The Calculation of Axisymmetric Duct Geometries for Incompressible Rotational Flow with Blockage Effects and Body Forces

In this paper a numerical algorithm is described for solving the boundary value problem associated with axisymmetric, inviscid, incompressible and irrota-tional flow with a circumferentially arranged cascade of aerofoils placed in the duct. The governing equations are formulated in terms of the stream function ψ(x,y) and the functionφ(x,y) as independent variables where for irrotational flow φ(x,y) can be recognized as the velocity potential function, for rotational flow φ(x,y) ceases being the velocity potential function but does remain orthogonal to the stream lines. A numerical method based on finite differences solving a Poisson type equation on a uniform mesh is employed. The technique described is capable of tackling the so-called inverse problem where the velocity wall distributions are prescribed from which the duct wall shape is calculated, as well as the direct problem where the velocity distribution on the duct walls are calculated from prescribed duct wall shapes. Results for the case of prescribing the radius i.e. the so called Dirichlet boundary conditions are given. A downstream condition is prescribed such that cylindrical flow, that is flow which is independent of the axial coordinate, exists. The numerical results are obtained by using Green's function for the Laplace equation on a rectangle. The presence of the blades has a bearing on the rate of mass flow and thus alters the usual equation of continuity.