Finite Element Analysis of Subsonic Flow over a Lifting Thin Airfoil

摘要

The finite element method was used to solve the flow field problem around a thin, lifting, flat plate, airfoil. The governing equation solved is the Laplace equation, which is valid for inviscid, irrotational, incompressible flow. The finite element equations were derived through the method of weighted residuals with weighting functions selected by the Galerkin method. For the purposes of analysis, the infinite flow field was replaced by a finite domain. Neumann type boundary conditions were imposed on the airfoil surface. Dirichlet boundary conditions were specified as required by the problem formulation for uniqueness. Three types of solution methods were used, for various treatments of the jump discontinuity required in the lifting problem. The first method was a superposition technique, which treated the potential along the upper and lower nodes of the branch cut as constant. The circulation was determined by applying the Kutta condition during the combination of the subproblems. The second method was an iterative technique where the circulation was varied until the Kutta condition was satisfied. This method also specified constant potential along the upper and lower branch cut nodes. The third method was also an iterative technique on circulation; however, only the ratio of potentials across the branch cut nodes were kept constant.