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 $R{e}_{\lambda }$. 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 ${\lambda }_3$ 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 ${\lambda }_1$, 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 ${\lambda }_1{\tau }_{\eta }$, where ${\tau }_{\eta }$ is the Kolmogorov time scale, is a decreasing function of $M_t$. Its dependence on $R{e}_{\lambda }$ is weak when the driving force is solenoidal, while it is an increasing function of $R{e}_{\lambda }$ when the solenoidal and compressible forces are comparable. Similar facts hold for $|{\lambda }_3|$, in contrast with well-studied short-correlated model; 2) The ratio of the first two Lyapunov exponents ${\lambda }_1/{\lambda }_2$ decreases with $R{e}_{\lambda }$, 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, ${\lambda }_1:{\lambda }_2:{\lambda }_3\approx 4:1:-5$ for $R{e}_{\lambda }>80$, which is consistent with incompressible turbulence studies; 4) The ratio of dilation-to-vorticity is a more suitable parameter to characterize LEs than $M_t$.

arXiv comment25 pages, 18 figures

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Cited by in corpus (1)

• Lagrangian stochastic integrals of motion in isotropic random flows