Standard logarithmic mean velocity distribution in a band-limited restricted nonlinear model of turbulent flow in a half-channel

Bretheim, Joel; Meneveau, Charles; Gayme, Dennice F.

doi:10.1063/1.4906987

Cited by 36 publications

(43 citation statements)

References 19 publications

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“…form a non-linear dynamical system that self-sustains S3T turbulence [13,16,40,49,50]. The quasi-linear structure of this system allows us to isolate the linear dynamics of the incoherent component of the turbulence, C b :…”

Section: Isolating the Linear Dynamics Of The Second Order Cumulantmentioning

confidence: 99%

“…Because S3T dynamics is recovered in the limit N → ∞ we can identify solutions of RNL ∞ systems with S3T. While the use in S3T dynamics of a time-dependent Lyapunov equation to advance the perturbation covariance in time allows direct solution for the second cumulant corresponding to the covariance obtained from a formally infinite ensemble, the great advantage of RNL N systems is in allowing extension of second order S3T statistical state dynamics methods to study turbulence at high Reynolds number [16,39,40].…”

Section: Introductionmentioning

confidence: 99%

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Statistical state dynamics-based analysis of the physical mechanisms sustaining and regulating turbulence in Couette flow

Farrell

Ioannou

2017

Phys. Rev. Fluids

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This paper describes a study of the self-sustaining process in wall-turbulence. The study is based on a second order statistical state dynamics model of Couette flow in which the state variables are the streamwise mean flow (first cumulant) and perturbation covariance (second cumulant). This statistical state dynamics model is closed by either setting the third cumulant to zero or by replacing it with a stochastic parameterization. Statistical state dynamics models with this form are referred to as S3T models. S3T models have been shown to self-sustain turbulence with a mean flow and second order perturbation structure similar to that obtained by direct numerical simulation of the equations of motion. The use of a statistical state dynamics model to study the physical mechanisms underlying turbulence has important advantages over the traditional approach of studying the dynamics of individual realizations of turbulence. One advantage is that the analytical structure of S3T statistical state dynamics models isolates the interaction between the mean flow and the 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 statistical state dynamics models of S3T form is that the analytical structure of S3T turbulence can be completely characterized. For example, the perturbation component of turbulence in the S3T system is demonstrably maintained by a parametric perturbation growth mechanism in which fluctuation of the mean flow maintains the perturbation field which in turn maintains the mean flow fluctuations in a synergistic interaction. Furthermore, the equilibrium statistical state of S3T turbulence can be demonstrated to be enforced by feedback regulation in which transient growth of the perturbations episodically suppresses streak growth preventing runaway parametric growth of the perturbation component. Using S3T to isolate the 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 advancing understanding of wall-turbulence.

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Section: Isolating the Linear Dynamics Of The Second Order Cumulantmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Statistical state dynamics-based analysis of the physical mechanisms sustaining and regulating turbulence in Couette flow

Farrell

Ioannou

2017

Phys. Rev. Fluids

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“…Refs. [19,[22][23][24][25][26]. The RNL model implementations of SSD comprise joint evolution of a coherent mean flow (first cumulant) and an ensemble approximation to the secondorder perturbation statistics which is considered conceptually to be an approximation to the covariance of the perturbations (second cumulant) although this covariance is not explicitly calculated.…”

Section: Introductionmentioning

confidence: 99%

“…Dashed lines are channel flow DNS data from Moser et al[58] Symbols are RNL data. This figure is from[25].…”

mentioning

confidence: 99%

A statistical state dynamics approach to wall turbulence

Farrell

Gayme

Ioannou

2017

Phil. Trans. R. Soc. A.

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One contribution of 14 to a theme issue 'Toward the development of high-fidelity models of wall turbulence at large Reynolds number' . This paper reviews results obtained 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 that retains only the interaction between the streamwise mean flow and the streamwise mean perturbation covariance. This closure restricts nonlinearity in the SSD to that explicitly retained in the streamwise constant mean flow together with nonlinear interactions between the mean flow 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 a finite ensemble of realizations of the perturbation equation share the same mean flow, provide tractable approximations to the SSD, which is equivalent to an infinite ensemble RNL system. This infinite ensemble system, referred to as the stochastic structural stability theory system, introduces new analysis tools for studying turbulence. RNL systems provide computationally efficient means to approximate the SSD and produce self-sustaining turbulence exhibiting qualitative features similar to those observed in direct numerical simulations despite greatly simplified dynamics. The results presented 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 'bandlimiting' can be used to improve quantitative accuracy of RNL turbulence. These results suggest that the SSD

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“…Even in the absence of stochastic forcing, certain measures of turbulence, e.g., the correct mean velocity profile, are maintained through interactions between the mean flow and a small subset of streamwise varying modes. Even though turbulence can be triggered with white-in-time stochastic forcing, correct statistics cannot be obtained without accounting for the dynamics of the streamwise averaged mean flow or without manipulation of the underlying dynamical modes (Bretheim et al 2015;Thomas et al 2015).…”

Section: Stochastic Forcing and Flow Statisticsmentioning

confidence: 99%

Colour of turbulence

2017

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In this paper, we address the problem of how to account for second-order statistics of turbulent flows using low-complexity stochastic dynamical models based on the linearized Navier-Stokes equations. The complexity is quantified by the number of degrees of freedom in the linearized evolution model that are directly influenced by stochastic excitation sources. For the case where only a subset of velocity correlations are known, we develop a framework to complete unavailable second-order statistics in a way that is consistent with linearization around turbulent mean velocity. In general, white-in-time stochastic forcing is not sufficient to explain turbulent flow statistics. We develop models for colored-intime forcing using a maximum entropy formulation together with a regularization that serves as a proxy for rank minimization. We show that colored-in-time excitation of the Navier-Stokes equations can also be interpreted as a low-rank modification to the generator of the linearized dynamics. Our method provides a data-driven refinement of models that originate from first principles and captures complex dynamics of turbulent flows in a way that is tractable for analysis, optimization, and control design.

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Standard logarithmic mean velocity distribution in a band-limited restricted nonlinear model of turbulent flow in a half-channel

Cited by 36 publications

References 19 publications

Statistical state dynamics-based analysis of the physical mechanisms sustaining and regulating turbulence in Couette flow

Statistical state dynamics-based analysis of the physical mechanisms sustaining and regulating turbulence in Couette flow

A statistical state dynamics approach to wall turbulence

Colour of turbulence

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