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.
Highlights
Turbulence is widely regarded as the primary exemplar of an essentially nonlinear phenomenon
The perturbation-perturbation nonlinearity, N4, does not configure the perturbations to extract more energy from the mean flow than they would in the absence of this term implying that N4 acts as a disruption to the parametric growth process supporting the Lyapunov vector rather than augmenting the perturbation maintenance by the frequently hypothesized mechanism in which perturbation-perturbation nonlinearity replenishes the subspace of perturbations configured to transfer energy from the mean flow to the perturbations.The fact that the mean NL flow has been adjusted to near neutrality indicates that the first Lyapunov vector should be a dominant component of the NL perturbation state
While support of both the energy and energetics is on a single feedback neutralized Lyapunov vector in the case of restricted non-linear system (RNL), in the case of NL the energy and energetics are not confined to a single Lyapunov vector but rather are spread by nonlinearity over the Lyapunov vectors
Summary
Turbulence is widely regarded as the primary exemplar of an essentially nonlinear phenomenon. We contrast the energetics of the Lyapunov vectors on the turbulent NL mean flow just shown with the corresponding energetics of the nx = 1 Fourier component of the state vector obtained from the turbulent NL simulation itself in order to determine whether the N4 term has the effect of influencing the perturbations to be in a more or less favorable configuration for extracting energy from the mean flow.
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