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Calculation of Free-Surface Flow around a Ship in ShaBow Water by Rankine Source Method H. Yasukawa Mitsubishi Heavy Industries Nagasaki, Japan Abstract Rankine Source Method was applied to the shallow water and channel problems, and cal- culations were made of free-surface flow around a ship moving in calm water and her wave-making resistance. The calculated results were compared with experiments and calculations by conventional linearized analytical method. It was shown that the Rankine Source Method promises an improvement in predicting free-surface flow around a ship in shallow water than the conventional linearized analytical method. 1. Introduction Flow around a ship traveling in shallow and restricted water is much different from that in deep and unrestricted water due to the in- fluence of sea bottom and bank walls. There- fore many studies have been carried out for investigation of performance and safe naviga- tion of the ship in the shallow and restricted water such as river, coast, harbor, channel and so on. The theory on ship waves and wave-making resistance in shallow water was presented by Havelock in 1922 ~13. Kinoshita and Inui cor- rected the Havelock's theory so as to satisfy the boundary condition of sea bottom exactly [2~. Further Inui extended its theory to the channel problems [34. By these studies the hydrodynamic characteristics of waves and wave-making resistances near the critical speed (F. h = l.0, where Fnh means Froude num- ber base] on water depth h) were made clear. Kirsch carried out calculations of wave-making resistance in various water depth and channel width for mathematical ship hull form and dis- cussed the shallow and channel effect t43. Bai calculated wave-making resistances in shallow channel by localized Finite Element Method t5~. Mueller compared experiments with calculations by conventional linear theory in shallow water, and indicated that the conven- tional linear theory could not predict free- surface elevation and pressure distributions with sufficient accuracy [63. Recently Mel and Choi presented the higher order theory on 643 a slender ship moving in channel t73. The other hand, there is Rankine Source Method developed by Gadd t8] and Dawson ~9], a kind of numerical method for solving steady free-surface potential flow in deep water. In this method the free-surface condition based on double body flow is employed, and the ac- curacy of predicting wave-making resistance is generally better than the conventional linear calculations. Further detailed informations for wave elevations, pressure distributions and velocity fields around ships can be ob- tained easily. In this paper the Rankine Source Method was applied to the shallow water and channel problems. Calculations of free-surface flows and wave-making resistance were made for Inuid S-201 ship hull form [10], and the calculated results were compared with experiments and calculations by conventional linearized analytical method. It was shown that the Rankine Source Method promises an improvement in predicting free-surface flow around a ship in shallow water than the conventional linearized analytical method. 2. Formulation of Shallow Water and Channel Problems by Rankine Source Method Let us consider a ship moving in center of channel in calm water. We assume that sec- tional shape of the channel is uniform in lengthwise direction. The coordinate system is defined as shown in Fig.1. x-axis coin- cides with the direction of steady uniform stream whose velocity U is identical with the ship speed. The origin of the axis is at the point of intersection of still water surface, midship section and center plane of the ship. y-axis is horizontal and normal to x-axis, and z-axis directing vertically upwards. Supposing a ship is in an inviscid, irrota- tional, incompressible fluid, the velocity potential , which represents flow around the ship and satisfies Laplace's equation V2 = 0, is introduced. ~ has to satisfy the followin& boundary conditions : ~xCx + ~y~y ~ ~z = 0 on z = `, ( 1 )

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Now Al is expressed as : 1 = OF + tH + (C, where OF : velocity potential representing free- surface, tH : added velocity potential representing ship hull due to the presence of free- surface, tC : added velocity potential representing channel due to the presence of free- surface. (F. tH and tC are represented by Rankine sources which are distributed on undisturbed free-surface SF, ship hull SH and channel sur- face Sc as follows : IF(P) = J.lsFCJF(Q)GF(P, Q)dx'dy', (8) tH(P) D.SHOH(Q )GH(P, Q')dSH, (9) Fig.1 Coordinate system tC( ) ~Sc3C(Q )GC(P' Q")dSc, (10) 1(~2+y+~2_U2) + go = ~ on z = i, (2, tnH = 0 on ship hull surface, (3) ARC = 0 on channel surface, (4) where ~ is elevation of free-surface, g the acceleration of gravity. nH and no mean the outward normal directions on ship full and channel surface respectively. Eqs.(l) and (2) represent the exact free-surface conditions. Eqs.~3) and (4) represent the boundary condi- tions of ship hull and channel surface. Here in order to simplify the calculation, the ex- act free-surface conditions (1) and (2) are linearized as : is expressed by following form : where (5) where 0 : velocity potential for double body flow, ~ : velocity potential for steady wavy flow. It is assumed that the double body flow is dominant in the flow field, and we neglect higher order terms with respect to ~ for eqs.(l) and (2~. Then the linearized free- surface condition based on double body flow is derived [9] as : tOSISS + 2~0SOSSIS + gIZ = -OSOSS on z = 0, (6) where the subscript S means partial derivative along the streamline of double body flow on still water surface. In this paper eq.(6) is employed for the free-surface condition. Double body potential 0 can be obtained by Hess and Smith's method t114. Therefore our concern becomes to solve the ~ so as to satisfy the boundary conditions. respectively. Here oF, oH and oC are strength of source distributions for free-surface, ship hull and channel respectively. P is field point (x, y, z). Q. Q' and Q " are source points for free-surface, ship hull and channel respectively, and are defined as follows : Q = (x', y', O ), Q = (xH, YH, ZH), Q" = (xc, Yc, ZC) GF is represented as : GF(P, Q) = 1/RF' (ll) RF = /(x-x' )2 + (y_y')2 + Z2. GH and GC are representing the double body flows for ship hull and channel as : GH(P, Q' ) = 1/RH + 1/RH', (12) Gc(P, Q " ) = 1/RC + 1/Rc', (13) where 644 RH = i(x-xH) + (Y-YH) + (Z-ZH) ~ RH' = /(x-xH) + (Y-Y8) + (Z+ZH) ' RC = /(X-xc) + (Y-YC) + (Z-ZC) ' RC' = i(X-xc) + (Y-YC) + (Z+zC)

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3. Numerical Procedure 4. Wave Height and Wave-Making Resistance Eqs.(3), (4) and (6) can be discretized by following procedures: (a) A finite area of the still water is divided into ME rectangular panels IF (j = 1 ~ ME), the hull surface into MH panels THE (A = 1 ~ MH) and the channel surface into MC panels ~ (n = 1 ~ Mc). (b) It is assumed that variance on each panel is represented by value on the panel, and the source strength is con- stant in each panel. The system of simultaneous equations with respect to the source strength oFJ (j = 1 MF), (7HdQ (A = 1 ~ MH) and C7Cn (n = 1 ~ Mc) are compose as follows : ij]~oF j: + ~BiQ]~GHQ) ~ tCin]~GCn} = {COii [Dkj ~ l3Fj ~ + [EkQ] l3HQ: + [Fkn] i()Cni [Gm; ] t(JFj ~ + [HmQ] {C>HQ: + [Imn] iC>Cn: where Aij = [ha (Aoi ;~2+Boi A,; )dx dy ]P=P 1~ ,] - 2~6ij ~ siQ = [JI~HQ(AOi~+sOi aS )dSH]P= C;n = [JI:~c (AOi~+Boi as )dsc]P= Dkj = rr7:F j anHdx dy ]P=Pk Eke = [ TITHE a nHdSH]P=Pk Fkn = [TI~cnanHdSc ] P=Pk ' Gmj = Errs j ancdx dy ]P=Pm ~ HmQ = t~r7HtanCdSHl P=Pm ' (14) (15) (16) (17) (18) (19) (20) (21) (22) ~ - circa anCdSC ] P=Pm Aoi = f osi/g (23) Boi = 2ositossi/g | (24) C0i = -~2sitossi/g J Si. is the Kroenecker delta function. For the calculations of a 2/3 S2 terms in eqs.(15) - (17), the 4 points upstream dif- ferencing was used so as to satisfy the radia- tion condition of free-surface numericaly t93. The C>Fj (j = 1 ~ ME), o~Q (A = 1 ~ Mp) and oCn (n = 1 ~ Mr) are obtained by solving the matrix of eq.(14). 645 Wave height ~ is represented by a linearized form of eq.(2) with respect to as: <; = 2t (U2-tOx~oy ~2ox~x~24'oy~y)~z=O' (25) Pressure p is represented by a linearized form of Bernoulli's equation as : P = 2p(U2-~2X_~2 _~2 -2~0xl~x-2loy~y~2oz~z), (26) where p is water density. Wave-making resistance Rw is evaluated by integration of pressure on ship hull as: Rw -5sHP nHx dSH, (27) where nHX is x-component of directional cosines of the ship hull surface. In this paper, p and Rw are represented by the following non-dimensional expressions : C p/lpU2 (28) Cw = RW/2PSO U2, (29) where S. is wetted surface area of ship hull in still water. 5. Results and Discussions For evaluation of the present method, cal- culations in deep and shallow water were carried out for Inuid S-201 ship hull form t10] and compared with experimental data t64~103. Further, calculations were made of wave-making resistance of the ship in channel. 5.1 Results in deep water First, calculations in deep and un- restricted water were carried out. Fig.2 shows panel arrangements for ship hull sur- face. We used 2 types of panels for valida- tion. Panel H-1 has 220 panels and panel H-2 has 440 panels. Fig.3 shows panel arrange- ments for free-surface. Panel F-1 has swept back [9], and panel F-2 has not. When we cal- culated the flow by using the free-surface panels with rapid change of panel width in lengthwise, oscillations occurred in the cal- culated results for free-surface source strength. So the free-surface panels with smooth change of the panel width were used to prevent the oscillations. Fig.4 shows varia- tion of calculated wave-making resistance with respect to the size of free-surface panel region. Wave-making resistance coefficient Cw converges around BFp/L = 0.4, where BFp is lateral width of free-surface panel region and L the ship length.

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Panel H-1 ~11..1 1 ~ ~ ~ ~ ~ ~ ~ ~ ~ 1 111 - Fig.2 Arrangements of hull surface panels ~ Panel F-1 Fig.3 Arrangements of free-surface panels Fig.5 shows comparison of wave-making resistance curves in deep water. In this figure, 'fixed coed.' means to take no account of sinkage and trim into the computations. The calculated results by using panel H-1 and F-1 show good agreement with the results by panel H-2 and F-1, so the panel H-1 seems to be sufficient for the number of hull surface panels. For the hump and hollow in Cw curve the results for panel H-1 and F-1 show better agreement with Inui's experiments [10] than the results for panel H-1 and F-2. There is a discrepancy between the present calculation and the Dawson's one [123. The reason for the discrepancy may be due to the difference of number of the panels. (Dawson used total 512 panels [123.) 5.2 Results in shallow water Next, calculations in shallow and un- restricted water with horizontal sea bottom were carried out. We calculated in hid = 2.389 and 3.413, where hid means the ratio of water depth and ship draft. For this case Mueller has made detailed experiments t63. Panel H-2, F-1 and the sea bottom panels as shown in Fig.6 were used for the computations. Fig.7 shows variation of calculated wave- making resistance for the free-surface and sea bottom panel regions in shallow water. The sea bottom panel region is larger than the free-surface region by 10% of ship length. It seems that the convergence does not achieve yet at BFp/L = 0.65. Thus we need larger 0.007 3 Panel H-1 & F-1 Deep water ~ | BFP ~ Fn=0.310 / ~\ / Ship hull Free-surface panels 0.006 f-o~o 1 1 1 1 1 1 0.2 0.3 0.4 0.5 0.6 0.7 BFP/L Fig.4 Variation of calculated wave-making resistance for free-surface panel region in deep water panel region than that in deep water. In the present calculations, however, the free-surface panels with BFp/L = 0.55 (BBp/L = 0.65, where BBp is lateral width of bottom surface panel region) were employed for saving computation time. Figs.8 and 9 show comparison of wave-making resistance curves in shallow water. The present calculations (in 'fixed cond.') are a little larger than Mueller's experiments t6] in low speed range, and the tendency of hump and hollow in Cw curve shows good agreement with the experiments. However, Froude number at the calculated maximum Cw is different from the experiment. The tendency of hump and hol- low in calculated C curve by conventional linear theory (Have~ock's integral) [6] is a little different from the experiments. How- ever the Cw values show good agreement with the experiments as well as the present calculations. Now, let us compare Cw curves in Figs.5, 8 and 9. In the present calculations, with decrease of water depth Cw near the critical speed increase and Froude number at the maxi- mum Cw becomes smaller. These tendencies agree with the experimental results. However, the Froude number at the maximum Cw is larger than the experiment and the differece of this Froude number becomes larger with decrease of water depth. The present method of calcula- tion does not take into account the effect of sinkage and trim. For reference, therefore, attempts were made of calculations of wave- making resistance by use of sinkage and trim estimated from Mueller's experiments [63. In this computation hull surface panels were rearranged. In Figs.8 and 9 'free coed.' means the calculation made by this way. Froude number at maximum Cw in 'free coed.' becomes smaller than that in 'fixed coed.' and comes closer to the experiment. Thus, it is suggested that for the improvement of the present method the effect of sinkage and trim should be included. 646

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0.025 0.020 Cal $~N 'it total pane' numbers 0.015 0.010 0.005 - 2< Cal. (panel H-1 & F-l ) 640 ~ - C1 Cal. (panelH-1 &F-2) 650 ! ~Cal. ( panel H-2 & F- 1) 860 | x Cal. by C.W.Dawson (12) 512 J O Exp. by T.Inui (10) Pa ~ fixed coed. Six ~- f~ ,( x JO max. Cw Exp.~ Cal. (fixed) OO00L' ~ ~ ~ ~ ~ ~ I I ~ I 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 En Fig.5 Comparison of wave-making resistance curves in deep water Fig.6 Arrangement of ship hull, free-surface and sea bottom panels (in/d = 2.389) 647

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Free-surface panels 0008: 0.007 _ 0.006 _ L \ / | FP | / 1 I '//i~/~ ' I ~ 1 7 ~ Ship hull Sea bottom Panel H-2 & F-1 panels h/d = 2.389, En = 0.310 I I I I I I 0.2 0.3 0.4 0.5 0.6 0.7 BFP / L l ~ 0.3 0.4 0.5 0.6 0.7 0.8 REP/L Fig.7 Variation of calculated wave-making resistance for free-surface and sea bottom panel regions in shallow water Fig.10 shows comparisons of wave profiles in shallow water (h/d=2.389~. Agreement be- tween the present calculations and Mueller's experiments is good as a whole. However, it can be pointed out that calculated wave height at the fore part is lower and tendency of wave profiles at the aft part is a little different from the experiments. The reason for this difference seems to be the nonlinear effect of free-surface condition and the viscous effect which are neglected in the present formula- tions. The conventional linear calculations are less satisfactory than the present calcu- lations. The difference of the calculated results in between 'fixed coed.' and 'free coed.' in small. Fig.ll shows comparisons of pressure dis- tributions on sea (tank) bottom in shallow water (h/d=2.389~. The present calculations show fairly good agreement with Mueller's ex- periments at the tank bottom below the hull surface y0/L = 0.0 where yO is lateral dis- tance from center line of ship. The conven- tional linear calculations show good agreement with the experiments also. The pressure coef- ficient at y0/L = 0.1667 is smaller than the experimental one near the negative peak value. The conventional linear calculations are also smaller than the experiments near the negative peak value. The difference of the calculated results in between 'fixed coed.' and 'free coed.' is small. Figs.12 and 13 show comparisons of prospect view of wave-pattern and wave contour around a ship for different water depth. Froude number in the computations is En = 0.410 (Fnh = 0.849 for shallow water case) and at its Froude num- ber the wave-making resistance increases remarkably in shallow water. From the figure it is found that the waves go down near the midship and much swell behind the ship hull in shallow water. Thus the effect of shallow water on the ship waves appears more remarkably near the stern part than at the fore part of the ship. It was shown that the present Rankine Source Method made an improvement of predict- ing the wave-pattern around a ship (wave profile), and we can predict the change of wave-pattern for various water depth as shown in Fig.12. However, the accuracy of predict- ing the wave-making resistance by the present method was same order as that by the conven- tional linear calculations. Thus, improvement of the present method may be required for bet- ter prediction. 5.3 Results in channel Finally, calculations in shallow channel with rectangular section were carried out. We calculated in hid = 2.048 and W/L = 2.0, where W/L means the ratio of channel width and ship length. For this case Inui has made the calculations by conventional linear theory [103. Panel H-2, F-1 and the channel panels as shown in Fig.14 were used for the computations. Fig.15 shows comparison of wave-making resistance curves in the channel. The conven- tional linear calculations by Inui [10] are larger than the present calculations as a whole, and particularly increase of the wave- making resistance due to channel effect is much larger near the critical speed. Further the shift of hump and hollow in Cw curve can be seen between both calculations. The dis- continuity of Cw curve at Fnh = 1.0 occurs in the conventional linear calculation, but it does not in the present calculation. Thus, it was showed that the present calculation of wave-making resistance was different from the conventional linear calculation. The Rankine Source Method was applied to the shallow water and channel problems. Cal- culations were made of wave-making resistance, wave-pattern around a ship (wave profile) and pressure distributions on sea bottom. The calculated results were compared with experi- ments and calculations by conventional linear analytical method. It was shown that the Rankine Source Method made an improvement of predicting the wave-pattern around a ship in shallow water. However, the accuracy of pre- dicting the wave-making resistance by the present method was same order as that by the conventional linear theory. Improvement of the present method may be required for better prediction. For example, the effect of sinkage and trim should be considered exactly. 648

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0.025 0.020 3 0.01 5 0.010 0.005 I I,- Expel |Cal.(fixed) l 0 0000.15 0.20 0.25 0.30 0.35 0.40 o.i5 0.50 0.55 0.60 0.65 O Cal. ~ fixed coed. ~ Cal. ~ free coed. ~ -- - - - - Exp. by E.Mueller (6) x Conventional linear theory ,' ~ Havelock's integral ~ (6) ,' h /d = 2.389, h /L= 0.233 ,' , i/ ,* ~ / /l'/; ,'~ ~~ if/ / ~ ~ max. Cw \ ~ \ \ 0.4 0.5 0.6 0.7 0.8 Fnh 0.9 1.0 1.1 1.2 1.3 Fig.8 Comparison of wave-making resistance curves in shallow water (in/d = 2. 389) u.u'~ 0.020 Cal O Cal. ~ fixed coed. Cal. ~ free coed. ~ Exp. by E.Mueller C63 0.015 0.010 0.005 x Conventional linear theory ,f/,/ ~ Havelock's integral ~ (6) ~_;, ~ ;/ ,, `,// h/d=3.413, h/L=0.333 \ max. Cw \ Exp. l | Cal; ~ fixed ~ v vvv 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 En 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Fnh Fig.9 Comparison of wave-making resistance curves in shallow water (in/d = 3.413) 649

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1.c ore 0.6 0.4 0.2 0.0 -0.2 -0.4 1.2 ~ '3 ~Ace, Fn = 0.289, Fnh = 0~599 -0.6 -0.4 -0.2 0.0 Fore x / ~ Aft 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -06 ~ 1 1 1 1 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 ~ ~ Fn = 0.332, Fnh = 0.688 ~ w~ l Fore x / ~ Aft O Cal. ~ fixed coed. Cal. ~ free coed. ~ Exp. by E.Mueller (6) 0.2 0.4 0.6 - - Conventional linear theory t6) h/d=2.389, h/~=0.233 Fig.10 Comparison of wave-profiles in shallow water (in/d = 2.389) 0.3 0.2 0.1 0.2 0.1 0.0 -0.1 Fn = 0.332, Fnh = 0.688 Yo/L=0.0 o.o~? , ~1 -O 1 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Fore x / 0.~! Yo/~=0.1667 Aft _? ., 1 1 1 1 1 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Fore x / ~ Aft Y Measured line on tankbottom Ship hull MU ~ L/2 0 . Yo . ~ ~ ' x L/2 h/d=2.389, h/L=0.233 Fig.ll Comparison of pressure distributions on sea bottom in shallow water (in/d = 2.389) 650

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Fn-0.410 ~ ~1~ shal low water Aid, ( had =2.389 / Fn =0.41 0 1.00f 0.80t 0.60t 0.40~ ~ on n Act ~ o. of Coo -0.20 -n 4n . -0.60 _ -0.80 _ -1 .00 _ it, o.o deep water ~ , ~ g.l2 Comparison of prospect view of wave- pattern around a ship for different water depth (factor of wave height is 1.5) 2~/L~ ~ / ~ ~ :~-o.o_,- 2.SO shal low water - ( hid =2.389 ) Fig.13 Comparison of wave contour around a ship for different water depth (contour interval A: is 10.0) 651

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Fig.14 Arrangement of ship hull, free-surface and channel panels (in/d = 2.048, W/L = 2.0) o.osa 0.040 3 0.030 0.02a o.o10 -~- Shallow O Channel ' h/~=2.048, h/~=0.2 - W/~=2.0 water ~ Present Cal. / water l Conventional - Channel J linear theory ~ 1 o) / ,, ~ ~ // .,,,, // / " '4 ~ -I 0 oca 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 En 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 13 1.4 Fnh Fig.15 Comparison of wave-making resistance curves in channel 652

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Computation including the sinkage and trim can be made by improvement of the present method as the following iteration procedures: (a) First, computation of the flow around the ship fixed is made, and the vertical forces (linkage force and trim moment) are calculated. (b) From the vertical forces, we determine the amounts of sinkage and trim needed to hydrostatic balance. (c) The panels for the ship hull surface are rearranged to take the sinkage and trim into account, and the flow is recomputed. Furthermore, the nonlinear effect of free- surface condition should be considered. Acknowledgments 10. Inui, T., "Study on Wave-Making Resist- ance of Ships", 60th Anniversary Series, The Society of Naval Architects of Japan, Vol.2, pp.173-355 (1957~. 11. Hess, J.L. and Smith, A.M.O., "Calcula- tion of Non-Lifting Potential Flow About Arbitrary Three-Dimensional Bodies", Report No.E.S.40622, Dauglas Aircraft Co. Ltd. (1962~. 12. Dawson, C.W., "Calculations with the XYZ Free Surface Program for Five Ship Models", Proceedings of the Workshop on Ship Wave- Resistance Computations, Vol.2, Bethesda, Maryland, pp.232-255 (1979~. Appendix Determination of wave-making resistance from Mueller's experiments The author would like to express his sin- Mueller provides total resistance and cere gratitude to Dr. E. Baba, Manager of Ship residual resistance curves for Inuid S-201 in Hydrodynamics Laboratory of Nagasaki Research shallow water [63. In this paper, the wave- and Development Center, MHI, and Dr. T. making resistance was determined from Nagamatsu, Research Manager of the same Mueller's measured total resistance by follow- laboratory, for their guidance and valuable ing formula : discussions. References 1. Havelock, T.H., "The Effect of Shallow Water on Wave Resistance", Proceedings of the Royal Society, A. Vol.100, pp.499-505 (1922). 2. Kinoshita, M. and Inui, T., "Wave-Making Resistance of a Submerged Spheroid, Ellip- soid and a Ship in a Shallow Sea", Journal of the Society of Naval Architects of Japan, Vol.75, pp.119-135 (1953), (presented in 1944). 3. Inui, T., "Wave-Making Resistance in Shal- low Sea and in Restricted Water, with Spe- cial Reference to its Discontinuities", Journal of the Society of Naval Architects of Japan, Vol.76, pp.l-10 (1954), (presented in 1946). 4. Kirsch, M., "Shallow Water and Channel Ef- fects on Wave Resistance", Journal of Ship Research, Vol.10, No.4, pp.164-181 (1966~. 5. Bai, K.J., "A Localized Finite-Element Method for Steady Three-Dimensional Free- Surface Flow Problems", 2nd International Conference on Numerical Ship Hydrodynamics, Berkeley, pp.78-87 (1977). 6. Mueller, E., "Analysis of the Potential Flow Field and of Ship Resistance in Water of Finite Depth", International Shipbuilding Progress, Vol.32, No.376, pp.266-277 (1985~. 7. Mei, C.C. and Choi, H.S., "Forces on a Slender Ship Advancing Near Critical Speed in a Shallow Channel", 4th International Conference on Numerical Ship Hydrodynamics, Washington D.C. (1985). 8. Gadd, G.E., "A Method of Computing the Flow and Surface Wave Pattern Around Full Forms", The Royal Institution of Naval Architects, Vol.18, pp.207-219 (1976). 9. Dawson, C.W., "A Practical Computer Method for Solving Ship-Wave Problems", 2nd Inter- national Conference on Numerical Ship Hydrodynamics, Berkeley, pp.30-38 (1977). Cw = Ct ~ Cfo~l+K), (30) where Ct : total resistance coefficient, Cfo : frictional resistance coefficient corresponding to the flat plat, K : form factor. Cfo was calculated by Hughes' formula. Form factor K was determined as : K = 0.265 for h/d = 2.389, K = 0.230 for h/d = 3.413. 653

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DISCUSSION by R.C. Ertekin I wish to make some comments on your paper. First of all, it is very surprising that you do not mention the existence of upstream waves in shallow water. There are 3 papers presented in this conference on the subject, where you can find all the related references. Your parameters in shallow water falls in the range we used in our experiments (Ertekin, Webster, Wehausen, 1984, 15th ONR Symp., Hamburg). I am not familiar with Mueller's paper but if he hadn't observed these waves the something is wrong with his observations. In any case, I will seriously question the results and data for Fnh larger than, say 0.6 or 0.7 in shallow water because linear theory is no longer valid, also steadiness, in general, is not possible. Even though we have not paid any attention to Thews and Landweber's (1935) work in the last 50 years, we now know that in very shallow water linear and steady results do not have much meaning. In shallow water, the angle that divergent waves make with the waterline of the ship must be much wider than the deep water case. Even the linear theory can predict this. Therefore, I do not think that your Fig.13 is accurate. It doesn't look right so it must be wrong. By the way, considering that you are using linear theory, could you explain why you need to distribute source on the sea floor (which is flat) and on the free surface? Thank you. Author's Reply Thank you for your comments. The present paper deals with steady wave-making problem, so we did not refer to the papers of unsteady wave-making problem in detail. The detailed review on the unsteady wave-making problem in restricted water is shown in ref.[7]. As expressed in Dr. Ogiwara's reply, we think that the influence of the solution is small for unrestricted shallow water. The present method is a kind of the panel method where Rankine sources are distributed on boundary surface, so we need to distribute the source on the free-surface and sea bottom surface. In case of flat sea bottom, there is the method which takes into account infinite image of the sources distributed on the free- surface and ship hull surface. However this method can not apply to non-horizontal sea bottom generally and has much time for calculation of the infinite image. For the above reasons we employed not the infinite image method but the source distribution method on the sea bottom. DISCUSSION by S. Ogiwara As referred by your paper, T.H. Havelock (1922) studied shallow water effect on ship wave resistance, in which he predicted significant feature of wave pattern that the angle of diverging wave increases as the water depth decreases. Could you simulate the same feature by the proposed Rankine source method? Moreover, considering the case of extremely shallow water, we can find the soliton which generates periodically forward upstream from the bow, as pointed out by T.Y. Wu and others. The method, you proposed here, is not able to treat such phenomena, because this method is involved in the framework of steady state. How do you think about the limitation of water depth (or Fnh=U/ ~ ), to which this method is able to apply? Finally, Resistance and Flow Committee of the 19th ITTC is carrying out the evaluation of shallow water effect on ship resistance and flow around a hull through the Cooperative Experimental Program. You are encouraged to conduct new experiments in order to verify the effectiveness of your numerical method, and to contribute to ITTC. Author's Reply Thank you for your discussions and comment. So far as we observed the comparison of calculated wave-height distributions for different water depth (see Fig.13), it does not seems that the remarkable change of diverging waves appears in the present calculation. The reason why the change does not appear may be due to adoption of linear free-surface condition based on double-body flow in the present method. The linear free-surface condition is employed in the present method, so predicting accuracy becomes poorer in high speed range. Actually we can not see that the tendency of wave-making resistance curve in the present calculations agrees with that in the experiments for water depth Froude number larger than 0.8 (see Figs.8 and 9). However, 654

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from the view point of practical use, the present method has sufficient accuracy for the Froude number smaller than 0.8. Needless to say, for better prediction non-linear effect of free-surface condition should be considered. It is well known that when a ship moves in shallow channel near the critical speed, solution generates periodically forward upstream from the bow and the flow around the ship becomes unsteady[Al][A2]. The generation of the solution is related with blockage coefficient of the channel (or towing tank) and the amplitude of the solution decreases as the blockage coefficient decreasestA3]. All calculations in the present paper except Fig.15 are for unrestricted shallow water taking no account of the channel walls. [Al] Huang, D.B., Sibul, O.J. and Wehausen, J.V.: Ships in Very Shallow Water, Festkolloquium zur Emeritierung von Karl Wieghardt, Institut fur Schiffbau der Universitat Hamburg, Bericht Nr.427, pp.29-49, 1982. [A2] Wu, D.M. and Wu, T.Y.: Three-dimensional Nonlinear Long Waves due to Moving Surface Pressure, Proc. of 14th Symp. on Naval Hydrodynamics, Ann Arbor, pp.103- 129, 1982. [A3] Ertekin, R.C., Webster, W.C. and Wehausen, J.V.: Ship-Generated Solitons, Proc. of 15th Symp. on Naval Hydro- dynamics, Hamburg, pp.347-364, 1984. 655

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