Page 440 - Practical Design Ships and Floating Structures

415

kinetic energy and the dissipation of turbulence, respectively, d is the normal distance from the wall,
and P is the production of turbulent kinematic energy.

The viscous stress tensors, ~ii (i, j = 1,2,3), are defined as
(:, E, : -)
Ty =(flu+,) -+---(v.v)
8,
where p=pv and = UT + -I + WE. v is the kinematic viscosity and 6,j is the Kronecker's symbol. p, is
the turbulent viscous coefficient determined by two turbulence models, the Baldwin-hmax (B-L)
turbulence model and the Chien's low Reynolds number k-E model, without a wall function in the
present case. The equation of the state of motion for a compressible fluid with the artificial
4 -
compressibility is given by vc*p, wherec = 5, u + v + w and s1 is the constant.

2.2 Initial and Boundary Conditions
2.2. i initial conditions

An uniform flow, a flat free surface and a constant distribution of the turbulent quantities are specified
2.2.2 Boundary conditions
ah ah ah
On the fiee surface: the wave height h (x, y, t) is evaluated by - + u - + v - = w . In the present
&hay
case, three components of the velocities (u, v, w) are determined with the extrapolation, and the
pressure w is updated by y = pgh . k and E on this surface are given with the mirror conditions.
On the wetted part of the hull surface: u = v = w = 0, k = k,,,,,, and p = are imposed, whereas
the pressure on this surface is obtained by the Neumann approach.
At the inlet: the uniform flow and the constant turbulence variables are given.
At the outlet: all variables are extrapolated with a zero-gradient approach.
On the centreline boundary and the external boundary: the mirror conditions for all variables are
implemented.

2.3 Numerical Solution
During the process of the resolution of the bulk RANS flow, the Roe's approach with the MUSCL type
and a central-difference approach are separately implemented for the inviscid fluxes and the viscous
fluxes; the solution vectors are updated by the DDADI-factorization with the local time step; and the
convergence rate is improved by the V-cycle of the multigrid method. Once the solutions for the bulk
RANS flow are given, the spatial derivative on the free surface is evaluated with a third-order upwind
differencdthe second-order nonessential oscillation (ENO) scheme; and the wave height is updated
with the second-order explicit Adams-Bashforth scheme. Thus, the pressure on the free surface is
adjusted. This procedure is repeated until the steady state is reached.

In the present approach, an asymptotical method for the treatment of the free surface is implemented.
First, a zone used for an extrapolation of the free surface is specified, and it should be close the hull
surface. Secondly, to maintain this zone as small as possible, the local filtering is applied, which
covers this zone. Finally, the free surface in this zone is determined by a linear least-square fitting. The
benefit is that problems, such as a high aspect ratio within the boundary layer and a numerical
singularity at the contact line, can be avoided. Note that at a transom stem, the transom is dry for the
KCS model and the HTC model using the dry-transom model but a partially wetted transom is
maintained for the DTMB 5415 model and the tanker model. This implies that the water is enforced to