Instantons in a Lagrangian model of turbulence

Grigorio, L. S.; Bouchet, Freddy; Pereira, R. M.; Chevillard, Laurent

doi:10.1088/1751-8121/aa51a3

Cited by 18 publications

(37 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2015 b ; Grafke, Grauer & Schäfer 2015 a ; Grigorio et al. 2017), splitting methods that copy rare event realisations to efficiently accumulate statistics (Allen, Warren & ten Wolde 2005; Giardinà, Kurchan & Peliti 2006; Cérou & Guyader 2007; Tailleur & Kurchan 2007; Nemoto et al. 2016; Teo et al.…”

Section: Discussionmentioning

confidence: 99%

“…However, in recent years, alternative methods have been developed that could address this problem. In particular, our understanding of the problem could benefit from studies using rare event sampling algorithms, such as a method calculating the instanton based on Freidlin-Wentzell theory (Chernykh & Stepanov 2001;Heymann & Vanden-Eijnden 2008;Grafke et al 2015b;Grafke, Grauer & Schäfer 2015a;Grigorio et al 2017), splitting methods that copy rare event realisations to efficiently accumulate statistics (Allen, Warren & ten Wolde 2005;Giardinà, Kurchan & Peliti 2006;Cérou & Guyader 2007;Tailleur & Kurchan 2007;Nemoto et al 2016;Teo et al 2016;Lestang et al 2018;Bouchet, Rolland & Simonnet 2019) and also a recently proposed method that relies on feedback control of the Reynolds number (Nemoto & Alexakis 2018). Such studies can help to overcome the difficulty caused by the extremely long computational time required to accurately describe the rare transition events close to the onset.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Rare transitions to thin-layer turbulent condensates

2019

View full text Add to dashboard Cite

Turbulent flows in a thin layer can develop an inverse energy cascade leading to spectral condensation of energy when the layer height is smaller than a certain threshold. These spectral condensates take the form of large-scale vortices in physical space. Recently, evidence for bistability was found in this system close to the critical height: depending on the initial conditions, the flow is either in a condensate state with most of the energy in the two-dimensional (2-D) large-scale modes, or in a three-dimensional (3-D) flow state with most of the energy in the small-scale modes. This bistable regime is characterised by the statistical properties of random and rare transitions between these two locally stable states. Here, we examine these statistical properties in thin-layer turbulent flows, where the energy is injected by either stochastic or deterministic forcing. To this end, by using a large number of direct numerical simulations (DNS), we measure the decay time τ d of the 2-D condensate to 3-D flow state and the build-up time τ b of the 2-D condensate. We show that both of these times τ d , τ b follow an exponential distribution with mean values increasing faster than exponentially as the layer height approaches the threshold. We further show that the dynamics of large-scale kinetic energy may be modeled by a stochastic Langevin equation. From time-series analysis of DNS data, we determine the effective potential that shows two minima corresponding to the 2-D and 3-D states when the layer height is close to the threshold.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Rare transitions to thin-layer turbulent condensates

2019

View full text Add to dashboard Cite

show abstract

“…As an example, the RFD approximation gives realistic results of homogeneous and isotropic turbulence (of Taylor-based Reynolds number R λ = 140) when τ η /T = 0.1, and deteriorate for τ η /T < 0.01 (Chevillard et al 2008). Subsequent numerical and theoretical developments in the framework of the RFD approximation indeed show that the model pointed in (1.5) fails at giving a realistic picture of turbulence at high Reynolds numbers (Afonso & Meneveau 2010;Moriconi et al 2014;Grigorio et al 2017). We may wonder whether it is possible to constrain the drift V ij and diffusion D ijkl terms in a different way to ensure, for instance, a basic property of turbulence (Frisch 1995) that is the finiteness of the average dissipation rate 2νE tr(S 2 ) when R e → ∞.…”

Section: Introductionmentioning

confidence: 98%

A multifractal model for the velocity gradient dynamics in turbulent flows

Pereira¹,

Moriconi²,

Chevillard³

2018

J. Fluid Mech.

107

View full text Add to dashboard Cite

We develop a stochastic model for the velocity gradients dynamics along a Lagrangian trajectory in isotropic and homogeneous turbulent flows. Comparing with different attempts proposed in the literature, the present model, at the cost of introducing a free parameter known in turbulence phenomenology as the intermittency coefficient, gives a realistic picture of velocity gradient statistics at any Reynolds number. To achieve this level of accuracy, we use as a first modelling step a regularized self-stretching term in the framework of the Recent Fluid Deformation (RFD) approximation that was shown to give a realistic picture of small scales statistics of turbulence only up to moderate Reynolds numbers. As a second step, we constrain the dynamics, in the spirit of Girimaji & Pope (1990), in order to impose a peculiar statistical structure to the dissipation seen by the Lagrangian particle. This probabilistic closure uses as a building block a random field that fulfils the statistical description of the intermittency, i.e. multifractal, phenomenon. To do so, we define and generalize to a statistically stationary framework a proposition made by Schmitt (2003). These considerations lead us to propose a non-linear and non-Markovian closed dynamics for the elements of the velocity gradient tensor. We numerically integrate this dynamics and observe that a stationary regime is indeed reached, in which (i) the gradients variance is proportional to the Reynolds number, (ii) gradients are typically correlated over the (small) Kolmogorov time scale and gradients norms over the (large) integral time scale (iii) the joint probability distribution function of the two non vanishing invariants Q and R reproduces the characteristic teardrop shape, (iv) vorticity gets preferentially aligned with the intermediate eigendirection of the deformation tensor and (v) gradients are strongly non-Gaussian and intermittent, a behaviour that we quantify by appropriate high order moments. Additionally, we examine the problem of rotation rate statistics of (axisymmetric) anisotropic particles as observed in Direct Numerical Simulations. Although our realistic picture of velocity gradient fluctuations leads to better results when compared to the former RFD approximation, it is still unable to provide an accurate description for the rotation rate variance of oblate spheroids.

show abstract

“…Instantons refer to the most probable trajectories in phase space that achieve a given rare event (in the limit of weak noise). Suitable numerical schemes can be used to evaluate instantons as well as the related probabilities of rare events (Chernykh & Stepanov 2001;Grafke, Grauer & Schäfer 2013;Bouchet, Laurie & Zaboronski 2014;Laurie & Bouchet 2015;Grigorio et al 2017). An example is the investigation of the physics of rogue waves (Dematteis, Grafke & Vanden-Eijnden 2018;Dematteis et al 2019).…”

Section: Introductionmentioning

confidence: 99%