Implicit time-dependent methods for the solution of the Euler equations. I - Analysis of the methods. II - Application to transonic flows

A detailed analysis of centered implicit methods is presented for the solution of the Euler equations. The analysis is begun in one space-dimension for a linear hyperbolic system and for a nonlinear hyperbolic system such as that formed by the Euler equations. The choice of an implicit scheme is then discussed for steady problems and unsteady problems. The effects of nonlinearities and of boundary conditions on the stability are considered as is the correct formulation of the method for the case of several space variables. An extension to the Navier-Stokes equations is outlined. Some basic ideas on the nature of the unsteady Euler equations are introduced, and two classical approaches are described for the formulation of implicit methods on a curvilinear mesh. The presentation is here restricted to the case of two space-variables. The implicit method of NASA-Ames for the solution of Euler equations in various steady and unsteady realistic flow problems is outlined. The second method, the implicit finite-volume method of Lerat, Sides, and Daru (1982, 1984), is described in detail. Both steady and unsteady transonic applications around airfoils are presented. Results are presented for NACA 0012, RAE 2822, and NLR 7301 airfoils.