A new relaxational treatment of the compressible two-dimensional flow about an aerofoil with circulation

Abstract

The incompressible two-dimensional flow about an aerofoil with circulation is calculated using relaxation on the square mesh formed by the incompressible velocity equipotentials (φ = constant) and the streamlines (ψ = constant). Log (l/q0) and θo, where (qo, θo) is the incompressible velocity vector on polar co-ordinates, are harmonic functions in the (φ, ψ)-plane, and can be found by well-known relaxation or squaring methods. Boundary conditions are specified in the (x, y) or physical plane, but starting from an assumption for the surface velocity, approximate boundary conditions can be found for the (φ, ψ)-plane, which then enable a more accurate value of the surface velocity to be calculated, and so on. The circulation is imposed on the field by having a smaller number of equipotential lines of the mesh cutting the lower surface of the aerofoil than cutting the upper surface. Non-linear compressible flow equations involving log (l/q) and θ, where (q, θ) is the compressible flow vector, are solved by relaxation on the (φ, ψ) grid. The results for a worked example are compared with experimental curves provided by the National Physical Laboratory for the same aerofoil at approximately the same angle of incidence. There is reasonable agreement. Supersonic patches were experienced and are not difficult to treat by relaxation, although the difference equations become poorly conditioned.