Wall shear stress fluctuations: Mixed scaling and their effects on velocity fluctuations in a turbulent boundary layer

The present work investigates numerically the statistics of the wall shear stress fluctuations in a turbulent boundary layer (TBL) and their relation to the velocity fluctuations outside of the near-wall region. The flow data are obtained from a Direct Numerical Simulation (DNS) of a zero pressure-gradient TBL using the high-order flow solver Incompact3D [S. Laizet and E. Lamballais, “High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy,” J. Comput. Phys. 228(16), 5989 (2009)]. The maximum Reynolds number of the simulation is $Re𝜃≈2000$⁠, based on the free-stream velocity and the momentum thickness of the boundary layer. The simulation data suggest that the root-mean-squared fluctuations of the streamwise and spanwise wall shear-stress components $τx$ and $τz$ follow a logarithmic dependence on the Reynolds number, consistent with the empirical correlation of Örlü and Schlatter [R. Örlü and P. Schlatter, “On the fluctuating wall-shear stress in zero pressure-gradient turbulent boundary layer flows,” Phys. Fluids 23, 021704 (2011)]. These functional dependencies can be used to estimate the Reynolds number dependence of the wall turbulence dissipation rate in good agreement with reference DNS data. Our results suggest that the rare negative events of $τx$ can be associated with the extreme values of $τz$ and are related to the presence of coherent structures in the buffer layer, mainly quasi-streamwise vortices. We also develop a theoretical model, based on a generalisation of the Townsend-Perry hypothesis of wall-attached eddies, to link the statistical moments of the filtered wall shear stress fluctuations and the second order structure function of fluctuating velocities at a distance y from the wall. This model suggests that the wall shear stress fluctuations may induce a higher slope in the turbulence energy spectra of streamwise velocities than the one predicted by the Townsend-Perry attached-eddy model.