Nested traveling wave structures in elastoinertial turbulence

Abstract

Elastoinertial turbulence (EIT) is a chaotic flow resulting from the interplay between inertia and viscoelasticity in wall bounded shear flows. Understanding EIT is important because it is thought to set a limit on the effectiveness of turbulent drag reduction in polymer solutions. Here, we analyze simulations of two dimensional EIT in channel flow using Spectral Proper Orthogonal Decomposition (SPOD), discovering a family of traveling wave structures that capture the sheetlike stress fluctuations that characterize EIT. The frequency dependence of the leading SPOD mode contains distinct peaks and the mode structures corresponding to these peaks exhibit well defined traveling structures. The structure of the dominant traveling mode exhibits shift reflect symmetry similar to the viscoelasticity modified Tollmien Schlichting (TS) wave, where the velocity fluctuation in the traveling mode is characterized by large scale regular structures spanning the channel and the polymer stress field is characterized by thin, inclined sheets of high polymer stress localized at the critical layers near the channel walls. The traveling structures corresponding to the higher frequency modes have a very similar structure, but are nested in a region roughly bounded by the critical layer positions of the next lower frequency mode. A simple theory based on the idea that the critical layers of mode $\kappa$ form the "walls" for the structure of mode $\kappa+1$ yields quantitative agreement with the observed wave speeds and critical layer positions, indicating self similarity between the structures. The physical idea behind this theory is that the sheetlike localized stress fluctuations in the critical layer prevent velocity fluctuations from penetrating them.