The mechanism by which nonlinearity sustains turbulence in plane Couette flow

TL;DR

This paper reveals that in plane Couette flow turbulence, nonlinear interactions sustain turbulence primarily through a parametric growth process mediated by the fluctuating mean flow, with Lyapunov vectors supporting turbulence.

Contribution

It identifies the parametric interaction with the mean flow as the key nonlinear mechanism maintaining turbulence, contrasting with perturbation-perturbation nonlinearity.

Findings

Perturbation energy is replenished by a non-normality mediated parametric growth process.

The fluctuating mean flow is adjusted to marginal Lyapunov stability.

Perturbation-perturbation nonlinearity does not significantly contribute to turbulence maintenance.

Abstract

Turbulence in wall-bounded shear flow results from a synergistic interaction between linear non-normality and nonlinearity in which non-normal growth of a subset of perturbations configured to transfer energy from the externally forced component of the turbulent state to the perturbation component maintains the perturbation energy, while the subset of energy-transferring perturbations is replenished by nonlinearity. Although it is accepted that both linear non-normality mediated energy transfer from the forced component of the mean flow and nonlinear interactions among perturbations are required to maintain the turbulent state, the detailed physical mechanism by which these processes interact in maintaining turbulence has not been determined. In this work a statistical state dynamics based analysis is performed on turbulent Couette flow at $R=600$ and a comparison to DNS is used to demonstrate that the perturbation component in Couette flow turbulence is replenished by a non-normality mediated parametric growth process in which the fluctuating streamwise mean flow has been adjusted to marginal Lyapunov stability. It is further shown that the alternative mechanism in which the subspace of non-normally growing perturbations is maintained directly by perturbation-perturbation nonlinearity does not contribute to maintaining the turbulent state. This work identifies parametric interaction between the fluctuating streamwise mean flow and the streamwise varying perturbations to be the mechanism of the nonlinear interaction maintaining the perturbation component of the turbulent state, and identifies the associated Lyapunov vectors with positive energetics as the structures of the perturbation subspace supporting the turbulence.