Geoscience Reference
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Figure 13.2 A Gaussian probability density made dimensionless with its standard
deviation (left) and the corresponding probability distribution (right). Adapted
from Lumley and Panofsky ( 1964 ).
13.3.2 Observed probability densities
We defined the probability density through ensemble averaging, but in dealing with
turbulence data we typically use time averaging of a stationary time series. One can
determine the probability density of a measured signal by dividing its amplitude
range into bins, and then, for each sample of the signal, adding a count to the proper
bin. As the number of samples becomes large, the result, when properly scaled
(Problem 13.1) , converges to the probability density. With fields from numerical
turbulence simulation one can average over statistically homogeneous directions
within a “snapshot” - a field at a given time. Averaging over a small ensemble of
such snapshots can be sufficient to generate reliable statistics.
13.3.2.1 Velocity derivatives
Figure 13.3 shows probability densities of the first two time derivatives of the
streamwise velocity fluctuation u(t) measured at a point in a laboratory flowof mod-
erate turbulence Reynolds number R t . We typically interpret these as proportional
to streamwise spatial derivatives, as suggested by G. I. Taylor ( 1938 ):
If the velocity of the air stream which carries the eddies is very much greater than the
turbulent velocity, one may assume that the sequence of changes in u at the fixed point are
simply due to the passage of an unchanging pattern of turbulent motion over the point, i.e.
one may assume that
u
=
f(t)
=
f(x/U),
(7)
where x is measured upstream at time t
=
0 from the fixed point where u is measured. In
the limit when u/U
0 Eq. (7) is certainly true.
 
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