7.5 Wall functions

CFD simulations may be used to calculate the forces on solid bodies exerted by the fluid, e.g. in aerodynamics. The wall shear stress is then calculated according to eqn. With turbulent boundary layers, the eqn calculation requires cells with very small lengths normal to the wall to be accurate. The resulting mesh is inevitably large, which carries a high computational cost. The problem for CFD is how to calculate eqn with sufficient accuracy, but at an affordable cost.

Wall functions provide a solution to this problem by exploiting the universal character of the velocity distribution described in Sec. 7.4 . They use the law of the wall Eq. (7.13 ) as a model to provide a reasonable prediction of eqn from a relatively inaccurate eqn calculation at the wall.

PICT\relax \special {t4ht=

Wall functions use the near-wall cell centre height eqn, i.e. the distance to the wall from the centre P of each near-wall cell. Typically when using wall functions, eqn should correspond to a eqn within the typical range of applicability of the log law Eq. (7.13 ), i.e. eqn.

With such a mesh, the calculated eqn is then significantly lower than its true value. Wall function models compensate for the resulting error in the prediction of eqn by increasing viscosity at the wall. The increase is applied to eqn at the wall patch faces, which would otherwise be eqn, corresponding to eqn.

Standard wall function

The standard wall function for a “smooth” wall calculates eqn for each patch face based on the near-wall eqn. No adjustment is made to eqn when eqn corresponds to the viscous sub-layer. When eqn corresponds to the inertial sub-layer, eqn is calculated by

 ! -----y+P----- + + t = ln(y+ )+ B 1 for yP > ytr: P \relax \special {t4ht=
The condition (eqn) that determines whether eqn lies within the inertial sub-layer corresponds to eqn at the intersection of Eq. (7.11 ) and Eq. (7.13 ), calculated iteratively as
y+ = ln(y+)= + B 11: tr tr \relax \special {t4ht=
eqn is calculated in each near-wall cell according to:
 1=2 ! y+ = c1= 4 k--y- : P P \relax \special {t4ht=
This expression is derived from Eq. (7.9 ) for eqn and Eq. (7.16 ). The subscript eqn denotes all properties are evaluated at P.

The wall function Eq. (7.17) is derived from the notion that eqn is calculated numerically (assuming a stationary wall) by:

-w- @ux- ux- = eff @y = ( + t) y : \relax \special {t4ht=
At the same time, combining Eq. (7.9 ) and Eq. (7.10 ) gives
-w- y+-ux- = u+ y : \relax \special {t4ht=
Comparing Eq. (7.20 ) and Eq. (7.21 ) gives eqn , which combines with Eq. (7.13 ) to yield Eq. (7.17).
Notes on CFD: General Principles - 7.5 Wall functions