Euler Equations

If turbulence and viscous effects do not need to be resolved (assuming these can be treated as weak effects in a given application), then the problem of gridding the compu­tational domain to extremely fine resolution is obviated to some extent, allowing compu­tational costs to be reduced by orders of magnitude. If the flow is assumed to be inviscid, then the Navier-Stokes equations reduce to a simplified set of equations, called the Euler equations, which can be solved instead. By simply dropping the viscous terms from the full Navier-Stokes equations, the Euler equations can be written in tensor differential form in Cartesian, nonaccelerating coordinates as

— – f = 0 (Mass conservation), (14.10)

31 dxj

3 (ph) d(phu;) dp dp

———– 1——– — =——— b U;–

31 dxj 31 3 3Xj

Alternately, by dropping the viscous terms from Eq. 14.8, the Euler equations can be written in conservation form as

In practice the Euler equations have been found to yield a very good approximation to the high Reynolds number flows around real helicopters. Although the viscous terms are neglected, numerical solutions to the Euler equations can provide considerable insight into rather complicated flows. Vorticity in an inviscid flow, such as that governed by the Euler equations, should convect freely without any dissipation or diffusion. But the process of discretizing the Euler equations still leads to nonphysical numerical dissipation and diffusion of vorticity, however. This is an artifact of discretizing the spatial derivatives using Taylor’s series approximations. Fortuitously, though, this behavior may improve, to some extent, the ability of numerical solutions to the Euler equations to mimic reality. For instance, airfoils and rotors modeled using the Euler equations should not generate lift, because the Euler equations by themselves do not allow circulation to be developed – this process takes place through the action of viscosity. In practice, however, the numerical solution of the Euler equations introduces an artificial source of viscosity, even an infinitesimal amount of which will allow the essentially viscous problem of flow over a lifting surface to be modeled using an inviscid set of equations. The Euler equations have been used with a great deal of success to predict aerodynamic loadings on rotor blades – see, for example, Roberts & Murman (1985), Sankar et al. (1986), and the reviews by Caradonna (1990) and Conlisk (2002).

Despite the limitations of using the Euler equations, Euler-based CFD methods have helped the helicopter manufacturing industry to gain new insight into many complex rotor problems and has allowed the limitations of simpler models (such as those used in compre­hensive analyses; see Section 14.11) to be better understood. Yet, the high computational costs and numerical problems encountered in preserving the concentrated vorticity within the rotor wake have hindered application to many practical problems that require an ac­curate, long term treatment of the rotor wake, say over several rotor revolutions. This is particularly problematic for problems in В VI noise prediction. The inability of most Euler – based computational techniques to preserve the vorticity in the wake from dissipation at the hands of artificial numerical diffusion continues to form an opportunity for good research. Alternative methods such as vorticity transport and vortex methods continue to offer a good compromise in this regard. Although Euler-based methods will eventually be adopted as the simplest set of equations yielding acceptably accurate predictions of the flow around helicopters, the fundamental limitations of the approach should be recognized.