Geoscience Reference
In-Depth Information
14
Isotropic tensors
14.1 Introduction
An
isotropic
turbulence field, a concept introduced by G. I. Taylor (1935), has
statistical properties that are independent of translation, rotation, and reflection of
the coordinate axes.
The covariance equations in
Chapter 5
include production terms involving gradi-
ents of the mean velocity and mean scalar fields. As we shall see, these mean-field
gradients are
anisotropic
and cause the energy-containing range of shear-driven
turbulence to be anisotropic as well. Likewise, we'll see that buoyancy-driven
turbulence is anisotropic. Thus it appears unlikely that equilibrium isotropic tur-
bulence exists naturally.
†
It can be produced through DNS, however, by adding a
random-force term to the Navier-Stokes equation
(Problem 14.14)
.
As we discussed in
Chapter 7
, the hypothesis of
local isotropy
,orisotropyofthe
smallest-scale structure of a large-
R
t
turbulent flow, is due to
Kolmogorov
(
1941
).
Because it greatly simplifies the specification of the fine structure of turbulent
flows, it is widely used in analytical, observational, and numerical studies. But
Kolmogorov pointed out that the arguments for local isotropy, which we'll present
in his words in
Section 14.5
, are largely physical ones. Today the observational
evidence against it is compelling, and the physics underlying local
anisotropy
is an
active area of research.
14.2 Cartesian tensors
Jeffreys
(
1961
) discusses basic concepts of cartesian tensors. If we have two sets
of rectangular axes with the same origin,
(Ox, Oy, Oz)
and
(Ox
,Oy
,Oz
)
,then
we can write the transformations
†
Betchov
(
1957
) did attempt to generate isotropic turbulence with a multiplicity of small, colliding air jets.
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