Quantitative theory for spikes and bubbles in the Richtmyer-Meshkov instability at arbitrary density ratios

To predict the growth rates of spikes and those of bubbles at a Richtmyer-Meshkov unstable interface between two fluids of arbitrary density ratios and over the entire nonlinear evolution stage is very important. So far most theories are applicable to bubbles only. There are no accurate theories available in the literature for spikes in systems with finite density ratios. In this paper, we present a theory that is applicable to both spikes and bubbles. Our theoretical predictions are in good agreement with numerical data for both spikes and bubbles over a wide range of density ratios and with various initial conditions. The theoretical predictions also agree well with experimental results.

Figure 1

Evolution of spikes and bubbles which characterize the development of the mixing layer in Richtmyer-Meshkov instability. The Layzer approximation approximates the shape of the interface near the tip of the finger as a parabola, shown as the blue curves in (d).

Figure 2

Comparisons between theoretical predictions given by Eqs. (18) and (21) (blue solid curves), theoretical predictions given by Goncharov [10] (red dashed curves), and numerical results from Ref. [16] (black circle marks) for $a_{0}k=0.125$ and various Atwood numbers $A=\pm 0.98,\pm 0.94,\pm 0.88,\pm 0.75,\pm 0.5,\pm 0.25$. Outsets: Growth rates for spikes. Insets: Growth rates for the corresponding bubbles.

Figure 3

Comparisons between theoretical predictions given by Eqs. (18) and (21) (blue solid curves), theoretical predictions given by Goncharov [10] (red dashed curves), and numerical results from Ref. [17] (black circle marks) for $a_{0}k=0.5$ and various Atwood numbers $A=\pm 1,\pm 0.7,\pm 0.3,0$. Outsets: Growth rates for spikes. Insets: Growth rates for the corresponding bubbles.

Figure 4

Comparisons between theoretical predictions given by Eq. (21) (blue solid curves) and experimental data (black circle marks). The data in (a) and (b) are from Ref. [19]. The data in (c) and (d) are from Ref. [20]. The parameters are listed in the individual subpanels.

Figure 5

Comparison for the spike growth rate between the results from the asymptotically matched solution given by Eqs. (18) and (21) (blue solid curve), from the asymptotic large-time solution in Ref. [13] (black dashed-dotted curve), from the small-time solution in Ref. [14] (magenta dotted curve), and the numerical results in Ref. [16] (black circle marks). The physical parameters are the same as in Fig. 2. The figure shows that the small-time solution, the asymptotic large-time solution, and the asymptotically matched solution presented in this paper agree with the numerical results at early times, at late times, and over the entire time domain, respectively.