Absolute stability of a Benard-von Karman vortex street in a confined geometry

P. Boniface,1 L. Limat,1 L. Lebon,1 and M. Receveur1

1Matiere et Systemes Complexes, CNRS and Universite Paris Diderot UMR 7057,Batiment Condorcet, 10 rue Alice Domon et Leonie Duquet, 75013 Paris, France

We have investigated the flow induced by a tape moving on a free surface in a rectangular geometry.Depending on the experimental parameters we observed three flow patterns: recirculation at thebottom of the container, lateral recirculation with a stable vortex street and irregular unstationnaryflow. Surprisingly, the vortex street is stable for a continuous range of configurations which contrastswith the von Karman unique stability condition. This is in agreement with ancient calculations fora confined geometry (Rosenhead 1929), this is one of its first experimental confirmation.

PACS numbers: 47.20.Ft, Instability of shear flows, 47.32.ck, Vortex streets, 47.20.-k, Flow instabilities

The Benard-Von Karman vortex street is a structurethat can be observed in the wake of an obstacle immersedinside a stationary unidirectional flow of large enough ve-locity [1]. It has been investigated experimentally for thefirt time by H. Benard [2] and modelled by von Karman[3][4] at the beginning of the XXth century, and since,it has been extensively investigated. This phenomenonis ubiquitous in meteorology, oceanography, naval engi-neering, vehicle design, modelling of the air flows aroundbuildings or piles of bridges, etc [5]. According to VonKarman [3], a vortex street of point vortices in a planeperfect flow can be stable only if the wavelength of thevortex street spatial periodicity, i.e. the distance betweentwo consecutive vortices in one array of the street 2b islinked to the width of the street 2a by

a

b= 1

πargcosh (

√2) ≈ 0.281. (1)

Though this condition is very well satisfied in exper-iments, theoretically, the precise nature of this uniquestable solution, is still under discussion after severalcontroversies [6][7][8][9]. Recently, it has been proposedthat the stability could be in fact only apparent and ofconvective nature, in the sense of Huerre, Chomaz andMonkewitz [10]. The whole vortex arrangement wouldbe in fact intrisically unstable in its own framework [11].

A very different situation arises when the bidimen-sional vortex street is confined between two lateral walls,separated by a distance 2c. Calculations performedon point vortices in the complex plane addressed thisquestion at the end of the 20’s, and found non trivialresults [12][13][14]. Rosenhead, established that in thisconfiguration, equation (1) had to be replaced in theplane (a/c, b/c) by a continuous stability tongue of finiteand non-zero extent. To our knowledge, no experimentalcheck of this result has been published yet, though recentattempts combined with numerical computations beginnow to appear [15][16]. Experimental investigationsof this phenomenon in this configuration is difficult :usual experiments combine the vortex emission behind

the obstacle with the wake growth and reorganizationdownstream. This growth involves complex questionsabout the absolute and convective nature of the vortexsystem stability [17][18].

In this letter, we present a new experimental studyusing a completely different strategy. We show a newway to create double rows of vortices with almostno mean velocity ; their stability can be investigatedfor various confinement degrees [19][20]. We induce alongitudinal flow inside a long rectangular pool usinga tape moving at high speed on a water layer freesurface. The liquid below the tape is dragged at thetape velocity, and a recirculation has to occur in theopposite direction because of mass conservation. If thisrecirculation occurs on the lateral sides of the tank, themixing layers between the two flows can be unstableand two coupled Kelvin-Helmholtz instabilities candevelop [21][22], leading to two counter-rotative vortexrows spatially out of phase, i.e a classic vortex street.The spatial properties of this street can be investigatedvarying the degree of confinement, using moving lateralwalls and varying the water layer depth. Because ofmass conservation the mean velocity calculated acrossa section of the container vanishes which implies thatthe double row of vortices is studied nearly in its ownframework without any mean flow superimposed to it,in absolute stability conditions. This flow geometryhas intrinsically a great technical interest, its geometrycan be viewed as an uncurved equivalent of Couder etal experiments [23] in which vortices are formed andobserved in the vicinity of concentric rotating disks.Both geometries have a great technical interest forapplications in which a thin plate or a disk is movingat high speed between static walls as it is the case formoving belts [24] and hard disks drives [25][26]. Ourgeometry has also, presumably, some implications forthe drag resistance of slender boats advancing in anarrow channel.

The experiment is represented on figure 1. An endlesstape of width 2lb is pulled with a constant speed at the

2

FIG. 1. (a) and (b), detailed drawing of our experiment, inwhich a belt is running at high speed on the central part ofthe free surface of a long slender tank of water. In (c) and(d), are represented the two main patterns of flow observedat large scale: in (c) the recirculation occurs at the bottomof the container and in (d) it occurs on the lateral sides.

free surface of a liquid layer in a long pool, of lengthL = 2 m and with a rectangular cross section : 70cm width versus 20 cm in height. The tape velocityV can vary between 1 to 140 cm.s−1. It is possibleto select smaller lateral length, 2lb < 2c < 70 cm, byusing adjustable walls, while the liquid depth h can beselected at will by removing or adding water before theexperiment. Two tape widths has been explored: lb = 25cm and lb = 5 cm. To investigate the effects of a very thinbelt, we used an endless rope with a diameter section 0.5cm. The container is made of transparent glass, in orderto allow us to use optical diagnostics and in particularparticle image velocimetry (PIV) ; Polyamid seedingparticles (PSP) with a diameter 50 µm were dispersedin the liquid before measurements, and their motion wasfollowed by a tunable laser sheet and a high speed videocamera. The particles motion is analyzed and gives theflow velocity field, its instantaneous streamlines or thevorticity distribution in a selected plan.

Depending on the tape velocity and the aspect ratios,three very different kinds of flow have been observed.The liquid immediately below the tape is dragged in onedirection while mass conservation under a closed channelimplies that the liquid must somewhere recirculate inthe opposite direction. At low tape velocity, all thefree surface follows the tape motion, while the liquidrecirculates by the bottom of the container (figure 1.c).Above a critical velocity, the flow structure changesdrastically: the recirculation occurrs mainly along thechannel walls (figure 1.d). A layer of high shear intensitydeveloped between the flow directly dragged by thetape and the flow that follows the lateral containerwalls, which gives rise to eddy formation via the fa-mous Kelvin-Helmholtz instability. Depending on the

FIG. 2. (a) Experimental measures of the Reynolds number atwhich we observe a transition from the recirculation by thebottom flow to the lateral sides. (b) Phase diagram of thethree flow patterns observed for h = 45 mm : recirculationby the bottom (blue), recirculation along the lateral wallswith double vortex row formation (green), recirculation alongthe lateral walls with no stable array (red). (c) and (d) showtypical instantaneous flow lines observed in the two last cases.

aspect ratios of the geometrical lengths involved (2lb,h, 2c), this eddy formation can either give rise to auniform turbulent state represented in figure 2.d, inwhich vortices of any sizes are observed with complexbehaviors (coalescence, appearance, disappearance...),or to a coherent state represented in figure 2.c withlarge structures adopting the geometry of a Benard-VonKarman vortex street, each row being placed on eachside of the floating tape in spatial phase oppositionwith each other. Beautiful streets were also obtainedwith the rope pulled at the free surface of the container(figure 3), but in this case, only this kind of flow wasobserved in our experimental conditions. In all the casesexplored, the vortex streets were static or nearly staticin the framework of the container. The drift velocitywas exactly equal to zero for the experiments with thesmallest values of lb, hence the rope, while a slight butnoticeable drift in the direction opposite to the tapemotion was noticed in the belt cases, with a maximalvelocity smaller than 8% of the tape velocity.

The critical velocity Vc at which the recirculation flowstops to occur at the bottom and starts around the lateralwalls of the container has been measured upon liquiddepth h for two different tape widths, lb = 25 cm andlb = 5 cm, and two different lateral confinements, lf =c − lb = 10 cm and 15 cm. The results are plotted on

3

FIG. 3. Typical flow observed in a horizontal plane, at themiddle heigh of the water layer, inside the container, usingthe rope. As the vortex street does not move, these are tem-poral means calculated over 30 seconds. (a) liquid velocitycalculated by PIV, (b) stream lines, (c) vorticity distribution.

figure 2.a, in which we have defined the Reynolds numberas Re = hV

ν. Slight variations of temperatures have been

taken into account in the estimation of the ν value [27].All the data seem to collapse on a single straight line,which implies that the Reynolds at the transition Retonly depends on the depth of the water h, and is ruledby the following empirical law

Ret = Ah −B. (2)

with A ≈ 1.3 × 105 m−1 and B ≈ 950. This correspondsto a belt velocity at the transition Vt = ν (A − B

h) who

tends empirically to Vt(h → ∞) = 13 cm.s−1, at 20○Cin water. The mechanism of this behavior remainsmisunderstood, though the order of magnitude of Vtsuggests a link with capillarity-gravity wave drag at thefree surface [28]. This could explain how the tape canentrain so easily the whole free surface in a certain rangeof velocity. Having however scaled the tape velocityon this empirical scale, we have built phase diagramsspecifying when each of the three flows were observed.A typical example is reproduced on figure 2.b, in whichthe ratio r = lb/c plotted on the horizontal axis measuresthe degree of confinement applied to the obtained array,as most often the distance between each rows seemed tobe close to lb. It appears that the double row of vorticesis unstable when the confinement is high, but becomesstabilized in its own framework, when the confinementis sufficiently low. These observations are consistentwith the ideas defended recently that the double row isin fact intrinsically unstable in an absolute framework,but can be stabilized apparently by convective effects,or can be really absolutely stabilized by the action of

FIG. 4. (a) Sketch of the vortex street with the definitionof the variables a, b and c. (b) Representation of the stabil-ity area calculated by Rosenhead. The lower part of stabilityzone, when 1/c tend to 0, coincides with von Karman condi-tion giver by equation (1), but progressively enlarges at largervalues of the b/c ratio. We plotted all our experimental data,with the rope setup or the belt setup. It appears that the sta-bility area predicted by Rosenhead is explored and satisfiedby the experimental measures.

walls confining laterally the flow [11].

The precise geometry selection of the vortex streetis a complex question. As suggested on fig. 4-a, thesystem can chose two typical lenghts : the width of thesreet 2a and its waveleght 2b. We have investigateda and b dependance upon lb and c. The experimentalvalues are highly fluctuating in time, and so is theresidual drift velocity of the street. In this paper we willonly focus on the possible correlation between a andb that should replace equation (1) in a confined geometry.

The stability of a vortex street in a bidimensionnalchannel has been extensively investigated at the end ofthe 20’s, in the framework of point vortices in the com-plex plane. The complex potential of the flow can becalculated exactly in the base state by adding the ve-locity flows induced by the two rows of vortices and alltheir successive images by the walls ; this guarantees theboundary condition of a vanishing normal velocity at thewalls. Following Rosenhead [12], the complex potentialreads:

w(z = x+iy) = − iκ2πlog⎡⎢⎢⎢⎣

ϑ1 ( z2b − ia2b∣ 2icb)ϑ3 ( z2b − i

a2b∣ 2icb)

ϑ2 ( z2b + ia2b∣ 2icb)ϑ4 ( z2b + i

a2b∣ 2icb)

⎤⎥⎥⎥⎦.

(3)

4

where x is the streamwise coordinate, y the transversecoordinate and κ the circulation of the vortex. The func-tions ϑi(u, τ) are the elliptic functions of Tannery andMolk [29], that read, for q = eiπτ :

• ϑ1(u∣τ) = 2∑n(−1)nq(n+ 12)2 sin((2n + 1)πu),

• ϑ2(u∣τ) = 2∑n q(n+12)2 cos((2n + 1)πu),

• ϑ3(u∣τ) = 1 + 2∑n qn2

cos(2πnu),

• ϑ4(u∣τ) = 1 + 2∑n(−1)nqn2

cos(2πnu).

Rosenhead investigated the stability of the vortex street,having the potential of equation (3). He found thatthe stability of this system was reached in a tongue offinite extent, in the plane (a/c, b/c) that is suggested onfigure 4.b. We plotted our fluctuating values of a andb, for stable vortex streets obtained with a belt and arope, on figure 4.b. All the data are dispersed insidethe Rosenhead stability area and are not accumulatingon the von Karman limit given by equation (1). Thisremarkable result indicates that the presence of thewalls modifies the stability of the street, allowing for acontinuous band of ratios b/a, instead of the ”sharp”selection observed on isolated wakes. As mentionedabove, a particularity of our experiment is that we arein conditions of zero superimposed mean velocity tothe array, which implies that our result holds for theabsolute stability of the double row.

To conclude, we realized a new experiment to studythe absolute stability of a Benard-Von Karman streetin its own framework. In contrast with the well-known selection given by (1), we observed that thesystem explores a continuous band of stability, in away consistent with Rosenhead’s calculations. Thisobservance is presumably the first and presently theunique experimental confirmation of this theory. Itwould be now interesting to check also what happensfor a wake behind an obstacle confined between twowalls. The present result is presumably important forthe diverse applications mentioned at the beginning ofthe paper, and presumably, the drag encountered bya moving slender boat inside a narrow channel, themomentum carried by the vortices being to be addedto the more classical wave drag due to surface waves [28].

Aknowledgments. We are indebted to stimulating dis-cussions with C. Arratia, J.-M. Chomaz, P. Ern, F.Galaire, C. Gondret, C.-T. Pham, M. Rabaud, L. Tuck-erman and J. E. Wesfreid.

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