|
Further on Laws of Averaging
- However, the fluctuating components do not bring about the bulk displacement of a fluid elements. The instantaneous displacement is u'dt, and that is not responsible for the bulk motion. We can conclude from the above
![](images/image002.gif) |
(7.1) |
Due to the interaction of fluctuating components, macroscopic momentum transport takes place. Therefore, interaction effect between two fluctuating components over a long period is non-zero and this can be expressed as
Taking time average of these two integrals and write
![](images/image006.gif) |
(7.2) |
and
![](images/image008.gif) |
(7.3) |
- Now, we can make a general statement with any two fluctuating parameters, say, with f ' and g' as
![](images/image010.gif) |
(7.4) |
![](images/image012.gif) |
(7.5) |
The time averages of the spatial gradients of the fluctuating components also follow the same laws, and they can be written as
![](images/image014.gif) |
(7.6) |
|