^{1}

^{2}

^{1}

^{1}

A detailed numerical simulation of a shock accelerated heavy gas (SF
_{6}) cylinder surrounded by air gas is presented. It is a simplified configuration of the more general shock-accelerated inhomogeneous flows which occur in a wide variety of astrophysical systems. From the snapshots of the time evolution of the gas cylinder, we find that the evolution of the shock accelerated gas cylinder is in some ways similar to the roll-ups of a vortex sheet for both roll up into a spiral and fall into a self-similar behavior. The systemic and meaningful analyses of the negative circulation, the center of vorticity and the vortex spacing are in a good agreement with results obtained from the prediction of vorticity dynamics. Unlike the mixing zone width in single-mode or multi-mode Richtmyer-Meshkov instability which doesn’t exist, a single power law of time owing to the bubble and spike fronts follow a power law of
t
^{θ}
with different power exponents, the normalized length of the shock accelerated gas cylinder follows a single power law with
θ
= 0.43
in its self-similar regime obtained from the numerical results.

When an impulsive acceleration impinges on the corrugated interface between two fluids of different densities, the instability at the interface will arise due to the deposited vorticity induced by the baroclinic torque production term

In the RM instability researches, an interesting configuration is a shock wave interacting with a cylindrical interface (circular interface in two dimensions) between two fluids. When a planar shock wave impacts on a heavy gas (e.g., SF_{6}) cylinder around by an ambient gas (e.g., air), a shock wave is reflected and a refracted shock wave transmits into the heavy gas cylinder. Because the heavy gas acoustic impedance exceeds that of the ambient gas, the refracted shock is slower than the incident shock wave, and a convergent shock refraction pattern occurs. Because of this and the curvature of the cylinder, the transmitted shock focuses at the downstream vertex. This focusing will induce a pressure rising that eventually leads to a cusp-like protrusion [

The propagation of a shock wave in an inhomogeneous medium, and the response of the medium to impulsive acceleration are of fundamental interest in astrophysical systems. The evolution of the interstellar medium in spiral galaxies is significantly influenced by the strong shock waves generated by supernovae explosion [

There are many experimental and numerical researches of the shock-cylinder interactions which concentrate on different subjects. Haas and Sturtevant [

As be mentioned just, there were some studies on the shock accelerated heavy gas cylinder, however, in the aforementioned papers no attention has been paid on a perhaps existing scaling law in contrast with the single mode or multi mode RM instability [

This paper applies our large eddy simulation code MVFT (multi-viscous flow and turbulence) [

The initial conditions are significantly important in numerical simulations, especially for the membrane less RM instability researches. Initially, a sharp interface [

In the present simulations, the initial conditions were adapted to the Mach 1.2 shock tube experiment of Tomkins et al. [

Gases | Density (kg/m^{3}) | Specific Heat Ratio | Kinematic Viscosity (10^{−6} m^{2}/s) | Prandtl Number | Diffusion Coefficient in Air (cm^{2}/s) |
---|---|---|---|---|---|

Air | 0.95 | 1.40 | 15.7 | 0.71 | 0.204 |

SF_{6 } | 4.85 | 1.09 | 2.47 | 0.90 | 0.097 |

shock wave with the interface is calculated using a path integration of velocity

where

with

where

The vorticity generated by a shock wave propagating through a circular cross-section has been studied by Picon and Boris [

where ^{3}. From these data, one can get the amount of circulation is

Additionally, one can calculate the vortex strength by applying the approximate model of the compact vortex. When a shock impacts on a heavy gas cylinder, the cylinder will stand here relative to the ambient gas because of its inertia. Then the cylinder will have a velocity

In two dimensional case, as analogous to center of mass, one can define the coordinate of center of vorticity,

The time evolution of the center of vorticity is presented in the

Here,

The vorticity distribution along y direction computed on the fine mesh resolution at seven different times is shown in

Considering the mole fraction

The left and right edge locations of the cylinder,

To the authors’ knowledge, there hasn’t been an investigation of the power law of the length for a shock accelerated gas cylinder, although the power law followed by the mixing zone width has been investigated in the single mode or multi-mode Richtmyer-Meshkov instability. Dimonte and Schneider [

with

In conclusion, we studied the evolution of a shock impinging heavy gas cylinder and the growth of the normalized length. A self-similar behavior is shown from the snapshots of the evolution of the gas cylinder. The two dimensional numerical results of the negative circulation, the center of vorticity, and the vortex spacing are in a good agreement with the results obtained from the analyses of Picon and Boris [

This work was sponsored by the National Science Foundation of China under Grants No. 11202195 and No. 11072228 and the Science Foundation of the China Academy of Engineering Physics under Grants No. 2011B0202005 and No. 2011A0201002.