Small amplitude theory of Richtmyer–Meshkov instability

Abstract

This paper presents a new analysis of small amplitude Richtmyer–Meshkov instability. The linear theory for the case of reflected rarefaction waves, a problem not treated in previous work, is formulated and numerically solved. This paper also carries out a systematic comparison of Richtmyer’s impulsive model to the small amplitude theory, which has identified domains of agreement as well as disagreement between the two. This comparison includes both the reflected shock and reflected rarefaction cases. Additional key results include the formulation of criteria determining the reflected wave type in terms of preshocked quantities, identification of parameter regimes corresponding to total transmission of the incident wave, discussion of an instability associated with a rarefaction wave, investigation of phase inversions and the related phenomenon of freeze-out, and study of the sensitivity of the numerical solutions to initial conditions.

Computing multi-mode shock-induced compressible turbulent mixing at late times

“…The accuracy of the model sensibly improves when the post-shocked quantities a + 0 and A + t are employed (Richtmyer 1960). A discussion on the agreement and disagreement between compressible linear theory, based on the linearization of the Euler equations in one space dimension, and the impulsive model was discussed in (Velikovich and Dimonte 1996;Yang et al 1994). Large eddy and direct numerical simulations can greatly benefit from comparing the numerical results with theoretical results, including zero-order, linear, weakly nonlinear and highly nonlinear theories, similar to Stanic et al (2012).…”

Stability of stagnation via an expanding accretion shock wave

“…42 The difference is that in the latter case, we have one unstable eigenmode that is regular at 0 t + → . There is a single radial eigenmode that describes the instability development, and all the studies of the classical RMI focus on it; [42][43][44] δ makes it possible to use the log-log scale, illustrating the rapid decay of these perturbations ensured by the large negative real parts of n σ . Frequency of the oscillations is seen to increase with n as a result of the increased imaginary part of n σ .…”

A method to simulate linear stability of impulsively accelerated density interfaces in ideal-MHD and gas dynamics

“…Such simulations exploit the fact that the base state is a self-similarly evolving solution with a finite number of waves (a transmitted shock, a contact discontinuity, and either a reflected shock or a reflected rarefaction). The reflected and transmitted waves usually bound the computational domain for computing the linear quantities [3]. Analytical extensions of the linear stability analysis when the base state is more complicated than the one considered by Richmyer and Yang et alhave not been performed.…”

“…Richtmyer analyzed with the interaction of a shock wave with a perturbed contact discontinuity separating gases of different densities and concluded that the perturbations on the contact discontinuity grew linearly with time. The linear analysis was further developed by Yang et al [3] in which they also considered the case when the reflected wave is a rarefaction. Computing the linear response for both the reflected shock and reflected rarefaction cases requires special consideration for each case, and a different set of equations must be solved.…”