rately account for all important parameters that may influence results. Zhao et al. (11) applied a two-dimensional method that accounts for splash effects to the water entry of a wedge and a flared ship section. For the ship section, numerical pressure predictions compared favorably with model test measurements. For the wedge, pressures were overpredicted, and significant three-dimensional effects were shown to be important. Schumann (12) used a finite volume method with a VOF approach to capture the free surface. He investigated bow flare slamming phenomena for different flare angles. Sames et al. (13) presented a complete slamming analysis, including ship motion perditions and water entry simulations. They found that the motion history of water entry significantly affected predicted pressures.

We computed water entry processes for a flared ship section and a wedge and compared numerical results with measurements. Our systematic two- and three-dimensional computations assessed the influence of initial and boundary conditions.

2 Solution Method

The conservation equations of mass, volume concentrations and momentum describe the behavior of a multi-fluid system:




Here, V is an arbitrary control volume bounded by a closed surface S, ρ is the density, v is the fluid velocity vector, n is the unit vector normal to the surface S and directed outwards, ci is the volume concentration of the ith fluid component, T is the stress tensor, and fb is the resultant body force. For Newtonian incompressible fluids considered here the stress tensor is related to the rate of strain tensor via Stokes’ law:


where D is the rate of strain tensor, μ is the dynamic viscosity, p is the pressure, and I is the unit tensor.

The conservation equations of the volume concentration of each fluid component are another form of the mass conservation equation; their sum equals the overall mass conservation equation (1).

The mixture of fluids is treated as a single effective fluid, whose physical properties can be expressed as a function of the volume concentrations and the physical properties of each fluid component:


where ρi and μi are the density and viscosity of the ith fluid component, respectively.

In order to solve the governing equations, the solution domain is first subdivided into an arbitrary number of contiguous control volumes (CVs) or cells. Control volumes can be of an arbitrary polyhedral shape (see Fig. 1) allowing for local grid refinement, sliding grids, and block-structured grids with non-matching block interfaces. This simplifies grid generation and enables a more effective distribution of grid points by clustering them in the regions of strong variation of variables. The values of all dependent variables are stored in the center of each CV. The

Figure 1: Polyhedral control volume.

governing equations are integrated over each CV. The volume and surface integrals are calculated using second-order approximations (linear interpolation, central differences, and midpoint rule integration). The method is fully implicit, i.e., the spatial integrals are evaluated at the new time level while the old values appear only in the approximations of the time derivative (linear or quadratic backward scheme). The pressure-correction equation derived from the discretized continuity and momentum equations (14) is used to determine the pressure. Two-equation models belonging to the k-∈ family yield the eddy viscosity in case of turbulent flow computations. The resulting system of nonlinear and coupled algebraic equations is solved iteratively in a sequential manner. The linearized equations are solved using solvers from the family of conjugate-gradient methods. Iterations are repeated within each time step

The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement