Analytical linear theory for Richtmyer-Meshkov instability in shock-flame interactions*

Shock-flame interactions play a critical role in various combustion applications, including flame acceleration and control in supersonic propulsion systems. In safety combustion technology, understanding shock-flame catch-up is particularly important as the post-interaction flow field can lead to deflagration-to-detonation transition. The increased flame surface and temperature resulting from the interaction contribute to enhanced flame speed. This work aims to quantify the deformation rate of the flame surface and burning velocity caused by a planar shock crossing a corrugated laminar flame from behind. During the interaction, a pair of corrugated transmitted and reflected shocks are generated, resembling the canonical Ricthmyer-Meshkov instability (RMI). At times significantly shorter than the characteristic flame burning time, the flame front deforms as an inert interface dominated by the RMI. To understand this phenomenon better, we employ a comprehensive, fully-analytical linear theory that explicitly describes the flame growth rate. Our analytical approach accounts for the unbalanced shock-generated tangential velocities across the flame interface, obtaining both transient and asymptotic growth rates and providing a comprehensive analysis of the flame evolution during the interaction process.