Compressible Euler Equations on a Sphere and Elliptic-Hyperbolic Property

TL;DR

This paper derives the equations governing supersonic conical flows on a sphere, revealing their elliptic or hyperbolic nature depending on the Mach number, and emphasizes coordinate-independent tensor calculus for numerical solutions.

Contribution

It systematically derives coordinate-free equations for conical flows on a sphere, enabling flexible numerical approaches and analysis of flow type based on Mach number.

Findings

Equations reduce 3D Euler equations to 2D on a sphere.

Flow type (elliptic or hyperbolic) depends on Mach number.

Tensor calculus facilitates coordinate-independent formulation.

Abstract

In this work we systematically derive the governing equations of supersonic conical flow by projecting the 3D Euler equations onto the unit sphere. These equations result from taking the assumption of conical invariance on the 3D flow field. Under this assumption, the compressible Euler equations reduce to a system defined on the surface of the unit sphere. This compressible flow problem has been successfully used to study steady supersonic flow past cones of arbitrary cross section by reducing the number of spatial dimensions from 3 down to 2 while still capturing many of the relevant 3D effects. In this paper the powerful machinery of tensor calculus is utilized to avoid reference to any particular coordinate system. With the flexibility to use any coordinate system on the surface of a sphere, the equations can be more readily solved numerically when a structured mesh is used by defining the mesh lines to be the coordinate lines. The type of the system of partial differential equations would be hyperbolic or elliptic based on whether the crossflow Mach number is supersonic or subsonic.