Previous work has shown that the one-dimensional (1D) inviscid compressible flow (Euler) equations admit a wide variety of scale-invariant solutions (including the famous Noh, Sedov, and Guderley shock solutions) when the included equation of state (EOS) closure model assumes a certain scale-invariant form. However, this scale-invariant EOS class does not include even simple models used for shock compression of crystalline solids, including many broadly applicable representations of Mie-Grüneisen EOS. Intuitively, this incompatibility naturally arises from the presence of multiple dimensional scales in the Mie-Grüneisen EOS, which are otherwise absent from scale-invariant models that feature only dimensionless parameters (such as the adiabatic index in the ideal gas EOS). The current work extends previous efforts intended to rectify this inconsistency, by using a scale-invariant EOS model to approximate a Mie-Grüneisen EOS form. To this end, the adiabatic bulk modulus for the Mie-Grüneisen EOS is constructed, and its key features are used to motivate the selection of a scale-invariant approximation form. The remaining surrogate model parameters are selected through enforcement of the Rankine-Hugoniot jump conditions for an infinitely strong shock in a Mie-Grüneisen material. Finally, the approximate EOS is used in conjunction with the 1D inviscid Euler equations to calculate a semi-analytical, Guderley-like imploding shock solution in a metal sphere, and to determine if and when the solution may be valid for the underlying Mie-Grüneisen EOS.

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...

Interactions between an evolving solid and inviscid flow can result insubstantial computational complexity, particularly in circumstances involving varied boundary conditions between the solid and fluid phases. Examples of such interactions include melting, sublimation, and deflagration, all of which exhibit bidirectional coupling, mass/heat transfer, and topological change of the solid-fluid interface. The diffuse interface method is a powerful technique that has been used to describe a wide range of solid-phase interface-driven phenomena. The implicit treatment of the interface eliminates the need for cumbersome interface tracking, and advances in adaptive mesh refinement have provided a way to sufficiently resolve diffuse interfaces without excessive computational cost. However, the general scale-invariant coupling of these techniques to flow solvers has been relatively unexplored. In this work, a robust method is presented for treating diffuse solid-fluid interfaces with arbitrary boundary conditions. Source terms defined over the diffuse region mimic boundary conditions at the solid-fluid interface, and it is demonstrated that the diffuse length scale has no adverse effects. To show the efficacy of the method, a one-dimensional implementation is introduced and tested for three types of boundaries: mass flux through the boundary, a moving boundary, and passive interaction of the boundary with an incident acoustic wave. These demonstrate expected behavior in all cases. Convergence analysis is also performed and compared against the sharp-interface solution, and linear convergence is observed. This method lays the groundwork for the extension to viscous flow, and the solution of problems involving time-varying mass-flux boundaries.

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