Lyapunov exponents and Lagrangian chaos suppression in compressible homogeneous isotropic turbulence

Abstract

We study Lyapunov exponents of tracers in compressible homogeneous isotropic turbulence at different turbulent Mach number $M_t$ and Taylor-scale Reynolds number $R{e}_{\hat{I}}»$. We demonstrate that statistics of finite-time Lyapunov exponents have the same form as in incompressible flow due to density-velocity coupling. Modulus of the smallest Lyapunov exponent $\hat{I}{»}_3$ provides the principal Lyapunov exponent of the time-reversed flow, which usually is wrong in a compressible flow. This exponent, along with the principal Lyapunov exponent $\hat{I}{»}_1$, determines all the exponents due to the vanishing of the sum of all Lyapunov exponents. Numerical results by high-order schemes for solving the Navier-Stokes equations and tracking particles verify these theoretical predictions. We found that: 1) The largest normalized Lyapunov exponent $\hat{I}{»}_1\ddot{I}{„}_{\hat{I}}\cdot$, where $\ddot{I}{„}_{\hat{I}}\cdot$ is the Kolmogorov time scale, is a decreasing function of $M_t$. Its dependence on $R{e}_{\hat{I}}»$ is weak when the driving force is solenoidal, while it is an increasing function of $R{e}_{\hat{I}}»$ when the solenoidal and compressible forces are comparable. Similar facts hold for $|\hat{I}{»}_3|$, in contrast with well-studied short-correlated model; 2) The ratio of the first two Lyapunov exponents $\hat{I}{»}_1/\hat{I}{»}_2$ decreases with $R{e}_{\hat{I}}»$, and is virtually independent of $M_t$ for $M_t\le 1$ in the case of solenoidal force but decreases as $M_t$ increases when solenoidal and compressible forces are comparable; 3) For purely solenoidal force, $\hat{I}{»}_1:\hat{I}{»}_2:\hat{I}{»}_3\approx 4:1:-5$ for $R{e}_{\hat{I}}»>80$, which is consistent with incompressible turbulence studies; 4) The ratio of dilation-to-vorticity is a more suitable parameter to characterize LEs than $M_t$.

25 pages, 18 figures