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Numerical Simulation of Ship Waves and Some Discussions on Bow Wave Breaking & Viscous Interactions of Stern Wave K.-H. Mori, S.-H. Kwag, Y. Doi (Hiroshima University, Japan) Abstract Numerical calculations are carried out to simulate the free-surface flows around the Wigley model and S-103 Inuid model. The N-S equation is solved by a finite difference method where the body-fitted coordinate system, the wall function and the triple-grid system are invoked. The numerical scheme being examined for the scheme to be accept- able for discussions, the calculations are extended to the turbulent high- Reynolds number flows with the aid of the O-equation model to discuss the Reynolds number dependency of the waves. The wave elevation at the Reynolds number of 104 is much less than that at 10 6 although the Froude number is the same. The numerical results are referred to predict the ap- pearance of the sub-breaking waves around bow and stern. The prediction is qualitatively supported by the ex- perimental observation. They are also applied to study precisely on the stern flow of S-103 as to which extensive experimental data are available. A1- though it is not yet made clear about the interaction between the separation and the stern wave generation, the ef- fects of the bow wave on the develop- ments of the boundary layer flows are concluded to be significant. The sub- breaking of the stern wave is also discussed. 1. Introduction _ Free-surface flow around ship is one of the most complicated flows where various nonlinear phenomena exist such as wave breaking, viscous interactions, free-surface tension and so on. Not only for the practical hull form design but also for academic interests, it is important to make clear their flow mechanism. They are worthy to be studied more intensively. There are pretty many experimen- tal studies even about the wave break 191 ing such as Duncan[1], Mori[2], Maruo[3], Grosenbaugh[4] and so on. In spite of their extensive experi- ments, however, the free-surface non- linear phenomena still remain unclear. Theoretical investigations are also at- tempted to explain the phenomena or to provide a suitable model. Dagan[5], Tanaka[6] and Mori[7] applied the in- stability analysis to predict the breaking. Some models for breaking waves are proposed after experiments. There are few studies on the free- surface tension; Maruo[3]. The stern flows with the free- surface show also important phenomena in ship hydrodynamics. Although Doi[8] and Stern[9] studied extensively about them, very little are made clear. Despite the viscous interactions are essential there, theoretical approaches are so limited and most of the ap- proaches are based on the simple flow models. On the other hand, there are some researches by the direct numerical simulations such as Miyata[10], Grosenbaugh[4], Shin[11] and so on. They have tried to make clear the mechanism of the Navier-Stokes equa- tions directly. Because the free- surface flows of our interests are strongly nonlinear and viscous effects are primary, the simulations by solving the Navier-Stokes equation can be a desirable tool for the study. They can provide any necessary data for the study once a calculation is carried out. However, the important point is on whether their codes are accurate enough for such studies. It may be possible to draw misleading conclusions from the results calculated by an un-proved code. The use of an insufficient grid scheme is likely to bring forth misun- derstanding for the phenomena. The as- sumptions of 2-dimensionality or laminar low Reynolds number flows are also possible sources for misun- derstanding. The present paper is a study along

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the approach lastly mentioned; the wave breaking and the viscous interaction of the stern waves are studied by making use of the results of numerical simula tions. The numerical scheme for the simulation i s based on the MAC method where the body-fitted coordinates and the non-staggered grid system are used. The convection terms are presented by the third-order upstream and differencings. The wall function is invoked to follow a steep veloci ty changes close to the hull in high Reynolds number flows. The turbulent stress terms are presented by a O equation model. Bearing in mind the above ment i oned danger s, the computing code is validated first by carrying out var ions computations to be convinced with, although it may not be enough due to the limitation of the computer. The f r e e - s u r f ac e f l ows around the Hi gl ey model at the Reynolds numbers of 104 and 1 o6 are used for this purpose . Although the theory is 2 d imen s i anal, the r e sul t s are us ed to predict the appearance of the sub breaking wave s . The vi s cons inte rac tion of the stern waves is also dis cussed . Hn = WAt + ( Rle + Vt ) V2w - ( un ax + vn aw + wn aw ) - ax{vt(aU +~Wx)}- ay{Vt(az + a-y)} - az{vt(2 aaZW)} an = p+ z Fn2 53) All the variables are on the car- tesian coordinates system(x,y, z) where x is in the uniform flow direction, y in the lateral, and z in the vertical direction respectively; u, v and w are the velocity components in the x-, y-, and z-directions, respectively. They are normalized by the model overall length L and the uniform velocity Uo. Subscripts denote the differentia- t ions wi th respect to the referred variables and superscripts the values at the referred time-step. The term At stands for the time increment, p the pressure and At the eddy viscosity. Rn and En are Reynolds and Froude numbers respectively based on L and Uo, and 2. Numerical Simulation of Shim Waves v2 a + a + a (4) 2.1 Basic Equation Numerical simulations of 3-D free- surface flows are carried out by solv- ing the N-S equation basically follow- ing to the MAC method. The velocity components u, v and w at (n+1 ) time step are determined by un+1 = ( En An ) At vn 1 = ( On ~ ) At ( 1 ) wn+1 = ( Hn An ) At where At + ( Re + at ) V2u - ( us ax + an ay + w" aaz ) - aX{vt(2 ax) }- t{Vt(a.y + ax) } az~Vt(az + axw)} n _ G At + ( Re + at ) V2v un aaX + V" aV + wn av ) ax{vt(aay + a3XV)}- aay{Vt(2~} (2) - a {vt(av + aw)} 192 Differentiating ( 1 ) with respect to x, y and z, we can have V ~ = F + G + H - ( u n+1 + v n+1 + w n+1 ) / AL (5) x y z The last term in ( 5) is expected to be zero to sat i sty the cant inui ty condition. ( 5) can be solved by the relaxation method. The new free-surface at the (n+1 ) th time-step is calculated by moving the marker particles by n+1 n n x = x + u At n+1 n n y = y + v lit n+1 n n z = z + w At (6) The oncoming flow is accelerated from zero to the given constant velocity. Third-order upstream dif- ferencing is used for the convection terms wi th the fourth-order truncation error, and for the central differenc- ings, 4- or 5-point central differenc- ings are used. It is desirable to introduce numerical coordinate transformations which simplifies the computational domain in the transformed domain and facilitates applications of the bound

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ary conditions. In the present study, a numerically-generated, body-fitted coordinate system is used, ~ = ((x,y,Z), n = ntx'Y, and ~ = (tx,y,z) (7) It offers the advantages of generality and flexibility and, most importantly, transforms the computa- tional domain into a simple rectangular region with equal grid spacing. Through transformations, (1) can be written for the velocity component q as qt+Uq;+Vqn+Wq; (8) =(Rl +vt)v2g- K-REYSF(:,n,`) where,U,V and W are the contravariant velocities and K is the pressure gradient. Their full expressions can be found in [12]. REYSF((, n, ~ ) repre- sents the terms transformed from the last three terms on RHS of (2). 2.2 TriPle-grid Method It is common to use a single grid system for the whole computation whose minimum size is determined for the numerical diffusion to be less than that by viscosity. However, the grid size for the calculation of the free- surface elevation must be determined by a different scale, the minimum wave length[131. In our simulation, three mesh systems are used whose sizes are different from each other depending on the characteristic of equations. We call it triple-grid method here. The first one is for the convective terms in the N-S equation, the second is for the Poisson equation, and the third is for the free-surface equation. The third grid system requires the finest one; a quarter of the first one in each direction. Because it is used only on the free-surface which is two- dimensional, the increment of the memory is modest. On the other hand, the second one can be coarser than the first one; here half of the first one is used. According to the results, the development of the free-surface eleva- tion is strikingly improved. The CPU time and the memory size of the present computation are rather reduced owing to the use of coarse meshes of the second mesh system for the Poisson equation and the diffusion term. 2.3 Body Surface Condition In the numerical solution for vis- cous flows, the no-slip condition for the solid surface is used by discretiz ing the region fine enough to the im- posed condition. This method, however, requires a large number of grid points to resolve the large gradients in the near-wall region especially for the high Reynolds number flows. This is the main barrier in the high Reynolds number calculations. In view of the complexity involved in resolving the near-wall flow, it is preferable to employ a simpler wall-function approach for the velocity profile which can be valid for the velocity profile in the near wall region. In the present study, the two- point wall-function approach is employed as by Chen and Patel[14]. The numerical solution is that the velocity at n=3(~=1 is on the wall) is provided and the wall-friction velocity us is updated with some iterations by requir- ing this velocity to satisfy the law- of-wall equation. The iterative two- point wall-function approach for a three-dimensional flow means that it provides the updated boundary condi- tions for the numerical solution and the procedure is iterated until the solution converges. Fig.1 Perspective view of grid scheme 2.4 Computational Results and Discus- sions Computations are performed for the flow fields around the Wigley model with free-surface at Rn=1o4 and 108 . Fig.1 shows the perspective view of the grid scheme used. For the numerical stability and efficiency, the grid scheme near the hull is required to be orthogonal to it and the grid size should change smoothly. The grid number is 74x29xl9. During the computations, the location of grids between the free- surface and the bottom is re- distributed proportionally to the free- surface elevation. By this scheme, it is expected that the free-surface con 193

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dition, i.e., the constant pressure condition can be directly applied without any interpolation. For the high Reynolds number flow, the two-layer al- gebraic Baldwin-Lomax model is used to make the eddy viscosity. The numerical results are compared with the ex- perimental data [15]. Drag coeff. (x10-3) 7.O 5.0 3.0 1.0 ~ '4igley Model Rn = 106 En = 0.316 Of Schoenherr's ~] friction line / at Rn=10**6 / ~ _ = = _ , _, -~~C Measured Cw (assuming K=0.07) / , . . . 1.0 1.5 2.0 2.5 3.0 Fig.2 Time history of drag coefficients 0.5ol rr 0.50 n 50- . 194 In order to check the convergence of the computations, the wave patterns and drag coefficients of Cp, Of and Cw are compared along the marching time step as shown in Figs.2 and 3, where Cp, of and Cw are the pressure, fric- tional and wave-making resistance coef- ficients respectively. Schoenherr fric- tion line and the measured wave-making resistance coefficient are shown for comparison. Although the wave seems still developing further, it can be as- sumed to be converged at T=3.0 where T is the non-dimensional time, which can be supported by the results shown in Fig.3. The calculated frictional resis- tance, which is directly derived by the difference of the velocities at the two points, is still larger than the Schoenherr. Fig.4 shows the logarithmic plot of velocity at x/L=0.835 in the format of the law-of-wall (q+ versus y+) using the friction velocity. Some plots are drawn at several points in the girth- and depth-wise directions; they are generally in good agreement. Fig.5 shows the comparison of the velocity distributions near the wall between that obtained by making use of AD ,~ ~ //O.2,,-J,,~ _ AP Fig.3 Time history of wave patterns for Wigley at Rn=106 and Fn=0.316

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20.0 . u 1o.a 3 ~ ~ ~ 3 8 ~ 8 ~ g ~ : z/D~O. 09 O : z/D=O. 52 O : z/O=O. 15 , . .. t. _ 1 10 1o2 103 104 y Fig.4 Log plot of velocity for Wigley model(x/L=0.835) the law-of-wall and directly. Because the expression by the wall-function is not valid any more for the separated flows, it is not applied in the stern 5% where the separation is suspected. There~the scheme is switched into the direct method. We can see that the 0.4 2y/B 0.2 0.4 2y/B 0.2 ~3 ~-at-W-~ -or-I ~1 FP AP AP Fig.6 Velocity vectors on free surface (a) Laminar flow at Rn=104 (b) Turbulent flow at Rn=106 0 _ 0 Go Q O Wigley0 I O O O Rn=lo6~ . ~_ Fn=0.316 ~x x x . . ~Go cn LO ~ cut 0 0 0 _ _ _ i ~ x x x/ it_ 0 _ O O Q Go ~ o ~ o ll ~ 11 J ~ _ , CX , J O . l ~Cal ~ CO ~cn 0 _` 0 0 0 o o o ~ ~ ~ X X _ XJ ~ Fig.5 u-velocity distribution on free-surface (a) without wall function (b) with wall function tall-function approach removes much of She dependency of the numerical solu- ~ion on the location of the two mesh points and the steep velocity changes are well followed even by limited size of grid. The usual logarithmic law of the wall can be reasonably used in favorable pressure gradients. Fig.6 shows the comparison of velocity vectors on free-surface be- tween the two Reynolds numbers: the one (b) (a) (b) iS 104 which is laminar flow while the other 1oB , turbulent. The laminar flow is subject to separation in the stern region and wider boundary layer thick- ness, while the turbulent flow is to larger velocity gradient near the hull by which we can guess a larger wall- friction on the body surface. Fig.7 shows the comparison of the wave pat- terns at Rn= 104 and 106. We can clearly see that the Reynolds number dependency 195

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0.50 (a) 0.50~ /~/~"/"~'' ,~ _ AP cat ~ 0 i. ~ 0 / ~0 ~0 ~O q FP \ 0 ~o/ Wi g1 ey Model Rn=1 0**6 Fn=0.316 i6( \~;AP - : Cal cul ated O: measured Fig.8 Wave profiles on hull surface 0.2 a SWL Fig.7 Wave patterns of Wigley model at Fn=0.316 (a) Laminar flow at Rn=104 (Mesh size: lOOx25x15) (b) Turbulent flow at Rn=10 6 (Mesh size:70x25x15) o.o 1 l AP Fig.9 Pressure contour on Wigley hull surface at Fn=0.316 - (a) Laminar flow at Rn=104 (b) Turbulent flow at Rn=106 of the wave. It may confuse us that even the second wave crest differs much in height, for we usually expect the Reynolds number effect on wave is not so significant there. Fig.8 shows the wave profile along the hull surface. The Reynolds number of the measurement is 3.59X106. It is well presented around the bow, but slight discrepancies are still observed in the aft half of the hull. Fig.9 shows the pressure contours on the hull surface at Rn=104 and 106. The pressure around the stern region is much recovered at Ace than the low Reynolds number flow. The pressure distribu- tion on the hull surface shows some wiggles in appearance at the bow and stern parts due to the use of still coarse mesh. However, the wave height on free-surface, which means the pres- sure here, shows no serious wrinkles because finer grid is used there. Fig.10 shows the comparison of wave patterns between the calculated and the measured. Although they can not be compared exactly due to difference of Reynolds number, we can say that the computed patterns are qualitatively reasonable for both the bow and stern waves. Fig.11 shows the velocity vectors on the transverse section near A.P. in which the vertical motion can be ob- served around the keel. It seems not so easy to calculate the cross flows accurately at the stern part consider- ing some aspects in the numerical point of views. First, the assumption of the symmetry or the steady flows should be pointed out. The unsteadiness and non-symmetry observed in experiment should be taken into account in numeri- cal simulation. Of course the grid 196

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Fig.10 Comparison of wave patterns at Fn=0.289 (a) Computed contour(Rn=lxlO 6 ) (b) Measured contour(Rn=3x106)[15] _. ~ - `` ~ _ _~` ~ i< i...~N ,~,,`~N ,. `~N ~ ~\ \~\~4 And As XiittitY ~ ~ t~ttti ~ ~ t ~ ' T=3.O Fig.11 Velocity vectors of Wigley model(Rn=106, Fn=0.316 and x/L=1.02) Ctx1 03 - 4.0 : measured [ 15 ] X : computed ( Rn=l o6 ) - 2.0 ---: measured(corrected at Rn=lO6) 0.25 0.28 0.31 F n Fig.12 Comparison of total drag coefficients between the computed and measured. 197 (a) used in the computing domain is still coarse and can be a reason of not being able to capture completely the details of the fluid motions. In Fig.12, comparison is made for the total drag coefficients between the calculated at Rn=1o6 and the measured. The Reynolds numbers of measurement are 2.84x106, 3.28x106 and 3.59X106 for the corresponding Froude number of 0.25, 0.289 and 0.316, respectively. For the more direct comparisons, the measured results are corrected at the same Reynolds number of 1o6 by use of the Prandtl-Schlichting's friction formula. The computed drag is still greater than the experimental data; We can not men- tion the reason for the difference con- clusively, but the accuracy of the velocity calculation close to the hull may not be enough which resulted in poor agreement in the frictional resis- tance. The computing time is abt. 90 hrs for one case by Apollo DN-10000 (abt.13 MFLOPS). 3. Detection of Sub-breaking Waves 3.1 Appearing Condition of Sub-breaking Waves Computed results are applied to detect the appearance of the appearing condition of sub-breaking waves around bow. The critical conditions for their appearance were studied in Mori[7]. There the Creakings at their infant stage are concluded as a free-surface turbulent flow. The flow mechanism is

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supposed that the surplus energy ac- cumulated around the wave crest by the increment of the free-surface elevation is dissipated through the turbulence production and free-surface could even- tually maintain itself without any overturnings or backward flows. An instability analysis fog 2-dimensional flows provides a critical condition for their appearance; M aM_ auS_ 1 anz , O (9) UshOs has nz has where M is the circumferential force given by k} = ( ~ Us - nz g ) nz, (10) s is the stream line coordinate along us M/Us a) cat to . o- 1 the free-surface and h is its metric coefficient, while n is the normal;nz the direction cosine of n to z. Us is the velocity component of basic flow in the s-direction; ~ is the curvature of the free-surface and g the gravity ac- celeration. Limiting ourselves to a narrow proximity to the wave crest, we assume nz=1 and a /h~s.a/3x ; then (9) can be reduced approximately into uS a M __ > 0 where, to o l M/U ~ 2 M = lC Us ~ g us 0 o 1 Fn=0.316 . . - ~. I X 0 . 4 (FP) (I. s. 9) 0 0 0 l O Fn=0,25 OO to . M/U S - LO O . to us cat to , to ~, -0. 5 (FP) -0.4 ( s . s . 9) u, cat 0, . to us to lo 1 it M/Us M/U s l -0.3 x (S.S.8) o up ~` E'n=0.20 1 ~ ~hi_ -0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 (FP) X (s.s.7) Fig.13 Variation of M/Us and free-surface elevation and lines analyses for bow wave breaking. 198 (11) (12) M/U s lo rut lo o to l

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Because M is always negative, the negative gradient of M/Us to x suggests the possibilities for the free-surface flow to be unstable. 3.2 Numerical Application for Bow Waves The appearing condition is numeri- cally simulated to predict the ship wave sub-breakings by (11). Although the flow for the Wigley model is not 2- dimensional, 3-dimensionality may not be so strong that we can expect it is applicable without serious errors. Fig.13 shows the variations of M/US vs x around the first bow wave crest. The analyses are made at three speeds of Fn=0.20, 0.25 and 0.316 along the curved lines indicated there; ~ is the free-surface elevation. At Fn=0.20, no steep negative gradient is seen, but the gradients at Fn=0.25 and 0.316 are significantly negative behind the wave crest. It may be suggested that the free-surface flows at Fn=0.25 and 0.316 are likely to be unstable behind the wave crest while that at Fn=0.20 is stable. Fig.14 shows the photographs of the free- surface flows taken at three cor- responding Froude numbers. There can be seen wrinkle-like waves behind the diverging waves at Fn=0.25 and 0.316; those at Fn=0.316 are much more inten- sive than those at Fn=0.25. On the con- trary, no such waves can be observed at Fn=0.20. This observation supports the instability analysis for the bow wave breaking. 4. Discussion on Stern Waves The stern wave of S-103 is studied to make clear the flow mechanism espe- cially on the viscous interaction by referring the computed results. S-103 is an Inuid model with the beam/ length ratio of 0.09 and extensive ex- periments have been carried out by Doi[8]. All the experimental data are referred from there. 4.1 Computation for S-103 Model Fig.15 is the computed wave con- tour at Fn=0.30 and Rn=106. The result is that at the time T=4.0, when the convergence is well assured. The grid size is 74x29x30; computations are carried out on another finer grid scheme to find no significant dif- ference in the resistance. The comput- ing domain is -1.4~x/1~2.0 and 0.0 OCR for page 191
l JO - '\\` on 0.05 0.03 - compu ted ---- measured 0.01 ,,'/ - 1 ' ,r~ . - O .f Hi_ ,, /y - O . () 3 Fig.16 Comparison of wave profiles of S-103 model between the calculated and measured on hull surface at Fn=0.30 profile shows a good agreement with the measured to conclude that the present numerical scheme works well and the results may endure for our purpose to discuss on the flow mechanism. 4.2 Review of Experiments Now let's refer the wave contours of S-103 from [8], shown in Fig.17. We can notice significant difference in the stern wave patterns although the Froude number changes modestly from 0.26 to 0.30; at Fn=0.27 no significant stern wave is observed compared with those at Fn=0.26 or 0.28. On the other hand, a wide "wake" zone is observed behind the hull at Fn=0.30. A careful observation of the "wake", as shown in 'I ~ / / l- \ //// //1/~ Fig.15 Wave pattern of S-103 at Fn=0.30 and Rn=106 (Contour interval is 0.02x2g;/Uo and dotted lines show negative values) Fig.17 Wave patterns of S-103 at four different Froude numbers Fig.18, tells us that the free-surface fluctuates intensively there. The free- surface of Fn=0.27 is completely dif- ferent where such free-surface fluctua- tion is not observed. This fluctuation of the free-surface is sub-breaking. It is reported in Doi[8] that the starting points of the stern waves could be easily and definitely deter- mined along the hull at Froude numbers other than 0.27. This is because the wave profile at Fn=0.27 is a little dif- ferent from that at other speeds. from the observed wave profiles 4.3 Discussion on Viscous Interaction Fig.19 shows the computed stern wave patterns at the three Froude num 200

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~- - -. Fig.18 Stern wave pictures of S-103 at Fn=0.27 and 0.30 Fig.19 Stern wave patterns of S-103 at Rn=lo6 (a) Fn=0.27 (b) Fn=0.28 (c) Fn=0'.30 (Contour interval is 0.02x2g;/Uo and dotted lines show negative values) hers of 0.27, 0.28 and 0.30. Comparing the first stern wave crests, we can see that the result of Fn=0.27 looks dif- ferent from the others; not so sharply developed. It differs from that of Fn=0.28 although the difference in the speed is not so large. The stern wave crest of Fn=0.27 is not clear. This may agree qualitatively with the ob- served. On the other hand, the crest of Fn=0.30 is rather sharp and large. The modest elevation of the stern wave at Fn=0.27 may be much related to the development of the boundary layer and separation. Fig.20 shows the velocity profiles in the boundary layer around the stern and close to the free-surface. The separation of Fn=0.27 takes place at more upstream position than that of Fn=0.30. This situation can be seen more clearly in the limit- ing streamlines shown in Fig.21. The separated region of Fn=0.27 is sig- nificantly wider than that of 0.30. The experiments by twin tufts show similar tendency; the separated region of Fn=0.30 close to the free-surface is due to the free-surface sub-breaking which has been shown in Fig.18. It is quite natural that a dull pressure recovery by separation may bring forth " '' '~'/~//,rK'--,','>~ AP 201 I(a) (b)

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modest wave elevation. On the other hand, at Fn=0.30, separation region is so limited that a steep pressure gradient may generate strong waves. Here we should remind that all the computations are carried out at the same Reynolds number of 106. This means that the flow fields are exactly the same in the sense of the viscous ef- fects. Then why such a difference in separation? The bow wave may be responsible; the phase of the bow wave can be a key factor for the separa- tion. The wave contour lines change peculiarly in the boundary layer and wake at all the speeds. A careful ob- servation of the free-surface, shown in Fig.18, suggests us complicated flows in the boundary layer, which may cor o.R 2y/B 2y/B n 4 n R] 0.4 ~ Fig.20 u-distribution for S-103 on free-surface (a) Fn=0.27 (b) Fn=0.30 ~t t t t t t Z/l - O _ _ _ _ --0.02 _ --0.04 _ <20mr~ `r ,: BEAD \~\ TWIN -TUFT respond to the computed peculiar con- tour curves. Fig.22 shows the velocity vectors at the two y-z sections. Compared be- tween the two Froude numbers, it is ob- viously seen that the viscous region of Fn=0.27 is much wider than that of Fn=0.30. A wider wake region made the stern wave elevation modest. The peculiar changes of the wave contour, pointed out in Fig.19, is assumed to appear in the viscous region. 4.4 Detection of Sub-Breaking Fig.23 shows the comparison of wave profiles and velocity vectors of Fn=0.30 between the measured and the calculated. The measured free-surface fluctuates intensively around the crest (indicated by I there). This fluctua- tion corresponds to the sub-breaking seen in Fig.18. Because no special at- tention to the sub-breaking is paid in the calculation, the calculated free-surface is steady, of course. It should be pointed out that the measured wave crest is in upstream compared with the calculated. As seen Fig.18, the crest angle of Fn=0.30 is much larger than the calculated. If we remind the good agreement in the wave profile along the hull surface, shown in Fig.16, we can say that the appearance of the free-surface fluctuation, i.e., sub-breaking makes the crest shift for- ward which is commonly observed in ex- periments. The detection of the appearance of sub-breaking waves is made for the stern waves at Fn=0.27 and 0.30. The values of M/US are calculated along FREE SURFACE ~ , _ I, ; ; ; ~_ _ _ _ _ - _ _ _ 1 - - , ., . ~ 1 _ _ _ x/l-o.g x/1-3.0 x/l=o.g x/l-l.o FREE SURFACE FREE SURFACE _ _ Xtl=O.9 X/l=1.0 X/l=O.9 A.P Fn=0.27 202 T.. Xll=1.0 Fig.21 Calculated(above) and observed(below) limiting streamlines at Fn=0.27 and 0.30

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y x/1=0.95 ~0.15 x/1=1.05 x/1=0.95 (b) y/l=O.O9 which are shown Similarly to Fig.13, the negative gradient of M/US to x suggests the oc- currence of unstable free-surface flows. A steep negative gradient is- seen in Fn=0.30 but not so much in Fn=0.27. This means that the wave crest of Fn=0.30 can be subject to sub- breaking but not at Fn=0.27. it\ \ \ ~ ~ i~, t`~W \ ~ ~ >\4 ~ ~ We can point out that the bow wave affects much on the separation and eventually the stern wave generation appreciably. The appearance of sub- breaking waves makes the flow field completely different and it may be necessary to include them in the com- putation. Me asured all o.o4 0.02 -0.02 Intensive Fluctuation ~ \ 1 _ o_ JO -0.04-, 1.O 1.1 1.2 Z/1 0.04 0.00 -0.04 Calculated Free Surf ace ._ Fig.23 Wave profiles and velocity vectors on x-z plane at y/l=0.15 for S-103 \ ~z ~\ Lr red O _ . o _ z Fig.22 Velocity vectors in y-z planes of S-103 (a) Fn=0.27 (b) Fn=0.30 1 - . . / / Fn=0.30 . - / M/Us M/U s \ (A 0_ O I I r I 1~2 1~4 1~6 x/l o O up o up r. o O o o o _ ~ o 1 M/Us M/U s Fig.24 Variation of M/Us and free- surface elevation and lines analyses for stern wave breaking of S-103 203

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5. Concluding Remarks A computational code is developed to simulate the high Reynolds number free-surface flows around ship by solv- ing the full Navier-Stokes equation. After validating the scheme, it is ap- plied for the studies on the Reynolds number effects, the detection of wave breaking and viscous interaction of the stern wave. Findings through the present study are summarized as fol- lows. (1) A numerical scheme is developed to simulate 3-D free-surface flows at high Reynolds number by which a steady- state solution can be obtained with monotonic convergency. The triple grid method is applied to get well-developed free-surface waves within a moderate computer's memory. The wall function is used to overcome still larger minimum grid spacing, which is confirmed to work well. (2) The Reynolds number effects on waves are significant especially on the stern wave and pressure distribution in the aft part of the hull. (3) The calculated resistance is greater than the measured. The estima- tion of the frictional resistance is still a source of over-estimation. The use of much finer grid is a possible improvement. (4) The criterion for the appearance of sub-breaking waves works well, and the present numerical scheme can be ap- plied to detect the occurrence of breaking waves of ships. The results are supported by the observation. (5) The stern wave is much affected by the separation of the boundary layer flow which may be under the influence of the bow wave. (6) The occurrence of sub-breaking changes the flow fields drastically, especially for the stern waves. It is important to introduce a model into the numerical calculation which is capable for the breaking. All the calculations are carried out by Apollo DN-10000 Computer at N.A.& O.K. Dep't of Hiroshima University. Acknowledgments The second author Mr.Kwag, who is now on leave from HMRI(Hyundai Maritime Re- search Institute, Korea), expresses ap- preciation to HMRI to allow him to make research at Hiroshima University. References [1] Duncan,J.H.: "The Breaking and Non- breaking Wave Resistance of a Two-dimensional Hydrofoil", Jour. Fluid Mech.,Vol. 126,1983 [2] Mori,K.,Doi,Y.:"Flow Characteris- tics of 2-Dimensional Sub-Breaking Waves, Turbulence Measurements and Flow Modeling", Hemisphere Pub. Co.,1985 [3] Maruo,H.,Ikehata,M.:"Some Discus- sions on the Free Surface Flow around the Bow", Proc. of 16th sump. on Naval Hydro.,1986 [4] Grosenbaugh,M.A.,Yeung,R.W.:"Non- linear Bow Flows - An Experimental and Theoretical Investigation",_Proc. of 17th Symp. on Naval Hydro.,1988 [5] Dagan,G.,Tulin,M.P. : "Two Dimen- sional Free-Surface Gravity Flow past Blunt Bodies", Jour. Fluid Mech., Vol.51,Part3, 1972 [6] Tanaka,M.,Dold,J.W., Peregrine,D. H. : "Instability and Breaking of a Solitary Wave", Jour. Fluid Mech., Vol.185,1987 [7] Mori,K., Shin,M.S. :"Sub-Breaking Wave:Its Characteristics, Appearing Condition and Numerical Simulation", Proc. of 17th Symp. on Naval Hydro., 1989 [8] Doi,Y., Ka~itani,H., Miyata,H., Kuzumi,S.:"Characteristics of Stern Waves generated by Ships of Simple Hull Form (1st Report)", Jour. of Soc. of Naval Arch. of Japan, Vol. 150,1981 [9] Stern, F. :"Influence of Waves on the Boundary Layer of a Surface- Piercing Body",Proc. of 4th Int'l Conf. on Numerical Shin Hydro.,1985 tlO] Miyata,H.,Ka;itani,H.,Shirai,M., Sato,T.,Kuzumi,S.,Kanai,M.:"Numerical and Experimental Analysis of Nonlinear Bow and Stern Waves of a Two- Dimensional Body(4th Report)", Jour. of Soc. of Naval Arch. of Japan, Vol. 157,1985 Ill] Shin,M., Mori,K.:"On Turbulent Characteristics and Numerical Simula- tion of 2-Dimensional Sub-Breaking Waves", Jour. of Soc. of Naval Arch., Vol.165,1989 [12] Kwag,S.H.,Mori,K.,Shin,M. :"Num- erical Computation of 3-D Free Surface Flows by N-S Solver and Detection of Sub-breaking", Jour. of Soc. of Naval Arch. of Japan, Vol.166, 1989 [13] Xu,Q.,Mori,K.,Shin,M. : "Double 204

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Mesh Method for Efficient Finite- Difference Calculations", Jour. of the Soc. of Naval Arch. of Japan Vol .166, 1989 [143 Chen,H.C., Patel,V.C. : "Cal- culation of Trailing-Edge, Stern and Wake Flows by a Time-Marching Solution of the Part ially-Parabolic Equations", IIHR Report No. 285 1985 , , [15] IHI, SRI,UT,YNU : "Cooperative Ex- periments on Wigley Parabolic Models in Japan ", p repare d f or the ITTC Re s i s - tance Committee, 2nd edition, Dec., 1983 DISCUSSION Fred Stern The University of Iowa, USA The authors' treatment of the free-surface boundary conditions is unclear. It appears that a MAC type method is implemented using a fixed grid that does not conform to the free surface since the terms necessary for a moving grid have not been included in the governing equations. A similar approach was presented by Hinc at the 5th International Conference on Numerical Ship Hydrodynamics. Please explain the differences between the present approach and that of Hino. The grids used for each of the triple grids should be clearly stated. It appears from the results that the grid for the convective terms in the NS equations is much too coarse to accurately resolve the viscous flow, e.g., Fig. 4 indicates very few points within the boundary layer and virtually no resolution of the logarithmic and outer wake-like region of the boundary layer. The reliability of the discussions of the results is uncertain due to these issues. AllTHORS' REPLY 1. The free-surface boundary conditions consist of the pressure and kinematic ones. Here, the pressure condition can be directly applied because the uppermost moving grid at each time step corresponds always to the free-surface. The kinematic condition, the fluid particles on the free-surface keep staying on it, is used to determine the free-surface elevation at each time step in which the velocities u, v, and w are extrapolated equally from the value at the lower grid points. The viscous condition, the tangential stress is zero on the free-surface, is not considered here. 2. In Hino's case, there are several grid points above the still water surface and the pressure boundary condition is applied at the intermediate point between grid points as used in the SUMMAC. But in our case, the pressure condition is directly applied without any interpolation. The time dependent term is, of course, taken into account when the grid is rearranged. 3. On the Triple Grid Method: We used three different schemes A, B. and C for the convective terms, the diffusion terms, and the free- surface condition respectively; A, B. and C are 74~29~19, 37~29~19, and 296~116xl in x, y, z directions. The point of the triple grid method is that the fourth order central difference scheme in the Poisson equation does not so much improve the accuracy even if finer grid is used, while, as you comment, the fine mesh system is necessary for the convective term. This is because the truncation errors which come from the dissipation terms, gradient of pressure and Poisson equation give little influence on the accuracy of the computations. The move of the free-surface particle to satisfy the kinematic free-surface condition requires fine grid. 4. Due to the limitation of computer, about 15 points are used around A.P. within boundary layer. It is the reason why we have introduced the wall function to compensate for the smaller number of grid points near the hull surface. DISCUSSION Hoyle Raven Maritime Research Institute Netherlands, The Netherlands When studying Reynolds number effects with a numerical method, one must be sure to have negligible numerical viscosity. This requires that the hull boundary layer is well resolved. How many grid points in normal direction did you use inside the boundary layer? It seems that your grid is not adapted to the boundary layer thickness and may therefore have no grid point at all inside the boundary layer near the bow. AUTHORS' REPLY Thanlc you for your kind interest in our paper. Fifteen grid points are used within boundary layer around A.P. In CFD, the grid requirement is not easy to satisfy in case of calculating the boundary layer. Especially around the bow, the boundary layer thickness is so thin that it is very hard to catch the larger velocity gradient with two or three grids. In order to overcome that problem, we applied the empirical wall function approach to the scheme except for some stern region where the separation is suspected. DISCUSSION Ronald W. Young University of California at Berkeley, USA The instability criteria mentioned in the paper applies to, or was derived on the basis of, 2-D flow. I find it rather skeptical that it can be used for oblique waves along the side of the ship. Perhaps some additional justification is helpful. AUTHORS' REPLY As the discusser pointed out, the criteria used here is that derived from 2-D instability analysis. Therefore, exactly speaking, it may not be applied to the 3-D ship waves. However, the obliqueness of the present case is so small that we expected to have some qualitative detection of sub-breaking; dependency of Froude number and so on. Through the present application, we have been much encouraged to provide a criteria based on 3-D analysis. 205

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