In this work, an exact inviscid solution is developed for the incompressible Euler equations in the context of a bidirectional, cyclonic flowfield in a right-cylindrical chamber with a hollow core. The presence of a hollow core confines the flow domain to an annular swirling region that extends into a toroid in three-dimensional space. The procedure that we follow is based on the Bragg–Hawthorne framework and a judicious assortment of boundary conditions that correspond to a wall-bounded cyclonic motion with a cylindrical core. At the outset, a self-similar stream function is obtained directly from the Bragg–Hawthorne equation under the premises of steady, axisymmetric, and inviscid conditions. The resulting formulation enables us to describe the bidirectional evolution of the so-called inner and outer vortex motions, including their fundamental properties, such as the interfacial layer known as the mantle; it also unravels compact analytical expressions for 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 for a fully flowing cylindrical cyclone. 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 devices that offer distinct advantages over conventional, non-swirling systems.
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On the generalized Beltramian motion of the bidirectional vortex in a conical cyclone
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.
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- Award ID(s):
- 1761675
- PAR ID:
- 10340987
- Date Published:
- Journal Name:
- Physics of Fluids
- Volume:
- 34
- Issue:
- 3
- ISSN:
- 1070-6631
- Page Range / eLocation ID:
- 036604
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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