Superpotential Solution for Jet-Engine External Potential and Internal Rotational Flow Interaction

Abstract

For flows with prescribed parallel shear far upstream, the vorticity generation term in the disturbance stream-function (Ψ) equation and the Bernoulli “constant,” both of which vary from stream surface to stream surface, are explicitly evaluated as power series in Ψ with curvature-dependent coefficients, for axisymmetric flows using some invariant properties of vorticity. By casting the linearized stream-function equation in conservation form, extended Cauchy-Reimann conditions are obtained, implying a “superpotential” φ* satisfying a “Laplace-like” equation useful for solving flows past prescribed shapes; the corresponding tangency and Kutta conditions, interestingly, take a “potential form,” so that simple changes to existing potential flow algorithms extend their applicability to strong oncoming shears with arbitrary curvature. The theory, which applies to duct flows behind actuator disks generating shear, is sketched for both “analysis” and “design” formulations; here, we address the interaction between external potential and internal rotational jet-engine flows occurring through both an assumed actuator disk and a trailing edge slipstream, and provide representative numerical calculations.