Turbulent transport for wall shear stress fluctuations

Statistical structure and the underlying energy budget of wall shear stress fluctuations are studied in both Poiseulle and Couette flows with emphasis on its streamwise component. Using a dimensional analysis and direct numerical simulation data, it is shown that the spectra of streamwise wall dissipation for $\lambda \lesssim 1000 \delta_\nu$ are asymptotically invariant with the Reynolds number ($Re$), whereas those for $\lambda \gtrsim \delta$ decay with $Re$ (here, $\lambda$ is a nominal wall-parallel wavelength, and $\delta_\nu$ and $\delta$ are the viscous inner and outer length scales, respectively). The wall dissipation increases with $Re$ due to the increasing contribution of the spectra at $1000 \delta_\nu \lesssim \lambda \lesssim \delta$. The subsequent analysis of the energy budget shows that the near-wall motions associated with these wall dissipation spectra are mainly driven by turbulent transport and are `inactive' in the sense that they contain very little Reynolds shear stress (or turbulence production). As such, turbulent transport spectra near the wall are also found to share the same $Re$-scaling behaviour with wall dissipation, and this is observed in the spectra of both the wall-normal and inter-scale turbulent transports. The turbulent transport underpinning the increase of wall dissipation with $Re$ is characterised by energy fluxes towards the wall, together with inverse energy transfer from small to large length scales along the wall-parallel directions.