The Life and Death of Euler,Bernoulli,and Navier—Stokes Equations and Associated CFD for So-Called Incompressible Fluid Flow

Abstract

Euler's equations, derived in 1755, are the basis for general fluid dynamics analysis. Despite their elegance, it is well known that fluid dynamics is a weird academic area, full of paradoxes and absurdities, for which viscosity is often the scapegoat. The Euler and sibling Navier–Stokes equations have long been ‘contaminated’ with artificial viscosity, upwinding and other measures in computational fluid dynamics (CFD). Although derived by the momentum principle, the Euler equations are not genuine momentum equations, as they are unable to accommodate any cases with energy loss. Then, should we blindly keep using these non-genuine momentum equations? In this paper, the culprit for this fatal non-genuineness is shown to be a seemingly innocent ‘artificial’ equalization of crosswise and streamwise fluid pressures, by unintentionally disregarding any possible energy loss in deriving the Euler equations.