Passage of a shock wave through inhomogeneous media and its impact on gas-bubble deformation

The paper investigates shock-induced vortical flows within inhomogeneous media of nonuniform thermodynamic properties. Numerical simulations are performed using a Eulerian type mathematical model for compressible multicomponent flow problems. The model, which accounts for pressure nonequilibrium and applies different equations of state for individual flow components, shows excellent capabilities for the resolution of interfaces separating compressible fluids as well as for capturing the baroclinic source of vorticity generation. The developed finite volume Godunov type computational approach is equipped with an approximate Riemann solver for calculating fluxes and handles numerically diffused zones at flow component interfaces. The computations are performed for various initial conditions and are compared with available experimental data. The initial conditions promoting a shock-bubble interaction process include weak to high planar shock waves with a Mach number ranging from $1.2$ to 3 and isolated cylindrical bubble inhomogeneities of helium, argon, nitrogen, krypton, and sulphur hexafluoride. The numerical results reveal the characteristic features of the evolving flow topology. The impulsively generated flow perturbations are dominated by the reflection and refraction of the shock, the compression, and acceleration as well as the vorticity generation within the medium. The study is further extended to investigate the influence of the ratio of the heat capacities on the interface deformation.

Figure 1

Numerical flux notations within a 2D quadratic mesh.

Figure 2

Schematic diagram representing notations at each computational cell for the piecewise linear variable states (dashed line) in the second order scheme.

Figure 3

Wave configuration at a cell boundary in the approximate HLL Riemann solver with right initial data $\left(U_R\right)$, left initial data $\left(U_L\right)$, and the star region.

Figure 4

Schematic diagram of a shock-bubble interaction flow field and different wave configurations (a) light bubble and (b) heavy bubble. The planar shock moves from right to left.

Figure 5

Schematic diagram of the initial state of shock-bubble interaction test problems.

Figure 6

Pressure (a) and density (b) distributions for shock-He bubble interaction along the tube center line at $t=410\mu \mathrm{s}$ for different mesh resolutions.

Figure 7

Pressure (a) and density (b) distributions for shock-Kr bubble interaction along the tube center line at $t=410\mu \mathrm{s}$ for different mesh resolutions

Figure 8

Incident shock (Inc) and transmitted shock (Tr) wave positions (left) and bubble points 1 and 2 locations (right) at $\text{Ma}=1.5$ for air-He [(a1), (a2)], air-${\mathrm{N}}_2$ [(b1), (b2)], and air-Kr [(c1), (c2)] constitutions. Comparison between present numerical results and experimental data [6].

Figure 9

Numerical schlieren images showing the interface evolution in time of the different air-bubble constitutions: air-He (left), air-${\mathrm{N}}_2$ (middle), and air-Kr (right) at $\text{Ma}=1.5$.

Figure 10

Sequence of schlieren images representing the ${\mathrm{SF}}_6$ bubble evolution as a result of its interaction with a planar shock wave. The time interval between successive images from the numerical simulation is $10\mu \mathrm{s}$.

Figure 11

Schematic diagram with the characteristic elements of the interaction of a shock with a ${\mathrm{SF}}_6$ bubble. Evolution stages: (a) early stage of the interaction, (b) a moment after the shock wave passed the bubble. Tracking points: Inc-incident shock, P1-upstream interface, P2-downstream interface, R1-reflected shock, Rr-refracted shock, Tr-transmitted shock, and Jet-head.

Figure 12

Recorded various shocks and interface positions during the interaction of a shock wave with a ${\mathrm{SF}}_6$ bubble. Symbols are linked to the tracking locations shown in Fig. 11. The present numerical results (filled symbols) are compared with the experimental (a) and the numerical predictions (b) of [8].

Figure 13

Numerical schlieren images for the various air-gas constitutions at time $60\mu \mathrm{s}$ and $\text{Ma}=1.5$.

Figure 14

Variation of the normalized transmitted shock velocity as a function of the Atwood number at time $60\mu \mathrm{s}$ and $\text{Ma}=1.5$.

Figure 15

Incident shock and bubble upstream point 1 positions for different air-gas constitutions at $\text{Ma}=1.5$ [(a1), (a2)] and $\text{Ma}=2.5$ [(b1), (b2)].

Figure 16

The velocities of point (1) for different gas bubbles as a function of time for (a) $\text{Ma}=1.5$ and (b) $\text{Ma}=2.5$.

Figure 17

Volume fraction contours of various gas-air constitutions and Ma numbers at time 236 $\mu \mathrm{s}$.

Figure 18

Compression ratio of ${\mathrm{N}}_2$ bubble as a function of the Mach number at time $t= 510\mu \mathrm{s}$.

Figure 19

Vorticity field contours of various gas-air constitutions and Ma numbers at time $166\mu \mathrm{s}$.

Figure 20

Time evolution of the circulation during the shock-bubble interaction process for different Mach numbers and gases: (a) air-He, (b) air-Ar, (c) air-Kr, and (d) air-${\mathrm{SF}}_6$ constitutions.

Figure 21

Volume fraction contours of the air bubbles at time $t=306\mu \mathrm{s}$ for different interaction scenarios: (a) $\gamma =1.2$, (b) $\gamma =1.4$, (c) $\gamma =1.67$, and (d) comparison of volume fraction profiles.

Figure 22

Relative change of the bubble size as a function of time (a) horizontal diameter, (b) vertical diameter, during the interaction scenarios for different heat capacity ratios $\gamma$.