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Figure 2.3 Realizations of effluent plumes contrasted with the ensemble-averaged
plume.
2.2.2 Does the ensemble-averaged field exist?
An instantaneous effluent plume from a point source
(Figure 2.3)
is sinuous and
irregular. Measurements show that near the source the effluent concentration is
roughly uniform within it and zero outside it. But the ensemble-averaged plume is
diffuse and smooth; in homogeneous, wind-tunnel turbulence its mean concentra-
tion profile is found to be Gaussian. This suggests that the ensemble-averaged field
is unlikely to exist in any realization of a turbulent flow, even for an instant.
2.2.3 The ensemble-averaging rules
The ensemble average (we shall use
average
and
mean
interchangeably) has
convenient properties that are commonly called the
Reynolds averaging
rules:
†
• The average of a sum is the sum of the averages (the
distributive
property):
+
b
+
b.
a
˜
= ˜
a
(2.5)
†
Reynolds
(
1895
) used a volume average but assumed it follows the rules for the ensemble average, so in the
turbulence community they now bear his name. We'll discuss the volume averaging rules in
Chapter 3
.
