A general formula for the growth rate of the Richtmyer-Meshkov instability ͓R. D. Richtmyer, Commun. Pure Appl. Math. 13, 297 ͑1960͒; E. E. Meshkov, Sov. Fluid Dyn. 4, 101 ͑1969͔͒ is derived within the framework of the impulsive model. It allows us to predict the growth rate in both heavy-light and light-heavy configurations. This formula is validated over more than 100 cases with various values of the shock strength and the adiabatic exponents. The range of validity of the impulsive model is also specified. Comparisons are performed against the results available in the literature. This expression may be reduced to the Richtmyer or the Meyer-Blewett ͓Phys. Fluids 15, 753 ͑1972͔͒ formulas in particular cases. Specific configurations are built in order to emphasize the differences between the latter prescriptions and the formula proposed in this paper.

A novel method to set highly accurate initial conditions has been designed in the context of shock tube experiments for the Richtmyer-Meshkov instability study. Stereolithography has been used to design the membrane supports which initially materialize the gaseous interface. The visualizations of both heavy-light and light-heavy sinusoidal interfaces were carried out with laser sheet diagnostics. Experiments are in very good agreement with theory and simulations for the heavy-light case, but probably due to the membrane effects, quickly deviate from them in the light-heavy configuration.

In this paper we present a drastic simplification of the perturbation method for the Richtmyer–Meshkov instability developed by Zhang and Sohn [Phys. Fluids 9, 1106 (1997)]. This theory is devoted to the calculus of the growth rate of the perturbation of the interface in the weakly nonlinear stage. In the standard approach, expansions appear to be power series in time. We build accurate approximations by retaining only the terms with the highest power in time. This simplifies and accelerates the solution. High-order expressions are then easily reachable. Furthermore, computations for multimode interfaces become tractable. The accuracy of this approach is checked against two-dimensional numerical simulations. The selection mode process is studied and the phase between modes is shown to be as important as the wave number or the amplitude. Inferences for the intermediate nonlinear regime are also proposed. In particular, a class of homothetic configurations is inferred; its validity is verified with numerical simulations even as vortex structures appear at the interface.

International audienceThe nonlinear evolution of 2D single-mode Richtmyer-Meshkov instabilities is investigated through experiments in shock tube and numerical simulations. In our shock tube, the interface is materialized by a thin membrane attached to a stereo-lithographed grid. The purpose of this study is to compare experimental and numerical results, verify that using a higher Mach number for the incident shock wave (M-isw) than in a previous study [C. Mariani, M. Vandenboomgaerde, G. Jourdan, D. Souffland, and L. Houas, ``Investigation of the Richtmyer-Meshkov instability with stereolithographed interfaces,'' Phys. Rev. Lett. 100, 254503 (2008)] drastically reduces the deleterious effects of the membrane remnants, explore the effect of a high initial amplitude at the interface on the growth of the perturbation, and understand the lack of roll-up structures in the nonlinear phase of the instability. Using grayscale gradient rather than gray level, a new processing of the raw pictures is developed. Numerical simulations run with the TRICLADE code show that adding short-wavelength (swl) perturbations to the single-mode interface is necessary to understand the morphology of the experimental pictures. Through these comparisons, a minimal description of the swl perturbations is proposed. The experimental growth rates are correctly predicted by the TRICLADE code. Positive agreements are also obtained by models only if they take into account the effect of a high initial amplitude. A first result is the agreement that we obtain between experiment, simulations, and models indicates that the membrane remnants no longer influence the global dynamics of the instability. This is due to the increase of M-isw from 1.15 to 1.45. Another result of this investigation is that swl perturbations disrupt the spikes which grow at the interface and prevent the appearance of roll-up structures. Numerical simulations show the importance of a diffuse transition layer at the initial interface as a stabilizing process of the swl perturbations. In our case, only such a layer would allow the roll-up structures to grow. Finally, our analysis confirms that, when they are present at the same time, the swl perturbations distort the long wavelength perturbations, and that the latter decreases the growth of the former. (C) 2014 AIP Publishing LLC