The Marker-density method in cartesian grids applied to nonlinear ship waves

A numerical simulation method is presented for nonlinear waves generated by advancing ships. The nonlinear free surface is determined by a modified Marker-density method. Pressures and velocities in body boundary cells and free surface cells are calculated through a simultaneous iterative method. The filtered Navier–Stokes equations and filtered continuity equation are used as governing equations. The equations are solved by using a finite difference method in a Cartesian grid system. The body boundary is defined by the line segments connecting the points where grid lines and body surface meet. Pressures are coupled with velocities through two-step projection method in the present approach. The governing equations are approximated in finite difference form using a forward time centered space (FTCS) scheme except convection terms. The convection terms are approximated in finite difference form employing third order upwind scheme in space and Adams–Bashforth scheme in time. For the verification of the present numerical simulation, a numerical computation is carried out with Series-60 cargo ship. For the grid convergence test, three grid levels are selected with a constant grid refinement ratio. For the purpose of understanding spilling type breaking waves near a ship bow, a wedge shape model is selected, and its numerical simulation is performed using the computational fluid dynamics program developed during the present research. For a specific case, plunging type breaking bow waves with scars which are frequently generated by a planing hull type patrol ship, are numerically simulated at several different speeds. In addition, the pressure resistance coefficients from the numerical computations are investigated in detail with respect to the residual resistance coefficients from the corresponding ship model experiments.

► A numerical method for ship making nonlinear waves was developed.
► Marker-density method was modified and applied for nonlinear waves.
► The order of accuracy of the method is 1.08 and the uncertainty is 8.58%.
► The present numerical method is applicable to the hull form design in preliminary design stage.