DOI 10.1140/epje/i2013-13020-5

Regular Article

Eur. Phys. J. E (2013) 35: 20 THE EUROPEANPHYSICAL JOURNAL E

Velocity field of the spiral vortex flow in the Couette-Taylorsystem

Nizar Abchaa, Olivier Crumeyrolle, Alexander B. Ezerskya, and Innocent Mutabazib

Laboratoire Ondes et Milieux Complexes, UMR 6294, CNRS-Universite du Havre, 53, rue de Prony, CS 80540, F-76058, Le HavreCedex, France

Received 26 April 2012 and Received in final form 26 October 2012Published online: 7 March 2013 – c© EDP Sciences / Societa Italiana di Fisica / Springer-Verlag 2013

Abstract. Spiral vortex flow in the counter-rotating Couette-Taylor system with a large aspect ratio andan intermediate gap has been investigated using Particle Image Velocimetry (PIV). From data of velocitycomponents, we have determined nonlinear properties (anharmonicity, mirror symmetry, axial and radialflow rates) of spiral vortices and compared them to those of Taylor vortices. The velocity field around aspatio-temporal defect has been measured. There is a good agreement between these experimental resultswith available results from numerical simulations.

1 Introduction

The Couette-Taylor system which consists of flow in thegap between two independently rotating coaxial cylindershas a rich diagram of states resulting from the successivebreaking of the flow symmetries [1–7]. Many studies inthis flow system have focused on the case when the outercylinder is fixed while rotating the inner cylinder. In thiscase, the transition scenario to chaos has been well iden-tified [4]: the base flow (circular Couette flow) bifurcatesto stationary axisymmetric Taylor vortex flow, which be-comes unstable to wavy vortex flow and then to modulatedwavy vortex flow before transition to turbulent Taylor vor-tex flow. In the general case when both the cylinders ro-tate, the transition scenario is more complex and dependson the value of the control parameters: the radius ratio,the aspect ratio and the Reynolds numbers correspondingto the rotating cylinders [1–3]. For sufficiently counter-rotating cylinders, the circular Couette flow undergoes aHopf bifurcation to spiral vortex flow (SVF) which con-sists of helicoidal nonlinear waves propagating along andaround the cylinder axis. The critical conditions of thespiral vortex flow in case of infinite cylinders have beendetermined in theoretical analysis by Krueger et al. [5]and Demay et al. [7] in the small gap approximation andby Langford et al. [8] for a finite gap size. The later workwas followed by the analysis of the convective and absolutenature of the spiral vortex flow [9]. In their work, Demayet al. [7] computed also the nonlinear coefficients of the

a Present address: Laboratoire de Morphodynamique Conti-nentale et Cotiere, UMR 6143 CNRS-Universite de Caen-BasseNormandie, 24, rue des Tilleuls, F-14000 Caen, France.

b e-mail: mutabazi@univ-lehavre.fr

complex Ginzburg-Landau equation which describes thebehaviour of the spiral vortex pattern near the onset. Za-leski et al. [10] analyzed the wave number selection mech-anism of the spiral vortex flow assuming a small vortexinclination. Knobloch and Pierce [11] have analyzed someproperties of the spiral pattern in the framework of thecoupled complex Ginzburg-Landau equation.

The first quantitative experimental characterization ofspirals regime was made by Schulz and Pfister [12] us-ing Laser Dopler Velocimetry (LDV) and Particle ImageVelocimetry (PIV) in a wide-gap Couette-Taylor systemwith an intermediate aspect ratio. Besides the determi-nation of critical parameters, they showed that the spiralfrequency decreases with the rotation velocity of the innercylinder for the fixed rotation velociy of the outer cylinder.These results were in agreement with the numerical simu-lations of the Navier-Stokes equations [13–15]. Recent the-oretical and experimental works [16–19] showed that spi-rals generated in the Couette-Taylor and Taylor-Dean sys-tems have an anomalous dispersion: the linear group ve-locity and phase velocity have opposite signs. The coupledcomplex Ginzburg-Landau equations with this anomalousdispersion offer a good theoretical framework to explainthe existence of a stable source between the two counter-propagating spirals in extended system [16]. The anoma-lous dispersion of spiral vortex flow was implicitly assumedin works [10,20].

In their numerical simulations of spiral pattern, Ed-wards et al. [21], Demay et al. [22], Marques and cowork-ers [23, 24] and Meseguer et al. [25, 26] imposed an axialpressure gradient in order to enforce a net zero axial flux,a situation encountered in experiments with closed ends.A similar approach was used by Ali and Weidman [27] for

Page 2 of 7 Eur. Phys. J. E (2013) 36: 20

the linear stability analysis of Couette-Taylor system withaxial sliding of the inner cylinder. More recently, Hoff-man et al. [14] performed a direct numerical computationin order to take into account the effects of finite aspectratio and have highlighted new properties of spiral vor-tex flow such as the anharmonicity and mirror symmetry.The present work aims partly to determine experimentallythese nonlinear properties of spiral vortices in a Couette-Taylor system with a large aspect ratio and an interme-diate radius ratio. We have measured the radial and ax-ial velocity components using Particle Image Velocimetry(PIV) technique. In particular, we have elucidated the flowstructure of spiral vortices in comparison with Taylor vor-tices and have computed, from velocity components, thedegree of anharmoncity of patterns, the axial and radialfluxes far from the end plates. The paper is organized asfollows: in sect. 2 we present the theoretical backgroundfor the counter-rotating Couette-Taylor system. The ex-perimental system and procedure are described in sect. 3.The results are presented in sect. 4 while sect. 5 containsdiscussion and conclusion.

2 Theoretical background

We consider the flow transition of an incompressible New-tonian fluid confined between two counter-rotating cylin-ders of radius a and b. For a given flow configuration witha given radius ratio η = a/b and aspect ratio Γ = L/d,the flow control parameters are the inner and outer cylin-der Reynolds number Ri = Ωiad/ν and Ro = Ωobd/ν,where L is the working fluid length, d = b − a is the gapsize, Ωi and Ωo = μΩi are the angular frequency of theinner and outer cylinder respectively, ν is the kinematicviscosity of the working fluid. The cylindrical coordinates(r, θ, z) are appropriate for the Couette-Taylor flow. In thecase of infinite length (Γ → ∞), the base state is the cir-cular Couette flow with an azimuthal velocity profile givenby [28,29]

V (r)=C1r +C2

r; C1 =Ωi

μ−η2

1−η2, C2 =Ωia

2 1−μ

1−η2. (1)

The flow has a nodal surface r0 = a[(1−μ)/(η2−μ)]1/2 =const. on which the velocity vanishes, i.e. V (r0) = 0. Ac-cording to Rayleigh circulation criterion [28, 29], the cir-cular Couette flow is potentially unstable to centrifugal-force–driven perturbations for a < r < r0 and stable forr0 < r < b. The first instability mode appears from thecircular Couette flow as a supercritical Hopf bifurcation inthe form of helicoidal nonlinear waves propagating alongand around the inner cylinder. The resulting pattern isknown as spiral vortex flow (SVF). The SVF breaks theazimuthal rotational symmetry (SO2) and the axial re-flection symmetry about the mid-height (O2) of the baseflow [30]. It oscillates in time and rotates in the azimuthwhile propagating in the axial direction. In finite lengthsystem, at the threshold, the spiral pattern appears in themiddle of the system. Its size increases with the controlparameter Ri until the total length of the system is filled

with spiral vortices. The bifurcated state may result intwo spiral patterns traveling in opposite directions andseparated by a stationary front [16]. For infinite lengthsystem, the spiral vortex pattern may be represented asfollows [5–8,31]:

I(t, r, θ, z) = F(r)[A(t, z)eiΦA + B(t, z)eiΦB

]+ c.c., (2)

where A(t, z), B(t, z) are the amplitudes of right-handedand left-handed traveling spirals, F(r) is the radial struc-ture function which is determined from linear stabilityanalysis, and c.c. represents the complex conjugate terms.The phases ΦA and ΦB combine the dependence on thetime t, the axial and azimuthal variables z and θ as fol-lows:

ΦA = kz + mθ − ωt; ΦB = −kz + mθ − ωt; (3)

ω is the pattern frequency, k and m are axial and az-imuthal wave numbers respectively. The complex ampli-tudes A(t, z) and B(t, z) in eq. (2) can be described bycoupled complex Ginzburg-Landau equations [11, 31, 32].The signs in the phase expressions as defined in eq. (3)have been chosen in a such way that k, m, and θ arepositive quantities. Because of the single-valued quantitiesdescribing the pattern in the azimuth, m takes integer val-ues. In unrolled cylindrical surface in the plane (θ, z), theisophase curves of spiral patterns are inclined with a slope−m/k for right-handed spiral and m/k for left-handed spi-ral. The axial phase velocity of the spiral is related to itsrotation frequency Ωs by c = ω/k = ±mΩs/k, where (+)corresponds to a right-handed spiral and (−) to a left-handed spiral. The rotation frequency Ωs = ω/m of thecritical spiral eigenmodes is related to the inner cylinderrotation Ωi by [14,33–35]

Ωs = C(m) +

r0∫

a

V (r) dr

/ r0∫

a

r dr

= C(m) + 2Ωiμ − η2

1 − η2

[12− 1 − μ

μ − η2

η20

1 − η20

ln η20

], (4)

where η0 = a/r0. The fit constant C(m) to be determinedexperimentally is the precession frequency relative to themean rotation. Hoffman et al. [14] have shown that theconstant C(m) was almost independent of the outer cylin-der rotation for m = 1. The spiral pattern rotates in thesame direction as the inner cylinder but with a differentangular frequency.

The spiral vortex flow has highly nonlinear propertiessuch as asymmetry and nonvanishing axial flow rate. Theasymmetry of the spiral vortex flow can be characterizedby a parameter introduced by Hoffman et al. [14]

P =

λ/2∫

−λ/2

|vr(z) − vr(−z)| dz

λ/2∫

−λ/2

|vr(z) + vr(−z)| dz

, (5)

Eur. Phys. J. E (2013) 36: 20 Page 3 of 7

where λ = 2π/k is the pattern wavelength in the axialdirection.

We have defined the flow rates across different sectionsas follows:

Qz = 2π

b∫

a

vzr dr,

Qθ =

λ/2∫

−λ/2

b∫

a

vθ dr dz,

Qr = 2πr

λ/2∫

−λ/2

vr dz.

(6)

The paper focuses on results extracted from veloc-ity measurements to quantify these nonlinear properties:asymmetry, axial and radial flow for some values of Ro

and Ri.

3 Experimental system and procedure

The experimental system consists of two vertical coax-ial cylinders immersed in a large rectangular plexiglassbox filled with water for thermal insulation. The innercylinder has a radius a = 4 cm, the gap between thecylinders is d = 1 cm and the cylinder working lengthis L = 45.9 cm. The radius ratio is η = 0.8 and the as-pect ratio is Γ = 45.9. Such an aspect ratio is sufficientlylarge for the system to be considered an extended one,i.e. for the end plates to have negligible effects on thepattern dynamics. The working liquid is deionized waterwith kinematic viscosity ν = 0.98 · 10−2 cm2/s at temper-ature T = 21.1 ◦C. Both cylinders rotate independentlyin opposite directions at angular frequencies Ωi and Ωo

for inner and outer cylinder. For the visualization of flowstructures, we have added 2% by volume of KalliroscopeAQ1000, which is a suspension of 1% to 2% of reflectiveflakes of a typical size of 30 μm × 6 μm × 0.07 μm [36,37].These flakes have a density ρ = 1.62 g/cm3 and a relativelylarge refractive index n = 1.85.

Their sedimentation remains negligible in horizon-tal or vertical configurations if the experiment lasts lessthan 10 hours. Viscosity measurements have shown thatKalliroscope flakes in suspension have negligible influenceon the viscosity of the working solution as long as theadded concentration is less than 5%.

Using a He-Ne laser sheet (1 mm wide beam, spread bya cylindrical lens), it was possible to visualize the cross-section of the flow in the (r, z)-plane (fig. 1a). To recordthe motion in the radial direction, we have used a 2-dCCD camera (A641f, Basler). We extracted the axial dis-tribution of the light intensity, at regular time intervalson the line r = a + d/2 and then plotted the space-timediagram I(z, t) (fig. 1b). We also recorded the intensityat different radial positions and then plotted the space-time diagram I(r, t) (fig. 1c). Although very convenient

Fig. 1. Spiral vortex flow for Ro = −299 and Ri = 212: a)Flow cross-section (r, z); b) space-time diagram I(z, t) alongthe axial direction at the mid-gap r = a + d/2; c) space-timediagram I(r, t) in the radial direction.

for the analysis of spatio-temporal properties of patterns,the space-time diagrams obtained from the reflected lightintensity do not provide any quantitative data of velocityor vorticity. The latter can be obtained by PIV and arevery important for the estimate of energy and momentumtransfer in different flow regimes.

For PIV measurement, the working fluid was seededwith spherical glass particles of diameter 8–11 μm anddensity ρ = 1.6 g/cm3, with a concentration of about1 ppm. The PIV system consists of two Nd-YAG lasersources, a MasterPiv processor (from Tecflow) and a CCDcamera (Kodak) with 1034×779 pixels. The time delay be-tween two laser pulses varies from 0.5 to 25 ms, dependingon the values of Reynolds numbers.

In the PIV technique, the flow in the test area of theplane (r, z) is visualized with a thin light sheet that il-luminates the glass particles, the positions of which canbe recorded at short time intervals. We have recorded 195pairs of images of size 1034 × 779 pixels. Each image ofa pair was sampled into windows of 32 × 32 pixels withan overlapping of 50%. The velocity fields were computedusing the intercorrelation function, which is implementedin the software “Corelia-V2IP” (Tecflow). From velocityfields, we have deduced the azimuthal component of thevorticity ωθ = (∂vr/∂z − ∂vz/∂r). We have performedPIV measurements in the circular Couette flow (CCF),in the Taylor vortex flow (TVF) and Wavy vortex flow(WVF) regimes in order to calibrate our data acquisitionsystem and to fit data available in the literature for theseregimes [38, 39]. The measured radial and axial velocitycomponents have been fitted by a polynomial function sat-isfying the nonslip condition at the cylindrical walls r = aand r = b. The PIV allows for the visualization of velocityand vorticity fields in the cross-section (r, z) (fig. 2a). Thevorticity field (fig. 2a) exhibits zones of positive vorticity

Page 4 of 7 Eur. Phys. J. E (2013) 36: 20

Fig. 2. a) Instantaneous velocity (arrows) and vorticity fields(color); the color varies from blue (minimal negative vorticity)to red (maximal positive vorticity). Space-time diagrams ofthe radial velocity component for Ro = −299 and Ri = 212: b)u(ζ, t), c) u(x, t); the color varies from blue (minimal negativevelocity) to red (maximal positive velocity).

Fig. 3. Scheme of the main flow positions: (1) outflow; (2) in-flow; (3) vortex core; (4) mid-gap (x = 0.5).

(red color) and zones of negative vorticity (blue color).From the instantaneous data, we have plotted the space-time diagrams of velocity components, as illustrated forthe radial component u in the axial (fig. 2b) and radial(fig. 2c) directions and extracted from them the ampli-tudes and phases using the complex demodulation tech-nique described in Bot et al. [18, 19]. Figure 3 illustratesthe main flow positions in the gap.

4 Results

We will describe results on spiral patterns observed in aCouette-Taylor flow system with a radius ratio η = 0.8,in the range of Reynolds numbers {−340 < Ro < −220}and {174 < Ri < 276}. The data will be presented in non-dimensional form using d as a length scale, Ωia as velocityscale and d2/ν as time scale. So the axial coordinate willbe denoted ζ = z/d and the radial coordinate r is scaled

Table 1. Critical values of the spiral pattern: f = ωd2/(2πν),q = kd.

Parameters

Ro = −299 Ro = −251 Ro = −230

Critical Ric = 212 Ric = 202 Ric = 174

values μ = −1.13 μ = −0.99 μ = −1.06

f 1.70 1.55 1.65

q 4.48 5.24 5.22

m 3 2 2

θ 4.23◦ 4.85◦ 4.88◦

Fig. 4. The velocity field in the plane (x, ζ): a) the TVF regimeat Ri = 125 and Ro = 0; b) the SVF regime at Ri = 214 andRo = −251, the vertical dashed line corresponds to the nodalsurface x = x0.

as follows: r = a + dx, where x = 0 at the inner cylin-der and x = 1 at the outer cylinder. The dimensionlessaxial wave number is q = kd = 2πd/λ and the dimen-sionless frequency is f = ωd2/(2πν). The velocity dataare scaled by the inner cylinder velocity Ωia as follows:vr = Ωiau, vz = Ωiaw. The critical values of the spiralpattern are given in table 1.

The spiral vortex flow has nonlinear properties whichare different from Taylor vortices. Following recent nu-merical studies [14,25,26], we have computed from exper-imental data, anharmonicity, asymmetry and flow ratesand made comparison whenever this was possible with theproperties of Taylor vortices.

From the axial profile of the radial velocity, we haveestimated the width of the inflow and that of the outflow(figs. 4 and 5). The axial variation of the radial veloc-ity component in mid-gap (x = 0.5) shows that the width(Δin) of the inflow is larger than that of the outflow region(Δout). The ratio α = Δin/Δout determines the degree ofanharmonicity of the vortex flow. The velocity profiles in

Eur. Phys. J. E (2013) 36: 20 Page 5 of 7

Table 2. Nonlinear properties for the SVF regime and the TVF regime for some values of Ri and Ro.

Flow Ri Ro Δin Δout α β P Qζ/(2π) Qx/(2π)

SVF 214−251 10.6 8.7 1.22 0.40 0.513 −0.99 −2.93

−235 10.4 9.3 1.12 0.37 0.487 −0.67 −1.76

−226 10.0 9.2 1.09 0.10 0.426 −0.40 −0.31

TVF125 0 10.0 7.7 1.30 − 0.028 0.02 0.035

105 0 10.2 8.1 1.26 − 0.008 − −

Fig. 5. Axial profiles of radial velocity u(ζ) taken at the mid-gap: a) in the TVF regime at Ri = 125 and Ro = 0; b) in theSVF regime at Ri = 214 and Ro = {−251,−235,−226}.

fig. 5 show that α > 1 for SVF (fig. 5a) as well as for TVF(fig. 5b). The degree of anharmonicity α increases withthe counter-rotation i.e. when Ro becomes more negative.We show in table 2 that the values of α for the SVF re-main lower than those of TVF [14]. The variation of theanharmonicity of SVF can be obtained also from Fourieranalysis and complex demodulation of velocity fields in themidgap. To that end, we have performed first Fast FourierTransform of the radial velocity field u(ζ, t) and then weapplied complex demodulation around the fundamentaland the second harmonic modes in order to separate the

amplitude Ai and phase Φi of each mode (i = 1, 2)

u(t, z) � Re {A1(ζ, t) exp(iΦ1) + A2(ζ, t) exp(iΦ2)} .

Figure 6 shows space-time diagrams of obtained ampli-tudes and phases for Ro = −299, Ri = 212. The ampli-tudes exhibit fluctuations in time and in space, thereforewe have computed their average values defined as follows:

〈|Ai|〉 =1

LT

L∫

0

T∫

0

|Ai(t, z)| dtdz

and L and T are the pattern length in the axial direc-tion and the pattern total acquisition time. The standarddeviation of these amplitudes is less than 15% of theiraveraged values. We have calculated the amplitude ratioβ = 〈|A2|〉 / 〈|A1|〉 as a function of Ro for fixed Ri. Thevalues of β show the same trend as α. This technique worksless accurately for TVF.

The critical TVF possesses axial mirror symmetryaround the position of maximal radial outflow at ζ =λTVF/2 as shown in fig. 7a. In order to measure the degreeto which this symmetry is broken in the spiral vortex flow(fig. 7b), we have calculated in the mid-gap (x = 1/2), theasymmetry parameter P defined in eq. (5). The obtainedvalues are given in table 2. We found that the SVF has alarger mirror symmetry breaking which increases when Ro

decreases. In fact, the asymmetry parameter of the SVFis an order of magnitude larger than that of the TVF.

From the axial and radial velocity components, wehave calculated the instantaneous axial and radial flowrates using formula (6) written in dimensionless form

Qζ = 2π

1∫

0

w(x)(1 + δx) dx,

Qx(x = 0.5) = 2π(1 + 0.5δ)

π/q∫

−π/q

u(ζ) dζ, (7)

δ = (1 − η)/η.

The azimuthal flow rate was not computed since the az-imuthal velocity component was not measured. The dataare presented in table 2. One observes that flow rates areimportant for spiral vortex flow and that they increasewith rotation of the outer cylinder. The presence of the

Page 6 of 7 Eur. Phys. J. E (2013) 36: 20

Fig. 6. Space-time diagrams of amplitudes and phases of the fundamental mode A1 and of the second harmonic mode A2 ofthe radial component u(ζ, t): a) amplitude |A1(ζ, t)|; b) phase Φ1(ζ, t); c) amplitude |A2(ζ, t)|; d) phase Φ2(ζ, t)). Phases aregiven in rad in the range [−π, π], amplitudes in arbitrary units.

Fig. 7. Axial profiles of radial velocity u(ζ) at mid-gap: a) inthe TVF regime at Ri = 125 and Ro = 0; b) in the SVF regimeat Ri = 214 and Ro = −226.

axial flow rate in spiral vortex flow is important for theaxial transport properties [40]. The TVF does not haveany axial flow rates near the threshold. The value of theaxial flow rate in the TVF for Ri = 125 is within experi-mental error of ΔQ/2π = ±0.02.

5 Discussion and conclusion

The experimental investigation of spiral vortex flow hasbeen performed in a counter-rotating Couette-Taylor sys-tem with a large aspect ratio. Visualization of patterncross-section by reflective particles in an aqueous suspen-sion allows us to construct space-time diagrams of in-tensity of light reflected by vortices and to extract withhigh precision frequency and wave number of pattern. PIVtechnique can be applied to a relatively short length (fewvortex pairs) and to a limited time interval. However, itgives quantitative data on velocity components that areessential for flow characterization. In fact, it is possibleto compute vorticity, flow rate deformation, kinetic en-ergy and dissipation rate [41]. Using both flakes visualiza-tion and PIV technique, we have obtained new quantita-tive properties that are important in the understanding ofthe spiral flow dynamics [41,42]. The velocity and vortic-ity fields show clearly that the spiral is composed of twocounter-rotating vortices of different size. The separatricesbetween vortices in spiral vortex pattern are inclined in themeridional plane (θ = const). Unlike the TVF whose vor-tex core is located in the mid-gap and which fills the wholeannular gap, the vortex core in SVF is located near theinner cylinder (fig. 4) in accordance with the Rayleigh cir-culation criterion, although the vortex structure extendsto potentially stable zone (penetrative instability).

The main objective of the present study was the de-termination of nonlinear properties of the spiral vortexflow (table 2). We have shown that spiral vortices havean anharmonicity degree that is comparable with that ofthe Taylor vortex flow while it has larger mirror symme-try breaking. Our results on the asymmetry parameter Pobtained for a Couette-Taylor system with η = 0.8 haveconfirmed recent numerical results of Hoffman et al. [14]

Eur. Phys. J. E (2013) 36: 20 Page 7 of 7

obtained for a Couette-Taylor system with different radiusratio (η = 0.5). This indicates that this parameter mightbe independent of the radius ratio. We have found nonzeroaxial and radial instantaneous flow rates in the spiral vor-tex flow. The nonzero axial flow rate of SVF is a purelynonlinear effect, linear stability analysis gives a zero axialflow [23, 24]. In finite-length systems, the total flow van-ishes since the nonzero axial flow of SVF is compensatedby the axial pressure gradient induced by endplates at thetop and bottom of the cylinders. The presence of a weakaxial flow rate in a spiral vortex flow has been pointed outby different groups in numerical simulations [21–24]. Thevalues found in our experiment are comparable with thoseof Antonijoan et al. [23]. The axial flow rate is a featurenot present in the Taylor vortex flow near the threshold.To our best knowledge, there are no other works whichhave focused on the detailed determination of these non-linear properties in the Couette-Taylor system.

In this experimental study, we have investigated thespatio-temporal properties of velocity and vorticity fieldsof spiral vortex flow in the counter-rotating Couette-Taylor system. The vortex structure and nonlinear prop-erties of the pattern have been well established. Theseresults which are in good agreement with numerical simu-lations results bring complementary information to thoseobtained previously by different authors [1,2,12] and opennew perspectives in the study of transport properties ofcounter-rotating Couette-Taylor flow.

This work has received a financial support from the CPER-Haute-Normandie under the program THETE (Thermo-

Hydrodynamique et Transferts d’Energie).

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