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

Within wall turbulence, there is a sublayer where the mean wall-normal flux of the streamwise momentum is constant and related to the logarithmic wall-normal profile of the mean streamwise velocity. This relation, i.e., the law 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 them theoretically to those of the streamwise velocity at the same position from the wall, we consider an orthogonal decomposition of the fluctuations on a plane of the streamwise and wall-normal velocities. Since a large timescale is expected for the component that would dominate the momentum flux, it is singled out by temporal smoothing. The resultant formula is consistent with time-series data of a boundary layer in a wind tunnel. We also extend the formula to thermally stratified cases.