Isotropic tensors - Turbulence in the Atmosphere - page 319

∂ 2 u 1
∂x 1 ∂x 1

2

δ ij δ kn δ lm +

∂u i
∂x k ∂x m

∂u j
∂x l ∂x n =

δ ij δ ln δ km + δ ij δ kl δ mn

6 δ il δ jn δ km +

1

− δ il δ kn δ jm + δ jl δ in δ km + δ jl δ kn δ im

+ δ kl δ in δ jm + δ kl δ jn δ im + δ ik δ jl δ mn + δ jk δ il δ mn + δ ik δ jn δ lm

δ jk δ ln δ im . + δ ik δ ln δ jm + δ jk δ in δ lm + (14.24)

∂u 1
∂x 1

3

δ ij δ kl δ mn −

3 δ ij δ km δ ln + δ ik δ lj δ mn + δ im δ jn δ kl

∂u i
∂x j

∂u k
∂x l

∂u m 4

∂x n =

6 δ ij δ kn δ lm + δ il δ kj δ mn + δ in δ kl δ jm

1

−

4 δ il δ kn δ jm +

δ in δ kj δ lm +

3

− δ il δ km δ jn + δ in δ km δ lj

δ im δ jk δ ln .

+

δ ik δ lm δ jn + δ ik δ ln δ jm + δ im δ jl δ kn +

(14.25)

14.5 Local isotropy
14.5.1 The concept
The concept of local isotropy is credited to Kolmogorov ( 1941 ). His short paper
appears in the Friedlander-Topper (1961) collection as a rather rough English trans-
lation. After defining locally isotropic turbulence in the formal terms of probability
statistics, he wrote:
It is natural that in so general and somewhat indefinite a formulation the just advanced
proposition cannot be rigorously proved. We may indicate here only certain general con-
siderations speaking for [it]. For very large R [Reynolds number] the turbulent flow may
be thought of in the following way: on the [ensemble] averaged flow… are superposed the
“pulsations of the first order” consisting in disorderly displacements of separate fluid vol-
umes, one with respect to another, of diameters of the order of magnitude ( 1 ) (where

is Prandtl's mixing [length]); the order of magnitude … of these relative velocities we
denote by v ( 1 ) . The pulsations of the first order are for very large R in their turn unsteady,
and on them are superposed the pulsations of second order with mixing [length] ( 2 ) < ( 1 )
and relative velocities v ( 2 ) <v ( 1 ) ; such a process of successive refinement of turbulent
pulsations may be carried until for the pulsations of some sufficiently large order n the
Reynolds number

=

(n) v (n)
ν
becomes so small that the effect of viscosity on the pulsations of the order n finally prevents
the formation of pulsations of the order n

R (n) =

1.
From the energetical point of view it is natural to imagine the process of turbulent mixing
in the following way: the pulsations of the first order absorb the energy of the motion and

+