Acoustic stability of nonadiabatic high-energy-density shocks

As predicted by D'yakov [Shock wave stability, Zh. Eksp. Teor. Fiz. 27, 288 (1954)] and Kontorovich [On the shock waves stability, Zh. Eksp. Teor. Fiz. 33, 1525 (1957)], an initially disturbed shock front may exhibit different asymptotic behaviors, depending on the slope of the Rankine-Hugoniot curve. Adiabatic and isolated planar shocks traveling steadily through ideal gases are stable in the sense that any perturbation on the shock front decays in time with the power $t^{-3/2}$ (or $t^{-1/2}$ in the strong-shock limit). While some gases whose equations of state cannot be modeled as a perfect gas, as those governed by van der Waals forces, may induce constant-amplitude oscillations to the shock front in the long-time regime, fully unstable behaviors are seldom to occur due to the unlikely conditions that the equation of state must meet. In this paper, it has been found that unstable conditions might be found when the gas undergoes an endothermic or exothermic transformation behind the shock. In particular, it is reported that constant-amplitude oscillations can occur when the amount of energy release is positively correlated to the shock strength and, if this correlation is sufficiently strong, the shock turns to be fully unstable. The opposite highly damped oscillating regime may occur in negatively correlated configurations. The mathematical description then adds two independent parameters to the regular adiabatic index $\gamma$ and shock Mach number ${\mathcal{M}}_1$, namely, the total energy added/removed and its sensitivity with the shock strength. The formulation in terms of endothermic or exothermic effects is extended but not restricted to include effects associated to ionization, dissociation, thermal radiation, and thermonuclear transformations, so long as the time associated to these effects is a much shorter time than the acoustic time.

Figure 1

Sketch of the Rankine-Hugoniot curve and the Rayleigh-Michelson line for a nonadiabatic shock. Different processes will place the final postshock state (orange point) in a different position.

Figure 2

DK parameter ${\mathrm{\Gamma }}_s$ as a function of the dimensionless energy $\mathcal{Q}$ for a strong shock ${\mathcal{M}}_1=10$ and $\gamma =1.4$, and for different values of energy sensitivities $\varepsilon$ and qualitative representation of ${\beta }_{\mathrm{RH}}$.

Figure 3

Sketch of the perturbed shock wave and the perturbation variables in the shock reference frame, with $\ell \ll {\psi }_{s0}\ll 2\pi /k$.

Figure 4

Asymptotic temporal evolution of the shock ripple amplitude ${\xi }_s\left(\tau \right)$ for $\gamma =1.4, {\mathcal{M}}_1=10, \mathcal{Q}=15$ and different values of the energy sensitivity parameter $\varepsilon$. Dashed lines correspond to asymptotic regression trends.

Figure 5

Left: Stability regions as a function of the energy release $q$ and DK parameter ${\mathrm{\Gamma }}_s$. Isocurves of $\varepsilon$ as shown in Fig. 2. Triangles correspond to the computations in Fig. 4. Right: Sketch of the distinguished regimes for both ${\mathrm{\Gamma }}_s$ and ${\sigma }_b$ parameters.

Figure 6

Function ${\varepsilon }_{\mathrm{uns}}$ as a function of $\mathcal{Q}$ for $\gamma =1.4$ (left) and $\gamma =1.01$ (right) and for different values of the shock Mach number ${\mathcal{M}}_1$.

Figure 7

Critical values of energy change sensitivity, ${\varepsilon }_{\mathrm{rad}}$ and ${\varepsilon }_{\mathrm{dam}}$, as a function of $\mathcal{Q}$ for different propagation Mach numbers ${\mathcal{M}}_1$ and for $\gamma =1.4$ (left) and $\gamma =1.01$ (right).

Figure 8

Energy removed by ionization $\mathcal{Q}$, associated energy sensitivity $\varepsilon$, and degree of ionization ${\alpha }_i$ as a function of the Mach number ${\mathcal{M}}_1$ for argon. Triangle: value of ${\alpha }_i$ according to Ref. [72].

Figure 9

Energy removed by dissociation $\mathcal{Q}$, associated energy sensitivity $\varepsilon$, and degree of dissociation ${\alpha }_d$ as a function of the Mach number ${\mathcal{M}}_1$ for ${\mathrm{N}}_2$ [10].

Figure 10

Energy loss by radiation $\mathcal{Q}$ and the associated energy sensitivity $\varepsilon$ as a function of the Mach number ${\mathcal{M}}_1$ for $\gamma =5/3$.

Figure 11

DK parameter ${\mathrm{\Gamma }}_s$ and limits ${\phi }_{\mathrm{rad}}$ and ${\phi }_{\mathrm{dam}}$ as a function of the Mach number ${\mathcal{M}}_1$ for adiabatic shocks (a), gaseous detonations as in Ref. [38] (b), ionization shocks in argon (c), dissociation shocks in molecular nitrogen (d), radiative shocks for $\gamma =5/3$ (e), and thermonuclear shocks for $\gamma =4/3$ (f).

Figure 12

Density jump ${\mathcal{R}}_s$ as a function of ${\mathcal{P}}_s$ (left) and ${\mathcal{T}}_s$ (right) for $\gamma =5/3, {\mathcal{M}}_1=(1,48)$. Conditions chosen to be similar to Ref. [6].

Figure 13

Density ${\mathcal{R}}_s$ (left) temperature ${\mathcal{T}}_s$ (right) jumps as a function of ${\mathcal{M}}_1$.