Vortex sinks with axial ﬂow: Solution and applications

Vortex sinks with axial flow: Solution and applicationsVladimir Shtern, Anatoly Borissov, and Fazle HussainDepartment of Mechanical Engineering, University of Houston, Houston, Texas 77204-4792

~Received 2 May 1996; accepted 9 June 1997!

In this paper we develop a new class of analytical solutions of the Navier–Stokes equations andsuggest ways to predict and control complex swirling flows. We consider vortex sinks oncurvedaxisymmetric surfaces with an axial flow and obtain a five-parameter solution family that describesa large variety of flow patterns and models fluid motion in a cylindrical can, whirlpools, tornadoes,and cosmic swirling jets. The singularity of these solutions on the flow axis is removed by matchingthem with swirling jets. The resulting composite solutions describe flows, consisting of up to sevenseparation regions~recirculatory ‘‘bubbles’’ and vortex rings!, and model flows in the Ranque–Hilsch tube, in the meniscus of electrosprays, in vortex breakdown, and in an industrial vortexburner. The analytical solutions allow a clear understanding of how different control parametersaffect the flow and guide selection of optimal parameter values for desired flow features. Theapproach permits extension to swirling flows with heat transfer and chemical reaction, and have thepotential of being significantly useful for further detailed investigation by direct or large-eddynumerical simulations as well as laboratory experimentation. ©1997 American Institute ofPhysics.@S1070-6631~97!02110-7#

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I. INTRODUCTION

Despite the long history of research in swirling flowsome problems crucial for applications remain unexplorand many are unresolved. The primary difficulty is thswirling flows are typically complicated, consisting of a ferecirculatory domains. Moreover, these flows have paradcal features, including such striking phenomena as collap1

~i.e., strong concentration of the axial and angular momen!,bistability2 ~i.e., two stable states!, the Ranque effect3 ofthermal separation, and vortex breakdown.4 While collapseand bistability are observed in swirl-free flows as well, tRanque effect and vortex breakdown arespecificto swirlingflows. Careful analyses are needed to predict and conthese effects in natural and technological applications. Hwe present a powerful new approach for such an analbased on the generalization of the classical vortex-sink fl

The planar vortex-sink~or vortex-source! flow is one ofthe simplest solutions of the Euler and Navier–Stokes eqtions. Such solutions~especially point vortices! are widelyused as building blocks of more complex motions andmodels of practical flows~see Ref. 4 for examples!. Here, theplanar vortex sink is generalized to cover axisymmetric votex sinks with anaxial flow. This is a new solution for aviscous incompressible fluid, incorporating an axial flowith a radial shear. The incorporation of the axial flow snificantly enriches the family of solutions, increasing tnumber of dimensionless parameters for the velocity fieldto five, and hence enabling the modeling of a larger variof flows.

Prior analytical solutions related to ours dealt withswirl-free flows. A solution describing a flow near an infinite prous cylinder with uniform suction was found independenby Wuest,5 Lew,6 Yasuhara,7 and Stuart8 ~Stuart also studiedflow along a corner!. Berman9 studied a similar flow, but ina porous annulus. Wang’s review10 and our own literaturesearch revealed no other works on this subject. Unfo

Phys. Fluids 9 (10), October 1997 1070-6631/97/9(10)/2941/

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nately, the results5–9 address very specific flows of limitepractical interest. In contrast, our solution addressesswirlingflows of widespread technological importance.

The vortex-sink relation for radial and swirl velocitiesa common feature of many swirling flows, due to~i! conser-vation of angular momentum in regions where viscous difsion is negligible; and~ii ! entrainment of ambient fluid bynear-axis jet-like flows. The jet axis serves as a line sinkambient fluid, as shown by Schlichting11 for swirl-free jetsand by Long12 for swirling jets. The vortex sink is, in particular, an asymptotic solution for Long’s jet, as the distanfrom the axis becomes large; for other examples see Ref

The vortex-sink region is observed not only in open balso in confined swirling flows. Far from boundaries, a flois typically oblivious of much of the constraint posed bboundary conditions. For instance, flows in vortex tube14

and vortex generators15 are strongly asymmetric near tangetial inlet nozzles, but become nearly axisymmetric, evendownstream distances comparable with the nozzle diameAlso, the no-slip condition on sidewalls does not influenthe main flow of interest, outside the boundary layer. Fthese reasons, the fine details of boundary conditions careasonably omitted for studies of robust features of swirlflows, as undertaken here.

Note that there are a few characteristics~governed byconservation laws! whose values are crucial for large-scaflow patterns, including the entrainment rate~here the radialReynolds number Re! and the angular momentum~the swirlReynolds numberG!. The reduction of detailed boundarconditions to a few integral characteristics is consistent wthe universality of the vortex-sink region observed in a vaety of swirling flows. Using just these characteristics—Rand G, for the radial and swirl velocity—we find that onlthree more parameters are required to specify the axiallocity in an axisymmetric flow. These parameters characize contributions to the velocity profile from an outer frestream, due to an axial pressure gradient, and radial con

294119/$10.00 © 1997 American Institute of Physics

license or copyright, see http://pof.aip.org/pof/copyright.jsp

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gence of the flow~as shown in Sec. II!. Thus, the goal of thispaper is to develop analytical solutions governing seveimportant features common to a wide range of swirliflows. As a first step we obtain here a generalized vortsink solution.

The generalized vortex sink has a compact algebraicresentation and a clear physical explanation. These feathelp to elucidate the intriguing mechanisms in the swirliflows listed above. In particular, the generalized vortex sis applied here to model the strong momentum concentrain whirlpools, tornadoes, and cosmic jets. Also, the newlution explains the general mechanism of vortex filament fmation.

A further extension of the generalized vortex sink isuse as an outer solution and the avoidance of its singbehavior on the axis by matching with inner solutions, suas swirling jets. The resulting composite solutions enamodeling of internal ~i.e., away from walls! separation,which is a rather subtle but common phenomenon in swirlflows. The separation causes the appearance of recirculadomains, which are either semi-infinite or compact~bubble-and torus-shaped! regions. The recirculatory domains are esential features of vortex breakdown, of the Ranque effand of vortex combustion.

The composite solutions describe complex flow patteincluding up to seven recirculatory domains and model hvortex breakdown in sealed cylindrical cans, vortex burneand Ranque–Hilsch tubes. Note that our analytical modecovers the entire range of control parameters. This advanenables classification of all possible flow regimes, as welidentification of the optimal parameter values for applictions. Of course, for parameter values thus selected, mdetailed flow features can be studied experimentally anddirect numerical simulation.

The remainder of this paper is organized as follows Tgeneralized vortex-sink solution is derived in Sec. II. A clasification of the stream surface shapes is presented inIII. Relevant inner solutions are discussed in Sec. IV amatched with the vortex sink in Sec. V, where a varietyflow patterns covered are shown. Examples of applicatiof the generalized vortex sink and composite solutionsgiven in Sec. VI and Sec. VII. Based on our new solutiowe discuss a general mechanism of vortex filament formtion in Sec. VIII. The key results are summarized in Sec.where possible extensions are elaborated on as well.

II. GENERALIZED VORTEX SINK

We consider a steady swirling flow, where the velocityindependent of the axial coordinatez. In this case, theNavier–Stokes equations for the radial velocity,n r , andswirl, nf , are decoupled from the equation for the axial vlocity, nz . Here,n r , nf , andnz are the velocity componentin cylindrical coordinates (r ,f,z).16 For axisymmetricn r

and nf ~nz may depend onf!, the continuity equation reduces tod(rn r)/dr50 ~i.e., rn r5const! corresponding tosink ~or source! flow:

n r5Q~2pr !215nr 21 Re. ~1a!

2942 Phys. Fluids, Vol. 9, No. 10, October 1997


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Here Q52pn Re is the flow rate per unit axial lengtthrough a cylindrical surface,r 5const; n is the kinematicviscosity. The equation for swirl reduces to

n r r21d~rnf!/dr5n@r 21d/dr~r dnf /dr !2r 22nf#.

Multiplying by r 2/n and introducingj5 ln(r/r0), with r 0 as alength scale, yields

Re~nf8 1nf!5nf9 2nf ,

where the prime denotes differentiation with respect toj.Note that the use ofj results in constant coefficients. A general solution is

nf5C1r Re111C2r 21. ~1b!

The second term corresponds to the potential swirl, andfirst term represents the solid-body rotation at Re50. In thecontext of this paper, we consider the particular solutC150 @the role of the first term in~1b! is discussed in SecIV B #. In this case,nf5K(2pr )21, whereK52pC2 is thecirculation along a circle withr 5const and z5const.Throughout our analysis, we use the dimensionless pareter G[rnf /n5K(2pn)21, which is the swirl Reynoldsnumber. The (n r ,nf) flow is the well-known vortex sink~vortex source! for Re,0 ~Re.0!.

The primary objective of our study is to generalize tvortex sink by adding an axial flow. Note thatz-independent axial flow does not affect the continuity eqtion and the momentum equations forn r andnf . The gov-erning equation fornz reduces to the form

Re Wj1GWf5Wjj1Wff , ~1c!

whereW5nzr 0 /n and the subscripts denote differentiatioIt is worth noting that one could also consider a nonaxisymetric axial flow with an axisymmetric vortex sink given b~1a! and ~1b!.

Since the coefficients in~1c! are constant, the axial velocity has normal-mode solutions,W5exp(aj1imf); theazimuthal wave numberm50,61,62,..., anda5a1 or a2

are the roots of the dispersion relation,a22a Re2m22 imG50. A general solution of~1c! is a superposition of normamodes, whose coefficients are determined by the boundconditions. However, the applications of such nonaxisymetric solutions are unknown to us, so that we focus herethe more practical axisymmetric case (m50). The axisym-metric solution isW5Wc1Wr(r /r 0)Re, whereWc and Wr

are constants.A further generalization involves the inclusion of a

axial pressure gradient, i.e.,]p/]z5constÞ0. For this case,a termP exp(2j) must be added to the left-hand side of~1c!,where P5r 0

3(rn2)21 ]p/]z is a dimensionless parametecharacterizing the axial pressure gradient;r is the density.Then, the solution forW becomesW5Wc1Wr(r /r 0)Re

1Wp(r/r0)2, whereWp5P/(422 Re); the third term repre

sents the contribution of the axial pressure gradient.For flow in an unbounded domain 0<r ,`, Wc is the

velocity on the axis for Re.0 or at infinity for Re,0 andWp50. In a moving Galilean frame, one can enforceWc

50. However, to model some flows~e.g., within vortextubes!, it is relevant to choose the frame such that zero ax

Shtern, Borissov, and Hussain


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velocity occurs neither atr 50 nor r 5`, but in-between~e.g., on the sidewall!. Thus, it is useful to retainWc as a freeparameter characterizing the uniform part of the axial floThe other parameters,Wp andWr , characterize the nonuniform shear of the axial velocity induced by the axial pressgradient and the radial advection, respectively. Thus, thelocity field,

n r5Re n/r , ~2a!

nf5Gn/r , ~2b!

nz5@Wc1Wp~r /r 0!21Wr~r /r 0!Re#n/r 0 , ~2c!

is the new generalized vortex-sink solution, which satisfithe Navier–Stokes equations and includesfive dimensionlessparameters: Re,G, Wc , Wr , andWp . The vortex sink~2a!,~2b!, which is a known solution, is generalized here byaxial flow ~2c!. The axial flow includes a parabolic part@thefirst two terms in~2c!#, which is similar to pipe flow and isindependent of the vortex-sink component. In contrast,last term in~2c! depends on the radial flow rate Re andinterpreted below as the entrainment effect of a near-axis

Since the velocity in~2! depends on the radial coordinaonly ~through power-law functions!, the streamline equationdr /ds5v can also be explicitly integrated to yield

z/r 05z0 /r 01a~r /r 0!21b~r /r 0!Re121c~r /r 0!4, ~3a!

f5f01S ln~r /r 0!, ~3b!

where a5Wc(2 Re)21, b5Wr@Re(Re12)#21, c5Wp(4 Re)21, and the swirl parameterS5nf /n r5G/Re.The velocity field is identical on axisymmetric stream sufaces~3a!, differing only by a shiftz0 alongz. Streamlinescurve around on these surfaces, and their projection oz5const planes are logarithmic spirals, as is clear from~3b!.The planar vortex sink is a particular case correspondinga5b5c50 in ~3!.

To ‘‘visualize’’ flow patterns for comparison with experiments, it is convenient to use the Stokes streafunction C; i.e., n r52r 21 ]C/]z and nz5r 21 ]C/]r .According to ~2!–~3!, the dimensionless streamfunction,c5C(nr 0 Re)21, has the form

c5a~r /r 0!21b~r /r 0!Re121c~r /r 0!42z/r 0 . ~4!

Finally, the pressurep for this solution family has the distribution

p5p`2 12r~n/r !2~Re21G2!1r~n/r 0!2Pz/r 0 . ~5!

According to~5!, the pressure decreases asr→0. Since, asshown below,G2@Re2 typically in practical applications, thepressure drop near the axis is mainly an effect of swirl.physical terms, this pressure drop reflects the mutual coteraction of the centrifugal force and the radial pressgradient—the so-called ‘‘cyclostrophic balance.’’ The presure can also decrease or increase in the axial directionpending on the sign ofP.

Thus, ~2!–~5! represent a new five-parameter solutifamily. This number of parameters being quite large,solution can capture a large variety of nontrivial flow paterns. A relevant adjustment of the parameters enables o

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approximate~with a relative ease! some important elementof practical flows. As mentioned in the Introduction, prisolutions exist for flows over a porous cylinder5–8 and in anannulus;9 these are particular cases of our generalized stion ~2! with G50. One can apply~2! without this restrictionfor a spiral flow in an annulus with porous walls that rotaand move along the axis. However, this use of~2! is overlyrestrictive and of limited interest. In this paper, we considsignificantly more broad and interesting applications of b~2! itself, and also of composite solutions resulting fromatching~2! and inner solutions. For the sake of these aplications, we interpret~2! as a generalized vortex sink osurfaces of revolution.

III. SHAPE OF THE SURFACES OF REVOLUTION

The family of surfaces of revolution is governed by~3a!.Depending on the parameter values, the stream surface hvariety of shapes. Figure 1 shows typical meridional strealines, i.e. the meridional sections of the stream surfaces.cause of symmetry with respect to thez axis, Fig. 1 showsonly the right-hand sides. The shapes differ by numberextrema~which increases from left to right in Fig. 1!, flowdirection, and the asymptotic behavior asr→0. The latterfeature is different for a strong sink@Re,22, Figs. 1~a!–~c!#, a weak sink~22,Re,0, Figs. 1~d!–~f!#, and a [email protected], Figs. 1~g!–~i!#. Table I shows the representative prameter values for Fig. 1~a!–~i!. To summarize, the streamsurface shape depends on the radial flow rate~Re! and pa-rametersa, b, andc, which characterize contributions to thaxial flow from theuniform stream (a), radial advection (b)and axial pressure gradient (c). The radial advection causeaccumulation and dispersion of the axial momentumRe,0 and Re.0, respectively. The accumulation by sinflows ~Re,0! leads to a singularity of the axial flow atr50; the character of this singularity is qualitatively differefor strong~Re,22! and weak~22,Re,0! sinks.

A. Strong sinks

For sink flows with Re,22, z→6` asr→0 along thesurface of revolution, see~3a!. Figures 1~a!–~c! correspondto different specific values ofa, b, and c, but the sameRe524 ~this value is of a special interest, as shown in SIV A !. However, the features discussed below are commfor anya, b, andc for Re,22. The shape of the surface orevolution depends on signs ofa, b, andc. If a.0, c.0,andb,0 in ~3a!, the axial flow is unidirectional. Streamlinestart from z5` at r 5` and go toz52` as r→0 @Fig.1~a!#; that is, a fluid particle descends on the surface of relution along a spiral path. The velocitiesn r and nf mono-tonically increase~in modulus! downstream, in contrast tonz , which has a~negative! maximum near the inflectionpoint of the curve in Fig. 1~a!, and tends to2` asr→0 andr→`.

If b.0 andc.0, then the surface has a shape like thshown in Fig. 1~b!. There is a single extremum~minimum! atany value ofa. A uniform flow influences only the radiaposition of the extremum. Sincenz is negative on the periphery but now positive near the axis, there is a counterflowcontrast to the unidirectional flow in Fig. 1~a!. The flow pat-

2943Shtern, Borissov, and Hussain


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FIG. 1. Shapes of stream surfaces. The number of local extrema increases from left to right.~a–c! The curves go to infinity near the axis for a strong sinRe,22. ~d–f! The curves terminate on the axis at a finitez for a weak sink,22,Re,0. ~g–i! The curves have a few zero derivatives on the axis forsource, Re.0. See Table I for specific values of parameters.

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tern is even more complex forc.0, b,0, a,0, and suffi-ciently largeuau, which describes a ‘‘vortex sink with a barrier’’ @Fig. 1~c!#. As a increases at fixedb and c ~i.e.,uniform downflow increases!, the upflow disappears and thstream surface becomes that shown in Fig. 1~a!.

Thus, the stream surface shape depends on the siga, b, andc, i.e. on the direction of each axial flow contr



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bution: uniform flow (a), radial advection (b), and axialpressure gradient (c). The effect of (c) dominates the othersat a large distance from the axis; see~2c! and~3a! asr→`.Therefore, the axial flow direction for larger is governed bythe sign ofc. Near the axis, (b) dominates; see~2c! asr→0for Re,0. Therefore, the axial flow direction for smallr isgoverned by the sign ofb. The uniform flow can dominate



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for intermediater @see the vicinity ofr 51 in Fig. 1~c!#. Weinterpret~in Secs. IV–V! the radial advection as a result oentrainment by a near-axis jet. This jet serves as a linefor ambient fluid, with specific values of the radial flow ra~Re! that differ for weak and strong swirls~Sec. IV!. Thisentrainment corresponds to a singularity in the contributof (b) to the axial velocity atr 50.

While the stream surfaces have a power-law singulaat r 50 for Re,22, the singularity becomes logarithmicRe522, and ~3a! transforms intoz/r 05z0 /r 01a(r /r 0)2

1b ln(r/r0)1c(r/r0)4. As Re further increases, the strea

surface becomes even more smooth.

B. Weak sinks

For sink flows with22,Re,0, z has a bounded valuat r 50; see~3a!. This feature distinguishes Figs. 1~d!–1~f!from Figs. 1~a!–1~c!, although the figures are otherwissimilar. Figures 1~d!–1~f! correspond to different specifivalues ofa, b, andc, but the same Re521 ~this value is ofa special interest, as shown in Sec. IV C!. While z isbounded atr 50, nz→6` asr→0 along the surface in Figs1~d!–1~f!, according to~2c!. At Re50, the singularity ofnz

becomes logarithmic, and~2c! transforms intonz5@Wc

1W ln(r/r0)1Wp(r/r0)2#n/r0.

C. Source

For source flows~Re.0!, nz is bounded atr 50. Ac-cordingly, the stream surfaces are tangential toz5constplanes atr 50, as Figs. 1~g!–1~i! show. Fluid particles moveaway from the symmetry axis, in contrast to Figs. 1~a!–1~f!.The present classification captures all possible shapes ostream surfaces. Simultaneous changes in the signs ofa, b, cresult only in the reflection of streamlines with respectplanes normal to the axis, thus doubling the numberstream surface patterns.

Since the velocity is singular atr 50, solution~2! is notuseful near the axis for practical flows. However, the vorsink can serve as an outer solution, to be matched withinner solution valid near the axis~i.e., nearr 50!. Below wediscuss jet-like flows that are appropriate candidates forinner solutions.

IV. INNER SOLUTIONS

An inner solution is needed, in particular, to makenz

bounded atr 50. On the other hand, the inner solution mu

TABLE I. Parameters for Fig. 1.

Figure Re z0 a b c

~a! 24 0.91 0.11 20.06 0.04~b! 24 0 20.25 0.12 0.19~c! 24 2.06 21.24 20.12 0.31~d! 21 0 0.25 0.25 0.03~e! 21 1 20.25 21 0.25~f! 21 0 23.12 3.75 0.37~g! 1 0 0.25 0.06 0.03~h! 1 1 21 0.12 0.25~i! 1 0 6.50 28 2.50



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rapidly decay asr increases to match the outer solutioTherefore,nz has its maximal value near the axis, with jelike flow in the inner~‘‘core’’ ! region. The core radius istypically small in comparison with that of the outer regiowhere~2! is valid. For this reason, one can apply the bounary layer approximation for the near-axis jet. The inner slutions can be swirl-free, weakly or strongly swirling jeteach of which is considered below.

A. Swirl-free jet

For a round swirl-free jet, the meridional flow inside thcore can be represented by the Schlichting solution:11

nz58nz21B@11B~r /z!2#22, ~6a!

n r54nrz22B@12B~r /z!2#@11B~r /z!2#22, ~6b!

c52Br2~zr0!21@11B~r /z!2#21, ~6c!

where the dimensionless constantB53J(64prn2)21 char-acterizes the axial momentum fluxJ through a plane normato the jet axis.

At fixed z, ~6b! indicates that rn r /n→24 asB(r /z)2→`. This exactly coincides with~2a! at Re524. Atfixed z, ~6a! indicates that

nz→8nB21z3r 24, as B~r /z!2→`. ~A1!

Compare with~2c!, which gives

nz5Wrnr 03r 24, ~A2!

at Wc5Wp50 and Re524. Relations~A1! and ~A2! havethe same power-law dependence onr , but asymptote~A1!also involvesz ~this difference is absent for strongly swirlinjets, as shown in Sec. IV C!. Although~A1! and~A2! are notmatched uniformly with respect toz, their contributions rap-idly decay asr→` and becomes negligible compared withe two first terms in~2c!. Thus,~2! and~6! are matched withrespect ton r but not with respect tonz .

Note that it is typical that inner and outer solutions anot matched with respect to all velocity components inleading terms. For example, the wall boundary layer aouter solutions are not matched with respect to the norvelocity ~while the longitudinal velocity is matched!. An ex-ample more closely related to our study is the Landaudiscussed in Sec. V A.

Concerning the streamfunction,~6c! yields theasymptotic expansionc52z/r 01z3(Br0)21r 221••• asr→` for fixed z. Note that the first term of the expansiocoincides with the last term in~4!. The second terms of theexpansion and~4! have the same power law with respect torfor Re524. The coefficient in the expansion depends onz@similar to ~A2!#, but the difference with~4! is insignificantbecause both the second terms tend to zero and becomeligible in comparison with2z/r 0 as r→`. Moreover, theother terms in~4! tend to infinity with increasingr .

Thus, the Schlichting jet is an appropriate inner solutiowhich can be matched with~2!–~4! at Re524. The Schlich-ting solution smoothes the singularity in~2! on the axis,z.0, with nz bounded atr 50, n r proportional tor , and c;r 2 in the vicinity of r 50. Note that the Schlichting jet is a



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common part~near the axis! of many conical flows2,13 whenthe Reynolds number based on the axial velocity is larHere we have shown the same feature for nonconical flas well.

B. Weakly swirling jet

Concerning swirling jets as inner solutions, one canternatively consider weak or strong swirl. By weak swirl, wmean that the meridional motion dominates the swirl, andboundary layer equations for the meridional motion arecoupled from that for the swirl. For this reason, the merional motion is governed by the Schlichting jet, and tboundary layer solution for the swirl is1,17

nf5Gnr 21@11z2/~Br2!#21. ~7!

The asymptotic expansion of~7! as r→` at a fixedz is nf

5Gnr 212GnB21z2r 231••• . The first term of the expansion coincides with the outer solution~2b!. Note that thesecond term of the expansion has the same power law fras the first term of~1b! at Re524. Therefore, the first termof ~1b! is implicitly included in the inner solution~7!. Forthis reason, we have omitted this term in the outer solut~2b!.

The inner solution~6! involves a single new parameterBcharacterizing the jet intensity, while the circulationG in ~7!is the same as that in the outer solution~2!. Since we use theinner solution to replace the second term in~3!, B plays therole of b in the combined solution studied in Sec. V A.

C. Long’s jet

The equations for the meridional motion and swirl acoupled when the swirl is strong, for which an appropriainner solution is Long’s jet.12 Unfortunately, there is no general analytical solution for Long’s jet. However, thasymptotic behavior of Long’s solution is known:12

n r→2nr 21, ~8a!

nz→nr 21G/A2, ~8b!

and

nf→nr 21G, ~8c!

asr→` at a fixedz. The asymptote for the swirl~8c! agreeswith ~2b!. Note that the next term in the expansion12 yieldsthe difference,nf2nr 21G, which vanishes exponentially ar→`; thus, the agreement is excellent.

Relations~8! show that the potential vortex is an aequate outer solution for Long’s jet. The asymptote forradial velocity ~8a! agrees with~2a! at Re521, which ex-actly corresponds to the weak-sink case@Figs. 1~d!–1~f!#.Concerning the axial velocity,~8b! coincides with the secondterm of ~2c! at Re521 and Wr5GA2. We conclude thatLong’s jet is an appropriate inner solution and canmatched with~2!–~4! at Re521. Note that the matching fonz is uniform with respect toz here, in contrast to the weaswirl case. Thus~2! and Long’s jet are matched with respeto all velocity components.



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D. Annular jet

A sufficiently strong swirl induces inversion of the neaaxis flow.12 For weak swirl, a maximum of the longitudinavelocity ~at fixed z! occurs on the symmetry axis. As thswirl increases, the maximum shifts away from the axis. Taxis is now the position of a localminimumof the longitu-dinal velocity. This minimum decreases as the swirlcreases and then becomes negative (nz,0), i.e. flow rever-sal occurs. With a further increase in swirl, thenz,0 domainexpands and the reversed flow becomes irrotational.2 Thenear-axis region of the irrotational flow is bounded byannular swirling jet2 @see Fig. 3~a! for z.0#. Although theflow pattern developed is rather complex, there are analytsolutions for large swirl, i.e.G@1. One of the solutions17

covers only a thin near-axis region, while the other13 encom-passes an outer~inviscid vortical! flow as well. The lattersolution describes three flow regions:~i! cp52cs(12x)/(12xj ) and nf[0, for xj,x< l ~near-axis flow!;~ii ! c j52cs tanhj and nf5Gnr 21(12tanhj)/2 in thevicinity of x5xj ~annular jet!; ~iii ! cn5cs$x@22(11xj )x/xj #/@xj (12xj )#%1/2 and nf5Gnr 21 for 0<x,xj

~outer flow!. Here cs52Gr 021Rxj@(12xj )/(11xj )#1/2, R

5(r 21z2)1/2, x5z/R, and j5Gxj (x2xj )/@2(11xj )(12xj

2)1/2#.The annular jet direction is given byxj5cosuj ; u j is a

polar angle of the annular jet. The above solution cancombined into the following compact form:

c5cp~11tanhj!/21cn~12tanhj!/2, ~9a!

nf5Gnr 21~12tanhj!/2; ~9b!

this is a uniform approximation for the entire range, 0<x<1. A new control parameter introduced in~9! is xj ; theother parameters are the same as those for the outer sol~2!. The 12xj value must be small in order that~9! can serveas an inner~near-axis! solution for matching with~2!. Sup-pose that 12xj→0 asG→`. Then the asymptotic relation~8! are valid for solution~9! as well and, therefore,~9! is wellmatched with~2!.

To summarize, a few solutions describing near-axis jcan serve as an inner solution for the generalized vortexto obtain a composite solution that is regular on the axissymmetry.

V. COMPOSITE VORTEX SINK

A. Matching approach

Here we match the exact NSE solution~2! obtained inSec. II and the exact boundary layer solutions listed in SIV to constructcomposite solutionsthat are uniform approxi-mations in both the inner and outer flow regions. The coposite solutions can be either multiplicative or additive18

Consider a one-dimensional problem with a boundary laat x5xb as Re→`. Denote the outer solution asy0(x) andthe boundary layer~inner! solution asyi(h), where the innercoordinate h is a scaledx. The matching condition isyi(`)5y0(xb). The multiplicative composite solution isym

5y0(x)yi(h)/yi(`) and the additive one isya5y0(x)1yi(h)2yi(`).



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It is instructive first to demonstrate how the matchitechnique works, using a simple exact solution. Consideexample closely related to our study—the Landau jet, whthe Stokes streamfunctionC has the representation

C5n~r 21z2!1/2y~x!,

y~x!5ye52 Rea~12x2!/@41Rea~12x!#,

x5z/~r 21z2!1/2.

Here Rea5zna /n, na is the velocity atr 50, and the subscrip‘‘ e’’ indicates thatye is the exact solution~to distinguishfrom the following approximations!.

As Rea→`, ye tends to the outer solutiony0(x)52(11x) for 12x5O(1) and to the inner solutionyi(h)54h/(41h), h5Rea(12x) in the vicinity of x5xb51.The inner solution is the Schlichting jet~6! ~with B5Rea/8!,and the outer solution describes a flow induced by the uform sink along the axis,x51, with yi(`)5y0(1)54@52Re in ~2a!#.

It is interesting that the multiplicative composite solutioym coincides withye ~for this particular example!. The addi-tive composite solution,ya54h/(41h)22(12x), doesnot coincide withye , but the error,ye2ya58(12x)/@41Rea(12x)#,8/Rea , uniformly tends to zero in21<x<1as Rea→`. Note that the outer and inner solutions, beimatched with respect ton r , are not matched with respectnz :nz052n/(r 21z2)1/2 for the outer solution providingnz0

52n/uzu at x51, while nzi→0 ash→`. Thus, this exampleshows that matching with respect to all velocity componeis not generic but exceptional. The Long and annular jetssuch exceptions becausenz , n r , andnf in these cases havthe same asymptotic behavior~;r 21 asr→`!, as shown inSecs. IV C and D.

Now we apply this matching technique to obtain coposite uniform approximations, using~2! as the outer solu-tion and the near-axis jets~Sec. IV! as inner solutions. Thefollowing additive composite solutions result from summrizing the outer and inner solutions and extracting commparts,18 which are here singular terms in the generalized vtex sink.

B. Weakly swirling combined flow

Solutions of Sec. IV are unidirectional jets originatingz5r 50 and flowing in the positivez direction, thus cover-ing only the half-axis. However, the generalized vortex sis singular on the entirez axis, thus necessitating an innsolution encompassing the entire axis. Therefore, to mathe vortex sink, one has to construct a solution describinjet flowing from its origin in both the positive and negativezdirections; this is thebipolar jet. For this goal, we apply~6!for both positive and negativez, substitute~6! for the secondterm in ~4!, and thus obtain

c5a~r /r 0!22Br2~zr0!21@11B~r /z!2#211c~r /r 0!4.~10!

The solution~10! remains singular at the origin,z5r 50. Toavoid this singularity, one needs to match~10! and a stagna-tion flow nearz5r 50. This regularization needs a specstudy that is out of the scope of this paper. Here we ignthe singularity at the origin because it is of minor influen



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for the global flow pattern. The swirl distribution~7! is aneven function ofz, so we apply~7! for the entire flow region.

It is striking that the simple composite solution~10! de-scribes rather complex flow patterns as Fig. 2 illustratwhere streamlines of the meridional motion and the corsponding profile ofnz(r ) at z52`, resulting from~10!, areshown for the parameter values listed in Table II.

First, consider the case where there is no uniform axcomponent in the outer flow~i.e., a50!, but there is a sheaflow induced by the axial pressure gradient (cÞ0). In thiscase, the bipolar jet extends to infinity in bothz directions@Fig. 2~a!#, a flow pattern termed by us a ‘‘bipolar jet inshear flow.’’

Figure 2~b! shows the case for uniform axial flow (c50). Since the uniform flow is directed upward, it stagnathe downward jet at some negativez @see the saddle point inthe lower panel of Fig. 2~b!#. As a result, a recirculatory‘‘bubble’’ develops between the saddle point and the origin Fig. 2~b!. In contrast, the upward jet extends to infinibecause the uniform flow acts to increase the jet velocThe flow pattern in Fig. 2~b! is thus a ‘‘bubble in a uniformflow.’’

If there is also a shear in the outer flow, the bubble thchanges its shape. When the uniform and shear parts oouter flow have the same direction, the upstream part ofbubble becomes narrow in comparison with the downstrepart @Fig. 2~c!#. The bubble shape depends on values oaandb @both are negative in Fig. 2~c!#, but the flow topologyremains the same in Figs. 2~b! and 2~c!.

The flow topology becomes significantly more complwhen the uniform and shear parts of the outer flow haopposite directions. The flow pattern in this case chanqualitatively with the ratioc/a. In Figs. 2~d!–2~f!, a(,0) isfixed butc(.0) increases. For sufficiently smallc, the me-ridional flow near the axis is similar to that atc50 @compareFigs. 2~d! and 2~b!#. However, the axial flow has an opposidirection far from the axis in Fig. 2~d!. Between the upwardand downward flows, there are two semi-infinite domainsrecirculatory flow. The existence and arrangement of reculatory domains are important for effective heat and mtransfer in practical flows~e.g., in combustion chambers, seSec. VII D!. These domains are separated by a saddle ci~denoted by crosses! in Fig. 2~d!. The flow in the upper do-main moves down at the periphery, turns around nearsaddle circle, and moves up for smallerr . The flow is similar~but opposite! in the lower domain. There are two streasurfaces passing through the saddle circle and separatinunidirectional and recirculatory flows.

As c increases, the saddle circle approaches, toucand destroys the bubble surface. The passing of the cthrough the surfaces is a boundary crisis or catastrophechanges the flow topology. The stream surfaces reconand the bubble@Fig. 2~d!# transforms into the vortex ring@Fig. 2~e!#. The semi-infinite recirculatory domains now etend up to the axis and the upward unidirectional flow dappears@compare Figs. 2~d! and 2~e!#. There is a gap be-tween the recirculatory domains where the vortex ringlocated in Fig. 2~e!. To classify flow patterns, we call th



FIG. 2. Meridional streamlines for the generalized vortex sink matched with the bipolar Schlichting jet.~a! No uniform outer flow.~b! Uniform outer flow.~c! Codirected outer uniform and shear flows.~d–f! Counterdirected outer uniform and shear flows; the shear flow part increases from~d! to ~f!. See TableII for parameter values.

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flow shown in Fig. 2~e! a ‘‘vortex ring in a counterflow,’’much like an axisymmetric jet flow.

As c increases further, the saddle circle mentioned abapproaches the other circle~the centerline of the vortex ring!.At somec, these two circles merge and cease to exist,the vortex ring disappears. For largerc, the flow pattern



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becomes that shown in Fig. 2~f!. Now, the outer downflowpenetrates the gap between the upper and lower recirculadomains. There is a bipolar jet near the origin as in Fig. 2~a!,but the downward jet exhibits a vortex breakdown-tystructure with a semi-infinite recirculatory domain. The floin Fig. 2~f! is thus a ‘‘bipolar jet with vortex breakdown.’



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As a→0 for fixed c ~or asB increases!, the lower recircu-latory domain in Fig. 2~f! descends, so the flow pattern aproaches that shown in Fig. 2~a!.

In summary, Fig. 2 shows a variety of flow patterns dscribed by~10!, for which the swirl distribution is similarand rather simple. According to~7!, the circulation is con-stant along the conical surfaces,r /z5const. As the polarangleu increases from 0° to 90°, the circulation grows frozero on the axis,r 50, to its maximum value on the planez50.

C. Strongly swirling combined flow

Despite the fact that strong swirl influences the merional motion, the global flow structure in this case is neathe same as that for weak swirl, unless the bipolar jetcomes annular. The small difference is in thenz distributionnear the axis; in particular, the maximumnz position canshift away from the axis~see Secs. IV C and IV D! forLong’s jet, which we choose not to address here. Consiation of Long’s jet is technically laborious due to the absenof an analytical solution, and also would not provide quatatively new flow patterns in comparison with those shownFig. 2. For this reason, we consider here the composite stion combining ~2! and the analytical solution~9! for theannular jet, providing new qualitative features in the flostructure.

First, we construct the bipolar jet starting from solutio~9! that is valid forz.0 only. Denoting the streamfunctio~9a! as c1 , we definec252c1(2z), and introduce thecombined inner solution,

c i5c1 , for z.0 and c i5c2 , for z,0,

which is an odd function ofz. Substitutingc i for the secondterm in ~4! yields the composite solution for the streamfuntion:

c5a~r /r 0!21c i1c~r /r 0!4. ~11!

Since the swirl is symmetric with respect toz50, the com-posite solution for the swirl is

nf i5nf1, for z.0 and nf i5nf2, for z,0,

wherenf1 is defined by~9b! and nf2(z)5nf1(2z) for z,0.

Figure 3 shows typical flow patterns~for particular val-ues in Table III! for the strongly swirling flows with annulajets @the profilenz(r ) at z5` is shown in the top panels#.

TABLE II. Parameters for Fig. 2.

Figure a B c c ~contour levels!

~a! 0 100 20.042 25, 22, 0, 2, 5, 10, 20~b! 20.4 4 0 25, 22, 20.5, 0, 0.1, 0.25, 0.5, 0.6~c! 20.4 4 20.1 25, 22, 20.5, 20.1, 0, 0.05, 0.25, 0.5~d! 20.4 4 0.005 210, 25, 22.2, 22, 20.5, 0,

0.1, 0.25, 0.5, 0.6, 2, 5~e! 20.4 4 0.012 210, 25, 22, 20.2, 0,

0.1, 0.36, 0.5, 0.6, 2, 5, 10, 20~f! 20.4 4 0.04 210, 25, 22, 20.2, 20.05, 0,

0.1, 0.36, 0.5, 2, 5, 10



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The matching procedure is valid if the polar angleu j of theannular jet is small. We use sufficiently largeu j for Fig. 3 tohighlight the flow patterns; one can imagine that the neaxis region with the annular jet is ‘‘blown up.’’

In Fig. 3~a!, the annular bipolar jet itself is shown without any outer flow, i.e.a5c50 in ~11!. Fluid flows to theorigin along both the axis and equatorial plane. This flocollision induces a bipolar jet that flows outward alongconical surface,u5u j . Such a flow can be induced bboundary conditions corresponding to a given vortex sinkthe planez50.13 The fact that the jet is annular, not consodated, is owing to the strong swirl.

Figure 3~b! shows the resulting flow due to the aboveand a uniform downflow~c50, a.0!. The uniform streamsimply increases the downflow near the upper [email protected]~a!#, because these flows are codirected. In contrast,upward annular jet is directed oppositely to the unifostream. Since the maximal jet velocity is proportional to 1rwith a uniform stream velocity, the uniform stream domnates the jet for sufficiently larger . For this reason, thestream turns down and inward, forming a vortex-ring flonear the origin. The vortex-ring domain is separated fromambient flow by a surface whose section resembles amoclinic orbit in Fig. 3~b!. This orbit, starting at the origin, isfirst directed upward along the conical surface, then tudownward and comes back to the origin tangentially alothe z50 plane.

The uniform stream and the jet-induced [email protected]~a!# are oppositely directed near the lower half-axis in F3~b!. Sincenz of the upflow decays proportionally to 1/z asz→2`, the uniform stream dominates the upflow sufciently far below, where the fluid flows downward. Howevethe upflow dominates the uniform flow near the origin, athe fluid flows upward. For this reason, there is a sadpoint on the negativez axis @Fig. 3~b!# and a ‘‘bubble’’ ap-pears.

Figure 3~c! shows the opposite case to that in Fig. 3~b!.That is, there is an outer downflow with shear, but withothe uniform component~i.e., a50, c.0!. The main differ-ence with the pattern in Fig. 3~b! is the bubble shape. Thbubble in the lower part of Fig. 3~c! extends to infinity be-cause the outer downflow now has zero velocity on the symetry axis. For this reason, the downflow cannot stagnatejet-induced upflow@see the near-axis region forz,0 in Fig.3~a!# for finite z. Since the upflow velocity decays as 1/z, thewidth of the upflow tends to zero asz→2`. Also, the widthof the recirculatory domain@positioned below the origin inFig. 3~c!# tends to zero asz→2`. In this case, there is arecirculatory ‘‘tail’’ @Fig. 3~c!# instead of a ‘‘bubble’’@Fig3~b!# for z,0.

In Fig. 3~d!, we show a flow pattern that is intermediabetween those in Figs. 3~b! and 3~c!, containing both theuniform and shear downflows~a.0 andc.0!. The differ-ence is that the lower part of the bubble is flattened in F3~b!, but rounded in Fig. 3~d!. The flat bubble shape in Fig3~b! is due to the fact that the upflow induced by the annujet @Fig. 3~a!# is nearly uniform with respect tor at a fixedz.This feature of the irrotational flow near the axis is evidefrom ~9!. Since the ambient flow is also uniform, the boun



FIG. 3. Meridional streamlines for the generalized vortex sink matched with the bipolar annular jet.~a! No outer flow.~b! Uniform outer flow.~c! Outer shearflow. ~d! Codirected outer uniform and shear flows.~e–f! Counterdirected outer uniform and shear flows; the shear flow part increases from~e! to ~f!. SeeTable III for parameter values.

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ary between the upflow and downflow in the lower partFig. 3~b! is nearly horizontal near the axis. This boundabecomes parabolic in Fig. 3~d!, owing to the shear outeflow. The pattern in Fig. 3~d! approaches that in Fig. 3~c! if



fa→0 with fixedc, and that in Fig. 3~b! if c→0 for fixeda.The flow patterns are significantly more complicated

the uniform and shear components of the outer flows hopposite directions (ac,0). Figures 3~e! and 3~f! show the



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Phys. Fluids, Vol. 9

Downloaded 14 Ap

TABLE III. Parameters for Fig. 3.

Figure a G c xj c31022 ~contour levels!

~a! 0 50 0 0.9 22, 21.5, 21, 20.5, 20.25, 0, 0.25, 0.5, 1, 1.5, 2~b! 5 50 0 0.9 20.4, 20.3, 20.1, 20.03, 0, 0.01, 0.25, 0.5, 1, 1.5,~c! 0 50 0.5 0.9 20.5, 20.2, 20.1, 20.05, 0, 0.25, 1, 3~d! 8 50 1 0.9 20.2, 20.05,20.02, 0, 0.02, 0.25, 0.5, 1, 2~e! 10 50 20.5 0.9 210, 25, 21, 20.015, 0, 0.05, 0.1, 0.5, 0.9, 1.4~f! 15 50 20.5 0.9 25, 21, 20.1, 20.01, 0, 0.05, 0.1, 0.5, 1, 1.5, 2.1

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patterns for the same shear upflow (c,0) but different uni-form downflows (a.0). For the slower [email protected]~e!#, the pattern forz.0 is rather simple. There is a neaaxis recirculatory domain separated from the ambient upflby the heavy lines in Fig. 3~e!. Inside the domain, there is thvortex ring near the origin. On the upper part of the vortering boundary, there is the saddle circle@denoted by the3symbols near the top of Fig. 3~e!#. Beyond the circle, there isa semi-infinite recirculatory region~not shown!.

Below the origin (z,0), there is a cone-shape bubbsurrounded by the wide vortex ring. Between the bubblering, there is a thin gap that separates also the ring fromouter upflow. The flow inside the gap starts atz52`,moves up, then around the ring, and finally down nearaxis. Between these regions of up- and downflows, theran additional recirculatory semi-infinite region, whictouches the ring along the saddle circle. A complex arranment and multiplicity of recirculatory domains, as shownFig. 3~e!, is typical in many technological swirling flowse.g. in the Ranque–Hilsch tubes~Sec. VII C! and vortexcombustion chambers~Sec. VII D!.

The qualitative difference between Fig. 3~f! and Fig. 3~e!is in the upper part of the flow (z.0). The saddle circle~the3 symbols! is located on the boundary separating the ouupflow and the recirculatory domains in Fig. 3~f!. In contrastto Fig. 3~e!, the outer flow does not extend up to the originFig. 3~f!. The ‘‘gap’’ recirculatory domain is attached to thupper vortex ring along the ‘‘heteroclinic trajectory,’’ whicstarts at the upper saddle and terminates at the [email protected]~f!#. The other domains are similar in Figs. 3~e! and 3~f!;the lower vortex ring is simply larger in Fig. 3~f!.

In summary, the results of this section show thatcomposite solutions resulting from matching of the geneized vortex sink with jet-like inner solutions describe a rivariety of rather complex flow patterns, typical of practicswirling flows and important for momentum, heat, and mtransfer. For example, there are seven different flow domin Figs. 3~e! and 3~f!. It is striking that such complicatedflows are encompassed by our rather simple analytical stions, which explicitly reveal the physical effects responsifor a variety of flow separation phenomena. Namely, radadvection, swirl, and counterflow in various combinatioinduce ‘‘bubbles,’’ vortex rings, and semi-infinite recircultory domains. Such flow elements are important featureswirling flows in nature and technology, as demonstratedSecs. VI–VII for particular applications.

, No. 10, October 1997

r 2006 to 129.7.158.43. Redistribution subject to AIP

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VI. APPLICATIONS OF THE GENERALIZED VORTEXSINK

A. Whirlpool

A well-known solution exists for a whirlpool in aninvis-cid fluid;19 our new results include a solution for aviscousfluid and an analogy drawn between outer flows of tSchlichting jet and whirlpools.

To model a whirlpool, e.g., on the ocean surface, omust account for gravity, which causes the last term in~5! tobe 2rgz. Here,g is the acceleration due to gravity, andz,0 below the ocean level. On the whirlpool surfacez5 f (r ), the pressure is constant and equal to the atmosphpressurep` . Substituting~3a! with z050 in ~5!, and satis-fying the requirement that the sum of the two last terms~5! is zero, we find a5c50, Re524, and b52 1

2(161G2)n2/(gr0

3). This requirement physically means that thdynamic head,2 1

2r(n/r )2(Re21G2), balances the hydrostatic pressure2rgz on the whirlpool surfacez/r 0

5b(r /r 0)22.If the swirl Reynolds number is sufficiently high, i.e

G2@16, thenb52K2/(8p2gr03) becomes independent o

viscosity. For example, fornf50.001 m/s atr 50.2 m andn51026 m2/s ~values relevant for a bathtub vortex!, G5200and the viscous contribution tob is negligible, even in thiscase. Since the centrifugal acceleration atr 5r 0 is gc0

5K2/(4p2r 03), the latter expression forb can be written as

b52 12gc0 /g. Therefore, the whirlpool shape is determin

by the centrifugal/gravitational acceleration ratio atr 5r 0 .This coincides with the corresponding solution for an invcid fluid.19

For a whirlpool to develop, the centrifugal acceleratimust be of the same order of magnitude asg. In Fig. 4, wechoosegc05 1

2g, i.e. b520.25, andG5200 (S5250) toillustrate a physically realizable whirlpool@employing ~2!–~4!#. The corresponding stream surfacesc50, 1, and 2~leftside! and the axial velocity profile~right side! are shown inFig. 4~a!.

In Fig. 4~b!, a streamline on the whirlpool surfacec50explicitly reveals the spiral character of the flow. The pitincreases downstream because the axial velocity dominthe swirl as the streamline approaches the axis, althoughvelocity components tend to infinity.

The velocity field follows from~2! with Re524, G524S, Wr58b, andWc50. The whirlpool attracts ambi-ent fluid to the axis, and the radial flow rate, Re524, isexactly the same as that for the Schlichting round jet~see



ce.
FIG. 4. Whirlpool model.~a! Meridional stream surfaces~left! and thenz profile ~right!. ~b! Three-dimensional view of a streamline on the whirlpool surfa
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Sec. IV A!. This coincidence of entrainment for the Schlicting jet and whirlpool is a new result that has a clear physbasis. The entrainment of the fluid remote from the axisthe same viscous effect, despite the different sources of amomentum in the Schlichting jet and whirlpool. Note thSchlichting’s solution describes the boundary-layer limita strong jet and also that the axial velocity on the whirlposurface tends to infinity asr→0. Therefore, in both casesthere is a strong axial flow attracting ambient fluid. Anothreason for the coincidence of Re for the whirlpool andSchlichting swirl-free jet is the fact that the axial velocidominates the swirl asr→0 along the whirlpool surface~be-causenf;r 21 and nz;r 24!, so that the swirl does not influence the entrainment rate.

Thus, our model captures the whirlpool shape, for whthe role of viscosity is typically negligible, but also the etrainment an ambient fluid—a viscous effect.

B. Cosmic jets

The formation of massive cosmic objects such as stgalaxy cores, and ‘‘black holes’’ generates far-range bipojets.20 The jets develop near accretion disks, which aregions of higher density than that of the ambient media. Tdisk matter moves as a planar vortex sink condensing omassive object. This motion drives the ambient mediuminduces normal-to-disk bipolar swirling jets, which advethe angular momentum far away from the disk. The axmomentum is focused by the converging flow, but disperby viscous diffusion.

A simple hydrodynamic model1 describes the farfield othese jets, where the dispersion dominates and the axialocity decays with distance. In the near field, the focusdominates and the axial velocity increases along thefrom zero on the disk plane,z50. In an intermediate domain, where the axial velocity reaches its maximum, thecusing and dispersion nearly balance each other, and



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axial velocity is nearly independent ofz. In Fig. 5, we showthat the solution~2! models the bipolar swirling jet in thisdomain, using the same parameter values as for Fig. 4.

The physical reason for specifyingc50 is that there isno axial pressure gradient in the ambient space; we taka50, to satisfy the condition thatnz tends to zero as streamlines approach the disk. Finally, we choose Re524 as themaximum entrainment rate. Cosmic jets are certainly turlent, but the induced flow of an ambient medium canlaminar far from the axis. Also, a uniform eddy viscosity issatisfactory model for the mean flow of turbulent jets, so tRe still equals24, but with a Re based on the eddy viscoity. The parametersb and S remain free and describe thgravity of the massive body and the angular momentumdisk particles, respectively.

The specific values chosen for Fig. 5 show that the saparameter values can illustrate different physical applitions. In particular, the streamlines of the meridional motifor the cosmic jet model are the same as shown in Fig. 4~a!,

FIG. 5. Model of a cosmic bipolar jet. Streamlines of the meridional moti



FIG. 6. Meridional streamlines in a model tornado near~a! the bulge and~b! the ground.

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but now the flow pattern is symmetric with respect to plaz50 ~Fig. 5!. The accretion disk corresponds to planez50, and the massive body is located at the frame originthis model.

In contrast to the whirlpool case, the pressure is not cstant on stream surfaces and the fluid occupies the espace. The common features are that stream surfacesverge to the axis downstream and the axial velocity increaalong streamlines~compare with the farfield of the jet,1

where stream surfaces diverge from the axis and the avelocity decays downstream!.

Thus, solution~2! models an ambient flow induced bystrong jet, regardless of how the jet is forced. The physmechanism of forcing is quite different for the whirlpool ancosmic jets, but the kinematic features are the same:~i! flowfocusing by the radial convergence and~ii ! irrotational char-acter of swirl in an outer flow. An additional illustration osuch universality is a tornado model, considered below.

C. Tornado

A solution similar to that shown in Figs. 4 and 5 can almodel a swirling flow near a wall outside a boundary layAs an example, consider the flow produced in a cylindricontainer by a rotating endwall.21 Our model approximateswell the flow near the axis in the vicinity of the fixed enwall, where the flow pattern is similar to that in tornadoeNote that our model is not valid near the rotating endwbecause it describes potential but not solid-body swirl. Aother example is a tornado itself or, more precisely, thegion adjacent to the ground where the tornado convergeits axis ~conically similar flows model the farfield of atornado13!.

It has been observed22 that a tornado funnel oftenabruptly expands into a wide bulge at some height fromground. Bulge formation in tornadoes can be caused byferent factors. Suppose, for example, that a tornado funenters a region of high atmospheric turbulence. Turbumixing decreases the near-axis swirl, thus increasingpressure by cyclostrophic balance. In turn, the induced p



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tive axial pressure gradient slows the axial flow and caustreamline divergence, i.e., positive values of the radiallocity.

In terms of our model, this pressure increase and divgence correspond toP.0 and Re.0, and solution~2! pre-dicts an abrupt widening of the funnel~Fig. 6!. Figure 6~a!shows the meridional streamlines, which model the expsion of the tornado funnel into a bulge; Fig. 6~b! illustratestornado formation near the ground. The parameters ara510, b510, c5210, Re54, andS5250 for Fig. 6~a! anda5c50, b520.25, Re524, and S550 for Fig. 6~b!.These values are chosen to satisfy the condition thatnz→0asr→` along streamlines approaching the ground, to obta strong jet, and to match the flow characteristics in Fi6~a! and 6~b!.

In summary, our solution~2!–~5! is able to approximateboth tornado focusing and expansion into a bulge. The laeffect leads us to the general problem of vortex breakdoFirst, we consider vortex breakdown without reverse flowhich we call a ‘‘vortex-breakdown wake.’’

D. Vortex-breakdown wake

The appearance of the tornado bulge is certainly a tof vortex breakdown. There exists a conical model2 for vor-tex breakdown at a tornado’s mid-height, with a reversflow inside the bulge. In contrast, there is no flow reversathe above model for the bulge. Vortex breakdown is indeobserved without reversed flow under some conditions.23 Inthis case, stream surfaces gradually diverge as in a far sder wake. At very high Reynolds numbers, a slender turlent wake develops just downstream of the vortex breakdopoint.24

For slender wakes, the longitudinal velocity has tsame sign on and far away from the axis, due to viscdiffusion of axial momentum. Thus, flow reversal, which ocurs inside a vortex-breakdown ‘‘bubble,’’ does not occura slender wake.

There is a recent experimental observation25 of vortexbreakdown in a cylindrical container with a thin rod on th



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axis rotating independently of the rotating endwall. In F7~a!, the rod is seen as a bright vertical strip. Near the rthe flow moves downward from the fixed endwall to trotating endwall. If the rod is fixed, then there is a recirclatory ‘‘bubble’’ similar to that appearing without the rod.21

FIG. 7. Flow induced by independent corotation of the lower endwallcentral rod in a sealed cylinder.~a! Photo visualizing the rod and slendewake near the axis. Models of the bulge surface~b! and a streamline on it~c!.



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However, if the rod rotates in the same direction as the ewall, the bubble transforms into a slender domain, startbehind the breakdown point and extending up to the endw

The rod ‘‘forces’’ the meridional flow to stagnate, buthe swirl is fixed and independent ofz near the rod due to theno-slip condition. Near the lower part of the rod, there acontributions to the meridional flow from two effects:~i! therotating disk generates a flow toward it near the axis, and~ii !the interaction of the rotating rod normal to the fixed wainducing a flow away from the wall~see Ref. 11, p. 93 and p214, respectively!. Since these effects oppose each other,expect a very slow meridional motion inside the slendermain.

Figure 7~a! shows a photo of the breakdown domain25

note the light diverging strips. If the rod corotates faenough~the rod/endwall angular velocity ratio is'6!, thebreakdown ‘‘bubble’’ is transformed into a slender wakWe model the domain shape with the help of solution~3! atthe parameter values chosen to approximate the experimobservation. Figures 7~b! and 7~c! show the meridional crosssection of the stream surface and a three-dimensional viea streamline on this surface that models the wake surfacFig. 7~a!. The parameters:a512, b510, c5210, Re54,and S5100, provide good qualitative agreement with texperimental breakdown shape~we cannot compare the velocity field because the corresponding experimental datanot yet available!. The experiment shows that the wake suface expands downstream of the breakdown point, deveinto a nearly cylindrical shape, and then again abruptlypands. Our model captures even this fine feature of the wsurface.

To summarize this section, even the~rather simple! gen-eralized vortex source itself is useful in modeling many prtical flows. In the next section, we apply our composite slutions to model flows that exhibit considerably mocomplex patterns.

VII. APPLICATIONS OF THE COMPOSITE SOLUTIONS

A. Vortex breakdown bubbles

Although there is no reverse flow in the above examplthe composite solutions~10! and ~11! enable one to modereversing swirling flows, including vortex-breakdow‘‘bubbles.’’

Figure 8~a! is a photo of the typical vortex breakdowbubble observed experimentally.21 In Figs. 8~b! and 8~c!, wecompare the flow resulting from~10! with B51 and for bothFigs. 8~b! and 8~c!, but witha520.08 andc50 for ~b!, anda520.08 andc520.0001 for Fig. 8~c!. These parametevalues provide the best agreement with the experimentasult.

B. Self-swirling flow in meniscus

An intriguing flow phenomenon that we model heresolution ~10! is the self-swirling observed in liquid meniscof electrosprays.26,27 A drop of an electrically conductingliquid at the edge of a capillary tube takes a conical form

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the presence of an axial electric field. Figure 9~a! is a sche-matic showing the capillary tubeT, meniscusM , and elec-tric field E.

Electrical shear stresses acting on the meniscus surf@t in Fig. 9~a!# induce a recirculatory motion of the liquidthat flows to the apex near the surface and away fromapex along the axis@C in Fig. 9~a!#. Under some conditions~a sufficiently strong electric field, and small viscosity anelectrical conductivity of the liquid!, swirl appears, althoughthere is no obvious swirlforcing. This swirl can become verystrong, so that the flow inside the meniscus resembles anado. Far away from the apex, the flow is satisfactorily aproximated by conical solutions.26,27 Near the apex, how-ever, the meniscus shape is not conical, and a jet erupts fthe tip @J in Fig. 9~a!#. These features are not covered by thconical solutions, but can be modeled with the help of sotion ~10!. Figures 9~b!–9~d! are obtained with~10! for a50,B51000,c520.001, andS550. Figure 9~b! shows the me-ridional streamlines, and Figs. 9~c! and 9~d! show the three-dimensional view of the streamlines. Figure 9~c! shows astreamline embedded in the meniscus and positioned at sdistance from the surface@C in Fig. 9~a!#. The formation ofa submerged swirling jet directed inside the meniscus is clin Figs. 9~b! and 9~c!. Figure 9~d! shows a streamline on thesurface of both the meniscus and the jet@M and J in Fig.9~a!#. The parameters are adjusted to obtain good agreemwith the experimentally observed flow patterns27 ~only visu-alization with the help of a microscope is currently availabfor the flow near the meniscus tip!. Note that the near-axisparts of the streamlines in Figs. 4 and 9 differ. As mentionabove, the pitch increases downstream in Fig. 4~b!, but is

FIG. 8. ~a! Vortex-breakdown ‘‘bubbles’’ observed in the experiment.21

~b!,~c! Models of the ‘‘bubbles’’ by~10!; see the text for parameter valuesc values on streamlines are~b! 212, 28, 24, 22, 21, 20.2, 20.05, 0,0.05, 0.3, 1, 2 and~c! 240, 220, 2100, 22, 20.2, 0, 0.5, 1, 2.



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nearly constant in Figs. 9~c! and 9~d!. The reason is thastream surfaces diverge downstream in the Schlichting~Fig. 9!, but converge for solution~2! ~Fig. 4!. The axialvelocity is bounded and decays as 1/z, thus influencing thepitch defined by thenz /nf ratio. These flow features in menisci are quite similar to those in vortex tubes, as discusin the next section.

C. Vortex tube

The Ranque–Hilsch vortex tube is in widespread usea compact, environmentally safe cooling device containno moving parts.28 Figure 10~a! shows the schematic ostream surfaces in a meridional cross section~not to scale!.Compressed air enters the tube tangentially through inleI ,generating a strong swirling flow. This flow then separainto two streams:14 ~i! streamC ~to become cold at the outflow! reverses direction near the stagnation pointS andleaves the tube through orificeOC , and~ii ! streamH ~thatbecomes hot at the outflow! is unidirectional and leaves thtube far downstream through the~typically peripheral! outletOH.

Our solution family ~2!–~3! enables one to modestreamsC, H, and the separating surfaceS through on ap-propriate choice of the parameters. Figure 10~b! shows themodel streamlines of the coldC, hot H, and separatingSflows for a50.5,c50, Re524, andS550. The streamlinesdiffer by values ofb520.25 (C), 0 (S), and 0.25 (H). Theb sign governs whether the flow is reversing, separatingunidirectional.

Solution~10! provides a better approximation of the floand captures all the streams simultaneously in a single flfield. Figure 10~c! shows streamlines of the meridional mtion calculated from~10! usinga50.5, B51000, andc50.The numerical values are chosen here just to demonstratability of our solution to model a flow with very differenscales in different regions. The goal of the present worknot to provide specific numerical data, but to clearly illumnate physical processes which are crucial for the RanqHilsch effect. Analysis of the flow field details and optimzation of the Ranque–Hilsch effect through paramechanges are the subject of a separate study.

Our model captures two important features of the flodynamics in the vortex tube:~i! focusing of the swirling flowto the axis along separatrixS, and ~ii ! a sharp decrease ipressure along streamlines according to~5!. The pressuredecrease provides adiabatic cooling of air (T/T`

5(p/p`) I 21/g, g is the ratio of specific heats!, while theswirl focusing generates a sharp peak of the axial vorticnear the axis.

The effects~i!–~ii ! are the basis for the next two:~iii !strong diffusion of the vorticity causing kinetic energy tranfer from near-axis to peripheral regions, and~iv! separationof near-axis and peripheral flows. Our view~based on theanalysis of the literature and our own preliminary studies! isthat features~i!–~iv! are the essence of the Ranque–Hilseffect. The present model describes all these featuresprovides a background for more detailed studies and optzation.



FIG. 9. ~a! Schematic of a meniscus and jet in electrosprays.~b! Meridional streamlines of a self-swirling flow withc5220, 210, 25, 0, 5, 10, 20, 30, 50.Three-dimensional view of streamlines:~c! inside the meniscus and~d! on the meniscus and jet surface.

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A solution for the heat equation~based on our results fovelocity and pressure! can also be analytically calculatedThis is beyond the scope of this paper, which addressesmotion only. Nevertheless, in the next section, we consione ‘‘thermal’’ application that does not necessitate a sotion of the heat equation.

D. Vortex burner

It is well known that swirling flows with vortex breakdown provide flame stabilization and are useful fcombustion.28 Figure 11~a! shows a schematic of an ABBEV-burner.29 Tangential injection of gaseous fuel and athrough slits in the conical sidewall generates a stronswirling flow inside the burner. Liquid fuel is injectethrough the central nozzle, then atomized into drops



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evaporated. Vortex breakdown induces the recirculatzone RZ ~near the exit!, which stabilizes the flame fronlocated slightly upstream and is shown as a thin parabshell. For more details concerning the burner design and flfeatures see Ref. 29. Here we present a simplified flow mobased on our solution to demonstrate how the flame frontbe calculated near the vortex breakdown ‘‘bubble.’’

Figure 11~b! shows a flow pattern obtained with~10! fora520.33,B55, andc50.02. The pattern is similar to thain Fig. 2~f! and is used here to model the flow in the burnTaking into account the turbulent character of the flow in tburner and interpreting the viscosity as some appropreddy viscosity, we disregard the no-slip condition andplace some streamlines by walls in Fig. 11~b!. The stream-lines c52 shown by a bold curve in Fig. 11~b! serve as the



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burner sidewall. The bold curvec50 in the left part of Fig.11~b! imitates the endwall; the other bold curvec50 locatednear the exit is the boundary of the recirculatory zoneRZ.There are also streamlines,c520.025 and20.05, insideandc50.01, 0.1, 0.5, and 1, outsideRZ. The flow patternsin Figs. 11~a! and 11~b! are roughly similar.

To model the flame front, we do not address here thheat and reaction-diffusion equations, but instead apply tfollowing approach. If the contribution to heat and mastransfer from diffusion is small in comparison with that fromadvection, then one can neglect the thickness of the flafront and consider the front as a surface. The temperatuand concentration are uniform and have the ignition valuon this surface. If the flame front is steady, the normal-tfront velocity of the flow nn is also uniform and equals2nc , where nc is the propagation speed of combustionTherefore, in such an approach, the flame front is a surfaof revolution,F(r ,z)50, governed by the equation30,31

n ]F/]r 1nz ]F/]z52nc@~]F/]r !21~]F/]z!2#1/2.~12a!

RepresentingF[z2Z(r ) and substituting in~12a! yields

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22n r2!. ~12b!

The flame front is expected to be axisymmetric and smoo@see Fig. 11~a!#. In this case,dZ/dr50 at r 50 and, sincen r50 at r 50, it follows from ~12b! thatnc5nz(0,Z0). Here

FIG. 10. ~a! Schematic of the meridional motion in the vortex tube. Flowenters through inletI , cold outflowC leaves the tube through outletOC,and hot outflowH leaves the tube through outletOH near valveV. SurfaceS separatesC andH. ~b! Three-dimensional views of streamlines inC andH, and onS. ~c! Meridional flow from the composite solution~10! givingstreamlines withc5250, 225, 25, 0, 5, 25, 50.



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Z05Z(0) corresponds to an intersection point of the flamfront with axisz. It is convenient to start the integration o~12b! this point. For physical reasons, this point mustlocated in the interval of thez axis downstream of the originr 5z50, but upstream ofRZ in Fig. 11~b!.

The flame propagates upstream~to the region wherethere is unburned fuel!, and the advection velocity must bpositive to stagnate the front. A particular value ofnc de-pends on combustion components. For example,nc

50.3 m/s for combustion of pure methane in air corresponto the marginal value of self-ignition.32 In our problem, agiven nc specifiesZ0 , andvice versa. Since thenc value issmall in comparison with the maximal velocity in the burnethe flame front is located slightly upstream of the bubbThe thick curve modeling the flame front in Fig. 11~b! startsat z5Z0515, while the bubble starts atz516.5.

To summarize, the analytic solutions~2!, ~10!, and~11!are clearly useful in modeling rather complex practical flowCertainly, each application considered above~along withmany others! requires more thorough treatment. Our aimSec. VI–VII has been to display samples of possible apcations; their detailed considerations are subjects for adtional studies. In the following discussion~Sec. VIII!, we tryto focus on some common features from the solutionstained and their applications.

VIII. FORMATION OF VORTEX FILAMENTS

Slender vortex filaments, observed in nearly all trantional and turbulent flows, play a significant role in vortedynamics and transfer processes. The filaments are oviewed as the ‘‘sinews and muscles’’ of turbulent flows. Fexample, cosmic jets involve large-scale motion in the intstellar and intergalactic medium20 in the form of ‘‘vortex

FIG. 11. ~a! Schematic of a EV- burner~ABB Power Generation, Switzer-land!. ~b! Our model based on solution~10!. The peripheral~c52! and left~c50! solid streamlines model the burner walls. The right streamline~c50!is the boundary of vortex breakdown region. The other parabolic bold cuis the flame front.



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filaments’’ ~i.e., slender regions of high vorticity!. Anotherwell-known example is the wing tip vortex that developsbehind aircraft.

Based on our solutions and examples of applications, wdiscuss here a general mechanism by which vortex filamencan form, driven by self-focusing of swirling flow~i.e., in-creases in the axial momentum and swirl velocity near thaxis! triggered by some external factor. Above a delta wingof an aircraft, for instance, the external factors include thvortex sheets that separate from the wing leading edge athe cross-flow, which stimulates rolling up of the sheets intovortices with axial flow. The upper part of Fig. 12~extractedfrom flow vizualization33! shows streamlines forming thecore of a delta-wing vortex~the white strips!. Measure-ments34 reveal that the core is an intense swirling jet with asharp peak in the longitudinal velocity. This jet results fromfocusing of the longitudinal momentum and swirl in the rollup process.

The lower part of Fig. 12 is a schematic of this proces~streamlines! drawn with the help of our solution~2! at a520.5, b50.25, c50, Re524, andS5230. The curvesdiffer only with the initial r andz values and model stream-lines separating from the leading edge. The curves collaponto the axis, which models the vortex core.

The mechanism of vortex core formation is as followsThe roll up of the separation surface leads to the generatioof swirl ~as shown by the streamlines in the lower part oFig. 12!. This swirl induces a pressure drop near the axisthus attracting other streamlines to the axis; this further focuses the swirl, therefore, further decreasing pressure in tvortex core. This positive feedback leads to strong accumulation of the axial vorticity and momentum, i.e., to vortexfilament formation.

In the present solution, the convergence of swirling flowis modeled by a sink. The potential-vortex component of thsolution corresponds to the conservation of angular mometum during the streamline convergence. The singularity othe axial velocity~see termWr(r /r 0)Re in ~2c! at Re,0!reflects a strong concentration of the axial momentum nethe axis by the converging flow. The significant accumulation of kinetic energy in a near-axis region is modeled by thfact that all velocity components and pressure are infinite o

FIG. 12. Visualization of the delta-wing vortex32 ~above! and our schematicexplaining the vortex formation~below!: streamlines separating from theleading edge are cumulated in the vortex core.



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the axis, according to~2! and ~5!. As noted above, the solution ~2!–~5! is not valid at the filament center, where reglarization ~or ‘‘cut off’’ ! of velocity and pressure occurs bviscous effects, modeled here by the composite soluti~10!–~11!. The radius of the vortex core where the soluti~2! must match an inner solution can serve as the length sr 0 used in~1!–~5!.

The viscous action causing vortex regularization in prtical flows is physically obvious. Solution~2! helps us toilluminate the focusing of swirling flows; this effect is lesclear ~but more important for applications! than the regular-ization. The focusing is similar to the accumulation effeoccurring in colliding swirl-free jets~as in cumulative artil-lery shells!, where a bipolar jet appears due to convergenof a conical annular jet that self-collides at the cone ape

Solution ~2! describes such a focusing of axial mometum, but also an additional effect a strong pressure dropthe axis owing to swirl concentration. Composite soluti~10! also captures the opposite effect: an abrupt broadenof the vortex core, i.e., vortex breakdown. Divergencestream surfaces is caused here by the interaction of a swijet with an ambient flow~Sec. V!. Thus, the generalizedvortex-sink solution@~2!–~5!# provides insight into the vor-tex filament formation, and the composite solution~10! illu-minates vortex breakdown.

IX. CONCLUDING REMARKS

We have derived a new analytical solution of thNavier–Stokes equations, representing a generalizationthe planar vortex sink, by adding an axial flow with radishear. This generalized five-parameter flow on surfacesrevolution has been applied to model elements of whirlpoocosmic jets, tornadoes, and recirculatory swirling motion isealed cylinder. We have extended this solution to the naxis region via matching with swirling jet flows, thus obtaiing composite solutions. The composite solutions have bapplied for vortex breakdown, the Ranque–Hilsch effeflows in menisci of electrosprays, and in vortex burners. Tmodeling helps us to explain the important physical featuof these vortex flows and serves a potentially quite usenew approach to parametric studies and optimization of ptical swirling flows. We anticipate other applications suchcumulative jets in the collapsing bubbles of cavitation, aheat and mass transfer in swirling flows.

The present results can be generalized even furtheinclude nonaxisymmetric flows. Recently, rather strikingsults have been obtained35–37 related to the instability andbifurcation of planar vortex-sink flows. Bifurcation of thvortex-sinks leads to Oseen’s solutions, correspondingnonaxisymmetric flow patterns with a few spiral branches35

To generalize Oseen’s solution via the addition of an axflow will require a laborious analysis, including both numecal simulations and asymptotic solutions. This is certainlyimportant issue for further study, but beyond the scope ofpresent paper, in which we focus on the generalized axismetric solution.



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ACKNOWLEDGMENTS

This research was founded by the Air Force OfficeScientific Research Grant No. F49620-95-1-0302. Thethors are grateful to Wade Schoppa for a review of the mascript.

1M. A. Goldshtik and V. N. Shtern, ‘‘Collapse in conical viscous flows,’’Fluid Mech.218, 483 ~1990!.

2V. Shtern and F. Hussain, ‘‘Hysteresis in swirling jets,’’ J. Fluid Mec309, 1 ~1996!.

3A. A. Borissov, P. A. Kuibin, and V. L. Okulov, ‘‘Calculation of theRanque effect in the vortex tube,’’ Acta Mech.4 Suppl., 289~1994!.

4P. G. Saffman,Vortex Dynamics~Cambridge University Press, England1992!.

5W. Wuest, ‘‘Asymptotische absaugegrenzschichten an langsangestrozylindrischen Korpern,’’ Ing. Arch.23, 198 ~1955!.

6H. G. Lew, ‘‘Asymptotic suction characteristics of the boundary layer oa circular cylinder,’’ J. Aeronaut. Sci.23, 895 ~1956!.

7M. Yasuhara, ‘‘On the asymptotic solution of the laminar compressboundary layer over a circular cylinder with uniform suction,’’ J. PhySoc. Jpn.12, 102 ~1957!.

8J. T. Stuart, ‘‘A simple corner flow with a suction,’’ Q. J. Mech. AppMath. 19, 217 ~1966!.

9A. A. Berman, ‘‘Laminar flow in an annulus with porous walls,’’ J. AppPhys.29, 71 ~1958!.

10C. Y. Wang, ‘‘Exact solutions of the steady-state Navier–Stokes eqtions,’’ Annu. Rev. Fluid Mech.23, 159 ~1991!.

11H. Schlichting,Boundary-Layer Theory~McGraw-Hill, New York, 1979!.12R. R. Long, ‘‘A vortex in an infinite viscous fluid,’’ J. Fluid Mech.11, 611

~1961!.13V. Shtern and F. Hussain, ‘‘Hysteresis in a swirling jet as a torna

model,’’ Phys. Fluids A5, 2183~1993!.14C. D. Fulton, ‘‘Ranque tube,’’ Refr. Eng.58, 473 ~1950!.15T. Sarpkaya, ‘‘Vortex breakdown in conical swirling flows,’’ AIAA J.9,

1792 ~1971!.16G. K. Batchelor,An Introduction to Fluid Dynamics~Cambridge Univer-

sity Press, England, 1967!.17M. R. Foster and F. T. Smith, ‘‘Stability of Long’s vortex at large flo

force,’’ J. Fluid Mech.206, 405 ~1989!.18M. Van Dyke, Perturbation Methods in Fluid Mechanics~Academic

Press, New York, 1964!.



fu-u-

ten

r

e

-

o

19H. Lamb, Hydrodynamics~Cambridge University Press, England, 1993!,p. 28.

20A. Konigl, ‘‘Stellar and galactic jets: Theoretical issues,’’ Can. J. Ph64, 362 ~1986!.

21M. P. Escudier, ‘‘Observations of the flow produced in a cylindrical cotainer by a rotating endwall,’’ Exp. Fluids2, 189 ~1984!.

22O. R. Burggraf and M. R. Foster, ‘‘Continuation or breakdown in tornadlike vortices,’’ J. Fluid Mech.80, 685 ~1977!.

23S. Backstein, ‘‘Experiments on vortex breakdown,’’ inVortex Flows inAerodynamics~RWTH Aachen University of Technology, Germany1994!.

24T. Sarpkaya, ‘‘Turbulent vortex breakdown,’’ Phys. Fluids7, 2301~1995!.25H. Husain, V. Shtern, and F. Hussain, ‘‘Control of vortex breakdown

Bull. Am. Phys. Soc.41, 1764~1996!; see also AIAA Paper No. 97-18791997.

26V. Shtern, M. Goldshtik, and F. Hussain, ‘‘Generation of swirl duesymmetry breaking,’’ Phys. Rev. E49, 2881~1994!.

27V. Shtern and A. Barrero, ‘‘Bifurcation of swirl in liquid cones,’’ J. FluidMech.300, 169 ~1995!.

28A. K. Gupta, D. G. Lilley, and N. Sared,Swirl Flows~Abacus, TunbridgeWells, 1984!.

29J. J. Keller, Th. Sattelmayer, and F. Thu¨ringer, ‘‘Double-cone burner forgas turbine type 9 retrofit application,’’Procedings of the 19th Interna-tional Congress On Combustion Engines, Florence, Italy, CIMAC 1991.

30G. H. Markstein,Unsteady Flame Propagation~Pergamon, New York,1964!, p. 8.

31F. A. Williams, Combustion Theory, 2nd ed.~Addison-Wesley, Reading,MA 1984!, p. 44.

32A. A. Borisov, P. A. Kuibin, and V. L. Okulov, ‘‘Flame shapes in swirflows,’’ Russ. J. Eng. Thermophys.3, 243 ~1993!.

33H. Werle, Hoiille Blanch28, 330~1963!; @the photo is reproduced from RL. Panton,Incompressible Flow~Wiley, New York, 1996!, p. 569#.

34P. B. Earnshow, ‘‘An experimental investigation the structure ofleading-edge vortex,’’ Aeronaut. Res. Council Rep.22, 876 ~1961!.

35M. Goldshtik, F. Hussain, and V. Shtern, ‘‘Symmetry breaking in vortesource and Jeffery–Hamel flows,’’ J. Fluid Mech.232, 521 ~1991!.

36M. Shusser and D. Weis, ‘‘Stability analysis of source and sink flowsPhys. Fluids7, 2345~1995!.

37M. Shusser and D. Weis, ‘‘Stability of source-vortex and doublet flowsPhys. Fluids.8, 3197~1996!.



Vortex sinks with axial ﬂow: Solution and applications

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Transcript of Vortex sinks with axial ﬂow: Solution and applications