External flow computations using global boundary conditions

We numerically integrate the compressible Navier-Stokes equations by means of a finite volume technique on the domain exterior to an airfoil. The curvilinear grid we use for discretization of the Navier-Stokes equations is obviously finite, it covers only a certain bounded region around the airfoil, consequently, we need to set some artificial boundary conditions at the external boundary of this region. The artificial boundary conditions we use here are non-local in space. They are constructed specifically for the case of a steady-state solution. In constructing the artificial boundary conditions, we linearize the Navier-Stokes equations around the far-field solution and apply the difference potentials method. The resulting global conditions are implemented together with a pseudotime multigrid iteration procedure for achieving the steady state. The main goal of this paper is to describe the numerical procedure itself, therefore, we primarily emphasize the computation of artificial boundary conditions and the combined usage of these artificial boundary conditions and the original algorithm for integrating the Navier-Stokes equations. The underlying theory that justifies the proposed numerical techniques will accordingly be addressed more briefly. We also present some results of computational experiments that show that for the different flow regimes (subcritical and supercritical, laminar and turbulent), as well as for the different geometries (i.e., different airfoils), the global artificial boundary conditions appear to be essentially more robust, i.e., they may provide far better convergence properties and much weaker dependence of the solution on the size of computational domain than standard external boundary conditions, which are usually based on extrapolation of physical and/or characteristic variables.