In this work, an exact inviscid solution is introduced for the incompressible Euler equation in the context of a bidirectional, cyclonic vortex in a right-cylindrical chamber with a hollow core. The presence of a gaseous core restricts the flow domain to an annular vortex region that extends into a toroid in three dimensional space. The procedure that we follow is based on the Bragg-Hawthorne framework, which is used in conjunction with a unique assortment of boundary conditions that mirror in large part those entailed in the derivation of a comparably complex-lamellar mean flow profile by Vyas and Majdalani (Vyas, A. B., and Majdalani, J., "Exact Solution of the Bidirectional Vortex," AIAA Journal, Vol. 44, No. 10, 2006, pp. 2208-2216). At the outset, a self-similar solution is retrieved from the Bragg-Hawthorne equation under the premises of steady, axisymmetric, and inviscid conditions, as opposed to the vorticity-streamfunction approach used previously. The resulting formulation is then utilized to describe the bidirectional evolution of the inner and outer vortex motions, including their fundamental properties, such as the interfacial layer known as the mantle, as well as the velocity, pressure, and vorticity fields, with particular attention being devoted to their peak values and spatial excursions that accompany successive expansions of the core radius. By way of confirmation, it is shown that removal of the hollow core restores the well-established solution in a fully flowing cylindrical chamber. Immediate applications of cyclonic flows include liquid and hybrid rocket engines, swirl-driven combustion devices, as well as a multitude of heat exchangers, centrifuges, cyclone separators, and flow separation contraptions that offer distinct advantages over conventional, non-swirling systems.
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