Page 184 - Bird R.B. Transport phenomena

168 Chapter 5 Velocity Distributions in Turbulent Flow

This result may be solved for /лЯ/д and the result can be expressed in terms of dimensionless
variables:

A dvjdy
I т [1 - (y/R)] - 1
о
f- dvjdy
[1 -

where y + = yv+p/fi and v + = vjv*. When у = R/2, the value of y + is

( 5 5 9 )
For this value of y , the logarithmic distribution in the caption of Fig. 5.5-3 gives
+
TT = Ш = 0-0052 (5.5-10)
=
Ш
4o5
dy
Substituting this into Eq. 5.5-8 gives

This result emphasizes that, far from the tube wall, molecular momentum transport is negli-
gible in comparison with eddy transport.

§5.6 TURBULENT FLOW IN JETS
In the previous section we discussed the flow in ducts, such as circular tubes; such flows
are examples of wall turbulence. Another main class of turbulent flows is free turbulence,
and the main examples of these flows are jets and wakes. The time-smoothed velocity in
these types of flows can be described adequately by using Prandtl's expression for the
eddy viscosity in Fig. 5.4-3, or by using Prandtl's mixing length theory with the empiri-
cism given in Eq. 5.4-6. The former method is simpler, and hence we use it in the follow-
ing illustrative example.

EXAMPLE 5.6-1 A jet of fluid emerges from a circular hole into a semi-infinite reservoir of the same fluid as
depicted in Fig. 5.6-1. In the same figure we show roughly what we expect the profiles of
Time-Smoothed tn z-component of the velocity to look like. We would expect that for various values of z
Velocity Distribution in fa e profiles will be similar in shape, differing only by a scale factor for distance and veloc-
e
a Circular Wall Jer ity. We also can imagine that as the jet moves outward, it will create a net radial inflow so
that some of the surrounding fluid will be dragged along. We want to find the time-
smoothed velocity distribution in the jet and also the amount of fluid crossing each plane of
constant z. Before working through the solution, it may be useful to review the information
on jets in Table 5.1-1.

1
H. Schlichting, Boundary-Layer Theory, McGraw-Hill, New York, 7th edition (1979), pp. 747-750.
2
A. A. Townsend, The Structure of Turbulent Shear Flow, Cambridge University Press, 2nd edition
(1976), Chapter 6.
3
J. O. Hinze, Turbulence, McGraw-Hill, New York, 2nd edition (1975), Chapter 6.
S. Goldstein, Modern Developments in Fluid Dynamics, Oxford University Press (1938), and Dover
4
reprint (1965), pp. 592-597.