On the compressible bidirectional vortex in a cyclonically driven Trkalian flow field

In this study, the Bragg-Hawthorne equation (BHE) is extended in the context of a steady, inviscid and compressible fluid, thus leading to an assortment of partial differential equations that must be solved simultaneously. A solution is pursued by implementing a Rayleigh-Janzen expansion in the square of the reference Mach number. The corresponding formulation is subsequently used to derive a compressible approximation for the Trkalian model of the bidirectional vortex. The approximate solution is compared to a representative computational fluid dynamics simulation in order to validate the modelling assumptions under realistic conditions. The latter is found to exhibit an appreciable steepening of the axial velocity profile, which is accompanied by an axial dependence in the mantle location that is somewhat reminiscent of the radial shifting of mantles reported in some experimental trials and simulations. In this context, flows with a strong swirl intensity do not seem to be significantly affected by the introduction of compressibility. Rather, as the swirl intensity is reduced the effects of compressibility become more noticeable, especially in the axial and radial velocity components. It may also be realized that imparting a progressively larger swirl component stands to promote the axisymmetric distribution of flow field properties, and these include an implicit resistance to dilatational effects in the tangential direction. From a broader perspective, this study provides a viable approximation to the Trkalian motion associated with cyclonic flows, while serving as a limited proof of concept for the compressible Bragg-Hawthorne procedure applied to a steady, axisymmetric and inviscid fluid.