A minimal model of self-sustaining turbulence

Thomas, Vaughan; Farrell, Brian F.; Ioannou, Petros J.; Gayme, Dennice F.

doi:10.1063/1.4931776

Cited by 51 publications

(96 citation statements)

References 67 publications

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“…Thomas et al [24] showed that band-limited RNL systems produce mean profiles and other structural features that are consistent with the baseline RNL system. Here we discuss only a subset of those results focusing on the particular case in which we keep only the kx = 3 mode corresponding to λ = 4π/3δ.…”

Section: Self-sustaining Turbulence In a Restricted Nonlinear Modelmentioning

confidence: 96%

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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Section: Self-sustaining Turbulence In a Restricted Nonlinear Modelmentioning

confidence: 96%

A statistical state dynamics approach to wall turbulence

Farrell

Gayme

Ioannou

2017

Phil. Trans. R. Soc. A.

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“…In S3T, and the related system referred to as CE2 (second-order cumulant expansion, Marston 2010), nonlinearity due to perturbation-perturbation advection is either set to zero or stochastically parameterized, so that the SSD is closed at second order. This second-order closure has proven useful in the study of coherent structure emergence in barotropic turbulence (Farrell & Ioannou 2007;Marston et al 2008;Srinivasan & Young 2012;Tobias & Marston 2013;Bakas & Ioannou 2013;Constantinou et al 2014;Parker & Krommes 2014;Bakas et al 2018), two-layer baroclinic turbulence (Farrell & Ioannou 2008, 2009aMarston 2010Marston , 2012Farrell & Ioannou 2017c), turbulence in the shallow-water equations on the equatorial beta-plane (Farrell & Ioannou 2009b), drift wave turbulence in plasmas (Farrell & Ioannou 2009;Parker & Krommes 2013), unstratified 2D turbulence (Bakas & Ioannou 2011), rotating magnetohydrodynamics (Tobias et al 2011;Squire & Bhattacharjee 2015;Constantinou & Parker 2018), 3D wall-bounded shear flow turbulence (Farrell & Ioannou 2012;Thomas et al 2014Thomas et al , 2015, and the turbulence of stable ion-temperature-gradient modes in plasmas (St-Onge & Krommes 2017). In the present work we place 2D stratified Boussinesq turbulence into the mechanistic and phenomenological context of the mean flow-turbulence interaction mechanism that has been identified in these other turbulent systems.…”

Section: Introductionmentioning

confidence: 99%

Statistical state dynamics of vertically sheared horizontal flows in two-dimensional stratified turbulence

Fitzgerald

Farrell

2018

J. Fluid Mech.

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Simulations of strongly stratified turbulence often exhibit coherent large-scale structures called vertically sheared horizontal flows (VSHFs). VSHFs emerge in both twodimensional (2D) and three-dimensional (3D) stratified turbulence with similar vertical structure. The mechanism responsible for VSHF formation is not fully understood. In this work, the formation and equilibration of VSHFs in a 2D Boussinesq model of stratified turbulence is studied using statistical state dynamics (SSD). In SSD, equations of motion are expressed directly in the statistical variables of the turbulent state. Restriction to 2D turbulence makes available an analytically and computationally attractive implementation of SSD referred to as S3T, in which the SSD is expressed by coupling the equation for the horizontal mean structure with the equation for the ensemble mean perturbation covariance. This second order SSD produces accurate statistics, through second order, when compared with fully nonlinear simulations. In particular, S3T captures the spontaneous emergence of the VSHF and associated density layers seen in simulations of turbulence maintained by homogeneous large-scale stochastic excitation. An advantage of the S3T system is that the VSHF formation mechanism, which is wavemean flow interaction between the emergent VSHF and the stochastically excited largescale gravity waves, is analytically understood in the S3T system. Comparison with fully nonlinear simulations verifies that S3T solutions accurately predict the scale selection, dependence on stochastic excitation strength, and nonlinear equilibrium structure of the VSHF. These results facilitate relating VSHF theory and geophysical examples of turbulent jets such as the ocean's equatorial deep jets.

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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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A minimal model of self-sustaining turbulence

Cited by 51 publications

References 67 publications

A statistical state dynamics approach to wall turbulence

A statistical state dynamics approach to wall turbulence

Statistical state dynamics of vertically sheared horizontal flows in two-dimensional stratified turbulence

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