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

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 $Re_{\tau}$ $\sim$ ${\mathcal{O}}$($10^3$) -- ${\mathcal{O}}$($10^4$). 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).