Geoscience Reference
In-Depth Information
x
=
l
1
x
+
m
1
y
+
n
1
z,
y
=
l
2
x
+
m
2
y
+
n
2
z,
(14.1)
z
=
l
3
x
+
m
3
y
+
n
3
z,
where the parameters
l, m,
and
n
are
direction cosines
- the cosines of the angles
between the old and new axes. (
l
1
is the cosine of the angle between
Ox
and
Ox
,
m
1
that between
Oy
and
Ox
,…,
l
2
the cosine of the angle between
Ox
and
Oy
,….)
We can also invert
Eq. (14.1)
to express
x,y,
and
z
in terms of the primed quantities.
In cartesian tensor notation we write these transformations more compactly as
x
j
a
ij
x
j
;
=
a
ij
x
i
,
i
=
a
ik
a
jk
=
δ
ij
.
(14.2)
Quantities (such as velocity
u
i
) that transform in this way are called
tensors of the
first order
,or
vectors
. A set of nine quantities
w
ik
referred to a set of axes and
transformed to another set by the rule
w
jl
=
a
ij
a
kl
w
ik
(14.3)
is called a tensor of the second order.
The covariance
s w
e meet in turbulence are tensors. Exa
mpl
es include the scalar-
scalar covariance
ab
, a zero-order tensor; the scalar flux
cu
i
and the
mean
scalar
gradient
∂C/∂x
i
, first-order tensors; and the kinematic Reynolds stress
u
i
u
j
and the
mean strain rate
∂U
i
/∂x
j
, second-order tensors. Yet-higher-order tensors appear in
the molecular destruction terms in the moment equations of
Chapter 5
.
These are all
single-point
tensors - tensors that involve only one spatial point.
Multi-point
tensors involve more than one spatial point.
14.3 Determining the form of isotropic tensors
An isotropic tensor is one whose components do not change under rotation, transla-
tion, or reflection of the coordinate axes. There are two ways to determine its form:
through the process described by Jeffreys, which uses the transformation rules; and
by a technique due to Robertson and described by
Batchelor
(
1960
). The first was
also used by
Taylor
(
1935
); it is quite tedious. We will use the latter.
14.3.1 Single-point tensors
Since a vector's components change under coordinate rotation, the only isotropic
vector is the zero vector. Put another way, a nonzero isotropic vector would have a
direction, and that would be in conflict with isotropy. Thus, the turbulent flux and