Analysis of a FEM/BEM coupling method for transonic flow computations

A sensitive issue in numerical calculations for exterior flow problems, e.g. around airfoils, is the treatment of the far field boundary conditions on a computational domain which is bounded. In this paper we investigate this problem for two–dimensional transonic potential flows with subsonic far field flow around airfoil profiles. We take the artificial far field boundary in the subsonic flow region. In the far field we approximate the subsonic potential flow by the Prandtl–Glauert linearization. The latter leads via the Green representation theorem to a boundary integral equation on the far field boundary. This defines a nonlocal boundary condition for the interior ring domain. Our approach leads naturally to a coupled finite element/boundary element method for numerical calculations. It is compared with local boundary conditions. The error analysis for the method is given and we prove convergence provided the solution to the analytic transonic flow problem around the profile exists. 1. Formulation of the problem 1.1. The boundary value problem. Let Ω ⊂ R be an open bounded domain surrounding a given simply connected wing section P ⊂ R. The boundary of Ω consists of three parts ∂Ω := Γ∞ ∪ ΓP ∪ Σ , (1.1) whose interiors are mutually disjoint and where Γ∞ and ΓP are disjoint closed Jordan curves connected by Σ. The curve Γ∞ ∈ C∞ is the artificial exterior boundary of Ω which is taken in order to obtain a bounded computational domain. The curve ΓP is the common boundary between Ω and the profile P , which has a corner, the trailing edge (TE), and is C∞ otherwise. We denote by Σ a slit in Ω, joining the trailing edge with Γ∞. The unbounded far field domain exterior to Γ∞ will be denoted by Ω = R\Ω ∪ P . (1.2) The prolongation of the slit Σ in Ω to infinity will be denoted by Σ. Without loss of generality, we assume that the travelling velocity is given by a constant vector field ~v∞ which is parallel to the x1–axis, see Figure 1. Received by the editor August 13, 1993 and, in revised form, September 18, 1995. 1991 Mathematics Subject Classification. Primary 65N30, 68N38, 76H05, 49M10, 35L67.