Turbulent transport for wall shear stress fluctuations

TL;DR

This work analyzes wall shear stress fluctuations in wall-bounded turbulence (Poiseuille and Couette flows) using direct numerical simulations and dimensional analysis within polar-log spectral coordinates. It demonstrates that inner-range dissipation spectra with $\lambda \lesssim 1000\delta_\nu$ are Reynolds-number invariant, while outer-range spectra with $\lambda \gtrsim \delta$ decay with $Re$, causing the total wall dissipation to rise with $Re$ due to intermediate scales $1000\delta_\nu < \lambda < \delta$. The near-wall motions responsible for these dissipation characteristics are shown to be inactive (negligible Reynolds stress) and are sustained by turbulent transport, with both wall-normal and inter-scale transport spectra sharing the same $Re$-scaling and exhibiting energy flux toward the wall and inverse transfer along wall-parallel directions. These findings provide a mechanistic link between outer-scale dynamics and near-wall dissipation, informing models of wall-bounded turbulence and improving understanding of wall-shear fluctuations in engineering applications.

Abstract

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 $λ\lesssim 1000 δ_ν$ are asymptotically invariant with the Reynolds number ($Re$), whereas those for $λ\gtrsim δ$ decay with $Re$ (here, $λ$ is a nominal wall-parallel wavelength, and $δ_ν$ and $δ$ are the viscous inner and outer length scales, respectively). The wall dissipation increases with $Re$ due to the increasing contribution of the spectra at $1000 δ_ν\lesssim λ\lesssim δ$. 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.

Figures (14)

• Figure 1: Two-dimensional spectral densities in polar-log coordinates of $u'^2$ and $\epsilon_\mathrm{w}$, $k_\mathrm{ref} = 1/ (50000\delta_v)$: (a) $E_{u'^2}^\#$ at $y^+ = 15$, $Re_\tau = 5200$, Poiseuille; (b) $E_{\epsilon_\mathrm{w}}^\#$, $Re_\tau = 5200$, Poiseuille; (c) $E_{u'^2}^\#$ at $y^+ = 15$, $Re_\tau = 500$, Couette; (d) $E_{\epsilon_\mathrm{w}}^\#$, $Re_\tau = 500$, Couette. Here, the contour levels are suitably chosen to reveal the similarities between the spectral densities of $u'^2$ and $\epsilon_\mathrm{w}$.
• Figure 2: Inner-scale normalised two-dimensional spectral densities in polar-log coordinates of $\epsilon_\mathrm{w}$ for Poiseuille flows, $k_\mathrm{ref} = 1/ (50000\delta_v)$.
• Figure 3: Outer-scale normalised two-dimensional spectral densities in polar-log coordinates of $\epsilon_\mathrm{w}$ for Poiseuille flows, $k_\mathrm{ref} = 1/ (100\delta)$.
• Figure 4: Dissipation rate of $\langle u'^2 \rangle^+$ at walls, $\epsilon_\mathrm{w}$ for Couettte ($\circ$) and Poiseuille ($\square$) flows at various $Re_\tau$: (a) total and inner-scale-filtered $\epsilon_\mathrm{w}$; (b) inner-scale-filtered $\epsilon_\mathrm{w}$, magnification of (a); (c) outer-scale-filtered $\epsilon_\mathrm{w}$ for Couette flow; (d) outer-scale-filtered $\epsilon_\mathrm{w}$ for Poiseuille flow. (e) difference between $\epsilon_\mathrm{w}$ and the sum of inner-scale-filtered $\epsilon_\mathrm{w}$ and outer-scale-filtered $\epsilon_\mathrm{w}$ (Poiseulle flow for $Re_\tau>1000$).
• Figure 5: Filtered budget terms near the wall of Poiseuille flows. (a,b) Production; (c,d) pressure transport; (e,f) turbulent transport; (g,h) viscous transport; (i,j) dissipation rate.