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Three-dimensional swirling flows in a tall cylinder driven by a
rotatingendwallJ. M. Lopez Citation: Phys. Fluids 24, 014101
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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.
014101-4 J. M. Lopez Phys. Fluids 24, 014101 (2012)
-
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. 4. (Color online) Contours of the axisymmetric components
of the vorticity for Re¼ 2900 and H/R¼ 4.6. There are15 positive
(solid/red) and 15 negative (dashed/yellow) contour levels in the
range [�1.0, 1.0] for each plot.
014101-5 Three-dimensional swirling flows in a tall cylinder
Phys. Fluids 24, 014101 (2012)
-
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
FIG. 5. (Color online) Contours of the axisymmetric components
of the vorticity for Re¼ 3100 and H/R¼ 5.3. There are15 positive
(solid/red) and 15 negative (dashed/yellow) contour levels in the
range [�1.0, 1.0] for each plot.
014101-6 J. M. Lopez Phys. Fluids 24, 014101 (2012)
-
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.
014101-7 Three-dimensional swirling flows in a tall cylinder
Phys. Fluids 24, 014101 (2012)
-
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).
014101-8 J. M. Lopez Phys. Fluids 24, 014101 (2012)
-
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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