Efficient Solutions of the Euler and Navier-stokes Equations for External Flows

This invited paper presents efficient solutions of the Euler and Navier-Stokes equations based on finite difference and finite volume formulations. They were obtained with several methods recently developed for the efficient analysis of steady and unsteady external flows. (i) A biased-flux method based on an explicit finite volume formulation is first presented for the solution of the Euler equations. A second-order scheme is used for the flux calculation with an upwind or downwind bias for the flow variables, according to the subsonic or supersonic character of the local flow. The method is first validated for compressible flows with expansions and shock waves in nozzles, and then used to obtain solutions for subsonic, transonic and supersonic flows past airfoils. (ii) Efficient NavierStokes solutions are also presented for the flows past airfoils at low Reynolds numbers, dominated by viscous effects, which are of interest for microaircraft and unmanned air vehicle applications.