Page 426 - Bird R.B. Transport phenomena

408 Chapter 13 Temperature Distributions in Turbulent Flow

Clearly T averages to zero so that T = 0, but quantities like v' T, v' T, and v' T' will not
z
y
x
be zero because of the "correlation" between the velocity and temperature fluctuations
at any point.
For a nonisothermal pure fluid we need three equations of change, and we want to
discuss here their time-smoothed forms. The time-smoothed equations of continuity and
motion for a fluid with constant density and viscosity were given in Eqs. 5.2-10 and 12,
and need not be repeated here. For a fluid with constant /л, p, C , and k, Eq. 11.2-5, when
p
put in the д I dt form by using Eq. 3.5-4, and with Newton's and Fourier's law included,
becomes

in which only a few sample terms in the viscous dissipation term — (T:VV) = дФ г; have
been written (see Eq. B.7-1 for the complete expression).
In Eq. 13.1-2 we replace T by T = T + T, v x by v x + v' , and so on. Then the equation
x
is time-smoothed to give

Щ

dy 2

( 1 3 Л 3 )
"
Comparison of this equation with the preceding one shows that the time-smoothed
equation has the same form as the original equation, except for the appearance of the
terms indicated by dashed underlines, which are concerned with the turbulent fluctua-
tions. We are thus led to the definition of the turbulent heat flux q ( 0 with components
qf = pC^T i#> = pC^T qf = pC ^f (13.1-4)
p
and the turbulent energy dissipation function Ф^:

dXiJ\3x,) + [dxjya

The similarity between the components of q (f) in Eq. 13.1-4 and those of т (0 in Eq. 5.2-8
should be noted. In Eq. 13.1-5, v[, v' 2f and v' are synonymous with v' , v' , and v' , and x u
x
3
y
z
x , and x 3 have the same meaning as x, y, and z.
2
To summarize, we list all three time-smoothed equations of change for turbulent
flows of pure fluids with constant д, р, C , and к in their D/Dt form (the first two were
p
given in Eqs. 5.2-10 and 12):
Continuity (V • v) 0 (13.1-6)
=
Motion p Dv = -Vp - [V • (T (13.1-7)
Di •')] + pg
Energy oC °^- -(V- (q-> ?'>)) + м (13.1-8)
p(
-r Dt +