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Register a new userStatistical state dynamics-based analysis of the physical mechanisms sustaining and regulating turbulence in Couette flow
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This paper describes a study of the self-sustaining process in wall-turbulence based on a second order statistical state dynamics (SSD) model of Couette flow. SSD models with this form are referred to as S3T models and self-sustain turbulence with a mean flow and second order perturbation structure similar to that obtained by DNS. The use of a SSD model to study the physical mechanisms underlying turbulence has advantages over the traditional approach of studying the dynamics of individual realizations of turbulence. One advantage is that the analytical structure of SSD isolates and directly expresses the interaction between the coherent mean flow and the incoherent perturbation components of the turbulence. Isolation of the interaction between these components reveals how this interaction underlies both the maintenance of the turbulence variance by transfer of energy from the externally driven flow to the perturbation components as well as the enforcement of the observed statistical mean turbulent state by feedback regulation between the mean and perturbation fields. Another advantage of studying turbulence using SSD models is that the analytical structure of S3T turbulence can be completely characterized. For example, turbulence in the S3T system is maintained by a parametric growth mechanism. Furthermore, the equilibrium statistical state of the turbulence can be demonstrated to be enforced by feedback regulation in which transient growth of the incoherent perturbations episodically suppresses coherent streak growth preventing runaway parametric growth of the incoherent turbulent component. Using S3T to isolate these parametric growth and feedback regulation mechanisms allows a detailed characterization of the dynamics of the self-sustaining process in S3T turbulence with compelling implications for understanding the mechanism of wall-turbulence.
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While linear non-normality underlies the mechanism of energy transfer from the externally driven flow to the perturbation field that sustains turbulence, nonlinearity is also known to play an essential role. The goal of this study is to better understand the role of nonlinearity in sustaining turbulence. The method used in this study is implementation in Couette flow of a statistical state dynamics (SSD) closure at second order in a cumulant expansion of the Navier-Stokes equations in which the averaging operator is the streamwise mean. The perturbations are the deviations from the streamwise mean and two mechanisms potentially contributing to maintaining these perturbations are identified. These are parametric perturbation growth arising from interaction of the perturbations with the fluctuating mean flow and transient growth of perturbations arising from nonlinear interaction between components of the perturbation field. By the method of comparing the turbulence maintained in the SSD and in the associated direct numerical simulation (DNS) in which these mechanisms have been selectively included and excluded, parametric growth is found to maintain the perturbation field of the turbulence while the more commonly invoked mechanism of transient growth of perturbations arising from scattering by nonlinear interaction is found to suppress perturbation growth. In addition to verifying that the parametric mechanism maintains the perturbations in DNS it is also verified that the Lyapunov vectors are the structures that dominate the perturbation energy and energetics in DNS. It is further verified that these vectors are responsible for maintaining the roll circulation that underlies the self-sustaining process (SSP) and in particular the maintenance of the fluctuating streak that supports the parametric perturbation growth.
Despite the nonlinear nature of turbulence, there is evidence that part of the energy-transfer mechanisms sustaining wall turbulence can be ascribed to linear processes. The different scenarios stem from linear stability theory and comprise exponential instabilities, neutral modes, transient growth from non-normal operators, and parametric instabilities from temporal mean-flow variations, among others. These mechanisms, each potentially capable of leading to the observed turbulence structure, are rooted in theoretical and conceptual arguments. Whether the flow follows any or a combination of them remains elusive. Here, we evaluate the linear mechanisms responsible for the energy transfer from the streamwise-averaged mean-flow ($bf U$) to the fluctuating velocities ($bf u$). We use cause-and-effect analysis based on interventions. This is achieved by direct numerical simulation of turbulent channel flows at low Reynolds number, in which the energy transfer from $bf U$ to $bf u$ is constrained to preclude a targeted linear mechanism. We show that transient growth is sufficient for sustaining realistic wall turbulence. Self-sustaining turbulence persists when exponential instabilities, neutral modes, and parametric instabilities of the mean flow are suppressed. We further show that a key component of transient growth is the Orr/push-over mechanism induced by spanwise variations of the base flow. Finally, we demonstrate that an ensemble of simulations with various frozen-in-time $bf U$ arranged so that only transient growth is active, can faithfully represent the energy transfer from $bf U$ to $bf u$ as in realistic turbulence. Our approach provides direct cause-and-effect evaluation of the linear energy-injection mechanisms from $bf U$ to $bf u$ in the fully nonlinear system and simplifies the conceptual model of self-sustaining wall turbulence.
Recent studies have brought into question the view that at sufficiently high Reynolds number turbulence is an asymptotic state. We present the first direct observation of the decay of turbulent states in Taylor-Couette flow with lifetimes spanning five orders of magnitude. We also show that there is a regime where Taylor-Couette flow shares many of the decay characteristics observed in other shear flows, including Poisson statistics and the coexistence of laminar and turbulent patches. Our data suggest that characteristic decay times increase super-exponentially with increasing Reynolds number but remain bounded in agreement with the most recent data from pipe flow and with a recent theoretical model. This suggests that, contrary to the prevailing view, turbulence in linearly stable shear flows may be generically transient.
This paper reviews results from the study of wall-bounded turbulent flows using statistical state dynamics (SSD) that demonstrate the benefits of adopting this perspective for understanding turbulence in wall-bounded shear flows. The SSD approach used in this work employs a second-order closure which isolates the interaction between the streamwise mean and the equivalent of the perturbation covariance. This closure restricts nonlinearity in the SSD to that explicitly retained in the streamwise constant mean together with nonlinear interactions between the mean and the perturbation covariance. This dynamical restriction, in which explicit perturbation-perturbation nonlinearity is removed from the perturbation equation, results in a simplified dynamics referred to as the restricted nonlinear (RNL) dynamics. RNL systems in which an ensemble of a finite number of realizations of the perturbation equation share the same mean flow provide tractable approximations to the equivalently infinite ensemble RNL system. The infinite ensemble system, referred to as the S3T, introduces new analysis tools for studying turbulence. The RNL with a single ensemble member can be alternatively viewed as a realization of RNL dynamics. RNL systems provide computationally efficient means to approximate the SSD, producing self-sustaining turbulence exhibiting qualitative features similar to those observed in direct numerical simulations (DNS) despite its greatly simplified dynamics. Finally, we show that RNL turbulence can be supported by as few as a single streamwise varying component interacting with the streamwise constant mean flow and that judicious selection of this truncated support, or band-limiting, can be used to improve quantitative accuracy of RNL turbulence. The results suggest that the SSD approach provides new analytical and computational tools allowing new insights into wall-turbulence.
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.
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