Reynolds averaging ($f=\overline{f}+f^{\prime },\overline{f^{\prime }}=0$)

Reynolds averaging separates a resolved mean from zero-mean fluctuations. The idealized averaging operation is linear and idempotent, commutes with the derivatives under consideration and treats resolved mean factors as constant inside its averaging scale. Products then satisfy $\overline{fg}=\overline{f}\overline{g}+\overline{f^{\prime }{g}^{\prime }}$. Actual finite spatial filters need scale separation or corrections when these rules are only approximate.