On the generalized Beltramian motion of the bidirectional vortex in a conical cyclone

Abstract

This work presents an exact solution of Euler's incompressible equations in the context of a bidirectional vortex evolving inside a conically shaped cyclonic chamber. The corresponding helical flowfield is modeled under inviscid conditions assuming constant angular momentum. By leveraging the axisymmetric nature of the problem, a steady-state solution of the generalized Beltramian type is obtained directly from first principles, namely, from the Bragg–Hawthorne equation in spherical coordinates. The resulting stream function representation enables us to fully describe the ensuing swirl-dominated motion including its fundamental flow characteristics. After identifying an isolated singularity that appears at a cone divergence half-angle of 63.43°, two piecewise formulations are provided that correspond to either fluid injection or extraction at the top section of the conical cyclone. In this process, analytical expressions are readily retrieved for the three velocity components, vorticity, and pressure. Other essential flow indicators, such as the theoretically preferred mantle orientation, the empirically favored locus of zero vertical velocity, the maximum polar and axial velocities, the crossflow velocity, and other such terms, are systematically deduced. Results are validated using limiting process verifications and comparisons to both numerical and experimental measurements. The subtle differences between the present model and a strictly Beltramian flowfield are also highlighted and discussed. The conically cyclonic configuration considered here is relevant to propulsive devices, such as vortex-fired liquid rocket engines with tapered walls; meteorological phenomena, such as tornadoes, dust devils, and fire whirls; and industrial contraptions, such as cyclonic flow separators, collectors, centrifuges, boilers, vacuum cleaners, cement grinders, and so on.