For isotropic turbulence, we can write
Using (16.4), (16.5) and (16.6) in Eqn. (16.3) we get,
 |
(16.7) |
For isotropic turbulence one can write (see page 189 of J. O. Hinze, Turbulence, Second Edition, McGraw Hill Book Company, 1987)
 |
(16.8) |
Where , f(r) is the longitudinal correlation function. The function depends on the relative position vector r . The function under consideration will depend on the scalar magnitude of r .
The general relation stated above in Eqn. (16.8) can be written for specific values of i, j, k and l
|
if  |
(16.9) |
|
if  |
(16.10) |
|
If  |
(16.11) |
| = 0 for all other combinations |
(16.12) |
Using the above specific relations one can evaluate the terms of Eqn. (16.7) in the following way
Invoking (16.13), (16.14) and (16.15) into the Eqn. (16.7) we obtain
Therefore, when turbulence tends to be isotropic, we can write
 |
(16.16) |
|
