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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 Nω including viscous corrections to define the near-wall values of ω, BC1; and ii) obtain ω 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 ‘hook 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: 5×106, 2×107, 108, 5×108 and 2×109, 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 U∞ 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: Nξ=161 and Nζ=41. The number of grid nodes in the normal direction, Nη, increases with Rn. Nη=81 for the lowest Rn and 10 grid lines are added each time Rn is increased, which leads to Nη=121 for Rn=2×109. The grid line spacing in the normal direction is defined by one-dimensional stretching functions, which are tuned to obtain a maximum value of y− 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: • Friction resistance coefficient, • Pressure resistance coefficient, • Wake fraction, Wf: • Maximum cross-stream velocity at x=0.989L, (Vw)max, with • 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 Ω 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.

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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 560 3.2 Wall Boundary Condition for To investigate the influence of the numerical implementation of the wall boundary condition of ω, we have calculated the flow at Rn=5×106 with Menter's version of the k-ω model with the two options considered: BC1, which obtains ω from the theoretical value for y−<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-ω model using different implementations of the ω 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 −0.075 −0.076 (vt)med×104 0.265 0.257 4 (vt)max×10 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-ω 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-ω model with different wall boundary conditions. These results show that the flow prediction is clearly dependent on the numerical implementation of the ω wall boundary condition. They suggest that the ω behaviour at the wall can hardly be seen as a strong point of the k-ω 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-ω 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. ∆ 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,

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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 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 ∆ 5×106 1.573 1.816 1.606 1.561 1.598 1.445 1.939 1.944 0.499 2×107 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 5×108 0.846 0.965 0.916 0.885 0.900 0.824 0.981 0.983 0.141 2×109 0.732 0.825 0.791 0.767 0.776 0.713 0.841 0.843 0.130 5×106 0.598 0.788 0.741 0.713 0.687 0.655 0.923 0.911 0.325 2×107 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 5×108 0.423 0.605 0.547 0.543 0.570 0.575 0.690 0.675 0.267 2×109 0.402 0.578 0.535 0.542 0.565 0.570 0.648 0.638 0.236 5×106 0.528 0.632 0.619 0.563 0.576 0.532 0.616 0.609 0.104 Wf 2×107 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 5×108 0.347 0.393 0.393 0.369 0.380 0.365 0.384 0.386 0.046 2×109 0.315 0.352 0.353 0.334 0.341 0.330 0.345 0.345 0.038 5×106 0.209 0.280 0.290 0.287 0.266 0.263 0.347 0.343 0.138 (Vw)max 2×107 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 5×108 0.206 0.294 0.252 0.229 0.256 0.244 0.395 0.369 0.189 2×109 0.208 0.263 0.228 0.227 0.229 0.228 0.336 0.331 0.128 5×106 −0.012 −0.059 −0.068 −0.028 −0.015 −0.007 −0.075 −0.075 0.063 2×107 −0.004 −0.055 −0.040 −0.007 −0.016 −0.011 −0.065 −0.064 0.061 108 0.000 −0.035 −0.001 0.000 −0.001 0.000 −0.041 −0.041 0.041 5×108 0.000 −0.005 0.000 0.000 0.000 0.000 −0.027 −0.034 0.034 2×109 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (vt)max ×104 5×106 2.327 1.627 1.259 1.113 1.095 1.046 1.492 1.582 1.281 2×107 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 5×108 1.374 0.917 0.711 0.660 0.708 0.603 0.763 0.862 0.800 2×109 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- ω models, KW and KWM. The calculated limiting streamlines at Rn=5×106, Rn=108 and Rn=2×109 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.

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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 562 results of the SA, KW and KWM models are very similar. At full scale, Rn=2×109, 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=5×106. 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, 5×106, 108 and 2×109, 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 ‘hook shape' does appear for the k-ω models, and to some extent, for the one-equation models5 SA and MT. However, at x=0.989L, the ‘hook shape' tends to disappear with the increase of the Reynolds number. At Rn=2×109, none of the predictions exhibits a ‘hook 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=2×109, 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.

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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 563 the stretching of the bilge vortex with the increase of Rn is clear for the three models. Figure 7: Limiting streamlines at Rn=2×109. Figure 6: Limiting streamlines at Rn=108. We should note that the present results do not imply that the typical ‘hook 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=2×109. At this location, the bilge vortex is almost axisymmetric and the U1 plot shows the typical ‘hook 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-ω models, we have pointed out the deficiencies of a widely accepted numerical implementation of the wall boundary condition of ω. The results of the calculation of the flow around the Dyne (Mystery) tanker at five Reynolds numbers, 5×106, 2×107, 8, 5×108 and 2×109, suggest the following conclusions: 10 • 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. • 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.

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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=5×106. 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

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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=2×109. x=0.989L obtained Figure 8: Axial velocity isolines at Rn=5×106. 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

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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=2×109. 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

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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 567 Figure 12: Velocity field at x=1.1L obtained with Menter's k-ω model at Rn=2×109. Above: axial velocity, U1, isolines. Below: cross-stream velocity field. • 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. • Although the performance of the k-ω models seems to be very encouraging, the predictions depend on the numerical implementation of the ω 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.—The Scaling of High Reynolds Number Viscous Flow Predictions Using CFD Techniques—Third Osaka Colloquium, Osaka, Japan. [2] Eça L., Hoekstra M.—Numerical Calculations of Ship Stern Flows at Full-Scale Reynolds Numbers Twenfirst Symposium on Naval Ship Hydrodynamics, Trondheim, June 1996. [3] Hoekstra M., Eça L.—An Example of Error Quantification of Ship-Related CFD Results—7th Numerical Ship Hydrodynamics Conference, Nantes, July 1999. [4] Eça L., Hoekstra M.—On the Numerical Verification of Ship Stern Flow Calculations—1st MARNET Workshop, Barcelona, November 1999. [5] Deng G.B., Visonneau M.—Comparison of Explicit Algebraic Stress Models and Second-Order Turbulence Closures for Steady Flows around Ships —7th Numerical Ship Hydrodynamics Conference, Nantes, July 1999. [6] Cebeci T., Smith A.M.O.—Analysis of Turbulent Boundary Layers.—Academic Press, November 1984. [7] Baldwin B.S., Lomax H.—Thin Layer Approximation and Algebraic Models for Separated Turbulent Flows—AIAA Paper 78–257, January 1978. [8] Spalart P.R., Allmaras S.R.—A One-Equations Turbulence Model for Aerodynamic Flows—AIAA 30th Aerospace Sciences Meeting, Reno, January 1992. [9] Menter F.R.—Eddy Viscosity Transport Equations and Their Relation to the k-ε Model—Journal of Fluids Engineering, Vol. 119, December 1997, pp. 876–884. [10] Chen H.C, Patel V.C.—Practical Near-Wall Turbulence Models for Complex Flows Including Separation.—AIAA 19th Fluid Dynamics, Plasma Dynamics and Lasers Conference, June 8–10, 1987. the authoritative version for attribution.

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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 568 [11] Chien K.Y—Prediction of Channel and Boundary-Layer Flows with a Low-Reynolds-Number Turbulence Model — AIAA Journal, January 1992, pp. 33–38. [12] Wilcox D.C.—Turbulence Modeling for CFD—DWC Industries 1993. [13] Menter F.R.—Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications—AIAA Journal, Vol. 32, August 1994, pp. 1598– 1605. [14] Gibson M.M., Dafa Alla A.A.—Two-Equation Model for Turbulent Wall Flow—AIAA Journal, Vol. 33, August 1995, pp. 1514–1518. [15] Baldwin S.B., Barth T.J.—A One-Equation Turbulence Transport Model for High Reynolds Wall-Bounded Flows—AIAA Paper 91–0610, 29th Aerospace Sciences Meeting, Reno Nevada, January 1991. [16] Hoekstra M.—Numerical Simulation of Ship Stern Flows with a Space-Marching Navier-Stokes Method—PhD Thesis Delft University, 1999. [17] Wolfshtein M.—The Velocity and Temperature Distribution in One-Dimensional Flow with Turbulence Augmentation and Pressure Gradient.— International Journal of Heat and Mass Transfer, Vol. 12, 1969, pp. 301–318. [18] Goldberg U., Peroomian O., Chakravarthy S.,—A Wall Distance-Free k-ε Model with Enhanced Near-Wall Treatment—Journal of Fluids Engineering, Vol. 120, September 1998, 457–462. [19] Hoekstra M., Eça L.—PARNASSOS: An Efficient Method for Ship Stern Flow Calculation—Third Osaka Colloquium on Advanced CFD Applications to Ship Flow and Hull Form Design, Osaka, Japan, 1998. [20] van der Ploeg A., Eça L., Hoekstra M.—Combining Accuracy and Efficiency with Robustness in Ship Stern Flow Computation —Twen-third Symposium on Naval Ship Hydrodynamics, September 2000. [21] Larsson L., Patel V.C., Dyne G. (eds.)—Ship Viscous Flow.—Proceedings of 1990 SSPA-CTH-IIHR Workshop, Flowtech International AB, Research Report Nº2, Gothenburg, June 1991. [22] Proceedings of CFD Workshop Tokyo 1994, Ship Research Institute Tokyo, March 1994. [23] Sorenson R.L.—Three-Dimensional Grid Generation about Fighter Aircraft for Zonal Finite-Difference Computations —AIAA 86–0429. AIAA 24th Aerospace Sciences Conference, 1986, Reno, NV. the authoritative version for attribution.