Collapsing Cavities and Converging Shocks in Non-Ideal Materials

Abstract

As modern hydrodynamic codes increase in sophistication, the availability of realistic test problems becomes increasingly important. In gas dynamics, one common unrealistic aspect of most test problems is the ideal gas assumption, which is unsuited to many real applications, especially those involving high pressure and speed metal deformation. Our work considers the collapsing cavity and converging shock test problems, showing to what extent the ideal gas assumption can be removed from their specification. It is found that while most materials simply do not admit simple (i.e. scaling) solutions in this context, there are infinite-dimensional families of materials which do admit such solutions. We characterize such materials, derive the appropriate ordinary differential equations, and analyze the associated nonlinear eigenvalue problem. It is shown that there is an inherent tension between boundedness of the solution, boundedness of its derivatives, and the entropy condition. The special case of a constant-speed cavity collapse is considered and found to be heuristically possible, contrary to common intuition. Finally, we give an example of a concrete non-ideal collapsing cavity scaling solution based on a recently proposed pseudo-Mie-Gruneisen equation of state. † boyd, schmidt, ramsey, and baty damage, sonoluminescence (7), and as a potentially challenging test problem for the quantitative verification of inviscid compressible flow (Euler) codes.Similarly, the problem of an infinitely strong, cylindrical or spherical shock wave converging into an ideal, compressible, inviscid fluid was first solved by Guderley (8) in 1942. The classical Guderley solution has subsequently been investigated by authors such as Stanyukovich (9), Zel'dovich and Raizer (10), and Chisnell (11). Much of the attention surrounding this problem has been related to increasingly precise evaluations of its solution, its connection to potential initiating events such as curvilinear shock tubes or pistons, and applications to astrophysical processes (12), laser fusion (13,14), and quantitative code verification (15).In the context of inviscid compressible flow, both of these problems feature a flow field driving a discontinuity (to avoid ambiguity, we will use the word "jump" to describe both processes) into a quiescent, zero-pressure material (see Fig. 1). The only physical difference between the two problems is the mass density in the undisturbed region: either zero or a non-zero constant for the collapsing cavity and converging shock problems, respectively.(Problems featuring powerlaw mass density variation in the undisturbed region have also been investigated. See, for example, Lazarus (6).) With this slight difference in mind, Lazarus unified the problems into one mathematical framework though the use of a logical variable, conducted an extensive phase plane analysis of the underlying differential equations, and constructed a variety of solutions.In his approach, Lazarus (6) also showed that both problems arise from scale-invariant, self-s...

Converging shock flows for a Mie-Grüneisen equation of state

“…( 1), (2), and ( 7) can be constructed except in highly-specialized cases featuring planar symmetry (k = 0). These universal symmetry solutions are discussed at length by Boyd et al [21][22][23] and Ramsey et al 20 , and will not be considered further in this work. Emphasis will instead be placed on Eqs.…”

“…From these developments has also arisen the related effort of determining the conditions under which the inviscid Euler equations may be expected to admit scale-invariant solutions. As demonstrated by Ovsiannikov 16 , Holm 17 , Axford and Holm 18 , Hutchens 19 , and most recently Ramsey et al 20 and Boyd et al [21][22][23] , the existence of these solutions essentially amounts to the EOS realizing a particular form. The ideal gas thus turns out to be but one of a general class of EOS instantiations that admit scale-invariant solutions, when coupled to the inviscid Euler equations.…”