Magnetohydrodynamic effects in a shock-accelerated gas cylinder

This work presents two-dimensional (2D) simulations on a cylindrical Richtmyer-Meshkov instability (RMI) in magnetohydrodynamics (MHD). Three studies are presented in an effort to quantify and qualify the evolution of the MHD RMI by varying the magnetic field orientation, strength of the magnetic field, and strength of the shock wave driving the instability. The orientations considered herein are either parallel or perpendicular to the shock wave motion. The second study varies the magnetic fields between 100, 250, and 500 gauss (G), while the third study considers incident shock wave Mach numbers $M=1.2,M=1.66$, and $M=2.2$. These parameter ranges were selected to be easily achievable in experiments while the interface perturbation was selected such that its evolution is independent of either the shock wave or magnetic field orientations independently. It was found that the MHD RMI evolution is dependent upon the magnetic field orientation relative to the shock transit direction as well as their individual magnitudes. This is because the mechanism of suppression, attributed to Alfven waves, is a function of the magnetic field strength and the orientation of the magnetic field, while the mechanism of RMI evolution, baroclinic vorticity deposition, is a function of the Mach number. Stronger magnetic fields were found to provide greater mixing suppression and have significant effects on RMI-like interface morphology. Finally, increasing the shock wave strength generated competing effects between higher RMI vorticity deposition and greater vorticity removal from the interface by faster Alfven waves.

Figure 1

Two-dimensional (2D) schematic of the cylindrical Richtmyer-Meshkov instability driven by a shock wave in (a) traditional hydrodynamics and (b) with magnetohydrodynamic effects.

Figure 2

A 2D schematic representing the effects of the magnetic field orientation relative to the shock wave direction on vorticity. (a) The magnetic field is perpendicular to the shock wave direction cause vortex translation normal to the shock wave. (b) The magnetic field is now parallel to the shock wave, causing vortex translation parallel to the shock wave. In both magnetic field orientations, the site of maximum baroclinic vorticity deposition is $\left|\alpha \right|=pi/2$.

Figure 3

Two-dimensional (2D) species pseudocolor representing all initial conditions for this work. Here the shock wave is traveling in the +$Y$ direction and the interface is considered as a 2-cm diameter cylinder of $N_2$ at $T=2500$ K. The magnetic field, if present, may be parallel (along $Y)$ or perpendicular (along $X)$ to the direction of propagation of the shock and is not shown here. The image is centered about $y=0$.

Figure 4

Magnetic orientation study; pseudocolor of species for the $M1.7B0,M1.7B_{\perp }50$, and $M1.7B_{\parallel }50$ cases with respect to $\tau$. The images are centered about $y$, respective to $\tau$ at $y(\tau \approx 3)=3.11\mathrm{cm},y(\tau \approx 7)=7.64\mathrm{cm}$, and $y(\tau \approx 15)=15.55\mathrm{cm}$.

Figure 5

Mixedness, $\mathrm{\Xi }$, with respect to $\tau$ for the magnetic orientation study.

Figure 6

Late-time pseudocolors for $\omega$ and $\mathrm{\Xi }$ for the magnetic orientation study. The images are centered about $y$, respective to $\tau$ at $y(\tau \approx 3)=3.11\mathrm{cm},y(\tau \approx 7)=7.64\mathrm{cm}$, and $y(\tau \approx 15)=15.55\mathrm{cm}$.

Figure 7

The total and reduced area (local) enstrophy for the $M1.7B0,M1.7B_{\perp }5$, and $M1.7B_{\parallel }5$ cases.

Figure 8

The correlation coefficient with respect to $\tau$ for the magnetic orientation study.

Figure 9

Magnetic strength study. Pseudocolor of species for the parallel magnetic field cases with respect to $\tau$. The images are centered about $y$, respective to $\tau$ at $y(\tau \approx 3)=3.11\mathrm{cm},y(\tau \approx 7)=7.64\mathrm{cm}$, and $y(\tau \approx 15)=15.55\mathrm{cm}$.

Figure 10

Magnetic strength study. Pseudocolor of species for the perpendicular magnetic field cases with respect to $\tau$. The images are centered about $y$, respective to $\tau$ at $y(\tau \approx 3)=3.11\mathrm{cm},y(\tau \approx 7)=7.64\mathrm{cm}$, and $y(\tau \approx 15)=15.55\mathrm{cm}$.

Figure 11

The mixedness for the magnetic strength study. Here $B0$ is included to highlight the effects of the magnetic field.

Figure 12

Late-time vorticity field for the magnetic strength study. The images are centered about $y(\tau \approx 15)=15.55\mathrm{cm}$.

Figure 13

The correlation coefficient, given as Eq. (16), for the magnetic strength study.

Figure 14

The late-time species and vorticity plots for the incident shock wave strength study. The species pseudocolors are all on the same scale shown on the right of the image while the vortices are Mach number specific and shown below each group. The images are centered about $y$, respective to $M$ at $y(M=1.2)=7.60\mathrm{cm},y(M=1.7)=15.55\mathrm{cm}$, and $y(M=2.2)=19.40\mathrm{cm}$.

Figure 15

The mixedness for the incident shock-wave study. Small dashed lines indicate $M=1.22$, solid lines indicate $M=1.66$, and large dashed lines indicate $M=2.2$, while black represents cases without a magnetic field and blue represents cases with a perpendicular magnetic field of 500 G.

Figure 16

The correlation coefficients for the incident shock wave study. Small dashed lines indicate $M=1.22$, solid lines indicate $M=1.66$, and large dashed lines indicate $M=2.2$, while black represents cases without a magnetic field and blue represents cases with a perpendicular magnetic field of 500 G.