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Study on the effect of Mach number and initial amplitudes on the evolution of a single-mode shock-induced hydro-dynamic instability
Abstract
In the present study the Mach number and the high-initial amplitudes effects on the evolution of the single-mode shock wave induced instability were investigated. To distinguish between the above-mentioned effects, two sets of shock-tube experiments were conducted: high-initial amplitudes with a low-Mach incident shock; and small amplitude initial conditions with moderate-Mach incident shock. In the high-amplitude experiments a reduction of the initial velocity with respect to the linear prediction was measured. The results were compared to those predicted by a vorticity deposition model and to previous experiments with moderate and high Mach numbers done by others and good agreement was found. The result suggested that the high-initial amplitude effect is the dominant one rather then the high-Mach number effect as suggested by others. In the small amplitude-moderate Mach numbers experiments a reduction from the impulsive theory was noted at late stages.
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Richtmyer–Meshkov instability is investigated for negative Atwood number and
two-dimensional sinusoidal perturbations by comparing experiments, numerical simulations
and analytic theories. The experiments were conducted on the NOVA laser
with strong radiatively driven shocks with Mach numbers greater than 10. Three
different hydrodynamics codes (RAGE, PROMETHEUS and FronTier) reproduce
the amplitude evolution and the gross features in the experiment while the fine-scale
features differ in the different numerical techniques. Linearized theories correctly calculate
the growth rates at small amplitude and early time, but fail at large amplitude
and late time. A nonlinear theory using asymptotic matching between the linear theory
and a potential flow model shows much better agreement with the late-time and
large-amplitude growth rates found in the experiments and simulations. We vary the
incident shock strength and initial perturbation amplitude to study the behaviour of
the simulations and theory and to study the effects of compression and nonlinearity.
Vorticity is deposited baroclinically by shock waves on density inhomogeneities. In two dimenslons, the circulation deposited on a planar interface may be derived analytically using shock polar analysis provided the shock refraction is regular . We present analytical expressions for Γ′, the circulation deposited per unit length of the unshocked planar interface, within and beyond the regular refraction regime. To lowest order, Γ′ scales as where M is the Mach number of the incident shock, η is the density ratio of the gases across the interface, α is the angle between the shock and the interface and γ is the ratio of specific heats for both gases. For α ≤ 30°, the error in this approximation is less than 10% for 1.0 < M ≤ 1.32 for all η > 1, and 5.8 ≤ η ≤ 32.6 for all M. We validate our results by quantification of direct numerical simulations of the compressible Euler equations with a second-order Godunov code.
We generalize the results for total circulation on non-planar (sinusoidal and circular) interfaces. For the circular bubble case, we introduce a ‘near-normality’ ansatz and obtain a model for total circulation on the bubble surface that agrees well with results of direct numerical simulations. A comparison with other models in the literature is presented.
Recent experiments by Meshkov have demonstrated that a shock‐accelerated perturbed surface separating two gases of different densities is unstable for shocks traveling from the lighter gas to the heavier gas and vice versa. Using a two‐dimensional Lagrangian hydrodynamic code, the authors show qualitative agreement with Meshkov's experiments. Experiment, numerical calculations, and linear analysis are compared. Good agreement was obtained between numerical calculations and linear analysis.
A potential flow model of Rayleigh–Taylor and Richtmyer–Meshkov bubbles on an interface between an incompressible fluid and a constant supporting pressure (Atwood number A=1) is presented. In the model, which extends the work of Layzer [Astrophys. J. 122, 1 (1955)], ordinary differential equations for the bubble heights and curvatures are obtained by considering the potential flow equations near the bubble tips. The model is applied to two-dimensional single-mode evolution as well as two-bubble competition, for both the Rayleigh–Taylor (RT) and the Richtmyer–Meshkov (RM) instabilities, the latter treated in an impulse approximation. The model predicts that the asymptotic velocity of a single-mode RM bubble of wavelength λ decays as λt−1, in contrast with the constant asymptotic velocity attained in the RT case. Bubble competition, which is believed to determine the multimode front evolution, is demonstrated for both the RT and RM instabilities. The capability of the model to predict bubble growth in a finite-thickness fluid layer is shown. Finally, the model is applied to the evolution of three-dimensional modes with an initial rectangular geometry. The model yields the aspect ratio dependence of the early nonlinear stages, in agreement with third-order perturbation theory. However, in the late nonlinear stage, the model predicts that the bubbles forget the initial geometry and attain the fastest growing shape, with a circular tip. The model results are in good agreement with full hydrodynamic simulations and analytic solutions, where available.
Shock-tube experiments were performed in order to verify recently developed theoretical models for the evolution of the shock-wave induced Richtmyer-Meshkov instability [Phys. Rev. Lett. 74, 534 (1995)]. Single-mode bubble and spike evolution and two-bubble interaction in both early and late nonlinear stages were investigated in a M approximate to 1.3 Air-to-SF6 shock-tube experiment. Experimental results for the single-mode and two-bubble cases, showing distinct bubble and spike evolution, were found to be in very good agreement with the theoretical model prediction as well as numerical simulations, verifying the keg elements of the bubble-merger model used for the prediction of the multimode bubble and spike front evolution.
The Richtmyer{endash}Meshkov instability is investigated with strong radiatively driven shocks (Mach{approx_gt}20, 5{times}compression) using the Nova laser. The target consists of a solid density ablator and lower-density plastics (Atwood number {ital A}{lt}0) in planar geometry to facilitate in-flight radiographic diagnosis. Perturbations {eta}={eta}{sub 0}cos{ital kx} are imposed at the interface to seed the instability. The experiments agree with full hydrodynamic simulations over a wide variety of conditions. For small initial amplitudes {parallel}{ital A}{parallel}{ital k}{eta}{sub 0}{lt}1, the growth rate agrees with a linear impulsive model using the average of the pre- and post-shock initial amplitudes. For {parallel}{ital A}{parallel}{ital k}{eta}{sub 0}{approx_gt}1, the growth rate is limited to the difference between the transmitted shock speed and the interface speed. {copyright} {ital 1996 American Institute of Physics.}
The nonlinear growth of the multimode Richtmyer-Meshkov instability in the limit of two fluids of similar densities (Atwood number A⃗0) is treated by the motion of point potential vortices. The dynamics of a periodic bubble array and the competition between bubbles of different sizes is analyzed. A statistical mechanics model for the multimode front mixing evolution, similar to the single-bubble growth and two-bubble interaction based model used by Alon et al. [Phys. Rev. Lett. 72, 2867 (1994)] for A=1, is presented. Using the statistical bubble merger model, a power law of t0.4 for the mixing zone growth is obtained, similar to that of the bubble front growth for the A=1 case and in good agreement with experiments and full numerical simulations.
The initial growth of irregularities on an interface between two ; compressible fluids is studied for impulsive (i.e., shock) acceleration. It was ; found that the ultimate rate of growth is roughly the same as that given by the ; incompressible theory, if the initial comnpression of the irregularities and of ; the fluids is taken into account. (auth)
The effect of the thin membrane on the time evolution of the shock wave induced turbulent mixing between the two gases initially
separated by it is investigated using two different sets of experiments. In the first set, in which a single-mode large-amplitude
initial perturbation was studied, two gas combinations (air/SF
and air/air) and two membrane thicknesses were used. The main conclusion of these experiments was that the tested membrane has a negligible
effect on the evolution of the mixing zone, which evolves as predicted theoretically. In the second set, in which similar
gas combinations and membrane thicknesses were used, small amplitude random-mode initial perturbation, caused by the membrane
rupture, rather than the large amplitude single-mode initial perturbation used in the first set, was studied. The conclusions
of these experiments were: (1) The membrane has a significant effect on the mixing zone during the initial stages of its growth.
This has also been observed in the air/air experiment where theoretically no growth should exist. (2) The membrane effect on the late time evolution, where the mixing
zone width has reached a relatively large-amplitude, was relatively small and in good agreement with full numerical simulations.
The main conclusion from the present experiments is that the effect of the membrane is important only during the initial stages
of the evolution (before the re-shock), when the perturbations have very small amplitudes, and is negligible when the perturbations
reach relatively large amplitudes.
Results are presented of an experimental study of the stability of the interface of two gases traversed by ashockwave. It is found that the interface is unstable both in the case of shock wave passage from the lighter to the heavier gas and for passage in the opposite direction. The interface disturbance grows linearly with time in the first approximation.
The material interface between two fluids of different densities is unstable under acceleration by a shock wave. This phenomenon is known as the Richtmyer-Meshkov instability. Theories have failed to provide quantitatively correct predictions for the growth rates of the unstable interface. Recently the authors have developed a quantitative theory based on the methods of Padé approximations and of asymptotic matching. In this letter, we extend our theory to the growth rates of the spike and bubble for the systems in two and three dimensions, and for systems with or without phase inversions. Our theoretical predictions are in excellent agreement with the results of full numerical simulations over the full time period from the initial linear (small amplitude) to moderately large amplitude nonlinear regimes.
The nonlinear evolution of large structure in Rayleigh-Taylor and Richtmyer-Meshkov bubble and spike fronts is studied numerically and explained theoretically on the basis of single-mode and two-bubble interaction physics at Atwood numbers A. Multimode Rayleigh-Taylor bubble (spike) fronts are found as hB = alphaBAgt2 [hs = alphasAgt2] with alphaB = 0.05, while Richtmyer-Meshkov bubble (spike) fronts are found as hB = aBtthetaB hs = astthetas\(A\) with thetaB = 0.4 at all A's. The dependence of these scaling laws and parameters on A and on initial conditions is explained.
Effects of high-Mach numbers and high initial amplitudes on the evolution of the single-mode Richtmyer-Meshkov shock-wave induced hydrodynamic instability are studied using theoretical models, experiments, and numerical simulations. Two regimes in which there is a significant deviation from the linear dependence of the initial velocity on the initial perturbation amplitude are defined and characterized. In one, the observed reduction of the initial velocity is primarily due to large initial amplitudes. This effect is accurately modeled by a vorticity deposition model, quantifying both the effect of the initial perturbation amplitude and the exact shape of the interface. In the other, the reduction is dominated by the proximity of the shock wave to the interface. This effect is modeled by a modified incompressible model where the shock wave is mimicked by a moving bounding wall. These results are supplemented with high initial amplitude Mach 1.2 shock-tube experiments, enabling separation of the two effects. It is shown that in most of the previous experiments, the observed reduction is predominantly due to the effect of high initial amplitudes.