7.4 Turbulent boundary layers

At solid walls, the tangential flow speed eqn increases rapidly across a thin boundary layer, as discussed in Sec. 6.4 . At high eqn, the velocity profile has a universal character shown below.

PICT\relax \special {t4ht=

The profile compares measured data5 in terms of a dimensionless velocity eqn and distance to the wall eqn, given by

u+ = u =u and y+ = u y= : x \relax \special {t4ht=
Both parameters in Eq. (7.9 ) are based on a friction velocity eqn which is related to the wall shear stress eqn by
 2 w = u : \relax \special {t4ht=
At the wall eqn. Close to the wall, eqn is suppressed, creating a region where flow is laminar eqn, known as the viscous sub-layer . The profile in this region is described by the relation
|-+----+-| -u--=-y--- \relax \special {t4ht=
Turbulence becomes significant through the buffer layer which describes the region eqn. Van Driest provides a model for the increase in mixing length through this region, by6
 + lm = y 1 exp( y =26) : \relax \special {t4ht=
Finally, in the inertial sub-layer for eqn, flow is turbulent and the velocity profile is described by the logarithmic law of the wall, often abbreviated to simply the log law, according to7
|------------------| | + 1 + | u = - ln(y ) + B| ------------------- \relax \special {t4ht=
The equation includes Kármán’s constant eqn and constant eqn. For a smooth wall, eqn8 – 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 eqn at the wall. In the viscous sub-layer, the shear stress is laminar, so
-w-= u2 = @ux-: @y \relax \special {t4ht=
This equation integrates with a zero constant of integration to give eqn, from which Eq. (7.11 ) is derived. In the inertial sub-layer, the shear stress is turbulent (laminar is negligible), so
 -w-= u2 = @ux- = l2 @ux- @ux-: t @y m @y @y \relax \special {t4ht=
Assuming eqn gives eqn , 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
u = (

5Joseph Kestin and Peter Richardson, Heat transfer across turbulent, incompressible boundary layers, 1963.
6Edward van Driest, On turbulent flow near a wall, 1956.
7Alternatively written eqn, where eqn.
8Hermann Schlichting and Klaus Gersten, Boundary-layer theory, 2017.

Notes on CFD: General Principles - 7.4 Turbulent boundary layers