Three-dimensional swirling flows in a tall cylinder driven by a...

Three-dimensional swirling flows in a tall cylinder driven by a rotatingendwallJ. M. Lopez Citation: Phys. Fluids 24, 014101 (2012); doi: 10.1063/1.3673608 View online: http://dx.doi.org/10.1063/1.3673608 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v24/i1 Published by the American Institute of Physics. Related ArticlesResonance phenomena and long-term chaotic advection in volume-preserving systems Chaos 22, 013103 (2012) Steady-state hydrodynamics of a viscous incompressible fluid with spinning particles J. Chem. Phys. 135, 234901 (2011) Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications Phys. Fluids 23, 123102 (2011) Inertial, barotropic, and baroclinic instabilities of the Bickley jet in two-layer rotating shallow water model Phys. Fluids 23, 126601 (2011) The instability of the boundary layer over a disk rotating in an enforced axial flow Phys. Fluids 23, 114108 (2011) Additional information on Phys. FluidsJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

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Three-dimensional swirling flows in a tall cylinder drivenby a rotating endwall

J. M. Lopeza)

School of Mathematical and Statistical Sciences, Arizona State University, Tempe,Arizona 85287, USA and Department of Mathematics, Kyungpook National University,Daegu 702-701, South Korea

(Received 2 September 2011; accepted 23 November 2011; published online 4 January2012)

The onset and nonlinear dynamics of swirling flows in relatively tall cylinders

driven by the rotation of an endwall are studied numerically. These flows are

distinguished from the more widely studied swirling flows in shorter cylinders; the

instability in the taller cylinders is direct to three-dimensional flows rather than to

unsteady axisymmetric flows. The simulations are in very good agreement with

recent experiments in terms of the critical Reynolds number, frequency, and

azimuthal wavenumber of the flows, but there is disagreement in the interpretation

of these flows. We show that these flows are indeed rotating waves and that they

have the same vorticity distributions as the flows measured using particle image

velocimetry in the experiments. Identifying these as rotating waves gives a direct

connection with prior linear stability analysis and the three-dimensional flows

found in shorter cylinders as secondary instabilities leading to modulated rotating

waves. VC 2012 American Institute of Physics. [doi:10.1063/1.3673608]

I. INTRODUCTION

There has been much recent interest in swirling flows in an enclosed cylinder driven by a

rotating endwall when the height-to-radius aspect ratio is greater than about 3, as the onset of

instability is to three-dimensional flow when the endwall rotation exceeds a critical value.1–7 The

most recent of these studies7 is curious as it analyzes particle image velocimetry (PIV) data of

these swirling flows from a multiple-helix point of view and makes the claim that the experimen-

tally observed flows for cylinder aspect ratios greater than about 3 are different from the three-

dimensional rotating wave states found at lower aspect ratios.8,9 If this is indeed the case, then it

begs the question as to what are these new states and how do they come about?

The experiments report that the new states appear directly from the steady axisymmetric basic

state as the endwall rotation, non-dimensionalized as a Reynolds number, is increased beyond a

critical value dependent on the aspect ratio. In the more recent paper,7 they analyze their experimen-

tal data in terms of helical modes. Their dye and air-bubble visualizations show very distinct helical

concentrations of either dye or air bubbles (they have conducted two sets of experiments with differ-

ent apparati, working fluids, and measurement and visualization techniques, and have obtained

consistent results), and hence their suggestion that these states are different from the rotating wave

states that bifurcate from the basic state. In an earlier paper,5 reporting on very similar experiments

they note a discrepancy between the experimentally observed critical Reynolds number and the

critical Reynolds number determined from the linear stability analysis for the corresponding

azimuthal wavenumber. In that earlier paper, they suggested that this may be due to a subcritical

bifurcation to three-dimensional flow, but have yet to investigate if this is the case. Up to now, no

nonlinear simulations have been reported to help address this issue. Our simulations reported here

show that the onset of three-dimensional flow is supercritical and that there is no hysteresis.

Equivariant dynamical systems theory10–15 states that when an SO(2) axisymmetric steadystate loses stability to three-dimensional perturbations as a single parameter is varied, and that if

a)Electronic mail: [email protected].

1070-6631/2012/24(1)/014101/9/$30.00 VC 2012 American Institute of Physics24, 014101-1

PHYSICS OF FLUIDS 24, 014101 (2012)

http://dx.doi.org/10.1063/1.3673608http://dx.doi.org/10.1063/1.3673608http://dx.doi.org/10.1063/1.3673608

the bifurcation is supercritical (energy of the three-dimensional perturbation growing linearly with

the bifurcation parameter and the frequency varying very weakly with the bifurcation parameter)

then the bifurcating state is a rotating wave. Rotating waves are three-dimensional structures that

are invariant in an appropriately rotating frame of reference (the types of states the authors7 claim

are not what they observe).

To date, the only nonlinear three-dimensional simulations of the flows in tall cylinders2 are

for aspect ratio H/R¼ 4.0 and the results are not consistent with the experimental observations.5,7The main discrepancy is that the numerical simulations first show an onset of axisymmetric time-

periodic flow undergoing a secondary instability to three-dimensional flow, whereas the experi-

ments report a direct transition to three-dimensional flow from the steady axisymmetric basic state

at a much lower Re. Our simulations presented here agree with the experiments in terms of thecritical Re, frequency, and azimuthal wave number at onset.

In this paper we address these issues, making direct comparisons with the PIV and laser

Doppler anemometry (LDA) data provided from the experiments and the corresponding linear sta-

bility analysis,5,7 and also relate our large aspect ratio results to lower aspect ratio experiments

and nonlinear simulations, as well as linear stability analysis, in order to place the tall cylinder

results in a proper content with regard to the results for shorter cylinders.

II. GOVERNING EQUATIONS AND NUMERICAL METHODS

Consider the flow in a stationary circular cylinder of radius R and height H, completely filledwith a fluid of kinematic viscosity �. The flow is driven by the rotation of the bottom endwall withan angular speed X. A schematic of the flow system is shown in Fig. 1. The Navier–Stokes equa-tions, non-dimensionalized using R as the length scale and 1/X as the time scale, are

ð@t þ u � rÞu ¼ �rpþ 1=Rer2u; r � u ¼ 0; (1)

where u¼ (u,v,w) is the velocity field in polar coordinates (r,h,z) [ [0,1]� [0,2p)� [0,H/R] and pis the kinematic pressure. The corresponding vorticity field is r�u¼ (n,g,f)¼ (1/r@hw� @zv,@zu� @rw, 1/r@r(rv)� 1/r@hu).

There are two governing parameters: the aspect ratio H/R and the Reynolds numberRe¼XR2/�. The boundary conditions are no-slip: at the stationary top, z¼H/R, and at the station-ary sidewall, r¼ 1, (u,v,w)¼ (0,0,0), while at the rotating bottom wall, z¼ 0, (u,v,w)¼ (0,r,0). Theresults presented here are for Re [ [2000, 3200] and H/R [ [3.5, 5.3].

FIG. 1. Schematic of the apparatus.

014101-2 J. M. Lopez Phys. Fluids 24, 014101 (2012)

The governing equations and boundary conditions are invariant under rotations through arbi-

trary angle / about the axis, R/, whose action on any solution u¼ (u,v,w)(r,h,z,t) is

R/ðu; v;wÞðr; h; z; tÞ ¼ ðu; v;wÞðr; hþ /; z; tÞ; (2)

i.e., the symmetry group of the system is SO(2). They are also invariant under arbitrary transla-tions in time s, whose action is

Usðu; v;wÞðr; h; z; tÞ ¼ ðu; v;wÞðr; h; z; tþ sÞ: (3)

Hence, the basic states, which depend on the two parameters Re and H/R, are steady andaxisymmetric.

The governing equations (1) have been solved using a second order time-splitting method,

with space discretized via a Galerkin–Fourier expansion in h and Chebyshev collocation in r andz. The spectral solver16 has recently been tested and used in a wide variety of enclosed cylinderflows.17–20 For the solutions presented here, we have used nr¼ 48, nz¼ 96 Chebyshev modes inthe radial and axial directions and nh¼ 13 Fourier modes in the azimuthal direction. The time-stepused was dt¼ 0.005.

III. RESULTS

We shall focus our efforts on 4 aspect ratios, three of these (H/R¼ 3.5, 4.6, and 5.3) correspondto those for which the experiments have focused on,6,7 and detailed onset experimental data are avail-

able,5 and the fourth, H/R¼ 4.0, is the case where there are discrepancies between the experimentsand numerical simulations.2 Table I summarizes our numerical results and the experimental results5

for these 4 values of H/R, giving the azimuthal wave number, m, of the bifurcating state, the criticalRe for onset (the experiments were conducted at discrete values of Re, using steps of DRe¼ 100, so inthe table we present a bracket bounded by the highest Re for sustained axisymmetric flow and the low-est Re for sustained three-dimensional flow at the given H/R, as reported), and the frequency of theflow oscillations at a point in space (numerically, we obtain this frequency from a time series of the

axial velocity at (r,h,z)¼ (0.5, 0.0, 0.5H/R), whereas the experiments21 determined the frequency fromlaser Doppler anemometry measurements of axial and azimuthal velocities at a number of points).

Along with the experimental results, they5 also determined the critical Re from numerical linear stabil-ity analysis, and they note that the experiments predict slightly lower critical Re for the m¼ 3 andm¼ 2 branches, but higher critical Re for the m¼ 4 branch. The critical Re from their linear stabilityanalysis is in better agreement with our nonlinear numerical estimates; but in any case, all three are in

very close agreement. They speculated that the slight disagreement between their experimental and

theoretical estimates may be due to subcritical bifurcations (which they did not investigate).

Our numerical estimates of the critical Re reported in Table I were determined from theenergy in the non-axisymmetric components of the flows. This was measured using the L2-normsof the azimuthal Fourier modes of a given solution,

Em ¼1

2

ðz¼H=Rz¼0

ðr¼1r¼0

um � u�m r dr dz; (4)

TABLE I. Critical Re, frequencies x and azimuthal wavenumber m at vari-ous H/R as determined by our nonlinear simulations and reported from

experiments.5

Numerical Experimental5

H/R m Rec x Rec x

3.5 3 2131.2 0.2967 (2000, 2100) 0.29

4.0 3 2433.9 0.2680 (2200, 2300) 0.27

4.6 2 2834.3 0.1350 (2600, 2700) 0.14

5.3 4 3035.6 0.4009 (3000, 3100) 0.38

014101-3 Three-dimensional swirling flows in a tall cylinder Phys. Fluids 24, 014101 (2012)

where um is the mth Fourier mode of the velocity field and u�m is its complex conjugate. For rotat-

ing wave states, Em are constant; and for a supercritical bifurcation, Em ! Re�Rec for(Re�Rec)/Rec� 1. Figure 2 shows how Em for the various branches of solutions in Table I varywith Re, clearly showing the onset of the three-dimensional states to be supercritical. Furthermore,

FIG. 2. Modal kinetic energies Em versus Re for the various rotating waves RWm at indicated aspect ratios H/R.

FIG. 3. (Color online) Contours of the axisymmetric components of the vorticity for Re¼ 2300 and H/R¼ 3.5. There are15 positive (solid/red) and 15 negative (dashed/yellow) contour levels in the range [�1.0, 1.0] for each plot.


our numerical results for H/R¼ 4.0 are completely consistent with the experimental findings, rais-ing questions about why the nonlinear simulations reported in Ref. 2 are in disagreement, given

the similarities in spectral solution technique and resolution used in the present study. We do note

that for these problems near the onset of three-dimensional flow, transients are very long, requir-

ing temporal evolutions of multiple viscous times.

Beyond comparing onset data between our simulations and the experiments, we now compare

the three-dimensional structure of the flow solutions with the PIV measurements from the experi-

ments.4,6,7 The axisymmetric components of the three components of vorticity, (n0,g0,f0), of thestates at (Re,H/R)¼ (2300,3.5), (2900,4.6), and (3100,5.3) are shown in Figs. 3–5, respectively.Being so close to the bifurcation, these are very close to the corresponding vorticity of the basic

state. These have features that are very much like those found in the basic states at lower H/R(compare with vorticity structures in Ref. 22 for H/R¼ 2.5 where the basic state loses stability toaxisymmetric time-periodic flow, which is subsequently unsteady to modulated rotating

waves,23–25 and with Ref. 26 where at H/R¼ 1.58 the basic state is simultaneously unstable tom¼ 0 unsteady axisymmetric flow and m¼ 2 rotating wave states). The point is that the basic statefeatures do not qualitatively change with H/R (the flow is essentially stretched in the axial direc-tion by a factor dependent on H/R), and they may have primary instability to either m¼ 0 orm= 0, or simultaneously to both. This was also borne out in the linear stability analysis of Ref. 8.

Instead of looking at the unstable basic state, we now consider the three-dimensional perturba-

tions of the bifurcating states. These are obtained by subtracting the axisymmetric components

(shown in Figs. 3–5) from the full three-dimensional solution. Very close to the onset of instability,

these perturbation fields are very close to the bifurcation eigenfunctions. Columns (a) and (b) of



Fig. 6 show the axial vorticity f and its three-dimensional perturbation f� f0 at z¼ 0.75H/R forRW3 and RW2 and at z¼ 0.5H/R for RW4. Column (a) should be compared with the experimentalPIV data taken at the same points in parameter space (see the top row of Fig. 8 in Ref. 7). So, one

must conclude given the qualitative and quantitative agreement on critical Re, frequency and wave-number as well as the spatial structure of the nonlinear three-dimensional states between our simula-

tions and the experiments that both are describing the same nonlinear flows. Our analysis shows that

these states are in fact rotating waves, of the same form as previously found by ourselves and others

in shorter cylinders. A fuller view of their perturbation three-dimensional structure is presented in

Fig. 7, showing isosurfaces of f� f0 at 60.01 for RW3, 60.005 for RW2, and 60.002 for RW2.We see that the perturbation axial vorticity is quite localized and comes in m pairs of spiraled struc-tures whose pitch angle varies considerably with axial distance, becoming more parallel to the axis

with increased distance from the stationary top endwall. These rotate uniformly without change of

shape (one complete turn in a period 2pm/x, where x is listed in Table I).In the analysis of their experimental data,4,7 they assume that the enclosed swirling flow is

made up of a combination of helical modes plus an axisymmetric “assigned” flow. It is not

yet clear if these helical modes form a compete basis, even for swirling flows unbounded in

the axial direction. Helical symmetry means that the flow is invariant to Hl,k whose action on thevelocity is



Hl;kðu; v;wÞðr; h; z; tÞ ¼ ðu; v;wÞðr; hþ k=l; zþ k; tÞ; (5)

where 2pl is the helix pitch. Helical symmetry requires an unbounded axial direction, and this isclearly not the case for the flows in the enclosed finite cylinder, and the question is whether this is

even true locally. We have seen above that the three-dimensional flows that result from the insta-

bility of the steady axisymmetric basic state, both in the experiments and the simulations, are

rotating waves, i.e., three-dimensional flows that are steady in an appropriate frame of reference.

At the bifurcation, the continuous SO(2) symmetry, R/ and Us, is broken and the bifurcating rotat-ing wave solution has discrete symmetry Zm¼R2p,R2p/2,…,R2p/m where m is the azimuthal wave-number of the solution. In time, the rotating wave is 2p/x-periodic, but when viewed in anappropriate rotating frame of reference it is stationary, i.e., for a rotating wave of azimuthal wave-

number m,

uRWmðr; h; z; tÞ ¼ uRWmðr; h� 2p=m; z; tþ 2p=xÞ: (6)

They4,7 assume that the observed three-dimensional instabilities are due to the instability of a con-

centrated central vortex, their so-called assigned flow, which they assume has vorticity of the form,

ðna; ga; faÞ ¼ 0;r

lfa;

C0pR2o

exp � r2

R2o

� �� ; (7)

where C0, R0, and l are fitting parameters representing the vortex circulation, its core radius and a he-lix pitch. The flow depends only on r, it is axisymmetric, axial translation symmetric, and steady.Note also that the radial vorticity na is assumed to be zero, but in the present problem the radial

FIG. 6. (Color online) Contours of (a) axial vorticity f, (b) f� f0, and (c) f� fa, for (top row) Re¼ 2300, H/R¼ 3.5 atz¼ 0.75H/R with f [ [�0.7, 0.7], f� f0 [ [�0.1, 0.1], and f� fa [ [�0.5, 0.5]; (middle row) Re¼ 2900, H/R¼ 4.6 atz¼ 0.75H/R with f [ [�0.5, 0.5], f� f0 [ [�0.06, 0.06], and f� fa [ [�0.5, 0.5]; and (bottom row) Re¼ 3100, H/R¼ 5.3at z¼ 0.5H/R with f [ [�0.36, 0.36], f� f0 [ [�0.024, 0.024], and f� fa [ [�0.8, 0.8]; red/solid contours are positive andyellow/dashed contours are negative. To obtain f� fa, we have used C0¼ 0.25 and Ro¼ 0.23.


vorticity of the basic flow is non-trivial, and its strength is only a little weaker than the azimuthal or

axial vorticity strength (see Figs. 3–5). They proceed to analyze their experimental PIV data by

selecting some values for C0, R0, and l (it is not clear what criteria is used to make this selection),and then at a specific axial location (z¼ 0.75H/R), they subtract fa from the PIV-measured f. Theresult shows at about mid-radius m vortex structures all of the same sign which are interpreted asbeing helical vortices, multiplets. If we perform the same exercise on our rotating wave solutions

(column (a) of Fig. 6), we obtain f� fa which are shown in column (c) of Fig. 6. These contours off� fa are in excellent agreement with the corresponding experimental contours shown in the bottomrow of Fig. 8 in Ref. 7. Similar experimental flows were earlier analyzed4 with similar results, but

the fitting parameters used were not reported. We have tried a variety of values of fitting parameters,

and the results are not too sensitive to their choice (within reason). What happens is that by subtract-

ing the assigned flow fa, which is a Gaussian distribution that approaches zero at about mid-radius,from the full f amounts to giving the three-dimensional perturbation fp ¼f� f0 a strong bias. Now,fp consists primarily of alternating patches of positive and negative axial vorticity of comparablestrengths (see column (b) of Fig. 6). By subtracting fa, the resulting field has a reduced positive per-turbation and an increased negative perturbation, and by a judicious selection of R0 and C0 one canlocate where this bias is applied in order to produce what looks like multiplets (column (c) of

Fig. 6). Note that in this exercise, the parameter l, which is related to the helix angle, plays no role.In their interpretation of instability, they assume that the assigned flow has a constant helix

angle, and further that the helix angles of the velocity v/w and the vorticity g/f are the same. Wenote that the ratio of these two angles plays a very important role in vortex breakdown.27

IV. CONCLUSIONS

All of the states examined in the experiments4,7 are rotating waves that bifurcate supercriti-

cally at symmetry-breaking Hopf bifurcations from the steady axisymmetric basic state. They

FIG. 7. (Color online) Isosurfaces of the perturbation axial vorticity for (a) Re¼ 2300, H/R¼ 3.3 with isolevelsf� f0¼ 0.01 (red/dark) and f� f0¼�0.01 (yellow/light), (b) Re¼ 2900, H/R¼ 4.6 with isolevels f� f0¼ 0.005 (red/dark) and f� f0¼�0.005 (yellow/light), (c) Re¼ 3100, H/R¼ 5.3 with isolevels f� f0¼ 0.002 (red/dark) andf� f0¼�0.002 (yellow/light).


have in fact verified that their experiments are consistent with this by detailed comparisons with

linear stability analysis.5 Now, we have provided fully nonlinear numerical solutions of these

bifurcating rotating waves and have shown that they agree very well with the experiments (using

direct comparisons of the axial vorticity at the axial location where the PIV data was taken, the

critical Re for their onset, and their frequency as compared to the experimentally measured fre-quency using LDA). There is no doubt that these rotating waves in large H/R are the same as thosedescribed in shorter cylinders with H/R� 1.5,26 and are related to the modulated rotating wavesresulting from three-dimensional instabilities of the axisymmetric time-periodic states at interme-

diate H/R.1,9,24,25

ACKNOWLEDGMENTS

This work was completed during a sabbatical visit to Kyungpook National University, whose

hospitality is warmly appreciated, and partially funded by the Korea Science and Engineering

Foundation WCU Grant No. R32-2009-000-20021-0.

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2E. Serre and P. Bontoux, “Vortex breakdown in a three-dimensional swirling flow,” J. Fluid Mech. 459, 347 (2002).3T. T. Lim and Y. D. Cui, “On the generation of a spiral-type vortex breakdown in an enclosed cylindrical container,”Phys. Fluids 17, 044105 (2005).

4V. L. Okulov, I. V. Naumov, and J. N. Sorensen, “Self-organized vortex multiples in swirling flow,” Tech. Phys. Lett. 34,675 (2008).

5J. N. Sorensen, A. Y. Gelfgat, I. V. Naumov, and R. F. Mikkelsen, “Experimental and numerical results on three-dimen-sional instabilities in a rotating disk-tall cylinder flow,” Phys. Fluids 21, 054102 (2009).

6I. V. Naumov, V. I. Okulov, and J. N. Sorensen, “Diagnostics of spatial structures of vortex multiples in a swirl flow,”Thermophys. Aeromechanics 17, 551 (2010).

7J. N. Sorensen, I. V. Naumov, and V. L. Okulov, “Multiple helical modes of vortex breakdown,” J. Fluid Mech. 683, 430(2011).

8A. Y. Gelfgat, P. Z. Bar-Yoseph, and A. Solan, “Three-dimensional instability of axisymmetric flow in a rotating lid-cylinder enclosure,” J. Fluid Mech. 438, 363 (2001).

9J. M. Lopez, “Rotating and modulated rotating waves in transitions of an enclosed swirling flow,” J. Fluid Mech. 553,323 (2006).

10J. D. Crawford and E. Knobloch, “Symmetry and symmetry-breaking bifurcations in fluid dynamics,” Annu. Rev. FluidMech. 23, 341 (1991).

11R. E. Ecke, F. Zhong, and E. Knobloch, “Hopf-bifurcation with broken reflection symmetry in rotating Rayleigh-Bénardconvection,” Europhys. Lett. 19, 177 (1992).

12E. Knobloch, “Bifurcations in rotating systems,” in Lectures on Solar and Planetary Dynamos, edited by M. R. E. Proctorand A. D. Gilbert (Cambridge University Press, Cambridge, U.K., 1994), p. 331.

13E. Knobloch, “Symmetry and instability in rotating hydrodynamic and magnetohydrodynamic flows,” Phys. Fluids 8,1446 (1996).

14G. Iooss and M. Adelmeyer, Topics in Bifurcation Theory and Applications, 2nd ed. (World Scientific, Singapore, 1998).15P. Chossat and R. Lauterbach, Methods in Equivariant Bifurcations and Dynamical Systems (World Scientific, Singapore,

2000).16I. Mercader, O. Batiste, and A. Alonso, “An efficient spectral code for incompressible flows in cylindrical geometries,”

Comput. Fluids 39, 215 (2010).17J. M. Lopez and F. Marques, “Centrifugal effects in rotating convection: nonlinear dynamics,” J. Fluid Mech. 628, 269

(2009).18J. M. Lopez, F. Marques, A. M. Rubio, and M. Avila, “Crossflow instability of finite Bödewadt flows: transients and spi-

ral waves,” Phys. Fluids 21, 114107 (2009).19J. M. Lopez and F. Marques, “Sidewall boundary layer instabilities in a rapidly rotating cylinder driven by a differentially

co-rotating lid,” Phys. Fluids 22, 114109 (2010).20J. M. Lopez and F. Marques, “Instabilities and inertial waves in a librating cylinder,” J. Fluid Mech. 687, 171 (2011).21J. N. Sorensen, I. Naumov, and R. Mikkelsen, “Experimental investigation of three-dimensional flow instabilities in a

rotating lid-driven cavity,” Exp. Fluids 41, 425 (2006).22J. M. Lopez, “Axisymmetric vortex breakdown: Part 1. Confined swirling flow,” J. Fluid Mech. 221, 533 (1990).23J. L. Stevens, J. M. Lopez, and B. J. Cantwell, “Oscillatory flow states in an enclosed cylinder with a rotating endwall,”

J. Fluid Mech. 389, 101 (1999).24H. M. Blackburn and J. M. Lopez, “Symmetry breaking of the flow in a cylinder driven by a rotating end wall,” Phys.

Fluids 12, 2698 (2000).25H. M. Blackburn and J. M. Lopez, “Modulated rotating waves in an enclosed swirling flow,” J. Fluid Mech. 465, 33

(2002).26F. Marques, J. M. Lopez, and J. Shen, “Mode interactions in an enclosed swirling flow: A double Hopf bifurcation

between azimuthal wavenumbers 0 and 2,” J. Fluid Mech. 455, 263 (2002).27G. L. Brown and J. M. Lopez, “Axisymmetric vortex breakdown: Part 2. Physical mechanisms,” J. Fluid Mech. 221, 553

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