Large-deviation joint statistics of the finite-time Lyapunov spectrum in isotropic turbulence

Abstract

One of the hallmarks of turbulent flows is the chaotic behavior of fluid particle paths with exponentially growing separation among them while their distance does not exceed the viscous range. The maximal (positive) Lyapunov exponent represents the average strength of the exponential growth rate, while fluctuations in the rate of growth are characterized by the finite-time Lyapunov exponents (FTLEs). In the last decade or so, the notion of Lagrangian coherent structures (which are often computed using FTLEs) has gained attention as a tool for visualizing coherent trajectory patterns in a flow and distinguishing regions of the flow with different mixing properties. A quantitative statistical characterization of FTLEs can be accomplished using the statistical theory of large deviations, based on the so-called Cramér function. To obtain the Cramér function from data, we use both the method based on measuring moments and measuring histograms and introduce a finite-size correction to the histogram-based method. We generalize the existing univariate formalism to the joint distributions of the two FTLEs needed to fully specify the Lyapunov spectrum in 3D flows. The joint Cramér function of turbulence is measured from two direct numerical simulation datasets of isotropic turbulence. Results are compared with joint statistics of FTLEs computed using only the symmetric part of the velocity gradient tensor, as well as with joint statistics of instantaneous strain-rate eigenvalues. When using only the strain contribution of the velocity gradient, the maximal FTLE nearly doubles in magnitude, highlighting the role of rotation in de-correlating the fluid deformations along particle paths. We also extend the large-deviation theory to study the statistics of the ratio of FTLEs. The most likely ratio of the FTLEs λ1 : λ2 : λ3 is shown to be about 4:1:−5, compared to about 8:3:−11 when using only the strain-rate tensor for calculating fluid volume deformations. The results serve to characterize the fundamental statistical and geometric structure of turbulence at small scales including cumulative, time integrated effects. These are important for deformable particles such as droplets and polymers advected by turbulence.

Quartic polynomial approximation for fluctuations of separation of trajectories in chaos and correlation dimension

“…It seems that this approximation works in the NS case. We prove this for the channel turbulence of [14] and observe that the fit would probably also work for the measurements of [13] which provide H(λ) that seemingly can be fit with a simple function. Since many of the physical examples provided above have weakly compressible flows then we also introduce an approximation scheme for D where the flow compressibility is considered as an expansion parameter.…”

“…For time reversible statistics, the quadratic approximation, holding in the Kraichnan model, is γ(k) = λ 1 k(k + d)/d which implies H(0) = λ 1 d/4. Inspection of the data of [13] for the motion of tracers in the NS turbulence demonstrates H(0) is appreciably larger than 3λ 1 /4. This is reasonable because the NS flow is neither time-reversible nor short correlated.…”

“…This property of a chaotic system have been studied for a long time, see e. g. [8][9][10], however recently there appeared new measurements where the chaotic system is formed by the motion of the fluid particles resolved below the smoothness (viscous [12]) scale of the Navier-Stokes (NS) turbulence. Thus γ(k) or H(λ) were obtained for the homogeneous [13] and channel [14,15] turbulent flows. The dependence on the Reynolds number in the case of homogeneous turbulence was considered in [16] who also studied γ(k) for the chaotic motion of inertial particles ( [13,15,16] used a somewhat different definition of γ(k) which coincides with ours at k > −1.…”

Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective

“…Large-deviation theory (Kifer, 1990;Touchette, 2009) is a powerful approach from statistical physics for estimating the probability of rare events with many applications. It has recently been applied to the behaviour of FTLEs at long integration times (Kuptsov and Politi, 2011;Laffargue et al, 2013;Johnson and Meneveau, 2015). In the following, largedeviation theory is briefly described in the form in which it is used in the present study.…”

Fluctuations of Lyapunov exponents in homogeneous and isotropic turbulence

“…In these contexts, chaos is used to characterize mixing and transport properties as well as determining Lagrangian coherent structures (LCS) [45]. Unlike Eulerian chaos, several statistical analyses of Lagrangian chaos have been done in the past [26,27,46].…”