## 7.7Turbulence near walls

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

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

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

The peak in causes a similar peak in , shown in the following diagram. To the left of the peak, turbulent energy is transported back towards the wall by diﬀusion The proﬁle of dissipation results from , obtained from Eq. (7.1 ). Very close to the wall (), 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 model described in Sec. 7.1 , boundary conditions must be speciﬁed for and at solid walls. The distribution of , non-dimensionalised as , close to the wall is shown below.

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

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

• zero gradient for ;
• calculated near-wall cell value for , according to:
 (7.26)
The calculation uses from the near-wall cell. The expression for is Eq. (7.4 ) with Eq. (6.24 ). The expression for uses the asymptotic condition for in Eq. (7.28 ).
Notes on CFD: General Principles - 7.7 Turbulence near walls