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

(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

(7.2)

and

(7.3)

• Now, we can make a general statement with any two fluctuating parameters, say, with f ' and g' as

	(7.4)
	(7.5)

The time averages of the spatial gradients of the fluctuating components also follow the same laws, and they can be written as

(7.6)