Ring bursting behavior en route to turbulence in quasi two-dimensional Taylor-Couette flows

3.7 MB·English

by  Sebastian Altmeyer

Ring bursting behavior en route to turbulence in quasi two-dimensional
Taylor-Couette flows
Sebastian Altmeyer,1,∗ Younghae Do,2,† and Ying-Cheng Lai3
1Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria
2Department of Mathematics, KNU-Center for Nonlinear Dynamics,
Kyungpook National University, Daegu, 702-701, Korea
3School of Electrical, Computer and Energy Engineering,
Arizona State University, Tempe, Arizona 85287, USA
(Dated: January 12, 2015)
We investigate the quasi two-dimensional Taylor-Couette system in the regime where the radius
5 ratio is close to unity - a transitional regime between three and two dimensions. By systematically
1
increasingtheReynoldsnumberweobserveanumberofstandardtransitions,suchasonefromthe
0
classical Taylor vortex flow (TVF) to wavy vortex flow (WVF), as well as the transition to fully
2
developed turbulence. Prior to the onset of turbulence we observe intermittent burst patterns of

n
localizedturbulentpatches,confirmingtheexperimentallyobservedpatternofveryshortwavelength
a
bursts (VSWBs). A striking finding is that, for Reynolds number larger than the onset of VSWBs,
J
a new type of intermittently bursting behaviors emerge: burst patterns of azimuthally closed rings
8
of various orders. We call them ring-burst patterns, which surround the cylinder completely but

] remainlocalizedandseparatedbynon-turbulentmostlywavystructuresintheaxialdirection. We
n useanumberofquantitativemeasures,includingthecross-flowenergy,tocharacterizethering-burst
y
patternsandtodistinguishthemfromthebackgroundflow. Thering-burstpatternsareinteresting
d
because it does not occur in either three- or two-dimensional Taylor-Couette flow: it occurs only
-
u in the transition, quasi two-dimensional regime of the system, a regime that is less studied but
l certainly deserves further attention so as to obtain deeper insights into turbulence.
f
.
s
c PACSnumbers: 47.20.Ky,47.32.cf,47.54.-r
i
s
y
h I. INTRODUCTION mensions [6]. To gain new insights into turbulence, it
p is of interest to study the transitional regime between
[ three and two dimensions. In such a regime, properties
Characteristics of turbulence in three- and two-
1 of both three- and two-dimensional flows are relevant,
dimensional flows are typically quite distinct. For ex-
v and one naturally wonders whether there are any unex-
ample, in three-dimension flows, the energy spectrum
2
pected features associated with, for instance, transition
0 of fully-developed turbulence obeys the well-known Kol-
to turbulence [7, 8]. The purpose of this paper is to
0 mogorov 1941 scaling law [1] of k−5/3, while in two
2 report a new phenomenon in a paradigmatic quasi-two-
dimensions the scaling [2, 3] is k−3. Although two-
0 dimensionalsystem: theTaylor-Couetteflow[9]withgap
. dimensionalflowsystemsoffergreatadvantagesfromthe
1 betweentheinnerandoutercylinderssonarrowthatthe
standpointofcomputationandmathematicalanalysisas
0 system is neither completely three-dimensional nor ex-
5 compared with three-dimensional flows, the former are
actly two-dimensional. In fact, this regime has not been
1 not merely a kind of toy model of turbulence. In fact,
investigated systematically previously. The new type of
: two-dimensional turbulence is relevant to the dynamics
v
intermittent dynamics occurs en route to turbulence as
i ofoceaniccurrents,originoftheozoneholethroughmix-
X the Reynolds number is increased.
ing of chemical species in the polar stratosphere, the ex-
r The Taylor-Couette system, a flow between two con-
a istenceofpolarvortex,strongeddymotionssuchastrop-
centric rotating cylinders, has been a paradigm in the
ical cyclones, and other large-scale motions of planetary
study of complex dynamical behaviors of fluid flows, es-
atmospheres [4, 5].
pecially turbulence [10–18]. The flow system can exhibit
Turbulence is arguably one of the most difficult prob-
a large variety number of ordered and disordered behav-
lems in science and engineering, and the vast literature
iors in different parameter regimes. For cylinders of rea-
on turbulence was mostly focused on three or two di-
sonablelength,theeffectivedimensionalityofthesystem
is determined by a single parameter - the ratio between
the radii of the two cylinders. If the ratio is markedly
∗Electronicaddress: [email protected] less than unity, the flow is three dimensional. As the ra-
†Electronicaddress: [email protected] tio approaches unity, the flow becomes two dimensional.
2
Most previous studies focused on the setting where the turbulence,signifying a new route to turbulenceuniquely
radius ratio is below, say about 0.9, the so-called wide- for quasi two-dimensional Taylor-Couette flows. To our
gap regime [13], or when the ratio approaches unity so knowledge, there was no prior report of ring bursts pat-
that the geometry is locally planar, resulting in an effec- terns or likes. This is mainly due to the fact that the
tively two-dimensional Couette flow [19]. Here, we are quasi two-dimensional regime is a less explored territory
interested in the narrow-gap case, where the radius ra- inthegiantlandscapeofturbulenceresearch. Itwouldbe
tio is close to unity but still deviates from it so that the interesting to identify precursors to turbulence in quasi
flow is quasi two-dimensional. To be concrete, we fix the two-dimensional flow systems in general.
radius ratio to be 0.99 and, without loss of generality, In Sec. II, we outline our numerical method and
restrict our study to the case where the outer cylinder describe a number of regular states in quasi two-
is stationary. In fact, regardless of whether the outer dimensionalTaylor-Couetteflows. InSec.III,wepresent
cylinder is rotational or stationary, transition to turbu- our main results: numerical confirmation of experimen-
lence can occur with increasing Reynolds number. In tally observed VSWBs and more importantly, identifi-
particular, for systems of counter-rotating cylinders, an cationandquantitativeconfirmationofintermittentring
early work [10] showed that transition to turbulence can burstsasprecursorstoturbulence. InSec.IV,wepresent
be sudden as the Reynolds number is increased through conclusions and discussions.
a critical point. For fixed outer cylinder, the transition
from laminar flow to turbulence can occur through a se-
quence of instabilities of distinct nature [14, 20]. II. NUMERICAL METHOD AND BASIC
DYNAMICAL STATES OF QUASI
For Taylor-Couette system of counter-rotating cylin-
TWO-DIMENSIONAL TAYLOR-COUETTE
ders, spatially isolated flow patterns, the so-called lo-
FLOW
calized patches, can emerge through the whole fluid do-
main and decay [13, 21]. Depending on the parameters,
A. Numerical method
the localized patches can be laminar or more complex
patterns such as inter-penetrating spirals. In the wide-
The Taylor-Couette system consists of two indepen-
gap regime (three-dimensional flow), numerical simula-
dently rotating cylinders of finite length L and a fluid
tions [22] revealed the existence of so-called G¨ortler vor-
confined in the annular gap between the two cylinders.
tices [23], small scale azimuthal vortices that can cause
Weconsiderthesettinginwhichtheinnercylinderofra-
streaky structures and form herringbone-like patterns
diusR rotatesatangularspeedΩandtheoutercylinder
near the wall. Localized turbulent behaviors can arise i
of radius R is stationary. The end-walls enclosing the
when the G¨ortler vortices concentrate and grow at the o
annulusintheaxialdirectionarestationaryandthefluid
outflow boundaries of the Taylor vortex cell [22].
in the annulus is assumed to be Newtonian, isothermal
For narrow gap (quasi two-dimensional) flows, there
andincompressiblewithkinematicviscosityν. Usingthe
was experimental evidence of the phenomenon of very
gap width d = R −R as the length scale and the ra-
o i
short wavelength bursts (VSWBs) [14]. One contribu-
dial diffusion time (DT) d2/ν as the time scale, the non-
tion of our work is clear computational demonstration
dimensionalized Navier-Stokes and continuity equations
of VSWBs. Remarkably, we uncover a class of solutions
are
in quasi two-dimensional Taylor-Couette flow en route
to turbulence. These are localized, irregular, intermit- ∂ u+(u·∇)u=−∇p+∇2u, ∇·u=0, (1)
t
tently bursting, azimuthally closed patterns that man-
ifest themselves as various rings located along the ax- where u = (u ,u ,u ) is the flow velocity field in cylin-
r θ z
ial direction. For convenience, we refer to the states as drical coordinates (r,θ,z), the corresponding vorticity is
“ring-bursts.” The number of distinct rings can vary de- givenby∇×u=(ξ,η,ζ),andr isthenormalizedradius
pending on the parameter setting but their extents in for fluid in the gap (0 ≤ r ≤ 1). The three relevant pa-
theaxialdirectionaresimilar. Theringburstscanoccur rametersaretheReynoldsnumberRe=Ω R d/ν,thera-
i i
on some background flow that is not necessarily regu- diusratioR /R =0.99,andtheaspectratioΓ≡L/d=
i o
lar. For example, in typical settings the background can 44. The boundary conditions on the cylindrical surfaces
be wavy vortex flows (WVFs) with relatively high az- are of the non-slip type, with u(r ,θ,z,t) = (0,Re,0),
i
imuthal wave numbers. Because of the coexistence of u(r ,θ,z,t) = (0,0,0), where the non-dimensionalized
o
complex flow patterns, to single out ring bursts is chal- inner and outer radii are r = R /d and r = R /d,
i i o o
lenging, a task that we accomplish by developing an ef- respectively. Theboundaryconditionsintheaxialdirec-
fective mode separation method based on the cross-flow tion are u(r,θ,±0.5Γ,t)=(0,0,0).
energy. We also find that ring bursts are precursors to We solve Eq. (1) by using the standard second-order
3
time-splitting method with consistent boundary condi- Re is increased from this value, additional vortices ap-
tions for the pressure [24]. Spatial discretization is done pear, which enter the bulk from near the lids until they
viaaGalerkin-FourierexpansioninθandChebyshevcol- finally fill the whole annulus for Re value about 435 (ex-
location in r and z. The idealized boundary conditions perimentally the value is about [14] 437). For example,
are discontinuous at the junctions where the stationary weobserveaTVFwith22pairsofvorticeswithinthean-
end-walls meet the rotating inner cylinder. In experi- nulus, with the characterizing wavenumber of k =3.427,
ments there are small but finite gaps at these junctions andanadditionalpairofEkmanboundarylayervortices
where the azimuthal velocity is adjusted to zero. To near the top and bottom lids.
achieve accuracy associated with the spectral method, a Secondaryinstabilities-wavyvortexflow. Furtherin-
regularization of the discontinuous boundary conditions creasing Re, the TVF becomes unstable and is replaced
is implemented, which is of the form by wavy vortex flow (WVF), a kind of centrifugal insta-
bility that appears via a supercritical Hopf bifurcation,
u (r,θ,±0.5Γ,t)=Re{exp([r −r]/(cid:15))+exp([r−r ]/(cid:15))},
θ i o which exhibits a wavy-like modulation but the vortices
(2)
remain azimuthally closed. We find that the onset of
where (cid:15) is a small parameter characterizing the physical
WVF occurs for Re ≈ 473 (experimentally the value is
gaps. We use (cid:15) = 6 × 10−3. Our numerical method
about [14] 475). The WVF, as shown in Fig. 1(a), is as-
waspreviouslydevelopedtostudytheend-walleffectsin
sociatedwithrelativelyhighvaluesoftheazimuthalwave
theTaylor-Couettesystemwithco-andcounter-rotating
number m. In general, WVF states consisting of many
cylinders[25,26]. Inthepresentworkweuseupton =
r azimuthalmodescanoccur. Forexample,theWVFpat-
50 and n = 500 Chebyshev modes in the radial and
z tern in Fig 1(a) contains m=39 modes, which we name
axialdirections,respectively,andupton =100Fourier
θ as WVF . We find that this particular WVF state is in
modesintheazimuthaldirection. Thetimestepischosen 39
factaglobalbackgroundflowstateinawiderangeofRe
to be δt=10−6.
values.
The TVF and WVF patterns occur regardless of
B. Qualitative description of basic dynamical states whether the flow is three dimensional or quasi two-
dimensional, although the critical values of Re for the
1. Low order instabilities onsets of these flow patterns depend on the system de-
tails.
Inthewide-gap,three-dimensionalTaylor-Couettesys-
tem, various flow patterns and their bifurcation behav-
iors are relatively well understood [10, 13]. In our quasi 2. High-order instabilities
two-dimensional setting, a number of known low-order,
non-turbulent instabilities still occur, which constitute Very short wavelength bursts and localized patches.
thebackgroundflowuponwhichanewtypeofringburst For the commonly studied [13], wide-gap, three-
structures emerge. Here we describe these low-order in- dimensional Taylor-Couette system, e.g., of radius ra-
stabilities. To visualize and distinguish qualitatively dif- tio 0.883, the typical sequence of solutions with increas-
ferent flow patterns, we use the contour plots of the az- ing Re values is as follows. As the stationary TVF be-
imuthal vorticity component η. comes unstable, time dependent WVF arises, followed
Primary instability - Taylor vortex flow. As the by the so-called modulated WVF [13], eventually lead-
Reynolds number Re is increased, the basic state, circu- ing to turbulent behaviors. However, in the narrow-gap,
larCouetteflow(CCF)becomesunstableandisreplaced quasi two-dimensional system, we have not observed the
by the classical Taylor vortex flow (TVF) that consists globaltransitionfromWVFtomodulatedWVF.Instead,
of toroidally closed vortices. TVFs with some increasing we find that very short wavelength bursts (VSWBs) ap-
numbers of vortices appear gradually over a large range pear directly after the onset of WVF without any other
ofRe, startingfromasingleTaylorvortexcellgenerated types of intermediate solutions. Onset of VSWBs is
bythemechanismofEkmanpumpingneartheendwalls. Re≈483,whichagreeswiththeexperimentallyobserved
Thisinitialcellcanappeareithernearthetoporthebot- onset value [14].
tom lid. In a perfectly circular geometry, initial Taylor As Re is increased, we observe localized patches (LPs)
vortex cells can occur simultaneously at both lids, but superposed on WVF. The pattern within LPs can be ei-
thisislesslikelyinrealisticsystemsduetotheinevitable ther wavy-like or turbulent, depending on the Re value
imperfections in the system. The onset of initial TVF (lower and high values for the former and the latter, re-
cellswasexperimentallyfoundforReabout[14]358. We spectively). In particular, LPs with wavy-like interior
find, numerically, that the onset value is about 356. As behaviors emerge, and for Re > 580 LPs with turbu-
4
(a) WVF for Re=500 therearecharacteristicdifferences. Forexample,VSWBs
39
appear randomly over the whole bulk fluid region with
seeminglyexpandingbehaviorsinalldirections, butring
bursts always remain localized in the axial direction. In
fact, the bursts are generated from the localized turbu-
lentpatchesthatgrowintheazimuthaldirectionasReis
increased. For sufficiently high values of Re, the patches
extend over one circumference, generating distinct ring
bursts that are separated from the flow patterns in the
rest of the bulk, as shown in Fig. 1(c). The character-
(b) Ring bursts (n=3) for Re=545 istics of the background flows in the regions surround-
ing the ring bursts depend strongly on Re. They can
range from wavy-like patterns (i.e., WVF ) to inter-
39
penetratingspiralsduetotheinteractionsamongvarious
azimuthalmodes,asshowninFig.1(c). Weobservering
burst patterns of different order n, as shown in Figs. 1
and 2). For example, we observe n ∈{1,2,3}. All these
states coexist but the probability for ring bursts with
larger values of n increases with Re. Along the axial
direction, the ring burst patterns can appear at any po-
(c) VSWB for Re=800 sition, except for the regions of Ekman vortices near the
lids, because strong boundary layer vortices prevent the
development of ring burst patterns.
While there is no apparent order associated with the
flow patterns within the ring bursts, the flow in the re-
gionsinbetweenexhibitaclearsignatureofWVF pat-
39
tern. The ring bursts can spontaneously break up and
disappear. Depending on the number of ring bursts, the
transienttimethatittakesforthebursttodecayintolo-
calized patches can be relatively long. In addition, there
can be transitions between patterns with distinct num-
bersofringbursts. Forexample, supposethereisaring-
FIG. 1: (Color online) Distinct flow patterns in quasi
burst pattern of order n = 3. If one ring disappears,
two-dimensional Taylor-Couette system. Contours of
a new ring-burst region can appear and grow. Transi-
the azimuthal vorticity component η with (left column,
tions between patterns with either increasing or decreas-
η ∈ [−400,400]) and without (right column, η(m (cid:54)= 0) ∈
[−200,200]) the axisymmetric component on an unrolled ing numbers of ring bursts have been observed.
cylindricalsurfaceatthemidgap(r=d/2)fordifferentflows For Re>850, we find that VSWBs fill the whole bulk
andvaluesofRe. Red(darkgray)andyellow(lightgray)col- fluid region, as shown in Fig. 1(c). In fact, for Re>900,
ors correspond to positive and negative values, respectively, the whole fluid region is saturated with VSWBs. These
and the black line indicates the zero contour.
numericalobservationsareinagreementwithexperimen-
tal findings [14]. Note, however, even when VSWBs fill
theinteriorofthebulkWVF-likepatternsalwaysappear
lent behaviors arise, which can either grow to VSBWs near the axial end-walls due to the fact that Ekman vor-
or decay slowly. The number of LPs depends on Re as tices can stabilize the boundary layers against turbulent
well. In general, the higher the value of Re the larger bursts. ThesizeoftheWVF-likeEkmanregionsdepends
the number of LPs appears. The patches are randomly on Re - for higher value of Re the smaller the WVF re-
distributed over the whole bulk length and are always gions near the lids appear, as can be seen from Fig. 1.
visibleexceptfortheregionnearthelidsinwhichstrong
Underlying flow patterns. The right panels in Fig. 1
Ekman vortices arise.
show contours for the same pattern as those for the left
Ring-burst patterns. When Re is increased above panels but without the underlying axisymmetric contri-
about 540, we discover a new type of localized, inter- bution. The resulting “reduced” flow pattern of WVF
39
mittent burst solution, the ring bursts that coexist with in Fig. 1(a) appears quite regular, indicating the exis-
VSWBs. While both types of solutions are localized, tence of the dominant wavy mode with m = 39 in the
5
(a) WVF (b) Ring bursts (c) Ring burst (d) VSWB modes, as shown in Fig. 1(b). The flow patterns in the
39 (n=3) (n=1)
Re=500 Re=545 Re=550 Re=800 central region (including that containing the three ring
bursts) exhibit a completely different behavior. In par-
ticular, with respect to contours η(m(cid:54)=0) the turbulent
ring bursts and the separating WVF pattern appear
39
indistinguishable. Thisfindingsuggestaquitestrongax-
isymmetric dominance of the flow even when ring bursts
arise, confirming that the bursting structure is ordered
in azimuthally closed rings around the cylinder with the
samem=0axisymmetry. ForturbulentflowsorVSWBs
in Fig. 1(c), none of the patterns has such an ordered
structure.
III. CHARACTERIZATION OF RING BURSTS
A. Angular momentum, azimuthal vorticity, and
modal kinetic energy
Figure 2 shows the isosurfaces of the angular momen-
tumrv(toprow)andazimuthalvorticityη(bottomrow)
for representative flow patterns at different values of Re.
The dominant axisymmetric contribution and high az-
imuthal wavenumber associated with WVF , as shown
39
in Fig. 2(a), serve as the background pattern for ring
bursts and VSWBs. The three turbulent, azimuthally
closed burst regions associated with the order-3 ring
burst pattern is distinctly visible, as shown in Fig. 2(b).
Within each burst region, both rv and η appear random
butthe(background)flowpatternsinbetweentheburst-
ing regions are remnant of the WVF pattern. With
39
increasing Re higher m modes emerge, but the separa-
tion between burst and non-burst regions persists. For
high values of Re, VSWBs arise, as shown in Fig. 2(d).
In this case, isosurface plots of rv and η exhibit random
flow patterns except for the regions near the lids due to
the Ekman boundary layers.
To better visualize and illustrate the similarity and
FIG. 2: (Color online) Angular momentum and vortic-
differences between ring burst and VSWB patterns, we
ity. Isosurfacesofrv (toprow)andη (bottomrow)forflows
presentinFig.3thebehaviorsoftheangularmomentum
atdifferentvaluesofRe(isolevelsshownforthetopandbot-
(top panels) and azimuthal vorticity (bottom panels) for
tom rows are rv = 80 and η = ±30, respectively). For clear
a segment of the bulk flow. The length of the segment
visualization,hereandinallfollowingthree-dimensionalplots
the radius ratio is scaled by the factor 100. is Γ/10. We see that, for the ring burst pattern (left
column), there is a dominant axisymmetric component,
but this is lost for VSWB. While the entire structure of
azimuthal direction. The black curves in the vertical the ring-burst pattern possesses an axisymmetry due to
direction represent the zero contour lines. At several itsazimuthaldominance(m=0),noapparentsymmetry
azimuthal positions, these lines narrow, indicating the exists in the interior of the burst regions. There are thus
emergence of a second but weaker mode, e.g., m = 8. two distinct spatial scales associated with ring bursts: a
For order 3 ring bursts, the flow patterns near the lids large scale determined by the axisymmetry and a small
are somewhat modified due to the presence of higher m scale present in the interior of the bursting regions.
6
(a) Ring bursts of order 3 (b) VSWB We next examine the modal kinetic energy defined as
Re=545 Re=800
(cid:88) (cid:90) 2π(cid:90) Γ(cid:90) ro
E := E = u u∗ rdrdzdθ, (3)
m m m
m 0 0 ri
where u (u∗ ) is the m-th (complex conjugated)
m m
Fourier mode of the velocity field. Due to time depen-
denceofthesolutionsitisnecessarytocalculatethetime
averagedkineticenergyE. Intheenergycharacterization
local quantities such as the radial velocity at mid-height
and mid-gap, u (d/2,0,Γ/2,t), are relevant.
r
Figures 4(a) and 4(b) show the time-averaged axisym-
metric energy E versus Re and its time evolutions for a
0
number of representative Re values, respectively. We see
that E increases monotonously with Re, regardless of
0
the nature of the bulk flow patterns. Likewise the time
FIG.3: (Coloronline)Three-dimensionalviewsofangu- evolution of the kinetic energy presents no clear indica-
lar momentum and vorticity. Magnitude of the angular tion of ring burst patterns. For example, for Re = 545,
momentum(toppanels)andazimuthalvorticity(bottompan- in the time interval [40,44] the flow pattern should be
els) for a flow segment of vertical length Γ/10 for ring bursts
order-3 ring burst, but the energy time evolution does
(left column) and VSWB (right column). The segment is
not show any signature of this burst pattern.
taken from the region containing the middle ring burst in
Fig. 2(b). Figures 5(a) and 5(b) show the power spectral den-
sity (PSD) of the radial velocity profile at the midgap
(a) 8 for WVF and ring burst, respectively. We see that the
PSD associated with the ring burst pattern (b) indicates
7
theexistenceofsignificantlyhighermodesthantheWVF
108E 6
0 pattern (a). In fact, the PSD of WVF in (a) shows a
5 39
strong peak at the frequency of about 19, which corre-
4
sponds to the dominant azimuthal mode (m=39). This
3
peakisstillpresentin(b)butitismuchbroadened,indi-
450 550 650 750
Re cating WVF as the background flow pattern for the ring
Re burst. Figure 5(c) shows the scaled PSD curves for sev-
(b) 4.3 br3 br2 br2 560i eral flow patterns for different values of Re. We see that
4.2
thePSDcurvesforthering-burstpatternsessentiallycol-
108E 4.1 550 lapse into one frequency band, but the PSD curve asso-
0 br br 545
2 3 ciatedwithWVF liesslightlybelowthoseofringburst
4 540 39
patterns. There is relatively large difference for small
535
3.9 frequencies but it becomes insignificant for higher fre-
0 10 20 30 40
quencies. This is further support for the role of WVF
39
t
in providing the skeleton structure for all ring burst pat-
terns.
FIG. 4: (Color online) Behaviors of the kinetic energy.
(a) Time-averaged (≈ 50 diffusion time) axisymmetric en-
ergy E versus Re and (b) time series of E for a number
0 0 B. Cross-flow energy
of representative values of Re. The time intervals in which
distinct ring-burst patterns last are indicated. For exam-
ple, for Re = 560, ring burst pattern of order 3 is present FromFig.4(b),weseethatthemodalkineticenergyis
for 2.5 < t < 6.9, and that of order 2 can be found for noteffectiveatdistinguishingringburstsfromotherbasic
9.2 (cid:46) t (cid:46) 15.8 and 42.6 (cid:46) t (cid:46) 47.1. For Re = 545, order-2 flow patterns. A suitable and commonly used quantity
ring burst pattern appears for 0.5 (cid:46) t (cid:46) 4.5 and the order-3 inthestudyoffluidturbulenceistheso-calledcross-flow
pattern is present for 40.3(cid:46)t(cid:46)44. However, the ring burst
energy,wheretypicallysmallvaluesindicateaflowbeing
patterns are not reflected in E (t).
0 laminar, large values corresponding to turbulence.
7
(a) (a)
(b)
(b)
(c)
FIG. 6: (Color online) Radial component of cross-flow
(c) energy. Spacetime plots of the radial component of the
cross-flow energy, Ecf,r(r,t) = (cid:104)u2 +u2(cid:105) , averaged over
r z A(r)
the surfaces A of a concentric cylinder at radius r for (a)
Re = 500, (b) Re = 545, and (c) Re = 560. Red (yellow)
color indicates high (low) energy with contours defined as
∆Ecf,r = 5×105. The maximum energy values in (a-c) are
approximately8.1×105,8.5×106,and1.2×106,respectively.
For better visualization the radial gap width is magnified by
a factor of 500 (the same for Fig. 7 below).
FIG. 5: (Color online) Power spectral density (PSD). gle ring burst pattern (order 1) exists for 43≤t≤47, as
PSDcurvescalculatedfromtheradialflowcomponentu =
d/2 shown in Fig. 6(c).
u(d/2,0,Γ/2,t)for(a)WVF (Re=500)and(b)ringburst
39 Plots of the cross-flow energy exhibit two features.
of order 3 for Re = 545. (c) PSD curves scaled by the re-
Firstly, presence of the order-1 ring burst pattern is ac-
spective Reynolds number for different flow patterns. As the
companied by a significant increase in the radial cross-
numberofringburstsisincreased,thelocalpeaksinthePSD
flow energy as compared with that associated with the
curves become more pronounced.
background flow, e.g., WVF , as indicated by the uni-
39
form red (dark gray) regions. Secondly, the profile of
1. Radial component of cross-flow energy Ecf,r(r,t) for any ring burst pattern is approximately
symmetric with respect to the middle of the gap. It is
Theradialcomponentofthecross-flowenergyisgiven thus plausible that the ring-burst patterns are similar to
by [27] the so-called turbulent WVF state [22, 28–30] (see Sec.
IIIC). The symmetry may be a consequence of the quasi
Ecf,r(r,t)=(cid:104)u2+u2(cid:105) , (4)
r z A(r) two-dimensional nature of the flow. In all cases, the oc-
where(cid:104)·(cid:105) denotesaveragingoverthesurfaceofacon- currence of ring burst patterns enhances the magnitude
A(r)
centric cylinder at radius r. The cross-flow energy com- of the cross-flow energy.
ponent Ecf,r(r,t) measures the instantaneous energy as- Figure 7(a) shows a magnification of a segment of
sociatedwiththeradialandaxialvelocitycomponentsat Fig. 6(b) for t ∈ [0,1.8], which contains the emergence
radial distance r. Figure 6 shows the spacetime plots of and disappearance of order-3 ring burst for Re = 545,
Ecf,r(r,t) over the time period of 50 DT for three values occurring at t≈0.45 and t≈1.35, respectively. Outside
of Re. We observe the temporal emergence and disap- thistimeintervaltheflowisWVF ,whichconstitutesa
39
pearance of various ring-burst patterns in the bulk. For nearly uniform background without any significant vari-
example, for Re = 545, the order-3 ring burst pattern ationinEcf,r(r,t). ThespatialdistributionofEcf,r(r,t)
exists for 40 ≤ t ≤ 44, and the order-2 pattern appears exhibits a kind of symmetry about r = 0.5. Figure 7(b)
for 0.5 ≤ t ≤ 4.5, as shown in Fig. 6(b). For Re = 560, shows the average cross-flow energy, (cid:104)Ecf,r(cid:105) , as a func-
r
an order-3 pattern appears for 2.5 ≤ t ≤ 7.0, and a sin- tionoftimet,correspondingtotheemergenceanddisap-
8
(a)
(a)
(b) 15
>r10
Ecf
<
60 5 (b)
1
0
0 0.5 1 1.5 2
t
FIG.7: (Coloronline)Magnifiedviewoftheradialcom-
ponent of the cross-flow energy. (a) Magnification of a
segmentofFig.6(b)fort∈[0,1.8]illustratingtheemergence
anddisappearanceoforder-3ringburstpatternforRe=545.
The contours are defined through ∆Ecf,r =5∗105, and the
maximumenergyvalueisabout8.288×106. (b)Averagedra-
FIG.8: (Coloronline)Onset of burst patterns. (a)Burst
dial cross-flow energy (cid:104)Ecf,r(cid:105) versus t for order-3 ring burst
r fraction f as a function of Re. A linear fit indicates that
b
forRe=545. ThenearlyconstantbackgroundflowisWVF
39 onset of VSWBs occurs for Re ≈ 480 (the experimental on-
with (cid:104)Ecf,r(WVF )(cid:105) ≈2×10−6.
39 r set[14]isRe≈483). (b)Themaximumaverageradialenergy
(cid:104)Ecf,r(cid:105) [Eq. (4)] versus Re for ring bursts of different order
r
(n∈{1,2,3}). The linear behavior suggests that ring bursts
pearance of the ring burst pattern. We observe a signifi- are result of a forward bifurcation, the onset of which occurs
cant enhancement of the average radial energy over that for Re ≈ 537. In this figure, the average energy of the un-
c
of the background flow ((cid:104)Ecf,r(cid:105)r ≈ 2×10−6. In fact, in derlayingbackgroundflow,(cid:104)Ecf,r(cid:105)r,hasbeensubtractedoff.
the time interval where the ring burst exists, the maxi-
mumvalueoftheaverageradialenergyisaboutoneorder
bursts does not depend on n. This is evidence that ring
ofmagnitudelargerthanthatofthebackgroundflow. In
bursts emerge after VSWBs.
general, the maximum energy depends on the number n
(order)ofringburstsintheannulus,wherealargervalue
of n corresponds to larger value of the maximum energy. 2. Axial component of cross-flow energy
The emergence and development of any type of burst
patterns with increasing Re can be conveniently charac- The axial component of the cross-flow energy is
terized using the quantity f , the percentage of the an-
B
nulus showing bursts in the spacetime plot. Figure 8(a) Emcf,z(z,t)=(cid:104)(ur)2m+(uz)2m(cid:105)A(z). (5)
shows f versus Re, where we observe approximately
B where A(z) stands for averaging over the radial and az-
a linear behavior for Re (cid:46) 900, and the transition to
imuthal variables on the surface of a disc at a fixed axial
bursts occurs for Re ≈ 480, in agreement with the on- cf,z
positionz. Figure9showsthetime-averagedvalueE
set of VSWB [14]. Onset of ring bursts can be revealed m
for the order-3 ring burst pattern for different azimuthal
by examining the maximum value of the average radial
cf,z
modes. TheaxisymmetriccomponentE isdominant.
energy, (cid:104)Ecf,r(cid:105) , versus Re, as shown in Fig. 8(b). Re- 0
r
Nearthecenteroftheringburstregionintheaxialdirec-
gardlessoftheordernofringbursts, thereisalinearin-
cf,z
crease in (cid:104)Ecf,r(cid:105)r with Re, suggesting a type of forward tion,thevaluesofE0 aresmallerthanthosearoundthe
edges. However,thecontributionsfromallhighermodes,
bifurcation. Calculations of the flow amplitudes show
cf,z
a square-root type of scaling behavior with increasing i.e., Em for m>0, have similarly small magnitudes as
cf,z
parameter difference from the “critical” point, providing compared with E , as shown in Fig. 9(b). In fact, for
0
furthersupportfortheforwardnatureofthebifurcation. highermmodes,theaxialcomponentsoftheircross-flow
Duetostronglocalizationoftheringbursts, theslopeof energies are randomly distributed over z and they are
the linear scaling behavior depends on the ring-burst or- not indicators of any appreciable difference between the
der n. Nonetheless, the onset value Re ≈ 537 of ring background flow and the ring burst pattern.
c
9
(a) (a)
(b) (b)
FIG. 9: (Color online) Axial component of cross-flow FIG. 10: (Color online) Characterization of axial cross-
energy. Associated with the order-3 ring burst pattern for flow energy based on mode separation. (a) Varia-
Re=545,(a)timeaveragedaxisymmetricenergycomponent tion of the axial cross-flow energy component Ecf,z(z) =
m
Ec0f,z and(b)energycontributionsEcmf,z fromselectedhigher Ebcf,z(z)+Ewcf,z(z)versusthecutoffwavenumbermc fortwo
azimuthal m modes. In (b), additional axial average energy axial positions in the burst region (z = 0.3Γ and 0.45Γ, cir-
values are indicated. cles) and in the background region (z = 0.15Γ and 0.35Γ,
squares). (b)SpatialvariationsofEcf,z(z)=(cid:80)mc−1Ecf,z(z)
b m=1 m
and Ecf,z(z)=(cid:80)M Ecf,z(z) with z for m =20 (circles)
3. Mode separation of axial cross-flow energy and 30w(squares).m=Rmegcionms above (below) eacch curve indi-
cate E (E ). The two horizontal dashed lines give the 80%
b w
To better characterize the ring-burst patterns in rela- threshold of the maximum values of [E (m = 20)] (lower)
w c
tion to the background wavy-like and general burst pat- and [Ew(mc =30)] (upper), respectively. In the calculations
terns,wedeviseamethodbasedontheideaofmodesep- thezeromodecontributionisexcluded. Duetonormalization
with E the sum must be unity.
aration. Since a burst pattern includes modes of higher M
azimuthal wave numbers, we can decompose the axial
component of the cross-flow energy Ecf,z(z,t) into two
m off wavenumber m . We observe that the curves for
c
distinct subcomponents:
z positions within the burst region (circles, e.g, at
m(cid:88)c−1 (cid:88)M z = 0.3Γ and0.45Γ, Figs. 1 and 9) are higher than
Ecf,z(z)+Ecf,z(z)= Ecf,z(z)+ Ecf,z(z), (6) those in the wavy-like background (squares, e.g., at
w b m m
m=1 m=mc z = 0.15Γ and 0.35Γ), where the former exhibit a rapid
whereEcf,z(z)andEcf,z(z)denotetheaxialcomponents decrease in the energy to collapse with the latter for mc
w b about39. ThevariationsofE andE alongtheannulus
of the cross-flow energy associated with the background b w
lengthfortwodifferentcutoffwavenumbers[m =20,30,
wavy-like and burst patterns, respectively, and m is c
c
asindicatedbytheverticallinesinFig.10(a)]areshown
some cutoff mode number. Note that the axisymmet-
in Fig. 10(b). Neglecting the differences in their mag-
ric component of the cross-flow energy is excluded be-
nitudes, the variations show qualitatively similar behav-
cause it is significantly larger than all other components
iors. Using a cutoff level with 80% of the maximum of
[Figs. 9(a,b)]. We choose the normalization factor to be
the background contribution [E (m )], we find a good
the total cross-flow energy for all modes except m=0: w c
agreementwiththevisibleenergythresholdsbetweenthe
M background and burst regions (cf. Fig. 1). We thus see
(cid:88)
Ecf,z = Ecf,z. (7)
M m that,throughsomeproperchoiceofthecutoffmodenum-
m=1 ber,theburstandnon-burstregionscanbedistinguished
Figure 10(a) shows the basic E and the burst contri- by examining the axial cross-flow energy variations asso-
w
bution E at different axial positions versus the cut- ciated with the cutoff mode.
b
10
(a) two-pair Taylor vortices that constitute four single vor-
texcells,whichholdsforallring-burstflowsthatwehave
succeeded in uncovering. Analogous to the behavior of
the cross-flow energy, this behavior is indicative of that
of turbulent WVFs [22, 28–30]. We note that the size of
onlyonepairofTaylorvortices(twocells)istoosmallto
account for the observed range of axial expansion. This
isconsistentwiththeformationprocessofringbursts. In
particular, any localized turbulent patch, after its gener-
ation, first expands in the axial direction (to four cells)
(b)
beforegrowingintheazimuthaldirection. Wheneverthe
burst region has expanded to a larger size in the axial
direction, the closed ring structure is destroyed, leading
to VSWBs.
IV. CONCLUSIONS AND DISCUSSIONS
This paper provides a comprehensive numerical study
of the quasi two-dimensional Taylor-Couette systems of
FIG. 11: (Color online) Axial spacing of ring bursts. (a)
radius ratio close but not equal to unity, a regime that
SPSD of the radial velocity u at the midgap for Re = 545,
r
has not been studied previously. The relevant control or
where k is the axial wave number. (b) Axial spacing of the
burstregionsfororder-nringbursts(n∈{1,2,3}),indicated bifurcation parameter is the Reynolds number Re. For
by the horizontal lines. The average spacing is about 0.079, small Re values, TVF initially arises near one of the lids
the axial spacing of two pairs of Taylor vortices constituting in a single cell, extends and finally fills the bulk inte-
four vortex cells. rior completely. As Re is increased the TVF loses its
stability and WVF emerges through a supercritical Hopf
bifurcation. WVFs with high azimuthal wavenumbers,
C. Axial spacing and localization e.g., m=39, constituteapersistentbackgroundflow, on
top of which more complex flow structures develop, such
The axial ranges of distinct ring burst regions are ap- as VSWBs that has been experimentally observed [14].
proximately identical. Figure 11(a) shows the spatial Themainresultofthispaperisthediscoveryofanew
power spectral density (SPSD) of the radial velocity u type of transient, intermittent state en route to turbu-
r
along a line at the midgap in the center of the bulk for lencewithincreasingRe: ringbursts. Theyemergewhen
the background flow and three types of ring-burst pat- localized turbulent patches grow and close azimuthally,
terns. We observe that the four curves coincide at the signifying a higher order instability. The ring bursts oc-
first sharp peak determined by the wave number associ- cur in various orders and the azimuthally closed burst
ated with the background flow WVF that consists of regions possess axial expansion surrounded by the WVF
39
24 vortex pairs in the axial direction. The correspond- backgroundflowormorecomplexflowsathighRevalues.
ing axial wavelength and wave number are λ ≈ 1.667 The ring burst patterns differ characteristically from the
and k ≈ 3.770, respectively. In addition, several broad- localized VSWB turbulent patches. The axial expansion
band peaks at higher wave numbers exist in the SPSD of ring burst patterns of different orders correlates well
of the ring burst patterns, corresponding to a number of with the size of the double-pair Taylor vortex structure,
short wavelength bursts within the respective patterns. providing plausible reason that the patterns are similar
The broadband nature at higher wave numbers are in- to the turbulent WVF structure [22, 28–30] that usu-
dicative of dominance of small-scale burst patterns. Fig- allyoccursthroughthewholeannulusathigherReynolds
ure11(b)showstheaxialspacingassociatedwithvarious numbers. We develop a mode separation method based
ring burst patterns as a function of Re, where each hor- on decomposing the cross-flow energy to distinguish be-
izontal dashed line represents the averaged axial spacing tween burst and non-burst patterns, where the former
for a particular ring burst pattern. The axial spacing andlatterareassociatedwithhigherandlowerazimuthal
is apparently independent of the value of Re and of the modes, respectively. We also find that the radial cross-
order n of the ring-burst pattern. The typical value of flow energy changes significantly in the presence of ring
axial expansion agrees well with the axial dimension of burst patterns. By examining the maximum value of