Geoscience Reference
In-Depth Information
16
Statistics in turbulence analysis
With some reasonable assumptions and the Fourier-Stieltjes representation we can
gain analytical insights into a wide range of turbulence problems. We'll discuss
several examples in this chapter.
16.1 Evolution equations for spectra
We'll begin with an idealized problem involving a passive, conserved scalar in a
field of isotropic turbulence. With the Fourier-Stieltjes representation we'll convert
the scalar conservation equation to an evolution equation for its power spectral
density, or spectrum. That will give insight into the maintenance of its inertial
subrange, and show what motivated
Obukhov
(
1949
) and, independently,
Corrsin
(
1951
) to propose a Kolmogorov-like similarity hypothesis for it. Then we'll extend
the analysis to a conserved scalar in a horizontally homogeneous turbulent boundary
layer, where scalar fluctuations are generated by the large-
R
t
turbulence acting on
a mean scalar gradient in the vertical direction.
16.1.1 A scalar in steady, isotropic turbulence
16.1.1.1 The spectral equation
Imagine a volume of steady turbulence of large
R
t
and
Co
t
(turbulence Corrsin
number,
Chapter 7
), as in
Chapter 6
,
Section 6.3
but now isotropic. The velocity
field, which has only a fluctuating part
u
i
(
x
,t)
, advects a scalar field
c
:
˜
c
,t
+
˜
(
cu
j
)
,j
˜
=
s(
x
,t)
+
γ
c
,jj
.
˜
(16.1)
s(
x
,t)
is a stochastic, zero-mean, homogeneous, stationary source term that fluctu-
ates between positive values (a source of
c
) and negative values (a sink). If we think
of the scalar as temperature, then the source term represents zero-mean stochastic
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