Stability of stagnation via an expanding accretion shock wave

Abstract

Stagnation of a cold plasma streaming to the center or axis of symmetry via an expanding accretion shock wave is ubiquitous in inertial confinement fusion (ICF) and high-energy- (2014)]. Some experimental evidence indicates that stagnation via an expanding shock wave is stable, but its stability has never been studied theoretically. We present such analysis for the stagnation that does not involve a rarefaction wave behind the expanding shock front and is described by the classic ideal-gas Noh solution in spherical and cylindrical geometry. In either case the stagnated flow has been demonstrated to be stable, initial perturbations exhibiting a powerlaw, oscillatory or monotonic, decay with time for all the eigenmodes. This conclusion has been supported by our simulations done both on a Cartesian grid and on a curvilinear grid in spherical coordinates. Dispersion equation determining the eigenvalues of the problem and explicit formulas for the eigenfunction profiles corresponding to these eigenvalues are presented, making it possible to use the theory for hydro code verification in two and three dimensions.

The impact of vorticity waves on the shock dynamics in core-collapse supernovae

“…In the latter low mode-number scenario (λ x λ y ), the problem reduces to the one-dimensional interaction of the shock with radial perturbation waves. Such stability analysis has been developed by Velikovich et al (2016) for the classical Noh's configuration in adiabatic conditions. The asymptotic far-field solution for the acoustic disturbances is also written in terms of harmonic functions, representing stable traveling fronts that occur only when the shock oscillation frequency is sufficiently high, ζ > 1.…”

Solution of the Noh problem with an arbitrary equation of state

“…In this case, the 1D background flow is a solution of a planar Riemann of shock-interface interaction, quite similar to the simple particular case of a rigid piston problem defining the planar Noh accretion-shock solution. Second, the physical stability of the classic Noh solution has been proven [40]. The long experience of successful verification of the 2D and 3D…”

“…Indeed, a deep connection can be traced between the physical picture of stagnation as modeled by the classic Noh solution and the RM instability. In our recent paper containing the perturbation analysis of the classic Noh solution [40], we discussed the analogy between the spectra of the perturbed Noh flow and that of an RM-unstable flow in planar geometry, see Ref.…”

“…The classic Noh accretion-shock flow does not involve a free surface or a material interface, where an RM or Rayleigh-Taylor instability could develop. Hence, all of its eigenmodes are stable [40]. This is one of the reasons why it is such an important test problem.…”

“…In Ref. 40, we have demonstrated that the physical stability of the classic Noh solution is ensured by the stability of a shock front in an ideal gas. In other words, the Noh accretion-shock flow is stable because the reflection coefficient for the sound waves incident upon the expanding shock front from the shocked gas is less than unity for an ideal gas with an arbitrary γ .…”