On the compressible bidirectional vortex. Part 1: A Bragg-Hawthorne stream function formulation

The Bragg-Hawthorne equation, also named Squire-Long, takes advantage of a requirement that the stagnation pressure head, H, and angular momentum, B, may be generally related to the stream function, Ψ. This reduces the Navier-Stokes equations to a single stream function representation, wherein H and B may be specified based on the application of interest. In this paper, the Bragg-Hawthorne equation (BHE) is extended in the context of a steady, inviscid, and compressible fluid. Then given an assortment of suitable assumptions, our approach leads to a pair of partial differential equations that must be treated simultaneously. Rather than solve the ensuing coupled equations numerically, we opt for a reduced-order model by implementing the Rayleigh-Janzen expansion method in which the square of the reference Mach number is employed as a perturbation parameter. The resulting linearized equations are retrievable to an arbitrary level of precision, thus leading to the establishment of a reliable, compressible BHE framework. From a practical standpoint, we find that a first-order correction is sufficient to capture the bulk compressible contribution in most physical settings. In the second part of this paper series, the procedure is tested to derive a compressible solution for the linear Beltramian model of the bidirectional vortex; this motion corresponds to a swirl-driven cyclonic flowfield with an axially reversing character that proves to be of particular interest to the development of an innovative, self-cooled, liquid rocket engine concept. It can therefore be seen that, although the application of interest remains focused on the bidirectional vortex motion, the framework that we offer retains sufficient generality to be useful in handling a wide range of axisymmetric problems, especially those that may be conveniently expressed in polar-cylindrical coordinates.