THE DESIGN OF AXISYMMETRIC DUCTS FOR INCOMPRESSIBLE FLOW WITH BLOCKAGE EFFECTS AND BODY FORCES

摘要

In this paper a numerical algorithm is described for solving the boundary value problem associated with axisymmetric, inviscid, incompressible and irrotational and irrotational flow with a circumferentially arranged cascade of aerofoils placed in the duct. The algorithm is capable of calculating the duct wall geometries from prescribed wall velocity distributions. The equations modeling the flow are derived using the stream function ψ(x,y) and the function Φ(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, the x and y are the usual axial and radial coordinates in cylindrical polar coordinates respectively. The technique described is capable of tackling the so-called inverse problem where the velocity wall distributions are prescribed from which the duct geometry is calculated, as well as the direct problem where the velocity distribution on the pressure and suction surfaces are calculated from prescribed geometries. The two different cases outlined in this paper are boundary value problems with Neumann and Dirichlet boundary conditions respectively with results for the Neumann boundary condition only included. The axial velocity and the swirl velocity are prescribed such that no vorticity is transported through the duct. The governing linear elliptic second order partial differential is coupled with a set of quasi-linear hyperbolic first order partial differential equations with characteristics parallel to the Φ and ψ axes, the numerical solution is thus obtained iteratively using finite differences to approximate the derivatives. The presence of the blades has a bearing on the rate of mass flow and thus alters the usual equation of continuity. The forces generated by the blades are resolved into components parallel and perpendicular to the flow direction, modeling respectively viscous effects and the guiding action of the blades.