7.8 Resolving the viscous sub-layer

Wall functions were introduced in Sec. 7.5 to avoid the need for a large mesh with cells small enough to resolve the boundary layer into the viscous sub-layer. They provide a reasonable prediction of eqn using the log law for the velocity distribution in the inertial sub-layer.

PICT\relax \special {t4ht=

A CFD simulation may alternatively use a mesh with sufficiently thin cells to resolve the flow through the viscous sub-layer, e.g. with near-wall cell centre height corresponding to eqn = 1, for a more accurate prediction of eqn. If so, the turbulence model must then be able to function reliably in viscous flow regions.

Such models are usually described as “low Reynolds number”. The expression does not refer to the eqn of the flow based on the characteristic scales of the problem, e.g. axial mean flow speed and diameter for a pipe. Instead it is a “turbulence” Reynolds number eqn based on the scales of speed eqn and size eqn of turbulent eddies and can be defined as

 2 Re = k--:
This definition is obtained from scale arguments introduced in Sec. 6.6 , in which . Since eqn represents fluctuations, eqn. Combining these expressions into a Reynolds number yields Eq. (7.27 ).

Asymptotic consistency

Low-eqn turbulence models pay attention to the behaviour of fluctuating velocities, e.g. eqn, eqn, in the limit that eqn at the solid boundary.

PICT\relax \special {t4ht=

They aim to capture the shape the profiles of and eqn as they approach eqn. Let eqn and eqn define the directions tangential and normal to the wall respectively. Profiles in the fluctuating velocities can be expressed by polynomials in eqn, i.e.

 0 2 ux = a0 + a1y + a2y + ::: u0 = b0 + b1y + b2y2 + ::: y \relax \special {t4ht=
where eqn, etc. are functions of space and time. The no slip condition implies eqn, so eqn to the lowest order in eqn. For eqn, it is eqn since, at the wall, eqn and by continuity eqn.

The turbulent properties are, to the lowest order in eqn, as follows.

It follows that models achieve asymptotic consistency when

k y2 and
in the limit that eqn.
Notes on CFD: General Principles - 7.8 Resolving the viscous sub-layer