Active and inactive components of the streamwise velocity in wall-bounded turbulence

Abstract

Townsend (1961) introduced the concept of active and inactive motions for wall-bounded turbulent flows, where the active motions are solely responsible for producing the Reynolds shear stress, the key momentum transport term in these flows. While the wall-normal component of velocity is associated exclusively with the active motions, the wall-parallel components of velocity are associated with both active and inactive motions. In this paper, we propose a method to segregate the active and inactive components of the 2-D energy spectrum of the streamwise velocity, thereby allowing us to test the self-similarity characteristics of the former which are central to theoretical models for wall-turbulence. The approach is based on analyzing datasets comprising two-point streamwise velocity signals coupled with a spectral linear stochastic estimation (SLSE) based procedure. The data considered span a friction Reynolds number range $R{e}_{\tau }$ $\sim$ $\mathcal{O}$($10^3$) -- $\mathcal{O}$($10^4$). The procedure linearly decomposes the full 2-D spectrum ($\Phi$) into two components, ${\Phi }_{ia}$ and ${\Phi }_a$, comprising contributions predominantly from the inactive and active motions, respectively. This is confirmed by ${\Phi }_a$ exhibiting wall-scaling, for both streamwise and spanwise wavelengths, corresponding well with the Reynolds shear stress cospectra reported in the literature. Both ${\Phi }_a$ and ${\Phi }_{ia}$ are found to depict prominent self-similar characteristics in the inertially dominated region close to the wall, suggestive of contributions from Townsend's attached eddies. Inactive contributions from the attached eddies reveal pure $k^{-1}$-scaling for the associated 1-D spectra (where $k$ is the streamwise/spanwise wavenumber), lending empirical support to the attached eddy model of Perry & Chong (1982).

arXiv comment22 pages, 8 figures, submitted to the Journal of Fluid Mechanics

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• Evidence that superstructures comprise of self-similar coherent motions in high $R{e}_{\tau }$ boundary layers
• Informative and non-informative decomposition of turbulent flow fields
• Intrinsic compressibility effects in near-wall turbulence
• Model-based spectral coherence analysis
• Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterisation of a high Reynolds number transition region
• Active and inactive contributions to the wall pressure and wall-shear stress in turbulent boundary layers
• Dissipation Scaling and Structural Order in Turbulent Channel Flows
• The eddies are attached, but it is all right
• On the wall-normal velocity variance in canonical wall-bounded turbulence