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Within wall turbulence, there is a sublayer where the mean wall-normal fluxof the streamwise momentum is constant and related to the logarithmicwall-normal profile of the mean streamwise velocity. This relation, i.e., thelaw of the wall, has been used to estimate the mean stress at the wall surface.However, the momentum flux exhibits large temporal fluctuations. To relate themtheoretically to those of the streamwise velocity at the same position from thewall, we consider an orthogonal decomposition of the fluctuations on a plane ofthe streamwise and wall-normal velocities. Since a large timescale is expectedfor the component that would dominate the momentum flux, it is singled out bytemporal smoothing. The resultant formula is consistent with time-series dataof a boundary layer in a wind tunnel. We also extend the formula to thermallystratified cases.

Momentum flux fluctuations in wall turbulence: a formula beyond the law of the wall