Time-evolution K–L analysis of coherent structures based on DNS of turbulent Newtonian and viscoelastic flows

Abstract

The turbulent dynamics corresponding to direct numerical simulation (DNS) data of Newtonian and viscoelastic turbulent channel flows is analyzed here through a projection of a time sequence of velocity fields into a set of Karhunen–Loeve (K–L) modes, large enough to contain, on the average, more than 90% of the fluctuating turbulence kinetic energy. Previous calculations of the K–L eigenstructure have demonstrated a dramatic decrease in the K–L dimensionality accompanying the presence of viscoelasticity in turbulent channel flows. Projection of turbulent velocity information into the most energetic K–L modes allows for a significant data reduction, most suitable for the study of the time-evolution dynamics. We exploit this feature here in analyzing the time series of the projection coefficients of the velocity field into K–L modes to gain further quantitative insights into the behavior of the overall flow dynamics. The presence of viscoelasticity induces significant changes to the turbulence dynamics as compared to the Newtonian one obtained at the same friction Reynolds number. Namely, it cuts in half the representational entropy constructed from the projection coefficients whereas it doubles the fluctuating kinetic energy. Moreover, dynamic correlation analysis of pairs of K–L coefficients showed a systematic increase in the time scales characterizing dynamic turbulent events with viscoelasticity. These correlations have distinct pulse behavior characteristic of intermittence and quasiperiodicity. Correlations between K–L modes corresponding to different wavenumbers along the principal flow direction depend sensitively on the velocity of the moving frame of reference with respect to which they are calculated, and resonate at characteristic velocities corresponding to the convective velocities of the coherent structures to which the two modes contribute. Based on that, we propose a new two-dimensional analysis of those correlations with respect to both the time and the moving frame velocity in order to detect coherent flow structures and their mean convective velocity. Those show considerable differences when they are calculated with and without the presence of viscoelasticity.