Page 435 - Bird R.B. Transport phenomena

§13.6 Fourier Analysis of Energy Transport in Tube Flow at Large Prandtl Numbers 417

ficients estimated from experiments. In this section we analyze a turbulent energy trans-
port problem without time-smoothing—that is, by direct use of the energy equation with
fluctuating velocity and temperature fields. The Fourier transform 1 is well suited for
such problems, and the "method of dominant balance" 2 gives useful information with-
out detailed computations.
The specific question considered here is the influence of the thermal diffusivity, a =
k/pC , on the expected distribution and fluctuations of the fluid temperature in turbulent
p
3
forced convection near a wall. This topic was discussed in Example 13.3-1 by an approx-
imate procedure.
Let us consider a fluid with constant p, C , and к in turbulent flow through a tube of
p
inner radius R = \D. The flow enters at z — — °° with uniform temperature T l and exits at
z = L. The tube wall is adiabatic for z < 0, and isothermal at T for 0 < z < L. Heat con-
o
duction in the z direction is neglected. The temperature distribution T(r, Q, z, t) is to be
analyzed in the long-time limit, in the thin thermal boundary layer that forms for z > 0
when the molecular thermal diffusivity a is small (as in a Newtonian fluid when the
Prandtl number, Pr = C fi/k = fi/pa, is large). A stretching function к(а) will be derived
p
for the average thickness of the thermal boundary layer without introducing an eddy
{l)
thermal diffusivity a .
In the limit as a —> 0, the thermal boundary layer lies entirely within the viscous sub-
layer, where the velocity components are given by truncated Taylor expansions in the
distance у = R — r from the wall (compare these expansions with those in Eqs. 5.4-8 to 10)
v = р у + О(у ) (13.6-1)
2
0 в
2
v = p y + O(y ) (13.6-2)
z z
Here the coefficients fi e and /3 are treated as given functions of 6, z, and t. These velocity
2
expressions satisfy the no-slip conditions and the wall-impermeability condition at у = 0
and the continuity equation at small y, and are consistent with the equation of motion to
the indicated orders in y. The energy equation can then be written as

2
2
Рв дТ , дТ\ (l^Pe. Wz\ У дТ д Т ~. .
R п А
2
with the usual boundary layer approximation for V T, and with the following boundary
conditions on T(y, в, z, t):
Inlet condition: atz = 0, T{y, в, 0, t) = T, forO <y< R (13.6-5)
Wallcondition: at у = 0, ПО, в, z, t) = T o forO < z < L (13.6-6)
The initial temperature distribution T(y, в, z, 0) is not needed, since its effect disappears
in the long-time limit.
To obtain results asymptotically valid for a —» 0, we introduce a stretched coordi-
nate У = у/к(а), which is the distance from the wall relative to the average boundary
layer thickness к(а). The range of У is from 0 at у = 0 to °° at у = R in the limit as a —» 0.

R. N. Bracewell, The Fourier Transform and its Applications, 2nd edition, McGraw-Hill, New York
1
(1978).
2
This method is well presented in С. М. Bender and S. A. Orzag, Advanced Mathematical Methods for
Scientists and Engineers, McGraw-Hill, New York (1978), pp. 435-437.
W. E. Stewart, AIChE Journal, 33, 2008-2016 (1987); errata, ibid., 34,1030 (1988); W. E. Stewart and
3
D. G. O'Sullivan, AIChE Journal (to be submitted).