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
.
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 is
|
(7.55) |
Another function, commonly used in thermal wall functions, is
,
where
is the function of
:
![h 3=4 i P = 9:24 (Pr ) 1 [1 + 0:28 exp ( 0:007Pr )]: \relax \special {t4ht=](img/index3375x.png)
|
(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.
