Rayleigh–Taylor stability for a normal shock wave–density discontinuity interaction

Abstract

The solution for the perturbation growth of a shock wave striking a density discontinuity at a material interface is developed. The Laplace transformation of the perturbation results in an equation which has a simple solution for weak shock waves. The solution for strong shock waves may be given by a power series. It is assumed that the equation of state is that of an ideal gas. The four independent parameters of the solution are the ratio of specific heat for each material, the density ratio at the interface, and the incoming shock strength. Properties of the solution which are investigated include the asymptotic behavior at large times of the perturbation velocity at the interface, the vorticity near the interface, and the rate of decay of the solution at large distances from the interface. The last is much weaker than the exponential decay in an incompressible fluid. The asymptotic solution near the interface, in addition to a constant term, consists of a number of slowly decaying discrete frequencies. The number is roughly proportional to the logarithm of the density ratio at the surface for strong shocks, and decreases with shock strength. For weak shocks the solution is compared with results for an incompressible fluid. Only interface perturbation velocities which tend to zero at large times lead to a limited deformation of the interface. It is found that these are possible only for density ratios less than about 1.5.

Richtmyer–Meshkov instability: theory of linear and nonlinear evolution

“…As the shocks separate away from the interface, their corrugation amplitude will oscillate and decrease in time. This well-known fact was predicted theoretically and observed in experiments/simulations and documented over more than 50 years of research on corrugated shocks, either in shock tubes or, recently, with high-power lasers ( Briscoe & Kovitz 1968;Fraley 1986;Velikovich & Phillips 1996;Wouchuk & Nishihara 1996). For an ideal gas EOS, it can be rigorously proven that the amplitude of those oscillations decreases asymptotically in time as t −3/2 for any shock with finite intensity (Zaidel 1960;Fraley 1986;Wouchuk & Cavada 2004).…”

“…The Richtmyer-Meshkov instability (RMI) was predicted by Richtmyer more than 50 years ago; since then it has attracted the attention of scientists worldwide because of its importance in different scenarios (Richtmyer 1960;Meshkov 1969;Fraley 1986;Holmes et al 1999;Zabusky 1999;Velikovich et al 2000Velikovich et al , 2007. Richtmyer studied the problem numerically for a planar shock refracted normally at a contact surface separating two different ideal gases, assuming that the material interface had an initial sinusoidal corrugation.…”

Startup process in the Richtmyer–Meshkov instability

“…2 Besides, numericists and experimentalists addressed the effect of shock proximity by using empirical corrections to the impulsive growth rate. [3][4][5] More complex, semianalytical studies have taken into account all relevant phenomena 6,7 and showed good agreement with numerical results obtained by linearizing the Euler equations between the perturbed interface and transmitted/reflected waves, 8 and with the linear interaction analysis at low Atwood numbers of Griffond. 9 In what follows, by modeling the proximity of the receding transmitted and reflected shocks, the analysis in Sec.…”

“…This latter ratio can be greater than unity in the heavy region 1 when the incident shock Mach number M I is very high but remains less than 10 as long as A + is not too close to unity and the incident shock is not too strong. For example, in the case of a "light air→ heavy SF 6 " shock interaction at M I = 8.0, whose Atwood number A + Ӎ 0.7 is quite high ͑1:5 density ratio͒ in the heavy region ⌬W / a 1 Ӎ 4.2.…”

“…Figure 16 in Ref. 8 ͑reflected shock case͒ shows that, for the air→ SF 6 case, which we chose to study in detail in Sec. IV, the disagreement with Richtmyer's mode grows as the incident shock strength increases.…”