Geoscience Reference
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From Eqs. (14.12 )and (14.18) we can write , the rate of dissipation of TKE,
under the assumption of local isotropy as (Problem 14.1)
15 ν ∂u 1
∂x 1
2
= νM ij ij
=
.
(14.19)
The variance of the streamwise derivative of u 1 is typically evaluated from the
variance of the time derivative through Taylor's hypothesis (Chapter 2) ,
∂u 1
∂x 1
2
U 2 ∂u 1
2
1
.
(14.20)
∂t
The rate of molecular destruction of u i u k , Eq. (14.12) , becomes with the local
isotropy form (14.18)
10 ν ∂u 1
∂x 1
2
2 νM ij kj
=
δ ik .
(14.21)
This says that under the assumption of local isotropy the velocity variances but not
the turbulent shear stress components are dissipated.
The rate of molecular destruction of scalar flux can be written as (Appendix,
Chapter 5 )
ν) ∂u i
∂x j
∂c
∂x j .
χ u i c =
+
(14.22)
As a contraction of a third-order, single-point tensor determined by the dissipative-
range structure, this vanishes under local isotropy.
The rate of destruction of scalar variance, Eq. (5.7) , involves a contraction of a
scalar gradient covariance. Under isotropy this covariance becomes
∂c
∂x 1
2
∂c
∂x i
∂c
∂x j =
δ ij .
(14.23)
14.4.3 Higher order examples
The number of terms in the expressions for isotropic, single-point velocity derivative
tensors increases rapidly with the order of the tensor. We saw that at fourth order
there are three terms; at sixth order there are 15. Finding the constraints needed to
evaluate the constants and doing the algebra can become quite tedious. As kindly
noted by Champagne ( 1978 ), the author worked out the isotropic forms of two
sixth-order tensors in the budget of vorticity variance (Problem 5.6) :
 
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