European Journal of Mechanics B/Fluids 31 (2012) 158–167

Contents lists available at SciVerse ScienceDirect

European Journal of Mechanics B/Fluids

journal homepage: www.elsevier.com/locate/ejmflu

Transient dynamics of the wavy regime in Taylor–Couette geometryRoland Kádár ∗, Corneliu BalanREOROM Group, Politehnica University of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania

a r t i c l e i n f o

Article history:Received 12 August 2010Received in revised form10 June 2011Accepted 14 July 2011Available online 23 July 2011

Keywords:Taylor–Couette flowWavy regimePreferred states

a b s t r a c t

The paper is concerned with experimental investigations of the Taylor–Couette flow between two finitelength large aspect ratio concentric cylinders, with asymmetric end disturbances. The fluid sample isNewtonian and the working conditions are isothermal. The applied experimental protocol consists of aconstant inner cylinder angular velocity ramp (the outer cylinder is at rest), followed by a period of 40minwith constant velocity. During this experimental time, the dynamics of thewavy regime is investigated bymeans of the average azimuthal amplitude, number of vortices, wave frequency and power spectra timedependences. The dimensionless ramp rate is chosen above critical rates for quasi-stationary flows usedin similar studies. The results feature transient patterns of unstable wavy vortices that return in time tothe simple Taylor regime or settle for two preferential states within the wavy regime. The investigationof WVF dynamics is of great importance due to the applications such as Taylor–Couette mixing-reactiondevices.

© 2011 Elsevier Masson SAS. All rights reserved.

1. Introduction

The Taylor–Couette flow is the motion generated betweentwo concentric cylinders in relative rotational motion (Fig. 1(a)).This type of flow exhibits a wide spectrum of hydrodynamicinstabilities in the transition from laminar to turbulent flow, as therelative angular velocity between the cylinders is increased. Eversince their discovery by Taylor [1], the Taylor–Couette instabilitiesare of perpetual interest for understanding the phenomenologyassociated to flow stability.

Flow stability between concentric cylinders is determined bythe magnitude of Taylor number (Ta) that can be defined as

Ta = Re · Ti =ρRΩδ

η·ρΩδ2

η=

ρ2RΩ2δ3

η2, (1)

where Re and Ti are the Reynolds and temporal numbers respec-tively, and result from the dimensionless form of the Cauchy mo-mentum equation ( see [2] and also [3]); ρ is the mass densityof the fluid, R is the radius of the inner cylinder, Ω is the angu-lar velocity of the inner cylinder, δ is the gap width and η is thedynamic viscosity. In particular, the Re number can be sought asthe ratio between the viscous, ρδ2/η, and the inertial, δ/RΩ , char-acteristic time scales. Finite length geometries are characterized

∗ Corresponding address: Institut für Technische Chemie und Polymerchemie,Karlsruher Institut für Technologie, Engesserstr. 18, 76128 Karlsruhe, Germany.Tel.: +49 15126143555; fax: +49 72160843153.

E-mail address: Roland.Kadar@kit.edu (R. Kádár).

0997-7546/$ – see front matter© 2011 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.euromechflu.2011.07.003

by the aspect and radius ratios: R/R1 and L/δ, with R1 and L beingthe outer cylinder (stationary) radius and the fluid column length,respectively. Phenomenologically, flow instability in isothermalTaylor–Couette motion, is due to increase of inertial forces, as therelative rotation of the cylinders is increased, against viscous forces(Re number increases). Therefore, at a critical value of the con-trol parameter (i.e. Re number), in order to maintain equilibrium,the flow reorganizes itself from the basic one dimensional Couetteflow (Fig. 1(b)) to a three dimensional pattern that consists of sta-tionary axisymmetric counter rotating toroidal vortices (Fig. 1(c)and (d)). These vortical structures – called Taylor vortices – repre-sent the first instabilitymode of the Taylor–Couette flow (i.e. TVF).By further increasing the relative rotation of the cylinders, a widevariety of instability modes can be observed, e.g. see [4]. For thesimple case in which the inner cylinder is rotating and the outercylinder is at rest, the main types of vortical structures are:(i) Taylor, (ii) wavy and, (iii) turbulent Taylor vortices, correspond-ing visualizations of the modes being presented in Fig. 2. Follow-ing TVF, due to excessive increase of the radial outward flow,the Taylor vortices deform both axially and radially [5], lead-ing to a non-axial symmetric complex flow wavy pattern—WVF(wavy vortex flow). From this point onward, the evolution of theflow can be characterized by the development of the character-istic frequencies of the flow (WVF is characterized by the az-imuthal wave frequency, fw). While the transition from TVF toWVF is due to a Hopf bifurcation (see [6]), within WVF thereare many bifurcation scenarios that have been observed, suchas Shil’nikov, homoclinic and SNIC (Saddle–node on invariantcircle) bifurcations; see [7–9]. The known wavy flow patterns

R. Kádár, C. Balan / European Journal of Mechanics B/Fluids 31 (2012) 158–167 159

a b c d

Fig. 1. (a) Schematic representation of the concentric cylinders used in the present experiments; (b) Schematic representation of the laminar Couette flow betweenconcentric cylinders; (c) Schematic representation of the primary instability mode, the Taylor vortex flow (TVF) (d) Experimental visualization (left) and numerical solution(right; solution obtained with ANSYS Fluent on a 3D periodic model) of TVF for the present investigations.

Fig. 2. Main instability modes observed in the present experiments: Taylor vortices, wavy vortices (wavy modes) and turbulent Taylor vortices. The diagrams represent themeasured torque (T ) and the number of vortices (N), as function of the angular velocity of the inner cylinder (Ω).

include modulated wavy vortices (MWV; additional characteris-tic frequency: fm, the modulation frequency), chaotic wavy vor-tices (CWV; multiple dominant characteristic frequencies) andwavy turbulent vortices (WTV; broad frequency spectra), e.g. see[10–13]. With emerging turbulence, CWV, the azimuthal ampli-tude of the waves diminishes until a Taylor-like axisymmetricmode is achieved, the turbulent Taylor vortex flow (TTV).

In the past decade, the transitory motions and bifurcations inthe Taylor–Couette flow were intensively studied for lower andhigher order transitions, using different imposed dynamics andboundary conditions, in relation to: (i) modification of WVF dueto rotation of the end plates [14,15], (ii) influence of heating onthe onset of TVF [16], (iii) influences of time harmonic modulationof the inner cylinder [17,18], (iv) characteristic time scale of theTVF [19], (v) the influence of counter-rotating outer cylinder—theso called Spiral mode [20,21], (vi) helical TCPF [22]. With respectto the time harmonic modulation problem, we note that the low-frequency limitmay be relevant to the problem investigated in thispaper, as complex dynamics arise in that regime; see also [23,24].

The hydrodynamic instabilities associated with transient TVFand WVF have been investigated by Abshagen et al. [14],

Manneville and Czarny [25] respectively Abshagen and Pfister [26],in relation to the imperfections generated by the influence ofthe so-called Ekman vortices produced at the ends of the innercylinder of the finite flow geometry. The authors studied thetransient dynamics at the onset, respectively decay, of the flowsaround the critical Reynolds numbers Recr1 ∼= 69 (TVF) andRecr2 ∼= 384 (WVF), for a wide gap set-up with the radius ratioof 0.5. One main conclusion of Abshagen and Pfister [26], withrelevance for the present study, is the alteration of the Hopfbifurcation by the experimental imperfections. A consequence ofthis effect is the appearance of spurious effects as the oscillatorycharacter of the wave amplitude and possible evolution of WVF tothe subharmonic small-jet mode. The phenomena was previouslyremarked by Jones [27], who mentioned that end-effects controlthe axial wavelength inWVF. For a given setup, the selection of theaxial wavelength and its dynamics for post-critical values of theReynolds numbers is also dependent on the applied experimentalprotocol, through the magnitude of the input step or rampincreasing/decreasing time of the inner cylinder velocity, for TVFsee [28].

160 R. Kádár, C. Balan / European Journal of Mechanics B/Fluids 31 (2012) 158–167

1.1. Ramped experimental protocols

In most of the previously mentioned studies, as well as others,the flow kinematic history is an important factor in studying thestability of flow between concentric cylinders. Different histories– i.e. ‘‘paths’’ to access a flow state – can change the criticalparameters and lead to different states under identical controlparameters (i.e. Reynolds numbers); see [29] and the comments ofWhite and Muller [30], Dutcher and Muller [13]. The experimentalramped input protocols used in the published literature wererecently reviewed by Dutcher and Muller [13] (see Table 1 intheir paper). Most studies typically apply experimental protocolsdesigned to ensure steady-state conditions before the beginningof the actual observations. These experimental protocols includevery low ramped input velocity signals followed by long periodsof stabilization at constant velocity. While it is quite clear thatappropriate ramp rates have to be used in order to createundisturbed flows, there is little emphasis on the influence of highramps on subsequent flow dynamics.

For experimental protocols with ramped input signals ingeometries with rotating inner and stationary outer cylinders, onecan define a dimensionless ramp rate dRe/dt, with Re being theReynolds number of the inner cylinder as control parameter, bynormalizing the time with a reference time scale of the flow.Common choices for the dimensionless time are t = t/(ρLδ/η),e.g. see [31,19], and t = t/

ρδ2/η

the dimensionless time with

ρδ2/η the viscous time scale. In this paper we represent data interms of the former, as [13] found an equivalence between thetwo quantities with respect to the critical ramp rates. The effectof ramp rate on the critical parameters was considered by Parket al. [31], Baxter and Andereck [32], Xiao et al. [33] and Dutcherand Muller [13]. The critical parameters obtained using differentramp rates were compared to theoretical results of the linearstability theory. Thus, Xiao et al. [33] found an increase in criticalvalues at dRe/dt > 10, for a geometry with radius ratio of 0.894and an aspect ratio of 94; Dutcher and Muller [13] observed thesame tendency for dRe/dt > 1 on a geometry having a radius ratioof 0.912 and an aspect ratio of 60.7 (i.e. the ratio between the radii ifthe inner and the outer cylinder and the ratio between the heightof the column and the gap). However, in most of the works citedlittle emphasis is given on the transient phenomena leading to thestabilized states on interest, and therefore many open questionsstill remain.

1.2. Goal of present investigations

In the stated framework, the present work is focused on theonset and transition from TVF to WVF and the time evolution ofthe wavy patterns, at different constant values of the Reynolds(Taylor) number. The input Re number signal consists of adimensionless ramp rate chosen above the critical rate forquasi-stationary flows used in similar studies. The experimentalmethod is straightforward and brings to focus the dynamicsimmediately after the rotational velocity is fixed. The flowdynamics is emphasized via the structural characteristics of theflow, namely the average azimuthal amplitude, number of vorticesin the fluid column and frequency spectra. The experimentalgeometry used is a high radius and aspect ratios geometry withasymmetric boundary conditions. The asymmetric particularity ofour experimental device, and mostly the free surface condition,is a de facto condition for all rheometer-based Taylor–Couettesystems. The imposed asymmetric boundary conditions i.e. thelack of reflection symmetry about the annulus half-length (L/2)can play a major role in the subsequent dynamics (e.g. see [34–36]for low aspect ratio geometries). We limit our tests to relativelylow Reynolds numbers, so that no deformation of the free surface

can be observed, and we concentrate our attention on the lowerboundary. As it is later shown, the lower boundary effects seemto influence mostly the wavy regime dynamics in the parameterrange studied.

The present work is part of an extensive study regarding thedynamics of the wavy regime subjected to ramped input signals inisothermal (angular velocity ramp), non-isothermal (temperatureand angular velocity ramps) and weakly elastic polymer flows(dimensionless ramp rates via polymer relaxation time) [3]. Whilethe paper does not constitute an exhaustive overview of thedynamics in steep ramped input signals it elaborates experimentsusing protocols not previously investigated and features novelparticularities of the wavy regime.

2. Experimental setup and procedures

The experimental setup consists of an Anton Paar PhysicaRheolab MC-1+ Rheometer, modified by the addition of atransparent stationary outer cylinder. The working geometryis sketched in Fig. 1(a), and is characterized by the followingparameters: the radius ratio R/R1 = 0.87 (R = 23.6 mm and R1 =

27 mm) and the aspect ratio L/δ = 33.5 (L = 114 mm and δ =

3.4 mm). The temperature was monitored using a temperaturesensor mounted on the lower end of the outer cylinder, thusbeing at distance l from the concentric cylinders. The maximumobserved temperature variation in the range (−1.5, +2.5) °Caround the operating temperature of 20°. The effect of the viscositygradient due to the temperature gradient is quantified by theNahme–Griffith number (Na), [30,37], and is defined as Na =

Br Γ , where Br = η(RΩ)2/kϑ is the Brinkman number (obtainedfrom the energy equation written in dimensionless form) andΓ = |(ϑ/η) (∂η/∂ϑ)|ϑ is the sensitivity of the viscosity tothe temperature gradient. While the temperature control errormay seem to be significant, in the present experiments, theNahme–Griffith number is in the range of Na ∈ [2.4 · 10−8, 3.4 ·

10−6] (values computed for the range of Ω investigated; the

water viscosity data used was taken from [38]). Therefore, thepresent data is well below the range of Na numbers for whichdestabilization was observed in the data of White and Muller [30].

For easier comparison with similar studies Ta, Re, as well as thereduced Reynolds number [5] defined as ε = Re/Recr1 − 1, werespecified whenever appropriate (here Recr1 is the critical value forthe onset of TVF).

The upper-end of the apparatus is opened to a free surface, acommon condition for rheometer based systems. The lower endof the rotated inner cylinder is followed by a closed zone of fixedlength l = 21 mm (see Fig. 1(a)) and then by a solid non-rotating wall. Thus, for the present configuration, the reflectionsymmetry is not valid. Asymmetric boundary conditions have beenstudied relatively broadly in the published literature, mainly byapplying at one or both ends a rotating plate, e.g. [39–42] orby forcing an oscillation at one of the ends, e.g. see [43,44]. Acomparison between symmetric and asymmetric end-conditionsTaylor–Couette flows with free surface was performed by Toyaet al. [45], Watanabe et al. [46,47] (radius ratio 0.67 and low aspectratios).

For flow visualization, water samples (η ∼= 10−3 Pa s at 20 °C)weremixedwith small amounts of reflective flakes (nomeasurableinfluence on the sample viscosity). The annulus was filled withthe mixture and the fluid was allowed to settle until no motioncould be visually observed, before the commencement of tests.Afterward, the velocity of the inner cylinder was increased quasi-steadily up to the desired value. All tests were performed using thesame acceleration (slope) of the ramped input signal, and namelydΩ/dt = 3.14 · 10−2 s−2, in terms of Re, dRe/dt = 2.52 s−1 ordRe/dt = 29.1 in terms of dimensionless ramp rate. The ramp

R. Kádár, C. Balan / European Journal of Mechanics B/Fluids 31 (2012) 158–167 161

Fig. 3. Description of experimental procedures: (a) the angular velocity input signals, (b) space–time plot (Ta = 5859) with themeasurement area for azimuthal amplitude,A = a/2 and (c) typical azimuthal amplitude result over 30 s (Ta = 36 619, Re = 504).

is achieved by a series of small steps in angular velocity of Ω =

0.03 rad/s over time intervals of1t = 1 s, values arbitrary chosen.The ramp rate used in the present experiments was chosen basedon preliminary investigations so that the flow thereby generatedwould not exhibit initial rapid azimuthal amplitude variationsthat would be missed by averaging. After reaching a desiredvelocity of the inner cylinder (corresponding to the establishedvalue of the Ta-number), velocity was maintained constant for aperiod of up to 2500 s, period which defines the experimentaltime (texp ∼= 40 min), (Fig. 3(a)). Within the experimental timethe measurements were performed over periods of 30 s at fixedintervals of time namely: tr ∈ [0, 30] s, tr ∈ [300, 330] s, tr ∈

[600, 630] s, tr ∈ [1200, 1230] s, tr ∈ [1800, 1830] s and tr ∈

[2400, 2430] s with tr = t − tinc being the relative time, t is theactual time elapsed since the beginning of the experiment and tincis the time required to increase the speed up to the desired Ta onthe given ramp. In these time intervals the following parameterswere considered: the azimuthal amplitude, the number of vorticesin the annulus and the azimuthal wave frequency. We obtain thefollowing ratios between the viscous dissipation time scale andthe observation times: (ρδ2/η)/30 ∼= 0.4 and (ρδ2/η)/2500 ∼=

0.005, respectively. The ratio between averaging time and the totalexperimental time is 30/texp ∼ 10−3. The measurements wereperformed at Ta in the range of [103, 105

] (Re ∈ [102, 103], ϵ ∈

[0.17, 7.1], Ω ∈ [1, 12] rad/s) which corresponds to less thanthe first third of the wavy regime as represented in Fig. 2. Thelimitation of Ta numbers investigated is mainly due to the videoacquisition frame rate (12 fps) and to avoid the penetration ofmajor disturbances via the free surface (a visual deformation ofthe free surface was observed in our experiments at around Ta ∼=

3.2 · 103 (Re ∼= 1500)).In the case of the azimuthal amplitude measurements, or axial

oscillation of the vortices, using the notations from Fig. 3, thequantity measured is the distance a (Fig. 3(b)) whilst the quantitytaken into consideration – the actual amplitude – is A = a/2.This amplitude was measured by extracting frames of all theamplitudes from movies recorded during the 30 s intervals (inFig. 3(b) represented as a spatio-temporal diagram). Due to thenon-symmetry of the end effects, the azimuthal amplitude of thewaves is function of the axial z-coordinate within the annulus of

length L. A zone in which the amplitude reaches maximum anddoes not vary significantly (within its vicinity) is located in thelower half of the annulus; see Fig. 3(b) (measurement zone). Theextracted frames were imported into a CAD software, where alinear dimension under zoomed conditions was assigned to theamplitude in each frame. A total of about 2500 frames have beenin this way processed. In order to investigate the time-dependentbehavior of the flow, the ratio of average values of A over the 30 sintervals to the gap was considered (see also [5,33])

Aaz :=Aaz

δ=

1δ

n∑i=1

Ai

n(2)

where n is the number of amplitude peaks recorded in 30 s.The ratio between the fluid column of length L and the gap

δ equals 33.53. Since the height of the Taylor vortex is 1z/2 ∼=

δ, the flow has the tendency to settle at 34 vortices. However,due to the presence of the lower-end zone, the 34th vortex iscontained therein, leaving a number of N = 33 vortices forthe Taylor–Couette fluid column. This is the reason why thenumber of vortices was used instead of the number of vortex pairs,i.e. cells, using the notations of Koschmieder [48]. The variation ofN with Re is of importance for applications such as Taylor–Couettemixers/polymerization reactors, where the mixing model impliesthe existence of N serially connected stirred tanks, [49,50].

Spatio-temporal diagrams were constructed by extracting avertical pixel line from each frame and the Fourier-transformanalysis of the reflected light intensity was performed. Thistechnique is standard for the stability analysis in various flowsystems, e.g. see [51,13,52]. In addition, the end zones werecompared to the measurement zone in terms of frequency spectrain order to emphasize the effect of the boundary effects. Theazimuthal wave frequency was also computed as fw = n/1t ,where n is the number of oscillations in the 1t = 30 s intervals.

3. Results and discussion

In this section, we present the experimental results structuredas follows. First, we analyze the onset of instabilities and theinfluence of end-effects. Thereafter, the transient behavior duringthe experimental time is discussed. The azimuthal amplitude,number of vortices and frequency spectra are therein analyzed.

162 R. Kádár, C. Balan / European Journal of Mechanics B/Fluids 31 (2012) 158–167

Fig. 4. Space–time plot showing the propagation of instabilities in the fluid column for tr ∈ [0, . . . , 30] s: (a) Ta = 2476, the fluid exhibits merging/splitting of vorticesmarking the transition toWVF; (b) Ta = 2871, there is no TVF prior to the filling of the columnwith oscillatory vortices and a local state selection is achieved in shorter time.

Fig. 5. The time dependent character of the flow patterns in the range of Taylor numbers investigated, as function of the relative time (tr). Starred critical Ta refers to the firstoccurrence of a certain regime at tr = 0 s whereas the unstarred ones for stationary flows. The small gray rectangles represent the 30 s measurement periods. The pictorialsequence represented at the top corresponds to Ta = 2000 (principal representation); the two lower positioned pictures correspond Tacr1 = 1800 (TVF) and Ta = 2.1 · 104

(WVF).

3.1. Onset and end-effects

In a finite Taylor–Couette geometry, transition from the laminarCouette flow to the TVF is first marked by the appearance of end-effects within the gap between the cylinders. As the flow drawsclose to the first critical point, the disturbances generated by theend vortices penetrate into the fluid column. The asymmetricend-conditions generate asymmetric end-effects which propagate

toward the interior of the annulus triggering the first instabilitymode, TVF, Fig. 4. The obtained Taylor critical number for theoccurrence of Taylor vortices in the whole annulus is Tacr = 1800(i.e. Recr = 112 at Ωcr = 1.4 rad/s), in good accordance with thelinear stability theory [53–55]. The good correspondence with thetheoretical results is due to the low Na numbers obtained in thesetypes of flows. For our geometry, the TVF regimewas reached aftertr = 29 s at constant Tacr1. For Ta = 2476, tr = 0 s (Fig. 4(a))

R. Kádár, C. Balan / European Journal of Mechanics B/Fluids 31 (2012) 158–167 163

Fig. 6. Comparison between experimental and numerical critical points (tr = 0). The numerical values are obtained from the torque of the inner cylinder while theexperimental points are based on visualizations.

Fig. 7. Power spectral analysis for end effects compared to the measurement area (Fig. 3), for Ta = 23 436, tr = 300, . . . , 330 s. Power density exhibits broader spectra dueto the free surface condition and less intensity due to the illuminating light gradient. Here, fΩ represents the frequency corresponding to the rotational speed of the innercylinder and fw the azimuthal wave frequency.

after a brief time in which the vortices fill the column, there is alsoa period characterized by TVF followed by a regime characterizedby unstable changes in the number of vortices, successive vortexsplitting/merging, that determines the appearance/disappearanceof a pair of vorticesmarking the transition toWVF.With increasingTa the time necessary for the development of Taylor regimedecreases, e.g. at Ta = 2109 the time elapsed being tr = 11 s. Inaddition, the splitting/merging of vortices becomes scarce, and theflow settles easily from LCF toWVF (Fig. 4). For Ta = 2871, tr = 0 s(Fig. 4(b)) there was no TVF observed in-between column fillingandWVF, a local state selection towardWVF being achieved faster.For Ta = 2109, however, at about tr ≈ 600 s, the flow changed to aWVF characterized by small azimuthal amplitude and a number ofwave crests per circumferencem = 1. Furthermore, at tr ≈ 2300 sthe azimuthal amplitudeA decreased to zero, settling again the TVFregime. In the interval Ta ∈ (3285, 4463) TVF regime is installedalmost instantaneously in the whole gap; however, after a finiteelapsed time (the onset time is decreasingwith increasing velocity)the pattern became WVF, which is maintained until the end ofthe experimental period; see Fig. 5. At Ta > 4463 the WVF isachieved immediately at the end of the ramp and is stable duringthe experimental time.

Owing to the steep ramp protocol, duplicate critical values ofthe Taylor number associated with TVF and WVF are obtained:overshooted values due to the unstable conditions generated by

Fig. 8. Experimental results of average azimuthal amplitude divided by the gap (2)and a scaling based on the Landau amplitude equation; see [56,15].

the ramp rate (tr = 0) and stabilized values (tr → texp). Forthe onset of TVF, Tacr1 = 1800 is a first critical value obtained

164 R. Kádár, C. Balan / European Journal of Mechanics B/Fluids 31 (2012) 158–167

Fig. 9. 3D plot of average azimuthal amplitude results as function of the angular velocity (Taylor number) and time. Ai, i = 1, . . . , 3 are preferential values/states of theaverage azimuthal amplitude.

for tr → texp and Ta∗

cr1 = 3285 for tr = 0. In the case ofWVF, at Ta∗

cr2 = 4463 for tr = 0, however the lowest Ta foroccurrence of WVF, Ta = 2019 ultimately ended in TVF at tr →

texp. Ta = 2476 is the lowest point with WVF at tr → texp.The dimensionless ramp rate used here is in the range of the dataconsidered by Dutcher and Muller [13] for Recr1 (Xiao et al. [33]considered Recr2 however, the dimensionless ramp rate consideredis in the range dRe/dt ∈ [0.1123, 2.247]). By computing thereduced Reynolds number, ϵ, one obtains the analogous values fordRe/dt = 20: ϵ = 0.26 in this case and ϵ = 0.28 for Dutcher andMuller [13] (approximative fitting; for similitudeRe∗

cr1 was countedfrom the appearance of a third vortex). The critical parametersfor primary and secondary transition, Tacr1 and Tacr2 have alsobeen numerically computed using the ANSYS FLUENT code on a3D periodic model of the experimental device (for more details,see [3]). Based on the torque variation during transition, one cansee a fairly good correspondence between the experimental andnumerical results; see Fig. 6.

By analyzing space–time plots one can learn the effect ofasymmetric boundary conditions on the wavy regime. Thus, it canbe seen that at the upper boundary the end vortices exhibit almostzero azimuthal amplitude because of relatively weaker couplingbetween the axial and azimuthal phase variables than in thebulk, [44]. Toward the bulk of the column the azimuthal amplitudeis increasing,with amaximum in the lower part followedby a smalldecrease toward the lower end. The power spectra at Ta = 23 436,for the end and azimuthal measurement zones is shown in Fig. 7.The diagram shows qualitatively similar spectra for zones 2 and 3.As expected, in zone 1, at the upper end, the signal is somewhatnoisier (high frequencies) due to the free surface end condition.One observes that from the phase dynamics point of view, theazimuthal amplitude distribution along the axis of the cylinders issimilar to results obtained using forced end-disturbances at one ofthe end-zones. By applying a forcedmodulation on one of the endsin TVFmode,Wu and Andereck [43] found an azimuthal amplitudevariation in the axial direction from zero to a maximum at themodulated end. While applying a similar protocol on aWVFmode,Wu and Andereck [44] found an azimuthal amplitude varying froma very low value to a maximum at the modulated end, for weakcoupling between azimuthal and axial phase variables.

The average azimuthal amplitude Aaz is shown in Fig. 8 asfunction of the Ta-number. The WVF is a regime characterized by

a non-monotonic variation of the azimuthal amplitude with Ta-number, from zero (TVF) to zero (TTV), reaching a maximum at acertain value of ϵ-parameter, which is dependent of the appliedprotocol. The maximum value for the azimuthal amplitude wasfound by Bust et al. [57] to be around ϵ = 1, while Wereley andLueptow [5] found a maximum at ϵ < 1, this difference beingattributed to experimental protocols by Wereley and Lueptow [5].In our case, as the input ramp is steeper than in the two previouslymentioned studies, it is expected that the maximum value ofazimuthal amplitude to be shifted to higher values of Ta-number.In our data, for tr > 300 s, the maximum of amplitude is foundaround Ta = 14 000, value which corresponds to ϵ ∼= 1.7.It is important to mention that for Ta < 6000, the measuredazimuthal amplitude is proportional to the square root of ϵ, whichis consistent with the scaling obtained from the Landau amplitudeequation; see [26].

3.2. Time dependences

In order to characterize the time dependence of the flowwithinthe wavy regime, the average azimuthal amplitude, the number ofvortices and the azimuthal wave frequency were considered in theexperimental time. The results concerning these measurementsare presented in Figs. 8–10. An overview of the average azimuthalamplitude as function of both Ta and time can be found inFig. 10. The measurements at Ta = 2476 (Re = 131) , Ta =

2871 (Re = 141) and even Ta = 5859 (Re = 202) showWVF regimes that decrease in amplitude over time. Based onthe previous findings, they can be considered wavy modes thatreturn to TVF (regime with zero azimuthal amplitude, A1 = 0).The dynamics of WVF and the selection of ‘preferred states’ isstrongly dependent upon the initial and boundary conditions,represented by: (i) the particular geometry of the setup, (ii)the applied experimental protocol, in particular the slope of theincreasing/decreasing ramp of the velocity [58,59]. The presentexperimental results of the amplitude evolution evidence theunstable character of WVF, most probably generated by the samemechanism described by Abshagen and Pfister [26]; see also theobservation from page 2 in [21]: ‘‘in the majority of publicationsWVF solution branch has to be seen to return to the TVF branchor to undergo higher order bifurcation, at larger driving’’. The

R. Kádár, C. Balan / European Journal of Mechanics B/Fluids 31 (2012) 158–167 165

Fig. 10. Experimental results of the number of vortices (N) and azimuthal wave frequency (fw): (a) number of vortices as function of angular velocity (Ta), (b) azimuthalwave frequency as function of angular velocity (Ta) for the time intervals considered; (c) number of vortices as function of time and (d) azimuthal wave frequency as functionof time, for the range of Ta investigated.

measurements at Ta = 3296 (Re = 151), Ta = 9155 (Re =

252), Ta = 13 183 (Re = 303) and Ta = 17 953 (Re = 353)show a settlement around an average azimuthal amplitude of A2 =

0.35, within the experimental time. For Ta = 23 436 (Re =

403), Ta = 29 662 (Re = 454), Ta = 36 619 (Re = 504) andTa = 71 774 (Re = 706) the average azimuthal settles around thevalue of A3 = 0.47. These three characteristic average amplitudevalues were used to identify preferred states of the flow: preferredstate A1 (being the TVF) and states A2, A3 two characteristic valuesassociated to WVF (see Fig. 10). Therefore, there are ranges of Tafor which the preferred state is the same rather than preferentialstates for each Ta studied. The term preferred state, as implied byour experiments, is to be considered in the sense of a preferentialvalue as used by Coles [29]. The existence of a discrete number ofpreferential states could indicate the presence of attractors in theflow dynamics. Thus, according to its kinematic history (i.e. initialvalues of the problem) the flowwill chose its correspondent out ofthe attracting set; see [60].

The variation of azimuthal amplitude (see Fig. 8) determines thechange in number of vortices N , as is shown in Fig. 10(a) and (c).

The time variation of discrete number N , for a given Re numberreveals a decreasing trend within most tests, as function of therelative time, tr . In general, this decrease took place in the timerange of tr ∈ [30, 300] s. Unstable regimes with high N at tr → 0could be reached followed, in time, by a decrease of up to 4 pairsof vortices (e.g. Ta = 29 662). Stable configurations were found forsmall Ta but also for higher Ta (Ta = 61 886 and Ta = 82 393,N =

27 and N = 25 respectively). Related to Ta-numbers that wereattracted to the first preferred state (A1 = 0), the results confirmthe previous ones: Ta = 2476, Ta = 2871 and Ta = 5859 settlefor the same value of the number of vortices, N = 33, whichwas found to correspond to Tacr1. For the case of the Ta numbersthat tend to the second preferred state (A2), there is a higher datascattering, (similar in pattern with the measurements of averageazimuthal amplitude): Ta = 3296 goes for N = 31, Ta = 9155for N = 29, and Ta = 13 183 and Ta = 17 953 both settle forN = 27. And finally, for the case of the Ta-numbers that tend to thethird preferred state (A3) the observations disclose the following:Ta = 23 436 and Ta = 29 662 settle for N = 25, Ta = 36 619 forN = 27 and Ta = 71 774 for N = 23. Within the results obtained,

166 R. Kádár, C. Balan / European Journal of Mechanics B/Fluids 31 (2012) 158–167

Fig. 11. Translation of vortices toward the lower-end boundary that allows the increase of azimuthal amplitude without modifying the number of vortices.

there is also evidence to suggest the downward translation of thevortices (given the nature of the lower-end condition, Fig. 1(a)) asa possible mechanism for azimuthal amplitude variation, Fig. 11.In the second case, the amplitude can vary independent of N , thuscontributing to the complex dynamics observed.

Regarding the characteristic frequencies, we emphasize thevariation of the azimuthal wave frequency, fw; see Fig. 10(b)and (d) for the Ta number and time dependences. The azimuthalwave frequency is increasing with Ta-number for all observationintervals, meaning that the increase in the number of wave crestsoutruns the increase in the angular velocity of the vortices. Asfunction of time, the azimuthal wave frequency is maintainedconstant for the most angular velocities studied. Moreover, themeasurements evidence three distinguishable groups. The first oneis formed by Ta = 2476, Ta = 2871 and Ta = 2396. The secondgroup contains Ta = 13 183, Ta = 23 436, Ta = 29 662, Ta =

36 619, Ta = 44 309, Ta = 118 646 and the third correspondsto Ta = 71 774, Ta = 93 745, Ta = 61 886, Ta = 82 329and Ta = 105 829. This grouping is slightly different from whatconstitutes the previously defined preferred states, and may wellbe an indication of the final outcome of the cases studied, in termsof preferential average azimuthal amplitude for t → ∞. The Tafor which fw is not constant in time, Ta = 23 436, Ta = 36 619,Ta = 44 309, Ta = 71 774 and Ta = 82 393 are due to a changein the number of waves per circumference (m). In addition to fw ,with increasing Ta lowmodulation frequencies could be identifiedin the frequency spectra.

4. Final remarks

In this paper, the complex dynamics of the wavy Taylor regimeunder a steep ramp input was considered in a Taylor–Couetteapparatus with asymmetric end-effects. During the experimentaltime of 40 min (Ta = ct.), the flow was characterized usingthe average azimuthal amplitude, number of vortices, azimuthalwave frequency and power spectra. The results show unstablebehaviors that lead to overshooting phenomena. Thus, transitionsfrom TVF to WVF and then back to TVF were observed in thevicinity of Tacr1. For higher Ta, within the experimental timelimits, the overshooting dynamics exhibits a complex behavior,showing tendency toward preferential values of average azimuthalamplitude. Overall, three preferential values were identified: A1 =

0 (TVF), A2 and A3 (A2 < A3). The number of vortices was foundto decrease in time for most Ta investigated. Usually, the decreaseoccurred at the beginning of experimental time and no later thantr = 1200s. A large decrease in the number of vortices was relatedto significant increase in azimuthal amplitude. The azimuthalwavefrequency remained constant throughout experimental time, withexceptions in the cases where a modification in the number ofwaves per circumference occurred. This process consists in thesplitting of a high amplitude wave into two small amplitudewaves. The changes in the number of vortices, azimuthal wavefrequency and the development of low frequency components inthe spectra could be mechanisms by which the flow tends toward

its preferred amplitude. The presence of length l between thefluid column and the stationary solid lower boundary seems tobe one of the important factors in the development of the flowtoward its preferred states. The investigated transition from TVFto WVF, and the dynamics within the WVF regime, evidence thepossible existence of somediscrete preferred stationary states. Thispattern of the dynamics evolution might be of great importancefor practical applications, in particular for establishing theworkingconditions and reaction control in Taylor–Couette reactors.

Acknowledgments

Roland Kádár acknowledges the financial support of CNCSISBD grant no. 349 for doctoral studies. The experiments have beenperformed at the BIOINGTEH platform, ‘‘Field-Matter Interaction’’laboratory, Politehnica University of Bucharest.

References

[1] G. Taylor, Stability of a viscous liquid contained between two rotatingcylinders, Phil. Trans. R. Soc. A CCXXIII (1923) 289–343.

[2] C. Balan, Lectii de Mecanica Fluidelor, Ed. Tehnica, Bucuresti, 2003.[3] R. Kádár, Flow pf pure viscous and viscoelastic fluids between concentric

cylinders with applications to rotational mixers, Ph.D. Thesis, PolitehnicaUniversity of Bucharest, Bucharest, 2010.

[4] C. Andereck, S. Liu, H. Swinney, Flow regimes in a circular Couette systemwithindependently rotating cylinders, J. Fluid Mech. 164 (1986) 155–183.

[5] S. Wereley, R. Lueptow, Spatio-temporal character of non-wavy and wavyTaylor–Couette flow, J. Fluid Mech. 364 (1998) 59–80.

[6] M. Golubitsky, V. Leblanc, I. Melbourne, Hopf bifurcation from rotating wavesand patterns in physical space, J. Nonlinear Sci. 10 (2000) 69–101.

[7] J. Abshagen, J.M. Lopez, F. Marques, G. Pfister, Bursting dynamics due to ahomoclinic cascade in Taylor–Couette flow, J. FluidMech. 613 (2008) 357–384.

[8] J.M. Lopez, F. Marques, Finite aspect ratio Taylor–Couette flow: Shil’nikovdynamics of 2-tori, Physica D 211 (2005) 168–191.

[9] J.M. Lopez, F. Marques, Small aspect ratio Taylor–Couette flow: onset of a very-low-frequency three-torus state, Phys. Rev. E 68 (2003) 036302.

[10] P. Fenstermacher, H. Swinney, J. Gollub, Dynamical instabilities and thetransition to chaotic Taylor vortex flow, J. Fluid Mech. 94 (1979) 103–128.

[11] M. Gorman, H. Swinney, Spatial and temporal characteristics of modulatedwaves in the circular Couette system, J. Fluid Mech. 117 (1982) 123–142.

[12] Y. Takeda, Quasi-periodic state transition to turbulence in a rotating Couettesystem, J. Fluid Mech. 389 (1999) 81–99.

[13] C. Dutcher, S. Muller, Spatio-temporal mode dynamics and higher ordertransitions in high aspect ratio Newtonian Taylor–Couette flows, J. FluidMech.641 (2009) 85–113.

[14] J. Abshagen, O. Meincke, G. Pfister, K. Cliffe, T. Mullin, Transient dynamics atthe onset of Taylor vortices, J. Fluid Mech. 75 (1) (2003) 335–343.

[15] J. Abshagen, M. Heise, C. Hoffmann, G. Pfister, Direction reversal of a rotatingwave in Taylor–Couette flow, J. Fluid Mech. 607 (2008) 199–208.

[16] U.A. Al-Mubaiyedh, R. Sureshkumar, B. Khomami, The effect of viscous heatingon the stability of Taylor–Couette flow, J. Fluid Mech. 462 (2002) 111–132.

[17] M. Avila, J. Lopez, F. Marques, W. Saric, Mode competition in modulatedTaylor–Couette flow, J. Fluid Mech. 601 (2008) 381–406.

[18] A.J. Youd, A.P. Willis, C.F. Barenghi, Reversing and non-reversing modulatedTaylor–Couette flow, J. Fluid Mech. 487 (2003) 367–376.

[19] O. Czarny, R. Lueptow, Time scales for transition in Taylor–Couette flow, Phys.Fluids 19 (2007) 054103.

[20] M. Heise, J. Abshagen, D. Küter, K. Hochstrate, G. Pfister, C. Hoffmann, Localizedspirals in Taylor–Couette flow, Phys. Rev. E 77 (2008) 026202.

R. Kádár, C. Balan / European Journal of Mechanics B/Fluids 31 (2012) 158–167 167

[21] C. Hoffmann, S. Altmeyer, A. Pinter, M. Lücke, Transitions between Taylorvortices and spirals via wavy Taylor vortices and wavy spirals, New J. Phys.11 (2009) 0532002.

[22] L.G. Raguin, J.G. Georgiadis, Kinematics of the stationary helical vortex modein Taylor poiseuille flow, J. Fluid Mech. 516 (2004) 125–154.

[23] F. Marques, J.M. Lopez, Taylor–Couette flowwith axial oscillations of the innercylinder: floquet analysis of the basic flow, J. Fluid Mech. 348 (1997) 153–175.

[24] F.Marques, J.M. Lopez, Spatial and temporal resonances in a periodically forcedhydrodynamic system, Physica D, Nonlinear Phenom. 136 (2000) 340–352.

[25] P. Manneville, O. Czarny, Aspect-ratio dependence of transient Taylor vorticesclose to threshold, Theor. Comput. Fluid Dyn. 23 (2009) 15–36.

[26] J. Abshagen, G. Pfister, Onset of wavy vortices in Taylor–Couette flow withimperfect reflection symmetry, Phys. Rev. E 78 (2008) 046296.

[27] C. Jones, The transition to wavy Taylor vortices, J. Fluid Mech. 157 (1985)135–162.

[28] J. Rigopoulos, J. Sheridan, M. Thompson, State selection in Taylor vortex flowreached with accelerated inner cylinder, J. Fluid Mech. 489 (2003) 79–99.

[29] D. Coles, Transition in circular Couette flow, J. Fluid Mech. 21 (3) (1965)385–425.

[30] J. White, S. Muller, Experimental studies on the stability of NewtonianTaylor–Couette flow in the presence of viscous heating, J. Fluid Mech. 4 (2002)133–159.

[31] K. Park, G.L. Crawford, R.J. Donnelly, Determination of transition in couetteflow in finite geometries, Phys. Rev. Lett. 47 (1981) 1448–1450.

[32] G.W. Baxter, C.D. Andereck, Formation of dynamical domains in a circularcouette system, Phys. Rev. Lett. 57 (1986) 3046–3049.

[33] Q. Xiao, T. Lin, Y. Chew, Effect of acceleration on the wavy Taylor vortex flow,Exp. Fluids 32 (2002) 639–644.

[34] F. Marques, J.M. Lopez, Onset of three-dimensional unsteady states in small-aspect-ratio Taylor–Couette flow, J. Fluid Mech. 561 (2006) 255–277.

[35] J. Abshagen, J.M. Lopez, F. Marques, G. Pfister, Mode competition of rotatingwaves in reflection-symmetric Taylor–Couette flow, J. Fluid Mech. 540 (2005)269–299.

[36] J. Abshagen, J.M. Lopez, F. Marques, G. Pfister, Symmetry breaking via globalbifurcations of modulated rotating waves in hydrodynamics, Phys. Rev. Lett.94 (2005) 074501.

[37] J. Pearson, Polymer flows dominated by high heat generation and low heattransfer, Polym. Eng. Sci. 18 (3) (1978) 222–229.

[38] A. Rao, Rheology of Fluid and Semisolid Foods, AN Aspen Publication,Gaithersburg, 1999.

[39] O. Meincke, C. Egbers, N. Scurtu, E. Bänsch, Taylor–Couette system withasymmetric boundary conditions, in: C. Egbers, G. Pfister (Eds.), Physicsof Rotating Fluids, in: Lecture Notes in Physics, vol. 549, Springer, 2000,pp. 22–36.

[40] T. Mullin, C. Blohm, Bifurcation phenomena in a Taylor–Couette flow withasymmetric boundary conditions, Phys. Fluids 13 (2001) 136–140.

[41] J. Abshagen, K.A. Cliffe, J. Langenberg, T. Mullin, G. Pfister, S.J. Tavener, TaylorCouette flow with independently rotating end plates, Theor. Comput. FluidDyn. 18 (2004) 129–136.

[42] J.M. Lopez, F. Marques, J. Shen, Complex dynamics in a short annular containerwith rotating bottom and inner cylinder, J. Fluid Mech. 501 (2004) 327–354.

[43] M. Wu, D. Andereck, Phase modulation of Taylor vortex flow, Phys. Rev. A 43(4) (1991).

[44] M. Wu, D. Andereck, Phase dynamics of wavy vortex flow, Phys. Rev. Lett. 43(4) (1991).

[45] Y. Toya, I. Nakamura, S. Yamashita, Y. Ueki, An experiment on a Taylor vortexflow in a gap with a small aspect ratio bifurcation of flows in an asymmetricsystem, Acta Mech. 102 (1994) 137–148.

[46] T. Watanabe, Y. Toya, I. Nakamura, Development of free surface flow betweenconcentric cylinders with vertical axes, J. Phys. Conf. Ser. 14 (2005) 9.

[47] T. Watanabe, H. Furukawa, Y. Toya, Transition of free-surface flow modes inTaylor–Couette system, J. Visualization 10 (2007) 309–316.

[48] E. Koschmieder, Bénard Cells and Taylor Vortices, Cambridge University Press,Cambridge, 1993.

[49] A. Racina, Vermischung in Taylor–Couette Strömung, Ph.D. Thesis, UniversitätKarlsruhe, TH, 2009.

[50] Z. Liu, R. Kádár,M. Kind, Hydrodynamic activation of the batch-polymerizationof methyl methacrylate in Taylor–Couette reactor, Macromol. Symp. (2010).

[51] J. White, S. Muller, Viscous heating and the stability of Newtonian andviscoelastic Taylor–Couette flows, Phys. Rev. Lett. 84 (22) (2000) 5130–5133.

[52] T. Tsukahara, N. Tillmark, P. Alfredsson, Flow regimes in a plane Couette flowwith system rotation, J. Fluid Mech. 648 (2010) 5–33.

[53] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, OxfordUniversity Press, Oxford, 1961.

[54] L. Landau, E. Lifschitz, Fluid Mechanics, 2nd ed., in: Course on TheoreticalPhysics, vol. VI, Taylor & Francis, London, 1959.

[55] A. Esser, S. Grossmann, Analytic expression for Taylor–Couette stabilityboundary, Phys. Fluids 8 (7) (1996) 1814–1819.

[56] G. Pfister, U. Gerdts, Dynamics of Taylor wavy vortex flow, Phys. Lett. A 83(1981) 23–25.

[57] G. Bust, B. Dornblaser, E. Koschmieder, Amplitudes and wavelengths of wavyTaylor vortices, Phys. Fluids 28 (5) (1985) 1243–1247.

[58] K. Park, Unusual transition sequence in Taylor wavy vortex flow, Phys. Rev. A29 (6) (1984).

[59] J. Antonijoan, J. Sanches, On stable Taylor vortices above the transition towavyvortices, Phys. Fluids 14 (5) (2002) 1661–1665.

[60] D.D. Joseph, Stability of Fluid Motions, Springer-Verlag, Berlin, Heidelberg,New York, 1976.