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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 553 Numerical Prediction of Scale Effects in Ship Stern Flows with Eddy-Viscosity Turbulence Models L.EÃ§a (Instituto Superior TÃ©cnico, Portugal) M.Hoekstra (Maritime Research Institute, Netherlands) ABSTRACT This paper presents a numerical investigation of scale effects on ship stern flows. The most popular algebraic, one- equation and two-equation eddy-viscosity turbulence models are successfully applied to the calculation of the flow around the Mystery tanker from model up to full scale Reynolds number. It is shown that the choice of the turbulence model does have an influence on the results (notably the near-wake field), but the differences caused by changing the turbulence model tend to diminish with the increase of the Reynolds number. 1 INTRODUCTION One of the main advantages of Computational Fluid Dynamics (CFD) over traditional model testing is the (potential) capability to predict Reynolds number effects on the flow field around a ship. But before this advantage can be exploited, two major tasks arise. The first is to change the potential capability to a true capability by making sure that the numerical method can cope with the extreme requirements posed by flow simulations at full scale Reynolds number. The second task is to go through the verification and validation processes. Only a few attempts to predict scale effects with CFD have been reported. One such attempt is [1], presenting results, however, which are not numerically convincing. The present authors have shown to be more successful in computing ship stern flows from model up to full scale Reynolds numbers, [2], with verification of numerical errors, [3] and [4]. Unfortunately, there are virtually no reliable experimental data available for full scale ship stern flows, so that the validation process is obstructed. The best thing to do is then to increase the level of confidence by validating at model- scale Reynolds number and showing that well-known trends for Rn increasing are systematically reproduced. In the last two decades, a huge effort has been made to validate CFD predictions at model scale Reynolds number. However, the present status is far from being completely satisfactory. Notably the accurate prediction of the axial velocity field in regions of high streamwise vorticity has proved to be difficult. Some success has been claimed for second moment closures, e.g. [5]. But if one aims at selecting a turbulence closure that is numerically robust at model and full scale Reynolds numbers, eddy-viscosity models are still the only reasonable choice. In this paper we present a numerical investigation into the prediction of scale effects with eddy-viscosity turbulence models, including algebraic, one-equation and two-equation models. Two main goals are considered: â¢ Investigate which turbulence models are numerically robust from model up to full scale Reynolds numbers, without the need of further tuning. â¢ Evaluate the influence of the Reynolds number on the differences between solutions obtained with different eddy- viscosity turbulence models. With achieving these goals, we expect to increase the confidence in the use of CFD at full scale Reynolds numbers, using turbulence models that have been originally developed for thin shear layers at model scale Reynolds numbers. The paper is organised in the following way: section 2 gives a brief description of the turbulence models and their numerical implementation. The results of application to the flow around the Mystery tanker at Reynolds numbers from model scale up to full scale Rn are presented and discussed in section 3. Section 4 summarizes the conclusions of the paper. the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 554 2 TURBULENCE MODELS In order to cover a broad spectrum of eddy-viscosity turbulence models, we have considered the following models: â¢ Algebraic models â Cebeci & Smith, [6], (CS) â Baldwin & Lomax [7], (BL) â¢ One-equation models â Spalart & Allmaras, [8], (SA) â Menter, [9], (MT) â¢ Two-equation models â k-Îµ models Two-layer model, [10], (KE-TL) Chien's low-Rn model, [11], (KE) â k-Ï models Standard, [12], (KW) Menter, [13], (KWM) â q-Î¶ model, [14], (QZ) The one-equation model of Baldwin & Barth, [15], and the SST version of Menter's k-Ï model, [13], were also tested. However, results of these models will not be included in the present paper. The Baldwin & Barth model showed a very poor behaviour in the initial test runs, while the SST version of Menter's k-Ï does not perform better than the other k- Ï models tested. It is possible to improve the quality of the predictions of ship stern flows with the one-equation and two-equation turbulence models using a simple correction to the production term of the transport equations, [16]. However, in this paper we will adopt the standard versions of the models. 2.1 Algebraic Models The two algebraic models are well-known and based on a two-layer definition of the eddy-viscosity, vt, where the eddy-viscosity is obtained from the minimum of its values in the two layers. In the inner-layer, both models use the mixing-length approach with the Van Driest damping function in the near-wall region. In the Cebeci & Smith model, the eddy-viscosity in the outer region is obtained from (1) where qe stands for the velocity at the edge of the viscous region and Î´* is the displacement thickness, which is an integral parameter defined for 2-D boundary-layer flows, and Î³k is the intermittency factor, which is given by (2) where Î´ is the thickness of the viscous region. In a ship stern flow calculation, performed in a curvilinear coordinate system, (Î¾, Î·, Î¶)1 some assumptions have to be made to compute qe, Î´* and Î³k. We determine these quantities from information along each grid line normal to the wall and the viscous layer thickness is calculated from the total head. Details on the calculation of qe, Î´* and Î³k are given in [16]. The main advantage of the Baldwin & Lomax model over the Cebeci & Smith model is the absence of Î´ and Î´* from the definition of the length scale in the outer region. In the Baldwin & Lomax model, (vt)0 is given by (3) where (4) Fmax is the maximum of the function (5) and ymax is the value of yn where Fmax occurs. Udif is the difference between the maximum and minimum values of q along an Î· grid line, |W| is the magnitude of the vorticity vector, Aâ=26 and is the nondimensional distance to the wall in wall coordinates. Fkleb is the equivalent of Î³k, and is given by the authoritative version for attribution. (6) Ccp=1.6, C2=0.25 and Ckleb=0.3. Although the calculation of Fwake and ymax seems to be straightforward, it is not. The values of these two parameters are directly related to the vorticity, which is inversely proportional to the grid line distance in the physical space. This dependency of F on the vorticity makes its numerical calculation extremely sensitive to 1Î¾ is a stream wise coordinate, Î· is a coordinate normal to the ship surface and Î¶ is a girthwise coordinate.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 555 the mean flow field, which depends on vt and is determined iteratively. To avoid undershoots and overshoots of the eddy- viscosity, the spatial variation of ymax must be limited. In the present implementation, the following limiters are adopted is the ymax value of a given Î¾ line at stream wise stationi and where stands for the ymax value on the previous upstream station for the same Î¾ line. It is our experience that it is very hard to converge the eddy-viscosity field when the limiters of ymax are turned off. On the other hand, if the limiters are always turned on there is a risk of ymax being determined by the upstream flow instead of the local flow quantities. Therefore, the apparent advantages of the Baldwin & Lomax model in the determination of (vt)0 are severely compromised by the numerical difficulties found in the determination of ymax. Details of the adjustment of the two models to make them suited for application to wakes are given in [16]. 2.2 One-Equation models 2.2.1 Spalart & Allmaras The new generation of one-equation turbulence models is based on a transport equation for the eddy viscosity rather than for the turbulence kinetic energy. The Spalart & Allmaras model proposed in [8] solves the following transport equation: (7) where (9) (8) Here and in the remainder of this paper S represents the rate of strain squared. The eddy-viscosity is obtained from The model constants are: The transport equation of (7), includes the term which does not contribute to the stability of its numerical solution. Therefore, equation (7) has been re-written as (10) In the wake, the distance to the wall is computed from (11) where yn is the distance measured along the normal grid lines and xâxte is the distance to the âtrailing edge' of the ship measured along the symmetry plane. 2.2.2 Menter The one equation model proposed by Menter in [9] derives the following transport equation from the k-Îµ model: (12) The eddy-viscosity is given by (13) and the authoritative version for attribution. (14) The model constants are:

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 556 2.2.3 Boundary Conditions In a ship stern flow calculation we need to specify boundary conditions at the six boundaries of the domain: inlet, outlet, ship surface, ship symmetry plane, free surface2 and external boundary. The boundary conditions are specified in the same way for both one-equation turbulence models. At the ship surface the turbulent quantities are zero and symmetry conditions are applied on the free surface and at the ship's symmetry plane. At the outlet, the streamwise derivative of the turbulent quantities is assumed to be zero. Dirichlet boundary conditions are imposed for the turbulent quantities on the external boundary and are fixed by the value on the border of the inlet plane. The present calculations were performed only for the aft part of the ship; the inlet plane is located at mid-ship, and so there is already a viscous region in the inlet plane. A standard straightforward procedure was adopted to define the turbulent quantities at the inlet plane: vt is calculated with the Cebeci & Smith algebraic model and the turbulent quantities are then derived from the known eddy- viscosity. 2.3 Two-equation Models 2.3.1 Two-layer k- Îµ The two-layer k-Îµ model presented by Chen and Patel in [10] solves two transport equations in the outer flow region. In the near-wall region only the equation for the turbulence kinetic energy, k, is solved. The value of Îµ is derived from an algebraic length scale. The near-wall model is equivalent to the one-equation model of Wolfshtein, [17]. The eddy-viscosity is obtained from (15) and k and Îµ for a steady flow are obtained from the solution of the equations: (16) and (17) In the near-wall region, Îµ is determined from (18) with (19) and (20) The fÂµ function defined by the Wolfshtein one-equation model is (21) The standard k-Îµ constants are cÂµ=0.09, C1=1.44, C2=1.92, Ïk=1 and ÏÎµ=1.3. The key feature of this two-layer model is the determination of the boundary between the inner and outer layers which is often defined by a criterion based on yâ. However, with yâ it is difficult to establish a criterion which is insensitive to the Reynolds number. In our approach, the inner-layer region is defined by the following criteria: The first criterion would be the natural choice to border the inner-layer region. However, in the iterative determination of the eddy-viscosity field it may lead to excessively large regions, which provoke numerical convergence problems. Therefore, we have added the second criterion which originates from the knowledge on flat plate boundary layers, where the âfully-turbulent' region starts at This approach does not guarantee that the fÂµ is close to 1 at the edge of inner-layer. Therefore, in the outer-layer the Îµ transport equation is solved but fÂµ is still obtained from (21). As in the one-equation models, (yn)*, defined by equation (11), is used in the wake to represent the distance to the wall. 2.3.2 Chien's k- Îµ model The low Reynolds k-Îµ model proposed by Chien, [11], does not distinguish between inner and outer layers and is directly applicable in the near-wall region. The eddy-viscosity is obtained from equation (15). The k and Îµ transport the authoritative version for attribution. equations of this model are: (22) 2In the present calculation we have used the double-model approximations and so the free surface is asymmetry plane.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 559 the skin friction coefficient for the flow on a flat plate. Therefore, only the first option is left to avoid the use of a grid- dependent boundary condition. In this paper we shall compare two alternatives: i) adopt equation (43) with NÏ including viscous corrections to define the near-wall values of Ï, BC1; and ii) obtain Ï at the wall from equation (44), BC2. 3 RESULTS AND DISCUSSION 3.1 General All calculations were carried out with the computer code PARNASSOS, [19], which solves the Reynolds Averaged Navier-Stokes equations in their complete form [20]. The test case in this paper is the flow around the Dyne (Mystery) tanker which has earlier been subject of comparative computations in the Gothenburg and Tokyo Workshops, [21] and [22]. Two main reasons justify this choice: the flow is sufficiently complex to test the accuracy of the turbulence models, and, in particular, it exhibits, at least at model scale Reynolds number, the so-called âhook shape' of the isolines of the axial velocity at the end of the stern, as a result of the existence of strong bilge vortices. A detailed numerical verification study has been performed with PARNASSOS for this test case, both at model scale Reynolds number, [3], and full scale Reynolds number, [4], which permits the selection of a grid with sufficient resolution. In the present study, five different Reynolds numbers have been considered: 5Ã106, 2Ã107, 108, 5Ã108 and 2Ã109, with the Reynolds number defined by A Cartesian coordinate system is introduced with the x axis along the undisturbed stream, the z axis vertical positive pointing upwards and y completing a right-hand system. The origin of the coordinate system is located on the forward perpendicular at the ship symmetry plane on the keel line. All the variables presented are made non-dimensional using Uâ and L as the velocity and length reference scales. The computational domain covers only the flow field near the stern. The inlet and outlet plane are x constant planes. The inlet plane is located at x=0.5L and the outlet plane at x=1.25L. The external boundary is an elliptical cylinder, given by: The remaining boundaries are the free surface, plane z=0.056L, the symmetry plane of the ship, y=0, and the hull surface. The volume grids were created with a proprietary elliptic PDE grid generator, based on the GRAPE approach [23]. The number of grid nodes in the streamwise and girthwise direction is the same for the five Reynolds numbers: NÎ¾=161 and NÎ¶=41. The number of grid nodes in the normal direction, NÎ·, increases with Rn. NÎ·=81 for the lowest Rn and 10 grid lines are added each time Rn is increased, which leads to NÎ·=121 for Rn=2Ã109. The grid line spacing in the normal direction is defined by one-dimensional stretching functions, which are tuned to obtain a maximum value of yâ at the first layer of grid nodes away from the ship surface of approximately 0.5. Five significant flow parameters were selected to compare the different numerical solutions: â¢ Friction resistance coefficient, â¢ Pressure resistance coefficient, â¢ Wake fraction, Wf: â¢ Maximum cross-stream velocity at x=0.989L, (Vw)max, with â¢ Minimum axial velocity component in the flow field, The maximum cross-stream velocity at x=0.989L is related to the bilge vortex intensity and identifies the existence of streamwise flow separation. The integrals included in the definitions of and Wf are evaluated with Gaussian quadrature rules assuming a bi-linear variation of the unknowns between the grid nodes. The area â¦ for the calculation of the wake fraction is the propeller disc, which has been located at x=0.989L with the axis of the propeller at z=0.0166L; the disc radius is R=0.015L, while a zero hub radius has been assumed. the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 560 3.2 Wall Boundary Condition for To investigate the influence of the numerical implementation of the wall boundary condition of Ï, we have calculated the flow at Rn=5Ã106 with Menter's version of the k-Ï model with the two options considered: BC1, which obtains Ï from the theoretical value for yâ<2.5 and BC2, which is based on an ad hoc definition of a finite value at the wall. Table 1 presents the five selected flow quantities and the mean and maximum values of vt obtained with BC1 and BC2. Table 1: Comparison of solutions obtained with Menter's k-Ï model using different implementations of the Ï wall boundary condition. Variable BC1 BC2 1.944 1.612 0.911 0.878 0.609 0.565 Wf 0.343 0.365 (Vw)max â0.075 â0.076 (vt)medÃ104 0.265 0.257 4 (vt)maxÃ10 1.582 1.504 The differences obtained between the two solutions are certainly not negligible. As one might expect, the friction resistance coefficient, exhibits the largest difference. However, the limiting streamlines of both calculations, which are depicted in figure 1, are similar. Figure 1: Limiting streamlines for Menter's k-Ï model with different wall boundary conditions. At the propeller plane, x=0.989L, there are significant differences between the isolines of the axial velocity, as shown in figure 2. With BC1 the speed is definitely lower in the inner wake than with BC2. Figure 2: Axial velocity isolines at x=0.989L for Menter's k-Ï model with different wall boundary conditions. These results show that the flow prediction is clearly dependent on the numerical implementation of the Ï wall boundary condition. They suggest that the Ï behaviour at the wall can hardly be seen as a strong point of the k-Ï models. From the results it is not clear which is the best choice, BC1 or BC2, however, as discussed above, the results of BC2 are inherently grid-dependent. Therefore, we will adopt BC1 for the remaining calculations with the k-Ï models. 3.3 Scaling Effects the authoritative version for attribution. The results of the five selected flow quantities and the maximum value of vt are given in table 2 for the five Reynolds numbers and for the various turbulence models tested. â stands for the maximum difference between the predictions of the different turbulence models at a given Reynolds number. The differences between predictions with different turbulence models come out as appreciable at model scale Reynolds number but tend to diminish with the increase of the Reynolds number. An exception is found in the maximum cross-stream velocity at the propeller plane; it is the only one of the selected flow variables which does not change monotonically with the Reynolds number. Figure 3 presents the friction resistance coefficients given in table 2. We have tried to compare the results with two friction lines, the ITTC line,

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 561 Table 2: Comparison of solutions obtained with the several turbulence models at different Reynolds numbers. Variable Rn CS SA MT KE-TL KE QZ KW KWM â 5Ã106 1.573 1.816 1.606 1.561 1.598 1.445 1.939 1.944 0.499 2Ã107 1.269 1.469 1.335 1.280 1.315 1.197 1.546 1.547 0.350 108 1.022 1.177 1.098 1.059 1.082 0.988 1.213 1.221 0.233 5Ã108 0.846 0.965 0.916 0.885 0.900 0.824 0.981 0.983 0.141 2Ã109 0.732 0.825 0.791 0.767 0.776 0.713 0.841 0.843 0.130 5Ã106 0.598 0.788 0.741 0.713 0.687 0.655 0.923 0.911 0.325 2Ã107 0.527 0.720 0.648 0.638 0.645 0.614 0.829 0.817 0.320 108 0.466 0.649 0.583 0.580 0.604 0.589 0.744 0.732 0.278 5Ã108 0.423 0.605 0.547 0.543 0.570 0.575 0.690 0.675 0.267 2Ã109 0.402 0.578 0.535 0.542 0.565 0.570 0.648 0.638 0.236 5Ã106 0.528 0.632 0.619 0.563 0.576 0.532 0.616 0.609 0.104 Wf 2Ã107 0.462 0.547 0.535 0.493 0.512 0.479 0.532 0.530 0.085 108 0.397 0.452 0.454 0.421 0.437 0.415 0.451 0.449 0.057 5Ã108 0.347 0.393 0.393 0.369 0.380 0.365 0.384 0.386 0.046 2Ã109 0.315 0.352 0.353 0.334 0.341 0.330 0.345 0.345 0.038 5Ã106 0.209 0.280 0.290 0.287 0.266 0.263 0.347 0.343 0.138 (Vw)max 2Ã107 0.199 0.301 0.289 0.286 0.284 0.271 0.378 0.367 0.179 108 0.203 0.302 0.276 0.262 0.284 0.268 0.396 0.377 0.193 5Ã108 0.206 0.294 0.252 0.229 0.256 0.244 0.395 0.369 0.189 2Ã109 0.208 0.263 0.228 0.227 0.229 0.228 0.336 0.331 0.128 5Ã106 â0.012 â0.059 â0.068 â0.028 â0.015 â0.007 â0.075 â0.075 0.063 2Ã107 â0.004 â0.055 â0.040 â0.007 â0.016 â0.011 â0.065 â0.064 0.061 108 0.000 â0.035 â0.001 0.000 â0.001 0.000 â0.041 â0.041 0.041 5Ã108 0.000 â0.005 0.000 0.000 0.000 0.000 â0.027 â0.034 0.034 2Ã109 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (vt)max Ã104 5Ã106 2.327 1.627 1.259 1.113 1.095 1.046 1.492 1.582 1.281 2Ã107 1.979 1.338 1.026 0.927 0.965 0.889 1.318 1.361 1.090 108 1.647 1.093 0.845 0.798 0.823 0.742 0.931 1.141 0.905 5Ã108 1.374 0.917 0.711 0.660 0.708 0.603 0.763 0.862 0.800 2Ã109 1.173 0.780 0.626 0.556 0.586 0.505 0.700 0.756 0.668 and the Schoenherr line, Since our computation domain covers the aft half of the hull only, we have estimated the equivalent plate friction of the aftbody as where Sw is the wetted surface of the ship included in the computational domain, which is assumed to be half of the total wetted surface. The predictions of all the turbulence models exhibit the correct trend with the increase of the Reynolds number, but there is a clear difference in slope. As a further relevant result, we have plotted the mean wake fraction, Wf, as a function of the Reynolds number in figure 4. It is interesting to note that in both figures 3 and 4 there is good agreement between the SA model and the two k- Ï models, KW and KWM. The calculated limiting streamlines at Rn=5Ã106, Rn=108 and Rn=2Ã109 are illustrated in figures 5 to 7 for the models CS, SA, MT, KW, KWM and KE. As in the previous results, the differences between the predictions of the various turbulence models tend to diminish with the increase of Rn. Once more, the the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 562 results of the SA, KW and KWM models are very similar. At full scale, Rn=2Ã109, the CS, MT and KE also show good agreement. Figure 3: Friction resistance coefficient of the aft-body, as a function of the Reynolds number for the various turbulence models tested. Figure 5: Limiting streamlines at Rn=5Ã106. Figure 4: Wake fraction, Wf, as a function of the Reynolds number for the several turbulence models tested. The axial velocity isolines at the propeller plane, x=0.989L, at the same three Reynolds numbers, 5Ã106, 108 and 2Ã109, are presented in figures 8 to 10. The turbulence models included are again the CS, SA, MT, KW, KWM and KE. The U1 isolines exhibit a drastic influence of Rn. At model scale, the typical âhook shape' does appear for the k-Ï models, and to some extent, for the one-equation models5 SA and MT. However, at x=0.989L, the âhook shape' tends to disappear with the increase of the Reynolds number. At Rn=2Ã109, none of the predictions exhibits a âhook shape' and differences between the results of the various models, including the algebraic CS model, are rather small. This effect of Rn is related to the stretching of the bilge vortex, generated within the ship boundary layer which reduces its thickness with the increase of Rn. Figures 11 to 13 illustrate the cross-stream velocity field at x=0.989L for the same three Reynolds numbers. The plots include the CS, MT and KWM models. Although there are some differences between the predictions of the three models even at Rn=2Ã109, the authoritative version for attribution. 5As mentioned before, these prediction can be easily improved with a simple correction to the production term of the transport equation of the turbulent quantity.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 563 the stretching of the bilge vortex with the increase of Rn is clear for the three models. Figure 7: Limiting streamlines at Rn=2Ã109. Figure 6: Limiting streamlines at Rn=108. We should note that the present results do not imply that the typical âhook shape' of the axial velocity isolines disappears with the increase of Rn, it just appears further downstream. Figure 14 presents the velocity field at x=1.1L obtained with the KWM model for Rn=2Ã109. At this location, the bilge vortex is almost axisymmetric and the U1 plot shows the typical âhook shape', which is understandbly weaker than at the propeller plane at model scale, because the bilge vortex has not only rolled up in the near wake but it has also diffused. 4 CONCLUSIONS We have presented results of a numerical investigation of scaling effects in ship stern flows using algebraic, one- equation and two-equation turbulence models. The turbulence models were all implemented without any special tuning dependent on the Reynolds number. For the two-equation k-Ï models, we have pointed out the deficiencies of a widely accepted numerical implementation of the wall boundary condition of Ï. The results of the calculation of the flow around the Dyne (Mystery) tanker at five Reynolds numbers, 5Ã106, 2Ã107, 8, 5Ã108 and 2Ã109, suggest the following conclusions: 10 â¢ It is possible to simulate numerically ship stern flows from model up to full scale Reynolds numbers with the most popular eddy-viscosity turbulence models, including algebraic, one-equation and two-equation models. â¢ In global terms, the predictions exhibit the same trend in the flow field with the increase of the Reynolds number for all the turbulence models. the authoritative version for attribution.

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. at Rn=5Ã106. x=0.989L obtained Figure 6: Axial velocity isolines at at x=0.989L obtained at Rn=108. Figure 7: Axial velocity isolines NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 564

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. at Rn=2Ã109. x=0.989L obtained Figure 8: Axial velocity isolines at Rn=5Ã106. Figure 9: Transverse velocity field at x=0.989L obtained at NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 565

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. Rn=108. Figure 10: Transverse velocity field at x=0.989L obtained at Rn=2Ã109. Figure 11: Transverse velocity field at x=0.989L obtained at NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 566

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 567 Figure 12: Velocity field at x=1.1L obtained with Menter's k-Ï model at Rn=2Ã109. Above: axial velocity, U1, isolines. Below: cross-stream velocity field. â¢ The discrepancies between flow fields obtained with different turbulence models at a given Reynolds number tend to decrease with the increase of the Reynolds number. â¢ Although the performance of the k-Ï models seems to be very encouraging, the predictions depend on the numerical implementation of the Ï boundary condition at a solid surface. Some implementations advised in the open literature are unacceptable. The present results reinforce the need for reliable experimental data at full scale Reynolds number for validation purposes. All eddy-viscosity turbulence models, used here, were essentially developed for boundary-layers at moderate Reynolds numbers. REFERENCES [1] Watson S.J.P., Bull P.W.âThe Scaling of High Reynolds Number Viscous Flow Predictions Using CFD TechniquesâThird Osaka Colloquium, Osaka, Japan. [2] EÃ§a L., Hoekstra M.âNumerical Calculations of Ship Stern Flows at Full-Scale Reynolds Numbers Twenfirst Symposium on Naval Ship Hydrodynamics, Trondheim, June 1996. [3] Hoekstra M., EÃ§a L.âAn Example of Error Quantification of Ship-Related CFD Resultsâ7th Numerical Ship Hydrodynamics Conference, Nantes, July 1999. [4] EÃ§a L., Hoekstra M.âOn the Numerical Verification of Ship Stern Flow Calculationsâ1st MARNET Workshop, Barcelona, November 1999. [5] Deng G.B., Visonneau M.âComparison of Explicit Algebraic Stress Models and Second-Order Turbulence Closures for Steady Flows around Ships â7th Numerical Ship Hydrodynamics Conference, Nantes, July 1999. [6] Cebeci T., Smith A.M.O.âAnalysis of Turbulent Boundary Layers.âAcademic Press, November 1984. [7] Baldwin B.S., Lomax H.âThin Layer Approximation and Algebraic Models for Separated Turbulent FlowsâAIAA Paper 78â257, January 1978. [8] Spalart P.R., Allmaras S.R.âA One-Equations Turbulence Model for Aerodynamic FlowsâAIAA 30th Aerospace Sciences Meeting, Reno, January 1992. [9] Menter F.R.âEddy Viscosity Transport Equations and Their Relation to the k-Îµ ModelâJournal of Fluids Engineering, Vol. 119, December 1997, pp. 876â884. [10] Chen H.C, Patel V.C.âPractical Near-Wall Turbulence Models for Complex Flows Including Separation.âAIAA 19th Fluid Dynamics, Plasma Dynamics and Lasers Conference, June 8â10, 1987. the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line NUMERICAL PREDICTION OF SCALE EFFECTS IN SHIP STERN FLOWS WITH EDDY-VISCOSITY TURBULENCE MODELS 568 [11] Chien K.YâPrediction of Channel and Boundary-Layer Flows with a Low-Reynolds-Number Turbulence Model â AIAA Journal, January 1992, pp. 33â38. [12] Wilcox D.C.âTurbulence Modeling for CFDâDWC Industries 1993. [13] Menter F.R.âTwo-Equation Eddy-Viscosity Turbulence Models for Engineering ApplicationsâAIAA Journal, Vol. 32, August 1994, pp. 1598â 1605. [14] Gibson M.M., Dafa Alla A.A.âTwo-Equation Model for Turbulent Wall FlowâAIAA Journal, Vol. 33, August 1995, pp. 1514â1518. [15] Baldwin S.B., Barth T.J.âA One-Equation Turbulence Transport Model for High Reynolds Wall-Bounded FlowsâAIAA Paper 91â0610, 29th Aerospace Sciences Meeting, Reno Nevada, January 1991. [16] Hoekstra M.âNumerical Simulation of Ship Stern Flows with a Space-Marching Navier-Stokes MethodâPhD Thesis Delft University, 1999. [17] Wolfshtein M.âThe Velocity and Temperature Distribution in One-Dimensional Flow with Turbulence Augmentation and Pressure Gradient.â International Journal of Heat and Mass Transfer, Vol. 12, 1969, pp. 301â318. [18] Goldberg U., Peroomian O., Chakravarthy S.,âA Wall Distance-Free k-Îµ Model with Enhanced Near-Wall TreatmentâJournal of Fluids Engineering, Vol. 120, September 1998, 457â462. [19] Hoekstra M., EÃ§a L.âPARNASSOS: An Efficient Method for Ship Stern Flow CalculationâThird Osaka Colloquium on Advanced CFD Applications to Ship Flow and Hull Form Design, Osaka, Japan, 1998. [20] van der Ploeg A., EÃ§a L., Hoekstra M.âCombining Accuracy and Efficiency with Robustness in Ship Stern Flow Computation âTwen-third Symposium on Naval Ship Hydrodynamics, September 2000. [21] Larsson L., Patel V.C., Dyne G. (eds.)âShip Viscous Flow.âProceedings of 1990 SSPA-CTH-IIHR Workshop, Flowtech International AB, Research Report NÂº2, Gothenburg, June 1991. [22] Proceedings of CFD Workshop Tokyo 1994, Ship Research Institute Tokyo, March 1994. [23] Sorenson R.L.âThree-Dimensional Grid Generation about Fighter Aircraft for Zonal Finite-Difference Computations âAIAA 86â0429. AIAA 24th Aerospace Sciences Conference, 1986, Reno, NV. the authoritative version for attribution.