Turbulent transport for wall-shear-stress fluctuations

Statistical structure and the underlying energy budget of wall-shear-stress fluctuations are studied in both Poiseuille 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 ( ${\textit{Re}}$ ), whereas those for $\lambda \gtrsim \delta$ decay with ${\textit{Re}}$ (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 ${\textit{Re}}$ 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 driven mainly 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 ${\textit{Re}}$ -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 ${\textit{Re}}$ is characterised by energy fluxes towards the wall, together with inverse energy transfer from small to large length scales along the wall-parallel directions.