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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
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