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

TL;DR

This work develops a Navier–Stokes–driven framework to quantify mean and fluctuating wall shear stress (WSS) in turbulent channel flow using direct numerical simulation data up to $Re_\tau \approx 2000$. By deriving exact plane-average and plane-fluctuating NS-based integral relations, the authors show that the mean WSS is captured by near-wall shear–acceleration and pressure–velocity terms within the buffer layer, with a $1/Re_\tau$-scaled pressure correction, and that the variance of WSS fluctuations is governed by covariances involving shear with acceleration and pressure gradient, also largely confined to the buffer layer. A Taylor’s frozen turbulence analysis reveals cancellations between Eulerian acceleration and mean advection contributions, while the near-wall peak in the fluctuations is traced to production-dominated sweeps (and to a lesser extent ejections). Overall, the buffer layer encodes sufficient information to estimate both the mean and fluctuating WSS from NS statistics, though outer-flow effects via large-scale structures remain relevant at higher Reynolds numbers. These insights provide a NS-consistent decomposition of skin friction and its fluctuations that can inform modeling of wall-bounded turbulence and related applications.

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.

Figures (10)

• Figure 1: Contribution to the square plane-average wall shear stress in equation (\ref{['eq:pavg']}).
• Figure 2: Percentage contribution from shear-acceleration plus shear-pressure-gradient to ($a$) plane-average and ($b$) fluctuating wall shear stresses. Results from five Reynolds numbers collapse rather well.
• Figure 3: Mean acceleration profile.
• Figure 4: rms of fluctuating wall shear stress and local shear at channel centre normalised by bulk quantities ($a,b$) and by inner quantities ($c,d$)
• Figure 5: Contributions to the square plane-fluctuating wall shear stress in equation (\ref{['eq:pfluc']}).