Exact Eulerian Solutions of the Cylindrical Bidirectional Vortex

Abstract

In this work, new types of Eulerian motions are investigated as potential flow candidates for describing the bidirectional vortex field in a right-cylindrical chamber. These basic solutions apply to industrial cyclone separators and to idealized representations of either liquid or hybrid rocket engines. The latter correspond to a bidirectional vortex chamber with weak sidewall permeability. As usual, we take the bulk motion to be isentropic along streamlines, with no concern for reactions, heat transfer, viscosity, compressibility, or unsteadiness. Our mathematical approach is anchored on the cylindrical Bragg-Hawthorne equation which is concurrently applied in its spherical form to the treatment of the conical cyclone (Barber, T. A., and Majdalani, J., "Exact Eulerian Solution of the Conical Bidirectional Vortex," AIAA Paper 2009-5306, August 2009). Among the unique characteristics of the new solutions, we cite the axial dependence of the swirl velocity, the Trkalian and Beltramian characters of the helical motions, the appreciable sensitivity to the outlet radius, the alternate location of the mantle, and the increased axial and radial velocity magnitudes, including the rate of mass transfer across the mantle, for which explicit approximations are obtained. Our results are compared to one another and to an existing, complex lamellar solution in which the swirl velocity collapses into a free vortex. In this vein, we find the strictly Beltramian flows to share virtually identical pressure variations and radial pressure gradients with those associated with the complex lamellar motion. By the same token, both families necessitate an asymptotic treatment to overcome their endpoint deficiencies caused by their dismissal of viscous stresses. From a broader perspective, the work delineates a logical framework through which self-similar, axisymmetric solutions to bidirectional and multidirectional vortex motions may be rigorously pursued. Furthermore, it illustrates the manner through which multiple configurations may be arrived at depending on the types of boundary conditions imposed. For example, both the slip condition at the sidewall and the inlet flow pattern at the headwall may be enforced or relaxed. Our analysis is carried out in the context of a right-cylindrical chamber first without and then with allowance for sidewall injection. The latter enables us to model the basic flow in the socalled Vortex Injection Hybrid Rocket Engine (VIHRE). Finally, the alternate mantle location and swirl velocity are verified in the light of existing measurements and numerical simulations performed with the Reynolds Stress-transport Model (RSM).

Sidewall Boundary Layers of the Bidirectional Vortex

“…Yet Bloor and Ingham's model remains, to this date, inviscid and singular at the vortex axis [20]. The same may be said of the class of Beltramian flows described in [27]. For this reason, the present article is aimed at developing an improved representation of the bidirectional vortex that secures the sidewall boundary layers in all three spatial directions: axial, radial, and tangential.…”

“…It was further extended to spherical geometry by Majdalani and Rienstra [25] and, to the treatment of conical cyclones, by Barber and Majdalani [26]. A complementary set of Eulerian solutions of the Beltramian type were later produced by Majdalani [27], who used the Bragg-Hawthorne equation as a starting point. These inviscid solutions resembled Bloor and Ingham's in exhibiting a singularity at the centerline.…”

Biglobal Instability of the Bidirectional Vortex. Part 2: Complex Lamellar and Beltramian Motions

“…8,9 Most recently, the Bragg-Hawthorne equation (BHE) 10 has been considered as the opening point for the discussion of new solutions for the bidirectional vortex. 11 Although the BHE exhibits an apparent resemblance to that found in the original analysis, the relation that it prescribes between the vorticity and streamfunction differs. At the outset, it gives rise to solutions of the Beltramian type where the Lamb vector is identically zero.…”

“…Being of the latter type, our coverage will encompass both linear and harmonic Beltramian models introduced in 2009. 11 The paper is organized to first introduce the stability of the complex-lamellar mean flow model, followed by the linear and finally, harmonic, Beltramian models. The respective results are discussed in full, and key stability characteristics are identified.…”

Unified Framework for Modeling Swirl Dominated Helical Motions

“…Then following Bloor and Ingham [47] in the choice of an angular momentum-streamfunction relation, they reconstructed the traditional Bloor-Ingham solution in spherical coordinates and managed to retrieve, in the process, exact expressions for the velocity, vorticity, pressure, and mantle location in a conical cyclone. Along similar lines, Majdalani [71,72] solved the Bragg-Hawthorne equation in polarcylindrical coordinates using two assortments of boundary conditions that were appropriate of either VCCWC or VIHRE concepts. He thus unravelled several new solutions of the Beltramian type that could be used to model the flowfield in cylindrically-shaped cyclonic chambers with (in the case of VIHRE) and without blowing at the sidewall (for the VCCWC).…”