Richtmyer-Meshkov instability of a flat interface subjected to a rippled shock wave

The Richtmyer-Meshkov (RM) instability of a nominally flat interface (${\mathrm{N}}_2/{\mathrm{SF}}_6$) subjected to a rippled shock, as the counterpart of a corrugated interface interacting with a planar shock, is studied experimentally in a vertical shock tube using both schlieren photography and fog visualization diagnostics. The nonplanar incident shock wave is produced by a planar shock diffracting around a rigid cylinder, and the flat interface is created by a membraneless technique. Three different distances $\eta$ (the ratio of spacing from cylinder to interface over cylinder diameter) are considered. Schlieren images indicate that the nonplanar incident shock can be divided into three different segments separated by two triple points. Fog visualization pictures show the formation of overall “$\mathrm{\Lambda }$” shaped interface structures and a ${\mathrm{N}}_2$ cavity at the center and two interface steps at both sides. With the increase of the dimensionless time, the dimensionless interface amplitude increases as well as the penetration depth of the cavity, and both curves exhibit reasonable collapse for different $\eta$ numbers. Through equating the preinterface perturbation of the rippled shock with a preshock perturbation of a corrugated interface, the growth rate of this instability is found to be noticeably smaller than that of the standard RM instability.

Figure 1

Sketch of the vertical shock tube showing the nonplanar incident shock formation and the technique to create the initial planar interface. $d$, cylinder diameter; $l$, distance from the center of the cylinder to the interface.

Figure 2

Schlieren photography (a) and fog visualization diagnostics (b).

Figure 3

Schlieren pictures displaying the wave pattern after the planar shock wave diffracts around a 10-mm-diameter rigid cylinder, The initial shock propagates from top to bottom, and $s$ denotes the distance from the cylinder to the planar incident shock front. IS, incident shock; RS, reflected shock; TP, triple point; MS, Mach stem.

Figure 4

(a) Schlieren pictures showing the shapes of the rippled incident shock and the interface. (b) Quantitative shape of the incident shock front extracted from the schlieren images. $a_0$, initial amplitude of rippled shock; $W_0$, initial width of rippled shock; ${\lambda }_0$, initial wavelength of rippled shock.

Figure 5

Fog visualization sequences showing the evolution of the planar interface impacted by the rippled shock for $\eta =2.0$ (a), $\eta =3.3$ (b), and $\eta =4.0$ (c). Solid circles indicate the ${\mathrm{N}}_2$ cavity and dashed circles indicate the steps. $a$: amplitude from top to bottom of the interface; $h$: height from top of the interface to bottom of the cavity; $A$: area of the cavity.

Figure 6

(a) Shape of the rippled incident shock wave; (b) schematic of the incident angle ($\theta$) of the shock front with the interface; (c) schematic of the pressure ($p$) distribution behind the rippled incident shock wave; (d) corresponding fog visualization image.

Figure 7

Time variation of the dimensionless geometrical sizes of the distorted interface for three cases. (a) Amplitude from top to bottom of the interface; (b) height from top of the interface to bottom of the cavity; (c) area of the cavity.