This study focuses on the development of two analytical models that describe the wall-bounded cyclonic flowfield in a hemispherical domain. The closed-form solutions that we pursue are motivated by the need to characterize the swirling bidirectional motion engendered in an upper stage thrust chamber, namely, the VR35K-A VORTEX® engine, conceived and developed by Sierra Nevada Corporation. Our analysis proceeds from the Bragg–Hawthorne formulation, which is quite effective in the treatment of steady, inviscid, and axisymmetric flows. In this work, we show that two rotational solutions may be derived for particular specifications of the stagnation head and tangential angular momentum expressions that appear in the Bragg–Hawthorne equation. Then, with the parental streamfunctions in hand, other properties of interest are deduced and these include the main velocity and pressure variations, vorticities, crossflow velocities, extensional and shearing strain rates, virtual energy dissipation rates, and both axial and polar mantle distributions; the latter consist of pairs of rotating, non-translating interfacial layers, separating the so-called inner and outer bidirectional and bipolar regions, respectively. More specifically, two Beltramian solutions are identified with mantles that appear at 50% and 61.06% of the chamber radius, respectively. By matching the outlet radius of the chamber to the mantle location in the equatorial plane, the outflow is permitted to exit the chamber seamlessly. In both models, the axial and radial velocities vary linearly with the injection speed and a characteristic inflow parameter consisting of a geometric ratio of the inlet area and the chamber radius squared.

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This work describes an inviscid solution for a cyclonic flowfield evolving in a hemispherical chamber configuration. In this context, a vigorously swirling motion is triggered by a gaseous stream that is introduced tangentially to the inner circumference of the chamber's equatorial plane. The resulting updraft spirals around while sweeping the chamber wall, reverses direction while approaching the headend, and then tunnels itself out through the inner core portion of the chamber. Our analysis proceeds from the Bragg–Hawthorne formulation, which proves effective in the treatment of steady, incompressible, and axisymmetric motions. Then using appropriate boundary conditions, we are able to obtain a closed-form expression for the Stokes streamfunction in both spherical and cylindrical coordinates. Other properties of interest are subsequently deduced, and these include the principal velocities and pressure distributions, vorticity, swirl intensity, helicity density, crossflow velocity, and the location of the axial mantle; the latter separates the outer annular updraft from the inner, centralized downdraft. Due in large part to the overarching spherical curvature, the cyclonic motion also exhibits a polar mantle across which the flow becomes entirely radial inward. Along this lower and shorter spherical interface, the polar velocity vanishes while switching angular direction. Interestingly, both polar and axial mantles coincide in the exit plane where the ideal outlet size, prescribed by the mantle position, is found to be approximately 70.7% of the chamber radius. We thus recover the same mantle fraction of the cyclonic flow analog in a right-cylindrical chamber where an essentially complex-lamellar motion is established.

This study focuses on the development of an internal potential flow solution in the context of a hemispherically bounded cyclonic chamber. The analysis proceeds from the Bragg–Hawthorne equation, which is quite effective in the treatment of steady, inviscid, and axisymmetric flows when expressed in terms of the streamfunction. Once the streamfunction is obtained, other flow properties are readily deduced; these include the principal velocity and pressure distributions, swirl intensity, crossflow velocity, and mantle location. Furthermore, given the overarching spherical geometry, two different types of mantles are identified and related to the coexistence of axially bidirectional and circularly bipolar regions. The first, axial mantle, which is traditionally used in the analysis of cylindrical and conical cyclone separators, consists of a rotating, non-translating interfacial layer along which the axial velocity vanishes. It thus separates the outer, vertical updraft, from the inner, swirling downdraft. The second, polar mantle, which arises in the context of a hemispherical flow configuration, coincides with the spherical interface along which the polar velocity vanishes. It hence partitions the flow domain into a much larger outer region, where the flow direction remains strictly counterclockwise, and a proportionally smaller inner region, where the outflow becomes clockwise. Despite their dissimilar structures, both axial and polar mantles meet in the exit plane at a fractional radius of 1/e2 or 13.53%. In this study, the unique characteristics of the resulting irrotational motion, which reduces to a continuously looping, hemispherically cyclonic potential vortex, are evaluated and discussed.