Page 168 - Bird R.B. Transport phenomena

Chapter 5

Velocity Distributions

in Turbulent Flow

§5.1 Comparisons of laminar and turbulent flows
§5.2 Time-smoothed equations of change for incompressible fluids
§5.3 The time-smoothed velocity profile near a wall

§5.4 Empirical expressions for the turbulent momentum flux
§5.5 Turbulent flow in ducts
§5.6° Turbulent flow in jets

In the previous chapters we discussed laminar flow problems only. We have seen that
the differential equations describing laminar flow are well understood and that, for a
number of simple systems, the velocity distribution and various derived quantities can
be obtained in a straightforward fashion. The limiting factor in applying the equations of
change is the mathematical complexity that one encounters in problems for which there
are several velocity components that are functions of several variables. Even there, with
the rapid development of computational fluid dynamics, such problems are gradually
yielding to numerical solution.
In this chapter we turn our attention to turbulent flow. Whereas laminar flow is
orderly, turbulent flow is chaotic. It is this chaotic nature of turbulent flow that poses
all sorts of difficulties. In fact, one might question whether or not the equations of
change given in Chapter 3 are even capable of describing the violently fluctuating mo-
tions in turbulent flow. Since the sizes of the turbulent eddies are several orders of
magnitude larger than the mean free path of the molecules of the fluid, the equations
of change are applicable. Numerical solutions of these equations are obtainable and
can be used for studying the details of the turbulence structure. For many purposes,
however, we are not interested in having such detailed information, in view of the
computational effort required. Therefore, in this chapter we shall concern ourselves
primarily with methods that enable us to describe the time-smoothed velocity and
pressure profiles.
In §5.1 we start by comparing the experimental results for laminar and turbulent
flows in several flow systems. In this way we can get some qualitative ideas about the
main differences between laminar and turbulent motions. These experiments help to de-
fine some of the challenges that face the fluid dynamicist.
In §5.2 we define several time-smoothed quantities, and show how these definitions
can be used to time-average the equations of change over a short time interval. These
equations describe the behavior of the time-smoothed velocity and pressure. The time-
smoothed equation of motion, however, contains the turbulent momentum flux. This flux

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