Page 425 - Bird R.B. Transport phenomena

§13.1 Time-smoothed equations of change for incompressible nonisothermal flow

§13.2 The time-smoothed temperature profile near a wall
§13.3 Empirical expressions for the turbulent heat flux
§13.4° Temperature distribution for turbulent flow in tubes
§13.5° Temperature distribution for turbulent flow in jets
§13.6* Fourier analysis of energy transport in tube flow at large Prandtl numbers

In Chapters 10 to 12 we have shown how to obtain temperature distributions in solids
and in fluids in laminar motion. The procedure has involved solving the equations of
change with appropriate boundary and initial conditions.
We now turn to the problem of rinding temperature profiles in turbulent flow. This
discussion is quite similar to that given in Chapter 5. We begin by time-smoothing the
equations of change. In the time-smoothed energy equation there appears a turbulent
(0
heat flux q , which is expressed in terms of the correlation of velocity and temperature
(0
fluctuations. There are several rather useful empiricisms for q , which enable one to pre-
dict time-smoothed temperature distributions in wall turbulence and in free turbulence.
We use heat transfer in tube flow to illustrate the method.
The most apparent influence of turbulence on heat transport is the enhanced trans-
port perpendicular to the main flow. If heat is injected into a fluid flowing in laminar
flow in the z direction, then the movement of heat in the x and у directions is solely by
conduction and proceeds very slowly. On the other hand, if the flow is turbulent, the heat
"spreads out" in the x and у directions extremely rapidly. This rapid dispersion of heat is
a characteristic feature of turbulent flow. This mixing process is worked out in some de-
tail here for flow in tubes and in circular jets.
Although it has been conventional to study turbulent heat transport via the time-
smoothed energy equation, it is also possible to analyze the heat flux at a wall by use of a
Fourier transform technique without time-smoothing. This is set forth in the last section.

513.1 TIME-SMOOTHED EQUATIONS OF CHANGE FOR
INCOMPRESSIBLE NONISOTHERMAL FLOW
In §5.2 we introduced the notions of time-smoothed quantities and turbulent fluctua-
tions. In this chapter we shall be primarily concerned with the temperature profiles. We
introduce the time-smoothed temperature T and temperature fluctuation V, and write
analogously to Eq. 5.2-1
T = f + Г (13.1-1)
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