7.4 Turbulent boundary layers
At solid walls, the tangential flow speed
increases rapidly across a thin boundary layer, as discussed in
Sec. 6.4
. At high 
, the velocity
profile has a universal character shown below.
The profile compares measured data in
terms of a dimensionless velocity 
and distance to the
wall 
, given by
|
(7.9) |
Both parameters in Eq. (
7.9
) are based on a
friction velocity 
which is related to
the wall shear stress
by
|
(7.10) |
At the wall
. Close to the wall,
is suppressed,
creating a region where flow is laminar
, known as the
viscous sub-layer . The profile in this
region is described by the relation
|
(7.11) |
Turbulence becomes significant through the
buffer layer which describes the
region
. Van Driest provides a model for the increase in mixing
length through this region, by
|
(7.12) |
Finally, in the
inertial
sub-layer
for
, flow is turbulent and the velocity profile is described by
the
logarithmic law of the
wall, often abbreviated to simply the
log law, according to
|
(7.13) |
The equation includes Kármán’s constant
and constant
.
For a smooth wall,
– 5.5 is commonly used. Both
Eq. (
7.11
) and Eq. (
7.13) can
be derived assuming a constant shear stress across the profile,
equating to
at the wall. In the viscous sub-layer, the shear
stress is laminar, so
|
(7.14) |
This equation integrates with a zero constant of integration to
give
, from which Eq. (
7.11
) is derived. In the inertial
sub-layer, the shear stress is turbulent (laminar is negligible),
so
|
(7.15) |
Assuming
gives
, which integrates to yield Eq. (
7.13). In
the inertial sub-layer,
as described in Eq. (
7.5
), which combines with
Eq. (
7.15
) and Eq. (
6.31
) to give
|
(7.16) |
