The mechanism by which nonlinearity sustains turbulence in plane Couette flow

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 energytransferring 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.

POD-based study of turbulent plane Poiseuille flow: comparing structure and dynamics between quasi-linear simulations and DNS

“…While this paper does not settle this question, ongoing work indicates that the parametric growth mechanism does dominate in DNS (Nikolaidis et al. 2018; Farrell et al. 2022 b ).…”

“…This question is of importance because, to the extent that parametric growth dominates the fluctuation dynamics in DNS turbulence, DNS turbulence inherits the analytic characterization of RNL turbulence. While this paper does not settle this question, ongoing work indicates that the parametric growth mechanism does dominate in DNS (Nikolaidis et al 2018;Farrell et al 2022b). 2)) and the covariance of the reflected flow about the x-z plane at the centre of the flow ( y = 1) as a function of the averaging time T av , for hk z = 2, 4, 6, 8 for DNS of NL100, where Ŝy is defined in (A8).…”

“…Of all the possible mechanisms that one might hypothesize, this mechanism is identified analytically in S3T-RNL dynamics to be modification of the time-dependent streamwise mean state by Reynolds stress feedback arising from the fluctuations specifically to bring the characteristic Lyapunov exponent of the linear fluctuation equation to the real number zero. Extensive study of DNS data has verified that the characteristic exponent of the DNS streamwise mean state corresponds to this parametric growth stabilization mechanism (Nikolaidis, Farrell & Ioannou 2018; Farrell et al. 2022 b ).…”

“…And finally, the statistical state of the turbulence is regulated by feedback from the second cumulant to bring the time-varying streamwise mean flow to neutral stability, in the sense that the characteristic Lyapunov exponent of the linear operator governing the second cumulant is exactly zero. As an illustrative example of the power and utility of being in possession of an analytic theory for the dynamics of wall turbulence, consider the problem of y/h = 0.17 y/h = 0.17 Extensive study of DNS data has verified that the characteristic exponent of the DNS streamwise mean state corresponds to this parametric growth stabilization mechanism (Nikolaidis, Farrell & Ioannou 2018;Farrell et al 2022b). It is worth noting that this mechanism of regulating turbulence to its statistical mean state by feedback regulation operating between the fluctuations and the mean state such as to stabilize the mean state to linear instability is similar to that hypothesis by Malkus (1956), who posited that the statistical state of turbulence is determined by feedback regulation to neutrality of the mean state's inflectional modes.…”

Globe valve in single phase flow: experimental and CFD analysis

“…The Navier-Stokes equations are nonlinear partial differential equations. The nonlinearity is the main contributor to the turbulence [4]. Solving the unsteady Navier-Stokes equations implies that one must take into account all the space-time scales of the solution for a result with maximum quality.…”