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, which corresponds to late-time Taylor-scale Reynolds numbers 250. In this expression, η 0 + represents the post-shock perturbation amplitude, 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 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.

Evolution of the density self-correlation in developing Richtmyer–Meshkov turbulence

“…λ L /λ v ≈ 1, despite the type of equilibrium observed in the evolution of the DSC. At Mach 1.2, this result is broadly consistent with the findings of Lombardini et al (2012) in a closely related flow, in which transition to fully developed turbulence (roughly corresponding to λ L /λ v = 10) was only observed for Mach 3.…”

“…These results suggest that perhaps the criterion λ L /λ v > 1 is not critical for this type of equilibrium to occur in the density field, but instead λ L /λ v ≈ 1 may be sufficient. These results are also broadly consistent with the findings of Lombardini et al (2012), who find in a large eddy simulation (LES) study of a shocked single interface that transition to fully developed turbulence occurs only for M 1.56, when measured by the presence of a −5/3 power law in the velocity spectra, and only for M 3.0, when measured by scale separation at late times. We may also consider the criterion developed by Zhou, Robey & Buckingham (2003a) and Zhou et al (2003b) for mixing transition in time-dependent flows.…”

On the Kolmogorov inertial subrange developing from Richtmyer-Meshkov instability

“…This finding is in contrast to that of Lombardini et al 16 where the authors found that the large scale spectrum, independently of Mach number, deviates from Saffman's spectrum and instead approaches a Batchelor type spectrum. However, the scaling in the limit of k → 0 may largely depend on the specific initial condition.…”

“…An estimate for the Taylor-scale Reynolds number in the inner mixing layer then follows Re λ ≈ 369, which is in good agreement with Lombardini et al 16 Taylor scale and Kolmogorov scale are related by λ η = 3 20…”

“…They also found a k −2 scaling as well as a classical Kolmogorov scaling after re-shock depending on the initial condition. Hill et al 15 and Lombardini et al 16 were able to verify the k −5/3 scaling by implicit large-eddy simulations at finite Reynolds numbers. However, the authors did not verify that their result is independent of grid resolution.…”