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On the Numerical Solution of the Turbulent Flow-Field past Double Ship Hulls at I,ow and High Reynolds Numbers G. D. Tzabiras and T. A. Loukakis National Technical University of Athens Athens, Greece ABS=ACr Turbulent flow calculations have been carried out for SSPA 720 double model at a low (Sx106) and a high (Sx108) Reynolds number. The partially parabolic algorithm was adopted to solve the complete momentum equations and k-e model was used for the Reynolds stress modeBing. At the low Reynolds number, results for a whole field solution are compared to those obtained by experimental input amidships. Comparisons are also made between low and high Reynolds calculations and conclusions concermug scaling laws are derived. 1. Introduction Advanced numerical methods, developed during the last few years [1] have been applied with encouraging results for the calculation of the turbulent flow-field past the stern of double ship models. Most of them are based on the simultaneous solution of the velocity and the pressure field (the latter being essential at the thick boundary layer region) and use zero, one or two-equation turbulence models. Although the next step seems to be the development of methods which take into account the free surface effect and the presence of a propeller, there is still a lot of useful numerical investigation to be made on double hulls. Two of the most important problems in this investigation are the simultaneous solution of the whole flow-field past a ship hull as well as the behaviour of the numerical solution at high Reynolds numbers, which is the case with real practical interest. In the present work both of the above problems have been worked out for the case of the SSPA 720 model, for which extended experimental information is available [2], [3].The model has been numerically tested at a low Reynolds number of SxlOb, which corresponds to the test conditions and at a high Reynolds number of Sx108, which corresponds to a full size ship At the low Re No numerics results at the stern region, obtained by either using input conditions amidships from experimental data or solving the complete flow field around the ship hull have been compared to experimental results. Moreover, the pressure coefficients at the bow region predicted by the viscous solver have been compared to measurements and to predictions from a potential flow solution. The purpose of these exercises is to provide some insight with regard to the applicability of the method to the actual problem of ship design, for which no experimental input data is available. The high Re No numerical experiment is obviously the more important from the practical point of view. Although experiments at such high Re Nos. do not exist, comparisons of the velocity profiles, skin friction and pressure coefficients to those predicted at the low Reynolds number can illustrate the trends of the differences between the two solutions. In this respect it is important to see if some flow phenomena(such as the strong cross flow reversal or the rapid changes in the velocity profiles around a stern frame) which occur at low Reynolds numbers are substantially lessintense at a high Reynolds number. Moreover comparisons can be made between the resistance components (viscous pressure and skin friction) in order to test various assumptions concerning the scale effect, when extrapolating model test data The method applied in the present work is basically the fully elliptic algorithm reported in [4], which solves the complete Reynolds equations on the physical 3-D space using a sequence of locally orthogonal Unilinear coordinate systems. The Reynolds stresses are modelled in this code by the standard two-equation k-e turbulence model. 2. The Numerical Method 2.1 Governing Equations For the numerical solution of the transport equations around the ship hull the computational domain is covered by a sequence of 2D orthogonal cunilinear grids as described in [5]. The latter are generated on transverse sections by the conformal mapping method [6] taking into acount the local non-orthogonality at the intersection of the section contour and the waterplane [71. Interpolated sections, needed for grid refinement along the ship, can be easily generated by cubic interpolation between the transformation coefficients of adjacent sections. Representative 2D grids used in the present calculations are shown in Fig. 1 for some sections along the model. In this figure the origin X=0 of the longitudinal axis coincides with the midship section. 539
2x / L= -.8 `~i\3,43 zx/L= .9 zx/ L=.987 Re 5X10 Re 5X108 Fig. 1 Orthogonal curvi linear grids at various sections 540
A local 3D orthogonal curvilinear co-ordinate system corresponds to each 2D grid generated as above, whose two co-ordinate lines coincide with the grid lines and the third one is normal to the section plane. In a general orthogonal collinear system (xi,xj,xl) with metrics (hi,hj,hl), where the indices i, i, 1 are in cyclic permutation, the complete Ui momentum (Reynolds) equation is written as C(ui) =- P hi aX + u3K3i+u~k,i-uiu3Ki3 uiu~Ki~ + (ail 033)K3i + (ail oD)K,i + coil (2Ki3 + Keg) + ',i~(2K 3~ + K 3~) + hi ~ + hi Dx, + hl axl (l) where C(ui) shows the convection terms of the Ui velocity component, that is C(ui)= h h h [ + ~ + 7~ ] The stress tensor components appearing on the right hand side of equation (l) are defined as rsIi = tic 2eii= He 2 [ham+ hih, aX, + hIh~ ax,] <,i3=~ e =p [ ]~( ,)+ hi a (ui)] (2) where the effective viscosity He is modelled according to the isotropic eddy viscosity concept, i.e. lte = ~ + lit (3) where ~ is the fluid viscosity and lit the eddy (or turbulent) viscosity. The curvature terms Kij are expressed as 1 phi K =~ If the xl axis of the adopted co-ordinate system is parallel to the ship symmetry axis (i.e. normal to the ship sections) the following simplifications are valid: hi = 1, Kl2 = K2l = K3l = Kl3 = 0 (4) As already mentioned, in the present investigation the standard k-e turbulence model [8] is employed for the modelling of the Reynolds stresses, that is the turbulent viscosity ,ut is calculated as ~ t PC D e (5) where k is the turbulence kinetic energy, ~ its dissipation rate and CD a constant equal to 0.09. Two more differential equations have to be solved in order to determine k and e. These equations can be cast in the following common form: div[ pOc - ~ t gradO ] = SO (6) m where~=kor e, ok= lam= l.3,Sk=G-Qc, S = 1 44 G £ _ 1 92 p £ and the generation tell11 G is expressed as G = 2pt[e2 + e jj + e2 + 1/ 2 (en + Hi + e il )] The complete equations (l) and (6) are solved numerically for all tranverse sections following the finite volume approximation. A staggered grid is employed and the differential transport equations are integrated in the corresponding control volume of each variable Op. resulting in an algebraic equation of the general form Apes = ANON + ASKS + ACHE + AW~W FADED +AUOu + ~ (8) where the notation P. N. S. E, W. D, U corresponds to grid points shown in Fig.2. Coefficients Ap, AN.... take into account the combined effect of convection and diffusion terms modelled according to the hybrid scheme of Spalding [9]. Equations (8) form a system of algebraic equations which is solved by succesive applications of the tndiagonal matrix algorithm. IN ~X2// 1 D WE 3 W P. , S Pi g .2 Speci fi cati on of gri d poi nts 541
2.2 Boundary Conditions The calculation domain around the ship can be divided in two sub-domains surrounding, respectively, the front and the rear part as shown in Fig. 3. In the front part domain (I), corresponding to the thin boundary layer region, relatively coarse grids can be used to model the viscous flow. Besides, high convergence rates of the numerical solution can be achieved due to the strong upstream convective influence and the existence of favorable pressure gradients over the major part of the body surface. The aft part calculation domain (II) covers the thick boundary layer region around the stem of the ship and extends in the near wake. The flow there is characterized by complex phenomena such as vortex formation, interaction between the boundary layer and the wake or adverse pressure gradients and finer grids must be applied in order to obtain reliable numerical results. Pi g .3 Def i n i ti on of sub-domai ns following conditions are valid: waterplane: The solution of the elliptic-type algebraic equations (8) requires specification of boundary conditions at ship symmetry plane: each bounty of the two sub-domains, that is at the inlet planes U. the external boundaries N. the exit planes D, the solid surface S and the syrrunetry planes of the ship (Fig. 3). The input boundary values for the velocity components at the inlet plane UI of the front part domain, located upstream the ship's bow, are calculated from the potential flow solution. The latter is obtained by the application of the classical Hess and Smith method [10] around the actual shape of the ship. The values of k and ~ at the same boundary are assumed to be equal to zero. The corresponding input boundary conditions for the velocity components and turbulence quantities at the inlet plane UII of the second calculation domain are determined from the front part flow solution by linear interpolation. At the same plane, input conditions can also be calculated by empirical formulae, whenever experimental data are available. At the exit planes D of the domains the flow is assumed to be fully developed, corresponding to the application of Neummann conditions for each variable, except the pressure. The latter is calculated by linear extrapolation from the computed values at the previous sections. The velocity components and the pressure at the external boundaries N are calculated from the potential flow solution, whereas for k and ~ the normal to the boundary derivatives vanish (Neummann condition). The turbulent flow near the solid boundary is modelled according to the standard wall function method [11] assuming that the velocity in the adjacent to the wall cells follows the logarithmic law u+ = 1/x In (Ey+) (9) where =0.42, E=9.79, y+ the non-dimensional distance from the wall and u+ the non-dimensional velocity parallel to it. Relation (9) is implicitly introduced in the momentum and k-e equations leading to a simplified set of boundary conditions for the corresponding variables. Finally at the two flow symmetry planes the U3 =0 , ant =0 , An= Ul'U2'P'~6 U2 = 0 , Ax' = 0 , ~ = Mu flak 2.3 The Solution Algorithm The existence of a dominant flow direction along the co-ordinate axis xl, which is parallel to the symmetry axis of the ship allows for a marching solution of the governing transport equations, known as the partially parabolic algorithm [12]. The method has been ordinally developed to solve the parabolized Navier-Stokes equations [13] but it can also be applied to the solution of the complete form of equations (1) and (6). According to the partially parabolic algorithm, a local numerical solution is performed in each transverse section of the calculation domain. Firstly the U3, u2 and u1 momentum equations are solved and then the pressure field and the velocity components are corrected to satisfy the continuity equation. Then the k-e equations are solved and the eddy viscosities are updated using relation (5). In this local solution two-dimensional in-core storage is essentially needed for various geometrical and flow parameters, permitting the use of fine grids even with conventional computers. After solution for every section of the domain is performed, a sweep is completed and calculations start again. Several sweeps are needed until both the velocity and pressure fields converge. The most crucial point in the application of the partially parabolic method is the treatment of the pressure field. In the present work the SIMPLE [14] algorithm has been adopted for the local correction of the pressure and the velocities. The application of this algorithm requires underrelaxation of variables during the iterative solution procedure, that is the updated value ~ of a variable is calculated as linear combination of its previous value NO and the solution En Of system (8), through relation ~ = r En + (1-r) To where r is the underrelaxation factor which is constant for every grid node of a transverse section. 542
Although SIMPLE and partially parabolic algorithms form the basis of the convergence procedure, it has been found that at the front part calculation domain (Fig.3) they can be combined in a different way than at the rear part domain. While at the stern region several iteration SIMPLE steps are needed to achieve local convergence, the existence of a thin boundary layer over the bow and the middle body of the ship allow a single step local solution at the front part domain. The latter results in a decrease of the computational cost by a factor of 30. A new approach has also been applied for the numerical solution at high Reynolds numbers. The high grid densities, required to model the turbulent flow near the wall at the above numbers, lead to the generation of computational cells having their normal to the wall dimension substantially lower that the other two dimensions. This geometrical property is quite unfavorable for the pressure correction methods applied in this case, especially at the stern region where steep longitudinal and transverse pressure gradients occur. Moreover, at the same region, it is difficult to obtain the necessary grid clustering near the wall by global grid generation methods. To overcome the aforementioned problems a special near wall treatment has been developed, as shown in Fig.4. The near wall computational cells corresponding to the initial mesh generation, can be automatically subdivided to any desired number of sub-cells and a second computational domain is created. Two different solutions are applied in the resulting internal and external domains. For the internal solution the pressure values within the normal to the wall generated cells is assumed to be equal to the "external" value at node N. This assumption valid near the solid boundary, is quite beneficial for the solution procedure followed: the US and u1 momentum and k-e equations are solved in the sub-domain as in the external domain, while the u2 component (normal to the wall) is calculated explicitly from the integrated continuity equation. For the external solution the SIMPLE procedure is followed. At the common boundary (B) of the two domains the boundary conditions for various variables are updated according to the adopted finite difference formulation for the convection and diffusion terms. Convergence in a transverse section is achieved after several successive internal and external solutions. ~ 4 IN ~ ,,,,,, ;: Fig.4 Near wal 1 treatment 543 3. The Numerical Tests As already mentioned in the introduction, calculations with the described methods were carried out for SSPA-720 double model. 3.1 Low Reynolds Number Computations For the front part calculations, a 32x20x61 grid was used where 32 is the number of grid nodes along the girth, 20 the number of nodes along the normal direction to the section contour and 61 the number of transverse sections. The inlet plane of the calculation domain was placed at 2X/I'-1.2 and the exit plane at 2X/I'0.30. Convergence of the numerical solution was achieved in 80 single-step sweeps of the domain and constant underrelaxation factors equal to 0.4 were used for each variable. The values of y+ in the adjacent to the wall cells ranged between 30 and 50, that is within the suggested region for the application of wall functions (30~150). In Fig. 6 results for the predicted Cp coefficient are compared to experiments as well as to potential flow calculations. The latter were obtained using 673 quadrilateral elements on the model surface. For the rear part calculation domain a 32x30x44 grid was used starting at 2X/I'O. 1 and extending up to 2X/L=1.4. Both types of input boundary conditions were tested, the first one corresponding to a whole field solution and the second to an experimental input. In the second case the velocities within the boundary layer were calculated according to the 1/n power law using the experimental data of Larsson [2], while the initial values for k and ~ were estimated by empirical formulae [5]. A total number of 35 sweeps was needed to obtain convergence in either case. An initial number of 15 iterative steps was required for local convergence in a transverse section, which reduced to one up to 5 steps during the last sweeps. The values of the underrelaxation factors at the rear part were constant and equal to 0.5 for every veriable. The values of y+ ranged between 30 and 170. In Fig.7 computational results for the streamwise (IJ/Ue) and crosswise (W/Ue) velocity components are compared to experiments for points 11 tol7 of station 2X/I'O.9 shown in Fig.5. In this Figure the vertical axis refers to the non-dimensionalized normal distance from the body surface with respect to the experimental [2] boundary thickness be and the horizontal axis to the non-dimensionalized velocities with respect to the velocity at the edge of the boundary layer. A special output program has been developed to compute the necessary variables along normal' to the body surface by linear interpolation among the stored values. It should be noted here that the experimental results were subject to blockage effects while no such effect has been accounted for in the calculations. In Fig. 8 the calculated, non-dimensionalized by the free stream velocity, CF coefficient is compared to the experimental data around the girth of the previous section. Results for the pressure distribution are also presented. 3.2 High Reynolds Number Calculations The same transverse sections and girthwise points as in the case of the low Reynolds number
calculations, have been used for the high Reynolds number tests. The grid density was different only along the normal direction, as shown in Fig. 1. A 32x30x61 grid was employed for the front part calculations and convergence of both the velocity and pressure fields was obtained in 150 sweeps. Constant underrelaxation factors equal to 0.3 were used for each variable. The values of y+ ranged between 100 and 300. A 32x30x44 grid was used for the stern part calculations. The underrelaxation factors were equal to 0.5 for all variables and convergence was achieved after 25 sweeps. Two grid types have been tested, that is a coarse near the wall grid with y+ varying from 100 to 1400 and a fine grid according to the method previously described. The latter was created by dividing the initial near wall cells in 10 sub-cells, that is a total of 40 grid nodes along the normal direction was used. Starting with the coarse grid solution, 15 more sweeps were needed to obtain convergence with the fine grid. The corresponding values of ye ranged between 15 and 150. Comparisons between the calculated results by the two grids as well as with the results for the low Reynolds number are presented in Figs. 9 to 12. /15 /13 112 ~9 '119 11 Fig.S Distribution of calculation points at 2X/L=O.9 4. Discussion of the Results 4.1 Low Reynolds Number The calculated pressures along the girth of venous sections of the fron part of the ship compare well with the experiment values, Figs. 6 a to 6 d, when corrected for blockage effects as proposed by Larsson [15]. The agreement is good both for the Viscous and the potential flow pressure calculations, with the exception of the lower part of the stern section with 2X/I'-0.93. In this case it is believed that the discrepancy is due to the insuffient accuracy of classical Hess and Smith method near the extremeties of the body. Cp .1 ~ .1 d of' ~ ~ 2SC/L=.93 ''/ / ~ pot. CDIC. / viscous calc. experiments a * fit . ~ b 2X / L=-.7 if\ __ ~ .._ ~ .~ * * ~< ~ ~ 2x/ L=-.4 . . . _ .1 In_ _ _ ~ _ _ ~ _ _ _ ~ _ _ a_ _K 2X/L- O L GIRTH % o 1 0 0 Fig.6 Comparisons of pressure coeffici ents at the fron part for Re=5xiO 6 544
lo Ax 1 14~, a) alc- =4~ can * CL o n * a, ,,_, C ~ _ Q ~ . _ ~ X X 1 ~ ~ -. 1 1 ~ 3 . .U ~ . ~ ~ L_. ~ m _ u' )* a) ` J `'~ : ~ - . ~ ~ *"it * U' . _ *- '~* '~* 'I,, lo - o a' \ * -I m _ -God *~ Go: ~ : \ I -' ' I ' ~ ~ ' ' L =_~ ~ Ir) I~ go o up 11 con J \ X _ ~ ~ Fir o _ Q Fir o > o ~ 0 Fir Q o : - : - n _ o --~ ~ 545 m Ir L
Therefore it may be concluded that the proposed Viscous flow calculations for the front part of the ship can be conveniently used in conduction with the same calculation method, which is widely used for the aft ·1 end of the ship. The advantages of such calculations are that they do not require any assumptions with regard to the upstream input boundary conditions, while they automatically generate input conditions for the aft part solution. Needless to say that the method is more expensive to run than simpler approaches. The calculated velocity profiles, using both the whole field solution and the aft part solution based on experimental input, are compared to measured values along the girth of section with 2X/L=O.9, Fig.7. The calculated results agree in general well with the -1 measurement for both velocity components. The results based on experimental input are somewhat better but, as it will be pointed out later, the overall difference in the total resistance prediction between the two methods is very small. The overprediction of the streamwise velocities near the surface at points 19 and 9 can be explained by observing from f~.7 that in this region the geometry of the hull surface is rapidly changing, a situation for which the k-e 4 turbulence model is known to overpredict [1], [16], [17]. It should be noted here that the comparison between measured and calculated velocity profiles is somewhat indirect because the measured results are affected by blockage effects. However numerical calculations for the same hull [4], taking into account blockage effects, have shown that the non-dimensional velaecity profiles remain practically the same. The predicted by both aforementioned methods values for CF along the girth of the same as above section are compared to experimental values in fig.8. The agreement is particularly good for both methods. The corresponding values for Cp, predicted by boath methods shown in the same Figure, are in close agreement. No experimental pressure values exist for this section. Finally, although there exist no measured values for the total resistance of the ship to be used for comparison purposes, it is interesting to note that the predicted by the whole field solution total resistance is 2.5% higher than the one predicted using experimental input for the stern part solution. 4.2 High Revnolds Number The results for the velocity profiles at station 2X/L=O.9 shown in fig. 9, allow us to conclude that the local and refinement produces no noticable effect, although it requires approximately 30% more computer time. The same conclusion is reached by observing Fig.10, where the girthwise results for Cp and CF at the same station are shown. This is a remarkable result showing that at high Reynolds numbers the wall function method is valid for a wide range of y+. However more numerical experiments should be made for various hull forms to establish this behaviour. The overall difference in the prediction of the total ship resistance using both methods is of the order of 1%. /~ GIRTH % ~ v / ~ ~ ~ . ~ ~ ~ ~ . it, . . . . . . . . Re 5~106 2 L 546 ~ * *I\\ \\\ \ I\ Cf~l00 whole field - exp. input experiments I , , GIRTH % Fig.8 Comparisons of pressure and friction coeffi ci ents at 2X/L=0.9 4.3 Comparison of the Low and High ReYnolds Number Cases 100 The profiles of the streamwise and crosswise velocity components are shown in Fig. 11 for the station with 2X/I'O.9. All calculations were performed using the whole field solution. The streamwise component is higher for the high Re. No., as expected, with larger differences near the keel. A more interesting conclusion is reached by observing the crosswise velocity profiles. At the high Re.No. the S shape of this profile, which exist at the low Re. No., is lost or is reduced. Consequently, the hull form is less prone to vortex formation at the high Re. No., a fact which has also been experimentally verified. Finally the girthwise distribution of the Cp and CF distribution for the same station are shown in Fig. 12 for both Re. Nos. In the same Figure the calculated values for pressure using potential flow are plotted. As expected the potential flow solution yields larger values than the viscous flow solutions, with the differences diminishing with increasing Re.No. The girthwise values of CF are more constant for the high Re.No., whereas CF has very high values near the keel for the low Re.No.
- - 11 o -- - -- lo 11~ o = o < + + o o o o - o a\ o x air r U O . ~ = 347 o C o air ~ . r
o n =\1 3 o ILL ~ 1 ~ a) \ 0 (D ~ O O ~ 1~ lto [Y tY , _ -"in ~ ~ `` \ \ `\ 1 1; _ 8; . _ ^8160 I Up ~ Q O ll \ X $r o Q _. o so Up o o o so -- ~ ~ " \ E "hi \ a " 8n ~ · o 't CL ' so L 548
cp [18], for which grid independency of the numerical solution was achieved. -41 .1 1\ ~ GIRTH ;: . . , ~ · ~ ~ ~ . . I VIellelI. R4r Axiom -Cf *1000 - y+=100....1400 ye = 15.. 150 Fi g . to Compari sons of pressure and fri cti on coeff i ci ents at 2X/L=0 .9 4.4 Scaling Laws for Ship Resistance Prediction With regard to the independency of the numerical results to the transverse grid parameters, previous calculations with a coarser grid [4] have yielded quite solar results for the low Re No case. Similarly for the high Re No. case, a more dense grid near the body surface has produced effectively the same velocity profiles. Therefore we can assume that the numerical results are reliable in this respect. However, no such examination was performed with respect to other grid parameters, which Night affect the numerical solution, e.g. the number of the hull cross sections used for the calculations especially for the low Re.No. case. Nevertheless, the numerical results seem to be good enough to allow for an attempt to demonstrate their impact on practical procedures for the prediction of the resistance of the ship. In this endeavour we are encouraged from similar trends for the case of a body of revolution As it is well known, ship resistance predictions are based on model experiments, on a flat plate friction line and on appropriate scaling laws. We now consider that the examined low Re.No case represents model tests with results shown In Table 1, which also contains the results of a corresponding full scale experiment. Since the free surface effect has been neglected, this pair of experiments is thought to be conducted at a Froude No equal to, say, 0.15. If we now use both the form factor (K) method and Froude's method, each in combination with both the I.T.T.C. and the A.T.T.C. friction lines, we can derive Table 2. Then, it can be concluded that none of the above combinations predicts the ship resistance adequatly aIld that the form factor method underpredicts, but it is closer to the calculated ship resistance than the Froude method, which overpredicts. Needless to say that the numerical methods presented herein can be easily used for the direct prediction of the ship resistance, a fact which necessitates full scale experiments to validate their accuracy. Finally, it should be mentioned that a complete set of calculations for a hull form require approximately 60 hours of computer time on a MicroVAX II machine, amrnount which can be reduced to about 2 hours on a modern RISC technology workstation. , GIRT X, , 100 REFERENCES 1. Patel, V.C., "Ship stern and wake flows: Status of experiment and theory", 17th ONR Symposium, The Hague (1988). 2. Larrson, L., "Boundary layers of ships (lbree- dimensional effects)", Ph.D.Thesis, Chalmers University of Technology, Goteborg (1975). 3. Lofdahl, L., "Measurement of the Reynolds stress tensor in the thick three-dimensional boundary layer near the stern of a ship model", Ph.D.Thesis, Chalmers University of Technology, Goteborg (1982). 4. Tzabiras, G., "On the calculation of the 3-D Reynolds stress tensor by two algorithms", Second International Symposium on Ship Viscous Resistance Goteborg (1985). 5. Tzabiras, G. and Loukakis, T., "A method for predicting the flow around the stern of double ship hulls", International Shipbuilding Progress, No 345, pp 94-105 (1983). 6. van Kerczek, C. and Tuck, E.O., "lye representation of ship hulls by conformal mapping functions", J.S.R., Vol.13, No.4, pp.284-298 (1969). 7. van Kerczek, C. and Stern, F., "The representation of ship hulls by conformal mapping functions: Fractional maps", J.S.R., vol.27, No.3, pp. 158-159 (1983). 549
8. Launder, B.E. and Spaldir~g, D.B., 'The numerical computation of turbulent flows", Comput. Methods in Appl. Mech. and Eng., vol.3, pp.269-289 (1974). 9. Spalding D.B., "A novel finite-difference 1 formulation for different expressions involving both first and second derivatives", Int.J. Numer. Methods in Eng.,vol.4, pp. 551-559 (1972). lO.Hess, J.L. and Smith, A.M.O., "Calculation of potential flow about arbitrary bodies" Prog.Aeronaut.Sci., vol.8, pp.1-138 (1966). 1 l.Rastogi, A.K. and Rodi, W., "Calculation of general three-dimensional boundary layers", AIAA J., vol.16, pp.151-159 (1978). 12.Pratap, V.S. and Spalding, D.B., "Numerical computations of the flow in curved ducts", Aeronaut. Q., Vol.26, pp.219-232 (1975). 13.Abdelmeguid, A M., Markatos, N.C.G. and Spalding, D.B., "A method of predict~ng three-dimensional turbulent flows around ship hulls", 1st Int.Symposium on Ship Viscous Resistance, Goteborg (1978). 14.Patankar, S.V. and Spalding, D.B., "A calculation procedure for heat, mass and momentum transfer ~n three-dimensional parabolic flows", Int.J. Heat and Mass Transfer, vol.15, pp.1787-1806 (1972). l5.Larsson, L. (editor), "SSPA-ITTC workshop on ship bondary layers", Goteborg, 1980. 16.Tzabiras, G., "Numerical and experurnental investigation of the flow field at the stern of double ship hulls", Ph.D. Thesis, N.T.U.A. (1984). 17.Tzabiras, G., "A numerical investigation of the turbulent flow-~:eld at the stern of a body of revolution", J.Appl.Math.Modelling, vol.l, pp.45-61, (1987). 18.Tzabiras, G. and Garofallidis, D., "Prediction of the resistance characteristics of an axisyrnmetric body with a propeller model", to be presented in PRADS-89 Conference, Varna. ,1 ~ 4 550 :\` ~ /; GIRTH ;: _r I L ~ . . I · I · 1~ , j , ~ ~ . . . _ ~ * * * * * * **** * ~. - Re- 5~108 Re-5xl°6 1` ~ Pot.flow \ \ Cf *1000 '~= ~ ~ ~ - _ , , , , , , {;IRT~ X, , keel 100 Fi g. 12 Compari sons of pressure and fri cti on coefficients at 2XiL=O.9
Tab l e 1: Cal cul ated resi stance coeff i ci ents Re = 5X10 CT 4.754xlO ~ Cp 1. 024x10 -3 CF 3.73 ~0-3 Re = 5x10,3 2.560 10 3 0.77 x1o ~ 3 1. 79XlO -3 Table 2: Calculation of ship total resistance coefficient by the form factor and Froude method . CTM x 10 3 CFMX10 3 K CRX10 3 CFSX10 3 CTsx 10 ~ CTsxlO 3 computed form factor pl us 4.754 3.397 0.399 _ 1.671 2.21 2.56 -13.6 I TTC fr~ct~on l ~ne form factor plus ATTC friction 1 ine 4.754 3.294 0.443 1.671 2.32 2.56 Froude s plhsd 4.754 j .397 1 1 1 35 1 1 671 1 3 OZ8 1 2.5 I TTC f . l . ;roude s T 4.754 1 ,.294 ~ _ ~ 1.46( 1 1.671 1 3~ 13 1 2.5~ : pl us ATTC f . l % Di ff . - 9.4 + 17 + 22 551
