The instantaneous stream-wise velocity of a turbulent flow is given as follows:

\(u\left( {x,\;y,\;z,\;t} \right) = \bar u\;\left( {x,\;y,\;z} \right) + u'\left( {x,\;y,\;z,\;t} \right)\)

The time-average of the fluctuating velocity u’ (x, y, z, t) is