Solution of 2D Euler Equations and Application to Airfoil Design

This paper deals with a numerical method for an airfoil design. It is shown how to create an airfoil from a given velocity distribution along a mean camber line. The method is based on searching a fixed point of a contractive operator. We need to have a fast solver of the Euler equations. The Newton method for solving implicit finite volume scheme is described. The resulting system of linear algebraic equations is solved by GMRES, the Jacobian-free version is described. Numerical results are presented. Introduction We present here a method how to get an airfoil from a given velocity distribution. This method can be used in the case of a subsonic inviscid compressible flow described by the Euler equations. The method was presented by [Pelant, 1998]. The main idea is to use a couple of a direct and an inverse operator. The direct operator represents an assignment of a velocity distribution to a shape of an airfoil and the second operator is its inversion. Since we are not able to construct an exact inversion we have to use some iteration method. We put the direct and the approximate inverse operator together to create a contractive operator and find its fixed point. The method can be used to modify existing airfoils. We get a velocity distribution of some airfoil, then we change it and we get a new airfoil with required quality. In the next sections these two operators are described. The theory of a numerical solution of flow dynamics can be found in [Feistauer et al., 2003]. Solution of Euler equations using Newton-Krylov method Our goal is to find a steady-state solution of the Euler equations describing inviscid compressible flow. The nonstationary equations can be written in the form ∂w ∂t + ∂f(w) ∂x + ∂g(w) ∂y = 0, (1) where w = (ρ, ρu, ρv,E) , f(w) = (ρu, ρu + p, ρuv, (E + p)u) , g(w) = (ρv, ρuv, ρv + p, (E + p)v) . The symbol ρ denotes the density, u,v denote the velocity components, E denotes the energy and p denotes the pressure. The Pressure p can be expressed by p = (γ − 1) ( E − 1 2 ρ ( u + v ) )