Abstract
The non‐linear, small‐disturbance, velocity‐potential equation for steady flow over a thin, symmetric, non‐lifting aerofoil is solved by the finite element method. Pressure distributions are computed for flow regimes ranging from incompressible flow to transonic flow with weak shocks. The governing equation is linearized by a sequence of linear equations which is solved iteratively using Galerkin's method. The infinite flow domain is replaced by a finite but sufficiently large domain which is discretized by bilinear rectangular elements. Boundary conditions of the Neumann type are imposed along an approximate boundary in accordance with classical thin‐aerofoil theory. Dirichlet conditions are imposed along the far‐field boundary from an asymptotic solution valid in the far‐field.
Convergence of solution algorithms occurs rapidly for all subsonic flows, but fails for transonic or mixed flows when the supersonic bubble is larger than one‐half on an element. For transonic flows a new upwinding scheme is used to modify the finite element formulations for elements within the supersonic bubble. The scheme, governed by two parameters, accounts for the proper zone of influence for those elements within the bubble by excluding the influence of iterative downwind forces on the solution at upwind nodes. The upwinding scheme permits convergence of the iterative solution algorithm and also captures the weak compression shock which forms automatically in the solution process without the need or use of shock elements.