The shock-cylinder interaction, which involves hydrodynamic instability, complex wave pattern and vortex dynamics, has been extensively studied because of its applications such as supersonic combustion ramjet, supernovae explosion and inertial confinement fusion (Ranjan et al. 2011). When impulsively accelerated by the single or multiple shock waves (Malamud et al. 2014), the cylinder undergoes a great deformation. The evolution of the volume is dependent on deposited vorticity, which drives the flow even if the shocks are away. The dominant source of the deposited vorticity is the baroclinic vorticity, because of misalignment of the pressure and density gradients in the flow. The effects of pressure gradients on the shock-cylinder interaction have been widely studied (Si et al. 2014), but there still exists a lack of studies on the effects of the density gradients, especially the density distribution of an inhomogeneous volume. The present work is intended to study how density distributions influence the hydrodynamic instability of the shocked gaseous cylinder, not only under the impact of the incident shock but also its reshock.
Figure 1(a) shows the configuration of numerical simulations which are performed by VAS2D (Sun et al. 1999). The Mach number of the incident shock is 1.23. The diameter of the SF6 cylinder D0 is 35 mm. The initial temperature and pressure are 288 K and 101 kpa, respectively. Figure 1(b) shows the different density distribution of the unshocked cylinders in sharp and Gaussian types. Figure 2 shows the numerical results of the schlieren sequences which start when the incident shock wave meets the cylinder. The dimensionless time is defined as τ= tD0/WS, where WS is the velocity of the incident shock. As the passage of the shock wave, the cylinder is deposited on a large number of vorticity, which gradually deforms the cylinder into a vortex pair as shown in τ=6.33. When the evolving cylindrical volume is impacted by the reshock, the vortical structures in the largest scale are quickly slowed down. At the same time, the small-scale vortical structures are intensified, and the gaseous mixing is enhanced.
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| Figure 1. The numerical configuration of the computations (a) and density distributions of the initial SF6 cylinder (b). Yi is the mass fraction of SF6 in the flow. |
As shown in Fig. 2, the evolution of the continuous cylinder in Gaussian type appears very different from that in the sharp type. The baroclinic vorticity is considered to be a dominant factor of the different behaviors. Because the density gradients in the sharp cylinder are larger than that in the continuous cylinder, more vorticity are deposited on the inhomogeneity as the passage of the incident shock wave over the cylinder. The vortex pair grows up more quickly and entrains more ambient gases. Although the deformation of the volume is strongly dependent on the type of density distribution, the propagation of the incident shock and reshock is weakly dependent on that, which can be found in Fig. 2. Further study is to quantitatively analyze the effects of density distribution on the shock-cylinder interaction.
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| Figure 2. Numerical results showing the interaction of multiple shocks with cylinders in sharp (a) and Gaussian (b) types. Numbers indicate the dimensionless time and ρ is the density of the flow. |
References
Ranjan D, Oakley J, Bonazza R. Shock-Bubble Interactions. Annu. Rev. Fluid Mech. 43, 117-140, 2011.
Malamud G, Leinov E, Sadot O, Elbaz Y, Ben-Dor G and Shvarts D. Reshocked Richtmyer-Meshkov instability: Numerical study and modeling of random multi-mode experiments. Phys. Fluids, 26, 084107, 2014.
Si T, Zhai ZG, Luo XS. Experimental study of Richtmyer-Meshkov instability in a cylindrical converging shock tube. Laser Part. Beams, 32, 343-351, 2014.
Sun M, Takayama K. Conservative smoothing on an adaptive quadrilateral grid. J. Comput. Phys. 150, 143-180, 1999.
