Evolution and stability of shock waves in dissipative gases characterized by activated inelastic collisions

Previous experiments have revealed that shock waves driven through dissipative gases may become unstable, for example, in granular gases and in molecular gases undergoing strong relaxation effects. The mechanisms controlling these instabilities are not well understood. We successfully isolated and investigated this instability in the canonical problem of piston-driven shock waves propagating into a medium characterized by inelastic collision processes. We treat the standard model of granular gases, where particle collisions are taken as inelastic, with a constant coefficient of restitution. The inelasticity is activated for sufficiently strong collisions. Molecular dynamic simulations were performed for 30 000 particles. We find that all shock waves investigated become unstable, with density nonuniformities forming in the relaxation region. The wavelength of these fingers is found to be comparable to the characteristic relaxation thickness. Shock Hugoniot curves for both elastic and inelastic collisions were obtained analytically and numerically. Analysis of these curves indicates that the instability is not of the Bethe-Zeldovich-Thompson or D'yakov-Kontorovich type. Analysis of the shock relaxation rates and rates for clustering in a convected fluid element with the same thermodynamic history ruled out the clustering instability of a homogeneous granular gas. Instead, wave reconstruction of the early transient evolution indicates that the onset of instability occurs during repressurization of the gas following the initial relaxation of the medium behind the lead shock. This repressurization gives rise to internal pressure waves in the presence of strong density gradients. This indicates that the mechanism of instability is more likely of the vorticity-generating Richtmyer-Meshkov type, relying on the action of the inner pressure wave development during the transient relaxation.

Figure 1

Temperature distribution for a thermally relaxing shock wave traveling at velocity $D$.

Figure 2

Evolution of shock structure for $u_p=20$, $u^{*}=10$, and $\varepsilon =0.95$.

Figure 3

Evolution of (a) temperature, (b) density, and (c) pressure on an $x$ vs $t$ plane, in the piston frame of reference, for $u_p=20$, $u^{*}=10$, and $\varepsilon =0.95$. Evolutions shown with select particle paths (solid white lines) and forward (solid black lines) and backward [dashed (blue) lines] running characteristics.

Figure 4

Evolution of shock morphology, for a single realization, at (a) $t=0.3$, (b) $t=1.5$, (c) $t=2.7$, and (d) $t=3.9$, with $u_p=20$, $u^{*}=10$, and $\varepsilon =0.95$.

Figure 5

Particle distribution and coarse-grained streamlines for a single realization (top), with ensemble and coarse-grained distributions of temperature and density (bottom) after $t=8.13$, with $u_p=20$, $u^{*}=10$, and $\varepsilon =0.95$.

Figure 6

Ensemble and coarse-grained 1D shock structure of (a) the density and (b) the kinetic energy after a piston displacement of $x_p=138.7{\lambda }_1$ for varying $u_p$ and $u^{*}$, with $u_p/u^{*}=2$, and $\varepsilon =0.95$.

Figure 7

Comparison of the shock evolution for temperature (top) and pressure (bottom) obtained from MD for (a) $u_p/u^{*}=1.0$, (b) $u_p/u^{*}=1.5$, and (c) $u_p/u^{*}=2.02$, with $u^{*}=10$, $\varepsilon =0.95$, and ${\eta }_1=0.012$. Selected particle paths (white) and forward running characteristics (black) are shown, where $\xi =x(t=0)$.

Figure 8

Ensemble and coarse-grain averaged 1D shock structure of (a) density and (b) temperature after a piston displacement of $x_p=138.7{\lambda }_1$ for different values of $u_p/u^{*}$, with $\varepsilon =0.95$.

Figure 9

Ensemble and coarse-grain averaged 1D shock structure of (a) density and (b) temperature after a piston displacement of $x_p=156.0{\lambda }_1$ for different values of $\varepsilon$, with $u_p/u^{*}=2.0$.

Figure 10

Evolution of shock front for varying $\varepsilon$, with $u^{*}=10$ and $u_p=20$.

Figure 11

Comparison of shock morphology for single realizations after a piston displacement of $x_p=156.0{\lambda }_1$ for different values of $u_p/u^{*}$ and $\varepsilon$, where $u^{*}=10$ and ${\eta }_1=0.012$.

Figure 12

Results for the shock Hugoniot of equilibrium states, plotted with the elastic Hugoniot for ${\eta }_1=0.012$ and the isotherm corresponding to $u^{*}=10$.

Figure 13

Results for the Rayleigh line (dashed) for $u_p/u^{*}=2.0$ and $\varepsilon =0.95$ plotted with the shock Hugoniot (solid line).

Figure 14

Relationship between velocity of shock wave $D$ and piston velocity $u_p$, for $u^{*}=10$, $\varepsilon =0.90$, and ${\eta }_1=0.012$.

Figure 15

Relationship between $u_p/u^{*}$ and the fraction of the impact energy involved in activated collisions behind the shock front, assuming elastic jump conditions across the shock wave.

Figure 16

Exponential time constant ${\tau }_R$ of cooling experienced by shocked particle paths for different values of $u_p/u^{*}$ and $\varepsilon$, where $\xi =x(t=0)$.

Figure 17

Mean exponential time constant ${\tau }_R$ of shocked particle paths for different values of $u_p/u^{*}$ and $\varepsilon$, plotted with the range in time to clustering instability for similar values of $\varepsilon$. Filled circles represent simulations where unstable structures are seen.