Atwood number effects on the instability of a uniform interface driven by a perturbed shock wave

The evolution of a uniform interface subjected to a perturbed shock wave has been experimentally studied over a range of Atwood numbers $0.22\le A\le 0.68$ and Mach numbers $1.2\le M\le 1.8$ using a vertical shock tube. The perturbed shock wave is produced by diffracting a planar incident shock over a rigid cylinder. The wave patterns of the perturbed shock are captured by high-speed shadowgraphy, while the evolution of the shocked interface is captured by planar Mie scattering. Besides the formations of a cavity and two steps, an apparent counter-rotating vortex pair emerges on the shocked interface due to the baroclinic vorticity deposition, as both the Atwood number and Mach number increase. Quantitatively, it is interesting to note that the amplitude growth rate of the shocked interface decreases with increasing the Atwood number, which is fundamentally different from the results related to the classical RM instability. This notable feature is explained by the approximation of an oblique shock hitting a uniform interface. For weak shock, a suitable time scaling is employed to collapse experimental data irrespective of the Atwood number difference.

Figure 1

Schematics of the test section of the shock tube with two visualization techniques: (a) shadowgraphy and (b) planar Mie scattering.

Figure 2

Shadowgraph pictures showing the wave patterns of the perturbed shock just before the shock impacts the uniform interface: (a) $M=1.2$; (b) $M=1.5$; (c) $M=1.8$. The perturbed shock propagates from top to bottom with amplitude $a_{\text{s}}$ and wavelength ${\lambda }_{\text{s}}$. Notation: IS, incident shock; RS, reflected shock; MS, Mach stem; TP, triple point.

Figure 3

Variations of the amplitude $a_{\text{s}}$ (a) and the wavelength ${\lambda }_{\text{s}}$ (b) of the perturbed shock with the distance between the incident shock (IS) and the cylinder center, $t{V}_{\text{s}}$. The dotted line indicates the location of the uniform interface.

Figure 4

Comparisons of the amplitude $a_{\text{s}}$ (a) and the wavelength ${\lambda }_{\text{s}}$ (b) of the perturbed shock from the present experiments with those from the numerical results in Ref. [16].

Figure 5

Mie scattering sequences showing the evolutions of the shocked interfaces with different initial Atwood numbers $A$ at Mach number $M=1.2$. Numbers indicate time in $\mu \mathrm{s}$.

Figure 6

Mie scattering sequences showing the evolutions of the shocked interfaces with different initial Atwood numbers $A$ at Mach number $M=1.5$. Numbers indicate time in $\mu \mathrm{s}$. Rectangle indicates the ${\mathrm{N}}_2$ cavity and circles indicate the steps. $a_{\text{i}}$ and ${\lambda }_{\text{i}}$ are the amplitude and wavelength of the shocked interface, respectively.

Figure 7

Mie scattering sequences showing the evolutions of the shocked interfaces with different initial Atwood numbers $A$ at Mach number $M=1.8$. Numbers indicate time in $\mu \mathrm{s}$. Circles indicate the vortex pair.

Figure 8

Schematic of the baroclinic vorticity deposition on the uniform interface. The perturbed shock propagates downward toward the uniform interface and applies a pressure gradient $\mathbf{\nabla }p$ across the density gradient $\mathbf{\nabla }\rho$, which deposits locally baroclinic vorticity on the interface.

Figure 9

Variations of the amplitude of the shocked interface $a_{\text{i}}$ with time $t$ for different initial Atwood numbers at Mach number (a) $M=1.2$, (b) $M=1.5$, and (c) $M=1.8$.

Figure 10

Schematic of the transmission and reflection of an oblique shock segment crossing a uniform interface initially at rest. ${\dot{a}}_{\text{s}}\left(t\right)$ and ${\dot{a}}_{\text{i}}\left(t\right)$ are the amplitude growth rates of the rippled shock and the shocked interface, respectively. $V_{\text{s}}$ and $u$ are the velocities of the shock and the shocked interface, respectively. Notation: IS, incident shock; RS, reflected shock; TS, transmitted shock; S, initial interface; ${\mathrm{S}}^{\prime }$, shocked interface.

Figure 11

Variations of the normalized amplitude of the shocked interface, $a_{\text{i}}/a_{\text{s0}}$, with the normalized time, $k\mathrm{\Delta }Vt$, for different initial Atwood numbers at Mach number (a) $M=1.2$, (b) $M=1.5$, and (c) $M=1.8$. $k=2\pi /{\lambda }_{\text{s0}}$ is the wave number.