Scale invariance of the homentropic inviscid Euler equations with application to the Noh problem

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, 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.

Figure 1

Flow field for example shock-free, homentropic solution with modified Tait EOS. Black, blue, and gray lines denote $n=0,1,$ and 2, respectively. The figures show the speed ($u$), density ($\rho$), pressure ($P$), and SIE ($I$) as a function of similarity variable ($\xi =r/t$).

Figure 2

Notional depiction of the Noh problem.

Figure 3

Shocked state of the classical 1D planar Noh problem for the modified Tait EOS. For a given value of $u_0$, the indicated value of each state variable holds throughout the entire shocked region, and the constant shock speed is as indicated. The dashed horizontal line near the top of each panel corresponds to the maximum pressure at which the modified Tait EOS is assumed to be valid (see Zel'dovich and Raizer [22]). The figures show the shocked density (${\rho }_2$), speed ($D_0$), pressure ($P_2$), and SIE ($I_2$) as a function of inflow velocity ($u_0$).