Abstract

In this paper a numerical algorithm is described for solving the boundary value problem associated with axisymmetric, inviscid, compressible and rotational and irrotational flow. The algorithm is capable of calculating the duct wall geometries from prescribed wall velocity distributions. The equations modeling the flow are expressed using the stream function <img height="12" src="http://www.ijapm.org/uploadfile/2014/0126/20140126020422478.png" width="13" />(x, y) and the function <img height="12" src="http://www.ijapm.org/uploadfile/2014/0126/20140126020441392.png" width="12" />(x, y) as independent variables, where for irrotational flow <img height="12" src="http://www.ijapm.org/uploadfile/2014/0126/20140126020441392.png" width="12" />(x, y) can be recognized as the velocity potential function, for rotational flow <img height="12" src="http://www.ijapm.org/uploadfile/2014/0126/20140126020441392.png" width="12" />(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 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 in fact 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 vorticity is transported through the duct. The governing second order partial differential is coupled with a set of quasi-linear hyperbolic first order partial differential equations with characteristics parallel to the <img height="12" src="http://www.ijapm.org/uploadfile/2014/0126/20140126020441392.png" width="12" /> and <img height="12" src="http://www.ijapm.org/uploadfile/2014/0126/20140126020422478.png" width="13" /> axes, the numerical solution is thus obtained iteratively using finite differences to approximate derivatives.