A Geometric Interpretation of the Euler Equations

The purpose of this work is to explore the notion of generalized similarity in the context of the Euler equations written in two-dimensional axi-symmetric coordinates. Leveraging previous work done by Coggeshall for the radiation hydrodynamics equations featuring an ideal gas equation of state, we show that the Euler equations are invariant under an optimal system of transformations including translations in time, translations in the azimuthal direction, and concurrent scaling in time and mass density. This symmetry algebra may be leveraged to construct a change of variables through which the Euler equations are reducible to a set of four coupled ordinary differential equations (ODEs), any solution of which possesses the same symmetries. After further constraining the ODEs using a physically motivated assumption on the velocity field, we obtain associated solutions for the density, pressure, specific internal energy, and entropy variables. These novel, closed-form solutions exemplify the generalization of classical similarity as encountered in geometry or intermediate asymptotics within the setting of the Euler equations interpreted as an 18-dimensional surface invariant under Lie point symmetries.