A Perron–Frobenius analysis of wall-bounded turbulence

Abstract

The Perron–Frobenius operator (PFO) is adapted from dynamical-system theory to the study of turbulent channel flow. It is shown that, as long as the analysis is restricted to the system attractor, the PFO can be used to differentiate causality and coherence from simple correlation without performing interventional experiments, and that the key difficulty remains the collection of enough data to populate the operator matrix. This is alleviated by limiting the analysis to two-dimensional projections of the phase space, and developing a series of indicators to choose the best parameter pairs from a large number of possibilities. The techniques thus developed are applied to the study of bursting in the inertial layer of the channel, with emphasis on the process by which bursts are reinitiated after they have decayed. Conditional averaging over phase-space trajectories suggested by the PFO shows, somewhat counter-intuitively, that a key ingredient for the burst recovery is the development of a low-shear region near the wall, overlaid by a lifted shear layer. This is confirmed by a computational experiment in which the control of the mean velocity profile by the turbulence fluctuations is artificially relaxed. The behaviour of the mean velocity profile is thus modified, but the association of low wall shear with the initiation of the bursts is maintained.