Compressible Euler equations on a sphere and elliptic–hyperbolic property

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 the steady supersonic flow past cones of arbitrary cross section by reducing the number of spatial dimensions from three down to two 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.

Ideal Magnetohydrodynamic Equations on a Sphere and Elliptic-Hyperbolic Property

“…Because the conical assumption has been used with such success and electrically conducting is an important topic for the future of aerodynamic design, we build upon previous work in conical flow by deriving and analyzing the type of the conical Ideal Magnetohydrodynamic equations. This work follows closely that by Sritharan and Holloway [7] on the conical Euler equations and likewise results in a system of equations which do not reference any particular coordinate system. A numerical method which solves these equations is developed in [8], and it can be seen that the coordinate free form has a natural compatibility with structured meshes.…”

Numerical Solution of Compressible Euler and Magnetohydrodynamic flow past an infinite cone

“…The conical equations, 1.1 and 1.2, involve the contracted covariant derivative (denoted (•) |β ) where the contraction is only performed over the components corresponding to the surface of the sphere (β ∈ {1, 2}). These forms of the equations are slightly different than the forms presented in [1] and [2], but they are still consistent, differing only by a factor of √ g. The forms considered for the analysis of these equations rely on the relationship…”

“…The conical Euler and MHD equations, which govern flow past an infinite cone of arbitrary cross section, are derived and analyzed in [1] and [2] respectively. These equations result from setting the corresponding system in a 3D Euclidean space covered by coordinates (ξ 1 , ξ 2 , r), where ξ β are defined on the surface of the sphere and r is the radial coordinate, and then setting the covariant derivative in the r direction equal to zero.…”

“…Such reductions in the complexity of problems ease analysis and accelerate the acquisition of numerical solutions. This is the precise motivation for the works in [1] and [2] which eliminate the radial dimension of supersonic flow problems governed by the Euler and Ideal Magnetohydrodynamic (MHD) equations respectively to acquire the conical versions of those equations. The Conical Euler equations:…”