Some Exact Solutions of the Flow Through Annular Cascade Actuator Discs

Exact solutions are presented for two types of flow of an incompressible inviscid fluid through an annular passage with a discontinuity in tangential velocity at one section, as would occur a t a cascade of airfoil blades of negligible chord. When stagnation conditions are constant throughout the flow region, it is shown that the product of the tangential vorticity component (77) and the radius (r) is constant along a stream line. The particular case when this product is a linear multiple of the stream function (^) is solved by dividing \p into persistent and perturbation components and separating the variables. A second exact solution is obtained when the product far) is a function of radius alone and when the stagnation conditions are allowed to vary. Two solutions of either of the above kinds can be exactly matched a t the discontinuity section, thus determining the complete flow pattern for arbitrarily selected conditions at infinity. The axial velocity at this section is almost exactly the mean of upstream and downstream values at the same radius. The family of exact solutions may be of use in estimating the accuracy of approximate solutions and, as shown in a numerical example, may match some desired flow conditions sufficiently closely to be of interest in design.