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### 8.1 Introduction

In a thin layer of fluid adjoining a wall, past which a turbulent stream is flowing, the effective viscosity of the fluid varies rapidly with distance from the wall.

Therefore the shear stress across the layer can not be correctly derived from the velocity difference and distance by use of the arithmetic- or harmonic-mean effective viscosity.

Instead, special functions are employed, which from the earliest days of CFD [Patankar and Spalding, 1967] have been called "wall functions".

The wall-function approach bridges the near-wall layer by employing empirical formulae to provide near-wall boundary conditions for the mean-flow and turbulence-transport equations.

These formulae therefore connect the wall conditions (e.g. the wall shear stress and heat flux) to the dependent variables (viz velocity and temperature) at the near-wall grid node.

This grid node is commonly presumed to lie outside the viscous sublayer in fully-turbulent fluid. The advantages of this approach are that it escapes the need to extend the computations right down to the wall.

Four different types of wall function are provided in PHOENICS, namely:

• equilibrium log-law wall functions (termed GRND2 wall functions in PHOENICS parlance);
• non-equilibrium log-law wall functions (termed GRND3 wall functions);
• equilibrium fully-rough log-law wall functions (termed GRND5 wall functions); and
• scalable wall functions, which can be applied to GRND2-type wall functions.

GRND3 wall functions should be used whenever it is necessary to compute accurately the turbulent transport of heat, and also species, at a reattachment point, where the velocity is close to zero.

The GRND2 and GRND3 wall functions include an option to permit wall-roughness effects to be simulated via the specification of an equivalent "sand-grain" roughness height.

GRND5 wall functions are intended primarily for use with atmospheric boundary layers where the roughness height is defined in terms of an aerodynamic roughness height, and optionally the displacement height.

The purpose of scalable wall functions is to avoid the problem of near-wall grid refinement invalidating the standard wall function. This is done by ensuring automatically that near-wall distance used by the standard wall function is not less than that implied by the intersection of linear and log-law profile.

The rest of this entry provides a detailed mathematical statement of these wall-function features.

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