Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow

Thomas, Vaughan; Lieu, Binh K.; Jovanović, Mihailo R.; Farrell, Brian F.; Ioannou, Petros J.; Gayme, Dennice F.

doi:10.1063/1.4898159

Cited by 59 publications

(142 citation statements)

References 72 publications

(86 reference statements)

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“…It should be highlighted that the solutions of the self-consistent model are obtained a priori without using any DNS or experimental data. Similar models where the mean flow is coupled to the linear perturbation equations through the Reynolds stress forcing can be found in the study of turbulent flows, as, for example, computing coherent structures that appear during the transition in a parallel shear flow [26] or by restricted nonlinear dynamics (RNL) of DNS with a linear fluctuation equation [25]. Another example is the stochastic structural stability theory (SSST) [22,23] where the ensemble average mean flow equation is coupled to the linear response formulated in terms of the covariance matrix and governed by the Lyapunov equation.…”

Section: Discussionmentioning

confidence: 99%

“…The present SC model, as well as SSST [22,23], RNL [25], and the model in Ref. [26], all rely on the common assumption of eliminating the nonlinearity arising from the cross-coupling between the different frequencies [right-hand side of (14b)] while maintaining solely the nonlinearity of the Reynolds stress retroaction onto the mean flow.…”

Section: Discussionmentioning

confidence: 99%

“…In semilinear models like SSST [23], RNL [25], and others [26], the fluctuation equation is approximated linearly, thus eliminating the right-hand side of (14b) while keeping the Reynolds stress nonlinearity [right-hand side of (14a)]. In the same spirit, our recent studies [19,32,33] seem to indicate that the nonlinear interaction of the fluctuation with itself gathered in the term −(u · ∇)u + (u · ∇)u has a negligible influence in the saturation process for certain flows.…”

Section: B Self-consistent Model For a Temporal Stochastic Forcingmentioning

confidence: 99%

“…This theory is able to describe sustained coherent structures that appear during the transition to turbulence in the 3D Couette flow [23] as well as in turbulent atmospheric flows [22,24]. As an alternative, the reduced nonlinear model (RNL) approach has been proposed [25] coupling the mean flow to a single realization of the stochastic linear response, avoiding the computation of the Lyapunov equation [23]. Another semilinear approach was recently proposed to describe the coherent structures appearing in the transition to turbulence for parallel shear flow [26].…”

Section: Introductionmentioning

confidence: 99%

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Saturation of the response to stochastic forcing in two-dimensional backward-facing step flow: A self-consistent approximation

Mantič-Lugo

Gallaire

2016

Phys. Rev. Fluids

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Selective noise amplifiers are characterized by large linear amplification to external perturbations in a particular frequency range despite their global linear stability. Applying a stochastic forcing with increasing amplitude, the response undergoes a strong nonlinear saturation when compared to the linear estimation. Building upon our previous work, we introduce a predictive model that describes this nonlinear dynamics, and we apply it to a canonical example of selective noise amplifiers: the backward-facing step flow. Rewriting conveniently the stochastic forcing and response in the frequency domain, the model consists in a mean flow equation coupled to the linear response to forcing at each frequency. This coupling is attained by the Reynolds stress, which is constructed by the integral in frequency of the independent responses. We generalize the model for a response to a white noise forcing δ-correlated in space and time restricting the flow dynamics to its most energetic patterns calculated from the optimal harmonic forcing and response of the flow. The model estimates accurately the response saturation when compared to direct numerical simulations, and it correctly approximates the structure of the response and the mean flow modification. It also shows that the response undergoes a selective process governed by the nonlinear gain, which promotes a response structure with an approximately single frequency and wavelength in the whole domain. These results suggest that the mean flow modification by the Reynolds stress is the key nonlinearity in the saturation process of the response to white noise.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: B Self-consistent Model For a Temporal Stochastic Forcingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Saturation of the response to stochastic forcing in two-dimensional backward-facing step flow: A self-consistent approximation

Mantič-Lugo

Gallaire

2016

Phys. Rev. Fluids

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“…5 More recently, the restricted nonlinear (RNL) model has been proposed. 9,10 The RNL model is similarly inspired by the prevalence of streamwise-coherent structures in wall-turbulence. However, it is more comprehensive than the 2D/3C model in that it describes the evolution of both a streamwise-constant velocity field, which for the purpose of this letter will be referred to as the "streamwise mean flow," as well as the evolution of a streamwise-varying perturbation field that interacts with the streamwise mean flow.…”

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confidence: 99%

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

2015

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Numerical simulations of wall-turbulence using the restricted nonlinear (RNL) model generate realistic mean velocity profiles in plane Couette and channel flow at low Reynolds numbers. The results are less accurate at higher Re, and while a logarithmic region is observed, its von-Kármán constant is not consistent with the standard logarithmic law. In half-channel flow we show that limiting the streamwise-varying wavenumber support of RNL turbulence to one or few empirically determined modes improves its predictions considerably. In particular, the mean velocity profiles obtained with the band-limited RNL model follow standard logarithmic behavior for the higher Reynolds numbers in this study.

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Spectral Energetics of a Quasilinear Approximation in Uniform Shear Turbulence

Hernández

Hwang

2021

Springer Proceedings in Physics

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The spectral energetics of a quasilinear (QL) model is studied in uniform shear turbulence. For the QL approximation, the velocity is decomposed into a mean averaged in the streamwise direction and the remaining fluctuation. The equations for the mean are fully considered, while the equations for the fluctuation are linearised around the mean. The QL model exhibits an energy cascade in the spanwise direction, but this is mediated by highly anisotropic small-scale motions unlike that in direct numerical simulation mediated by isotropic motions. In the streamwise direction, the energy cascade is shown to be completely inhibited in the QL model, resulting in highly elevated spectral energy intensity residing only at the streamwise integral length scales. It is also found that the streamwise wavenumber spectra of turbulent transport, obtained with the classical Reynolds decomposition, statistically characterizes the instability of the linearised fluctuation equations. Further supporting evidence of this claim is presented by carrying out a numerical experiment, in which the QL model with single streamwise Fourier mode is found to generate the strongest turbulence for L x /L z = 1 ∼ 3, consistent with previous findings (L x and L z are the streamwise and spanwise computational domains, respectively). Finally, the QL model is shown to completely ignore the role of slow pressure in the fluctuations, resulting in a significant damage of pressure-strain transport at all length scales. This explains the anisotropic turbulence of the QL model throughout the entire wavenumber space as well as the inhibited nonlinear regeneration of streamwise vortices in the self-sustaining process. * x,z,t = 1,(2.15a) * 10 −3 and −Φ uv * 10 −4 . This is in sharp contrast to the spectra of QL model. First of all,

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Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow

Cited by 59 publications

References 72 publications

Saturation of the response to stochastic forcing in two-dimensional backward-facing step flow: A self-consistent approximation

Saturation of the response to stochastic forcing in two-dimensional backward-facing step flow: A self-consistent approximation

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

Spectral Energetics of a Quasilinear Approximation in Uniform Shear Turbulence

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