7.13 Thermal boundary layers

In a turbulent boundary layer, the distribution of temperature is similar to velocity, with the viscous and intertial sub-layers, separated by a buffer layer, as discussed in Sec. 7.4 .

PICT\relax \special {t4ht=

By analogy with eqn, Eq. (7.10 ), we define friction temperature as

T = -qw- y-; Pr y+ \relax \special {t4ht=
The wall layer is then described by a dimensionless temperature
 + T----Tw- T = T ; \relax \special {t4ht=
where eqn is the fluid temperature at the wall. Ignoring heat generation by viscous stresses, the profile in the viscous sub-layer is described by the relation
|-----------| | + + | T---=-Pr-y-- \relax \special {t4ht=
The profile in the inertial sub-layer is commonly described by a log law for eqn
|--------------------| | + Prt + | T = - -- lny + BT | --------------------- \relax \special {t4ht=
The derivation of Eq. (7.51 ) and Eq. (7.52) assumes a constant heat flux across the profile, equating to eqn at the wall. In the viscous sub-layer, the heat flux is laminar so eqn and
 1 @T T = -----+: Pr @y \relax \special {t4ht=
This equation integrates between eqn at a distance eqn from the wall to eqn at the wall, to yield Eq. (7.51). In the inertial layer, the heat flux is assumed turbulent eqn and
 t 1 @T T = --Pr- @y+-: t \relax \special {t4ht=
Combining Eq. (6.24), Eq. (7.9) and Eq. (7.15 ) yields the ratio eqn. Substituting in Eq. (7.54 ) and integrating then leads to Eq. (7.52 ) where eqn is the constant of integration.

The constant eqn is generally considered to be a function of eqn. A reasonable approximation for this function is20

BT = 13:7 Pr2=3 7:5: \relax \special {t4ht=
Another function, commonly used in thermal wall functions, is eqn, where eqn is the function of eqn :21
 h 3=4 i P = 9:24 (Pr ) 1 [1 + 0:28 exp ( 0:007Pr )]: \relax \special {t4ht=
The expression for eqn uses the coefficient eqn from Eq. (7.11). These constants of integration are sometimes subsumed within the log function as a coefficient “eqn” in the log law expressions, as the footnote on page 483 explains.
20Hermann Schlichting and Klaus Gersten, Boundary-layer theory, 2017.
21Chandra Jayatilleke, The influence of Prandtl number and surface roughness on the resistance of the laminar sub-layer to momentum and heat transfer, 1966.

Notes on CFD: General Principles - 7.13 Thermal boundary layers