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 
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) |
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
).
