in t10~. Hence we cite here only the results. In the far field, the geometry of the tank affects the propagation of waves, but the generation mech- anism of the waves is not known. The wave field is described by a homogeneous two-dimensional KdV or KP equation [14] 2 6(ZZ2 + 20 / ~VdX . (15) It shows a balanced interplay between the nonlinear and the two-dimensional dispersion. It is a three- dimensional counterpart of the KdV equation, be- cause it contains the lateral dispersion as well as the longitudinal dispersion. In the above, the variables are made dimensionless by X = LX~, Y-WY~, ~ = A(', t = 2~ ~ (16) For the sake of brevity, the primes are dropped here- after. In the near field, i.e. in the flow region closely around the ship, the kinematic boundary condition on the ship surface should be invoked. For a slender body, Eq.~2) can be replaced by l~n ~-(U + ~Z)R2~1 + (R`/R)2~-~12 ~ (17) where the normal derivative on the ship surface is approximated by that on its transverse plane. Here RO stands for the circumferential derivative of the cross section. The presence of a ship and its mo- tion can be represented by source distributions. By applying the law of mass conservation to a fluid do- main surrounded by the ship surface, the free surface and a control surface located far away from the ship, but still within the near field, the source strength is readily determined q= 2pSzfx) With '= _SB4, (18) where S(x) is the longitudinal distribution of the cross sectional area of ship. 75

OCR for page 173
In the intermediate region, solutions of the far and the near fields are matched. As a result, the boundary condition for v at y = 0 turns out to be IVY=, 0, r) = - 12,BSzztx) . (19) And we have the leading-order hydrodynamic pres- sure P = /`,2psh; + (113) . (20) The wave response can be computed by the KP equation given in Eq.(15) with the boundary conditions given in Eq.(4) and Eq.(19). The hydro- dynamic forces and moment can be estimated based on the slender body approximation A. To do it, a simple explicit finite difference scheme is imple- mented for the KP equation, in which forward dif- ferences are chosen for time derivatives and central differences for spatial derivatives. But at the wall and the centerline of the tank, one-sided differences are used in order to incorporate with boundary con- ditions. Unidirectional Sommerfeld-type radiation conditions are imposed on both open boundaries. Neverthless, a relatively large computation domain ahead of the ship is provided to avoid numerically reflected waves from the open boundary. Then the computation domain is gradually enlarged in both directions as the computation proceeds. Based on numerical experience, the grid size and time incre- ment are chosen as Ax = 0.1, I\r = 0.00002 and Ay is so taken as the ratio of Ax/~\y remains unity in the physical plane for better resolution of dispersion. The Series 60 with Cb = 0.8 is numerically modelled in terms of the longitudinal distribution of its cross-sectional area. However, the portion of both ends has been slightly modified by a parabolic distribution in order to satisfy the slender body as- sumption. Finite Element Method The finite element method has been success- fully applied to nonlinear water-wave problems, for example, Washizu et al.~15], Ikegawa t16], Nakayama & Washizu t17], Washizu & Nakayama [18], Betts & Assaat [19], Bai, Kim & Kim t7] (hereafter referred to as BKK), Bai, Kim & Lee t20], Bai, Kim & Lee t21] and Kim & Bai t22~. The finite element method is based on Luke's variational principle in [16~-~19~. However, in the present paper, the variational func- tional given in Miles [8] is used as the basis of the finite element computations. This variational form is simply a direct application of the classical Hamil ton's principle to the nonlinear water-wave problem. For the problem at hand we can define the functional J and the Lagrangian L as follows: rt* JO L aft, (~21) /SF ) (by + U(Z~) dS-U /s n2 ~ dS 2 /D ~ dV 2 /- ~ dS r`22') - -/ P ~ dS, p so whereSF is the projection of SF on the Oxy plane and t* is the final time. ~ denotes the velocity poten- tial on the free surface, i.e. (x, y, t) = flex, y, a, t). By taking the variations on J with respect to the unknown functions, ~ and , we obtain /SF{(' 6~)~=~* - (g) [~)~=0) dS (~23) /0 /sp (~t + USA +2~V~2 + gz + P)Z=! Use dS aft, /o ~ /SF (O + U<2 - n in) bI dS - /S bin+ Unz)[if dS + /D V2 {' den. (24) Here [J = [Jo + [Jo,. Equation (,23`J shows that the dynamic free-surface boundary condition is re- covered from the stationary condition on J for the variation of ~ at each time step. The wave eleva- tions at t = 0 and t* are supposed to be specified as the constraints. Equation (24) shows that the kine- matic condition on SF and the governing equation in the fluid domain are recovered from the stationary condition on J for the variation of . In the numerical procedure for the applica- tion of the finite element method, we discretize the fluid domain into a number of finite elements. Then we approximate ~ in N-dimensional function space whose basis is continuous in D. We denote the basis of this trial space by {NiJ`=,,...,N. It is convenient to introduce another set of basis function, denoted by {Mk~k=l,...,NF, which is defined only on the free sur- face. By the introduction of these basis functions, one can represent , ~ and ~ as 76

OCR for page 173
+(x, y, z, t) = ~i(t)Ni(=, Y. a; a), (25) i(x, y, t) = (k(t)Mk(X, a), (26) ((x, y, i) = (k(t)Mk(X, A), (27) where Mi(x, y) = Nit (x, y, z; a) A=, (28) k= 1,...,NF. Here NF is the total number of nodes on SF and it is the nodal number of the basis function N`, of which the node coincides with that of the free-surface node k. Summation conventions for the repeated indices are used here. It should be noted that the basis function {Ni Ji=t,...,N is dependent on the free-surface shape z = `(x,y,t) but its restricition on SF is the function of (x, y) and independent of <. This special property of {Mk~k=~, ,NF is maintained here since new nodal points in D are shifted only along the z-axis at each time step. The tensors Kit, Pi' are the kinetic and po- tential energy tensors and Tk' is the tensor obtained from the free-surface integral, which can be inter- preted as a tensor related to the transfer rate be- tween these two kinds of energy. It is of interest to note that in Eq.(29), Pi' = gTki. However, Tk' will be defined differently from this in the present computation by introducing the lumping scheme. The stationary condition on J = r Ldt is equiv- alent to the following Euler-Lagrange equation Tii dt ` = - UCi`~' (30) - -Pi-JO -Pi! ~-Pi ~ Tk'-~ = - UCi`0 (31) Once the trial function is represented by using dt the above basis function, the Lagrangian L can be + K`,~jIj + A,, written as Kiwi = - fi for i ~ it- (32) L = ~kTk! dt ~ + UkCk' ~(29) - 2`K`i~i-fists` - 2 (kPkI ~-Pl 0, where Tk! = ~ MkM' dS, Ci`=; Mi-dS, SF [3X Kij = / VNi VNj dV, Pal = 9 JO MkM' dS, SF Here Eq.(30) and Eq.(31) are the nonlinear ordi- nary differential equation for Ink, ~k}k=l,---,NF and Eq.(32) is the algebraic equation for {i}iii,t which is the constraint for the above two equations. Here it should be noted that the second term on the right- hand side of Eq.(31) is computed by the volume in- tegral as originally defined, whereas BKK used the surface integral reduced from the original volume in- tegral. This change is made in the present work since the previous computation in BKK is found to be less accurate in the conservation of energy compared to the present scheme from our numerical test. Eq.(30) through Eq.(32) are less advantageous in computations with respect to the numerical sta- bility. To remedy this difficulty, we introduce the unwinding and local lumping schemes, which are often used in a wide class of computational fluid dynamics. Following these common steps, we ob- tain the final set of the reduced ordinary differential equations as follows: fi = Uis ri~Ni dS, d i = _ UT~-lCm`~' (33) T-1 (1~', Kits, + p + ) Pi = p is P(~, y, t)Mi d S. dd ~ = _ UT.-l Cm`~` (34) + Tkm (6imjij + film) ~ 177

OCR for page 173
Kijjj = - fi for i := id, where (35) Tk' = ~ (Mi + ~= ~ Mi) M: dS (36) ISF ( 2 a=Mi) COME dS, (37) {omits' ,k=l; Tki= ~ ~ 0 ~ otherwise. (38) Here car is the upwinding parameter as defined in Hughes & Brooks t234. In Equations (33) and (34), a consistent upwinding scheme (also known as Petrov- Galerkin method) is employed as discussed in t23], whereas an inconsistent upwinding scheme was used in BKK. We leave out the detail procedures here since one can find them in BKK. In the finite element computations, two numer- ical models are treated, i.e. a Series 60 ship model with block coefficient Cb=0.8 and a pressure patch on the free surface equivalent to the above ship model To simplify the finite-element grid generation, the Series 60 ship model is replaced by an vertical wall- sided ship which has the same cross-section area and the constant draft of the original model along the ship length. This means that the equivalent numer- ical ship model has rectangular cross section with a constant draft. The finite element subdivision un- der the ship's bottom is unchanged while the other finite element subdivision is changed at each time step to accomodate the new location of the free sur- face. An eight-node isoparametric element is used and the integration is carried out analytically along the vertical direction. In the computations for the pressure patch on the free surface, a single finite element is taken along the depth with a higher-order polynomial basis func- tion which satisfies the bottom condition. In this case the integration along the vertical direction is also carried out analytically. This is the so-called Aversion while the former case is the in-version in the adaptive finite element method. NUMERICAL RESULTS & DISCUSSIONS To simulate the tank tests t11], the Series 60 ship with Cb = 0.8 is chosen as the numerical model. Its length, beam and draft are 1.52m, 0.23m and 0.075m respectively. The water depth is 0.15m. The tank widths are 1.22m, 2.44m, 4.88m. In addition to these, a much wider tank is considered to exam- ine the effect of the tank width on the formation and propagation of upstream waves by the method based on the KP equation. But in the finite ele ment computations, an equivalent pressure patch is treated by increasing the width up to 480 times the water depth, since this is easier to compute than a ship model. Throughout the computations, the motion is assumed to start with a prescribed constant speed as a step function. In presenting the two sets of computed results, we denote those obtained by the KP equation with a slender body approximation by KP, and those of the finite element method by FEM. The physical quantities with dimensions are shown as functions of the real time for the convenience to compare with the earlier experimental work t11~. Accordingly the wave resistance and the lift force are given in Newton and the moment in Newton-meter. The numerical results of the forces and mo- ment for the experimental condition with the tank width of 2.44 m are shown in Fig.2 through Fig.5. In these figures, the solid line corresponds to FEM, while the dotted line to KP, if not indicated other- wise. Fig.2a-2c display the time histories of the wave resistance for the depth Froude numbers Fh = 0~9, 1.0, and 1.1, respectively. For Fh = 0.9, the com- puted values by both methods coincide fairly well after 5 seconds. However, the discrepancies are con- siderable for Fh=1.0 and 1.1. It is of interest to note that the magnitude of the oscillatory behav- ior becomes large in the case of FEM for Fh = 1.1. To investigate this rather large discrepancy, two dif- ferent expressions for the pressure are used for the computation of forces and moment in FEM; the ex act Bernoulli's equation and linearized Bernoulli's equation based on the linear shallow water approxi- mation. These two sets of computations are given in Fig. 3a-3c, where the solid lines indicate the exact form and the dotted lines the linear approximation. These figures explain partly the source of the dis- crepancies observed in Fig. 2b and 2c, since the KP equation is an approximate solution to the problem. Furthermore, the ship has been assumed to be slen- der and only the leading-order pressure has been taken into account. It is to mention that the both methods give greater wave resistance than the ex- perimental measurement in all the cases studied. Fig. 4a-4c are the computed results of the lift force for Fh = 0.9, 1.0, and 1.1. Contrary to the wave resistance discussed above, the lift forces by FEM are consistently smaller than those by KP. FEM predicts a longer oscillatory period than KP. Presumably this is due to the difference in the dis- persion in the two methods. In the similar fashion, the moment with re- spect to the origin is illustrated in Fig.5a-Sc. The moment estimated by KP is almost twice as large as that by FEM for Fh = 0.9 in Fig 5a. The results of KP contain considerable oscillatory components, whereas those obtained by FEM are nearly constant 178

OCR for page 173
after 5 seconds. The time histories of the forces and moment at the critical speed for the tank widths of 1.22m, 2.44m and 4.88m, denoted by 1, 2, 3 in this order, are depicted in Fig.6 through Fig.8. The wave resis- tance is given in Fig.6, and the lift force and moment are in Fig.7 and Fig.8, respectively. The results ob- tained by KP are indicated by (a) and FEM by (b) in these figures. As the tank width becomes larger,the wave resistance and the moment decrease and they seem to converge to some limit values. But the lift increases as the tank width increases. The evolutions of the surface elevation evalu- ated at 90cm ahead of the bow (Gauge No. 3 in t11~) for the tank width of 2.44m are shown in Fig.9 a-c. The asterisk marks correspond to the experi mental measurements, but the starting time is only of qualitative meaning due to the different nature of the initial conditions in the computations and the experiments. For the subcritical speed, Fh = 0~9, the measured profile looks closer to FEM, but the trend is opposed for Ah = 1.0 and 1.1. Over the three speeds, the generation period of the upstream solitons is shorter in the results calculated by KP compared to those by FEM. It is observed that the mean water level is slightly higher in KP for Fh = 0.9. In Table 1, the amplitude and propagation speed of the first soliton, and the generation period be- tween first two solitons are listed together with the experimental measurements. Here the tank widths are indicated by the ratios to the ship length, namely W/L = 0.8, 1.6 and 3.2 for 2W = 1.22m, 2.44m and 4.88m, respectively. The amplitude measured by experiments is the smallest among three for W/L = 0.8, but it is somewhat between the two com- puted results for W/L = 1.6 and 3.2. It is inter- esting to note that KP predicts consistently higher amplitudes for all cases. For the propagation speed, the overall behaviour is similar as in the amplitude. But there is no clear-cut trend in the case of the generation period. The period measured in the ex- periment is longer than the numerical predictions for Fh = 0.9, whereas the result obtained by FEM is the longest for Fh = 1.1. Based on his systematic experiments, Ertekin t11] concluded that the characteristics of upstream- advancing solitons are determined primarily by the blockage coefficient and the detailed geometry of disturbances is of secondary importance. Recently Pedersen [24] and Ertekin & Qian [2] investigated the influence of parameters other than the blockage coefficient on the generation mechanism of solitons. Along this line, we carried out additional computa- tions by KP for a slender ship with a parabolic dis- tribution of cross sectional area (Cb = 0.667) with the same blockage coefficient as the Series 60 ship (Cb = 0.8~. The result for W/L = 1.6 is given in Ta- ble 2. Comparing it to the corresponding result for the Series 60 ship in Table 1, we can recognize that the amplitude and the period have been remarkably changed. The amplitude is decreased by about 0.06 times the water depth and the period is increased by about 20% for all three speeds considered. We may conjecture that the hullform, which may be properly represented in terms of ship's block coefficient, plays a significant role on the generation of solitons. Fig.10 and Fig.11 are the snapshots of wave contour around the ship in a tank of width 4.88m at the critical speed. Due to the different nondimen- sionalization, the time instances are slightly shifted in two figures. The solid lines represent a constant positive surface elevation and the dotted lines denote a negative surface level. Two adjacent lines differ the surface elevation by 0.04 times the water depth. The wave contours obtained by KP looks more com- plicated and upstream waves propagate faster than those obtained by FEM. It is partly due to the dif- ference in the propagation speed, as shown in Table 1 (b). In these figures, we can observe the formation of stem waves at the wall, when the waves generated from the bow are reflected there. The stem waves are further developed, as time elapses. It suggests that the formation of straight crestlines is associated with the stem waves, which supports the conclusion made by Pedersen t24~. The perspective views of the wave fields for the above case are illustrated in Fig.12 for KP and Fig.13 for FEM, respectively. Cautions should be paid that the vertical displacement is exaggerated by 5 times compared to the horizontal scales. The up- stream solitons and three-dimensional downstream waves are clearly shown. Fig.14 shows the wave resistance computed by FEM by systematically increasing the tank width for the pressure patch. In this case the pressure distribution is specified to have the same blockage coefficient as the ship model in the earlier exper- imental condition. The pressure is assumed by a trapezoial distribution in both x- and y- directions and constant along the length of the parallel middle body in the x-direction. The length of the pressure patch is taken to be same as the ship length. Along the y-axis the pressure distribution is assumed to be constant along 0.8 times the water depth and changes linearly to zero at the edges of the patch. The width of the patch is taken 2.4 times the wa- ter depth. The number indicated to the lines cor- responds to the tank, whose width is consecutively doubled starting from 2W = 1.22m upto 5. The line 6 is the case for the tank width of 72m. The oscillatory component in the 179

OCR for page 173
wave resistance is pronounced for small tank width, whereas it becomes insignificant as the tank width becomes very large. However, the mean value of the wave resistance remains nearly constant after 5 sec- onds. It is to note that the values of wave resistance for the pressure patch are smaller than those for the ship (see Fig.6b). For the above pressure patch, the computation domain is continuously enlarged up to 2W/h = 480. The maximum tank width treated here may be re- garded as a case of infinite width at that time. Be- cause the tank width is kept sufficiently large by in- creasing it at every time step so that the disturbance near the side walls is not felt in the computations. Fig.15 is three-dimensional wave profile at Ut/h = 320 obtained from FEM. It is to observe that two diverging waves have already advanced upstream. The first crestline is plotted on a logarithmic scale in Fig.16, where xc is referred to the x loca- tion of the crestline at the centerline of the tank. Although the slope in this scale varies slightly with time, it is approximately 0.5. It suggests that the crestline is almost a parabola, which was also dis- cussed by Redekopp t25] and Lee & Grimshawi26~. ACKNOWLEDGEMENTS This work has been supported by the Korean Science & Engineering Foundation under the Non- linear Ship Hydrodynamics Program, Grant No. 87020703. REFERENCES t1 ~ Wu, T.Y., Generation of Upstream-Advancing Solitons by Moving Disturbances, Journal of Fluid Mechanics, Vol.184, 1987, pp.75-99. t2 ~ Ertekin, R.C. and Qian, Z.-M., Numerical Grid Generation and Upstream Waves for Ships Moving in Restricted Waters, Proceedings of the 5th International Conference on Numeri- cal Ship Hydrodynamics, 1989, pp.421-437. t3 ~ Mei, C.C. and Choi, H.S., Forces on a Slen- der Ship Advancing Near the Critical Speed in a Wide Canal, Journal of Fluid Mechanics, Vol. 179, 1987, pp.5~76. t4 ~ Katsis, C. and Akylas, T.R., On the Exci- tation of Long Nonlinear Water Waves by a Moving Pressure Distribution.Part 2: Three Dimensional Effects, Journal of Fluid Mechan- ics, Vol.177, 1987, pp.49-65. t5 ~ Wu, D.-M. and Wu, T.Y., Precursor Soli- tons Generated by Three- Dimensional Dis- turbances Moving in a Channel, Proceedings of IUTAM Symposium on Nonlinear Water Waves, 1987, pp.69 -76. to ~ Ertekin, R.C., Webster, W.C. and Wehausen, J.V., Waves Caused by a Moving Disturbance in a Shallow Channel of Finite Width, Jour- nal of Fluid Mechanics, Vol.169, 1986, pp.275- 292. t7 ~ Bai, K.J., Kim, J.W. and Kim, Y.H., Nu- merical Computations for a Nonlinear Free Surface Flow Problem, Proceedings of the 5th International Conference on Numerical Ship Hydrodynamics,1989 pp.40~420. t8 ~ Miles, J.W., On Hamilton's Principle for sur- face waves,J. Fluid Mech., 83, pp. 395-387. t9 ~ Luke, J.C., A Vatriational Principle for a Fluid with- a Free Surface, Journal of Fluid Mechanics, Vol.27, 1967, pp.39~397. t10 ~ Choi, H.S. and Mei, C.C., Wave Resistance and Squat of a Slender Ship Moving Near the Critical Speed in Restricted Water, Proceed- ings of the 5th International Conference on Numerical Ship Hydrodynamics, 1989, pp.43 454. t11 ~ Ertekin, R.C., Soliton Generation by Mov- ing Disturbances in Shallow Water: Theory, Computation and Experiment, Ph.D. Thesis, University of California, Berkeley, 1984. t12 ~ Lee, S.-J., Yates, G.T. and Wu, T.Y., Experi- ments and Analysis of Upstream- Advancing Solitary Waves Generated by Moving Distur- bances, Journal of Fluid Mechanics, Vol.199, 1989, pp.56~593. t13 ~ Mei, C.C., Radiation of Solitons by Slender Bodies Advancing in a Shallow Channel, Jour- nal of Fluid Mechanics, Vol.162, 1986, pp.53- 67. t14 ~ Kadomtsev, B.B. and Petviashvili, V.I., On the Stability of Solitary Waves in Weakly Dis- persive Media, Soviet Physics- DOKLADY, Vol.15, NO.6, 1970, pp.53~541. t15 ~ Washizu,K., Nakayama, T. and Ikegawa, M., Application of Finite Element Method to Some Free Furface Fluid Problems, Finite Elements in Water Resources, Pentech Press, London, 1977, pp.4.247-4.246. t16 ~ Ikegawa,M., 'Finite Element Analysis of Fluid Motion in a Container, Finite Element Meth 180

OCR for page 173
oafs in Flow Problems, UAH Press, Alabama, 1974, pp.737-738. t17 ~ Nakayama, T. and Washizu, K., Nonlinear Analysis of Liquid Motin in a Container Sub jected to Forced Pitching Oscillation, Interna tional Journal for Numerical Methods in En gineering, Vol.15, 1980, pp.1207-1220. t23 t18 ~ Washizu, K., Nal~ayama, T., Ikegawa, M., Tanaka, Y. and Adachi, T., Some Finite Ele- ment Techniques for the Analysis of Nonlinear Sloshing Problems, Finite Elements in Fluids Vol.5, John Wiley & Sons, 1984, pp.357-376. t19 ~ Betts, P.L. and Assaat,M.I., Larg - Amplitude Water Waves, Finite Elements in Fluids, Vol.4, John Wiley & Sons, 1982, pp.10~127. t20 ~ Bai, K.J., Kim, J.W. and Lee, H.S., A Numerical Radiation Condition for Two Di- mensional Steady Waves, Proc. Workshop on Nonlinear Mechanics, Korea Soc. Theoretical and Applied Mechanics, Seoul, Korea, 1990, pp.11~132. t21 ~ Bai, K.J., Kim, J.W. and Lee, H.S., An Application of the Finite Element Method to a Nonlinear Free Sruface Flow Problem, The ,/_ y = W ~ \ ~ tax \ a} (~////////////////////////////////////~////////~.2 ~ y=-W ~ elf ~ z = -h Fig.1 Definition Sketch - z - 10 O- ~I ~I 0 5 10 15 Time (See) Second World Congress on Computational ME chanics, Stutgart, Germany, 1990. t22 ~ Kim, J.W. and Bai, K.J., A note on Hamil- ton's Principle for a Free Surface Flow Prob- lem, Journal of Society of Naval Architects of Korea (in Korean), 1990, (in print). ~ Hughes, T. J. R. and Brooks, A. A Them retical Framework for Petrov-Galerkin Meth- ods with Discontinuous Weighting Functions: Application to the Streamlin - Upwind Proce- dure, Finite Elements in Fluids, Vol. 4, John Wiley & Sons, 1982, pp. 47-65. t24 ~ Pedersen, G., Thre - Dimensional Wave Pat- terns Generated by Moving Disturbances at Uanscritical Speeds, Journal of Fluid Mechan- ics, Vol.196, 1988,pp.39-63. t25 ~ Redekopp,L.G.,Similarity Solutions of Some Tw - Spac - Dimensional Nonlinear Wave Evm lution Equations, Studies in Applied Mathe- matics, Vol.63, 1980, pp.185-207. t26 ~ Lee, S.-J. and Grimshaw, H.J., Upstream- Advancing Waves Generated by Thre - Dimen sional Moving Disturbances, Physics of Fluid A2~2),1990, pp.l94-201. TV ~ 1.' 11 l v- o (a) Fh= 0.9 I n 20 25 v 181 5 1 1 1 10 15 Time (See) (b) Fh= 1.0 1 1 25 / I ~ 1 ~i ~I 10 15 20 25 Time (See) (C) Fh= 1.1 Fig.2 Wave Resistance (2W = 2.44m) : FEM,---------: KP

OCR for page 173
1 10 10 80 10 15 20 2'5 0 Time (See) O (a) Fh= 0.9 0 5 5 10 15 20 25 Time (See) (a) Fh= 0.9 One 1 10 15 Time (See) (b) Fh= 1.0 , 1 ~1 O 20 25 60 ~/ ~Z20 ~ 0d -2 10 15 20 25 Time (See) (c) Fh= 1.1 Fig.3 Exact and Linear Approximate Wave Resis- tances by FEM (2W = 2.44m). : Exact,---------: Linear ap- proximation '1 , 1 1 0 5 1 1 1 10 15 Time (See) (b) Fh= 1.0 , I , I 20 25 , ~ / \ ~ it: : 1 ' 1 1 1 5 10 15 Time (SecJ (C) Ah = 1.1 Fig.4 Lift Force (2W = 2.44m) : FEM,---------: KP 182 1 1 1 20 25

OCR for page 173
- z; - 1~ o o ~ 1~ o u ~ l r~ - - - ~ ~1 o ~ - 5 10 15 Time (See) (b) Fh= 1.0 1 1 - 1 20 25 ~' 1 1 1 ' 1 ' I ' 1 0 5 10 15 20 25 Time (See) (c) Fh= 1.1 Fig.5 Trirruning Moment (2W = 2.44m) : FEM,---------: KP ~_ , 81~ ,'' '''. V\, ' ~I ~, , I , I , . I ( ) 5 10 15 20 25 Time (See) (a) Fh= 0~9 ~} , ~ ~. ~3 10 15 20 25 Time (See (a) KP 1~ l ~h = 1.0 ' 1 0 5 10 15 20 25 Time (See) (b) FEM Fig.6 Wave Resistances for Three Different Tank Widths ~ 1: 1.22m, 2: 2.44m, 3: 4.88m ) at the Critical Speed R(h _ 60 - -40 2 l / ' /~{ ' / ' ~ 't ~' ~ ~' Fh = 1.0 l 0 5 10 15 Time ( Sec (a) KP ' 1 1 20 25 60 ~' \ '- ~ 1 1 5 o - 1 ' 1 ' 1 ' 1 10 15 20 25 Time (See) (b) FEM Fig.7 Lift Forces for Three Different Tank Widths ~ 1: 1.22m, 2: 2.44m, 3: 4.88m at the Critical Speed 183

OCR for page 173
~ 1~ o 1~ ~ r _ o 0~ _ _ ~1~ o ~ ~\~/~ 1 c,$ ~_ ~ 2 =8 ~4 1 1 1 1 1 ' 1 5 10 15 20 25 Time (See) (a) KP g~>V~~N 5 10 15 Tirne (See) (b) FEM ~ 20 25 Fig.8 1Yimming Moment for Three Different Tank Widths ~ 1: 1.22m, 2: 2.44m, 3: 4.88m ) at the Critical Speed ~_ v - ~8 o .. - ct _ 4 , ~- ~ _` v ~c o . - o4 p ~C ~ )*'- ~ 10 1~s ~D O. 1 5 10 15 20 25 Time (See) (a) Fh= 0.9 ''''' 0 5 10 15 20 25 Time (See) (b) Fh= 1.0 ~0 . ,\ ~ * ~ *'7 0~ ~' *********5 10 15 - 20 Time (See) (c) Fh= 1.1 Fig.9 Eree-Surface Elevation at the Guage (9Ocm Ahead of the Bow) : FEM, ---------: KP, *******: EXP 6 ~' 20 -10 0 10 o (a) Ut/h = 19.65 -2n -1n 1C (b) Ut/h = 39.3 Fig.10 Wave Contour at the Critical Speed (KP, 2W = 4.88m) 184

OCR for page 173
(a) Ut/h = 20.0 -20 -10 0 10 I ~ ~/~ ! i ,if ~ (b) Ut/h = 40.0 Fig.ll Wave Contour at the Critical Speed (FEM, 2W = 4.88m) Ut/h = 78.6 R 185 Ut/h = 157.2 Ut/h = 196.5 Fig.12 Wave Evolution at the Critical Speed (KP, 2W -4.88m)

OCR for page 173
Ut/h = 40.n Ut/h = Born Ut/h = 120.0 ~ 1~ 1~ Z 1: ~ 7;.\_/ , an. _ 3 Fh = 1.0 1 1 1 1 ' 1 10 15 Time (See) (Y ' ' ' ' ' ! ' ~' ' ' 1 ' 1 0 5 10 15 20 25 30 35 40 45 50 Time (See) 2 20 25 Fig.14 Wave Resistance for Various Tank Widths with Ut/h = 160.0 Pressure Patch (Fh = 1.0) Ut/h = 200.0 Fig.13 Wave Evolution at the Critical Speed (FEM, 2W = 4.88m) 186

OCR for page 173
- ~ ~ ~ ~ ~ - Fig.15 Wave Field for ~ Wide Tank (Ut/h-320) 5 4 ~3 a 2 1 .J~ I/ ~ An' *: ~ * */ *****t = 5.0 ***** t - 19.8 *~* t - 39.6 Sec Sec Sec _ ~ ~ ~ r ~ I ~ l ' I " " " ~ I ~ T I I I I l l l l r I I I I I ~ I ~ I I I I I ~ I ~ i I 1 -O 1 2 3 4 Log (x-xc) Fig.16 Plot of the Crestline of the First Upstream Diverging Wave 187

OCR for page 173
Table 1 The Amplitude and Speed of the First Soli- ton, and the Generation Period Between First Two Solitons for Series 60 (Cb = 0.8) (a) Amplitude A/h W/L=0.8 W/L= I.6 W/L=3.2 KP .480 .315 .202 Fh = 0.9 FEM .384 .234 .134 EXP .367 .273 .143 KP .623 .445 .322 Fh = 1.0 FEM .566 .397 .285 EXP .551 .438 .303 KP .785 .625 .490 Fh = 1.1 FEM .686 .566 .475 EXP .608 .585 .480 (b) Speed C/ ~W/L=0.8 W/L=1.6 W/L=3.2 KP 1.218 ~1.155 1.078 Fh = 0.9 FEM 1.175 1.100 1.050 EXP 1.170 1.100 1.060 KP 1.293 1.216 1.153 Fh = 1.0 FEM 1.250 1.175 1.125 EXP 1.240 1.190 1.130 KP 1.367 1.304 1.227 Fh = 1.1 FEM 1.300 1.250 1.200 EXP 1.280 1.260 1.210 (c) Generation Period UT,/b W/L=0.8 W/L= 1.6 W/L=3.2 KP 20.04 31.83 41.26 Fh = 0.9 FEM 30.20 37.40 53.60 EXP 32.70 48.10 65.10 KP 24.89 39.29 60.26 Fh = 1.0 FEM 35.40 47.30 88.40 EXP 37.80 49.80 85.20 KP 33.14 54.75 90.78 Fh = 1.1 FEM 44.20 57.40 128.70 EXP 39.00 50.11 103.60 Table 2 The Amplitude and Speed of the First Soli- ton, and the Generation Period Between First Two Solitons for a Slender Ship (Cb = 0.667) A/h C/~fik UT9/h Fh = 0-9 .253 1.123 38.91 Fh= 1.0 .384 1.216 48.47 Fh=1.1 .565 1.275 66.28 188

OCR for page 173
DISCUSSION William C. Webster University of California at Berkeley, USA In reference [6], we presented numerical results based on Green- Naghdi theory for the same problem. Did you compare your results with these computations? AUTHORS' REPLY We are well aware of your excellent paper coauthored with Profs. Ertekin and Wehausen, where valuable numerical results are contained based on the Green-Haghdi directed-sheet model. We have already compared these results with ours based on the Finite Element Method and the KP equation in two separate papers, both presented at the 5th International Conference on Numerical Ship Hydrodynamics in references [7,10]. We did not include these comparisons here since we concentrated only on the effect of the tank width on the wave responses by using the two different methods. DISCUSSION Theodore Y. Wu California Institute of Technology, USA This paper, delivered lucidly by Prof. Hang Choi, is of basic interest and bears significance in that two theoretical models of quite different approach and of different orders in accuracy are here applied to provide results on this valuable comparative study. I hope the authors can clarify whether their FEM-method is indeed equivalent to the exact Euler flow model on theoretical basis, notwithstanding numerical errors. Of particular interest would be a further exploration on the asymptotic behavior of these two models in two special limits: (i) as the body-length-to-channel width ratio tends to zero, (ii) as the velocity of forcing approaches the upper or the lower bound at which the upstream emission of solitary waves would evanesce. I would like to encourage the authors to continue their excellent efforts in these directions to cast new lights on this very interesting phenomenon. AIlTHORS' REPLY We highly appreciate the discussion raised by Prof. Wu. To the first question whether our FEM-method is equivalent to the exact Euler flow model on theoretical basis, we would like to stress that the basis of our FEM-method, i.e., the variational principle in our paper is equivalent to the exact inviscid irrotational flow with a free surface. In the procedure of the FEM-method, the unknown free surface is also represented as a part of solutions and solved numerically through iterative scheme. In this sense, the present FEM-method is equivalent to the exact potential flow model, except discretization of continuous functions. Concerning with the comment on the case of laterally infinite tank, we tried to numerically follow the similarity solution of Redekopp [25]. But due to the limited computing capacity, we are able to show only an intermediate result which indicates the crestline of the first diverging waves being nearly a parabola. Based on the present computations, it is hard to predict the upper and lower bounds of forcing speed at which the quasi-periodic emission of upstream-advancing solitons evanesces. DISCUSSION John V. Wehausen University of California at Berkeley, USA As I understand the authors, the calculations based upon the KP equations for a vertical strut just touching the bottom and with a profile determined by the section-curve of a Wigley hull or of Series 60, CB = 0.80, whereas those based upon the Laplace equation are for a ship with the same overall dimensions as the latter hull but with an altered section-area curve appropriate to a wedge-shaped hull; the blockage coefficients are the same. One is tempted to conjecture that the differences in results may be due as much to the different geometries as to the differenct methods of computation. Table 2 may support this conjecture for the KP equations. Ertekin's (11) conclusion that blockage coefficient is the most important parameter in determining properties of the solitons was based upon experiments with only one hull shape. Later computations for struts, using T. Y. Wu's generalized Boussinesq equations, have shown also some dependence upon hull form. Would the authors care to comment? By chance, calculations have been given in Ertekin, Qian and Wehausen (Engineering Science, Fluid Dynamics. A Symposium in Honor of T. Y. Wu. World Scientific Publishing Co., pp. 29-43, Table 1, line 1) for the same configuration as in Table 2, but only for Fh = 1.0. The generalized Boussinesq equations were used. The values obtained were A/h = 0.36, cl(gh)'h = 1.14, UTB/h = 61. The agreement for the first two values is perhaps not unsatisfactory, but this does not seem to be true for the third. The discrepancies could be a result of numerical error or of the different equations used. AUTHORS' REPLY Thank you very much for your nice comments. To the first comment, we agree with you. However, in the FEM computations we used not only the same length and draft but also the same sectional area with slightly reduced beam. To the comments in the second paragraph, we also agree with you. The discrepancies in the numerical results by different methods are due to the differences in the numerical procedures as well as the governing equations. 189

OCR for page 173