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On the Numerical Solution of the Total Ship Resistance Problem under a Predetermined Free Surface G. Tzabiras, T. Loukakis, G. Garofallidis (National Technical University of Athens, Greece) ABSTRACT The free surface around a ship model moving at constant speed was determined experimentally. The resistance components of the model were computed numerically by solving the Reynolds equations beneath the predetermined free surface. The calculations were made using the finite volume approach and the partially parabolic procedure. The standard k-e turbulence model was used for the Reynolds stresses. Calculated and measured values for the total resistance have been compared and the applicability of the method is discussed. NOMENCLATURE Al finite difference coefficients C(~) convection term of Cp pressure coefficient Cf skin friction coefficient G generation term of k g gravitational acceleration hi metrics h vertical distance kij curvature tensor k turbulence kinetic energy deformation tensor - 1J P pressure PI pressure (P+ggh) RF frictional resistance Rp pressure resistance RT total resistance velocity components collinear co-ordinates ui xi S.`:, source terms Greek symbols dissipation of k ~ fluid viscosity lit eddy viscosity he effective viscosity Q fluid density oil stress tensor variable Dept. of NAME, NTUA, 42, 28th Oktovriou Str., Athens 10682, GREECE 789 INTRODUCT ION The degree of accuracy at which a physical problem should be solved depends, obviously, on the application of the solution. In this respect, in ship design and construction, the most important problem pertaining to hydrodynamics is the accurate prediction of the ship speed and the corresponding propeller revolutions and shaft horsepower. This is because these quantities are specified in the contract of a newbuilding and the shipyard has to pay penalties if the ship propulsion performance is found inferior during the ship delivery trials. Therefore, the efforts of untold numbers of marine hydrodynamicists and mathematicians during the last century or so to solve analytically the ship propulsion problem have been well directed. However, and from the practical point of view, these efforts have been largely unsuccessful. Thus, in current engineering practice, only the prediction of the propeller performance in a prescribed wake field is thought to be trustworthy enough to be actually used on a routine basis. The prediction of pressures and shear stresses around the hull of a ship, moving at constant speed in calm water, remains an elusive goal. The same is tree for the wake field behind the hull and for the propeller hull form interactions. This state of the art is especially bothersome if the availability of virtually unlimited computational power is taken into account. On the other hand, more difficult but less important problems, from the point of view of contractual obligations, such as the dynamic behaviour of the ship in waves can be and are successfully treated using simple theories of moderate accuracy. The simplest of the yet unsolved problems in ship hydrodynamics is that of the ship resistance in calm water and at constant speed. In this case it is the belief of the authors, that for the vast majority of practical applications this problem should be treated from the beginning as a viscous flow problem. That is, it is believed that the incomplete modeling of the flow as inviscid, in order to obtain the wavemaking resistance separately, will not lead to a successful solution of the real life problem.

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In recent years the availability of computer codes solving the complete Reynolds equations and of powerful computers has yielded promising results in the case of computing the flow field around three dimensional, shiplike bodies in the absense of a free surface. A recent survey of this area of research is presented by Patel (11. But even for this simplifed case some problems seem to remain as can be deduced from the scarcity of computations for the high Reynolds numbers of the ship scale (2~. Very recently, results have been presented for the solution of the complete problem of a ship moving in a viscous fluid, e.g. Hino (3~. Although the results presented seem reasonable when compared to experimental values, no result for the ship resistance is given. Moreover, the method does not seem at this stage to include the computation of the equilibrium position of the ship under way. That is the ship is assumed to move at its static equilibrium position, which is a common assumption when trying to solve the ship resistance problem. However, as it is well known from experimental results, the dynamic equilibrium position is an important factor for the determination of the ship resistance, for a ship moving at constant speed and weight. From this discussion, one might conclude that, although many steps have been made towards the prediction of the ship resistance, no practical solution of the problem is in sight. This conclusion is strenghtened by the fact that no model of turbulence, required for the Reynolds equations, has been derived with free surface flows in mind. In view of the above it was decided that a meaningful intermediate step, in the long route necessary before ship resistance can be analytically computed, was to remove as many uncertainties as possible and to treat a simpler case with the method described in (4~. Thus, a three meter model of a liner ship was tested in a Towing Tank and its dynamic equilibrium position 9~nx4.65mx3.0m. as well as the wave pattern around it were measured together with its total resistance. In this manner, and looking at the ship from below, the actual solid and liquid boundaries for the flow were determined. It was thed straightforward to run the NTUA viscous flow computer code for this prescribed fluid region, applying appropriate conditions at the boundaries and obtain the total ship resistance, as the sum of the pressure and the wall shear stress forces. The viscous flow computer code of NTUA uses the standard k-e turbulence model which has been applied with relative success but it is unknown if it is valid near a free surface. For axisymmetric fully submerged bodies, the method gives good pressure predictions and a small overestimation of the velocities (5~. The integrated results, that is the force predictions, are good. For double hull ship forms, the method gives again good results for pressures and wall shear stresses although it overestimates the velocities in some areas (2~. However, when the method was used to obtain the integrated resistance force for the case of a landing ship running at a low Froude number, the computed results seemed to overpredict the measured resistance by almost 7% (6~. 790 The results of the present research effort, which is modest in scope as it is unsponsored, can be of value since they focus on the capabilities of the numerical solution of the viscous flow problem in a predetermined domain. As will be seen in the sequel, the analytical prediction of the total ship resistance, although reasonable, does not compare well with the experimental value. Therefore, more research is necessary as will be explained in the conclusions. For this reason, the idea of using the predetermined free surface to compute the ship resistance for the Reynolds number of the full size ship was abandoned as premature. DESCRIPTION OF THE EXPERIMENTS The 1:50 scale model of a liner ship was selected for the purposes of the investigation. This model had been tested previously at the Towing Tank of N.T.U.A. For the same ship, another model at a 1:30 scale had been tested at N.T.U.A. and a third model, at a 1:22.43 scale at the Bulgarian Ship Hydrodynamics Center. Thus, enough data were available for the determination of the form factor according to the lllC methods and definitions. The principal chacracteristics of the 1:50 model are shown below and the body plan of the model is shown In Fig.1. Length on Waterline LWL= 3.083 m Beam B = 0.429 m Draught T = 0.179 m Block Coefficient CB = 0.575 Prismatic Coefficient Cp = 0.601 Midship section Coefficient CM = 0.956 Wetted Surface S = 1.694 m Displacement ~ = 133 kp The dunmensions of the Towing Tank are Figure 1. The body plan of the model.

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Resistance Measurements The model was attached to a dynamometer-heave rod-pitch bearing assembly, which can measure the model resistance, parallel sinkage and running trim as the model is towed at constant speed. The model was tested repeatedly at a speed of 1.346 m/sec., which corresponds to a Froude Number of 0.245. At this speed and for a water temperature of 22C, the measured resistance and parallel sinkage were: Speed Resistance Parallel Sinkage 1.346 (mls) 0.682 (kp) 0.6 (cm) The measured valued of running trim was practically zero. With regard to the resistance measurements in particular and the test conditions in general, two remarks are in order. Firstly, the blockage effect on this model is negligible, about 0.4% when expressed as a speed correction and secondly that, although the model speed is quite constant, the resistance force is not. This can be explained by the fact that resistance measurements are obtained by measuring the deflection of an elastic member connecting the model to the towing carriage. During the run this spring - mass system can be excited to oscilate along the longitudinal axis of the model. For the particular model, an analysis of the time history of the resistance force during a typical 30 seconds run showed an oscillation of approximately +10% about the mean value. However, since the actual logitudinal deflections of the system are extremely small, the oscillation of the model is not expected to affect the steady wave pattern. Form Factor Determination During previous tests with the 1:50 and 1:30 scale models, it has been determined that the value of the form factor for - this hull form is 0.14. That is the total viscous resistance is CV=1.14CF, where CF is the frictional resistance of a flat plate according to the 1TTC 1957 formulation. The same value for the 1:22.43 scale model is 0.16. Needless to say that the ITT C method for deterring these form factors is approximate, as one tries to establish Usually at what speed the wavemaking resistance practically disappears and what is the corresponding model resistance, at a low speed region around Fr.No = 0.12, where the experimental results show no negligible scatter. Measurements of the Steady Wave Pattern The wavy free surface necessary for the numerical calculations was obtained in a mixed manner. Different methods were used for the intersection of the free surface and the hull surface, for the wave region near the hull surface and for the wave region away from the hull surface. Measarements of the Free Surface-Hull Intersection An auxiliary grid was painted on the surface of the model about the waterline. The dimensions of the grid were lem for the waterlines and 2cm for the transverse sections. It was then tried to determine the intersection photographically. The resolution of this procedure was not satisfactory, in particular near the bow where the measurements are most important. The required intersection was finally obtained by scratching several points at the side of the model during the run and then taking the model out of the water and drawing a faire d line through these points. Measurement of the Wave Region Near the Hull Surface This region was defined to extend from the centerline of the model to a distance of 38.9cm sideways. It is reminded here that the maximum half breath of the model is 21.45cm. For this region the wave pattern was obtained using photogrammetric methods. The methodology applied and the results obtained have been described in (7) and will only brieflfy discussed here. Two non-metric motor driven cameras, Hasselblad ELM with normal angle lenses of 8~nm principal distance, were used. The cameras were positioned on a rigid base, which could slide along a specially constructed guiding rail bolted on the towing bridge. In this way the relative position of the cameras remained undisturbed and the coverage of the ship model with stereoscopic models became possible. The two cameras were electronically synchronized and a powerful flash was attached to the system in order to take care of the poor lighting conditions under the bridge. The cameras were positioned approximately 200mm apart, at a distance of lm from the waterline and with an inclination of approximately 35 grad In this way a favourable base-to-distance ratio was ensured, while at the same time maximum possible coverage of the object was obtained. The problem of providing detail points on the water surface was solved by spraying, just before the shutters were fired, yellow paper-tape punch of lmm diameter on the water surfing. For the basic control of the orientations, an aluminum bar bearing two retro-targets at a distance of 283mm was hung on the model. The stereoscopic models were levelled, by taking two pairs of pictures, one at rest and one underway for each case. Nine pairs of stereoscopic models were taken in order to cover the whole legth of the model and the necessary area behind it. The photogrammetric processing of the photography was carried out on a ZEISS Stereocard G2 connected via a DIREC 1 unit to a desk top computer. Analytical processing of the stereocard data was used. An accuracy of about lmm for the wave heights can be obtained using this method. 791

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Measurement of the We're Region Away from the Hull Surface The wave region extending from a distance of 38.9cm from the centerline of the model to a distance of 218.9cm was covered by taking 95 longitudinal cuts of the wave surface. The longitudinal cuts were obtained by the repeated use of commercially available wave monitors of the resistance type. A specially constructed overhang beam was used for the attachment of five wave monitors at predetermined distances from the centerline of the tank. The probes were stationary and they were recording the wave elevation as the model was passing by. One probe was always positioned at a distance of 28.9cm from the centerline and it was used to "allign" the other probes longitudinally. Thus, four longitudinal cuts were obtained per run and 24 runs were necessary to obtain 96 longitudinal cuts. The adignement of the wave cuts was based on the crest of the first wave of the 28.9 cm cut. The same point was used to allign the whole system of the longitudinal cuts to the model using the results of the stereoscopic model of the bow region. An additional cut at a distance of 33.9 cm was used to check the results of the photogrammetric procedure. The details of the longitudinal cut measurements are given in (8). The accuracy of the wire probe measurements is of the order of one milimiter, on the basis of their static calibration curves. How this accuracy is affected during dynamic measurements is not known. Analytical Determination - Cuts of the Ware Pattern of Transverse curvilinear orthogonal grid is created using the method of singular distributions, as described in the sequel. In Fig. 2 eight orthogonal curvilinear meshes are shown at various sections along the ship model and the wake. The boundary S (Fig. 2a) is the model section contour and the boundary W is the free surface intersection with the corresponding transverse plane. The generation of an orthogonal grid in the 2D domain defined by the boundaries N,S, E, W shown in Fig. 2a is based on the incompressible potential flow solution (9~. A singularity distribution on the four boundaries is assumed, i.e., a source distribution on boundaries N. S and an eddy distribution on E. and W. Using rectilinear elements, the unknown distributions are calculated to satisfy the boundary conditions of a potential function A, that is at ~ =o an N,S at 1 =0 as E,W whre n is the normal direction on N or S contours and s the direction tangential to E or W boundaries. After the element source or eddy distributions are computed, the grid nodes can be specified as intersections of equi-potential and equi-stream function lines, following the iterative procedure described in (9~. The velocity components and the other flow variables refer to local orthogonal curvilinear co-ordinate systems coinciding in two dimensions with the grid lines x1 = const and x2 = const, while their third direction X3 is always parallel to the ship longitudinal axis. These systems vary, in general, along the ship as the geometry of the frames and the free surface changes. It should be noticed here that the co-ordinate system is always orthogonal, while the numerical grid is non-orthogonal in the X3 - direction. Based on the model speed and the frequency at which the probe signals were digitized, a minimum distance of 1.034 cm between successive transverse cuts could be used to determine the free surface in the outer field. Since the The Governing Equations measured points near the hull were located at random, an interpolation procedure was firstly applied to estimate the wave elevation in the inner field, on the transverse cuts determined at the outer field. Actually, all points within a bandwidth of 1.034 cm were used to determine the wave contour on the mid-plane of the band. Finally, a second order smoothing method was used to generate the waveform in the combined ul-momentum inner and outer field domain of each transverse cut, as shown in Fig.2. In this manner 300 transverse cuts were generated along the ship length, of which 150 were used for the numerical computations. DESCRIPTION OF THE NUMERICAL METHOD The co-ordinate system In a local orthogonal cirvilinear co-ordinate system, described as above, with metrics hi, h2, ho end curvatures k12' k21, the time averaged Navier-Stokes (Reynolds) equations can be written as in (101: Crud) = - h aX + punks - pu~u2k,2 + ~ ~t 622)k2~+2~2k~2+h .0X +h~ ~ ~2 +ht i3 u2-momentum The transport equations describing the flow C(u)=-h aX +pUlkl2-pulu2k2l+t022 all) 12 around the ship are solved numerically in the ~ 9~ aft physical space. The calculation domain consists of ~ 22 1 ~2 ~ 23 transverse sections and on each section a +2k2l~l2 + hi ~x2 + h Dxl ha dx3 792

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w "~! (I?.P) a . c the Main. Wear gods along 793 b

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art g re 2 (conffnued) 794 e l

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US momentum in* C(U 3) = - h OX +kl2(~13 + 23) 1 ~ 33 1 ~ 23 1 - 13 + + ~ ha DX 3 ha DX 2 hi dX1 where C(ui) shows the convection terms, i.e.: (1) C(Ui )= h P [I i + ~ ~ i] + p;_ (2) and the stress tensor cij includes the viscous stresses and the double velocity correlations. The components of cij are expressed as: Ail =2ple[h SEX + U2k~2]= he Eli 22 2~e[h tax + Ulk21]= He e22 Ou3 =2~e0X = Pee33 C7~2 Em 1~ OXEN + h2 OX2 U2k21 Ulkl2]steely 3=~e[h Ox + Ox ]=Reel3 623=~e~ i~x3 + dX2]=~ee23 (3) * The value of p appearing on the right hand side of momentum equations (1) is equal to p+ggh, where h is the vertical distance from a fixed level. The effective viscosity Be in expressions (3 is calculated according to the standard k-e turbulence model (11) as follows: 2 ~e= it+ ~t=~+0~09pk / (4) where ,ut is the eddy viscocity, k the turbulence kinetic energy and ~ its dissipation rate. The values of k and ~ are determined by solving two more differential equations, which in the orthogonal curvilinear system under consideration are written as: k-equation C(k)= h h [aX <~th1 Act ) Ala th2 Aft ] + 00 (~c Bilk )+G- pe e- equation )= x ~ h x )+ x ~h x ~ 1 1 1 2 2 2 ] t t}x (~, fix )+1.44G k-1.92p (5) where o~=1.3 and the generation term G is expanded as: G =2 p~ he 1~ + e222 + e333 + 2 (e212 + e223 + e223~3 The Reynolds equations (1) as well as the turbulence model equations (5) are discretized according to the finite volume approach using a staggered node arrangement (10). The resulting algebraic equations have the general form 6 Ap4>p = ~Aid~i + S `:,, (6) 1 where dip stands for the velocity components, the turbulence kinetic energy and its discipation rate and Hi are the values at the neighbouring nodes of P. Central differences are used to model Al along x1 and x2 directions while the corresponding coefficients on upstream ala downstream planes are calculated by the hybrid scheme (12). Boundary Conditions The boundaries of the calculation domain shown in Figs. 2a and 3 are the inlet U and outlet D planes, the external boundary N. the solid surface S. the free surface W and the flow symmetry plane E. The elliptic form of equations (6) requires specification of boundary conditions (Dirichlet or Neummann type) on each of these boundaries. [. ~ Figure 3. Definition of boundaries. 795

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At the inlet plane U and external boundary N the values of the velocity components u1, u2 U3 and the pressure are calculated by the potential flow solution under the predetermined free surface. The latter is performed by the classical Hess and Smith method (131. The ship hull and the free surface are covered by quadrilateral panels. Once the wave elevations are a priori known, the source distribution on each panel is calculated by satisfying the unique boundary condition un=O, where n is the normal to the hull or the free surface. The panel arrangement on the free surface region which has been used for the computations is shown in Fig.4. The external NP and the upstream UP boundaries of this region are located in the undisturbed free surface part in order to avoid as far as possible errors due to end effects (14~. It should be noticed here that the external boundary N for the viscous flow calculations is almost five times closer to the ship hull than NP and, therefore, it is expected that the calculation of the velocity components at N will not be practically affected by the aforementioned effects. A total of 1400 panels has been used to model the ship hull and 2000 panels to model the free surface. These numbers refer to the one half of the whole domain, since the flow has one symmetry plane. At the same boundaries U and N the values of k and ~ are assumed to be equal to zero. For viscous flow computations the inlet plane was placed at x = -0.2 m and the external boundary almost 25 cm (in the mean) apart from the solid surface. Neglecting surface tension, the dynamic boundary condition on the free surface can be written as on-p~c=0, where on and ~ are the nonnal and the shear stresses respectively. It is easy to show that these conditions, together with the elimination of the convective terms on the free surface (due to the kinematic condition) result in the application of Neummann type boundary conditions for the u2 and U3 velocity components on the W-boundary (Fig.5~. The same conditions are assumed to hold for k and (i.e. ~klDn~/0n=0), while the normal to the surface ul-component is set equal to zero. L/8 1 _ ~ Unto L ~1~ U2= U3 Figure 5. The free surface boundary. The wall function method (10), (11) has been employed to model the flow characteristics near the solid boundary. The values of y+ ranged between 30 and 180 in any case. On the symmetry plane the following conditions are valid: u = 0 ~ =0 ~ =U1,U3,~ ~ Finally, at the outlet plane D the flow is assumed to be fully developed, corresponding to the application of Neummann conditions for each variable. This plane was placed at x = 3.6 m. The Solution Procedure The solution of the transport equations (6) together with the determination of the pressure are made according to the partially parabolic algorithm (151. An initial guess of the pressure field is made, based on the calculated pressure values at the external boundary N. Then the solution proceeds by solving the momentum, pressure correction and k-e equations in each transverse section successively. The pressure is corrected according to the SIMPLE (16) algorithm so that continuity is satisfied in each cell of the domain. Once the free surface is known, a Dirichlet-type boundary condition for the pressure correction cannot be applied at this boundary, since it leads to overdetermination of the problem and prevents the satisfaction of the continuity equation in the adjacent to the surface cells. 1 L/4 Figure 4. The free surface panel arrangement. 796

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During the application of the partially parabolic algorithm only two-dimensional in-core storage is essentially needed for the various geometrical and flow parameters and, therefore, fine grids can be used. The upstream and downstream values of different variables are constant when calculations are performed at a certain station. After the solution for every section of the domain is obtained, a sweep is completed and the calculations start again. Several sweeps are needed until both the velocity and the pressure field converge. The use of an orthogonal grid in two dimensions proved to be quite successful with respect to convergence. This is due to the rather simple way that velocity corrections are coupled to changes of the corresponding pressure gradients, the latter being of crucial importance when the SIMPLE approach is followed. It has been found that relatively high underrelaxation factors can be used for the solution of the momentum and k-e equations even if only one SIMPLE step is performed in each station. Underrelaxation, necessary to obtain convergent solutions, is applied for every variable as: ~ bran + (I - redo where r is the underrelaxation factor (constant throughout the calculation domain), En the solution of (6) and NO the previous value of the variable. A 178 x 32 x 31 grid was used for the computations in any case, where 178 is the number of transverse sections, 32 the nodes girthwise and 31 along a normal. Constant underrelaxation factors equal to 0.5 were applied for all variables exept the pressure correction for which the value of r = 0.3 was adopted. Convergence was achieved in 400 single-step sweeps of the domain. RESULTS AND DISCUSSION As mentioned earlier, the predetermined boundary used for the computer runs consisted of the measured wave pattern plus the actual wetted surface of the hull. In this case the model sustained a parallel sinkage of 6 mm but no running trim. As a first result it should be mentioned that the numerical calculations under the aforementioned boundary exhibited good behaviour, i.e. they converged always. The running time for the computations was about 24 hours on a 2.6 Plops workstation for the grid described in the previous section. The corresponding time calculations was one the same machine. In order to gain some more insight in the relative magnitudes of the different components of ship resistance, it was decided to obtain similar results for a double model of the tested hull, even keeled but with the draught increased by 6 mm (3.35%~. This reasonable chaise turned out to be very meaningful! from the point of view of the wetted surface. That is whereas the static for the potential flow hour for 3500 elements on equilibrium wetted surface of the model was 1.694 me, the wetted surface underway was 1.738 ma, that is increased by 2,6%, and the wetted surface of the double model was 1.734 ma, very close to the actual wetted surface. In addition, the pressure resistance of the potential flow for the actual wetted surface and the measured wave pattern was computed. Because of the differences in the wetted surface, it was decided to present all calculated and measured resistance values in terms of force (lip), and not in the form of non-dimensional coefficients. The results of calculations and measurements are shown in Table 1, whereas the non-dimensional, integrated resistance force along the length of the ship are shown in Figs 6 and 7. In both the Table and the Figures the subscripts T. F & P correspond to the total force, the shear stress component and the pressure component respectively. This rule does not apply to the RF value computed from the experimental results, which is thought to contain both the shear stress and the viscous pressure component of the resistance. This value of RF was computed on the basis of the llTC friction line and a form factor of 1.14 determined from the experiments. Ache _~_ Calculated for actual free surface Calculated for double model Calculated for potential flow & actual free surface Measured R T & Calculated Radon the basis of form factor method R (kp~ 0.766 R F (lip) 0.604 Rp (kp) 0.162 0.694 0.572 0.122 0.682 0.596 0.128 Table 1. Comparison of resistance componets. From the contents of Table 1, it can be concluded that the method used overpedicts the resistance. This is obvious from the double model calculations which yield a total resistance value slightly higher than the experimental one, which contains a wavemaking resistance component. The same trend has been noticed in previous calculations for another ship shown in (6~. In the present case the overprediction for the total resistance is 12.5%, which renders the use of the results doubtful for practical applications. 797

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There are however some useful conclusions to be drawn from the contents of Table 1 and Figs 6 and 7. Firstly, if the difference between the calculated total resistance for the ship and the double model is taken to represent the wavemaking resistance, its value is Rw = 0.766 - 0.694 = 0.072 lip. This is significantly smaller than the wavemaking (or pressure) solution. If one then adds this value to the experimentally determined RF of 0.596, a value of RT = 0.668 is obtained, which is very close to the experimental value. Secondly, the value of RF along the length of the model is slightly different for the actual free surface and the double model, as the shape of the free surface seems to increase the shear flow resistance. And, finally, that the accuracy of the computations at the aftermost part of the ship for the pressure component of the resistance is very important due to the large slope of the curves, just before the final result is obtained. From the point of view now of possible improvements of the calculation procedure, there are at least two areas which can lower the predicted value of the resistance. The first such area is connected with an artificial blockage effect, which is inserted to the procedure by imposing the potential flow velocities as boundary conditions relatively close to the surface of the body. Obviously, this is done to reduce the time of computations and our experience a, shows that if this effect is eliminated, the predicted value of the resistance will be reduced by 2-3%. The second area of possible improvements is of a more fundamental nature as it questions the accuracy of the wall function approach in turbulent flow calculations. As it has been shown in (17) and (18), if a direct solution of the Reynolds equations is used all the way up to the solid boundary, better values for the wall shear stresses are predicted. Unpublished results of NTUA indicate that in this case a reduction of the frictional resistance by 5,5% is obtained for the aft part of a tanker hull. Unfortunately, this improvement is accompanied by a 25% increase in computing time. Nevertheless the up to now discussion of the results should also be seen in the light of the unavoidable shortcomings of any such experimental-numerical investigation. In this respect, the authors cannot guarantee the degree of accuracy of the free surface measurements and subsequent interpolation. Also, no attempt was made to achieve a grid independent solution, although the grid is fine enough according to our experience, but not necessarily so in the aftermost part of the hull surface. Finally, we recall the discussion made in the introduction about the shortcomings of the k-e turbulence model, which need to be further investigated. ACKNOWLEDGMENTS The authors would like to thank Dr. D. Lyrides for the careful measurements of the free surface, while he was carrying out his Diploma Thesis. Many thanks to our colleagues from the Leboratory of Photogrammetry for their companion measurements of the free surface near the model. Finally, we appreciate the help of Dr. S. Voutsinas and Dr. Y. Glekas for the analytical determination of the free surface. REFERENCES 1. Patel, V.C., "Ship Stern and Wake Flows: Status of Experiment and Theory", Proceedings of 17th ONR Symposium on Naval Hydrodynamics. The Hague, 1988. 2. Tzabiras, G.D. and Loukakis, T.A., "On the Numerical Solution of the Turbulent Flow-Field past Double Ship Hulls at Low and ~ Reynolds Numbers", Proceedings of 5th International Conference on Numerical Ship Hydrodynamics. Hiroshima, 1989, pp. 395-408. 3. Hino, T., "Computation of a Free Surface around an Advancing Ship by the Navier-Stokes Equations", Proceecings of 5th International Conference on Numerical Ship Hydrodynamics, Hiroshima, 1989, pp. 69-83. 4. Tzabiras, G.D., "On the Calculation of the 3-D Reynolds Stress Tensor by two Algorithms", Proceedings of 2nd International Symposium on Ship Viscous Resistance. Goteborg, 1985. 5. Tzabiras, G., Hytopoulos, F. and Nassos G., "On the Numerical Calculation of the Turbulent Flow-Field around Bodies of Revolution at Zero Incidence", Proceedings of Marine and Offshore Computer Applications Conference, Southampton, 1988, pp. 31-47. 6. Tzabiras, G.D. and Loukakis, T.A., "Reynolds Number Effect on the Resistance Components of 3D Bodies", Proceedings of 18th I.T.T.C., vol.2, 1987, pp 7~71. 7. Georgopoulos, A., Ioannidis, C., Potsiou, C. and Badekas, 1., "Photogrammetric Wave Profile Determination", Proceedings of XVI ISPRS Congress, vol. 27, part V.II, commision 5, Qyoto, 1988. Lyridis, D.B., "Experimental and Numerical Investigation of the Wave Pattern around a Ship Moving with Cosntant Speed", Diploma Thesis, Dept. of N.A.M.E., N.T.U.A., 1987. Tzabiras, G., Vaftadou, M. and Nassos, G. "A Numerical Method for the Generation of 2-D Orthogonal Curvilinear Grids" Proceedings of 1st Conference on Numerical Grid Generation in Computational Fluid Dynamics. 1986, pp. 183-195. 10. Tzabiras, G.D., "Numerical and Experimental Investigation of the Turbulent Flow-Field at the Stern of Double Ship Hulls", Ph.D. Thesis, N.T.U.A., 1984. . Launder, B.E. and Spalding, D.B., "The Numerical Computation of Turbulent Flows", Computer Methods in Applied Mechanics and Engineering. vol. 3(3), 1974, pp. 269-289. 798

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12. Spalding, D.B., "A Novel F~te-Difference Formulation for Different Expressions Evolving both First and Second Derivatives", Bit. J. of Numerical Methods in Engineering. vol.4, 1972, pp. 551-559. 13. Hess, J.L. ad Smith, A.M.O., "Calculation of Potential Flow about Arbitrary Bodies", Progress in Aeronautical Sciences. vol.8, 1966, pp. 1-138. 14. Kim, K.J., "Ship Flow Calculations and Resistance Minimization", Ph. D. Thesis, (palmers University of Technology, 1989. 15. Pratap, V.S. and Spalding D.B., "Numerical Computations of the Flow in Cunred Ducts", Aeronautical Quarterly, vol. 26, 1975, pp. 219-232. 0.~30 i i / 0.0020 ~ / o.oolo ~ / ~\ // \ / \ 0.0000 - - - -0.0010 ~1 J 3 Distance from F.P. in meters. -~.~.0 ~ i-',,,,, ~ ~' i,, ,- ', ~ ', ~ i,, i,,,,,, i, ',,, i,, ~ i,, ~ ~ ', r, ~ I ~ ~ I,,,, r, ~ ~ I,, rr~T1 - _ \ 16. Patankar, S.V. and Spalding, D.B., "A Calculation Procedure for Heat, Mass and Momentum Transfer in 3D Parabolic Flows", Int. 1. of Heat and Mass Transfer vol.15, 1972, pp. 1787-1806. ~~7. Chen, H.C. and Patel, V.C., "Practical Near-Wall Turbulence Models for Complex Flows Including Separation", Proceedings of AIAA 19th Fluid Dynamics, Plasma Dynamics and Lasers Conference Honolulu, 1987. 18. Tzabiras, G.D., "A Numerical Investigation of the Turbulent Flow-Field at the Stern of a Body of Revolution", J. of Applied Mathematical Modelling, vol.11, 1987, pp. 45-61. a/ .,~ ~ Have / ~\ ' , \< 1 ' Double model _ _ _ _ _ _ ~, Rp o.no 0.50 1.00 1.50 2.00 2.50 3.00 3.bc) Figure 6. Non-dimensional integrated pressure force. 0.00~ - O.0030 - 0.0020 0.0010 - Caere ~ An, /' / 'K' ~ a' Double model i ''~ ~ ''' Distance from F.P. in meters. 0~3~0 ~rl,, I I r, I,,, I ~ ,-,, I I ',,,,,,, I I I I 1,, I,,, I, I,,, . I I ~ I,,,, r~~tTr,, ~ r-rr~r 0.00 0.50 1.00 1.50 2.00 2.50 Figure 7. Non-dimensional integrated frictional force. 799 3.00 .~.50

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