The Richtmyer-Meshkov instability (RMI) is the unbounded growth of any perturbations initially present on an interface between fluids of different acoustic impedance, consequent to the impulsive acceleration of the interface (e.g. by a shock wave). As the shock traverses the interface, two categories of phenomena take place: the refraction of the shock, involving the formation of a transmitted and a reflected disturbance and their geometrical distortion associated with focusing and defocusing effects; and the baroclinic generation of vorticity at the interface, consequent to the cross product of the density gradient (associated with the interface) and the pressure gradient (associated with the shock wave). It is this vorticity generation that is responsible for the amplification of the interface perturbations, and in this sense the RMI is the “impulsive analog” of the Rayleigh-Taylor instability (RTI), which develops at an interface subjected to sustained acceleration.
The simplest case to study is that of an infinitely thin (discontinuous), two-dimensional interface containing only a single-mode sinusoidal perturbation. Upon shock acceleration, the perturbation grows: initially (for as long as the amplitude remains less than about 10% of the wavelength) maintaining the original sinusoidal symmetry (linear stage); then asymmetrically, with narrow “spikes” of the heavy fluid penetrating the lighter fluid and broad “bubbles” of the light fluid entering the denser one (non-linear stage). Eventually, secondary instabilities develop on the interface, any well-defined shape is lost, the two fluids begin to mix, the flow approaches a turbulent state, and the interface is termed a “mixing layer”. More realistic initial conditions are diffuse (finite thickness), three-dimensional, with a spectrum containing many wavelengths. In this case, transition from the linear to the non-linear stages proceeds from the high- to the low-wavenumber modes and it involves strong modal interactions, but the interface still eventually transitions to a mixing layer.
Flows of this type occur in a variety of situations spanning enormous time, length, and energy ranges: from laser-driven experiments in the pursuit of inertial confinement fusion (ICF), to proposed configurations for hypersonic combustion engine inlets, to supernovae explosions. The shock-induced mixing has very negative effects in the case of ICF while it holds great potential to improve the combustion processes in the case of hypersonic engines.
Two of the fundamental parameters that govern these phenomena are the Atwood number (measuring the original density contrast) and the Mach number. A great variety of experiments, analytical model and numerical simulations have been developed over more than five decades since the pioneering work by Richtmyer [1] and Meshkov [2], initially concentrating on macroscopic properties (like the time histories of the individual amplitudes and the overall thickness of the mixing layer), and later extending to entire fields (concentration, velocity, vorticity), their spectral properties, and the parameters (degree of mixedness, mixing rate) that could be inferred from them.
In this overview, I will attempt to summarize the progress achieved over the course of 55 years by a large number of scientists from many parts of the world.
[1] R. D. Richtmyer, Taylor instability in shock acceleration of compressible fluids, Comm. Pure Appl. Math. 8, 297 (1960).
[2] Y.Y. Meshkov, Instability of the interface of two gases accelerated by a shock wave, Fluid Dyn. 4, 101 (1969) [Izv. Akad. Nauk SSSR Mekh. Zhdik. Gaza 5, 151 (1969)].
