Navier-Stokes-driven analysis of mean and fluctuating wall shear stress in turbulent channel flow

Abstract

We propose a Navier-Stokes-driven analysis of the mean and fluctuating wall shear stress (WSS) applied to turbulent channel flow data from direct numerical simulations at friction Reynolds numbers up to $Re_τ\approx 2000$. Starting from the streamwise momentum equation, we derive exact integral equations that relate the square plane-average and the square fluctuating WSS to wall-normal integrals of terms combining shear with acceleration, shear with pressure-radient, and shear with viscous diffusion. The square plane-average WSS can be well approximated by the product of plane-average shear and plane-average acceleration integrated over the buffer layer with corrections from the mean pressure gradient which diminish as the reciprocal of the Reynolds number. The square fluctuating WSS is similarly well approximated by the shear-acceleration and shear-pressure-gradient covariances integrated over the buffer layer, but the latter increases in magnitude with Reynolds number and is therefore not negligible. The acceleration fluctuations around the plane-average acceleration consist of a local Eulerian fluctuating acceleration, an advective acceleration and a term which gives rise a turbulence production contribution to the shear-acceleration covariance. By Taylor's frozen turbulence hypothesis the Eulerian acceleration and the streamwise mean advection part of the advective acceleration cancel each other. The shear-acceleration covariance is characterised by a near-wall peak which results from turbulence production and, more specifically, sweeps.