On imploding cylindrical and spherical shock waves in a perfect gas

Abstract

The problem of a cylindrically or spherically imploding and reflecting shock wave in a flow initially at rest is studied without the use of the strong-shock approximation. Dimensional arguments are first used to show that this flow admits a general solution where an infinitesimally weak shock from infinity strengthens as it converges towards the origin. For a perfect-gas equation of state, this solution depends only on the dimensionality of the flow and on the ratio of specific heats. The Guderley power-law result can then be interpreted as the leading-order, strong-shock approximation, valid near the origin at the implosion centre. We improve the Guderley solution by adding two further terms in the series expansion solution for both the incoming and the reflected shock waves. A series expansion, valid where the shock is still weak and very far from the origin, is also constructed. With an appropriate change of variables and using the exact shock-jump conditions, a numerical, characteristics-based solution is obtained describing the general shock motion from almost infinity to very close to the reflection point. Comparisons are made between the series expansions, the characteristics solution, and the results obtained using an Euler solver. These show that the addition of two terms to the Guderley solution significantly extends the range of validity of the strong-shock series expansion.

Converging cylindrical shocks in ideal magnetohydrodynamics

“…A similar calculation using the first equation of (18) gives that the gas velocity immediately behind the shock in the laboratory frame of reference approaches the finite value u → −4 A/3 when r → 0. These results are consistent with (12). Figure 7 illustrates u − c characteristics obtained by integration of the equation dr/dt = u − c using a range of initial conditions and with (u, c) obtained from the numerical solutions.…”

“…11 When applied to the symmetrically converging shock of the Guderley type, GSD provides an extremely accurate approximation to power-law exponents and in fact can describe an almost universal shock collapse process from an infinitesimal wave at infinity to a strong-shock state near the point of convergence. 12,13 Interest in the effect of a magnetic field on the shock convergence process has been heightened by the knowledge that the presence of a sufficiently strong magnetic field can inhibit the RichtmyerMeshkov 14,15 instability that occurs when a shock-wave impacts and impulsively accelerates a perturbed density interface, thereby depositing vorticity and leading to rapid interface growth. 16,17 Suppression of the Richtmyer-Meshkov instability, while retaining the effects of shock heating, could be expected to have potentially important consequences for the realization of inertial confinement fusion (ICF).…”

Linear Simulations of the Cylindrical Richtmyer-Meshkov Instability in Hydrodynamics and MHD

“…But his theory was only valid for strong shocks. Ponchaut et al [13] adopted the method of series expansion and developed a numerical, characteristics based solution which was also valid for weak shocks. In the context of the RMI, Lombardini and Pullin [14] developed the theory for asymptotic growth rate in density interface for a three dimensional cylindrical geometry.…”

Mean free path effects in the shock-implosion problem

“…For finite strength shock waves, the approximate theory known as geometric shock dynamics ͑see Ref. 3͒ is most useful, though a more exact solution describing a generalized, "universal" shock implosion has been obtained by Ponchaut et al 5 These authors used various methods including a series expansion solution in which the Guderley solution forms the leadingorder term. This generalized implosion is asymptotic to the Guderley solution ͑in space and time͒ as time increases and the shock strength increases.…”