Figures 2 and 3 show the convergence of various iterative algorithms to solve linear problems with the zero machine accuracy. For practical calculations we reduce the linear residuals by two orders.

In the present paper only steady state is compared with experiments so only one non-linear iteration is made at each time step. Concerning the obtaining of steady state, free surface flow problem is much more complicated than zero Froude number problem. Because of the propagation of wave field, convergence on the whole calculation domain takes a long time. In fact we test only the convergence on a sphere near the full (R/1=1) and we reduce the non-linear residuals by two order of magnitude (non-linear residuals are obtained after the matrix and source terms are updated).

The multigrid algorithm converges up to the machine accuracy at a constant rate on elementary case but in 3-D for the fully coupled system the convergence depends strongly of prolongation and restriction operators and of smoothing operator. A bad smoothing leads to the divergence. Here the small wave lengths of the error induced by prolongation procedure are very difficult to smooth under the saturation level.

The grid refinement is not a “substantial refinement” but the solutions on the two grids are very different (figure 14). The conclusion is that small variation of grid size can induce strong differences on the solution due certainly to turbulence behavior.

W.W.Schultz

University of Michigan, USA

The rolling motion (tangency condition) you require at the free surface singularity makes matters worse for the second fluid (air in this case). It has been shown that the singularity has to be relieved in a different way. E.B.Dussan [(1979) Ann. Rev. Fluid Mech.] shows that slip is observed and material on the surface is mapped into the interior! Ting & Perlin [(1995) J. Fluid Mech.] shows similar behavior for high Reynolds number, oscillatory flow.

The purpose of our work is to propose a numerical formulation to solve Navier-Stokes equations coupled with classical boundary conditions (no-slip conditions on the wall, kine-matic and dynamic conditions on the free surface). It is well known that the wetting problem cannot be solved by these macroscopic equations. The physics does not follow the continuity hypothesis, and particularly the kinematic condition seems to be unverified. The problem is not to predict the meniscus and the dynamic contact angle at a very small scale, but to ensure the contact line progression with classical boundary conditions. In this case, mathematically, free surface has to be tangent to the wall.

H.Raven

Maritime Research Institute, The Netherlands

Your figures 10 and 11 compare calculated and measured wave patterns. The bow wave at the hull is well predicted, the wave profile is fair, but the diverging bow wave system is completely absent. Using a nonlinear free surface condition and a paneling like shown in figure 4, a panel method would probably give a better result for the waves. Does the use of Navier-Stokes equations have a greater amount of numerical damping than a panel method and does it therefore require a finer free surface discretization? Or are there other effects leading to this difference.

We have compared Navier-Stokes formulation with Rankine panel method for the same grid refinement (except in the boundary layer zone). It appears clearly that Navier-Stokes formulation has a greater amount of numerical damping. In fact to obtain similar results we have to use more or less a two times finer grid in each direction in the Navier-Stokes formulation. The explanation is that in the panel method a part of the solution (the mass conservation, that is to say the continuity equation) is mathematically exact and does not proceed from a discretization as in the Navier-Stokes formulation.