Mach number effect on the instability of a planar interface subjected to a rippled shock

The Richtmyer-Meshkov (RM) instability of a planar interface (${\mathrm{N}}_2-{\mathrm{SF}}_6$) subjected to a sinusoidal rippled shock, as the variant of a sinusoidal interface impinged by a planar shock, is investigated through high-order compressible multicomponent hydrodynamic simulations. The rippled shock is generated by a planar shock penetrating through a single-mode interface ($\mathrm{He}-{\mathrm{N}}_2$), and its propagation characteristic agrees reasonably with Bates' analytical solution. Evolution of the flat contact surface impacted by the rippled shock is found to be heavily dependent on the rippled shock phase, and it can be well explained by the impulsive perturbation and continuous perturbation regimes. Various rippled shocks with different Mach numbers ranging from 1.15 to 1.80 are considered. It is found that the influence of the shock strength on the instability growth behaves differently for rippled shocks at different phases. In the case that the shock-interface collision happens when the rippled shock amplitude vanishes for the first time, as the shock strength increases, the impulsive perturbation (i.e., amplitude growth caused by the impulsive shock impact) plays an increasingly more important role in the instability growth than the continuous perturbation (i.e., amplitude growth induced by the disturbed postshock pressure field). In contrast, in the case that the impingement occurs when the rippled shock amplitude becomes zero for the second time, the instability development contributed by the impulsive perturbation is a certain percentage of the total instability growth regardless of the shock strength. The role of the impulsive perturbation in the present nonstandard RM instability within the single-mode framework can be reasonably predicted by an empirical formula combined with the model of Ishizaki et al. [Phys. Rev. E 53, R5592 (1996)].

Figure 1

Schematic of the flow configurations at (a) the initial state, (b) the time moment when the incident shock (IS) completely passes across the contact interface (CI), and (c) the moment when the transmitted shock (TS) encounters the planar interface (PI). RS is the reflected shock.

Figure 2

Evolution of the ratio between the generated shock amplitude and its initial value as well as dimensionless variations of the amplitude growth rate of a rippled shock versus time under a Mach number of (a) $M_S=1.15$, (b) $M_S=1.34$, (c) $M_S=1.47$, and (d) $M_S=1.80$. The time is scaled as $\tau =t{W}_{\mathrm{TS}}/\lambda$ where $\lambda$ is the wavelength of the perturbation. The dash-dot and dash lines denote, respectively, the variations of the shock amplitude and growth rate predicted by Bates [30], and the open circle represents the present numerical result. The blue and red horizontal lines indicate a zero value of $a_S$ and $d{a}_S/d\tau$, respectively.

Figure 3

Density contour images at $\tau =10$ for a rippled shock at (a) phase 1, (b) phase 2, (c) phase 3, and (d) phase 4. BP and MP are, respectively, the boundary and middle points of the deformed planar interface (PI). The dash and solid lines denote the initial locations of the planar interface and contact interface (CI), respectively.

Figure 4

Dimensionless variations of the amplitude (a) and the amplitude growth rate (b) of the PI versus time for different shock phase cases. The dimensionless time follows the same normalization method as that in Fig. 2.

Figure 5

Density contour images at $\tau =10$ for the phase-1 rippled shocks with different Mach numbers: (a) $M_S=1.15$, (b) $M_S=1.34$, (c) $M_S=1.47$, and (d) $M_S=1.80$. The dash and solid lines are the initial positions of the PI and CI, respectively.

Figure 6

Dimensionless variations of the amplitude (a) and the amplitude growth rate (b) of the PI versus time for a rippled shock at phase 1 with different strengths.

Figure 7

Pressure distributions along the symmetry axis for the PI subjected to a rippled shock of $M_S=1.34$ at (a) phase 1, (b) phase 2, and (c) phase 3. The inserted figures show the pressure gradients in the vicinity of the evolving PI at $\tau =0.1$.

Figure 8

The distributions of the pressure (a) and pressure gradient (b) along the symmetry axis in the vicinity of the evolving PI for the different Mach number cases at $\tau =0.1$.

Figure 9

Dimensionless variations of the normalized amplitude (a) and growth rate (b) versus the time for a rippled shock at phase 3 with different Mach numbers.

Figure 10

Variation of the parameter $\varepsilon$ versus the shock Mach number. The dash-dot line denotes the empirical prediction of Eq. (8), and the gradients and deltas refer to the present numerical results under a rippled shock at phases 1 and 3, respectively.

Figure 11

Comparison of the asymptotic growth rate between the present numerical results and the impulsive model prediction for the phase-1 (a) and phase-3 (b) cases. The delta denotes the asymptotic growth rate obtained in Figs. 6 and 9. The gradient stands for the theoretical results calculated based on the impulsive model, and the square for the ratio of the above two growth rates $\delta =\left[{(d{a}_I/dt)}_{t\to \infty }\right]/\left[{(d{a}_I/dt)}_{\mathrm{Richtmyer}}\right]$.