## 7.14Thermal wall functions

Wall functions were introduced in Sec. 7.5 in order to improve the calculation of wall shear stress when cells are too large near a wall to resolve accurately. The same problem exists with heat ﬂux and an under-predicted . As before, the universal character of the boundary layer can be exploited, this time to improve the calculation of .

The temperature distribution is characterised by Eq. (7.51 ) for the viscous sub-layer, and the log law Eq. (7.52 ) for the inertial sub-layer. The transition for occurs at the intersection of the two equations, i.e. when

 (7.57)
While the transition between these regimes is ﬁxed at by Eq. (7.18 ) for the proﬁle, the (iterative) solution of Eq. (7.57 ) is dependent on and .

Using and Eq. (7.55 ) for , for air at with . For water under the same conditions, and the corresponding .

A wall function can be derived which adjusts the turbulent conductivity , in a similar manner to in the standard wall function in Sec. 7.5 . The model calculates for each patch face based on the near-wall cell .

No adjustment is made to when corresponds to the viscous sub-layer. When corresponds to the inertial sub-layer, is calculated as

 (7.58)
where denotes the laminar thermal conductivity, i.e. from Eq. (2.54 ), to distinguish it from Kármán’s constant . Eq. (7.58 ) uses from Eq. (7.19 ), as in the standard wall function.

The wall function is derived based on adjusting to improve the numerical calculation of by

 (7.59)
where represents a value close to the wall, e.g. in a near-wall cell. By comparison, Eq. (7.49 ) and Eq. (7.50 ) combine to give
 (7.60)
The heat ﬂux is consistent between Eq. (7.59 ) and Eq. (7.60 ) when
 (7.61)
When the log law Eq. (7.52 ) is then substituted in Eq. (7.61 ), it provides the thermal wall function model which adjusts according to Eq. (7.58 ).
Notes on CFD: General Principles - 7.14 Thermal wall functions