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

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}⁺ } {Δ} u/ ν ){0.167em} ({η }_{{0}⁺ } / {L}ρ ){0.167em} {A}⁺ / √{1{-} {A}^{{ { + } }² } } 1. 53{×} 1{0}⁴ / {C}² $ , which corresponds to late-time Taylor-scale Reynolds numbers ${ }250$ . In this expression, ${η }_{{0}⁺ } $ represents the post-shock perturbation amplitude, $ {Δ} u$ 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 SF6 (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 ~ k⁴ , 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 Aₑ , seem to indicate the existence of low- and high-Mach-number regimes.