skip to main content

Title: 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 more » 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. « less
Award ID(s):
Publication Date:
Journal Name:
Physics of Fluids
Page Range or eLocation-ID:
Sponsoring Org:
National Science Foundation
More Like this
  1. This work focuses on the use of a finite-volume solver to describe the wall-bounded cyclonic flowfield that evolves in a swirl-driven thrust chamber. More specifically, a non-reactive, cold-flow simulation is carried out using an idealized chamber configuration of a square-shaped, right-cylindrical enclosure with eight tangential injectors and a variable nozzle size. For simplicity, we opt for air as the working fluid and perform our simulations under steady, incompressible, and inviscid flow conditions. First, a meticulously developed mesh that consists of tetrahedral elements is generated in a manner to minimize the overall skewness, especially near injectors; second, this mesh is converted into a polyhedral grid to improve convergence characteristics and accuracy. After achieving convergence in all variables, our three velocity components are examined and compared to an existing analytical solution obtained by Vyas and Majdalani (Vyas, A. B., and Majdalani, J., “Exact Solution of the Bidirectional Vortex,” AIAA Journal, Vol. 44, No. 10, 2006, pp. 2208-2216). We find that the numerical model is capable of predicting the expected forced vortex behavior in the inner core region as well as the free vortex tail in the inviscid region. Moreover, the results appear to be in fair agreement with the Vyas–Majdalani solution derivedmore »under similarly inviscid conditions, and thus resulting in a quasi complex-lamellar profile. In this work, we are able to ascertain the axial independence of the swirl velocity no matter the value of the outlet radius, which confirms the key assumption made in most analytical models of wall-bounded cyclonic motions. Moreover, the pressure distribution predicted numerically is found to be in fair agreement with both theoretical formulations and experimental measurements of cyclone separators. The bidirectional character of the flowfield is also corroborated by the axial and radial velocity distributions, which are found to be concurrent with theory. Then using parametric trade studies, the sensitivity of the numerical simulations to the outlet diameter of the chamber is explored to determine the influence of outlet nozzle variations on the mantle location and the number of mantles. Since none of the cases considered here promote the onset of multiple mantles, we are led to believe that more factors are involved in producing more mantles than one. Besides the exit diameter, the formation of a multiple mantle structure may be influenced by the physical boundary conditions, nozzle radius, inlet curvature, and length. In closing, we show that the latter plays a significant role in controlling the development of backflow regions inside the chamber.« less
  2. 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 andmore »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.« less
  3. Many species of fish gather in dense collectives or schools where there are significant flow interactions from their shed wakes. Commonly, these swimmers shed a classic reverse von Kármán wake, however, schooling eels produce a bifurcated wake topology with two vortex rings shed per oscillation cycle. To examine the schooling interactions of a hydrofoil with a bifurcated wake topology, we present tomographic particle image velocimetry (tomo PIV) measurements of the flow interactions and direct force measurements of the performance of two low-aspect-ratio hydrofoils ( A R = 0.5 ) in an in-line and a staggered arrangement. Surprisingly, when the leader and follower are interacting in either arrangement there are only minor alterations to the flowfields beyond the superposition of the flowfields produced by the isolated leader and follower. Motivated by this finding, Garrick’s linear theory, a linear unsteady hydrofoil theory based on a potential flow assumption, was adapted to predict the lift and thrust performance of the follower. Here, the follower hydrofoil interacting with the leader’s wake is considered as the superposition of an isolated pitching foil with a time-varying cross-stream velocity derived from the wake flow measurements of the isolated leader. Linear theory predictions accurately capture the time-averaged liftmore »force and some of the major peaks in thrust derived from the follower interacting with the leader’s wake in a staggered arrangement. The thrust peaks that are not predicted by linear theory are likely driven by spatial variations in the flowfield acting on the follower or nonlinear flow interactions; neither of which are accounted for in the simple theory. This suggests that unsteady potential flow theory that does account for spatial variations in the flowfield acting on a hydrofoil can provide a relatively simple framework to understand and model the flow interactions that occur in schooling fish. Additionally, schooling eels can derive thrust and efficiency increases of 63-80% in either a in-line or a staggered arrangement where the follower is between two branched momentum jets or with one momentum jet branch directly impinging on it, respectively.« less
  4. One-quarter of the world’s tropical cyclones (TCs) occur in the Indian Ocean (IO) basin.The mechanisms for TC initiation in the IO are varied, but one recently discovered process involves the flow around the steep topography of Sumatra.  When the low-level flow impinges on Sumatra, it is blocked and the flow splits under typical environmental stratification.  As a result, wake vortices commonly develop at northern and southern island tips of the island. For the case of easterly flow, these circulationssubsequently move downstream over the IO.  The wake vortices emanating from the island tips are counter-rotating, but since Sumatra straddles the equator, the circulations are cyclonic in both hemispheres and thus have the potential for TC development. Using data from2.5yearsof observations from DYNAMO and YOTC, it is found that approximately 25% of the TCsthat occurred overIO basin during that periodwere initiated by Sumatra-induced wake vortices.Additional analysis of vortex statistics for the period 2008-17 has found that vortex counts are highest near Madden-Julian Oscillation (MJO) phase 1 when low-level easterlies are strongest across southern Sumatra. A secondary peak in vortex formation occurs during MJO phase 4 when low-level westerlies exist near the equator west of Sumatra. The latter finding suggests that MJO-related, low-levelmore »westerly surges on the equator impinging on Sumatracontribute to an increase in wake vortex development.  Numerical simulations have shown that circulations farther upstream such aswestern Pacific remnant TCs and the Borneo vortex can influence the development of Sumatra wake vortices and their growth into TCs over the IO.« less
  5. We vary the inflow properties in a finite-volume solver to investigate their effects on the computed cyclonic motion in a right-cylindrical vortex chamber. The latter comprises eight tangential injectors through which steady-state air is introduced under incompressible and inviscid conditions. To minimize cell skewness around injectors, a fine tetrahedral mesh is implemented first and then converted into polyhedral elements, namely, to improve convergence characteristics and precision. Once convergence is achieved, our principal variables are evaluated and compared using a range of inflow parameters. These include the tangential injector speed, count, diameter, and elevation. The resulting computations show that well-resolved numerical simulations can properly predict the forced vortex behavior that dominates in the core region as well as the free vortex tail that prevails radially outwardly, beyond the point of peak tangential speed. It is also shown that augmenting the mass influx by increasing the number of injectors, injector size, or average injection speed further amplifies the vortex strength and all peak velocities while shifting the mantle radially inwardly. Overall, the axial velocity is found to be the most sensitive to vertical displacements of the injection plane. By raising the injection plane to the top half portion of the chamber, themore »flow character is markedly altered, and an axially unidirectional vortex is engendered, particularly, with no upward motion or mantle formation. Conversely, the tangential and radial velocities are found to be axially independent and together with the pressure distribution prove to be the least sensitive to injection plane relocations.« less