Reynolds decomposition

Remarks

From the rules, $\overline{f'} = \bar f - \bar{\bar f} = 0$, and $$\overline{f g} = \bar f \, \bar g + \overline{f' g'}.$$ The last term, the correlation of fluctuations, is in general not zero; for velocity components it gives the Reynolds stresses.

Example: the ensemble average over a probability space of flow realisations satisfies all three rules, since expectation is linear and commutes with derivatives of smooth fields. Non-example: a spatial filter $\bar f = G * f$ with a smooth kernel $G$, as used in large eddy simulation, is linear and commutes with derivatives. But in general $\bar{\bar f} \neq \bar f$, so it fails rule 2 and $\overline{f'} \neq 0$. An average over a finite time window $T$ satisfies the rules only approximately, when $T$ is long compared with the fluctuations and short compared with changes of the mean.