Rayleigh-Taylor Unstable Flames: the Effect of Two-Mode Coupling

In the classical Rayleigh-Taylor (RT) instability, initial conditions are forgotten and the growth of the mixing layer becomes self-similar when short wavelength modes couple to generate longer wavelength modes. In this paper, we explore how adding a reaction at the unstable interface affects this inverse cascade in wavenumber ("inverse k-cascade"). We simulate a 2D, Boussinesq, premixed model flame perturbed by a large amplitude primary mode ($k_1$) and a smaller amplitude secondary mode ($k_2$). Early on, the modes are uncoupled and the flame propagates as a metastable traveling wave. Once the secondary mode has grown large enough, the modes couple. The traveling wave is destabilized and the flame front bubbles rapidly grow. This inverse k-cascade, driven by two-mode coupling, ultimately generates a long wavelength mode with wavenumber GCD$(k_1,k_2)$, where GCD is the greatest common divisor. We identify five distinct flame growth solution types, and show that the flame may stall, develop coherent pulsations, or even become a metastable traveling wave again depending on GCD$(k_1,k_2)$. Finally, we compare our results with two-mode coupling in ablative and classical RT and show that all three systems may follow the same mode coupling dynamics.