Chaotic Properties of a Turbulent Isotropic Fluid

Abstract

By tracking the divergence of two initially close trajectories in phase space in an Eulerian approach to forced turbulence, the relation between the maximal Lyapunov exponent λ, and the Reynolds number Re is measured using direct numerical simulations, performed on up to 2048 3 collocation points. The Lyapunov exponent is found to solely depend on the Reynolds number with λ ∝ Re 0.53 and that after a transient period the divergence of trajectories grows at the same rate at all scales. Finally a linear divergence is seen that is dependent on the energy forcing rate. Links are made with other chaotic systems.In Press Physical Review Letters 2018 Using the Eulerian approach, we track the divergence of fluid field trajectories, which initially differ by a small perturbation. We do a model independent analysis, evolving the Navier-Stokes equations for three dimensional homogeneous isotropic turbulence (HIT) using direct numerical simulation (DNS). The Eulerian approach to the study of the chaotic properties of turbulence has received only limited numerical tests prior to this Letter. Amongst approximate models, there have been EDQNM closure approximations [18] and shell model studies [19][20][21]. Amongst exact DNS studies, there have been some in two dimensions [22][23][24] and single runs in three dimensions at comparatively small box sizes [25,26], all more than a decade and a half ago. This Letter tests the theory of Ruelle [27] relating the maximal Lyapunov exponent λ and Re in DNS of HIT in a Eulerian sense. The paper also examines the time history of the divergence and finds a uniform exponential growth rate across all scales at an intermediate time and to show a linear growth for late time in three dimensional HIT. The simulations are also the largest yet for measuring the Eulerian aspects of chaos in HIT for DNS, performed on up to 2048 3 collocation points and reach an integral scale Reynolds number of 6200. This allows a more accurate measurement of the Re dependence of λ.For a chaotic system, an initially small perturbation |δu 0 | should grow according to |δu(t)| ≃ |δu 0 |e λt where t is time. It is theoretically predicted that the Lyapunov exponent should depend on the Reynolds number according to the rule [27,28] The Holder exponent, h, is given by |u(x + r) − u(x)| ∼ V l h , where V is the rms velocity, l the size of the eddy, Re = V L/ν the integral scale Reynolds number, L = (3π/4E) (E(k)/k)dk the integral length scale, E the energy, ν the viscosity, T 0 = L/V the large eddy turnover time, τ = (ν/ǫ) 1/2 the Kolmogorov time scale, and ǫ the dissipation rate. In the Kolmogorov theory, h is predicted to be 1/3 and so α is predicted to be 1/2 [27][28][29].Some of the new results found in this Letter from the Eulerian approach are inaccessible to the Lagrangian approach, such as the linear growth rate of the divergence at late times which has no direct Lagrangian counterpart. The paper also highlights different results from the two approaches. For instance, within the Lagrangian approach, the relation...

Fluctuations of Lyapunov exponents in homogeneous and isotropic turbulence

“…Once the values for λ, σ λ ,Re and σ Re are obtained, a weighed least squared fit is performed as shown in Figure 7 to Equation (2), obtaining a value for α = 0.53 ± 0.02 and D = 0.079 ± 0.007. These results are in agreement with those previously found in [14] in which the same forcing scheme is used but instead of FTLE, the direct method is used.…”

“…This analysis is done for a hydrodynamic flow evolved using the incompressible NSE . A discrepancy is noted between previous measurements by different authors [14,15]. This is resolved by considering corrections given by the dimensionless dissipation rate [50] (III D).…”

Information production in homogeneous isotropic turbulence

“…Our data shows the Re scaling exponent is less than expected when following the simplified prediction, although only slightly. Intermittency is often invoked to explain such differences, however, the multi-fractal model predicted deviations [46,47] in the opposite direction than that observed in DNS [13,14] for the related largest Lyapunov exponent. As such, it is hard to make the claim that this is an adequate explanation for our results, although we cannot rule it out definitively.…”

“…For a given system, there exist as many Lyapunov exponents as degrees of freedom, which for real world Eulerian fluid turbulence is presumably infinite. To date the majority of such work, at least in the case of direct numerical simulation (DNS), has been concerned with the calculation of * Electronic address: ab@ph.ed.ac.uk † Electronic address: daniel-clark@ed.ac.uk only the largest Lyapunov exponent [13,14]. It is, however, possible to compute multiple exponents, leading to a partial Lyapunov spectrum, to obtain a more in-depth understanding of the chaotic properties of the system.…”

“…Deviations from K41 have been well studied in the literature [31][32][33][34], with both intermittency and finite Reynolds number effects claimed to be possible causes [35][36][37][38]. In studies of the maximal Lyapunov exponent [13,14] for HIT, different scaling behavior than that predicted by both K41 [39] and the multi-fractal model [40] was observed. As such, exact agreement with this simplified prediction for the KS entropy would be very unexpected, and indeed our results also display a conflicting scaling behavior.…”