Journal of Fluid Mechanics

Papers

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

M. Lombardinia1 c1, D. I. Pullina1 and D. I. Meirona1

a1 Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA

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 $\ensuremath{-} 5/ 3$ 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 $({\eta }_{{0}^{+ } } \mrm{\Delta} u/ \nu )\hspace{0.167em} ({\eta }_{{0}^{+ } } / {L}_{\rho } )\hspace{0.167em} {A}^{+ } / \sqrt{1\ensuremath{-} {A}^{{ \mathop{ + }\nolimits }^{2} } } \gtrsim 1. 53\ensuremath{\times} 1{0}^{4} / {\mathscr{C}}^{2} $, which corresponds to late-time Taylor-scale Reynolds numbers ${\gtrsim }250$. In this expression, ${\eta }_{{0}^{+ } } $ represents the post-shock perturbation amplitude, $ \mrm{\Delta} u$ the change in interface velocity induced by the shock refraction, $\nu $ the characteristic kinematic viscosity of the mixture, ${L}_{\rho } $ the inner diffuse thickness of the initial density profile, ${A}^{+ } $ the post-shock Atwood ratio, and $\mathscr{C}({A}^{+ } , {\eta }_{{0}^{+ } } / {\lambda }_{0} )\approx 0. 3$ for the gas combination and post-shock perturbation amplitude considered. The initially perturbed interface separating air and SF6 (pre-shock Atwood ratio $A\approx 0. 67$) 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 ${\eta }_{{0}^{+ } } $ is fixed at about 25 % of the dominant wavelength ${\lambda }_{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 $n\approx 1. 4$ and large-scale ($k\ensuremath{\rightarrow} 0$) energy spectrum $\ensuremath{\sim} {k}^{4} $, and a molecular mixing fraction parameter, $\Theta \approx 0. 85$. An appropriate time scale characterizing the Taylor-scale Reynolds number decay, as well as the evolution of mixing parameters such as $\Theta $ and the effective Atwood ratio ${A}_{e} $, seem to indicate the existence of low- and high-Mach-number regimes.

(Received March 08 2011)

(Reviewed July 12 2011)

(Accepted September 25 2011)

(Online publication November 21 2011)

Key Words:

  • shock waves;
  • transition to turbulence;
  • turbulent mixing

Correspondence:

c1 Email address for correspondence: manuel@caltech.edu

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