Notes on CFD: General Principles

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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 $\text{[math]}$ . With turbulent boundary layers, the $\text{[math]}$ 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 $\text{[math]}$ 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 $\text{[math]}$ from a relatively inaccurate $\text{[math]}$ calculation at the wall.

$PICT\relax \special {t4ht=$

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

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

Standard wall function

The standard wall function for a “smooth” wall calculates $\text{[math]}$ for each patch face based on the near-wall $\text{[math]}$ . No adjustment is made to $\text{[math]}$ when $\text{[math]}$ corresponds to the viscous sub-layer. When $\text{[math]}$ corresponds to the inertial sub-layer, $\text{[math]}$ is calculated by

! -----y+P----- + + t = ln(y+ )+ B 1 for yP > ytr: P \relax \special {t4ht=

(7.17)

The condition ( $\text{[math]}$ ) that determines whether $\text{[math]}$ lies within the inertial sub-layer corresponds to $\text{[math]}$ at the intersection of Eq. (7.11 ) and Eq. (7.13 ), calculated iteratively as

y+ = ln(y+)= + B 11: tr tr \relax \special {t4ht=

(7.18)

$\text{[math]}$ is calculated in each near-wall cell according to:

1=2 ! y+ = c1= 4 k--y- : P P \relax \special {t4ht=

(7.19)

This expression is derived from Eq. (7.9 ) for $\text{[math]}$ and Eq. (7.16 ). The subscript $\text{[math]}$ denotes all properties are evaluated at P.

The wall function Eq. (7.17) is derived from the notion that $\text{[math]}$ is calculated numerically (assuming a stationary wall) by: