7.14 Thermal wall functions

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

PICT\relax \special {t4ht=

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 eqn for eqn occurs at the intersection of the two equations, i.e. when

y+ = Prt 1-ln y+ + -1B tr Pr tr Pr T \relax \special {t4ht=
While the transition between these regimes is fixed at eqn by Eq. (7.18 ) for the eqn profile, the (iterative) solution of Eq. (7.57 ) is dependent on eqn and eqn.

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

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

 ! Pr y+P + + t = ` ------+--------- 1 for yP > ytr; PrtlnyP = + BT \relax \special {t4ht=
where eqn denotes the laminar thermal conductivity, i.e. eqn from Eq. (2.54 ), to distinguish it from Kármán’s constant eqn. Eq. (7.58 ) uses eqn from Eq. (7.19 ), as in the standard eqn wall function.

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

 @T- T----Tw- qw = eff@y = ( t + `) y ; \relax \special {t4ht=
where eqn 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
 -y+ T----Tw- qw = T + y Pr `: \relax \special {t4ht=
The heat flux eqn is consistent between Eq. (7.59 ) and Eq. (7.60 ) when
 + ----`-- + T = t + ` Pr y : \relax \special {t4ht=
When the log law Eq. (7.52 ) is then substituted in Eq. (7.61 ), it provides the thermal wall function model which adjusts eqn according to Eq. (7.58 ).
Notes on CFD: General Principles - 7.14 Thermal wall functions