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...

Direct Numerical Simulation is performed of the forced Navier-Stokes equation in four spatial dimensions. Well equilibrated, long time runs at sufficient resolution were obtained to reliably measure spectral quantities, the velocity derivative skewness and the dimensionless dissipation rate. Comparisons to corresponding two and three dimensional results are made. Energy fluctuations are measured and show a clear reduction moving from three to four dimensions. The dynamics appear to show simplifications in four dimensions with a picture of increased forward energy transfer resulting in an extended inertial range with smaller Kolmogorov scale. This enhanced forwards transfer is linked to our finding of increased dissipative anomaly and velocity derivative skewness.

Abstract

In the context of the analysis of the chaotic properties of homogeneous and isotropic turbulence, direct numerical simulations are used to study the fluctuations of the finite time Lyapunov exponent (FTLE) and its relation to Reynolds number, lattice size and the choice of the steptime used to compute the Lyapunov exponents. The results show that using the FTLE method produces Lyapunov exponents that are remarkably stable under the variation of the steptime and lattice size. Furthermore, it reaches such stability faster than other characteristic quantities such as energy and dissipation rate. These results remain even if the steptime is made arbitrarily small. A discrepancy is also resolved between previous measurements of the dependence on the Reynolds number of the Lyapunov exponent. The signal produced by different variables in the steady state is analyzed and the self decorrelation time is used to determine the run time needed in the simulations to obtain proper statistics for each variable. Finally, a brief analysis on MHD flows is also presented, which shows that the Lyapunov exponent is still a robust measure in the simulations, although the Lyapunov exponent scaling with Reynolds number is significantly different from that of magnetically neutral hydrodynamic fluids. * richard.ho@ed.ac.uk † andres.armua@ed.ac.uk ‡ ab@ph.ed.ac.uk 2

We study the properties of homogeneous and isotropic turbulence in higher spatial dimensions through the lens of chaos and predictability using numerical simulations. We employ both direct numerical simulations and numerical calculations of the eddy damped quasi-normal Markovian closure approximation. Our closure results show a remarkable transition to a non-chaotic regime above the critical dimension, $d_c$ , which is found to be approximately 5.88. We relate these results to the properties of the energy cascade as a function of spatial dimension in the context of the idea of a critical dimension for turbulence where Kolmogorov's 1941 theory becomes exact.