Transition to turbulence in shock-driven mixing: a Mach number study

Abstract

Large-eddy simulations of single-shock-driven mixing suggest that, for sufficiently high incident Mach numbers, a two-gas mixing layer ultimately evolves to a late-time, fully developed turbulent flow, with Kolmogorov-like inertial subrange following a -5/3 power law. After estimating the kinetic energy injected into the diffuse density layer during the initial shock–interface interaction, we propose a semi-empirical characterization of fully developed turbulence in such flows, based on scale separation, as a function of the initial parameter space, as (η_(0^+)Δu/ν)(η_(0^+)/L_ρ)A^+/√(1-A^(+)^2) ≳ 1.53 × 10^4/C^2, which corresponds to late-time Taylor-scale Reynolds numbers ≳250. In this expression, η_(0^+) represents the post-shock perturbation amplitude, Δu the change in interface velocity induced by the shock refraction, ν the characteristic kinematic viscosity of the mixture, L_ρ the inner diffuse thickness of the initial density profile, A^+ the post-shock Atwood ratio, and C(A^+, η_(0^+)/λ_0)≈0.3 for the gas combination and post-shock perturbation amplitude considered. The initially perturbed interface separating air and SF_6 (pre-shock Atwood ratio A ≈ 0.67) was impacted in a heavy–light configuration by a shock wave of Mach number M_I = 1.05, 1.25, 1.56, 3.0 or 5.0, for which η_(0^+) is fixed at about 25% of the dominant wavelength λ_0 of an initial, Gaussian perturbation spectrum. Only partial isotropization of the flow (in the sense of turbulent kinetic energy and dissipation) is observed during the late-time evolution of the mixing zone. For all Mach numbers considered, the late-time flow resembles homogeneous decaying turbulence of Batchelor type, with a turbulent kinetic energy decay exponent n ≈ 1.4 and large-scale (k⟶0) energy spectrum ~k^4, and a molecular mixing fraction parameter, Θ ≈ 0.85. An appropriate time scale characterizing the Taylor-scale Reynolds number decay, as well as the evolution of mixing parameters such as Θ and the effective Atwood ratio A_e, seem to indicate the existence of low- and high-Mach-number regimes.

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