Inherent Numerical Discretization Dilemma Analysis of Euler Equations

Abstract

Euler equations, derived in 1755, are among the most important equations for general fluid analysis. Although this fundamental governing set of equations looks simple at first glance for incompressible, inviscid fluid flow, it does lead to a crucial numerical discretization dilemma related to existence, uniqueness, stability and oscillation problems. Take gradual pipe expansion flow, for instance: in the process of deriving Euler equations, the seemingly innocent artificial equalization of crosswise and forward-biased streamwise pressure will give serious discretization error compared with use of the Bernoulli equation. Here, the wall pressures, P W = P + d P, disappear from the Euler equations but later bring about troublesome problems, and even violate the fundamental energy principle. Various numerical measures have been devised to overcome such problems, such as artificial viscosity and streamline upwinding, but these are awkward and have high risk of flow field contamination in an attempt to somehow manipulate some energy dissipation when dealing with incompressible fluid flow. In this research, for the sake of finding the real culprit, we re-scrutinize the original derivation of Euler equations and compare the discretized Euler equation with a factorized Bernoulli equation, in order to find the actual pathology that leads to these serious numerical discretization dilemmas with the Euler equations.