Figure 13: Mystery tanker, Re=2.0×10⁹. Comparison of limiting streamlines for calculations performed at full scale and model scale Reynolds numbers.

The comparison between results at model scale and full scale Reynolds numbers is presented in figures 10 to 13. In general, the comparison of results at model and full scale Reynolds numbers is similar to the one obtained for the HSVA tanker.

The isolines of U¹, Cp and ω¹ are plotted in figure 10, where the decrease of the boundary layer thickness with the increase of Reynolds number is clear. The transverse velocity fields at x=0.989 are presented in figure 11. The bilge vortex location of the two calculations is not equal and the cross-stream velocities are larger at Re=2.0×10⁹ than at 5.0×10⁶. The isobars on the ship surface of the two calculations are compared in figure 12. The effect of the Reynolds number in the surface pressure distribution is evident.

The limiting streamlines of the two calculations are plotted in figure 13. There is a significant difference between the two flows. At model scale, a small streamwise flow separation region is predicted at the end of the stern close to the free surface. No streamwise flow separation is predicted at full scale Reynolds number.

The feasibility of the calculation of ship stern flows at full scale Reynolds numbers with direct application of the no-slip condition at the hull surface has been proven. The numerical method with which this result has been obtained is based on the Reduced form of Reynolds averaged Navier-Stokes equations, which allows the use of large numbers of grid nodes, required by the calculation of the near-wall region, at acceptable computing time.

The grid dependency studies performed for the flow around the HSVA tanker suggest that it is possible to obtain a grid-independent solution without the use of wall-functions. The number of grid nodes required in the normal direction is within the acceptable limits of the present method. This is fortunate because further increase of the number of grid nodes per streamwise station may require a more robust—but also more expensive—solver.

In the present calculations for the HSVA tanker, it was found that the solution was independent of the near-wall grid density for distances of the first grid node to the wall smaller than

The comparison of full scale and model scale Reynolds numbers calculations for the HSVA tanker and the Mystery tanker confirmed a strong dependence of the flow field at the end of the stern on the Reynolds number. The limiting streamlines showed a systematic change in position of the confluence of limiting streamlines with increasing Reynolds number. On the two test ships streamwise flow separation was delayed at high Re.

[1] Larsson L., Patel V.C., Dyne G. (eds.)— Ship Viscous Flow.—Proceedings of 1990 SSPA-CTH-IIHR Workshop, Flowtech International AB, Research Report N^o 2, Gothenburg, June 1991.

[2] Proceedings of CFD Workshop Tokyo 1994, Ship Research Institute Tokyo , March 1994.

[3] Ju S., Patel V.C.—Stern Flows at Full-Scale Reynolds Numbers.—Journal of Ship Research, Vol. 35, N^o 2, June 1991, pp. 101–103.

[4] Raven H.C., Hoekstra M.—A Parabolised Navier-Stokes Solution Method for Ship Stern Flow Calculations.—2^th International Symposium on Ship Viscous Resistance, Göteborg Sweden, March 1985.

[5] Hoekstra M., Raven H.C.—Application of a Parabolised Navier-Stokes Solution System to Ship Stern Flow Computation .—Osaka International Colloquium on Ship Viscous Flow, Osaka Japan, October 1985.