Investigations of counter-rotating Taylor-Couette flow (TCF) in the narrow
gap limit are conducted in a very large aspect ratio apparatus. The phase
diagram is presented and compared to that obtained by Andereck et al. The
spiral turbulence regime is studied by varying both internal and external
Reynolds numbers. Spiral turbulence is shown to emerge from the fully turbulent
regime via a continuous transition appearing first as a modulated turbulent
state, which eventually relaxes locally to the laminar flow. The connection
with the intermittent regimes of the plane Couette flow (pCf) is discussed

Chate, H

Dauchot, Olivier

Gregoire, G

Prigent, A

Saarloos, W,

English

arXiv

Saarloos, W, Van

HAL-CEA

Large-scale finite-wavelength modulation within turbulent shear flows

Investigations of counter-rotating Taylor-Couette flow (TCF) in the narrow gap limit are conducted in a very large aspect ratio apparatus. The phase diagram is presented and compared to that obtained by Andereck et al. The spiral turbulence regime is studied by varying both internal and external Reynolds numbers. Spiral turbulence is shown to emerge from the fully turbulent regime via a continuous transition appearing first as a modulated turbulent state, which eventually relaxes locally to the laminar flow. The connection with the intermittent regimes of the plane Couette flow (pCf) is discussed

Prigent, A.

"Barber pole turbulence" in large aspect ratio Taylor-Couette flow

HAL: Hyper Article en Ligne

Arnaud Prigent

Guillaume Grégoire

Hugues Chaté

Olivier Dauchot

Wim van Saarloos

Crossref

Physical Review Letters

Large-Scale Finite-Wavelength Modulation within Turbulent Shear Flows

We show that turbulent "spirals" and "spots" observed in Taylor-Couette and plane Couette flow correspond to a turbulence-intensity modulated finite-wavelength pattern which in every respect fits the phenomenology of coupled noisy Ginzburg-Landau (amplitude) equations with noise. This suggests the existence of a long-wavelength instability of the homogeneous turbulence regime.Theoretical Physic

Grégoire, G.

Chaté, H.

Dauchot, O.

Saarloos, W. van

Leiden University Scholary Publications

VOLUME 89, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 1 JULY 2002014501Large-Scale Finite-Wavelength Modulation within Turbulent Shear FlowsArnaud Prigent,1 Guillaume Grégoire,1 Hugues Chaté,1 Olivier Dauchot,1 and Wim van Saarloos21CEA–Service de Physique de l’État Condensé, Centre d’Études de Saclay, 91191 Gif-sur-Yvette, France2Instituut-Lorentz, Universiteit Leiden, Postbus 9506, 2300 RA Leiden, The Netherlands(Received 2 October 2001; published 14 June 2002)We show that turbulent “spirals” and “spots” observed in Taylor-Couette and plane Couette flow cor-respond to a turbulence-intensity modulated finite-wavelength pattern which in every respect fits thephenomenology of coupled noisy Ginzburg-Landau (amplitude) equations with noise. This suggests theexistence of a long-wavelength instability of the homogeneous turbulence regime.DOI: 10.1103/PhysRevLett.89.014501 PACS numbers: 47.20.Ft, 47.27.Cn, 47.54. +r, 83.60.WcOne of the fascinating phenomena in fluid mechanicsand pattern formation is the coexistence in space and timeof regions where the flow is disordered with others whereit is laminar. Already noted by Feynman as an outstand-ing problem [1], the origin and structure of these turbulentpatches (spots, slugs, etc.) commonly observed in shearflows at moderate Reynolds numbers R [2–4] remainslargely unknown. Particularly striking is the so-called“spiral” or “barber pole turbulence” regime of the Taylor-Couette flow (TCF), in which typically one helical stripeof disordered fluid motion appears in the otherwise laminarflow between two counterrotating cylinders [5–7]. Simi-larly, while plane Couette flow (PCF) between two platesis linearly stable at all R, it exhibits localized turbulentspots of intriguing dynamical nature [8–10]. In many sys-tems, the first appearance of patterns is associated with aninstability of a homogeneous, essentially noiseless stateto a spatially periodic state; the formation and dynam-ics of such deterministic patterns just above onset can bedescribed in terms of so-called amplitude or Ginzburg-Landau equations.In this Letter, we report a remarkable empirical obser-vation: in our large aspect ratio experiments on both PCFand TCF, localized turbulent patches actually appear as theresult of a well-defined transition to a finite-wavelengthturbulent pattern, which in every respect fits the phe-nomenology of amplitude equations with noise. Sincethe patterns correspond to a spatial modulation of theturbulence intensity, our findings yield empirical evidencethat homogeneous turbulence in PCF and TCF can exhibitan “instability” to a modulated state which, at the phe-nomenological level, is the noisy analog of the primarybifurcation in noiseless pattern-forming systems.Our PCF and TCF experimental setups have alreadybeen described elsewhere [8,9,11]. The PCF results havebeen obtained with a gap d  1.5 mm and aspect ratiosGx  385 along the flow direction and Gz  170 perpen-dicular to it. The TCF cylinders (inner and outer radiiri  49.09 mm and ro  49.96 mm are separated by agap d  0.872 mm giving azimuthal and axial aspect ra-tios Gu  357 and Gz  430. In both experiments, op-tical techniques allow for the visualization of the whole-1 0031-90070289(1)014501(4)$20.00flow seeded with Kalliroscope flakes. In the following, theReynolds number is defined as R  Udn where n is thekinematic velocity of the fluid (water 1 Kalliroscope). Forthe PCF, 6U is the velocity of the planes, while for theTCF the “inner” and “outer” Reynolds numbers Ri,o arebased on U  ri,oVi,o with Vi,o as the inner/outer angu-lar velocity.In Figs. 1a and 1b we show typical snapshots of the flowpatterns of interest here: oblique stripes form a fairly regu-lar pattern with wavelengths of the order of 40 2 60d. Thelargeness of this scale reveals why only a single “turbulentspiral” could be observed in previous TCF experiments[5,6]. Similarly, previous PCF studies could produce onlyone or two turbulent “spots” taking sometimes the shape ofan inclined stripe [8,9]. Laser Doppler velocimetry (LDV)measurements in the TCF case show that both the localvelocity and its fluctuations —which can be taken as ameasure of the turbulent intensity — are modulated withthe same wavelength as the light intensity (Fig. 1c). Thelighter regions are more turbulent. Moreover, the ampli-tude of the modulations of the velocity fluctuations and ofthe light intensity are proportional as the Reynolds num-ber is varied (Figs. 1d). The spiral turbulence regime thuscorresponds to a large-but-finite wavelength modulation ofthe strength of turbulence. In our experiments, the regularpattern was observed for 680 , R , 830 in the PCF, andin a fairly large region of the Ri, Ro parameter plane inour TCF apparatus (Fig. 2a).The similarity between the PCF and TCF patterns goesbeyond the above qualitative description. We first notethat, in the TCF case, the stripe pattern rotates in thelaboratory frame with the mean angular velocity 12 Vi 1Vo. When both the TCF and PCF patterns are station-ary (i.e., when Vi  2Vo for the TCF), they exhibit thesame wavelengths (in units of d) at any Reynolds number(Fig. 2b).As the PCF experiment is rather difficult to control forsmall gaps, we hereafter focus on our TCF data; neverthe-less, all our observations suggest that the essential findingsare common to both experiments.In most of the relevant region of the Ri, Ro parameterspace, the azimuthal wave number is quantized by the© 2002 The American Physical Society 014501-1VOLUME 89, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 1 JULY 2002110 120time00.5v rms (×10)-0.30.00.3〈v〉 (×10)-0.100.10.2v(c)0 0.02 0.04 0.06 0.08∆ I00.030.06∆ν rms(a) (b)(d)FIG. 1. Snapshots of the PCF [(a) R  716] and TCF [(b)Ri  703, Ro  2699]. The flow is in the horizontal directionin both cases. (b) The two vertical lines are due to the opticalsystem. (c) Local axial velocity time series yt (in ms, time ins) measured at midgap and midheight with LDV in the TCF ford  1.85 mm, Ri  670, and Ro  2850. Top: raw signal;middle: running average of yt over windows of 0.4 s; bottom:rms of previous average. The dotted lines indicate that thefluctuations are smallest where the average velocity increasesthrough zero. (d) Dyrms, amplitude of the modulation of the rmsof yt [calculated as in (c)], vs DI , amplitude of the modulationof the light intensity I for various Ri between 640 and 730 [otherparameters as in (c)].system circumference and hence constant: we observe sixstripes. In view of this, it is sufficient to record only thelight intensity I as a function of the z coordinate alonga line parallel to the axis of the cylinder. If we keep Rofixed and decrease Ri the modulation amplitude of thelight intensity and concomitantly the modulation ampli--1000 -500Ro400600800Ri(a)700 800R0204060λzPCFTCF(b)turbulencemodulatedturbulenceintermittencylaminar flowFIG. 2. (a) Phase diagram of our TCF experiment;(b) wavelength of the pattern in the transversal/axial di-rection as a function of R for PCF and TCF (in units of the gapd). For the TCF, Ri  2hRo and R  Ri1 1 h.014501-2tude of the velocity fluctuations grow in a continuousfashion reminiscent of a supercritical bifurcation uponcrossing the transition from the fully turbulent regime atthe top of Fig. 2a to the intermittent regime (we quantifythis below in Fig. 4). Regular patterns such as thoseshown in Fig. 1 break the z ! 2z parity symmetry. Theyare often reached after a transient during which “1z” and“2z” domains, separated by wandering domain walls orfronts, compete (Fig. 3a). However, near the instabilitythreshold (at high Ri), domains of both types fluctuateand new ones get spontaneously “nucleated” (Fig. 3b).These observations lead us to analyze the emergence ofmodulated turbulence stripes within the usual frameworkof amplitude equations by writing Iz, t in terms of twoslowly varying complex fields A1 and A2 [12]:Iz, t  A1z, t expik0z 2 v0t1 A2z, t expi2k0z 2 v0t 1 c.c.,where k0 and v0 are the basic scales of the pattern. Stan-dard demodulation techniques [13] applied to our experi-mental data then yield the amplitudes.We now proceed to a detailed analysis of our Iz, t TCFdata. All results presented below are for an outer Reynoldsnumber Ro  21200 and varying Ri. Similar conclusionswere reached at different Ro values.First we record the variation of A  jA1j 1 jA2j,the mean modulus of the amplitude(s), with the Reynoldsnumber. Figure 4a shows that A2 ~ Rc 2 Ri over somerange; this is the scaling for the pattern amplitude that onenormally finds for a stationary pattern just above the onsetof instability, but do note that the linear law here breaksdown close to Rc: For Ri larger than some value Rnucl 	750, new domains constantly “nucleate” (Fig. 3b). Thisfeature, which inhibits a meaningful measure of the ampli-tudes, is a direct consequence of the fact that we probea transition in a strongly fluctuating (turbulent) regime.This motivates us to model the transition to modulatedturbulence by the usual coupled amplitude equations thatFIG. 3. Space-time diagram of Iz in the TCF at Ro  2850.Time is running upward for 40 s, the full z domain (37.5 cm)is shown horizontally. (a) Ri  740: transient domain com-petition. (b) Ri  775: regime with incessant “nucleation” ofdomains.014501-2VOLUME 89, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 1 JULY 2002600 800Ri00.40.8(ξeff)2/g3(c)0.2 0.25k0.0370.038|A|(b)500 600 700 800 900Ri00.0010.002 |A|2(a)RcRnuclk00FIG. 4. Analysis of the amplitude A of TCF data at Ro 21200. (a) Open triangles: A2  jAj2 vs Ri measured fromspace-time regions containing various 1 and 2 domains. Thisvariation is linear for Ri less than Rnucl, the value beyondwhich spontaneous “nucleation” of 1 and 2 domains is ob-served. Filled triangles: jAjk0 vs Ri determined from analy-sis of monodomain regions. The linear fit yields the criticalthreshold Rc  857 6 5. (b) jAj as a function of wave numberk  ≠zf for Ri  640 from the demodulation of data from amonodomain region. The line is a parabolic fit. (c) jeff0 2g3as a function of Ri, as extracted from the curvature of parabolicfits like in (b).described the emergence and dynamics of patterns justabove onset in deterministic systems, but with an additivenoise term included to model the fluctuations:t0≠tA6  ´ 1 j20≠2z 2 g3jA6j2 2 g2jA7j2A61 ah6z, t. (1)Similar amplitude equations with noise have, e.g., beenused for liquid crystals [14]. Here t0 and j0 are thecharacteristic time and space scales of the modulations ofthe amplitudes, ´  Rc 2 RiRc is the reduced thresh-old, a is the noise strength, and h6z, t is a delta-correlated white noise. Without noise, the above equationsare the standard real Ginzburg-Landau equations fortwo coupled, symmetric, stationary patterns. The basic 1and 2 stripe patterns are taken to be stationary becauseextensive analysis of the data on both PCF and TCF [15]shows that they do not move in the frame in which themean flow vanishes. Apart from parameters like t0 andj0 which set the scales, the coupled amplitude equationscontain essentially two nontrivial parameters. The ratiog2g3 determines the relative suppression of one type ofpattern by the other: if g2g3 . 1 the dynamics drivesthe systems to states with domains where only one of theamplitudes is nonvanishing [12]. Our empirical experi-mental observation that we have either 1 or 2 domains014501-3therefore requires us to take g2g3 . 1 [12]. As we shallsee, the value 1.2 gives a reasonable and self-consistentdescription of the dynamical behavior we observe.The effective noise strength in our simulations of (1) isdetermined as follows. In the noiseless case a  0, theamplitudes in (1) always scale as jAj2 ~ ´. With noise,the noise-averaged amplitudes in our simulations do ex-hibit this scaling for large enough ´, as one would ex-pect, but this linear law breaks down near threshold. Forsmall ´, domain “nucleation” fluctuations become impor-tant, just as we see in our experiments [Figs. 3b and 4a].The crossover value Rnucl depends on the noise strength a[16]. We determine a by the requirement that it yields thesame ´ value for the onset of spontaneous nucleation ofdomains as in the experiment. This gives a  3 3 1023[17] in dimensionless units obtained by writing z and t inthe amplitude equations in units of j0 and t0.We now extract the length and time scales j0 and t0in Eqs. (1) from our experimental data. For deterministicpattern-forming systems, this can be done straight-forwardly from the temporal growth of the amplitudeand the dependence of A6k on wave number k withinsufficiently large domains, but for our case this has tobe done in parallel with numerical simulations, as thefluctuations renormalize the parameters. Thus the “bare”parameters j0 and t0 entering Eq. (1) are renormalizedto effective length and time scales jeff0 and teff0 whichcapture the behavior of simulations and experiments inthe fitting procedure described below.For each set of parameter values in the range whereno nucleation of opposite-amplitude domains is observed,several (typically four) single-domain space-time regionsare selected and the data are demodulated [13]. Thevariation of the local modulus jAj and phase gradient≠zf with the local wave number k  ≠zf is then calcu-lated from all points in each of these space-time domains.The parabolic shape of jAkj2 (Fig. 4b) is characteris-tic of Ginzburg-Landau equations such as Eqs. (1): with-out noise, the amplitude of (near) periodic patterns obeysjA6j2  ´ 2 j20 k 2 k02g3, and we checked numeri-cally that the parabolic shape of jAkj2 is preserved inthe noisy equations. Hence a parabolic fit yields estimatesof ´g3  maxkjAj2, j20g3, and k0. We find that coeffi-cient g3 is only slightly renormalized by the noise, whereasjeff0 can be much smaller than its bare value.Repeating the above data treatment at various Ri val-ues in the region of interest, we find the expected lineardependence of maxkjAj2  ´g3 on Ri (Fig. 4a). In fact,this provides another way of determining the threshold Rc,as well as a precise estimate of the (effective) nonlinearcoefficient g3  156 6 20. The same series of data alsoreveals a variation of jeff0 2g3 with ´ (Fig. 4c). As g3 isconstant and not significantly renormalized by the noise,it is the effective coherence length jeff0 which decreaseswith ´. Numerical simulations of Eqs. (1) reveal that inthe presence of noise the effective coherence length jeff0does vary precisely like in the experiments (Fig. 5a). This014501-3VOLUME 89, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 1 JULY 20020 0.025 0.05ε00.020.04ε/τ0eff(b)0 0.025 0.05ε00.20.40.60.81(ξ0eff)2(a)FIG. 5. Numerical simulations of Eqs. (1), varying ´ forg2g3  1.2, and a  0.003. (a) Effective squared coherencelength jeff0 2 as a function of ´. Note the similarity withFig. 4c with ´ ~ Rc 2 Ri. (b) Effective temporal growth rate´teff0 of jAj following small-range random initial conditions,compared to the noiseless growth rate ´t0 (dashed line).is an important confirmation of the overall relevance of ourapproach in terms of noisy amplitude equations.We have also estimated the effective values of the re-maining coefficients t0 and g2. During quench experi-ments where the system is suddenly brought down froma fully turbulent state, we fitted the growth of the modu-lation to an exponential form. The corresponding growthrate provides a rough estimate of ´teff0 . Our data (notshown) do show a linear variation with ´ which extrapo-lates to zero at a value which is fully consistent with theearlier determination of the threshold, so teff0 can be ex-tracted from the slope. In numerical simulations of thedynamics of Eqs. (1) following small-range random ini-tial conditions, we measure a constant effective time teff0somewhat different from the bare value t0 (Fig. 5b), butthe shift is typically small compared to the experimentalerror bars [15].We also took advantage of the domain walls or frontsseparating A2 and A1 domains to fit the profile of jAjagainst an exponential form jAj  expljzj. In the noise-less amplitude description, j20l2 	 ´g2g3 2 1. Ourfits are consistent with the value g2g3 2 1  1.2 whichaccording to our simulations reproduces all experimentalobservations, but the result is too imprecise to extract asystematic variation with ´. An exploration of fronts insimulations of Eqs. (1) show that the effective value of g2is generally smaller than its bare value [15].In summary, we have presented two sets of remark-able observations: (i) turbulent spirals and spots are essen-tially the same in PCF and TCF, and correspond to a finitewavelength modulation of the turbulence intensity; (ii) allour observations are fully captured by the dynamics ofcoupled amplitude equations with noise. Taken together,these results suggest the possibility of a large-wavelength014501-4instability within fully turbulent shear flows. The preciseorigin of such an instability is at present completely un-known. The work by Hegseth [18] suggests it is related tothe emergence of vortex type structures in the streamwisedirection. Direct numerical simulations or flow visualiza-tion might help to clarify this. Our work also suggests newways of looking at similar issues in other shear flows suchas the Blasius boundary layer, where it has long been ob-served [2,3] that turbulent spots can take fairly regular “V”shapes reminiscent of the inclined stripes reported here, orat the turbulent Taylor vortex flow where phase dynamicsconcepts have already been shown to be relevant [19].We thank Arnaud Chiffaudel for fruitful discussions.[1] R. P. Feynman, Lecture Notes in Physics (Addison-Wesley,Reading, MA, 1964), Vol. 2.[2] H. W. Emmons and A. E. Bryson, J. Aerosol Sci. 18, 490(1951).[3] M. Gad-El-Hak, R. F. Blackwelder, and J. J. Riley, J. FluidMech. 110, 73 (1981).[4] F. Alavyoon, D. S. Henningson, and P. Alfredsson, Phys.Fluids 29, 1328 (1986).[5] D. Coles, J. Fluid Mech. 21, 385 (1965); C. Van Atta,J. Fluid Mech. 25, 495 (1966); D. Coles and C. W. VanAtta, Phys. Fluids Suppl. S120 (1967).[6] C. D. Andereck, S. S. Liu, and H. L. Swinney, J. FluidMech. 164, 155 (1986).[7] J. Hegseth, C. D. Andereck, F. Hayot, and Y. Pomeau, Phys.Rev. Lett. 62, 257 (1989).[8] F. Daviaud, J. Hegseth, and P. Bergé, Phys. Rev. Lett. 69,2511 (1992).[9] O. Dauchot and F. Daviaud, Phys. Fluids 7, 335 (1995).[10] S. Bottin and H. Chaté, Eur. Phys. J. B 6, 143 (1998).[11] A. Prigent and O. Dauchot, Phys. Fluids 12, 10 (2000).[12] See, e.g., M. van Hecke, C. Storm, and W. van Saar-loos, Physica (Amsterdam) 134D, 1 (1999), and referencestherein.[13] V. Croquette and H. Williams, Physica (Amsterdam) 37D,300 (1989).[14] L. Kramer and W. Pesch, Annu. Rev. Fluid Mech. 27, 515(1995).[15] A. Prigent, G. Grégoire, H. Chaté, and O. Dauchot (to bepublished).[16] In fact, the numerically observed scaling defines an effec-tive threshold A2 ~ ´ 2 ´. The shift ´ depends ona, but for realistic values it remains negligible [12].[17] In principle, a could be estimated from more quantitativemeasurements [12].[18] J. J. Hegseth, Phys. Rev. E 54, 4915 (1996).[19] M. Wu, C. D. Andereck, and H. R. Brand, Europhys. Lett.19, 587 (1992).014501-4

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We show that turbulent "spirals" and "spots" observed in Taylor-Couette and plane Couette flow correspond to a turbulence-intensity modulated finite-wavelength pattern which in every respect fits the phenomenology of coupled noisy Ginzburg-Landau (amplitude) equations with noise. This suggests the existence of a long-wavelength instability of the homogeneous turbulence regime

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"Barber pole turbulence" in large aspect ratio Taylor-Couette flow

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