The occurrence and removal of indeterminacy from flow calculations in turbomachines

Abstract

The flow physics for inviscid flow on an arbitrary two-dimensional streamsurface for application to turbo-machines are re-examined without specialising to a particular type of surface. This re-examination introduces streamfunction as a function of three space dimensions which is two dimensional in the locality of the prescribed streamsurface and does not involve the concept of a force normal to the surface as introduced by Wu and used by Marsh and Smith. A special principal equation of motion is developed (designated the N-principal equation) which is applicable to any streamsurface. This N-principal equation has several advantages over the principal equations formerly used, the most powerful being that rotation of axes as carried out by Marsh is no longer necessary and solutions become possible for annular ducting of increased geometrical complexity including flow around a toroidal surface as in a fluid coupling. Numerical advantages of the N-principal equation are a saving in computer store and improved numerical accuracy in certain regions of the solution. The co-ordinate form of the N-principal equation is developed for application to hub-to-shroud streamsurfaces and for blade-to-blade streamsurfaces of revolution. The N-principal form of equation is compared to the principal forms of Marsh and Smith where it is shown that the N-principal form gains its advantage chiefly by its freedom from indeterminacy which inevitably appears with other forms. The latter are shown to be the equation of motion resolved in certain prescribed planes whereas the N-principal form is always resolved normal to the stream direction. However the N-principal form can be recovered from other forms if the equations are further developed. Other forms of indeterminacy and computational failure may arise from the choice of co-ordinate system used to express the derivatives and the manner in which the molecule grid points are chosen. Other failures or restrictions relate to the way in which the grid is set up. These are discussed and proposals made for ensuring their avoidance.