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 numbers Mt and Taylor-scale Reynolds numbers Reλ. We demonstrate that statistics of finite-time Lyapunov exponents have the same form as that in incompressible flow due to density-velocity coupling. The modulus of the smallest Lyapunov exponent λ3 provides the principal Lyapunov exponent of the time-reversed flow, which is usually wrong in a compressible flow. This exponent, along with the principal Lyapunov exponent λ1, determines all the exponents due to 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 λ1τη, where τη is the Kolmogorov timescale, is a decreasing function of Mt. Its dependence on Reλ is weak when the driving force is solenoidal, while it is an increasing function of Reλ when the solenoidal and compressible forces are comparable. Similar facts hold for |λ3|, in contrast to well-studied short-correlated model; (2) the ratio of the first two Lyapunov exponents λ1/λ2 decreases with Reλ and is virtually independent of Mt for Mt≤1 in the case of solenoidal force but decreases as Mt increases when solenoidal and compressible forces are comparable; (3) for purely solenoidal force, λ1:λ2:λ3≈4:1:−5 for Reλ>80, which is consistent with incompressible turbulence studies; and (4) the ratio of dilation-to-vorticity is a more suitable parameter to characterize Lyapunov exponents than Mt.