The stability of expanding reactive shocks in a van der Waals fluid

Abstract

Despite the extensive literature accumulated since the pioneering works of D'yakov and Kontorovich in the 1950s, the stability of steady shocks is still an open question when realistic boundary conditions are accounted. The consideration of a supporting mechanism, which is indeed a necessary condition for shock steadiness, modifies the perturbation shock dynamics in the unstable range. The Noh problem is a suitable example to form steady expanding shocks. This configuration is of great interest to the high-energy-density-physics community because of its direct application to inertial confinement fusion and astrophysics, for which the stagnation of a supersonically converging material via an accretion shock front is ubiquitous. In this work, we extend the generalized Noh problem, both base-flow solution and linear stability analysis, to conditions where endothermic or exothermic transformations undergo across the shock. Within the spontaneous acoustic emission conditions found for a van der Waals gas [J. W. Bates and D. C. Montgomery, “The D'yakov-Kontorovich instability of shock waves in real gases,” Phys. Rev. Lett. 84, 1180 (2000)], we find that cylindrical and spherical expanding shocks become literally unstable for sufficiently high mode numbers. Counterintuitively, the effect of exothermicity or endothermicity across the shock is found to be stabilizing or destabilizing, respectively.

On the stability of piston-driven planar shocks

“…It is found that the initial value problem associated with the corrugated planar-geometry case is qualitatively different from that corresponding to cylindrical and spherical steady expanding shocks associated with the Noh problem, addressed in Velikovich et al (2016), Huete et al (2021) andCalvo-Rivera et al (2022). It was demonstrated there that the shock exhibits an unstable power-law evolution when h > h c and the perturbation mode number is sufficiently high.…”

“…(2021) and Calvo-Rivera et al. (2022). It was demonstrated there that the shock exhibits an unstable power-law evolution when and the perturbation mode number is sufficiently high.…”

“…2009); for ionising shock waves in inert gases (Mond & Rutkevich 1994; Mond, Rutkevich & Toffin 1997); for shock waves dissociating hydrogen molecules (Griffiths, Sandeman & Hornung 1976; Bates & Montgomery 1999); for Gbar and Tbar pressure range shocks in solid metals, where the shell ionisation affects the shapes of Hugoniot curves (Rutkevich, Zaretsky & Mond 1997; Das, Bhattacharya & Menon 2011; Wetta, Pain & Heuzé 2018); for shock fronts producing exothermic reactions, such as a detonation (Huete & Vera 2019; Huete et al. 2020; Calvo-Rivera, Huete & Velikovich 2022). The latter is found to occur only when net heat release is positively correlated with the shock intensity or in non-ideal EoS.…”

“…The first example of a realistic EoS satisfying the neutral instability criterion was discovered by Bushman (1976) near copper's liquid-vapour transition. Other examples have been found since for condensed materials near the liquid-vapour transition, including water (Kuznetsov & Davydova 1988), a fluid approximated by the vdW EoS (Bates & Montgomery 2000) and magnesium (Lomonosov et al 2000;Konyukhov et al 2009); for ionising shock waves in inert gases (Mond & Rutkevich 1994;Mond, Rutkevich & Toffin 1997); for shock waves dissociating hydrogen molecules (Griffiths, Sandeman & Hornung 1976;Bates & Montgomery 1999); for Gbar and Tbar pressure range shocks in solid metals, where the shell ionisation affects the shapes of Hugoniot curves (Rutkevich, Zaretsky & Mond 1997;Das, Bhattacharya & Menon 2011;Wetta, Pain & Heuzé 2018); for shock fronts producing exothermic reactions, such as a detonation (Huete & Vera 2019;Huete et al 2020;Calvo-Rivera, Huete & Velikovich 2022). The latter is found to occur only when net heat release is positively correlated with the shock intensity or in non-ideal EoS.…”

The effects of gravitational and magnetic fields on the propagation of cylindrical strong shock waves in a van der Waals gas

“…Pandey [27], and Hermet et al [28], studied the propagation of shock waves with and without including the overtaking disturbances on the downstream flow inside a cross-sectional tube or channel for different gases. Recently, A. Calvo-Rivera et al [29], analyzed in detail the stability of a steadily expanding shock-front and its application to a reactive shock using Rankine-Hugoniot equations in a van der Waals gas. Further, Rivera and Huete [30] revisited the MHD shockfront stability using ideal gas equation of state (EoS), van der Waals EoS and also for some metals such as aluminum.…”