The instantaneous stream-wise velocity of a turbulent flow is given by $u(x,y,z,t)=\bar{u}(x,y,z)+u'(x,y,z,t)$. The time-average of the fluctuating velocity $u'(x,y,z,t)$ is

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zero. In Reynolds decomposition, the instantaneous velocity is split into a mean part and a fluctuation part. By definition, the time average of the fluctuation is zero.

The instantaneous stream-wise velocity of a turbulent flow is given by \(u(x,y,z,t)=\bar{u}(x,y,z)+u'(x,y,z,t)\). The time-average of the fluctuating velocity \(u'(x,y,z,t)\) is

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