Page 433 - Bird R.B. Transport phenomena

§13.5 Temperature Distribution for Turbulent Flow in Jets 415

§13.5 TEMPERATURE DISTRIBUTION FOR
TURBULENT FLOW IN JETS 1

In §5.6 we derived an expression for the velocity distribution in a circular fluid jet dis-
charging into an infinite expanse of the same fluid (see Fig. 5.6-1). Here we wish to ex-
tend this problem by considering an incoming jet with temperature T higher than that
o
of the surrounding fluid TV The problem then is to find the time-smoothed temperature
distribution T(r, z) in a steadily driven jet. We expect that this distribution will be monot-
one decreasing in both the r and z directions.
We start by assuming that viscous dissipation is negligible, and we neglect the con-
(0
tribution q (y) to the heat flux as well as the axial contribution to q . Then Eq. 13.1-8 takes
the time-averaged form

Then we express the turbulent heat flux in terms of the turbulent thermal conductivity
introduced in Eq. 13.3-1:

5
дг * V дг " pL P (0 ~fr ( 1 3 - " 2
Pr
When Eq. 13.5-1 is written in terms of a dimensionless temperature function
5 ~ < 1 3 5 3
1 l
о " i
it becomes

Vr dr z dz) Р г ш r dr V dr j { l ^ V
Here it has been assumed that the turbulent Prandtl number and the turbulent kinematic
viscosity are constants (see the discussion after Eq. 5.6-3). This equation is to be solved
with the boundary conditions:
B.C.I: at 2 = 0, 0 = 1 (13.5-5)
B.C. 2: at r = 0, 0 is finite (13.5-6)
B.C. 3: at r = °o, 0 = 0 (13.5-7)
Next we introduce the expressions for the time-smoothed velocity components v r and v z
in terms of a stream function F(f), as given in Eqs. 5.6-12 and 13, and a trial expression
for the dimensionless time-smoothed temperature function:

©(££) = 7 Д£) (13.5-8)

{i)
Here f = r/z and £ = {pv /w)z, where w is the total mass flow rate in the jet. The pro-
posal in Eq. 13.5-8 is motivated by the expression for v z that was found in Eq. 5.6-21.
When these expressions for the velocity components and the dimensionless temper-
ature are substituted into Eq. 13.5-1, some terms cancel and others can be combined, and
as a result, the following rather simple equation is obtained:
1 Л

1
J. O. Hinze, Turbulence, 2nd edition, McGraw-Hill, New York (1975), pp. 531-546.