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We investigate the inviscid compressible flow (Euler) equations constrainedby an "isentropic" equation of state (EOS), whose functional form in pressureis an arbitrary function of density alone. Under the aforementioned condition,we interrogate using symmetry methods the scale-invariance of the homentropicinviscid Euler equations. We find that under general conditions, we can reducethe inviscid Euler equations into a system of two coupled ordinary differentialequations. To exemplify the utility of these results, we formulate two examplescale-invariant, self-similar solutions. The first example includes ashock-free expanding bubble scenario, featuring a modified Tait EOS. The secondexample features the classical Noh problem, coupled to an arbitrary isentropicEOS. In this case, in order to satisfy the conditions set forth in theclassical Noh problem, we find that the solution for the flow is given by atranscendental algebraic equation in the shocked density.

Scale Invariance of the Homentropic Inviscid Euler Equations with Application to the Noh Problem