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

TL;DR

This paper explores the scale invariance of homentropic inviscid Euler equations with arbitrary isentropic EOS, deriving reduced ODE systems and applying them to self-similar solutions including the Noh problem.

Contribution

It demonstrates the scale invariance of the equations under general conditions and formulates self-similar solutions for specific flow scenarios, including the Noh problem.

Findings

Reduced the Euler equations to coupled ODEs under scale invariance.

Derived self-similar solutions for shock-free and shock-involving flows.

Provided algebraic solutions for the Noh problem with arbitrary EOS.

Abstract

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