7.7 Turbulence near walls

The characteristic velocity distribution in turbulent boundary layers in Sec. 7.4 , provided wall functions expressed as boundary conditions for eqn in Sec. 7.5 and Sec. 7.6 . Boundary conditions also need to be specified for turbulence fields at solid walls.

The turbulence generation eqn influences the distribution of turbulence fields near a wall. At the wall, eqn. In the inertial sub-layer, eqn from Eq. (7.5 ) with: eqn, obtained by combining Eq. (6.21 ) and Eq. (7.15 ); and, Eq. (6.24 ).

Since eqn decreases with increasing eqn in the inertial sub-layer, eqn passes through a peak within the buffer layer (at eqn).

PICT\relax \special {t4ht=

The peak in eqn causes a similar peak in eqn, shown in the following diagram. To the left of the peak, turbulent energy is transported back towards the wall by diffusion eqn The profile of dissipation results from G + Dk @2k=@y2 , obtained from Eq. (7.1 ). Very close to the wall (eqn), diffusion is predominately molecular, such that it non-zero at the wall. The dissipation also has a non-zero value at the wall.

Wall functions and turbulence fields

When using a turbulence model, such as the k model described in Sec. 7.1 , boundary conditions must be specified for eqn and at solid walls. The distribution of eqn, non-dimensionalised as eqn, close to the wall is shown below.

PICT\relax \special {t4ht=

At the wall, eqn but it rises quickly to a peak at eqn before levelling off at eqn as eqn.

With wall functions, the height of the centre of each near-wall cell should correspond to eqn within the range eqn. Viewed at that scale, the eqn profile appears flat. For , there is no such simple profile shape. These observations lead to the boundary conditions for eqn and when using wall functions:

  • zero gradient for eqn;
  • calculated near-wall cell value for , according to:
 (c3 =4k3=2= yP for y+ > y+;
The calculation uses eqn from the near-wall cell. The expression for eqn is Eq. (7.4 ) with Eq. (6.24 ). The expression for eqn uses the asymptotic condition eqn for in Eq. (7.28 ).
Notes on CFD: General Principles - 7.7 Turbulence near walls