## 7.13Thermal 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 .

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

 (7.49)
The wall layer is then described by a dimensionless temperature
 (7.50)
where 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
 (7.51)
The proﬁle in the inertial sub-layer is commonly described by a log law for
 (7.52)
The derivation of Eq. (7.51 ) and Eq. (7.52) assumes a constant heat ﬂux across the proﬁle, equating to at the wall. In the viscous sub-layer, the heat ﬂux is laminar so and
 (7.53)
This equation integrates between at a distance from the wall to at the wall, to yield Eq. (7.51). In the inertial layer, the heat ﬂux is assumed turbulent and
 (7.54)
Combining Eq. (6.24), Eq. (7.9) and Eq. (7.15 ) yields the ratio . Substituting in Eq. (7.54 ) and integrating then leads to Eq. (7.52 ) where is the constant of integration.

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

 (7.55)
Another function, commonly used in thermal wall functions, is , where is the function of :21
 (7.56)
The expression for uses the coeﬃcient from Eq. (7.11). These constants of integration are sometimes subsumed within the log function as a coeﬃcient “” 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