Paper

On the existence of self-similar converging shocks for arbitrary equation of state

Boyd, Zachary M, Baty, Roy S, Ramsey, Scott D

2017

Overview

We extend Guderley's problem of finding a self-similar scaling solution for a converging cylindrical or spherical shock wave from the ideal gas case to the case of flows with an arbitrary equation of state closure model, giving necessary conditions for the existence of a solution. The necessary condition is a thermodynamic one, namely that the adiabatic bulk modulus, \\(K_S\\), of the fluid be of the form \\(pf(\\rho)\\) where \\(p\\) is pressure, \\(\\rho\\) is mass density, and \\(f\\) is an arbitrary function. Although this condition has appeared in the literature before, here we give a more rigorous and extensive treatment. Of particular interest is our novel analysis of the governing ordinary differential equations (ODEs), which shows that, in general, the Guderley problem is always an eigenvalue problem. The need for an eigenvalue arises from basic shock stability principles -- an interesting connection to the existing literature on the relationship between self-similarity of the second kind and stability. We also investigate a special case, usually neglected by previous authors, where assuming constant shock velocity yields a reduction to ODEs for every material, but those ODEs never have a bounded, differentiable solution. This theoretical work is motivated by the need for more realistic test problems in the verification of inviscid compressible flow codes that simulate flows in a variety of non-ideal gas materials.

• Publisher: Cornell University Library, arXiv.org
• Subject: Bulk modulus / Compressible flow / Computer simulation / Convergence / Cylindrical waves / Differential equations / Eigenvalues / Equations of state / Ideal gas / Ordinary differential equations / Self-similarity / Shock waves / Spherical waves / Stability