**6. Flow field around a series-60 hull**

Additional verification of the numerical code is presented for the Series-60 hull for a low Reynolds number of 1.0x103. Figures 5, 6, and 7 show a plane of the computational grid generated around the hull at the level of the undisturbed free surface. The grid has 240x160x160 points and the smallest element is a cube with dimension of 0.0075Lpp. The grid has 11Lpp in the y direction, and 7.2 Lpp in the *x* and *z* directions.

A Three-Dimensional Numerical Simulation of the Free Surface Flow Around a Ship Hull 403

Figure 8 illustrates the pressure contour around the Series-60 hull at the same plane of the undisturbed free-surface. In this simulation, the free-surface is not allowed to deform. A stagnation point (in red) is observed at the bow of the hull and a low pressure region (in

Fig. 7. Grid generated around the Series-60 hull (bottom view).

Fig. 8. Pressure field around the series-60 hull, *RL*=1.0x103, *Fn*=0.25.

blue) is observed at the side walls of the body.

Fig. 5. Grid generated around the Series-60 hull.

A top view and a bottom view of the hull can be seen in Fig. 6 and Fig. 7, respectively. Close to the body, the grid is uniform and sufficiently refined to capture the viscous effects close to the hull surface. An exponential stretching is used to increase the size of the grid elements away from the body, where property gradients are small.

Fig. 6. Grid generated around the Series-60 hull (top view).

A top view and a bottom view of the hull can be seen in Fig. 6 and Fig. 7, respectively. Close to the body, the grid is uniform and sufficiently refined to capture the viscous effects close to the hull surface. An exponential stretching is used to increase the size of the grid elements

Fig. 5. Grid generated around the Series-60 hull.

away from the body, where property gradients are small.

Fig. 6. Grid generated around the Series-60 hull (top view).

Fig. 7. Grid generated around the Series-60 hull (bottom view).

Figure 8 illustrates the pressure contour around the Series-60 hull at the same plane of the undisturbed free-surface. In this simulation, the free-surface is not allowed to deform. A stagnation point (in red) is observed at the bow of the hull and a low pressure region (in blue) is observed at the side walls of the body.

Fig. 8. Pressure field around the series-60 hull, *RL*=1.0x103, *Fn*=0.25.

A Three-Dimensional Numerical Simulation of the Free Surface Flow Around a Ship Hull 405

Fig. 10. Free-surface elevation around the Series-60 hull, *RL*=1.0x103, *Fn*=0.25.

Fig. 11. Free-surface elevation around the Series-60 hull, *RL*=1.0x103, *Fn*=0.25 and velocity

field.

Figure 9 illustrates the pressure contour and the velocity field around the hull. The velocity profile inside the boundary layer and at the wake can be observed at the body surface and behind the hull, respectively.

Fig. 9. Pressure and velocity fields around the Series-60 hull, *RL*=1.0x103, *Fn*=0.25.

Figure 10 shows the free surface elevation after allowing the free surface to deform and after the steady state is obtained. The diverging wave formation can be observed at the port and starboard of the ship hull. Figure 11 shows the velocity field around the hull and at the level of the free surface. The velocity profile inside the boundary layer along the side walls of the hull and at the wake of the body can be observed in yellow arrows.

Figure 9 illustrates the pressure contour and the velocity field around the hull. The velocity profile inside the boundary layer and at the wake can be observed at the body surface and

Fig. 9. Pressure and velocity fields around the Series-60 hull, *RL*=1.0x103, *Fn*=0.25.

hull and at the wake of the body can be observed in yellow arrows.

Figure 10 shows the free surface elevation after allowing the free surface to deform and after the steady state is obtained. The diverging wave formation can be observed at the port and starboard of the ship hull. Figure 11 shows the velocity field around the hull and at the level of the free surface. The velocity profile inside the boundary layer along the side walls of the

behind the hull, respectively.

Fig. 10. Free-surface elevation around the Series-60 hull, *RL*=1.0x103, *Fn*=0.25.

Fig. 11. Free-surface elevation around the Series-60 hull, *RL*=1.0x103, *Fn*=0.25 and velocity field.

A Three-Dimensional Numerical Simulation of the Free Surface Flow Around a Ship Hull 407

An upwind and TVD numerical scheme was implemented to solve the unsteady slightly compressible Navier-Stokes equations for the free-surface flow around ship hulls. The physical domain is discretized in a Cartesian grid and the boundary condition on the body

The implemented code is parallelized using MPI to be run in an arbitrary number of computers of a cluster. The numerical code was verified for the flow around a sphere, and a

The results obtained for the sphere were compared to numerical and experimental data from literature showing the good quality of the numerical results. The numerical results obtained for the ship hull were not compared to other numerical and experimental data because of the difficulty to find those data for lower Reynolds number. However, the

validation of the numerical code for higher Reynolds numbers and configurations of practical interest, such as, resistance to motion, moonpool – free decay and forced motion,

This research was sponsored by the Brazilian Innovation Agency – FINEP under Grant

[1] Alessandrini, B., and Delhommeau, G., 1994, "Simulation of Three-Dimensional

[2] Campregher, R., Mansur, S. S., and Silveira-Neto, A., 2005, "Numerical Simulation of the

[4] Ciortan, C., Guedes, C., and Wanderley, J., 2007, "Assessment of Free Surface Treatment

Mechanics and Arctic Engineering, June 10-15, San Diego, California, USA. [5] Hino, T., 1987, "Numerical Simulation of a Viscous Flow with a Free Surface around a Ship Model", Journal of the Society of Naval Architects of Japan, Vol. 161. [6] Ratcliffe, T., 1998, "Validation of the Free Surface Reynolds-Averaged navier-Stokes and

[7] Rider, W.J., and Kothe, D. B., 1997, "Reconstructing Volume Tracking", Journal of

Unsteady Viscous Free Surface Flow around a Ship Model", International Journal

Flow Around a Sphere Using the Immersed Boundary Method for Low Reynolds Number", Proceedings of the Sixth International ERCOFTAC Workshop on Direct and Large-Eddy Simulation, held at the University of Poitiers, September, 12-14. [3] Ciortan, C., Wanderley, J., and Guedes, C., 2007, "Turbulent Free-surface Flow around a

Wigley Hull Using the Slightly Compressible Formulation", Ocean Engineering, V.

Techniques and Turbulence Models Influence Using the Slightly Compressible Flow Simulation", Proceedings of the 26th International Conference on Offshore

Potential Flow Codes", Proceedings of the 22nd ONR Symposium on Naval

turbulence model and

surface is implemented using the Immersed Boundary Method (IBM).

Next phase of development will include the implementation of the *k-*

for Numerical Methods in Fluids, Vol. 19, pp.321-342.

numerical results agree qualitatively well to experiments.

wave run-up and air gap, and wake and shadow flows.

**7. Conclusions** 

Series-60 hull.

**8. Acknowledgments** 

34, pp. 1383-1392.

Hydrodynamics.

Computational Physics, Vol. 141, 112-152.

0106067200.

**9. References** 

Figure 12 presents the pressure coefficient contour around the ship hull at the level of the free surface. The low pressure regions can be seen in blue at the wave crests of the divergence wave and the high pressure regions can be seen in red at the bow of the ship and at the wave trough.

Fig. 12. Free-surface elevation around the Series-60 hull, *RL*=1.0x103, *Fn*=0.25 and pressure field.

Figure 13 shows the total (in red), frictional (in black), and pressure (in blue) drag coefficients on the ship hull. After time=4, the steady state is obtained and the drag coefficients are constant.

Fig. 13. Drag coefficient on the Series-60 hull, *RL*=1.0x103, *Fn*=0.25.
