On the compressible bidirectional vortex. Part 2: A Beltramian flowfield approximation

In the first part of this paper series, a compressible Bragg-Hawthorne framework is developed in the form of a density-stream function formulation that may be applied to a wide class of steady, axisymmetric flow problems. In this sequel, the procedure is implemented with the aim of retrieving an approximate compressible solution to describe the helical motion observed in a cyclonic, bidirectional vortex chamber. Our approach centers on sequentially solving the linearized density-stream function relations to the extent of producing a closed-form expression for the compressible, swirl-dominated motion. This effort begins by conceiving a judicious set of boundary conditions that may be paired with the Bragg-Hawthorne procedure established previously. Then using a Rayleigh-Janzen expansion of the resulting system of partial differential equations, a solution is achieved at the leading and first orders in the injection Mach number squared. At the leading order, the incompressible approximation of the linear Beltramian flowfield is recovered, with compressibility effects relegated to the first order. In moving to higher orders, our approach requires the numerical integration of several groups of Bessel functions. These are specified individually as special functions that enable us to retain the analytical character of the first-order correction. Among significant findings predicted by this model, an appreciable steepening of the axial velocity profile is captured, thus mirroring a similar mechanism observed in solid rocket motors (SRMs). Furthermore, the mantle location, which is ordinarily committed to a single radial location in the absence of compressibility, gains an axial dependence that is reminiscent of the radial shifting of mantles reported in some experimental trials and numerical simulations. This sensitivity becomes more pronounced at higher injection Mach numbers and ratios of specific heats. The sensitivity of the solution to variations in κ, the inflow swirl parameter, is also investigated. We find that increasing κ leads to relative growth in both the incompressible and compressible velocities in the axial and radial directions. Conversely, at small values of κ, i.e. when the axial and radial velocities are overwhelmingly dominated by the tangential motion, the compressible solution approaches the leading-order result. Albeit counter-intuitive at first, imparting a progressively larger swirl component stands to promote the axisymmetric distribution of flowfield properties, and these include an implicit resistance to compression in the tangential direction, lest axisymmetry is violated. As for the density, its largest excursions occur near the centerline, and these become more appreciable at higher Mach numbers and ratios of specific heats. From a broader perspective, this study not only provides a viable approximation to the linear Beltramian motion associated with the classic cyclonic flowfield, it also offers a proof-of-concept of the procedure introduced by the authors in their companion article. Specifically, the present analysis confirms the validity of the newly established compressible Bragg-Hawthorne framework in the treatment of swirl-driven and other axisymmetric fluid motions.