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 (η0+1u/ν) (η0+/Lρ)A+/ √ 1− A+ & 1.53 × 104/C 2, which corresponds to late-time Taylor-scale Reynolds numbers &250. In this expression, η0+ represents the post-shock perturbation amplitude, 1u the change in interface velocity induced by the shock refraction, ν the characteristic kinematic viscosity of the mixture, Lρ the inner diffuse thickness of the initial density profile, A+ the post-shock Atwood ratio, and C (A+, η0+/λ0) ≈ 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 MI = 1.05, 1.25, 1.56, 3.0 or 5.0, for which η0+ is fixed at about 25 % of the dominant wavelength λ0 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 lowand high-Mach-number regimes.