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VI[13], employing a boundary-fitted curvilinear coordinate system that is advantageous for resolving viscous flows near body boundaries. Both methods use the density-function method to implement the nonlinear kinematic free surface condition. Some applications of those methods to ship wave problems are explained.

NUMERICAL METHOD
Finite-difference Method

The modified marker-and-cell method called TUMMAC-VIII[12] is constructed by combining the free surface treatment of the TUMMAC-VI[ 9] method for a two-layer flow and the no-slip body boundary treatment and the porosity expression for geometries of the TUMMAC-VII method[ 14]. A rectangular grid system with variable spacing is employed and the velocity and pressure points are defined in a staggered manner.

The governing equations for the two-layer flow are the following incompressible Navier-Stokes equations and the continuity equation.

(1a)

(1b)

· u=0 (2)

where,

a = v2u + f (3)

Here, the subscripts 1 and 2 denote the fluids below and above the interface respectively. In the present study 1corresponds to the water region and 2 the air region. u is the velocity, p is the pressure, t is the time, v is the kinematic viscosity and f is the external force including the gravitational acceleration. The surface tension is ignored here since its effect can be considered to be very small in the problems described in this paper.

The above equations for each layer are quite separately solved at each time step of time-marching following the MAC-type algorithm. Firstly the pressure field in the air region is obtained by solving the Poisson equation and secondly in the water region. The Richardson 's method is used for the solution of the Poisson equation. The configuration of the interface is determined by the free surface condition described in the subsequent section.

For the time derivative of velocity forward differencing is used and second-order centered differencing is for the space derivatives excluding the convective terms for which third-order upwind differencing is used.

Finite-volume method

The WISDAM-VI method employs the coordinate system that is fitted to the body boundary but not to the free surface, so that the boundary layer around the body of arbitrary form and large free-surface deformation is simultaneously simulated. Since the numerical schemes of the WISDAM-VI method are based on the WISDAM-V method[15] except for the free-surface treatment, the computational method is briefly explained here.

The governing equations are the Navier-Stokes equation and the continuity equation in conservative form.

(4)

where,

(5)

Here I and Re is the unit matrix and Reynolds number respectively and is the normalized pressure without hydrostatic component defined as

(6)

where Fn is the Froude number. The continuity equation for incompressible fluid is written as

div u=0. (7)

Third order upwind-biased flux interpolation scheme for the convective term and second-order central interpolation scheme for the other terms is used for the finite-volume discritization of Eq.(4) and Eq.(7).

Since the MAC type solution algorithm is employed, the pressure term is separated from the other terms and thus, the velocity field of the (n+1) time level is written as

un+1=unt·div T–Δt · , (8)

where, the superscript n denotes the time level. Taking the divergence of Eq.(8), following Poisson equation for is obtained.

(9)

Eq.(9) is solved iteratively by use of the SOR method.



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