Page 428 - Bird R.B. Transport phenomena

410 Chapter 13 Temperature Distributions in Turbulent Flow

holds at the junction with the viscous sublayer. The latter expression will involve both /x
and k; hence С will necessarily contain \x and k, and will therefore include the dimen-
sionless group Pr = C fi/k. If, in addition, we introduce the dimensionless coordinate
p
yv*/v, then Eq. 13.2-2 can be rewritten as

) l for^>l (13.2-3)
KpC p v*
in which /(Pr) is a function representing the thermal resistance between the wall and the
inertial sublayer. Landau and Lifshitz (see Ref. 1 on page 409) estimate, from a mixing-
3/4
length argument (see Eq. 13.3-3), that, for large Prandtl numbers,/(Pr) = constant - Pr ;
2/ 3
however, Example 13.3-1 implies that the function/(Pr) = constant • Pr is better. Keep
in mind that Eq. 13.2-3 can be expected to be valid only in the inertial sublayer and that it
should not be used in the immediate neighborhood of the wall.

§13.3 EMPIRICAL EXPRESSIONS FOR THE TURBULENT HEAT FLUX
In §13.1 we saw that the time-smoothing of the energy equation gives rise to a turbulent
heat flux q . In order to solve the energy equation for the time-smoothed temperature
(0
profiles, it is customary to postulate a relation between q (/) and the time-smoothed tem-
perature gradient. We summarize here two of the most popular empirical expressions;
more of these can be found in the heat transfer literature.

Eddy Thermal Conductivity
By analogy with the Fourier law of heat conduction we may write

iO
qf = -^ j- (13.3-1)
in which the quantity k U) is called the turbulent thermal conductivity or the eddy thermal
conductivity. This quantity is not a physical property of the fluid, but depends on posi-
tion, direction, and the nature of the turbulent flow.
ш
The eddy kinematic viscosity v {t) = д /р and the eddy thermal diffusivity a {t) =
{t)
k /pC p have the same dimensions. Their ratio is a dimensionless group
,.(0
Pr ( 0 = —- (13.3-2)
or
called the turbulent Prandtl number. This dimensionless quantity is of the order of unity,
(0
values in the literature varying from 0.5 to 1.0. For gas flow in conduits, Pr ranges from
1
0.7 to 0.9 (for circular tubes the value 0.85 has been recommended ), whereas for flow in
jets and wakes the value is more nearly 0.5. The assumption that Pr (0 = 1 is called the
Reynolds analogy.

The Mixing-Length Expression of Prandtl and Taylor
According to Prandtl's mixing-length theory, momentum and energy are transferred in
turbulent flow by the same mechanism. Hence, by analogy with Eq. 5.4-4, one obtains
dT (13.3-3)
dy dy

1 W. M. Kays and M. E. Crawford, Convective Heat and Mass Transfer, 3rd edition, McGraw-Hill,
New York (1993), pp. 259-266.