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tion are much more severe and in our opinion unacceptable in the end.

The present work investigates therefore the feasibility of performing full scale Reynolds number calculations without the use of wall functions. The numerical method is the present version of the computer code PARNASSOS, [4] to [7], which has been originally developed at MARIN and more recently extended and improved in cooperation with IST, [8]. The method is based on the reduced form of the Reynolds-averaged Navier Stokes, (RANS) equations, [9]. An eddy-viscosity algebraic turbulence model based on the formulation of Cebeci and Smith, [10], completes the mathematical model. The reduced form of the RANS equations allows the use of large numbers of grid nodes even with modest computer resources, both in memory and c.p.u.

The direct application of the no-slip condition at the wall and the use of the reduced form of the RANS equations make grid generation one of the major difficulties of full scale Reynolds number calculations. The present grid generation procedure combines elliptic and algebraic grid generation techniques. A large number of grid nodes is required in the hull-normal direction. It is important to investigate the sensitivity of the solution to the grid and to verify the grid requirements in the near-wall region.

The present paper presents the main features of the numerical method in section 2. The grid generation methodology is described in section 3. The grid dependency studies and an investigation of the influence of the distance of the first grid node to the wall in the direct application of the no-slip condition follow in section 4. Also included in section 4 is a comparison between numerical predictions at full scale and model scale Reynolds numbers for the two test cases of the Tokyo Workshop, [2]: the HSVA tanker and the Mystery tanker. The conclusions of the paper are summarized in section 5.

2
Computational Method
2.1
Mathematical Formulation

The Reynolds-averaged Navier-Stokes equations for steady flow of an incompressible fluid consist of equations expressing mass and momentum conservation, supplemented with a turbulence model. The conservation equations are written here for a general curvilinear coordinate system ξi (alternatively denoted as the ξ, η, ζ system) in contravariant form with the cartesian velocity components, Ui, as the dependent variables:

(1)

(2)

with

(3)

(4)

and

(5)

The tensorial summation convention applies; are the contravariant base vectors:

p is the pressure, ρ the fluid mass density, µ the fluid effective viscosity, is the Jacobian of the transformation between the two systems and gij is the contravariant metric tensor. A partial parabolisation is obtained by choosing ξ1 as the main-stream direction and by neglecting diffusion in that direction, i. e. the terms with j=1 in the viscous terms of momentum equations (2). The elliptic character of the equations is retained in the pressure field. All diffusion terms are dropped in the momentum equation in the normal direction. The present set of equations is classified by Rubin et al. in [9] as the Reduced Navier-Stokes (RNS) equations.

The fluid effective viscosity, µ, is obtained with an isotropic eddy-viscosity algebraic turbulence model, [10].

The use of the RNS equations implies that a physical meaning is attached to the grid, since diffusion is neglected in the streamwise direction, while that direction is determined by the grid. To take advantage of the roughly flow-conforming



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