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.

## I. INTRODUCTION

Rotating flows, of which helical motions constitute a subset,^{1} are often manifested in nature and man-made devices as they encompass an extensive spectrum of phenomena and lengthscales. Their existence predates human evolution as it surely must have accompanied the formation of geophysical flows such as water spouts, fire whirls, dust devils, tornadoes, and hurricanes.^{2–4} An unbounded tornado, which is relevant to the subject of this study, represents a classical geophysical vortex that evolves over a Meso-*δ* scale.^{5,6} Water spouts, fire whirls, and dust devils also constitute naturally driven vortices that are spread over a Micro-*β* scale. In fact, swirling motions may be seen to extend from the finest spinning trajectories of electrons in orbit, through the Meso and Macro scale activities ascribed to tropical hurricanes and synoptic cyclones, to the Mega scale motions of planetary excursions, galactic pinwheels, and even black holes.^{7–14} In this context, planetary vortex dynamics, as a science, focuses on investigating the Coriolis effects in rotating systems through the analysis of barotropic flows, polar vortices, beta-plane approximations, Ekman spirals, Stewartson layers, Rossby waves, and flow stratification.^{15–17} Vortices also emerge in astrophysics, specifically, in the modeling of magnetohydrodynamics^{18,19} and star formations;^{20} they also arise in biological organisms, particularly, in the modeling of cavity flows in the aortic sinuses,^{21} where swirl has proven to facilitate blood flow and valve regulation throughout the circulatory system.

It can therefore be seen that the understanding of rotating flow physics remains an important subject of research because of the multitude of favorable attributes that accompany swirl-dominated motions. These include, by way of illustration, increased mixing, prolonged residence time, extended stability margins, elevated wall pressures, enhanced regression rates, improved heat transfer, induced suction, and many such effects, depending on the application in question. In fact, the addition of swirl has proven to be effective at augmenting heat transfer characteristics in a variety of cylindrical chambers and ducts.^{22–32} In most applications, rotating flows give rise to centrifugally driven low and high pressure zones. Naturally, the low pressure zones can be employed as strong attractors to draw substances from the surrounding high pressure regions by way of suction. As a result, the favorable suction potential of vortices has often been incorporated into manufacturing applications such as flow filtration, isotope separation, cement casting, and clean handling processes in the semiconductor and electronics industries.^{33–35} Vortex-induced motions have also been exploited in the context of bluff body flow separation in combustion chambers.^{36,37} In this vein, the so-called “trapped vortex combustor” has been shown to be substantially helpful at improving flame stability, reducing pressure losses, and modulating emission characteristics in gas turbine engines. Examples include the Step-Swirl Gas-Turbine Combustor by Durbin *et al.,*^{38} the Trapped-Vortex Combustor by Katta and Roquemore,^{39} the Plasma Assisted Tornado Combustor by Matveev,^{40} the Triple Vortex Combustor by Matveev *et al.,*^{41} and the Hybrid Solar Receiver Combustor (HSRC) by Long *et al.*^{42}

In the propulsion community, the use of a bidirectional vortex motion to promote stability, efficiency, and self-cooling capabilities of chemical rocket and air-breathing engines has been the subject of several dedicated studies. On this count, one may cite the so-called Vortex Combustion Ramjet,^{43} the Vortex Combustion Combined Cycle (VCCC) Engine,^{44} the Cool-Wall Vortex Combustion Chamber (CWVCC),^{45,46} the MAELSTROM-G25 thruster,^{47} the Vortex Combustion Cold-Wall (VCCW) chamber,^{48–50} and the VR-3A Vision Engine, which have been pioneered by Orbital Technologies Corporation (ORBITEC^{TM}); these have culminated in the development of a volumetrically efficient configuration motivated by ongoing engine developments at Sierra Space Corporation, a subsidiary of Sierra Nevada Corporation. More specifically, they have evolved into a compact, self-cooled, and highly efficient VR35K-A VORTEX^{®} engine, which is intended for use on upper stage vehicles. The corresponding chamber is driven by wall-tangential injection, thus leading to the establishment of a cyclonic flowfield that is reminiscent of the flow in cyclone separators. To improve performance and propellant regression, a similarly operated cyclonic flowfield has been successfully incorporated into a cylindrically shaped hybrid rocket engine known as the Vortex Injection Hybrid Rocket Engine (VIHRE).^{51–53} These major developments have been accompanied by a series of theoretical and computational analyses aimed at characterizing the fluid dynamics of confined cyclonic motions in different geometric configurations including those pertaining to cyclone separators.

Among the studies based on computation, one may enumerate those that rely on Reynolds-averaged Navier–Stokes solvers, large-eddy simulations, finite element, and finite difference techniques. Related works extend to those by Hsieh and Rajamani,^{54} Ogawa,^{55} Hoekstra *et al.,*^{56,57} Derksen and van den Akker,^{58} Zavala Sansón *et al.,*^{59} Hu *et al.,*^{60} Rom,^{61} Zhu *et al.,*^{62} Molina *et al.,*^{63} Majdalani and Chiaverini,^{50} Maicke and Talamantes,^{64} Wang *et al.,*^{65} Rajesh *et al.,*^{66} and Sharma and Majdalani.^{67} The latter shows that, based on a finite-volume solver and suitable choices of nozzle inlet size and curvature, the frequently encountered backflow or central recirculation zones can be fully mitigated, thus facilitating the computational characterization of the stationary motion in a cylindrically shaped cyclonic chamber configuration.

Among the studies that devote themselves to the development of theoretical models, the most germane to this work include, for right-cylindrical and hemispherical chambers, essentially complex-lamellar helical profiles developed for both cylindrical^{68} and spherical configurations,^{69,70} a set of Trkalian and Beltramian solutions,^{71,72} and a general Beltramian motion with arbitrary headwall injection for cylindrical enclosures.^{73} These particular motions constitute elemental subsets of the helical class of rotating flows.^{74,75} More specifically, families of so-called “complex-lamellar”^{68} and Trkalian–Beltramian profiles^{71} are first derived for wall-bounded cyclonic fields in right-cylindrical chambers and, in subsequent studies, their flow analogues in hemispherical chamber configurations are developed systematically.^{70,72} The search for both linear and nonlinear solutions of cyclonic motions is also undertaken by Majdalani and Rienstra^{69} based on a spherical representation of the vorticity-stream function approach. At the outset, Hill's spherical vortex and the quasi-complex-lamellar profile by Vyas and Majdalani^{68} are recovered as special cases and, in this process, the existence of additional solutions for right-cylindrical chambers is demonstrated. Complementary viscous and dilatational corrections are also attempted^{76,77} including a unifying review and generalization^{78} that are mainly focused on right-cylindrical chambers. For conical configurations, however, two particularly relevant investigations may be cited, and these include an approximate formulation by Bloor and Ingham^{79} and its corresponding exact expression.^{80} The latter gives rise to an exact inviscid Beltramian solution that helps to elucidate several fundamental flow attributes associated with conical cyclones. It also serves to facilitate the extraction of several closed-form analytical approximations for basic parameters that affect cyclonic flow modeling. In this follow-up article, a complementary model will be developed and identified as a viable member of the generalized Beltramian class of solutions, specifically, which forms a subset of the larger family of helical motions whose mathematical properties^{81,82} and rich topological structures^{83–85} continue to attract interest in a variety of steady, periodic, and viscous flow applications.^{86–89}

## II. PROBLEM FORMULATION

### A. Geometry and flow conditions

Our geometry corresponds to the typical cylinder-on-cone cyclone separator, which is ubiquitously used in industry for flow separation and processing. In this case, a cylinder is attached above the pinnacle of the cone as illustrated in Fig. 1. In this study, the analysis is limited to the conical section only. The cylindrical portion in an industrial separator helps to promote swirl using a tangential injection inlet section.

Following tradition, the flow may be assumed to enter tangentially to the cone through an inlet area *A _{i}* and an average speed

*U*. As shown graphically, a spherical coordinate system $(R\xaf,\varphi ,\theta )$ may be anchored at the cusp of the cone; the latter diverges from a point at the bottom to a maximum radius of $a,$ thus prescribing a half-divergence angle of

*α*. The swirling motion is introduced at the top section of the cone where, as shown in Fig. 2, an axial velocity drives the flow downwardly. In order to reproduce the three-dimensional physical nature of the separator flowfield, both the tangential and axial flow entry boundary conditions need to be prescribed. The motion in the conical chamber begins as an outer downward spiral called “downdraft.” Once the height of the cone is traversed, the motion reverses itself and the flow is returned in an upward spiral coined “updraft” [see Fig. 1(b)]. A volume flow rate, $Q\xafi$, determines the mass inflow into the conical chamber. Similarly, an exit radius,

*b*, allows the returning fluid to exit the chamber seamlessly. Figure 1(a) displays the distinct vortex regions within an industrial cyclone separator with both underflow and overflow sections, whereas Fig. 1(b) depicts the conical geometry and terminology used to define the basic flow parameters. In this work, the underflow is not considered. Instead, the cone is assumed to extend to the origin of the coordinate system. The spherical coordinates can also be transformed into a cylindrical coordinate system $(r\xaf,\theta ,z\xaf)$, which is concurrently used throughout this work.

### B. Spherical equations and underlying assumptions

In conformance with ideal flow modeling assumptions [68], the flow is considered to be (i) steady, (ii) inviscid, (iii) incompressible, (iv) rotational, and (v) non-reactive. In accordance with the inviscid theory of swirling flows,^{90,91} the absence of friction enables us to justify the use of axisymmetry about the vertical axis. Then, using overbars to denote dimensional quantities, the fundamental equations for mass and momentum conservation may be reduced to

and

with vorticity being given by

### C. On the Beltrami–Gromeka–Lamb equation and Beltrami flow definition

To make further headway, Euler's equation may be transformed into the Beltrami–Gromeka–Lamb form; the latter brings vorticity to the forefront as a principal flow variable, namely, by linking it explicitly to the velocity $u\xaf$ and stream function $\psi \xaf$. Using standard nomenclature, the vector form of Euler's momentum equation may be written as

Next, a vector identity may be employed to morph Euler's equation into the Gromeka–Lamb equation, named as such after Ippolit Stepanovich Gromeka and Horace Lamb for their significant contributions to the fluid dynamics discipline.^{92–95} The Gromeka–Lamb equation has also been attributed to Eugenio Beltrami in view of his independent work on helical motions.^{96–98} The resulting Beltrami–Gromeka–Lamb equivalent^{97} can be readily deduced using Lamb's vector identity, $(u\xaf\xb7\u2207\xaf)u\xaf=12\u2207\xaf(u\xaf\xb7u\xaf)\u2212u\xaf\xd7\omega \xaf,$ which transforms Eq. (6) into

In the above, $H\xaf(\psi \xaf)$ denotes the fluid head or Bernoulli function. Actually, Eq. (7) represents the steady, inviscid, and incompressible form of the Beltrami–Gromeka–Lamb equation (also known as Crocco's entropy-free equation^{99} or the Crocco–Vazsony equation^{100}). It may be further reduced using the Lamb or swirl vector,^{101–103}^{,}$\u2113\xaf=\u2212u\xaf\xd7\omega \xaf$, to simply $\u2207H\xaf=\u2212\u2113\xaf$. On the one hand, Beltramian motions (also dubbed “Beltrami” or “screw” fields^{104–106}) must satisfy $\u2113\xaf=0$ or $u\xaf\xd7\omega \xaf=0.$ On the other hand, generalized Beltramian fields,^{105} which encompass a wider collection of solutions, must observe the less restrictive requirement prescribed by $\u2207\xaf\xd7u\xaf\xd7\omega \xaf=0.$

### D. Boundary conditions

Following similar models of conically shaped cyclonic chambers,^{79,80} the stream function may be suppressed along the centerline as well as the outer wall. This requirement translates into

Additionally, inlet conditions on the flow profile may be specified as shown in Fig. 2 and described in Sec. II A, namely, where *a* denotes the largest radius at the top of the cone and *b* <* a* represents the outflow radius of the updraft. In previous work, uniform axial injection at the inlet is implemented by setting^{80}

where *U* and *W* represent the average tangential and axial velocities at entry as illustrated in Fig. 2(a). Note that inflow and outflow conditions near the top section of the wall correspond to either a downward or an upward pointing *W*. In this work, the axial velocity profile at entry will be defined by the semi-parabolic curve that varies in the radial direction from $r\xaf=b$ to *a* as shown in Fig. 2(b).

### E. Inlet stream function formulation

The semi-parabolic curve from Fig. 2 may be used to define a generic axial profile of the form,

where *C*_{0} and *C*_{1} are pure constants. For further clarity, the use of axisymmetric cylindrical coordinates $(r\xaf,z\xaf)$ will be occasionally relied upon. Accordingly, the two fundamental boundary conditions on the axial velocity may be imposed at the inner and outer boundaries of the conical domain, specifically, $u\xafz(b,L)=\u2212W$ and $u\xafz(a,L)=0$ for an inflow near the top section of the conical wall. These produce

The particular stream function at the inlet becomes

Then, using $\psi \xafi(a)=0$, one is able to deduce, for the inlet condition,

where the negative sign corresponds to an upward pointing, outflow velocity profile. The incoming stream function that satisfies Eq. (10) is thus at hand. One gets

### F. Bragg–Hawthorne equation equivalence

In the context of steady axisymmetric motion, Eq. (7) may be further reduced to the less frequently employed Bragg–Hawthorne equation (BHE). Also referred to as the “Long–Squire” equation, the BHE emerges in the modeling of axisymmetric flows as an elliptical, second-order, nonlinear partial differential equation of the stream function.^{107,108} Although the BHE is generally named for either Bragg and Hawthorne or, alternatively, Long and Squire,^{109–111} some sources^{112–115} attribute its origins to Meissel.^{116} In practice, it may appear under various domain assumptions including those associated with stratified flow.^{117} Using spherical coordinates, the BHE may be written as

where $H\xaf(\psi \xaf)$ stands for the Bernoulli function and $B\xaf(\psi \xaf)$ denotes the angular momentum or circulation function.^{118,119} In seeking exact solutions to this problem, the right-hand side (RHS) of Eq. (15) may be expanded into a polynomial series in terms of $\psi \xaf$. Using

one gets

From an asymptotic perspective, the stream function may be chosen in such a manner to never exceed a value of unity. At the outset, higher-order terms in Eq. (17) may be neglected. By keeping the leading-order RHS terms, the BHE reduces to

At this juncture, it may be instructive to note that, in the absence of the first RHS term, Eq. (18) reproduces the BHE expression already considered by Bloor and Ingham^{79} and, subsequently, Barber and Majdalani.^{80} Specifically, one recovers

This particular form assumes a $\psi \xaf\u2212$invariant Bernoulli function and a variable swirl function that satisfies

along with

In the present study, we seek a different solution that retains the first RHS term in Eq. (18), namely,

Here, *a*_{0} can be determined from the variable inlet flow requirement using a constant angular momentum and a variable head that are consistent with the underlying assumption of an ideal fluid. As such, we have

in addition to

As shown in Appendix A, one may use Euler's equation to fully determine the coefficient $a0=2W/(a2\u2212b2)$ that appears in Eq. (22). As confirmed through Eq. (A10), we thus arrive at

## III. EXACT ANALYTICAL SOLUTION OF THE GENERALIZED BELTRAMI TYPE

In this section, Eq. (25) will be solved after reducing it to a set of ordinary differential equations (ODEs). First, however, we find it effective to normalize all quantities of interest.

### A. Normalized variables and nondimensional parameters

Using the maximum chamber radius and average tangential velocity as reference values, all variables may be normalized by dropping their overbars and rewriting them as

Note that, through proper normalization, the nondimensional conical swirl number *σ _{c}*, which serves as one of the main control parameters in conically bounded cyclonic flowfields,

^{80}emerges naturally, as detailed in Appendix B. In this vein, and pursuant to Figs. 1(b) and 2(b), it is possible to define the cylindrical radii $r\xaf\varphi $ and $r\xaf\alpha $ at the respective polar angles of $\varphi $ and

*α*, where $r\xaf\varphi $ denotes the horizontal polar radius taken at any fixed elevation $z\xaf$, and $r\xaf\alpha $ represents the full horizontal distance extending from the

*z*axis to the inclined wall. Thus, using the subscript to denote the polar angle at which the point is taken, the horizontal fraction of the radius $r\varphi (z)$ may be determined in any polar plane using

By the way of confirmation, the outflow fraction at $z=L/a=l$ may be determined to be $r\beta =r\xaf\beta /r\xaf\alpha =b/a$. Then, based on the foregoing definitions, compact forms of most variables may be obtained, starting with Eq. (14). In nondimensional form, the inlet stream function becomes

where the outflow corresponds to a positive, upward pointing axial velocity *W* in Fig. 2(b). Clearly, in view of the conical swirl number relation to the inflow parameters [e.g., Eq. (B3)], one can put

Next, we find it useful to define the conical inflow parameter, *κ _{c}*, by setting

This enables us to reduce the nondimensional inlet stream function into

Moreover, Eq. (25) becomes

### B. Homogeneous solution using separation of variables

The first step to reducing Eq. (33) into a more manageable form consists of assuming a product solution in the spherical radius and polar angle, that is, $\psi (R,\varphi )=F(R)G(\varphi )$. It may be further inferred that $F(R)=R4$ so that the stream function may be reconstructed from $\psi (R,\varphi )=R4G(\varphi )$. Substituting this ansatz into the BHE leads to

Note that the spherical radius may be wholly eliminated to the extent of converting Eq. (33) into an ODE in the polar angle provided that the assumptions made so far are maintained. Specifically, when higher-order terms in Eq. (17) are discarded and the stream function is restricted to a value of unity, one gets

Further rearrangements yield

where the prime denotes differentiation with respect to $\varphi $. At this juncture, an exact solution to Eq. (36) may be achieved by decomposing the equation and then recombining both parts using $G(\varphi )=Gh(\varphi )+Gp(\varphi )$, where the subscripts refer to the problem's homogeneous and particular expressions. As shown in Appendix C for an inflow, a suitable homogeneous solution that remains finite along the cone axis may be retrieved after introducing a variable transformation, $x=cos\u2009\varphi $, and converting Eq. (36) into a Gegenbauer equation. After some algebra, one arrives at

The corresponding stream function becomes

As for the particular part, it is described next.

### C. Particular and total solutions for the stream function

With Eq. (38) in hand, it is reasonable to express the particular solution as $\psi p(R,\varphi )=R4Gp(\varphi )$, where $Gp(\varphi )=Cp\u2009sin2\u2009\varphi $. The latter may be substituted into Eq. (36) to obtain

This expression enables us to extract $Cp=\u22134\kappa c/5$ and put

At this stage, we may recombine $G(\varphi )=Gh(\varphi )+Gp(\varphi )$ and write

By virtue of $\psi (R,\varphi )=R4G(\varphi )$, the stream function may be similarly expressed as

It may be readily verified that the condition at the centerline, $\psi (R,0)=0$ or $G(0)=0$, is identically satisfied by Eq. (42). As detailed in Appendix C, the attendant behavior may be viewed as the result of preventing unboundedness and thus eliminating the Legendre function of the second kind in the homogeneous part of the solution. The remaining constant, *C _{h}*, may be determined from the wall boundary condition, $\psi (R,\alpha )=0$ or $G(\alpha )=0$. This requirement enables us to retrieve

where several identical forms of the same characteristic parameter may be collected, namely,

Note that an isolated singularity in Eq. (44) emerges at a critical value of $\alpha crit=sin\u22121(2/5)\u22481.107\u2009\u2009rad$ or $63.43\xb0$, specifically, where $4\u22125\u2009sin2\u2009\alpha =0$. The emergence of this pole, however, is justifiably immaterial because the taper angles of the vast majority of cyclonic chambers do not reach $\alpha crit$. Beyond this point, however, the value of *λ* becomes positive, which makes it applicable to a problem with an outflow condition near the wall. In the meantime, realizing that the majority of cone divergence half-angles correspond to $\alpha \u226a\alpha crit$, one may conveniently use an asymptotic series approximation that pinpoints the first set of dominant terms in *λ*. One finds

Bearing these factors in mind, including the sign switching of $\lambda (\alpha )$ past the critical divergence half-angle, one can rearrange the stream function and write, for the inflow problem, the piecewise solution

where the superscript stands for “inflow.” Conversely, for the flow configuration that exhibits an outflow near the wall, one may simply use

where the superscript alludes to “outflow.” Both inflow and outflow solutions may be shown to satisfy Eq. (36) identically while vanishing both at the sidewall and the cone axis. For the reader's convenience, they are illustrated in Fig. 3 at two specific divergence half-angles of $55o$ and $65o$ that fall slightly below and above the critical angle. Clearly, the piecewise representations given by Eqs. (46) and (47) capture the expected streamline patterns quite well.

In view of the motivating application behind this study, the focus will be placed hereafter on the piecewise expression of $\psi (i)$ that corresponds to an influx near the sidewall over the practical $\alpha \u2264\alpha crit$ range. In this process, it is useful to consolidate some of the trigonometric functions using simple interchangeable identities such as $\u2009sin2\u2009\varphi =12[1\u2212cos\u2009(2\varphi )]$. One finds, for example,

Along similar lines, the stream function may be expressed as

The stream function's dependence on *λ* may be further eliminated by recasting Eq. (49) into one of several equivalent forms that include, for $\alpha \u2264\alpha crit$,

Due to the relative simplicity of the numerator in the last member of Eq. (50), the latter will be relied upon in the forthcoming analysis that will be carried out while dropping the superscripted symbol in $\psi (i)$. Naturally, the isolated singularity affecting *λ* may be seen to convey to the stream function rather identically; here too, one recovers the same $\alpha crit=12\u2009cos\u22121(\u221235)=cos\u22121(15)\u22481.107\u2009rad$.

## IV. RESULTS AND DISCUSSION

### A. Velocity field

Given the parental role of the stream function, the spherical radial and polar velocities may be readily generated from the velocity–stream function relations. One gets

and

Note that *κ _{c}* in the foregoing analysis may be interpreted as an off-swirl Ekman number.

^{78}As for the tangential velocity component, it may be determined in accordance with the constant angular momentum condition imposed through Eq. (23) via

Despite its simplicity, Eq. (53) represents a free vortex motion that secures the tangential inflow requirement at the wall.

In the interest of clarity, polar-cylindrical forms of the foregoing expressions may be readily constructed using the coordinate transformations provided in Appendix D. In short, using $\zeta \u2261cot\u2009\varphi =z/r$ and $\xi \u22611+\zeta 2\u2212\zeta ,$ a compact assortment of stream function and velocity components may be expressed solely as a function of the polar–cylindrical coordinates (*r*, *z*). We arrive at

with the spherical companions

or, with the polar-cylindrical companions,

For the reader's convenience, a comparison between the properties associated with the present model and those of the Beltramian profile obtained by Barber and Majdalani^{80} is provided in Table I. A side-by-side comparison of their velocity components in both spherical and radial coordinates is also furnished in Table II. Note that the subscript in $\xi \varphi \u2261csc\u2009\varphi \u2212cot\u2009\varphi $ stands as a reminder that *ξ* may be equivalently expressed in terms of the polar angle in lieu of *ζ*, although both expressions lead to identical values.

Term . | Generalized Beltramian $(\u2207\xd7u\xd7\omega =0)$ . | Beltramian $(u\xd7\omega =0)$ . |
---|---|---|

$\psi i(r,\theta ,z)$ | $\u2212\kappa c(1\u22122r2+r4)$ | $\u221212\kappa c(1\u2212r2)$ |

$dHd\psi $ | $\u22128\kappa c$ | 0 |

$BdBd\psi $ | 0 | κ _{c} |

$F(R)\u2009G(\varphi )$ | $R4G(\varphi )$ | $R2G(\varphi )$ |

$\psi (R,\varphi ,\theta )$ | $45\kappa cR4\u2009sin4\varphi [\lambda (15\u221212csc2\varphi )\u2212csc2\varphi ]$ | $12\kappa cR2\u2009sin2\u2009\varphi (\lambda \u2212ln\u2009\xi \varphi \u2212\xi \varphi csc\u2009\varphi )$ |

$\psi (r,\theta ,z)$ | $45\kappa cr4[\lambda (3\u221212\zeta 2)\u22121\u2212\zeta 2]$ | $12\kappa cr2(\lambda \u2212ln\u2009\xi \u2212\xi 1+\zeta 2)$ |

$\lambda (\alpha )$ | $(15\u2009sin2\u2009\alpha \u221212)\u22121$ | $\xi \alpha csc\u2009\alpha +ln\u2009\xi \alpha $ |

κ _{c} | $[2\pi \sigma c(1\u2212r\beta 2)]\u22121$ | $\pi \sigma c$ |

Term . | Generalized Beltramian $(\u2207\xd7u\xd7\omega =0)$ . | Beltramian $(u\xd7\omega =0)$ . |
---|---|---|

$\psi i(r,\theta ,z)$ | $\u2212\kappa c(1\u22122r2+r4)$ | $\u221212\kappa c(1\u2212r2)$ |

$dHd\psi $ | $\u22128\kappa c$ | 0 |

$BdBd\psi $ | 0 | κ _{c} |

$F(R)\u2009G(\varphi )$ | $R4G(\varphi )$ | $R2G(\varphi )$ |

$\psi (R,\varphi ,\theta )$ | $45\kappa cR4\u2009sin4\varphi [\lambda (15\u221212csc2\varphi )\u2212csc2\varphi ]$ | $12\kappa cR2\u2009sin2\u2009\varphi (\lambda \u2212ln\u2009\xi \varphi \u2212\xi \varphi csc\u2009\varphi )$ |

$\psi (r,\theta ,z)$ | $45\kappa cr4[\lambda (3\u221212\zeta 2)\u22121\u2212\zeta 2]$ | $12\kappa cr2(\lambda \u2212ln\u2009\xi \u2212\xi 1+\zeta 2)$ |

$\lambda (\alpha )$ | $(15\u2009sin2\u2009\alpha \u221212)\u22121$ | $\xi \alpha csc\u2009\alpha +ln\u2009\xi \alpha $ |

κ _{c} | $[2\pi \sigma c(1\u2212r\beta 2)]\u22121$ | $\pi \sigma c$ |

Term . | Generalized Beltramian . | Beltramian . |
---|---|---|

$uR(R,\varphi ,\theta )$ | $85\kappa cR2\u2009cos\u2009\varphi [\lambda (30\u2009sin2\u2009\varphi \u221212)\u22121]$ | $\kappa c[(\lambda \u2212ln\u2009\xi \varphi )cos\u2009\varphi \u22121]$ |

$uR(r,\theta ,z)$ | $85\kappa crz1+\zeta 2[\lambda (301+\zeta 2\u221212)\u22121]$ | $\kappa c(\zeta \lambda \u2212ln\u2009\xi 1+\zeta 2\u22121)$ |

$u\varphi (R,\varphi ,\theta )$ | $\u2212165\kappa cR2\u2009sin\u2009\varphi [\lambda (15\u2009sin2\u2009\varphi \u221212)\u22121]$ | $\kappa c[(ln\u2009\xi \varphi \u2212\lambda )sin\u2009\varphi +\xi \varphi ]$ |

$u\varphi (r,\theta ,z)$ | $\u2212165\kappa cr21+\zeta 2[\lambda (151+\zeta 2\u221212)\u22121]$ | $\u2212\kappa c(\lambda \u2212ln\u2009\xi 1+\zeta 2\u2212\xi )$ |

$u\theta (R,\varphi ,\theta )$ | $1R\u2009sin\u2009\varphi $ | $1R\u2009sin\u2009\varphi 1+(\kappa cR\u2009sin\u2009\varphi )2(\lambda \u2212ln\u2009\xi \varphi \u2212\xi \varphi csc\u2009\varphi )$ |

$u\theta (r,\theta ,z)$ | $1r$ | $1r1+(\kappa cr)2(\lambda \u2212ln\u2009\xi \u2212\xi 1+\zeta 2)$ |

$ur(R,\varphi ,\theta )$ | $45R2\kappa c[(12\lambda +1)sin\u20092\varphi ]$ | $\u2212\kappa c\xi \varphi $ |

$ur(r,\theta ,z)$ | $45\kappa crz(1+12\lambda )$ | $\u2212\kappa c\xi $ |

$uz(R,\varphi ,\theta )$ | $45\kappa cR2[(1\u221218\lambda )cos\u20092\varphi \u22123\u22126\lambda ]$ | $\kappa c(\lambda \u2212ln\u2009\xi \varphi \u22121)$ |

$uz(r,\theta ,z)$ | $45\kappa c[(6\lambda \u22122)r2\u2212(1+12\lambda )z2]$ | $\kappa c(\lambda \u2212ln\u2009\xi \u22121)$ |

Term . | Generalized Beltramian . | Beltramian . |
---|---|---|

$uR(R,\varphi ,\theta )$ | $85\kappa cR2\u2009cos\u2009\varphi [\lambda (30\u2009sin2\u2009\varphi \u221212)\u22121]$ | $\kappa c[(\lambda \u2212ln\u2009\xi \varphi )cos\u2009\varphi \u22121]$ |

$uR(r,\theta ,z)$ | $85\kappa crz1+\zeta 2[\lambda (301+\zeta 2\u221212)\u22121]$ | $\kappa c(\zeta \lambda \u2212ln\u2009\xi 1+\zeta 2\u22121)$ |

$u\varphi (R,\varphi ,\theta )$ | $\u2212165\kappa cR2\u2009sin\u2009\varphi [\lambda (15\u2009sin2\u2009\varphi \u221212)\u22121]$ | $\kappa c[(ln\u2009\xi \varphi \u2212\lambda )sin\u2009\varphi +\xi \varphi ]$ |

$u\varphi (r,\theta ,z)$ | $\u2212165\kappa cr21+\zeta 2[\lambda (151+\zeta 2\u221212)\u22121]$ | $\u2212\kappa c(\lambda \u2212ln\u2009\xi 1+\zeta 2\u2212\xi )$ |

$u\theta (R,\varphi ,\theta )$ | $1R\u2009sin\u2009\varphi $ | $1R\u2009sin\u2009\varphi 1+(\kappa cR\u2009sin\u2009\varphi )2(\lambda \u2212ln\u2009\xi \varphi \u2212\xi \varphi csc\u2009\varphi )$ |

$u\theta (r,\theta ,z)$ | $1r$ | $1r1+(\kappa cr)2(\lambda \u2212ln\u2009\xi \u2212\xi 1+\zeta 2)$ |

$ur(R,\varphi ,\theta )$ | $45R2\kappa c[(12\lambda +1)sin\u20092\varphi ]$ | $\u2212\kappa c\xi \varphi $ |

$ur(r,\theta ,z)$ | $45\kappa crz(1+12\lambda )$ | $\u2212\kappa c\xi $ |

$uz(R,\varphi ,\theta )$ | $45\kappa cR2[(1\u221218\lambda )cos\u20092\varphi \u22123\u22126\lambda ]$ | $\kappa c(\lambda \u2212ln\u2009\xi \varphi \u22121)$ |

$uz(r,\theta ,z)$ | $45\kappa c[(6\lambda \u22122)r2\u2212(1+12\lambda )z2]$ | $\kappa c(\lambda \u2212ln\u2009\xi \u22121)$ |

### B. Generalized Beltramian vorticity and pressure fields

Other characteristic flow properties, such as the vorticity and pressure, can be derived for the generalized Beltramian profile and compared side-by-side to their strictly Beltramian counterparts that are obtained for a uniform, rather than a semi-parabolic injection pattern at entry.^{80} For the sake of brevity, these findings are summarized in Table III for the vorticity and Table IV for the pressure field. These are further discussed in Sec. IV F below.

Term . | Generalized Beltramian . | Beltramian . |
---|---|---|

$\omega R(R,\varphi ,\theta )$ | 0 | $\kappa cuR/B$ |

$\omega R(r,\theta ,z)$ | 0 | $\kappa cuR/B$ |

$\omega \varphi (R,\varphi ,\theta )$ | 0 | $\kappa cu\varphi /B$ |

$\omega \varphi (r,\theta ,z)$ | 0 | $\kappa cu\varphi /B$ |

$\omega \theta (R,\varphi ,\theta )$ | $8\kappa cR\u2009sin\u2009\varphi $ | $\kappa cu\theta /B$ |

$\omega \theta (r,\theta ,z)$ | $8\kappa cr$ | $\kappa cu\theta /B$ |

$\omega r(R,\varphi ,\theta )$ | 0 | $\kappa cur/B$ |

$\omega r(r,\theta ,z)$ | 0 | $\kappa cur/B$ |

$\omega z(R,\varphi ,\theta )$ | 0 | $\kappa cuz/B$ |

$\omega z(r,\theta ,z)$ | 0 | $\kappa cuz/B$ |

Term . | Generalized Beltramian . | Beltramian . |
---|---|---|

$\omega R(R,\varphi ,\theta )$ | 0 | $\kappa cuR/B$ |

$\omega R(r,\theta ,z)$ | 0 | $\kappa cuR/B$ |

$\omega \varphi (R,\varphi ,\theta )$ | 0 | $\kappa cu\varphi /B$ |

$\omega \varphi (r,\theta ,z)$ | 0 | $\kappa cu\varphi /B$ |

$\omega \theta (R,\varphi ,\theta )$ | $8\kappa cR\u2009sin\u2009\varphi $ | $\kappa cu\theta /B$ |

$\omega \theta (r,\theta ,z)$ | $8\kappa cr$ | $\kappa cu\theta /B$ |

$\omega r(R,\varphi ,\theta )$ | 0 | $\kappa cur/B$ |

$\omega r(r,\theta ,z)$ | 0 | $\kappa cur/B$ |

$\omega z(R,\varphi ,\theta )$ | 0 | $\kappa cuz/B$ |

$\omega z(r,\theta ,z)$ | 0 | $\kappa cuz/B$ |

Term . | Generalized Beltramian . | Beltramian . |
---|---|---|

$\u2202p\u2202r$ | $r\u22123\u221212825\kappa c2r3(36\lambda 2\u22129\lambda \u22121)$ | $r\u221231+\zeta 2+\kappa c2r[\zeta 2\u2212\zeta 31+\zeta 2+\zeta (\lambda \u2212ln\u2009\xi \u22121)1+\zeta 2]$ |

$\u2202p\u2202z$ | $\u221212825\kappa c2z3(1+12\lambda )$ | $\kappa c2r1+\zeta 2(\zeta 2\u2212\zeta 1+\zeta 2\u2212\lambda +ln\u2009\xi +1)$ |

$p(r,z)$ | $\u221212r2\u22123225\kappa c2\tau $ | $\u221212r2+12\kappa c2\tau $ |

$\tau (r,z)$ | $(36\lambda 2\u22129\lambda \u22121)r4+(144\lambda 2+24\lambda +1)z4$ | $\zeta +\zeta 31+\zeta 2\u2212\zeta 2\u2212ln2\xi \u2212(2\lambda \u22121)\u2009ln\u2009(\xi +2\zeta )$ |

Term . | Generalized Beltramian . | Beltramian . |
---|---|---|

$\u2202p\u2202r$ | $r\u22123\u221212825\kappa c2r3(36\lambda 2\u22129\lambda \u22121)$ | $r\u221231+\zeta 2+\kappa c2r[\zeta 2\u2212\zeta 31+\zeta 2+\zeta (\lambda \u2212ln\u2009\xi \u22121)1+\zeta 2]$ |

$\u2202p\u2202z$ | $\u221212825\kappa c2z3(1+12\lambda )$ | $\kappa c2r1+\zeta 2(\zeta 2\u2212\zeta 1+\zeta 2\u2212\lambda +ln\u2009\xi +1)$ |

$p(r,z)$ | $\u221212r2\u22123225\kappa c2\tau $ | $\u221212r2+12\kappa c2\tau $ |

$\tau (r,z)$ | $(36\lambda 2\u22129\lambda \u22121)r4+(144\lambda 2+24\lambda +1)z4$ | $\zeta +\zeta 31+\zeta 2\u2212\zeta 2\u2212ln2\xi \u2212(2\lambda \u22121)\u2009ln\u2009(\xi +2\zeta )$ |

### C. Duality of experimental and theoretical mantles

The flowfield associated with the bidirectional vortex within a conical cyclone requires the existence of an area along which the velocities in the axial (*u _{z}*) or spherical radial (

*u*) directions will locally vanish in order for the motion to switch polarity. Originally, Bradley and Pulling

_{R}^{120,121}observed flow reversal features in their experiments on hydrocyclones, while Binnie and Teare

^{122}reported flow reversals in their experiments on swirling water across a nozzle. Using ink-dye injection with water as the working fluid, both groups were able to visualize in their experiments the specific regions where the axial velocities vanished.

^{121,122}Similar flow reversals of swirling motions have indeed been reported in configurations such as sudden expansions,

^{123–125}convergent–divergent nozzles,

^{126–128}and swirling pipes.

^{129–134}The proper characterization of vortex flow reversal may thus be viewed as a crucial aspect of cyclonic flow analysis, including work related to bidirectional vortex chambers

^{135–137}and cyclone separators.

^{138–146}

In this context, the locus of zero axial velocity, which separates the downdraft from the updraft, is often referred to as the mantle (see Fig. 1); it is discussed by both Binnie and Teare^{122} and Bradley and Pulling.^{120} The mantle is also dubbed the locus of zero vertical velocity (LZVV) or the “conical classification surface” in the cyclonic flow community.^{142–146} Since the area of flow reversal determines particle size separation criteria, the region demarcating the mantle constitutes a fundamental parameter in the design of cyclone separators.^{143,144} In this vein, particles residing outside the LZVV are deemed to possess a high probability of extraction through the bottom opening (coined “spigot” or underflow), whereas particles that are confined to the inner side of the LZVV are more susceptible to removal through the so-called “vortex finder” (or overflow).

In practice, dissimilar definitions of flow reversal and circulatory regions continue to appear in the literature, especially that the flow separation interfaces and mantles do not always extend over the entire swirl chamber. The mantle and reverse flow regions can actually differ depending on the flow configuration. For this reason, it may be helpful to briefly discuss the possible flow reversal regions associated with swirling pipes, nozzles, and cyclonic chambers, in order to clarify some of their prevalent differences. According to Nuttall,^{129} Binnie,^{147} Nissan and Bresan,^{130} Gore and Ranz,^{127} Escudier *et al.*,^{133} and Mattner *et al.*,^{134} several possible configurations exist for unidirectional swirling pipe flows. At low swirl, the evolution of the so-called regime I is characterized by uniform and unidirectional spiraling motion. As the swirl number is increased, the so-called regime II develops, thus giving rise to a flow reversal region, where part of the axial stream reverses direction, most frequently near the centerline. Finally, at certain swirl and flow parameters, the so-called regime III can appear along with axial profiles that exhibit alternating flow reversal regions. In this case, the velocities near the centerline and the wall will simultaneously revert back to, say, positive values, while the annular stream separating the wall from the core flow will convect in the opposite or negative direction. For unidirectional swirling tube and nozzle flows, the annular region in regime III does not usually extend over the entire length of the device. Instead, the reversal zone may be rather characterized by an eddy-type motion that occupies only a small portion of the tube or nozzle.^{122} The eddy-type motion can also be observed in conjunction with a fully bidirectional vortex configuration.^{120} By a fully bidirectional vortex, one alludes to the motion that evolves in such devices as cyclone separators,^{143,144} vortex-fired engines,^{71} and reverse vortex chambers.^{136,137} An ideal bidirectional vortex will therefore experience flow reversal through the entire length of its container. This behavior is not to be confused with the mechanisms of vortex instability and breakdown, which have been extensively explored in the context of unidirectional swirling motions.^{148–162} Vortex breakdown can also give rise to recirculatory zones of the types that have indeed been captured by Binnie and Teare^{122} and other dedicated researchers.^{127–133} The various recirculation zones that Binnie and Teare^{122} have examined along with many other researchers could also have been instigated by flow separation stemming from fluid motion over forward and backward steps. It can thus be seen that a wide array of geometric and flow conditions can affect the type of flow reversal that often accompanies swirl-induced flowfields. This could partly explain the need for persistent exploration of swirl phenomena where universal standards are yet to be established and where several enigmatic mechanisms remain unresolved.

A comparison between the results provided by Bradley and Pulling^{120} and Kelsall^{138} may help to explain the diverse perspectives of the mantle description in cylindrical and conical configurations and how the mantle is actually quantified experimentally [see Figs. 4(a) and 4(b)]. A diagram by Bhattacharyya^{140} of the flowfield in a hydrocyclone suggests that the LZVV remains cylindrically shaped in the close vicinity of the core throughout the entirety of the device [see Fig. 4(c)]. However, while the illustrations by Bhattacharyya^{140} differ from those by Bradley and Pulling,^{120} the graphical representation by Bradley and Pulling^{120} conveys similar flow physics, where the inner vortex appears as a thin cylindrical column that remains delineated from the outer vortex by the mantle interface. Along similar lines, Rietema,^{139} Dietz,^{163} Pericleous and Rhodes,^{141} and Pericleous^{164} depict the LZVV as a cylindrical layer near the core. In contrast, numerous other investigators such as Kelsall,^{138} Mikhaylov and Romenskiy,^{165} Pervov,^{166} Mothes and Löffler,^{167} Luo *et al.,*^{168} Zhou and Soo,^{169} Griffiths and Boysan,^{170} Peng *et al.,*^{171} and Derksen^{172} describe a conical surface of revolution of zero axial velocity that is inclined at a given divergence angle.

Naturally, the proper identification of the mantle location remains vital to our understanding of cyclonic flow dynamics. We recall that the present analytical framework is developed first in spherical coordinates with the aim of providing a simple stream function expression. Coordinates and corresponding velocities are then transformed into their cylindrical counterparts for better visualization and physical interpretation. The ensuing analysis gives rise to two types of mantles. The first and theoretically preferred mantle definition corresponds to the locus of the zero spherical radial velocity, $uR=0$, where the radially incoming flow switches polarity and begins to head outwardly. The second mantle relates to the experimentally preferred LZVV, which provides a simpler avenue for measuring the locus of zero axial velocity, $uz=0$, irrespective of the chamber geometry, using both flow visualization and advanced optical diagnostics. In this work, the present model takes advantage of the spherical coordinate system by depicting the mantle at a constant polar angle (*β _{R}* and

*β*) for a given divergence angle

_{z}*α*. This may be accomplished by taking

These conditions enable us to specify

or, asymptotically, using a four-term approximation

In practice, using radial fractions in horizontal cuts offers alternative means to quantify mantle locations, namely, by reconstructing each mantle vertically according to its horizontal radial fraction $r\beta R$ or $r\beta z$ (for $uR=0$ or $uz=0$). In view of Eq. (28), we have

The radial fraction that locates the LZVV in any horizontal plane is therefore fixed. Arriving at a constant radial fraction $r\beta z$, which seems surprising at first, may be ascribed to a rather unusual trigonometric identity

Tables V–VII display the characteristics of the generalized Beltramian solution along with those attributed to the strictly Beltramian profile discussed previously.^{80} To be practical, the operating ranges of *α* are limited here to the interval $[0,\pi /4]$ given that cyclone separators rarely exhibit divergence half-angles exceeding a value of $10\xb0$. Tables V and VI specifically depict the auxiliary constant *λ* and the characteristic mantle parameters *β _{R}*, $\beta R/\alpha $,

*β*, and $\beta z/\alpha $. Along similar lines, Table VII provides the fraction of the mantle radius, be it $r\beta R$ or $r\beta z$, in a horizontal plane taken at the top of the cone. This table also highlights the differences between the spherical and axial mantles in degrees and percentages for both the generalized and strictly Beltramian solutions.

_{z}. | Generalized Beltramian . | Beltramian . | ||||
---|---|---|---|---|---|---|

α
. | λ
. | β
. _{R} | β
. _{z} | λ
. | β
. _{R} | β
. _{z} |

$5\xb0$ | −0.084 | $3.5\xb0$ | $3.5\xb0$ | −2.63 | $3.0\xb0$ | $3.0\xb0$ |

$10\xb0$ | −0.087 | $7.1\xb0$ | $7.1\xb0$ | −1.92 | $6.1\xb0$ | $6.1\xb0$ |

$15\xb0$ | −0.091 | $10.5\xb0$ | $10.7\xb0$ | −1.52 | $9.1\xb0$ | $9.2\xb0$ |

$20\xb0$ | −0.098 | $13.1\xb0$ | $14.4\xb0$ | −1.22 | $12.1\xb0$ | $12.6\xb0$ |

$25\xb0$ | −0.107 | $17.4\xb0$ | $18.2\xb0$ | −0.982 | $15.2\xb0$ | $15.7\xb0$ |

$30\xb0$ | −0.121 | $20.7\xb0$ | $22.2\xb0$ | −0.781 | $18.2\xb0$ | $19.1\xb0$ |

$35\xb0$ | −0.142 | $23.9\xb0$ | $26.3\xb0$ | −0.605 | $21.2\xb0$ | $22.7\xb0$ |

$40\xb0$ | −0.172 | $27.0\xb0$ | $30.7\xb0$ | −0.445 | $24.2\xb0$ | $26.5\xb0$ |

$45\xb0$ | −0.222 | $30.0\xb0$ | $35.3\xb0$ | −0.296 | $27.2\xb0$ | $30.6\xb0$ |

. | Generalized Beltramian . | Beltramian . | ||||
---|---|---|---|---|---|---|

α
. | λ
. | β
. _{R} | β
. _{z} | λ
. | β
. _{R} | β
. _{z} |

$5\xb0$ | −0.084 | $3.5\xb0$ | $3.5\xb0$ | −2.63 | $3.0\xb0$ | $3.0\xb0$ |

$10\xb0$ | −0.087 | $7.1\xb0$ | $7.1\xb0$ | −1.92 | $6.1\xb0$ | $6.1\xb0$ |

$15\xb0$ | −0.091 | $10.5\xb0$ | $10.7\xb0$ | −1.52 | $9.1\xb0$ | $9.2\xb0$ |

$20\xb0$ | −0.098 | $13.1\xb0$ | $14.4\xb0$ | −1.22 | $12.1\xb0$ | $12.6\xb0$ |

$25\xb0$ | −0.107 | $17.4\xb0$ | $18.2\xb0$ | −0.982 | $15.2\xb0$ | $15.7\xb0$ |

$30\xb0$ | −0.121 | $20.7\xb0$ | $22.2\xb0$ | −0.781 | $18.2\xb0$ | $19.1\xb0$ |

$35\xb0$ | −0.142 | $23.9\xb0$ | $26.3\xb0$ | −0.605 | $21.2\xb0$ | $22.7\xb0$ |

$40\xb0$ | −0.172 | $27.0\xb0$ | $30.7\xb0$ | −0.445 | $24.2\xb0$ | $26.5\xb0$ |

$45\xb0$ | −0.222 | $30.0\xb0$ | $35.3\xb0$ | −0.296 | $27.2\xb0$ | $30.6\xb0$ |

. | Generalized Beltramian . | Beltramian . | ||||
---|---|---|---|---|---|---|

α
. | λ
. | $\beta R/\alpha $ (%) . | $\beta z/\alpha $ (%) . | λ
. | $\beta R/\alpha $ (%) . | $\beta z/\alpha $ (%) . |

$5\xb0$ | −0.084 | 70.7 | 70.7 | −2.63 | 60.7 | 60.6 |

$10\xb0$ | −0.087 | 70.5 | 71.1 | −1.92 | 60.6 | 61.0 |

$15\xb0$ | −0.091 | 70.3 | 71.5 | −1.52 | 60.6 | 61.4 |

$20\xb0$ | −0.098 | 70.0 | 72.2 | −1.22 | 60.6 | 62.0 |

$25\xb0$ | −0.107 | 69.6 | 73.0 | −0.982 | 60.6 | 62.8 |

$30\xb0$ | −0.121 | 69.0 | 74.0 | −0.781 | 60.5 | 63.8 |

$35\xb0$ | −0.142 | 68.4 | 75.3 | −0.605 | 60.5 | 64.9 |

$40\xb0$ | −0.172 | 67.6 | 76.7 | −0.445 | 60.5 | 66.4 |

$45\xb0$ | −0.222 | 66.7 | 78.4 | −0.296 | 60.4 | 68.0 |

. | Generalized Beltramian . | Beltramian . | ||||
---|---|---|---|---|---|---|

α
. | λ
. | $\beta R/\alpha $ (%) . | $\beta z/\alpha $ (%) . | λ
. | $\beta R/\alpha $ (%) . | $\beta z/\alpha $ (%) . |

$5\xb0$ | −0.084 | 70.7 | 70.7 | −2.63 | 60.7 | 60.6 |

$10\xb0$ | −0.087 | 70.5 | 71.1 | −1.92 | 60.6 | 61.0 |

$15\xb0$ | −0.091 | 70.3 | 71.5 | −1.52 | 60.6 | 61.4 |

$20\xb0$ | −0.098 | 70.0 | 72.2 | −1.22 | 60.6 | 62.0 |

$25\xb0$ | −0.107 | 69.6 | 73.0 | −0.982 | 60.6 | 62.8 |

$30\xb0$ | −0.121 | 69.0 | 74.0 | −0.781 | 60.5 | 63.8 |

$35\xb0$ | −0.142 | 68.4 | 75.3 | −0.605 | 60.5 | 64.9 |

$40\xb0$ | −0.172 | 67.6 | 76.7 | −0.445 | 60.5 | 66.4 |

$45\xb0$ | −0.222 | 66.7 | 78.4 | −0.296 | 60.4 | 68.0 |

. | Generalized Beltramian . | Beltramian . | ||||||
---|---|---|---|---|---|---|---|---|

α
. | $r\beta R$ . | $r\beta z$ . | $|\beta R\u2212\beta z|$ $(%)$ . | $|\beta R\u2212\beta z|\xb0$ . | $r\beta R$ . | $r\beta z$ . | $|\beta R\u2212\beta z|$ $(%)$ . | $|\beta R\u2212\beta z|\xb0$ . |

$5\xb0$ | 0.706 | 0.707 | 0.0 | $0\xb0$ | 0.606 | 0.606 | 0.1 | $0.0\xb0$ |

$10\xb0$ | 0.702 | 0.707 | 0.6 | $0\xb0$ | 0.602 | 0.606 | 0.4 | $0.0\xb0$ |

$15\xb0$ | 0.694 | 0.707 | 1.2 | $0.2\xb0$ | 0.597 | 0.605 | 0.8 | $0.1\xb0$ |

$20\xb0$ | 0.685 | 0.707 | 2.2 | $1.3\xb0$ | 0.590 | 0.604 | 1.4 | $0.5\xb0$ |

$25\xb0$ | 0.672 | 0.707 | 3.4 | $0.8\xb0$ | 0.580 | 0.603 | 2.2 | $0.5\xb0$ |

$30\xb0$ | 0.655 | 0.707 | 5.0 | $1.5\xb0$ | 0.568 | 0.601 | 3.3 | $0.9\xb0$ |

$35\xb0$ | 0.634 | 0.707 | 6.9 | $2.4\xb0$ | 0.553 | 0.598 | 4.4 | $1.5\xb0$ |

$40\xb0$ | 0.608 | 0.707 | 9.1 | $3.7\xb0$ | 0.535 | 0.595 | 5.9 | $2.3\xb0$ |

$45\xb0$ | 0.577 | 0.707 | 11.7 | $5.3\xb0$ | 0.514 | 0.592 | 7.6 | $3.4\xb0$ |

. | Generalized Beltramian . | Beltramian . | ||||||
---|---|---|---|---|---|---|---|---|

α
. | $r\beta R$ . | $r\beta z$ . | $|\beta R\u2212\beta z|$ $(%)$ . | $|\beta R\u2212\beta z|\xb0$ . | $r\beta R$ . | $r\beta z$ . | $|\beta R\u2212\beta z|$ $(%)$ . | $|\beta R\u2212\beta z|\xb0$ . |

$5\xb0$ | 0.706 | 0.707 | 0.0 | $0\xb0$ | 0.606 | 0.606 | 0.1 | $0.0\xb0$ |

$10\xb0$ | 0.702 | 0.707 | 0.6 | $0\xb0$ | 0.602 | 0.606 | 0.4 | $0.0\xb0$ |

$15\xb0$ | 0.694 | 0.707 | 1.2 | $0.2\xb0$ | 0.597 | 0.605 | 0.8 | $0.1\xb0$ |

$20\xb0$ | 0.685 | 0.707 | 2.2 | $1.3\xb0$ | 0.590 | 0.604 | 1.4 | $0.5\xb0$ |

$25\xb0$ | 0.672 | 0.707 | 3.4 | $0.8\xb0$ | 0.580 | 0.603 | 2.2 | $0.5\xb0$ |

$30\xb0$ | 0.655 | 0.707 | 5.0 | $1.5\xb0$ | 0.568 | 0.601 | 3.3 | $0.9\xb0$ |

$35\xb0$ | 0.634 | 0.707 | 6.9 | $2.4\xb0$ | 0.553 | 0.598 | 4.4 | $1.5\xb0$ |

$40\xb0$ | 0.608 | 0.707 | 9.1 | $3.7\xb0$ | 0.535 | 0.595 | 5.9 | $2.3\xb0$ |

$45\xb0$ | 0.577 | 0.707 | 11.7 | $5.3\xb0$ | 0.514 | 0.592 | 7.6 | $3.4\xb0$ |

Overall, it may be seen that the mantles based on $uz=0$ always match or exceed in magnitude those corresponding to *u _{R}* = 0, irrespective of the Beltramian model used. Specifically, as

*α*is increased from 0 to $\pi /2$,

*β*may be seen to vary from 0 to $\pi /4$, while

_{R}*β*is stretched all the way to $\pi /2$. This renders

_{z}*β*more physical from a theoretical standpoint, although

_{R}*β*remains more popular among experimentalists. According to Eq. (59),

_{z}*β*can become twice as large as

_{z}*β*for the limiting case of $\alpha =\pi /2$. In practice, both mantles start evenly at $\beta z=\beta R=\alpha /2$ and then diverge with successive increases in

_{R}*α*. As for their $\alpha \u2212$ normalized values, both $\beta z/\alpha $ and $\beta R/\alpha $ start at $1/2$, which coincides with the mantle radius associated with a quasi-complex-lamellar helical motion in a right-cylindrical cyclonic chamber.

^{71}Subsequently, $\beta R/\alpha $ slowly decreases, unlike $\beta z/\alpha $, which increases more rapidly as

*α*is incremented. For example, within the 0–$\pi /6$ range in Table VI, it may be seen that $\beta R/\alpha $ lingers at approximately 0.7 for the generalized Beltramian profile, and at 0.6 for the strictly Beltramian case; in contrast, $\beta z/\alpha $ may be seen to extend over the ranges of $[0.71,0.74]$ and $[0.61,0.64]$ for the two models in question. As for the relative differences between the use of

*u*= 0 and $uR=0$ in prescribing the mantle, discrepancies between the two approaches are more appreciable when using the generalized Beltramian profile. Additionally, in what concerns the radial fractions entailed in each approach, interesting dissimilarities may be noted in Table VII. Therein, it may be seen that both fractions start evenly at $r\beta R=r\beta z=1/2$; however, $r\beta R$ begins to measurably decrease monotonically unlike $r\beta z$, which remains fixed at $1/2$. For the strictly Beltramian profile, both fractions decrease although $r\beta z$ remains quite close to its initial value of 0.6 despite a relatively sizable variation in

_{z}*α*.

To further explore the mantle sensitivity to the cone angle, characteristic variations in *λ* and *β* are evaluated and showcased in Fig. 5 using several distinct line weights that depict different models and mantle types. In Fig. 5(a), the geometric constants from the two Beltramian models are contrasted, thus confirming their negative values in the prescribed range of *α*. As for Fig. 5(b), the sensitivity of the mantle inclination to the cone's divergence half-angle is illustrated by providing the mantle orientation in degrees as a function of *α*. It may be easily inferred that at one extreme, *β _{z}* associated with the generalized Beltramian model, which is designated by the chained curve, remains the highest. At the other extreme,

*β*of the strictly Beltramian model, which is represented by a dashed line, stands out as the lowest of these curves. As for the

_{R}*β*of the present formulation, it starts above its counterpart

_{R}*β*of the former model, then crosses over at a breakeven point; the latter may be calculated to be $28.51\xb0$ at $\alpha =0.741$ or $42.46\xb0$. The generalized Beltramian

_{z}*β*also predicts the same value of its Beltramian counterpart, namely, $41.15\xb0$ at $\alpha =68.53\xb0$.

_{R}### D. Vector fields and streamlines

The streamline patterns computed at six illustrative cone half-angles are showcased in Fig. 6 using vertical *r*-*z* slices along which the relevant ratio of *u _{r}* and

*u*, which controls the flow pattern in a vertical plane, becomes independent of

_{z}*κ*or

_{c}*σ*. These are provided for a unit

_{c}*κ*and $\alpha =15\xb0,\u200920\xb0,\u200930\xb0,45\xb0,\u200960\xb0,$ and 75°. They are followed in Fig. 7 by polar

_{c}*r*-

*θ*cuts taken midway through the chamber using $z=12l=12cot\u2009\alpha ,$ a fixed angle of $\alpha =30o,$ and four successive values of $\kappa c=1,\u20095,\u200910,$ and 50, which span nearly two orders of magnitude. It should be noted that in plotting the streamlines in Fig. 6, the sign of the stream function and, therefore

*u*and

_{r}*u*, will have to be switched past the singularity point, as $\alpha >63.435\xb0$, in order to maintain the same inward flow directionality. Such is the case for $\alpha =75\xb0$. Otherwise, the flow becomes outward and upward near the wall in lieu of the core. As for Fig. 7, several observations can be made. First, when the inflow parameter

_{z}*κ*increases, so does the significance of

_{c}*u*and

_{r}*u*relative to $u\theta $. As a result, the motion becomes gradually more dominated by the inward radial flow except near the cone axis; as $r\u21920$, the inviscid tangential speed will always overtake both of its axial and radial counterparts, irrespective of

_{z}*κ*, unless viscous corrections are accounted for. Because of the steeply increasing $u\theta $, a bathtub vortex may be seen to develop around the centerline for all values of

_{c}*κ*. Conversely, a radially dominated flow may be seen to evolve with successive increases in

_{c}*κ*as the conical wall is approached.

_{c}For a more quantitative model-to-model comparison, the streamlines corresponding to both Beltramian profiles are depicted in Fig. 8 at three different cone divergence half-angles of $\alpha =15\xb0,\u200930\xb0,$ and $45\xb0$. These are shown in an *r*-*z* plane using full lines for the present model and broken lines for the strictly Beltramian profile. In addition, the spherical velocity mantle angle, *β _{R}*, is shown distinctly in Fig. 8 for each Beltramian motion; therein, the larger angle corresponds to the generalized Beltramian profile, while the smaller angle refers to the strictly Beltramian model, as long as $\alpha \u226468.53\xb0$. Beyond this value, the roles fulfilled by these models are reversed. The corresponding behavior may be graphically discerned through the chained and dotted curves, respectively. It may be interesting to note that both Figs. 8(a) and 8(b) show a deeper penetration of streamlines near the top section of the cone using the Beltramian solution as a reference relative to the generalized Beltramian profile. Conversely, and except for the top curve, all of the lower curves in the cone display the opposite effect, namely, that of a deeper penetration of the generalized Beltramian streamlines. Figure 8(c) presents the same effects at a larger cone half-angle of $45\xb0$, where the region characterized by deeper penetration of the generalized Beltramian streamlines is widened. This behavior continues to set the trend with successive increases in

*α*.

### E. Velocity distributions

Figure 9 depicts the spherical radial, polar, axial, and cylindrical radial velocities, i.e., *u _{R}*, $u\varphi $,

*u*, and

_{z}*u*, for a divergence half-angle of $\alpha =45\xb0,\u2009\kappa c=12$, and four axial locations of $z/l=0.25$, 0.5, 0.75, and 1. As usual, this is accomplished using both Beltramian models under consideration. At first glance, it may be surmised that the velocities conform to the streamline behavior observed earlier in Fig. 8. Particularly, the velocity magnitudes corresponding to the generalized Beltramian profile seem to exceed those of their strictly Beltramian counterparts in the top portion of the conical cyclone. Naturally, the opposite effect occurs when

_{r}*z*is reduced. Another observation is that the curves delineating the spherical radial and axial velocities bear a striking resemblance irrespective of the formulation used. However, it should be noted that the present solution retains an

*R*

^{2}term in both

*u*and

_{R}*u*, while the strictly Beltramian model only depends on the polar angle $\varphi $ (see Table II). The presence of an

_{z}*R*

^{2}term enables both

*u*and

_{R}*u*to remain finite as the axis of rotation is approached, thus overcoming a recurrent core singularity, which has often been noted to affect swirling motions in the absence of viscosity. In the strictly Beltramian formulation, this singularity arises as $r\u21920$ or $\varphi \u21920$, in fulfillment of angular momentum conservation, or $B=ru\theta =$ constant for an ideal fluid. On this count, the generalized Beltramian solution seems to offer a favorable model for a conical cyclone. As for the polar and radial velocities, they may be graphically confirmed to preserve their parabolic and nearly linear forms in both formulations, respectively. The negative sign that prevails in both $u\varphi $ and

_{z}*u*is consistent with an inward motion that is directed toward the axis of rotation. Here too, $u\varphi $ may be seen to pass through an extremum in Fig. 9(b), where the absolute value of the polar velocity reaches its peak. Based on Eq. (52), one may retrieve the angle $\varphi min$ at which $|u\varphi |min$ is achieved. Starting with $u\u2032\varphi =0$, one extracts

_{r}The other important figure of merit consists of the crossflow velocity, which is responsible for the constant mass transfer from the outer vortex into the inner vortex across the interfacial layer at which *u _{R}* = 0. Based on Eq. (59), the value of $\varphi cross=\beta R=12\u2009cos\u22121(\u2009cos2\u2009\alpha )$, and one can deduce

It may be easily shown that the ratio $\varphi min/\varphi cross=cos\u22121[13(cos\u20092\alpha +2)]/\u2009cos\u22121(\u2009cos2\u2009\alpha )$ varies across the interval $[(23)1/2,2\pi \u2009cos\u22121(13)]$, that is, between 0.8165 and 0.7837, for $\alpha \u2208[0,\pi /2]$. Because $\varphi min<\beta R$, the polar velocity reaches its peak value in the inner vortex region at roughly 80% of the mantle inclination; this extremum is located closer to the cone axis than the mantle's crossflow velocity $(u\varphi )cross$, thus showing that the polar velocity continues to increase in magnitude after crossing the mantle interface.

As far as experimental and computational verifications are concerned, a discussion may be warranted. In the absence of universal standards for modeling swirling flows, it has proven rather challenging to identify a sufficient body of experimental and numerical results that could be judiciously compared to theoretically meaningful models, such as those derived from first principles; one contributing factor could be the fact that most experiments and simulations have actually predated the development of theoretically rigorous solutions. For example, some of the new formulations have unraveled closed-form exact and asymptotic solutions that have, in turn, revealed valuable similarity parameters that do not match those used formerly to guide the experiments and simulations themselves. For this reason, it is not unusual for existing measurements and numerical simulations to require rescaling in order to compensate for the absence of parameters that appear in the analytical solutions and that yet remain unreported in the experiments or simulations themselves.

Bearing these challenges in mind, Fig. 10 is used here to compare the present solution for $u\theta $ and *u _{z}* to the strictly Beltramian model as well as experimental and numerical data derived from Hsieh and Rajamani

^{173}and Monredon

*et al.*

^{174}These are carried out in conical hydrocyclones that possess air cores. Using $\alpha \u224810\xb0,$ our results are compared to the empirical values acquired at $z/l\u22480.7886$ for Hsieh and Rajamani

^{173}and 0.854 for Monredon

*et al.*

^{174}In this comparison, the tangential velocity associated with the present model shows general agreement only, being limited to a free-vortex formulation. As for the axial velocity, the present solution appears to be unique in its ability to terminate with a finite value at the cone axis, where $(uz)max=uz(R,0)=16\kappa cR2\u2009sin2\u2009\alpha /(5\u2009cos\u20092\alpha +3)$ depends on the cone half-angle

*α*, the distance

*R*, and the inflow parameter $\kappa c$; this behavior is different from that of other models and predictions where

*u*continues to increase as the flow approaches the cone axis (e.g., for the strictly Beltramian model) or the air core (for simulated hydrocyclones). In our case, $(uz)max/(\kappa cR2)$ varies sequentially according to ${0,0.1462,0.728,2.66,24}$ for divergence half-angles of ${0,15,30,45,60}$°. The rapid increase in the axial velocity along the cone axis is commensurate with the progressive increase in the diverging area and, in turn, the corresponding increase in surface area over which the parabolic profile is applied at entry. The speed is therefore increased to compensate for the added mass flow into the cone.

_{z}In view of the new capabilities that accompany the generalized Beltramian profile, it may be regarded as a complementary solution to the existing repertoire of engineering approximations for conical cyclones.

### F. Pressure and vorticity distributions

Finally, Fig. 11 illustrates the pressure drop and vorticity distributions associated with the two models under examination. This comparison is provided at four axial locations of $z/l=0.25$, 0.5, 0.75, and 1, $\kappa c=0.5$, and divergence half-angles of $\alpha =15\xb0$ and $45\xb0$ in the left and right columns of the composite graph, respectively. Figures 11(a) and 11(b) confirm that the pressure drop for the generalized Beltramian model follows closely that of its predecessor. In both cases, the dominating term, $12r\u22122$, controls the slope of the pressure curve. As for the vorticity, Figs. 11(c) and 11(d) are used to display its net magnitude using, as usual, full lines and broken lines to denote the generalized Beltramian formulation and its predecessor.

In what concerns vorticity characterization, the trends associated with the two models are somewhat reversed. On the one hand, the vorticity of the Beltramian model exhibits its highest point near the axis of rotation and diminishes outwardly as the wall is approached. On the other hand, the vorticity of the generalized Beltramian profile vanishes at the cone axis and then increases linearly as $r\u21921$. In all cases, the vorticity that accompanies the generalized Beltramian model seems to be much less appreciable than that of its predecessor. This behavior can be explained by revisiting Table III, where all vorticity components of the present model may be seen to be null except in the tangential direction, where $\omega \theta =8\kappa cR\u2009sin\u2009\varphi =8\kappa cr$ can only vary linearly in *r* independently of *z*.

## V. CONCLUSIONS

In this study, an exact inviscid solution is derived and explored as a viable alternative to describe the bidirectional swirling motion in a conical cyclone. In previous work, another cyclonic model was presented for a conical chamber, thus leading to a strictly Beltramian motion through which velocity and vorticity vectors remained strictly parallel over the entire fluid domain. Starting with the incompressible Bragg–Hawthorne equation and a judicious choice of physical assumptions and boundary conditions on the angular momentum and stagnation head, an ideal fluid profile is unraveled here in the form of an ultraspherical or Gegenbauer function. Following several transformations, the resulting function is reduced to a Legendre polynomial with unique physical characteristics. In contrast to the former Beltramian flowfield, which satisfies the $u\xd7\omega =0$ condition, the Gegenbauer-based model is accompanied by a curl-free Lamb vector, namely, $\u2207\xd7u\xd7\omega =0$. From this perspective, one may infer that the defining criterion of the preceding model forms a subset of the generalized Beltramian class of solutions. Presently, the velocity and vorticity vectors do not need to be parallel. It is sufficient for their cross-product to be either a constant or an irrotational Lamb vector.

In practice, the generalized Beltramian solution enables us to predict several quantities of merit in an exact manner, and these include the various velocity components, vorticities, and pressure distributions. It also reveals the presence of an off-swirl Ekman number, which we have coined the conical inflow parameter, *κ _{c}*. At the outset, precise closed-form expressions are obtained for the crossflow velocity, $(u\varphi )cross$, peak polar velocity, $(u\varphi )min$, and maximum axial velocity at the cone axis, $(uz)max$; these are accompanied by a full characterization of both theoretically and experimentally preferred mantle inclinations and radial fractions. These are found to be generally higher than their strictly Beltramian counterparts, as confirmed through asymptotic expansions and limiting process verifications. Furthermore, mantle inclinations associated with a vanishing

*u*, which are easier to measure experimentally, are found to slightly overpredict the divergence angle of the theoretical mantle based on a vanishing

_{z}*u*. This observation is found to be true for both helical models and suggests that the experimentally preferred basis for measuring the mantle in conical cyclones will invariably lead to a case of over-estimation.

_{R}In comparison with the strictly Beltramian model, the present formulation is seen to exhibit larger velocities near the wall, but rather smaller values near the center of rotation, where the generalized Beltramian vorticity vanishes identically. The vorticity vector here is found to possess only one non-vanishing component in the tangential direction, where $\omega \theta =8\kappa cr$ increases linearly with outward excursions in *r*. Moreover, despite its tangential speed being limited to a free vortex tail, the axial velocity of the generalized Beltramian profile bears a striking resemblance to its counterpart in cylindrical cyclones, where $(uz)max$ occurs at the center of rotation. As for its pressure distribution, it is equally dominated by the traditional $r\u22122$ term, which is responsible for the rapid depreciation of the pressure in the chamber core.

In closing, it may be useful to note that the inviscid model provided here increases our growing repository of stationary solutions for swirl-driven helical motions that satisfy both continuity and Euler's equation. It thus enhances our ability to describe the character of the bidirectional vortex in a wall-bounded conical chamber. Our models have all been procured from the Bragg–Hawthorne equation, which, when manipulated in conjunction with a realistic assortment of boundary conditions, can turn into a systematic framework for producing a multitude of helical flow profiles under various geometric configurations and physical settings. In fact, it is hoped that additional solutions obtained through this framework will be presented in forthcoming work, including extensions that capture the effects of viscosity and compressibility on the mean flow character.

## ACKNOWLEDGMENTS

This work was supported partly by Sierra Space Corporation (SSC), through Contract No. G460817, partly by the National Science Foundation, through Grant No. CMMI-1761675, and partly by the Hugh and Loeda Francis Chair of Excellence. The author further expresses his deepest appreciation of Martin J. Chiaverini, Donald Benner, Brian Pomeroy, and Arthur Sauer, for numerous technical exchanges and for providing the motivation, inspiration, and continual support of his cyclonic flow investigations.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author reports no conflicts of interest.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: MODEL CLOSURE USING SEMI-PARABOLIC INLET PROFILE

In order to determine *a*_{0}, one can start with the radial momentum equation for steady, axisymmetric, and inviscid flow. One has

At the inlet, we recall that $u\xafr(a,L)=0$ and simplify Eq. (A1) into

This enables us to deduce the pressure dependence on the angular momentum $B\xaf$, namely,

where *ρ* is treated as a pure constant. Next, the Bernoulli function may be evaluated at the inlet by putting

On the one hand, one may substitute the semi-parabolic form of the axial velocity at entry to obtain

which prompts us to divide through by $2C0$ and collect

On the other hand, the stream function at the inlet may be integrated based on the specified axial velocity to retrieve

By virtue of Eq. (24), one may substitute $H\xaf=a0\psi \xaf+\psi \xaf0$ to recover $a0=2C0$. This enables us to confirm that the semi-parabolic inlet profile is congruent with the assumed BHE formulation. Furthermore, one may substitute the resulting relation into Eq. (A5) and use $\psi \xaf(a,L)=0$ to extract

and so, at length, we arrive at

where the same constants, *C*_{0} and *C*_{1}, are found as before in Sec. II E. The BHE to be solved at present collapses into

In the above, the choice of a minus sign corresponds to an upward pointing *W* that may be associated with an outflow near the wall.

### APPENDIX B: CONICAL SWIRL NUMBER

Given the significance of the modified swirl number^{68} in the context of a right-cylindrical cyclonic chamber, a similar form may be specified for the conical flow analogue. The modified swirl number in a cylindrical chamber appears as

where the geometric swirl number *S _{g}* may be correlated with our basic parameters using

Here, $r\beta (z)$ represents the horizontal fraction of the radius in a polar plane as defined by Eq. (28). In the present configuration, the inlet volumetric flow rate may be calculated from

where we have used $\psi \xafa=0$ and evaluated $\psi \xafb$ from Eq. (A9), thus recovering $\psi \xafb=14W(a2\u2212b2)$. As for the modified conical swirl number, *σ _{c}*, it is specified as

^{80}

or, in terms of cone angles,

In practice, *σ _{c}* will be positive for injection because of the negative sign associated with a downward pointing

*W*in Fig. 2.

### APPENDIX C: HOMOGENEOUS SOLUTION OF THE REDUCED GEGENBAUER EQUATION

This section describes the procedure leading to an exact solution of the homogeneous form of Eq. (36), namely,

A useful transformation may be achieved by introducing the auxiliary variable $x=cos\u2009\varphi $. It follows that:

and so

where the subscript *x* implies functions and derivatives with respect to *x* in lieu of $\varphi $. Upon closer look, Eq. (C3) may be written in the form^{175,176}

The resulting ODE happens to be a special case of the ultraspherical differential equation, also known as Gegenbauer's differential equation,^{175}

This family of ODEs is named after a 19th century Austrian mathematician, Leopold Bernhard Gegenbauer, who managed to extract valuable polynomial solutions to Eq. (C5). In general, these may be expressed as

which, in view of our particular parameters, simplify into

where $Pnm(x)$ and $Qnm(x)$ stand for the associated Legendre functions at order *μ* of the first and second kind, respectively. When *m* = 0, the associated Legendre equation and polynomials reduce to the more common and regularly known Legendre equation and polynomials. Based on the relation between the associated and regular Legendre polynomials, one can put

or

where *C*_{5} and *C*_{6} are the negatives of *c*_{5} and *c*_{6}. Along similar lines, the Legendre derivatives may be related to the regular Legendre functions of the first and second kind using

Alternatively, the solution to Eq. (C4) may be formulated in terms of the ultraspherical or Gegenbauer functions of the first and second kind (so long as $n\u22652$). The consolidated solution becomes

Note that the superscript $\u22121/2$ is deliberately omitted in the second member of Eq. (C13), being inherently implied in the Gegenbauer function. Here, the superscript alludes to the degree of the function while *n* represents the order; the latter permits the function to be interchanged with a polynomial in the case of a whole integer. The families of polynomials named after Jacobi *et al.* constitute some of the classical orthogonal polynomials. In this work, a relation between Legendre and Gegenbauer functions may be anticipated because Legendre's equation appears when solving Laplace's PDE and Legendre's differential equation proves to be a special case of Eq. (C5) for $\mu =\u22121/2$. Since $\mu =\u22121/2$ in Eqs. (C6) and (C7), Legendre's equation may be readily restored. In fact, the Gegenbauer polynomials may be transformed straightforwardly into Legendre polynomials since $Jn\u22121/2(x)=Pn(x)$ whenever $n\u2208\mathbb{N}*$ and here *n* = 4 according to Eq. (C4). It follows that, using:

or

one may convert Eq. (C13) into

or, with no loss of generality,

where $C7=C\xaf7/12$ and $C8=C\xaf8/12$. Then, by the way of Eq. (C11), we arrive at

This outcome is reassuring as it proves to be identical to that of Eq. (C12) with the substitution of $C7=\u2212C5$ and $C8=\u2212C6$. In the wake of this dual confirmation of the analytical formulation, we may proceed to expand the Legendre functions using

These expressions may be inserted into Eq. (C18) and collected to obtain

At this juncture, *C*_{8} must be eliminated to secure a finite solution at the axis of the cone, particularly, as $x\u21921$ or $\varphi \u21920.$ Mathematically, this condition implies suppressing the singular Legendre functions of the second kind, $Q2(x)$ and $Q3(x)$, from becoming unbounded in Eq. (C18). Reverting back to $x=cos\u2009\varphi $, one regains the homogeneous part

As a windfall, the physical requirement $Gh(0)=0$ is readily secured by Eq. (C21).

### APPENDIX D: SPHERICAL-TO-CYLINDRICAL COORDINATE TRANSFORMATIONS

For added clarity, our velocities are converted from spherical to cylindrical coordinates according to the following transformations:

Next, the connecting relations between spherical and cylindrical coordinates may be revisited using

Apart from these definitions, we have the following identities:

## References

_{2}-H

_{2}propulsion applications

_{2}-GH

_{2}simulations of a miniature vortex combustion cold-wall chamber