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

Abstract

Townsend (J. Fluid Mech., vol. 11, issue 1, 1961, pp. 97-120) 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 two-dimensional (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 analysing datasets comprising two-point streamwise velocity signals coupled with a spectral linear stochastic estimation based procedure. The data considered span a friction Reynolds number range - . The procedure linearly decomposes the full 2-D spectrum into two components, and, comprising contributions predominantly from the inactive and active motions, respectively. This is confirmed by exhibiting wall scaling, for both streamwise and spanwise wavelengths, corresponding well with the Reynolds shear stress cospectra reported in the literature. Both and 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 -scaling for the associated one-dimensional spectra (where is the streamwise/spanwise wavenumber), lending empirical support to the attached eddy model of Perry & Chong (J. Fluid Mech., vol. 119, 1982, pp. 173-217).