Derivation and characteristics analysis of an acoustics-convection upstream resolution algorithm for the two-dimensional Euler and Navier-Stokes equations

Abstract

SUMMARYThe ÿrst of a two-paper series, this paper introduces a new decomposition not of the hyperbolic ux vector but of the ux vector Jacobian. The paper then details for the Euler and Navier-Stokes equations an intrinsically inÿnite directional upstream-bias formulation that rests on the mathematics and physics of multi-dimensional acoustics and convection. Based upon characteristic velocities, this formulation introduces the upstream bias directly at the di erential equation level, before the spatial discretization, within a characteristics-bias governing system. Through a decomposition of the Euler ux divergence into multi-dimensional acoustics and convection-acoustics components, this characteristics-bias system induces consistent upstream bias along all directions of spatial wave propagation, with anisotropic variable-strength upstreaming that correlates with the spatial distribution of characteristic velocities.

Finite element and implicit Runge-Kutta implementation of an acoustics-convection upstream resolution algorithm for the time-dependent two-dimensional Euler equations

“…With implied summation on repeated subscript indices, these equations can be abridged as the non-linear parabolic system @q @t + @f j (q) @x j − @f j @x j = 0 (1) which reduces to the Euler hyperbolic system when the uid-viscosity ux f j identically vanishes. For three-dimensional formulations, 1 6 j 6 3, and with R denoting the real-number ÿeld, the independent variable (x; t), x ≡ (x 1 ; x 2 ; x 3 ), in (1) …”

“…This continuum upstream-bias formulation derives from a characteristics-bias integral statement associated with (1). With reference to (2), the characteristic-bias integral is then deÿned as…”

“…As Reference [1] details, the non-discrete ux Jacobian decomposition (FJD) upstream-bias approximation is developed for the Euler and Navier-Stokes equations. With implied summation on repeated subscript indices, these equations can be abridged as the non-linear parabolic system @q @t + @f j (q) @x j − @f j @x j = 0 (1) which reduces to the Euler hyperbolic system when the uid-viscosity ux f j identically vanishes. For three-dimensional formulations, 1 6 j 6 3, and with R denoting the real-number ÿeld, the independent variable (x; t), x ≡ (x 1 ; x 2 ; x 3 ), in (1) varies in the domain D ≡ ×[t o ; t f ], [t o ; t f ] ⊂ R + , ⊂ R 3 .…”

“…IANNELLI approximations of the Euler and Navier-Stokes equations and subsequent computational implementation. The ÿrst part [1], also presented in this journal, synthesizes and analyses via characteristics the acoustic-convection upstream resolution algorithm. Detailing an integral formulation, a ÿnite element discretization, and an implicit Runge-Kutta time integration, this paper presents computational results for inviscid two-dimensional subsonic, transonic, and supersonic ows with shock re ections and interactions.…”

“…Associated with the Euler and Navier-Stokes equations, the non-discrete formulation results in a 'companion' characteristics-bias system that relies on a decomposition of the multi-dimensional Euler Jacobian into acoustics and convection components. For any Mach number, the characteristics-bias system induces consistent upwinding along every direction radiating from any ow-ÿeld point [1]. The formulation induces an anisotropic variable-strength upstream bias that directly correlates with the multi-dimensional spatial distribution of characteristic velocities.…”