ABSTRACT INTRODUCTION

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1 Wave Resistance Prediction of a Cataaran by Linearised Theory M.INSEL Faculty of Naval Architecture and Ocean Engineering, Istanbul Technical University, TURKEY A.F.MOLLAND, J.F.WELLICOME Departent of Ship Science, University of Southapton, SO9 5NH, UK ABSTRACT The far field wave syste and far field wave coefficients (Eggers coefficients) of a Kelvin source are presented using linearised theory including the effects of a canal depth and width. The wave resistance due to the distribution of such sources is derived using a far field wave energy approach. The tri and sinkage effects of the hull have been included by supplying the dynaic hull surface in the calculation schee. The effect of transo stern has also been investigated by alternative ethods.the developed code has been tested on a atheatically defined hull for and a series of high speed transo stern hull fors. Cataarans with separation ratios of.2 to.5 are used in the coparisons of the ethod with experiental results. INTRODUCTION Use of linearised wave resistance theory for the predictions of wave resistance of cataarans can be justified due to the slender for of the deihulls. A theoretical approach with extensions for transo stern, running tri and sinkage corrections is described to calculate the wave pattern and the wave resistance of a cataaran. In the current work, the far field wave syste of a Kelvin source in a shallow water canal is presented. The wave resistance due to the distribution of these sources is given using a far field wave energy approach. Far field wave coefficients, i.e. Eggers coefficients (1), are also presented. The hulls have been represented by centerplane Kelvin source distribution confiring to thin ship

2 assuption. The tri and sinkage effects on the wave resistance have been included by supplying the dynaic hull surface into the calculation schee. The effect of a transo stern has also been investigated by alternative ethods. The developed code has been tested on a atheatically defined hull for. Various progra paraeters are investigated with this for. A series of high speed transo stern hull fors varying in length to bea ratio are utilised to investigate the results of the predictions on practical hull fors. Finally the code has been applied to cataaran fors to calculate the wave interference between the deihulls. A separation ratio range fro.2 to.5 was utilised as well as deihulls in isolation. The nuerical results and the experiental results are copared in detail. BASIC ASSUMPTIONS AND BOUNDARY CONDITIONS The initial assuptions for the atheatical odel of the proble are: the fluid is ideal; incopressible and hoogenous; the flow is steady and irrotational; the surface tension can be neglected; the wave height is sall copared with the wave length. A cartesian coordinate syste oving with the odel and having its origin on the free surface at the odel centre is used. Ox is in the direction of otion, Oy and Oz are to the starboard and vertical upwards respectively (Figure 1). As the odel advances in the positive direction in the x-axis at a constant speed U in a canal having width of W and depth of H, the velocity potential can be expressed as a suation of the free strea and flow disturbance potentials. Φ(x, y,z)= -Ux+ φ(x, y,z) (1) Assuing the free surface is expressed as z = ζ (x, y) 2 and the underwater geoetry of the ship is represented by y = f(x, z) (for -L/2<x<L/2 and -T<z<), the following linearised boundary conditions ust be satisfied; i) Laplace equation φ φ φ φ + + = (2) = x y y ii) Free surface conditions a) Dynaic free surface condition gζ +U φ = at z = x b) Kineatic free surface condition U ζ - = at z = x φ z or by cobining both conditions φ xx + K φ = at z = z iii) Botto condition φ z = at z = _ H iv) Radiation condition (3) (4) (5) (6)

3 li φ = { 2 2 ( x + y ) o(1) for x < for x > v) Hull surface condition Uf x + φ y = at y = f(x, z) (8) The solution of this proble can be satisfied by representing the hull by a distribution of sources satisfying the boundary conditions. FAR FIELD WAVE SYSTEM OF A SOURCE IN A SHALLOW CANAL The velocity potential of a source with density of µ and located at ( x, y,z ) in shallow water with the depth of H and bounded by a canal having vertical walls with width of W (Figure 1) can be obtained by considering iages of the source due to tank walls at y = y + 2nW and +(2n+ 1)W y = _ y for n = _... and the iage due to tank botto at z2 = -(2H + z ). The far field velocity potential of a source can then be found and is given by INSEL (2) as; g -η ξ 1 φ FF = [ ( x)+ ( x)] U Σ Cos ω Sin ω = - β α K Cosθ (9) Cos( πy/w) for even Cosh( K (z + H)) Cosh( K H) Sin( πy/w) for odd (7) where ξ η = µ τ S Cos ( π y /W) (1) α Cos( ω x ) = µ τ ( π y/w) dx dy β Sin( ω x ) τ S 16πU Wg for even Sin (11) -K H e Cosh( K (H + z )) ( K K (1- K H Sech ( K H)+ Sin θ ) for odd = 2 Cos θ ) (12) ω = K Cosθ (13) The far field wave elevation can be found fro free surface condition. This corresponds to Eggers wave series and is given by INSEL (2) as;

4 ζ ξ η Cos( πy/w) [ Cos( ω x)+ Sin( ω x)] α β Sin( πy/w) = Σ = for even for odd For practical calculations can be truncated at a finite nuber M (typically 1). Equation 14 gives the far field wave elevation of an arbitrary source-sink distribution. ξ, η, α, β are the wave coefficients. ξ and η represent the syetric wave distribution relative to the tank centerline, eanwhile α and β represent the asyetric wave pattern. Hence for a syetric hull for relative to the canal centerline α, β are zero. By substituting n=2, (M=2N) the wave elevation (equation 14) becoes: N ζ = [ ξ ( ω n x)+ η Sin( ω n x)] Cos( 2nπy/W) n= n n (14) Cos (15) WAVE RESISTANCE OF A SHIP MODEL IN A SHALLOW CANAL The wave resistance of a ship odel can be obtained fro considerations of energy changes as given in INSEL (2) : (16) In equation 16, ξ and η and α and β represents wave resistance due to syetric wave pattern and asyetric wave pattern resepectively. NUMERICAL EVALUATION OF THEORETICAL WAVE RESISTANCE Based on the ethod described above, a coputer progra (WAVE2DF) for the deterination of theoretical wave resistance for a ship odel in a shallow water canal has been developed. The hulls are discretised into quadrilaterals. Each quadrilateral is odelled by a surface source distribution over the projection of the quadrilateral on the centerplane. µ 1234 U = - 2π A1234 dy da dx (17) A test case was conducted by using a atheatically defined hull for, WIGLEY HULL (2). The hull for is given by: B z 2 x 2 y = _ + (1-( ) )(1-( ) ) 2 T L/2 (18) Effects of Tri and Sinkage Preliinary calculations and experiental results of the Wigley hull indicated that tri and sinkage of the hull affects wave resistance. Hence tri and sinkage ust be included in the calculations. Progra WAVE2DF has a subroutine which refors the input hull shape into running hull shape by taking the tri and sinkage into account. The sections are represented by splines and are offset by the

5 tri and sinkage. Then the waterlines are interpolated fro the sections. Finally, panels are fored fro these sections and waterlines. Effect of Transo Cataaran deihulls due to their high speed capabilities feature with transo sterns. Flow at the transo stern displays different properties at the low speeds and high speeds on the resistance. At the low end of the speed range, the transo stern is wet and features a dead water zone-rotational flow. However at the high end of the speed range, the transo stern is dry. Resistance features of the wet and dry flow are also different. Various ethods are introduced to calculate transo stern drag. The ethods reviewed here are : a) Hydrostatic ter approach-dry Stern (by Chang 3 ): It assues that the transo is running copletely dry (fully ventilated). Transo drag is given by: = ρg z dzdy (19) R TS ATS b) Base drag approach-wet Stern (by Hoerner 4 ): Fro the tests on projectiles and fuselages a drag coponent known as base drag was easured and expressed as : CTS =.29/ ( C *WS/ ATS ) (2) TS F c) Sink line distribution at transo stern-dry Stern (by Yi 5 ): Yi has suggested that transo stern could be represented by a sink line at the botto of the transo. Sink strength is given by : U µ ( xts, y,z )= _ z (21) 2π As explained above soe of the ethods are applicable to only wet transo stern while others are applicable to only dry transo stern flows. FAR FIELD WAVE SYSTEM AND WAVE RESISTANCE OF A CATAMARAN IN A SHALLOW CANAL As a special case a cataaran ade up two syetrical deihulls is investigated as it is the ain concern for practical applications. For such a craft two hulls will be located at y1 = -S/2, y2 = S/2. Equation 15 and 16 becoe (2) : N 2nπy ζ = [2ξ nc s Cos ( ω n x)+ 2η nc s Sin( ω n x)] Cos( ) (22) W n= (23) where C S = Cos( K n S/2 Sinθ )= Cos( πns/w) n (24)

6 It is noted that ξ n and η n can be obtained either theoretically using equation 1 or by applying experiental values. COMPARISONS OF EXPERIMENTS AND NUMERICAL CALCULATIONS Extensive coparisons of the nuerical ethod with experiental results, given by Insel and Molland (6), were conducted. Firstly coparisons are carried out with atheatically defined for (Figure 2) in fixed to tri and sinkage conditions. As this condition eliinates the probles of transo stern, running tri and sinkage, nuerical results can be assessed. Figure 3 copares the experiental and theoretical results. The nuerical results are rather oscillatory for the fixed odel. This effect is ainly due to exaggeration of the interferences of bow waves and stern waves. In general, the theoretical calculations are higher than the experiental results. Secondly in order to investigate the running tri and sinkage effects the atheatically defined for was tested free to tri and with sinkage. The results are given in Figure 4. These results are very siilar to the fixed results. The oscillatory nature of the theoretical calculations is doinant. A round bilge series (Figure 2) (derived fro NPL round bilge series by changing L/B ratio to 7,9 11 for C3, C4, C5 respectively) of cataaran fors were used to investigate the cataaran interference for which the experiental results are given in Reference 2 and 6. In Figure 5, 6 and 7 the experiental results (CW=CT-(1+k)CF) are copared with theoretical calculations with and without transo corrections for C3, C4 and C5 deihulls respectively. Wave resistance calculations are corrected by transo corrections with a: indicating corrections by Chang's ethod, b: indicating corrections by Hoerner's ethod, c: indicating corrections by Yi's ethod. For the low speed end, the base drag approach by Hoerner is sufficiently good. For the high speed end Yi's ethod gives good results. The hydrostatic ter approach by Chang always overpredicts the wave drag. Hence a cobination of base drag and Yi's ethod offers the best approach. However lack of inforation for ventilation speed akes this approach very difficult, hence only Yi's transo stern sink ethod has been utilised for the cataaran calculations. Experiental results for cataaran cobinations (deihull, separation to length ratio of.2,.3,.4 and.5) are presented in Figure 8a, 9a and 1a. The calculated wave resistance with transo correction is given in Figures 8b, 9b and 1b. As noted earlier, errors in low speeds are high (for Fn<.4). However for the higher speeds, the effect of interactions are found in the calculations. Theoretical calculations for Froude nuber over.7 are correct in agnitude as well. This would indicate Yi's ethod is really applicable to high speeds with the current calculation schee. CONCLUSIONS

7 A nuerical approach to ultihull wave resistance has been developed including the effects of the canal wall and shallow water. The current approach is suitable for any ultihull provided the hulls are syetric relative to the centerplane and fairly thin. The wave pattern and wave resistance of cataarans is presented as a special case. The effect of tri and sinkage is deonstrated as an iportant variable for high speed ultihulls. The transo effects for low speed and high speed is investigated. The deterination of ventilation speed is required for an effective calculation schee. Coparisons between nuerical and experiental results for a round bilge series indicated that interference effects are predicted satisfactorily for high speeds (over Fn=.4) when a sink line distribution is used for the transo stern. REFERENCES 1. EGGERS, K.W.H., & SHARMA S.D., WARD L.W. An assesent of soe experiental ethods for deterining the waveaking characteristics of a ship for, Transactions of SNAME, INSEL, M., A investigation into the resistance coponents of high speed cataarans, Ph.D. Thesis, Departent of Ship Science, University of Southapton, CHANG, M.S., Wave resistance predictions using a singularity ethod, Proceedings of the Workshop on Ship Wave Resistance Coputations, Maryland USA, Vol:II, HOERNER, S.F., Fluid-Dynaic drag, Published by the author, YIM, B., Analyses of waves and the wave resistance due to transo stern ships, Journal of ship Research, June INSEL, M., & MOLLAND A.F., An investigation into the resistance coponents of high speed cataarans, Transactions of RINA, 1992

8 Figure 1: Axis syste and canal diensions Figure 2: Matheatical and conventional odel hull fors Figure 3: Coparison of experiental and theoretical results (C2-Fixed) Figure 4: Coparison of experiental and theoretical results (C2-free) Figure 5: Coparison of experiental and theoretical results (C3-Deihull) Figure 6: Coparison of experiental and theoretical results (C4-Deihull)

9 Figure 7: Coparison of experiental and theoretical results (C5-Deihull) Figure 8a: Experiental results for hull C3 Figure 8b: Theoretical results for hull C3 Figure 9a: Experiental results for hull C4 Figure 9b: Theoretical results for hull C4 Figure 1a: Experiental results for hull C5 Figure 1b: Theoretical reults for hull C5