A parameterization of the Richtmyer–Meshkov instability on a premixed flame interface induced by the successive passages of shock waves

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The interactions of a sinusoidal premixed flame interface with successive shock waves are studied by numerically solving the two-dimensional, reactive Navier–Stokes equations with high resolution schemes. The focus is the development of the perturbations on the flame interface due to the Richtmyer–Meshkov (RM) instability. A single irreversible chemical reaction for a C2H4+3O2+4N2 mixture is used to represent the combustion process during the interactions. Three cases of interactions, with different initial incident shock wave strengths (Ma=1.2, 1.7 and 2.2), are studied. The multiple interactions between the flame interface and the shock and its reflection waves lead to interesting new observations that have not been presented before. To parameterize the interactions, we introduce a time scale decoupling assumption, and propose to use two decoupled time scales, i.e., the decoupled chemical reaction time scale τC and the decoupled RM instability time scale τRM. Based on this assumption, τC is considered to be decoupled from τRM, so that both time scales can be defined and determined independently. Specifically, τRM is defined as the reciprocal of vorticity magnitudes that can be obtained by performing the computations of two-dimensional RM instability in a chemically frozen flow that removes the effect of chemical reactions. On the other hand, τC is defined as the ratio of reaction zone width of an un-stretched, laminar, premixed flame to propagating speed of the flame, during the one-dimensional laminar flame propagation without considering two-dimensional RM instability. Using these definitions, our calculations show that both τC and τRM decrease with the successive interactions of the shock waves with the flame interface in all three cases, and that τC is larger than τRM at early stages of interactions and then rapidly decreases to a level comparable with τRM at later stages of interactions. We compare the two-dimensional visualizations of the perturbation development on the interface in reactive flows with those in chemically frozen flows. The comparison shows that RM instability plays an important role in early evolution, but chemical reactions dominate later, and that the evolution is correlated with the variations of τC and τRM for all cases studied here. The quantitative correlations between the ratio of τC to τRM and chemical heat release rate in all three cases are calculated. The good correlations indicate the advantage of the approaches to determine τC and τRM proposed in present study. The observation that the ratio of τC to τRM is independent of the initial strength of incident shock wave suggests that the ratio reflects the nature of reactive RM instability, hence provides a new dimensionless parameter for understanding such flows.