F Giorgiutti

L Laurent

F Daviaud

Coupling of rotating water jets by surface waves

Keywords: number͑s͒: 47.20.Ϫk, 68.15.ϩe, 68.10.Ϫm

Above a critical flow rate, a single jet impinging from below on a water-air interface oscillates while arrays of jets exhibit collective behaviors. The aim of the paper is to explain the physics governing such an array. First, a series of experiments shows that the instability mechanism leading to the oscillation of a single jet is due to a feedback effect between surface distortion and the unstable jet as in the ''jet-edge'' system and that several jets are coupled through surface waves. Then a modelization is proposed in which jets are considered as oscillators close to their limit cycle coupled by delayed action to take into account the wave propagation. A good agreement is obtained between the predictions of the model and experiments on sets of two or four jets. The long range of the surface waves and the large number of oscillators render this jet configuration very similar to idealized phase coupled nonlinear oscillators investigated in theoretical models but with a not instantaneous coupling ensured by traveling waves. This allows in particular the existence of various synchronization modes, which appear as geometrical patterns. ͓S1063-651X͑98͒07407

I. INTRODUCTION

A lot of theoretical work has been done on large populations of coupled oscillators because of their interest to represent various physical or biological systems with a large number of degrees of freedom ͑see, for instance, ͓1-3͔͒. On the experimental side, several attempts have been made to design extended systems that can be considered as such a set of oscillators. Some of them use fluids perturbed by a spatially periodic action that to some extent behave as coupled oscillators such as coupled wakes ͓4͔, air jets ͓5͔, magnetically driven vortices ͓6͔, and to a lesser extent coupled thermal boundary layers ͓7͔. Modelization of these systems has often been done by using a nonlinear amplitude equation modified to take into account the coupling with the neighboring oscillators. The exact nature of the coupling term depends on the underlying physics and various schemes have been used such as an additional drive depending linearly on neighboring amplitude ͓4,6͔, an additional nonlinear term ͓6͔, or a delayed modification of the nonlinear saturation term ͓5,7͔. Another interesting system consists in using vertical jets close to a water-air interface. It can be easily extended to configurations with a large number of oscillators. When they are isolated, such jets can rotate or oscillate at constant amplitude ͓8,9͔. When they are associated in arrays, phase locks and a wide variety of phase patterns are observed that depend on the symmetries of the network ͓10͔.

Up to now, the understanding of the jet rotary instability was only partial, in particular the observed frequency dependence ͑decreasing with the jet velocity͒, which is quite unusual for standard jet instability. Moreover, no quantitative model for the coupling and the phase pattern has been proposed. The two aims of this paper are ͑i͒ to provide a clearer understanding of the jet rotation and ͑ii͒ to present the first model able to predict quantitatively the coupling between jets. In the next section, after a brief presentation of the experimental setup, experiments concerning the single jet instability ͓11,12͔ are presented. A model of the oscillation is discussed based on the ''jet tone'' concept ͓13͔, which provides the right scaling of the frequency. Then, in Sec. III, simple arrays of two or four jets are investigated experimentally and a model is presented to explain the observed phase patterns. In Sec. IV, the results are discussed. The model is tested by comparing the predicted and observed frequency and modes of oscillation for various sizes and flow rates. Then to explain the unlocking of jets when they are separated by a too large distance, the effect of a slight detuning of the jets is investigated. supplied from an elevated reservoir ͑to avoid any vibration due to a pump͒ through a flow meter and a regulation valve. Its top part is a removable 6-mm-thick perforated plate located a few millimeters below the surface. Various types of plates can be used. For the present paper, square networks of holes with a radius bϭ0.5 mm are used and flow velocities at the hole, and U lies between 0.5 and 0.7 m s Ϫ1 leading to the Reynolds numbers (Ub/) in the range 250-350. The whole device is immersed in a 110 l plexiglass constant level water tank. In some experiments the surface tension was varied in the range 50-75ϫ10 Ϫ3 Nm Ϫ1 using soap. To control the depth h of the jets below the surface ͑typically 6 mm͒, the vertical position of the jet nozzles can be adjusted with a micropositioner. More details can be found in ͓10͔. When one of the jets reaches the water-air interface, there is a stationary axisymmetric bump. Its radius bЈ is determined by the equilibrium between the jet pressure UЈ 2 ͑where is the water volumic mass and UЈ is the jet velocity at the bump͒ and the surface resistance, which scales as /bЈ ͑ is the superficial tension͒. When U is larger than a threshold U f so that this equilibrium is no longer possible, i.e., UЈ is so large that equilibrium would require bЈ smaller than b, the surface breaks and the system looks like a ''small fountain.''

II. PROPERTIES OF THE JET AS AN OSCILLATOR

A. Experimental setup

B. Observations

Above a critical velocity UϭU c the axisymmetric equilibrium of the bump becomes unstable and it starts rotating around the jet axis ͑Fig. 2͒, while generating surface waves with a spiral geometry ͑Fig. 3͒. The frequency range of these oscillations lies between 10 and 50 Hz, and measurements can be made either with stroboscopic lighting for visualization or intensity of a light beam reflecting on the bump towards a photodiode for time evolution. Experimental results about frequency can be summarized in the following way: ͑i͒ for a given geometry the frequency decreases with the jet velocity ͓Fig. 4͑a͔͒, ͑ii͒ it increases with surface tension ͓Fig. 4͑b͔͒, ͑iii͒ when the depth, or U, is varied, the most relevant parameter seems to be the bump height bЈ ͑where primes denotes quantities related to the bump͒ as shown in Fig. 4͑b͒ of ͓10͔.

It appears that the jet rotation can be synchronized when excited with surface waves in the range of frequencies corresponding to the free rotation one. This was shown in a series of experiments, which present some similarities with the oscillating edge experiment by Staubli and Rockwell ͓13͔. Surface waves were generated on the water tank through a blade connected to the coil of a loudspeaker. The evolution of the bump oscillation with the forcing frequency has been followed. Two phase-locking areas have been seen for forcing frequencies close to the bump frequency or in a region at lower frequencies. For other frequencies both oscillations, i.e., forcing waves and bump oscillation coexist ͑see Fig. 5͒.

C. Discussion

The observed rotation of the bump suggests that the jet is subjected to a helical instability. This is compatible with the Reynolds number range. Moreover, previous works demonstrated that a few diameters away from the nozzle, when the shear layer has widened, this mode is the one with the largest growth rate ͓11,12͔. However, for the free jet, the most unstable are of order of 0.5U/b, which corresponds to frequencies of the order of 100 Hz for the present experiment, i.e., typically three times the experimental ones. In addition, the observed frequency decreases with jet velocity instead of increasing. An additional frequency-selection mechanism must be invoqued. The water-air interface, which is not in the standard theory, provides this ingredient. The helical instability distorts the surface and this causes a pressure fluctuation along the jet path, which can be amplified by the jet. Such an amplification is well known and examples are given in ͓14,15͔. Indeed the strength of the feedback will depend on the amplitude of the surface distortion. When the helical instability distorts the bump, surface tension tends to restore the initial shape, which corresponds to the equilibrium between the surface tension and the pressure. The spring force scales as bЈ, i.e., the product of the surface tension by a characteristic length. To restore the initial shape, this force has to modify the momentum of the water inside the bump, which scales as UЈbЈ 3 . The characteristic time of the bump relaxation is the ratio of these two quantities, i.e., ϭUЈbЈ 2 /. Thus the feedback between the bump and the jet is expected to be optimum when the latter rotates at an angular velocity 0 , so that 0 r ϭ1, i.e., 0 is proportional to /bЈ 2 UЈ.

Using the flow rate conservation bЈ 2 UЈϭb 2 U, the scaling of frequency should be 0 ϭG(U)/b 2 U, where G(U) is an unknown dimensionless number that takes into account the numerical factor and the actual geometry of the flow, which may vary with the jet velocity. In Fig. 6, b 2 U 0 / ͓i.e., the expected G(U)͔ is plotted as a function of the velocity of the jet for various water velocities and depth. The fact that this quantity is almost constant ͑except for high U, i.e., low 0 ͒ shows that the model for oscillation is correct and moreover that geometry distortion due to variations of U is not too large. In conclusion, taking into account the value of G deduced from Fig. 6 one gets 0 ϭ͑0.9Ϯ0.1͒/b 2 U. ͑1͒

It is also possible to eliminate UЈ in the relation 0 Ϸ/bЈ 2 UЈ using the equilibrium discussed in Sec. II A between the stress UЈ 2 and superficial tension /b, i.e., UЈ is proportional to b Ϫ0.5 so that 0 should approximatively scales with b Ϫ1.5 . This curve is compatible with experimental results ͓10͔ although the exponent cannot be determined accurately ͑between Ϫ2 and Ϫ1.5͒.

As discussed at the beginning of the section, these frequencies are two or three times lower than the optimum for the amplification by the jet. However, taking a reasonable velocity profile across the jet, the linear theory still predicts an instability with Re(k z ) and Im(k z ) of the order of 1000 and 100 m Ϫ1 , respectively ͓12͔. The amplification takes place on distances l lying between bЈ and h so that its value e Im(k z )1 will be in the range 1-1.5. If the coupling between the pressure modification due to the bump distortion and the jet is not too lossy, the system can oscillate. To explain the rotation start at UϭU c , the following scenario is likely: when U is low, the Reynolds number is too low for an amplification process to take place. When U increases above U c , Im(k z ) reaches a value such that the oscillation is possible. The most favored frequency is 0 since the feedback on pressure is optimum at the bump resonance.

The loudspeaker experiment can be interpreted along the same lines. When the incoming waves distorts the bump efficiently i.e., when its frequency is close to 0 it can lock in phase the jet instability. This is not contradictory with the presence of another region at lower frequency, where there is a phase locking that could correspond to the direct excitation of the jet by the surface wave.

III. COUPLING OF OSCILLATORS IN SIMPLE GEOMETRIES A. Experimental results with simple arrays

Results of Sec. II suggest that when the system contains several jets, they can be coupled. This has been investigated for two types of jet configurations.

͑i͒ The first one consists in two jets separated by distances 6ϫ10 Ϫ3 ϽaϽ12ϫ10 Ϫ3 m, the plate being 6 mm deep. When a is too large ͑above 12ϫ10 Ϫ3 m͒ the coupling disappears. The flow rate was varied, limited on the lower side by the threshold for jet oscillation, and on the upper side by the fountain regime. This corresponds to the single jet frequency range: 240Ͻ 0 Ͻ300 s Ϫ1 ͓Eq. ͑1͔͒. The two bumps oscillate at the same frequency, with the same amplitude ͑note that the photodiode technique does not allow an accu-rate amplitude measurement͒ and the same direction of rotation. There are two differences with the single jet case: ͑i͒ the critical flow rate for the bump oscillation appearance is lower for two bumps than the one for a single one, ͑ii͒ the frequency is different from the single jet one 0 as shown in Fig. 7. The results obtained during 0 scans ͑i.e., U scans since 0 decreases as U increases͒ for various values of jet spacing a, are presented in Table I. The second column represents the phase difference between the motion of the two jets. Two collective modes are observed: an acoustic mode where the two bumps oscillate in phase ͑phase difference of 2͒, and an optical mode where the two bumps oscillate in phase opposition ͑phase difference of ͒. When there is a transition, the symbol at left ͑right͒ represents the mode at low ͑high͒ 0 . In this case the value of 0 at the transition is displayed in the third column. A transition between two modes is accompanied by a phase and frequency jump ͑always positive when 0 increases͒, which is indicated in the last column. Around the transition there is a domain where the system hesitates between the two modes with a typical half-width ⌬ 0 ϭ10 s Ϫ1 . Similar results are obtained for lines of five jets.

͑ii͒ The second network is made of four jets displayed on squares of different sizes. The plate is still at 6 mm from the surface. For this network the onset of the threshold velocity is lower than for two jets so that the range of 0 that can be explored is larger, between 240 and 310 s Ϫ1 . As for the twojet network, bumps oscillate at the same frequency at qualitatively the same amplitude and the same direction of rotation. The two modes are recovered: in the optical mode the two adjacent jets on the square oscillate in phase opposition while in the acoustic mode all jets are in phase. Again there are transitions and a perturbed area in the transition zone. The results are presented in Table II.

B. Modelization

A natural model to fit the data should be a network of jets oscillating all at the same frequency with the same direction of rotation and coupled through surface waves. One considers first jet m. It is defined by its position Z m ϭX m ϩiY m , and the bump displacement is z m ϭx m ϩiy m . When the stationary state is reached, z m ϭ␣ m e i 0 t with ␣ m its slowly varying complex amplitude and m ϭarg(␣ m ) represents the phase of the motion. When jet m is alone, the rotation occurs with a pulsation 0 so that the argument of ␣ m does not depend on time. This behavior is well represented by the following equation:

d␣ m dt Ϫ␥͑U͒͑1Ϫ␣ m ␣ m *͒␣ m ϭ0,

͑2͒

where an asterisk denotes complex conjugate. The second term represents the effect of the jet instability, which amplifies the motion at a rate ␥, ␥ is real ͑positive when UϾU c ͒, and which saturates for ͉␣ m ͉ϭ1 ͑amplitudes have been normalized to the free jet saturation amplitude͒.

One considers now an array of jets, coupled through surface waves. This can be more easily studied when the bumps rotate in stationary conditions at the frequency , i.e., when ␣ m (t)ϭ␣ 0 e i(Ϫ 0 )t . Note that Ϫ 0 Ӷ 0 so that ␣ m (t) is a slowly varying function. Surface waves have a welldefined wave number k(), which is the solution of the surface wave dispersion relation for deep water ͓in our case the product k()ϫ(depth) is about 5͔ ͓16͔:

k͑ ͒gϩk͑ ͒ 3 gl c 2 Ϫ 2 ϭ0 with l c 2 ϭ g . ͑3͒
A given jet n radiates a spiral capillary wave, which depends on the flow velocity U and on the bump displacement. Dur-FIG. 7. Evolution of the pulsation for two coupled jets with aϭ11 mm ͑squares͒, and for one single jet ͑circles͒ as a function of the jet velocity. ing its propagation between jets n and m separated by a distance r mn , the wave is attenuated by geometrical effect ͑energy flux proportional to 1/r in cylindrical geometry͒ and other dissipation mechanisms. The total attenuation will be included in a function f (r). The wave experiences also a phase shift given by k()r mn . The simplest coupling term in the amplitude equation compatible with surface waves leads to the following equation:

i͓Ϫ 0 ͑ U ͔͒␣ m Ϫ␥͑U ͒͑ 1Ϫ␣ m ␣ m *͒␣ m ϩ͑U ͒ ͚ n m
f ͑ r mn ͒e Ϫik͑͒r mn ␣ n ϭ0. ͑4͒

The function 0 (U) is the free jet frequency, which can be measured directly by single jet experiments ͓see Fig. 4͑a͔͒ and is the frequency of the motion in the array. The coupling constant (U) contains the wave amplitude and the physics of the forcing effect of a wave on the helical jet instability. This model allows us also to recover the fact that the threshold is lower when there are several jets. For instance, if one considers now in Eq. ͑4͒ a small amplitude oscillation ͑instead of a saturated one͒, the real part reads

Im ϭϪ␥͑U͒ϩRe ͩ ͚ n m f ͑ r mn ͒e Ϫik͑͒r mn ͪ ϭ0. ͑5͒
Even for UϽU c so that ␥ is negative, the growth rate can be positive in accordance with the observations described in Sec. III A.

There are two parameters in our dynamical system: ␥(U) the growth rate of the free jet instability, the coupling constant (U). In addition, the function f (r) must be adjusted. For the sake of simplicity, it will be taken equal to e Ϫr . The parameters ␥,, do not depend on the geometry of the jet lattice ͑a dependence in the nozzle diameter and depth is not excluded but all experiments have been made taking these two parameters constant͒. In Sec. III D, experiments will be simulated fitting ␥(U), (U), and by simple number. Thus the parameters and ␥ will be taken independent of U, i.e., the notation ''function of U'' will be skipped.

C. Analytic determination of phase patterns of two-or four-jet arrays

Two-jet network

The system is described by a set of two equations, which admit obviously solutions with ͉z 1 ͉ϭ͉z 2 ͉ϭ͉␣͉. A mode is defined by the phase shift between both jets ⌬ϭ 2 Ϫ 1 . The system ͑4͒ becomes i͑Ϫ 0 ͉͒␣͉Ϫ␥͑ 1Ϫ͉␣͉ 2 ͉͒␣͉ϩe Ϫa e Ϫik͑͒a ͉␣͉e i⌬ ϭ0, ͑6a͒ i͑Ϫ 0 ͉͒␣͉Ϫ␥͑ 1Ϫ͉␣͉ 2 ͉͒␣͉ϩe Ϫa e Ϫik͑͒a ͉␣͉e Ϫ⌬ ϭ0. ͑6b͒

This system has solutions if Eqs. ͑6a͒ and ͑6b͒ have the same as solution. Two modes are allowed, ⌬ϭ0 or ⌬ϭ. Their frequencies and amplitudes are obtained from the real and imaginary parts:

Ϫ 0 ϩ͉͉e Ϫa Im͓e i͓Ϫk͑ ͒aϩarg͑ ͔͒ ͔ϭ0, ͑7a͒ Ϫ␥͑1Ϫ͉␣ ͉ 2 ͒ϩ͉͉e Ϫa Re͓e i͓Ϫk͑ ͒aϩarg͑ ͔͒ ͔ϭ0 ͑7b͒
with equal to Ϫ1 or 1, respectively, for the optical and the acoustic mode. From Eq. ͑7a͒ it is possible to deduce the frequency and amplitude of both modes. A simple approximation can be obtained using Ϫ 0 Ӷ 0 , i.e., taking the coupling terms evaluated for ϭ 0 :

FIG. 8. Functions ( 0 ) and ␣ ( 0 ) defined by Eq. ͑8͒; dotted line ͑acoustical mode͒, full line ͑optical mode͒. The numerical values chosen here are aϭ8.5 mm and e Ϫa ϭ15e Ϫ3i/4 . →2 at 0 ϭ300 →2 at 0 ϭ320 →2 at 0 ϭ311 →2 at 0 ϭ300 ϭ 0 Ϫ͉͉e Ϫa sin͓Ϫk͑ 0 ͒aϩarg͑ ͔͒, ͑8a͒

͉␣ ͉ 2 ϭ1Ϫ ␥ ͉͉e Ϫa cos͓Ϫk͑ 0 ͒aϩarg͑ ͔͒. ͑8b͒
For 0 , i.e., U values so that Ϫk( 0 )aϩarg()ϭ/2 ͑mod ͒, the two modes have the same amplitude ͑equal to 1͒. At these points the frequency difference between the two modes is 2͉͉e Ϫa . In the Appendix it is shown that only the mode that has the largest amplitude is stable. This means that experimentally only this mode should be seen. An example of this prediction is shown in Fig. 8, which illustrates one of the computations discussed in Table III. The functions ␣ ( 0 ) and (0) are shown for aϭ8.5 mm and ϭ15e Ϫ3i/4 . According to the prescription that the observed mode is the one with the largest amplitude, a jump from acoustic to optical mode should be observed at 0 around 275 s Ϫ1 .

Four-jet network

One considers now a network with four jets on a square. Introducing the experimental mode structure, i.e., ͓0000͔ and ͓00͔ in Eq. ͑4͒, one obtains the equation equivalent to Eq. ͑7͒:

͑ Ϫ 0 ͒ϩ͉͉Im͑ 2e Ϫa e Ϫik͑͒aϩi arg͑ ͒ ϩe Ϫa& e Ϫik͑͒a&ϩi arg͑ ͒ )ϭ0, ͑9a͒ Ϫ␥͑1Ϫ͉␣ ͉ 2 ͒ϩ͉͉Re͑ 2e Ϫa e Ϫik͑͒aϩi arg͑ ͒
ϩe Ϫa& e Ϫik͑͒a&ϩi arg͑ ͒ ͒ϭ0, ͑9b͒

with equal to 1 or Ϫ1, respectively, for the acoustic and the optical mode. These equations provide the evolution of the amplitude and the frequency of each mode with the natural frequency 0 . Again, within the approximation Ϫ 0 Ӷ 0 , the two modes have the same amplitude for FIG. 9. Pulsation of the motion of jets displayed on a square (aϭ11 mm) as a function of the free jet pulsation ( 0 ). Experimental results for a depthϭ6 mm ͑squares͒, solution of Eq. ͑8͒ ͑circles͒, numerical simulation ͑triangles͒, with ϭ120, ␥ϭ60, ϭ33.3e Ϫiϫ3/4 . The lines with double arrows on the graph represent the transition areas between the acoustic and the optical mode. Ϫk( 0 )aϩarg()ϭ/2 ͑mod ͒, while the frequency difference at the transition between the two modes is 4͉͉e Ϫa .

IV. COMPARISON WITH THE EXPERIMENTAL DATA

A. Mode transition

In Table III, one shows a comparison between the experimental data and the ones obtained with the analytical treatment for values of arg() in an optimum range, i.e., around Ϫ3/4. The second column of Table III contains experimental data from Sec. III A while the last three columns give the modes with the largest amplitudes obtained from Eqs. ͑7͒ or ͑9͒. arg() has been determined by trying to fit all the observed transitions ͑their location depends on ͉͉ only through corrections of order (Ϫ 0 )/ 0 , as can be seen in Table IV͒, or at least the observed modes when the values of jet spacing a correspond to windows without transition. The computations are made in the window 250Ͻ 0 Ͻ325 s Ϫ1 for two and four jets. The predicted modes are in quite good agreement with the experimental ones, especially if one takes into account the error bars about ϩ/Ϫ10 s Ϫ1 on 0 at transition. In the following arg()ϭϪ3/4 will be taken.

B. Strength of the coupling

In a second step, ͉͉ and can be found from expression ͑7a͒ using the two-jet experiment. For both modes, experimental frequencies should follow the relation

lnͯ 2 Ϫ 0 2 cos"k͑ ͒aϩ/4… ͯϭl n͉͉Ϫa. ͑10͒
If the value of can be measured as a function of a, it should be possible to guess ͉͉ and . However, due to the small range of a and the low value of attenuation, the accuracy is poor. Several combinations of and are possible provided ͉͉e Ϫa stays in the same range. This is shown in Table IV, which displays a comparison between the five experimental mode transitions given in Tables I andII and the simulated ones taking two combinations of ͉͉ and .

To summarize, from Tables III andIV, it appears that the amplitude, nature, and location of modes can be reasonably fitted by the model especially if one takes into account the small number of parameters to be fitted: One ͓arg()͔ to fit all the transition ͑location and direction͒ and two ͓modu-lus͑͒ and ͔ for the values of frequencies ͑with little effect͒. The value of ␥ plays almost no role ͑it has been taken equal to 60͒. An example of comparison theory-experiment of ( 0 ) for a four-jet system is shown in Fig. 9. The transition has been simulated in two ways. First there is the largest amplitude mode given by Eq. ͑9͒. One admits then that the system cannot choose between the two modes if it exists a difference of less than 10% between the amplitudes of the modes. This assumption allows us to determine a small transition zone ͑''analytical transition''͒ where the two modes, like in the experiments, are unstable. This is possibly due to desynchronization as discussed in the next section. Second, there is the result obtained by evolving in time the jet motion using a scheme described elsewhere ͓17͔.

C. Effect of a scattering of oscillator properties

When a increases beyond 12 mm, the synchronization becomes poor and eventually is lost. Another experimental feature is that, in the transition zone ͑when the amplitudes of the two modes are close together͒ the system is unstable. A possible explanation could be the unavoidable dispersion in the 0 values, which tends to hamper the synchronization. This is a well known effect for coupled oscillators ͓1͔. This effect has been investigated more quantitatively for a two-jet system. One assumes that there is some scatter in the 0 values, i.e., that they become 1,2 ϭ 0 Ϯ⌬. In the presence of coupling, the motion of jet m can be expressed as ␣ m (t)e Ϫi 0 t . Provided it is slowly varying, the evolution of ␣ m (t) obeys the following equation:

d␣ m dt Ϫi͑ m Ϫ 0 ͒␣ m Ϫ␥͑1Ϫ␣ m ␣ m *͒␣ m ϩ ͚ n m
e Ϫa e Ϫik͑ n ͒r mn ␣ n ϭ0. ͑11͒

Equation ͑11͒ has been solved using an iterative scheme to determine the n values to be used in the coupling coefficients ͓17͔. For the system chosen ͑aϭ8.5 mm as in Fig. 8͒ there is a predicted transition acoustic→optical at 0 ϭ291 s Ϫ1 ͑observed at 280 s Ϫ1 ͒. For ⌬ϭ0, these modes are found by the evolutive computation. When ⌬ is nonzero, depending on the e Ϫa value, there exists a zone ⌬ Ͻ⌬ max where a synchronization is possible. The value of ⌬ max depends on k( 0 )aϩarg() and ͉e Ϫa ͉. This domain is displayed in Fig. 10͑a͒ for 0 ϭ240 s Ϫ1 ͑far from the mode transition͒ and 295 s Ϫ1 ͑close to the mode transition͒. The synchronization domain depends on k( 0 )a, i.e., the phase of the interaction between the jets. In particular, close to the transition point, the system becomes very sensitive to any defect, i.e., is hard to synchronize. This is expected since at this location the eigenvalues associated to the mode stability analysis are close to zero ͑see the Appendix͒. This is shown in Fig. 10͑b͒, which displays the value of ⌬ max for ͉e Ϫ ͉ϭ15 as a function of 0 . This simple model explains two experimental findings. ͑i͒ The synchronization becomes difficult for aϾ12 mm. According to Table IV, the values of e Ϫa at this point range between 16 (ϭ120) and 32 ( ϭ50). For such coupling parameters, according to Fig. 10͑b͒, the desynchronization occurs for ⌬ about 10-20 s Ϫ1 , i.e., an 0 scatter in the range 5-10 %, i.e., according to Eq. ͑1͒ a scatter in the hole diameter of order of 5%. ͑ii͒ The width of the unstable domain in Fig. 10͑b͒ is compatible with the width of the transition zone in experiments.

V. CONCLUSION

The mechanism for the oscillation of a single vertical jet impinging on an interface has been investigated. It turns out that a feedback mechanism between surface waves and the unstable jet is at the origin of the oscillations of the bump to the surface. An analogy can be made with the ''jet-edge'' system if considering a ''resonant edge'' mechanically attached to springs. Indeed, the stiffness of the springs is here equivalent to the surface tension of the fluid. In a dimensional model, the frequency is found to be proportional to the surface tension and conversely proportional to the jet velocity. These results are in good agreement with the experimental results obtained when varying the surface tension of the fluid, the velocity of the jet, and the depth of the nozzle.

Considering simple ensembles of jets, a phase-locking mechanism has been observed. When the jet velocity, i.e., the resonant frequency is modified and two modes are observed, namely an acoustic mode and an optical one. A modelization in which each bump is a nonlinear oscillator coupled with its neighbors through surface waves gives reasonable agreement with the experimental results, especially if one takes into account the scatter of jet frequencies due to imperfections. Generalization of this model to infinite networks is possible at the expense of a more complex formalism, which will be presented in a forthcoming paper ͓17͔. It appears that systems of jets are very close to the large population of coupled oscillators investigated by theoreticians. There is, however, one important difference, the coupling is ensured not by instantaneous interaction ͑reflected by a coupling constant͒ but by traveling waves propagating at finite velocity. This plays a significant role in the formation of phase patterns observed in large regular networks.

APPENDIX: STABILITY ANALYSIS OF THE LIMIT CYCLE IN THE TWO JET CONFIGURATION

Following the notation of Sec. III C, the two jet system has two limit cycles described by Eq. ͑6͒, the acoustic mode with an amplitude at saturation ␣ 1 and a frequency 1 and the optical mode with an amplitude ␣ Ϫ1 and a frequency Ϫ1 . The phase shift between the bumps of a given mode is , which is equal to zero for ϭ1 and otherwise. The equations governing the eigenmodes are i͑ Ϫ 0 ͒Ϫ␥͑ 1Ϫ␣ ␣ *͒ϩe ϪaϪik͑ ͒a e i ϭ0.

͑A1͒

One considers that the system is at the limit cycle for the mode ͑ϭϩ or Ϫ1͒ so that the envelope of rotation of bump m (mϭ1,2) is close to ␣ e i( Ϫ 0 )tϩim . A small perturbation corresponding to a mode Ј is added which is of the form f (t)e i( ЈϪ 0 )tϩim Ј . Thus, the envelope of the position of bump m is ␣ m ϭ␣ e ͓i( Ϫ 0 )tϩim ͔ ϩ f (t)e i( ЈϪ 0 )tϩim Ј and is as follows: After multiplication by e Ϫi ЈtϪim Ј , an evolution equation for f (t) can be deduced from the linearized equation ͑A1͒,

d f dt ϩi͑ ЈϪ 0 ͒ f ͑t͒Ϫ␥ f ͑t͒ϩ2␥␣ 2 f ͑t͒ ϩ f ͑t͒e Ϫa e Ϫik͑ Ј͒aϩi Ј ϭϪ␥␣ 2 f *͑t͒e 2i͑t͒ .

͑A3͒

Using Eq. ͑A1͒ expressed for the limit cycle of mode Ј, Eq. ͑A3͒ can be simplified. Taking the complex conjugate, a similar equation can be obtained for f *(t),

d f ␥dt ϭ͑␣ Ј 2 Ϫ2␣ 2 ͒ f ͑t͒Ϫ␣ 2 f *͑t͒e 2i͑t͒ ,

Figure 1

 1 Figure1describes the experimental setup. The jets flow out of a tranquilization chamber containing a 5-cm-thick porous medium. The chamber is filled by pressurized water

FIG. 2 .

 2 FIG.2. Picture of a rotating bump for UϾU c . The jet diameter is equal to 1 mm, the height and width of the bump are of the same order, about 3 mm.

  FIG. 4. ͑a͒ Evolution of 0 with the jet velocity U. ͑b͒ Effect on 0 of a change in surface tension obtained by adding soap to water.

  FIG. 5. Effect on the bump pulsation 0 of a launched surface wave ͑blade connected to a loudspeaker excited by a sinusoidal tension͒. The natural value of 0 is 126 s Ϫ1 .

  d␣ m dt Ϫ␥͑1Ϫ␣ m ␣ m *͒␣ m ϩe ϪaϪi 0 t ͓e Ϫik͑ ͒a ␣ e i͑ tϩ ͒ϩe Ϫik͑ Ј ͒a f ͑t͒e i͑ Јtϩ Ј ͒ ͔ϭ0. ͑A2͒The zero-order equation gives the equation of the limit cycle of mode . The terms can then be developed to first order in f . The linearized terms ared␣ m dt → ͩ d f dt ϩi͑ ЈϪ 0 ͒ f ͑t͒ ͪ e i͑ ЈϪ 0 ͒tϩim Ј , ␣ m ␣ m *␣ m →␣ 2 e i͑ ЈϪ 0 ͒tϩim Ј ͓2 f ͑t͒ϩ f *͑t͒e 2i͑t͒ ͔with (t)ϭ( Ϫ Ј)tϩm( Ϫ Ј ).

FIG. 10 .

 10 FIG.10. ͑a͒ Synchronization domain for 0 ϭ240 s Ϫ1 ͑crosses͒ and 295 s Ϫ1 ͑stars͒ for a two-jet system with aϭ8.5 mm in the space Xϭ͉e Ϫ ͉ and Y ϭ⌬; ͓arg() is kept equal to Ϫ3i/4͔. ͑b͒ Value of Y ϭ⌬ max as a function of Xϭ 0 for e Ϫ ϭ15e Ϫ3i/4 and aϭ8.5 mm.

TABLE I .

 I Modes in a two-jet array for 240Ͻ 0 Ͻ300 s Ϫ1 .

	a (m) Phase difference	0 at transition (s Ϫ1 )	jump at transition (s Ϫ1 )
	0.006	→2	275	220→220ϩ80
	0.0085 2→	280	226→226ϩ56
	0.009	2→	270	226→226ϩ56
	0.011

TABLE II .

 II Modes in a four-jet array for 240Ͻ 0 Ͻ310 s Ϫ1 .

	a (m)	Phase difference ͑neighboring jets͒	0 at transition (s Ϫ1 )	jump at transition (s Ϫ1 )
	0.0085 2→	295	250→250ϩ40
	0.009
	0.011 →2	300	250→250ϩ56

TABLE III .

 III Experimental modes and simulated ones for three values of arg͑͒.

			Result of Eqs. ͑7͒ or ͑9͒ for ͉͉ϭ33.33 and ϭ120
	a (m) jets	Transition at 0 (s Ϫ1 )	arg()ϭϪ5/8	arg()ϭϪ3/4	arg()ϭϪ7/8
	0.006 2J	→2 at 0 ϭ275	→2 at 0 ϭ277	→2 at 0 ϭ256	2
	0.0085 2J	2→ at 0 ϭ280	2→ at 0 ϭ310	2→ at 0 ϭ291	2→ at 0 ϭ273
	0.009 2J	2→ at 0 ϭ270	2→ at 0 ϭ287	2→ at 0 ϭ270	2→ at 0 ϭ255
	0.011 2J				→ 2 at 0 ϭ300
	0.0085 4J	2→ at 0 ϭ295	2→ at 0 ϭ295	2→ at 0 ϭ273	2→ at 0 ϭ255
	0.009 4J		2→ at 0 ϭ271
	0.011 4J

TABLE IV .

 IV Mode transitions for three values of ͉͉ and and arg()ϭϪ3/4.

	a (m)	Experimental
	number of jets	jump (s Ϫ1 )

Solution of Eqs. ͑7͒ ͑two jets͒ or ͑9͒ ͑four jets͒ ͉͉ϭ60 ϭ50 ͉͉ϭ100 ϭ120 ͉͉ϭ320 ϭ250 0.006 2J 220→220ϩ80 230→230ϩ70 220→220ϩ80 210→210ϩ90 at 0 ϭ275 s Ϫ1 at 0 ϭ255 s Ϫ1 at 0 ϭ255 s Ϫ1 at 0 ϭ255 s Ϫ1 0.0085 2J 226→226ϩ56 250→250ϩ55 250→250ϩ60 250→250ϩ60 at 0 ϭ280 s Ϫ1 at 0 ϭ293 s Ϫ1 at 0 ϭ290 s Ϫ1 at 0 ϭ291 s Ϫ1 0.009 2J 226→226ϩ56 240→240ϩ55 240→240ϩ65 240→240ϩ60 at 0 ϭ270 s Ϫ1 at 0 ϭ270 s Ϫ1 at 0 ϭ270 s Ϫ1 at 0 ϭ269 s Ϫ1 0.0085 4J 250→250ϩ40 250→250ϩ90 240→240ϩ80 230→230ϩ80 at 0 ϭ295 s Ϫ1 at 0 ϭ262 s Ϫ1 at 0 ϭ258 s Ϫ1 at 0 ϭ258 s Ϫ1 0.011 4J 250→250ϩ56 250→250ϩ90 260→260ϩ90 260→260ϩ80 at 0 ϭ300 s Ϫ1 at 0 ϭ310 s Ϫ1 at 0 ϭ310 s Ϫ1 at 0 ϭ312 s Ϫ1

Since (t) is a slowly varying function, the system can be considered as a linear operator on f . The eigenvalues are

given by the zeros of the determinant 2 Ϫ2(␣

2 and ␣ Ј 2 Ϫ␣ 2 . Both roots have a negative real part, i.e., the system is stable if and only if the limit cycle corresponds to the mode with the largest amplitude.


Daviaud

S Bottin

O Dauchot

F Daviaud

Intermittency in a Locally Forced Plane Couette Flow

Keywords: numbers: 47.20.Dr, 47.15. -x, 47.20.Ky, 47.27.Cn

In spite of impressive progress, the problem of the transition to chaos and turbulence still offers numerous open issues. Both manifest themselves as irregular regimes. Temporal chaos is associated with systems where few degrees of freedom are active owing to confinement effects, for which scenarios have been identified [START_REF] Schuster | Deterministic Chaos[END_REF]. By contrast, turbulence involves many spatial and temporal scales. In between one finds spatiotemporal chaos where the dynamics of ordered structures remains an important ingredient [START_REF] Cross | [END_REF].

In trying to identify steps in the transition process, a first distinction can be made from the nature of the primary bifurcation. Supercritical transitions are characterized by continuous growth of the distance to the basic state in the direction of some known unstable mode. Rayleigh-Bénard convection is a typical example. A scenario then develops, for which classical tools of (weakly) nonlinear analysis are available. Another, strongly different, situation occurs when the primary bifurcation is subcritical. In that case, our understanding is much more limited and entirely relies on our capability to determine branching solutions that become stable at finite distance of the basic state. In the realm of hydrodynamics, the plane Couette flow (PCF) corresponds to this situation since it is known to be linearly stable for all Reynolds numbers [3], whereas transition to turbulence is observed in experiments [4]. The PCF therefore seems to be a good prototype for studying "globally subcritical" transitions to turbulence [5], and has been the subject of numerous studies accordingly.

In order to reach those nonlinear branching solutions the most direct way is to excite the flow by an instantaneous localized perturbation which serves as the initial condition [6][7][8]. Turbulent spots generated in that way present a core of small scale turbulence surrounded by streaky structures which are also observed when they relax [9]. Another possibility has been explored by considering slight modifications of the original PCF configuration numerically [10]. Experimentally, this has also been done by introducing a thin wire parallel to the spanwise direction by Dauchot and Daviaud [11] who succeeded in stabilizing streamwise counterrotating vortices. Further investigations have con-firmed their role in the transition at the limit of the pure PCF [9]. In this context, we first study the transition scenario involving the breakdown of the streaks along the wire leading to a quasi-1D spatiotemporal regime. Then, in an attempt to understand it, we consider a modified experiment where the localization of perturbations around a bead placed in the middle of the flow yields a temporally intermittent dynamics.

Our PCF setup is made of an endless transparent plastic band guided by rollers achieving a shear flow without mean advection (see [6] for details). The Reynolds number is defined as R Uh͞n, where U is the speed of the wall, h 3.5 mm is half of the gap, and n is the kinematic viscosity of water. The accuracy on the determination of R is within 3%. Geometrically, we define the streamwise direction as x, the normal to the wall direction as y, and the spanwise direction as z. The test section is about 1 m long and using 2h as the length scale, the spanwise aspect ratio is G z ഠ 35 and the streamwise aspect ratio is G x ഠ 190. The flow can be forced in two ways. First, as in [11] a thin wire of radius r͞h [ ͓0.0035, 0.1714͔, parallel to the spanwise direction, is introduced in the central plane. Second, as shown in Fig. 1, a little bead of radius r͞h 0.35 is placed in the central plane and chosen as the origin ͑0, 0, 0͒ of the reference frame. The bead is held by a thin wire parallel to the streamwise direction that does not disturb the flow, as shown in [9]. The ͑ y, z͒ structure of the flow close to the perturbation is revealed Its temporal behavior is studied using another laser sheet in the ͑x, z͒ plane, illuminating the flow seeded with iriodin. Figure 2 displays the bifurcation diagram of the flow perturbed by the thin wire when varying r͞h. Upon increasing R, the first bifurcation of the basic state, observed at R R 0 , is towards streamwise vortices extending all around the wire. For sufficiently large r͞h typically .0.02 the following sequence is observed: At a larger R R 1 , a second bifurcation takes place, where the streamwise vortices are destabilized by localized turbulent bursts. A complex intermittent spatiotemporal regime sets in: A turbulent burst develops at some place, lasts for a while, interacts strongly with the streamwise vortices, and then relaxes. This process repeats itself elsewhere later. Such a regime is usually called spatiotemporal intermittency, at least in a loose sense [START_REF] Chaté | Turbulence: A Tentative Dictionary[END_REF]. For R . R 2 , the turbulent bursts merge to form a sustained band of turbulence all along the wire. Now, decreasing R from above R 2 , the band of turbulence remains sustained down to R 3 Ӎ R 2 . By contrast for smaller r͞h, the transition from streamwise vortices to turbulence is essentially direct and R 1 Ӎ R 2 , while hysteresis is observed for the reverse transition as R is decreased ͑R 3 Ӎ 325 6 5͒. A striking feature of this bifurcation diagram is that, throughout the r͞h range, sustained turbulence can be observed above R R 3 Ӎ 325 which is precisely the value found for the minimal Reynolds number beyond which turbulent spots can be generated in pure PCF [8].

The perturbation introduced by the wire preserves translational invariance in the spanwise direction. As a consequence, a quasi-one-dimensional regime of (provisionally called) spatiotemporal intermittency, is obtained. By contrast, using the bead, i.e., a pointwise perturbation, gives rise to spatially localized processes. Below some For R . R 0 0 , four pairs of stationary counterrotating streamwise vortices are indeed stabilized: two on each side of the bead. Figure 3(b) displays a ͑ y, z͒ cut through the two pairs of vortices on the right hand side of the structure in Fig. 3(a). These vortices are similar to those observed along the wire and described in previous studies [9,11]: They occupy the whole gap, their separation is of the same order as the wavelength of the streamwise vortices, and their symmetries are comparable. The pairs of vortices are labeled (a, b, c, and d) as shown in Fig. 3(a). Pairs a and b are symmetrical with respect to a reflection about the ͑x, y͒ plane. Pairs a and c are axially symmetrical with respect to the z axis.

For R . R 0 1 ͑Ӎ300͒ the vortices become unstable and bursts of turbulence are observed around the bead. However, turbulence is not sustained but intermittent. Preliminary observations indicate complex processes involving vortex merging, creation, and breakdown which are still under study. Turbulent bursts interrupting laminar intermissions are spatially similar to turbulent spots generated by instantaneous perturbations [6][7][8]. They can develop to become as wide as the span L z . Turbulent domains so created then collapse (relaminarization) homogenously all over their surface. A relaxation towards a brief stage of featureless flow is observed. Afterwards the pairs of streamwise vortices, henceforth called the laminar state, soon reappear.

For R . R 0 2 325, when a turbulent burst occurs, it no longer decays but turns into a sustained turbulent spot. No hysteresis is observed when decreasing R ͑R 0 3 R 0 2 ͒. The global phenomenon is very robust and does not seem to be sensitive to boundary effects. Several experiments have been performed with different layouts, varying the bead diameter, its position, and the noise level. Whereas the thresholds were slightly dependent on the experimental conditions, the intermittent behavior could be observed on roughly the same range of Reynolds numbers. Quantitatively, the intermittent regime has been characterized from the spatiotemporal record of the light diffused by iriodin particles along a line x͞h 4 in the central plane y 0. An excerpt of the signal is displayed in Fig. 4(a), corresponding to about 10 min. Thanks to the z to 2z symmetry, only half ͑z . 0͒ of the signal needs to be presented and further processed.

Eight experiments of 1 h each have been recorded at a rate of 5 lines per sec, for R 300, 306, 310, 314, 320, and 330. Information has been compressed by detecting the spanwise location z max of the maximum of the average intensity I͑z͒ ͑1͞z͒ P i1,i max I͑z i , t͒, where i [ ͓1, 256͔ is the index of the pixel in the line corresponding to a given instant of time t. The temporal signal z max ͑t͒ is then processed in order to detect the laminar-turbulent alternation. The mean lifetime of laminar and turbulent events ͗t lam ͘ and ͗t turb ͘ are then computed, as well as the fraction of time spent in the turbulent state F t , called the turbulent fraction. For R 300 and 330 less than 20 events were observed; the flow is still too close to the two thresholds R 0 1 and R 0 2 . For R 306, 310, 314, and 320 the number of events varies between 40 and 100. ment accuracy for R, as well as for the turbulent fraction, whereas it represents the standard deviation for the laminar and turbulent mean lifetime distribution. The large value of the standard deviation is due to the low number of events at R 300, 330 and also reveals the proximity of the thresholds R 0 1 and R 0 2 . For the other Reynolds numbers, it is more representative of the width of the distribution since there are enough samples to study the statistics. A detailed study of the distribution is in progress, but we can already mention that two kinds of events can be distinguished: long and short ones. As expected, F t increases with R, and its rapid variation allows a good determination of the intermittency threshold R 0 1 slightly below 300. ͗t lam ͘ increases rapidly while ͗t turb ͘ decreases to the finite value Ӎ240 sec upon decreasing R to R 0 1 . When increasing R, F t saturates to its maximum value F t 1 for R 0 2 325 that we recognize as the value beyond which turbulence around the wire becomes sustained.

The destabilization of the streaks around the bead, analyzed here as a temporal process, thus shares strong similarities with the spatiotemporal behavior observed around wires of sufficiently large diameter (see Fig. 2). Whereas turbulent bursts were nucleated at random places along the wire, they are now attached to the bead. In some sense, the structure is spatially frozen and spatiotemporal intermittency is converted into temporal intermittency, which suggests to interpret phenomena within the theory of lowdimensional dynamical systems. In this framework, several routes to chaos have been recognized [START_REF] Schuster | Deterministic Chaos[END_REF]; however features characteristic of known scenarios (e.g., [START_REF] Pomeau | [END_REF][14][15]) have not been clearly identified up to now. Some enlightening is expected from the detailed study of the streak breakdown and relaminarization steps that is currently underway [16]. Interestingly, the same kind of process has been observed by Coughlin [17] in numerical simulations of the PCF in confined geometry and can be understood in terms of homoclinic bifurcation of the kind present in simple models such as that derived by Waleffe [18]. The dynamics observed around the bead can then be interpreted as the result of a spatial freezing of the complex spatiotemporal dynamics observed along the wire resulting from such a mechanism. The obtained regime of temporal intermittency, typical of low-dimensional systems, in fact develops in an extended system. As such it keeps a trace of the large number of degrees of freedom that can be easily excited in the spatiotemporal regime. This observation explains why it has been empirically difficult to make a connection with standard scenarios but also suggests implications for the study of the transition to turbulence in other weakly confined systems.

FIG. 1 .

 1 FIG. 1. Modified Couette flow apparatus.

FIG. 2 .

 2 FIG. 2. Evolution of the phase diagram with r͞h for the wire.

FIG. 3 .

 3 FIG. 3. Photograph of the flow displaying (a) four streaks or pairs of streamwise vortices in the ͑x, y͒ plane close to the bead. (b) Section in the ͑ y, z͒ plane of two pairs of counterrotating streamwise vortices.

FIG. 4 .

 4 FIG. 4. (a) Excerpt from a spatiotemporal diagram (see text for detail). (b) The corresponding time series z͑t͒.

Figure 5

 5 Figure5displays the evolution of these quantities with the Reynolds number. The error bars indicate the measurement accuracy for R, as well as for the turbulent fraction, whereas it represents the standard deviation for the laminar and turbulent mean lifetime distribution. The large value of the standard deviation is due to the low number of events at R 300, 330 and also reveals the proximity of the thresholds R 0 1 and R 0 2 . For the other Reynolds numbers, it is more representative of the width of the distribution since there are enough samples to study the statistics. A detailed study of the distribution is in progress, but we can already mention that two kinds of events can be distinguished: long and short ones. As expected, F t increases with R, and its rapid variation allows a good determination of the intermittency threshold R 0 1 slightly below 300.

FIG. 5 .

 5 FIG. 5. Evolution of the turbulent fraction (a), the mean lifetime of laminar ( b), and turbulent (c) events, with the Reynolds number.

We thank P. Manneville for fruitful discussions and helpful remarks, G. Rousseau and J. F. Pinton for their help in data treatment, and C. Gasquet for technical assistance.


Angel Garcimartı ´n

Nathalie Mukolobwiez

Franc ¸ois Daviaud

Origin of waves in surface-tension-driven convection

Keywords: number͑s͒: 47.20.Dr, 47.20.Bp, 47.27.Ϫi

Waves appear in a liquid layer with a free surface if a sufficiently high horizontal temperature gradient is imposed. These waves have been compared to the hydrothermal waves predicted by a linear stability analysis of a parallel flow. However, depending on the experimental configurations, significant differences with theory are found. We show that there exists another kind of wave that cannot be explained by previous analysis. Our aim is to investigate which is the mechanism leading to this instability. Differential interferometry is used to obtain quantitative information on the temperature field. Experimental evidence is presented suggesting that these waves are the result of a boundary layer instability: the roll near the hot wall begins to oscillate, and the perturbations are dragged and amplified downflow. This mechanism could explain discrepancies between theory and some experimental observations. ͓S1063-651X͑97͒04108-1͔

I. INTRODUCTION

A fluid layer heated from one side ͑and cooled from the opposite͒ gets into motion, no matter how small the imposed temperature difference ⌬T between the end walls is. Two unbalanced forces set up a global flow. The first one arises from the change of the surface forces with temperature, also called the Marangoni effect. The second one is gravity, acting upon the density variations with temperature. The two main experimental parameters, the temperature difference ⌬T and the depth of the fluid layer h, can be expressed in terms of two nondimensional numbers: the Marangoni number Ma and the Rayleigh number Ra. We use the following definitions for Ra and Ma:

Raϭ

␣g␤h 4 , Maϭ

d dT ␤h 2 , ͑1͒
where ␣ is the thermal expansion coefficient, g is the gravity, ␤ is the imposed ͑horizontal͒ temperature gradient, h is the layer depth, is the kinematic viscosity, is the thermal diffusivity, is the surface tension, and is the dynamic viscosity. The ratio Ra/Ma, sometimes called the dynamic Bond number, depends on h 2 , and gives the relative importance of the gravity and surface tension forces. Thus for small h the Marangoni effect dominates, while for large depths gravity effects overcome the surface tension forces.

There is still another nondimensional number: the Prandtl number Pr, defined as Prϭ/, that accounts for the relative importance of thermal conduction and viscous dissipation. The geometry of the container is defined by two aspect ratios, namely, ⌫ x and ⌫ y . The first one corresponds to the distance between hot and cold walls divided by the height of the fluid layer; the second to the transversal dimension of the cell divided by the height.

In this work, we examine shallow layers, where surface tension effects are more relevant than gravity. The liquid used has a moderate Pr number, moderate meaning between 10 and 100, so the velocity follows the temperature field. Considerable interest in this configuration comes from several industrial processes taking place in similar situations: the fabrication and purification of high-quality crystals, electron-beam vaporization of metals and laser welding, for instance. Experiments are being carried out to explore the possibility of manufacturing crystals in space, where gravity is negligible and only surface tension forces are relevant. The knowledge of the mechanisms leading to instabilities as the control parameter ⌬T increases should help to improve these industrial procedures.

An analytical expression for this basic flow can be found, and experiments ͓1͔ showed that this description is accurate. Except near the end walls, the flow is horizontal, from hot to cold near the surface, and in the other sense near the bottom. It is therefore two dimensional in the core. Starting from this situation, several studies-both theoretical and experimental-have tried to describe the flow structure as ⌬T is increased. As soon as the basic flow becomes unstable, different kinds of phenomena have been observed in several experiments, some of which differ from the theoretical analyses.

Smith and Davis ͓2͔ performed a stability analysis for an infinite fluid layer with a free surface, where a constant horizontal temperature gradient was imposed. They did not take into account neither gravity nor heat exchange to the atmosphere. They found the most dangerous modes for a range of Pr numbers and provided the corresponding instability thresholds. For a range of small to moderate Pr numbers, they found that the instability was oscillatory, coining the term ''hydrothermal waves.'' Some experiments, carried out independently by several groups ͓1,3-5͔ indeed showed the existence of waves. But some intriguing features were also found that could not be *Present address: Laboratoire de Physique, Ecole Normale Supe ´r- ieure de Lyon, 46 Alle ´e d'Italie, F-69364 Lyon Cedex 07, France. Electronic address: agarcima@physique.ens-lyon.fr explained within the theoretical results. Both Schwabe et al. ͓4͔ and Ezersky et al. ͓5͔ observed a different sequence of states as they increased the control parameter ⌬T in a large ⌫ x configuration. In fact, before waves showed up, stationary corotative rolls were observed with their axes oriented perpendicularly to the temperature gradient. This new state had not been predicted. Villers and Platten ͓1͔ performed careful velocity measurements on a narrow channel (⌫ y Ϸ1) differentially heated at the ends, and they also found stationary corotative rolls that have been studied in new experiments with large ⌫ x ͓6͔. While the basic flow was found to be correctly described by theory, these stationary corotative rolls are clearly a depart from it. Daviaud and Vince ͓3͔ used a cell with a large ⌫ y and small ⌫ x . They found rolls for large depths. In this case, the rolls are counter-rotating and oriented in the same direction as the temperature gradient.

Other theoretical and numerical studies followed, starting from a different hypothesis. Gravity, for example, was taken into account ͓7,8͔, and the Biot number-which gives the heat loss to the air-was assumed to depart from zero ͓9͔. It was consistently found that instabilities were waves propagating with a given angle against the temperature gradient, or stationary counter-rotating rolls aligned with the temperature gradient, depending on the value of the parameters ͓9͔.

In the experiments it is observed that starting from rolls a further increase of ⌬T leads to the appearance of waves, which were found and described experimentally by Daviaud and Vince ͓3͔, Schwabe et al. ͓4͔, and Ezersky et al. ͓5͔ in different geometries. Oscillations were also found by Villers and Platten ͓1͔, but the constraints of their experimental setup did not allow a detailed study. Some features of the waves-such as frequency-seemed to fit the theoretical descriptions, but there were, however, significant discrepancies. In some cases, waves were found to travel at an angle and/or direction different to the predicted ͓10,4͔. In other cases ͓3͔, waves resembled more closely the theoretical studies ͓9͔, traveling at a certain angle against the temperature gradient.

In summary, theoretical analyses reproduce some features, but none can explain all the observed behaviors. Here we address the question of the physical mechanism lying at the origin of the waves, in an effort to elucidate these discrepancies. In order to do that, we have set up an experiment to observe the hot end of a fluid layer. We chose interferometry because it is a noninvasive yet sensitive method.

In Sec. II, experimental procedures are described. Results are presented in Sec. III. A brief discussion-where we venture a possible explanation of the phenomenon-is elaborated upon in Sec. IV. In Sec. V, some conclusions are provided.

II. EXPERIMENTAL SETUP

Three liquids have been used: silicone oils 47V5 and 47V0.65 from Rho ˆne-Poulenc, and decane. Their Prandtl numbers are respectively 30, 10, and 15 at 20 °C. The phenomenology presented in this paper is basically the same for these three liquids. In the following, we will just give one relevant figure of any of the liquids. It should be understood that the physical parameters for which the same phenomenon is observed in the other liquids may change.

The box used in this experiment is made of glass of good optical quality whose surfaces are polished to one wavelength or better. It is 10 cm long and 1 cm wide. The width of the cell cannot be increased significantly without producing so large a deflection in the laser beam that interferometry becomes impractical. Two metal prismatic pieces, whose section is 1ϫ1 cm 2 , are placed inside the box, at the two ends, to provide isothermal heating and cooling. Water is circulated through these pieces from thermal baths where temperature is fixed with a precision better than 0.05 °C. A thermocouple is attached to each metal piece near the bottom to register the temperature at the end walls. The air temperature is also monitored, although the heat exchange to the atmosphere cannot be controlled.

Various interferometry setups are commonplace. As typical temperature gradients have an order of magnitude of 1 °C/cm, an extremely sensitive apparatus would produce a lot of interference fringes for our system. Instead, a differential interferometry, in one of the variations of the Jamin interferometer ͓11͔, has been used. We provide a sketch of our setup in Fig. 1. A polarized He-Ne laser of 5 mW is filtered and collimated. The expanded beam is sent horizontally to the convective box, perpendicularly to the temperature gradient. After going through the liquid, the beam arrives at a glass flat polished to one wavelength, where the beam is reflected in both the first and second surfaces. These two reflections interfere. The interferogram is captured with a standard video camera and sent to a computer for acquisition and processing. A video recorder connected to the system allows one to register a movie of dynamic events.

In our setup, the two interfering beams have both passed through the liquid. In other interferometers, for instance in the Mach-Zender interferometer, the beam is split and one of the beams travels outside the object under study, thus providing an absolute reference when recombining with the beam that has gone through the object. This is not the case here. The information that can be obtained from our interferometer concerns the temperature difference between the zones crossed by the rays that interfere. If they are close enough, one can take the local temperature gradient instead of the temperature difference. If we call k the interference

͑2͒

In this formula, l is the length of fluid crossed by the laser beam, i.e., the transversal dimension of the layer; ␦e is the spacing between two interfering rays when entering the cell, sometimes called the lateral shear of the apparatus, and its value can be calculated from the thickness of the optical flat and its inclination; is the light wavelength; dn/dT is the variation of the refractive index with temperature; and dT/dx is the local temperature gradient in the direction of the shear introduced by the interferometer. In our setup, l is 1 cm; , the wavelength of the He-Ne laser, is 0.6348 m, and the lateral shear ␦e is 3 mm and has the same orientation than the temperature gradient ͑see Fig. 1͒. It follows that an interference fringe, whose k is constant, is also a line where the local temperature gradient is constant. All the factors multiplying the temperature gradient can be calculated from the properties of the interferometer except dn/dT. We have obtained this value by measuring n with an Abbe refractometer for the liquids we use. The values of dn/dT obtained for 47V5 and 47V0.65 silicone oils and decane are, respectively, Ϫ3.7ϫ10 Ϫ4 °CϪ1 ,Ϫ6.3ϫ10 Ϫ4 °CϪ1 , and Ϫ4.9 ϫ10 Ϫ4 °CϪ1 . The relative precision of the coefficient of ␦(dT/dx) in Eq. ͑2͒ is about 10%.

III. RESULTS

Our primary interest is to characterize the fluid layer destabilization. In particular, we want to study the waves which emerge above a temperature gradient threshold. We first turn to the basic flow, which is a well-characterized state, to test our experimental method. To understand the meaning of the interferogram, we need to obtain a relation between the temperature field and the interference order k. This is an inverse problem for which the general solution is not easy to find. But we can attempt to extract some quantitative information from the interferometry if we include previous knowledge, gathered by other means, on the temperature field. We indeed know that corotative rolls appear for given values of the control parameters; if we take a temperature field sinusoidally varying along both x and y, and calculate the constant gradient lines, we find a pattern of concentric fringes. Things are not so simple in the real system, where the rolls are inclined, but every pair of these concentric patterns can be identified with a roll ͑Fig. 2͒.

Temperature measurements with thermocouples provide an independent test for the interferometry. However, we have estimated that thermocouple probes-even if they are very small-may yield a systematic error as high as 0.2 °C. Taking this remark into account, we present, in Fig. 3, the temperature profile along the cell obtained with a thermocouple at the depth where the amplitude of the rolls is larger. After subtracting the constant temperature gradient between the two walls, the roll at about 2.5 cm from the hot side is found to have an amplitude of about 0.3 °C.

The roll shown in Fig. 2 has been obtained under the same conditions, i.e., the same Ma and Ra, and at the same place.

Assuming a constant background temperature gradient-note that a constant temperature gradient gives no fringes-we can calculate the roll amplitude. We proceed in the following way. We take the light intensity along a line going through approximately the middle of the roll ͓see Fig. 2͑b͔͒. For every two consecutive maxima of light intensity in the image, ␦(k) is equal to 1. The temperature gradient for these points is given by Eq. ͑2͒. The gradient for every intensity minimum can also be obtained, but an interpolation for all the curves is not reasonable because of the nonlinearities in the detection system. Once a set of temperature gradients is obtained at different points along the line, a numerical integration is performed on their interpolation, and the temperature along that line is obtained: see Fig. 2͑c͒. The amplitude of FIG. 2. ͑a͒ Interferogram obtained at 2.4 cm from the hot wall ͑left side͒. The dimensions of the zone displayed are 3 mm high ͑the whole liquid layer͒ by 7 mm. Note that at the points marked with X the temperature gradient is the same than in the background. ͑b͒ Light intensity along a segment passing through the two points marked with X in ͑a͒. The two points marked with X are indicated here with arrows. To find the amplitude of the roll, superimposed on a constant temperature gradient, the temperature gradient is taken to be zero at those points for the calculation. A temperature gradient can be assigned to each maximum and minimum of this line after Eq. ͑2͒. ͑c͒ The temperature profile resulting from the above-mentioned calculation ͑the origin of temperatures is arbi-trary͒. One can see that a concentric set of fringes is equivalent to half a roll.

the roll is about 0.25 °C, which is in agreement with the thermocouple measurements. Assuming the temperature field will adopt some known configuration, one can therefore extract quantitative information from the interferometry.

Above a temperature gradient threshold, the flow becomes unstable, and the rolls begin to oscillate. Indeed, we observed waves propagating from the hot side to the cold side. So we tried to visualize the region just near the hot wall, imposing a temperature difference such as to produce waves in the fluid layer. A roll whose amplitude is much larger than that of the other rolls in the cell can be seen there. This roll begins to oscillate: above a certain temperature threshold, waves are released. They are detached from the roll near the surface ͑Fig. 4͒. The tip of the roll observed in Fig. 4 moves back and forth. Each time it retracts, it releases a ''drop'' of hot fluid near the surface; cold fluid is seen to rise from the bottom and to join the hot ''drop.'' This can be seen in the interferogram as two concentric sets of fringes. The ensemble is dragged downflow, and constitutes one wave ͑in Fig. 5, one of these waves is shown at the middle of the cell͒. The picture we get from this scenario is quite similar to that observed by Dubois and Berge ´in Rayleigh-Be ´nard convection ͓13͔. They identified the oscillation mechanisms with boundary layer instabilities. In our case, these instabilities cause the release of a perturbation with a perturbation that is dragged by the flow.

The frequency of the waves has been described elsewhere ͓4,10͔. It is interesting to check that the rotation period of the roll adjacent to the hot wall is consistent with the frequency already reported. We measured this period in two ways. First, aluminum particles were seeded and tracked ͑see Fig. 6͒. In the second method, a fine resistive wire was placed inside the roll and a short temperature pulse was released. With a thermocouple placed elsewhere in the roll, the time of travel can be measured. This latter method is obviously more perturbative than the former, but timing is much easier. With both methods a period of about 1 s is obtained in a situation where waves were measured to have a frequency of 1.2 Hz.

It should be noted that in this roll the velocity is not at all constant. Near the hot wall the fluid accelerates, reaching a maximum speed near the surface. At that point the fluid turns abruptly and leaves the hot wall parallel to the surface. The return travel to the hot wall is much slower. Therefore the roll is not symmetric.

A measurement similar to that presented in Fig. 2 can be carried out for waves, which are in fact small traveling rolls. The temperature amplitude for the wave shown in Fig. 5 is given in Fig. 7. The value obtained is in agreement with measurements obtained from thermocouples ͑Fig. 8͒. It can be seen that the waves are much stronger near the surface, not only from thermocouple measurements, but from interferometry as well: fringes are more compressed in the upper part. Some amplification mechanism, maybe due to a surface instability, must be present, as shown in Ref. ͓10͔.

We complemented these observations with the shadowgraph method ͑Fig. 9͒. The cell is shown from the side, and an arrow marks the position where waves detach from the first roll. The first roll oscillates at a frequency of about 1 Hz, and the waves are produced accordingly at this frequency. FIG. 3. Temperature profile along the cell, obtained with a thermocouple for the same conditions as Fig. 2. Rolls are seen over a constant background temperature gradient of about 0.7 °C/cm. The amplitude of the roll located at about 2.4 cm from the hot wall is about 0.3°, which is consistent with the result shown in Fig. 2. The error bar is the standard deviation from the mean of a large number of consecutive measurements; the precision, however, might be much smaller due to the systematic perturbation introduced by the thermocouple.

FIG. 4. The tip of the firs roll is shown in these two shots at two stages of the oscillation. In ͑a͒ the tip ͑the protruding set of lines at the left upper half͒ has advanced and it is about to release a hot ''drop.'' In ͑b͒ the hot ''drop'' has been released, and the tip of the roll is retracted. The concentric set of fringes that can be seen in ͑a͒ at the right half of the picture is the cold ''drop'' of fluid that has risen from the bottom and has joined the previous hot ''drop.'' ͓The hot wall is at the left, and the photographs cover the same area as Fig. 2͑a͒.͔

IV. DISCUSSION

In this section we show that the waves we observed differ from the hydrothermal waves predicted by theory. We then provide some qualitative arguments in favor of another mechanism that might explain the experimental results presented in Sec. III.

There exist notable differences between these waves and hydrothermal waves. First, there is the parameter space where they show up. Hydrothermal waves appear only at small depths of fluid and for smaller temperature differences ͓3͔ than the waves we report here. Second, the direction of propagation is different. Hydrothermal waves go from the cold to the hot side, with the wave vector oriented at a given angle to the temperature gradient ͓2,7-9͔. In our case, the waves travel from the hot to the cold end, with the wave vector perpendicular to the temperature gradient. We propose that the physical origin of both waves is different. Additional work on the subject is in progress ͓14,15͔. Waves traveling in the same direction as the temperature gradient seem to be nothing but the result of the oscillation of the first roll that the flow carries down and amplifies. The observations we carried out strongly suggest that this could be the underlying process that gives rise to the waves traveling from the hot to the cold wall.

As we have stated, the first roll-and accordingly the waves-oscillate at about 1 Hz. We propose an underlying mechanism that leads to the observed characteristic time of the instability, namely, the formation of a vertical thermal boundary layer. The time that it takes for a thermal boundary layer to develop along a vertical hot wall can be obtained from quite general assumptions ͓16͔. If a sudden temperature step is applied to the wall, the characteristic growth time of the boundary layer is found to be ͓17͔ ϭ h 2 Ra 1/2 . ͑3͒

Scho ¨pf and Patterson carried out an experiment to study the transient regime in a side-heated cavity, and found that the characteristic time for the development and destabilization of the vertical boundary layer agrees with this formula. In our experiment, the situation is different: the vertical wall is kept at a constant temperature, and the surface is free. But we suspect nevertheless that the mechanism might be quite similar. Formula ͑3͒ gives 1 s for our parameters, which is in agreement with the measured frequencies. But we are not able to provide further evidence in support of it. In the formula, does not depend on h, and in the experiments the dependence of on h is very weak indeed within the limits of the experimental error ͓5͔. In Eq. ͑3͒, depends on ⌬T and the square root of the viscosity, but only small changes are accessible in the experiments. It should be remarked, however, that the tendency of is in agreement with formula ͑3͒ when the viscosity or the temperature difference are slightly varied ͓5͔.

The oscillations of the thermal boundary layer along a vertical hot plate is a well-known mechanism ͓18͔, and waves have been found in experiments carried out in closed cavities heated from the side ͓19,20͔. We venture there are two processes that give rise to waves in a lateral container heated from the side: the hydrothermal instability, and the vertical boundary layer instability. This could explain the different propagation schemes observed in experiments.

V. CONCLUSIONS

We have produced evidence showing that there is a mechanism different from the hydrothermal instability giving rise to waves in a shallow liquid layer with a free surface heated from the side. The situation we presented can be viewed as a thermal boundary layer instability along the vertical hot wall. This instability can create a perturbation which is dragged downflow. This picture shares some features with the description provided by Berge ´and Dubois of Rayleigh-Be ´nard convection ͓13͔, even if the experiments are not analogous. This alternative mechanism may explain the differences between the propagation schemes observed in some experiments that did not fit in theoretical descriptions.

FIG. 1 .

 1 FIG. 1. Sketch of the experimental setup. L: laser; SF: spatial filter; C: collimating lens; B: convective cell; GF: glass flat; S: screen; CCD: camera; ␦e: lateral shear; PC: computer.

FIG. 5 .

 5 FIG.5. A wave at about the middle of the cell ͓the dimensions of this picture are the same as in Fig.2͑a͔͒. The hot and cold ''drops'' have joined, forming a roll that moves toward the cold side ͑at the right͒. It can be seen that the amplitude of the wave is larger near the surface.

ORIGIN OF WAVES IN SURFACE-TENSION-DRIVEN . . .

ORIGIN OF WAVES IN SURFACE-TENSION-DRIVEN . . .

ACKNOWLEDGMENTS

M. Dubois and H. Mancini suggested many of the ideas contained here. Helpful suggestions, aid, and comments from C. Pe ´rez-Garcı ´a, J. Burguete, and D. Maza have improved this paper. P. Elizalde and B. Ozenda provided us technical help. Financial support from the European Union ͑Contract No. ERBCHRXT940546͒, from the Spanish Government ͑Contract No. PB95-0578͒, from the University of Navarre ͑PIUNA͒, and from Integrated French-Spanish Action Picasso No. 93024 is acknowledged.

FIG. 9. Shadowgraph pictures of the cell, taken from one side. The liquid layer is 3.1 mm high and 70 mm long. The hot side is at the left. In the upper picture, stationary rolls are seen. Their amplitude decreases as the distance from the hot wall increases. In the picture at the bottom, the temperature gradient is larger, and the first roll begins to oscillate. Apart from the first roll, all the other structures that can be seen in the shadowgraph are moving to the right, i.e., they are waves traveling toward the cold side. An arrow marks the position where waves detach from the first roll by the mechanism shown in Fig. 4.


Instabilities of a liquid layer locally heated on its free surface E. Favre and L. Blumenfeld We report experimental results concerning patterns in a model experiment built to study buoyant-thermocapillary-driven flows. The fluid is situated in a cooled cylindrical container and locally heated on its free surface. The resulting temperature gradient induces a basic flow which draws the surface fluid from the hot center toward the cold boundary. When the gradient is increased and depending on the height of liquid, the basic flow destabilizes into different stationary patterns. Above a second threshold, the patterns become time-dependent. These different instabilities are characterized and compared to recent theoretical results. © 1997 American Institute of Physics. ͓S1070-6631͑97͒02105-3͔

In the last decade, the behavior of buoyancy and thermocapillary-driven flows has received much attention both on fundamental and applied aspects. A linear stability analysis was carried out on thermocapillary flows, first without gravity, [START_REF] Smith | Instabilities of dynamic thermocapillary liquid layers[END_REF][START_REF] Davis | Thermocapillary instabilities[END_REF] then in the presence of buoyancy. [START_REF] Hadid | Buoyancy and thermocapillary-driven flows in differentially heated cavities for low Prandtl number fluids[END_REF][START_REF] Gershuni | On the stability of plane-parallel advective flows in long horizontal layers[END_REF][5][START_REF] Mercier | Buoyant-thermocapillary instabilities of differentially heated liquid layers[END_REF] These works show that the basic flow becomes unstable toward longitudinal rolls or hydrothermal waves. Experiments have been performed on fluid layers with free surface subjected to a horizontal temperature gradient both in rectangular [START_REF] Villers | Coupled buoyancy and Marangoni convection in acetone: experiments and comparison with numerical simulations[END_REF][START_REF] Daviaud | Traveling waves in a fluid layer subjected to a horizontal temperature gradient[END_REF] and cylindrical [START_REF] Schwabe | Instabilities of shallow thermocapillary liquid layers[END_REF][START_REF] Ezersky | Hydrothermal waves in Marangoni convection in a cylindrical container[END_REF][START_REF] Kamotani | Oscillatory convection due to combined buoyancy and thermocapillarity[END_REF] containers. Some of the theoretical predictions have been recovered, such as the existence of hydrothermal waves, but many points still need to be clarified. This fundamental research is supported by the need for understanding the thermal and hydrodynamical properties of such flows in many applications such as floating zone crystal growth, [START_REF] Xu | Convective thermocapillary instabilities in liquid bridges[END_REF] electron-beam vaporization [START_REF] Schiller | Electron Beam Technology ͑Wiley[END_REF] or laser welding. However, the real configurations are different from the theoretical ones. In particular, in the last two applications, the liquid is not driven by a horizontal thermal gradient on all its height but only on its surface. In this paper, we report new experimental results concerning the destabilization of a flow locally heated on its free surface. First, we describe the experimental set-up. Then we characterize the different instabilities which depend on the height of liquid and on the temperature difference. Finally, we discuss the experimental results and compare them with available theoretical studies.

The apparatus consists of a cylindrical container made of copper ͑diameter 2R 0 ϭ10 cm and height h 0 ϭ4 cm; see Fig. 1͒. The bottom is made of sapphire to allow visualization. The container is thermally regulated by circulating water and a Plexiglass plate is situated a few millimeters above the free surface to reduce evaporation. It is filled with silicone oil ͑viscosity ϭ0.0065 Stokes͒ of Prandtl number Pϭ10 to a height h that is measured with a precision of 0.02 mm. A meniscus is present at both boundaries. With the lid, h remains constant to 1% over an experimental run. The liquid is locally heated on its surface by a cylindrical heater ͑diameter 2r 0 ϭ3 cm͒. The two parameters which control the experiment are h and the temperature difference ⌬T between the heater and the boundaries. The flow is characterized by temperature ͑thermocouples͒ and velocity ͑LDV͒ measurements and by shadowgraphy. The patterns are visualized by shadowgraphic imaging: A parallel vertical light beam crosses the container from bottom to top and forms a picture on a screen, due to surface deformations at the oil-air interface and to horizontal temperature gradients in the fluid. The spatio-temporal evolution of the structures is recorded using a video camera and the images are digitized along a circle to follow the azimuthal evolution of the patterns. [START_REF] Daviaud | Traveling waves in a fluid layer subjected to a horizontal temperature gradient[END_REF] The structure of the flow and the different observed patterns are shown on the stability diagram of Fig. 2. Each experiment is performed with h fixed and ⌬T variable. The thermocapillary and the buoyancy effect can be characterized by the Marangoni number MaϭϪ(ץ/ץT)⌬Th/ 0 and the Rayleigh number Raϭg␣⌬Th 3 /, where g is the gravitational acceleration, ␣ the thermal expansion coefficient, the thermal diffusivity, 0 the density of the fluid, the kinematic viscosity and the surface tension. Whatever the height of liquid h, a basic flow is present when ⌬T is increased above zero ͑domain BF in Fig. 2͒. This flow is due to the thermocapillary effect and draws the surface fluid from the hot center towards the cold boundary. For large values of h (hϭ15 mm͒, Fig. 3 gives the radial velocity V r as a function of the depth z. The velocity is zero for zϽϪ8 mm and temperature measurements show horizontal isotherms in this zone. The thermocapillary effect creates a shear flow for Ϫ3ϽzϽ0 mm and a return flow is present for Ϫ8ϽhϽϪ3 mm. The basic flow thus appears as a torus and is in fact two dimensional ͑2D͒.

When ⌬T is increased above a first threshold, whose value varies with h, the basic flow becomes unstable and stationary patterns are observed. Depending on h, three types of patterns are displayed. For small h (hϽ2.8 mm͒, a multiroll state made of co-rotating rolls is encircled by the basic flow and the resulting flow remains 2D ͓cf. domain CR in Fig. 2 and Fig. 4͑a͔͒. These rolls are present near the heater. The number of rolls decreases with increasing h ͑3 rolls for hϭ2 mm͒ and their wavelength is Ӎ2.5h. For intermediate h (3.2ϽhϽ5 mm͒, the increase of ⌬T leads to a pattern with stationary ''petals'' ͓cf. domain SP in Fig. 2 and Fig. 4͑b͔͒. The radial extension of the petals is always less than 2R 0 /3 and their number decreases with h from 14 for hϭ3.2 mm to 11 for hϭ5 mm. The azimuthal invariance is broken and the resulting flow is 3D. The transition from co-rotating rolls to petals displays intermediate structures near hϭ3 mm, where RaϳMa. For large h (hϾ5.3 mm͒, stationary radial structures appear ͓cf. domain SR in Fig. 2 and Fig. 4͑c͔͒. For hϭ10 mm, the threshold corresponds to Maϭ1.7 ϫ 10 5 and Raϭ2.1 ϫ 10 6 . Contrary to petals, the rays have a small azimuthal extension, they reach the cold vertical boundary and they do not extend over the whole depth. Their number decreases from 11 for hϭ5.3 mm to 9 for hϭ30 mm. The rays correspond to counter-rotating rolls ͑with radial axis͒ which are superimposed on the basic flow, giving rise to a fully 3D state. The amplitude of the variation of V r increases with ⌬T above the threshold. [START_REF] Favre | [END_REF] The patterns become time-dependent above a second threshold. For small h, azimuthal waves are observed very near the heater ͑cf. domain HW in Fig. 2͒. They consist of a source emitting a clockwise and an anti-clockwise rotating wave towards a sink. For hϭ2 mm, 17 rotating structures are evidenced at threshold and the associated frequency is 2/h 2 . The co-rotating rolls are slightly distorted by these waves. For higher ⌬T, radial pulsations are observed. For intermediate h, the stationary petals propagate azimuthally ͓cf. domain RP in Fig. 2 and Fig. 5͑a͔͒. For hϭ4.3 mm, the azimuthal velocity of the pattern is 8.4/h at rϭ19.3 mm. For large h, the stationary rays oscillate azimuthally ͓cf. domain OR in Fig. 2 and Fig. 5͑b͔͒. For hϭ10 mm, the threshold corresponds to Maϭ4.0 ϫ 10 5 and Raϭ5.0 ϫ 10 6 . The frequency of oscillation f 0 increases with ⌬T. For hϭ6.3 mm and ⌬Tϭ11 K ͑resp. hϭ15.2 mm and ⌬Tϭ7.5 K͒ f 0 ϭ2.9/h 2 ͑resp. f 0 ϭ14.6/h 2 ). When the temperature difference is further increased, the dynamic becomes more complex and the system exhibits spatio-temporal chaos.

These results raise several interesting hydrodynamic issues. When h is large, the basic toric flow develops only in the upper part of the liquid layer and for hϽϪ8 mm, the fluid is at rest. This is due to a stable vertical stratification of the thermal field in the bottom of the cell. Experimental 15 ͑resp. numerical [START_REF] Kamotani | Oscillatory convection due to combined buoyancy and thermocapillarity[END_REF] ͒ results have displayed this stratification when Aϭ1 ͑resp. Aу0.75). In our configuration, this fact becomes important when A exceeds 0.3. Numerical simulations [START_REF] Hadid | Buoyancy and thermocapillary-driven flows in differentially heated cavities for low Prandtl number fluids[END_REF][START_REF] Villers | Coupled buoyancy and Marangoni convection in acetone: experiments and comparison with numerical simulations[END_REF] and experiments [START_REF] Villers | Coupled buoyancy and Marangoni convection in acetone: experiments and comparison with numerical simulations[END_REF][START_REF] Schwabe | Instabilities of shallow thermocapillary liquid layers[END_REF] for small h and various P have shown the multi-roll state, but the understanding of the origin of this state is still lacking. We interpret these rolls as being due to the dynamic interaction between the shear flow and the bottom. For typical critical values of ⌬T, the shear layer extends over the first 3 mm which exceeds the depth of fluid: The return flow of the basic torus cannot settle, hence the multi-rolls flow.

The rays correspond to rising hot fluid. This pattern appears when the height of liquid is quasi-infinite. However, the effect of the condition at the bottom is still present for Ϫ10ϽhϽϪ5 mm, as shown by the reduction of their number ͑from 11 to 9 when h increases͒ and by the decrease of the second critical curve of Fig. 2. For hу10 mm ͓i.e., for aspect ratios Aϭh/(R 0 Ϫr 0 )у0.3], ⌬T c is independent of h. This result is in agreement with the results of Kamotani et al. [START_REF] Kamotani | Oscillatory convection due to combined buoyancy and thermocapillarity[END_REF] for AϾ0.75 and extends them. The rays could be related to an inversion of the vertical temperature profile near the free surface and the cold wall, leading to a Be ´nard-Marangoni instability type, with a critical Marangoni number Ma c Ӎ600. This number is far greater than in the pure case where Ma c Ӎ80, because of the presence of the basic flow. Stationary rolls of Marangoni type have been found in some models, but associated with a linear flow 1 or never critical. [START_REF] Mercier | Buoyant-thermocapillary instabilities of differentially heated liquid layers[END_REF] In the light of the interaction between the basic flow and the condition at the bottom, we propose the petals to be the pattern which develops when the bottom is situated in the extension of the upper part of the return flow ͑i.e., Ϫ5ϽhϽϪ3 mm͒.

For small heigths, the dynamic states can be quantitatively compared to hydrothermal waves as defined in Refs. 1 and 6 and observed in Refs. 8 and 9. In fact, our results are closer to those of a recent study [START_REF] Mercier | Buoyant-thermocapillary instabilities of differentially heated liquid layers[END_REF] which takes into account the influence of thermal boundary conditions and geometry in the selection of the instability. The radial pulsations exhibited for higher ⌬T are similar to those observed in Ref. 10 and could be due to a different instability. [START_REF] Favre | [END_REF] For larger heights, the observed time-dependent regimes do not correspond to available theoretical results and further work is needed. Let us note that the study of the hysteresis phenomenon of the transition to dynamic states for petals or rays reveals the subcritical nature of the bifurcation, whereas the primary instability is supercritical whenever h. The amplitude of the oscillations of the rays as well as the velocity of the petals increases with the distance from threshold.

In a liquid layer locally heated on its free surface, the basic 2D flow destabilizes into different patterns depending on the aspect ratio. These patterns have been experimentally characterized, as well as their transition to dynamical regimes. The structures observed at small height are similar to those observed in pure horizontal thermal gradient configurations both in experiments and theories. On the other hand, the patterns observed for larger heights appear to be new and fully 3D. Work is under progress in this direction.

  DCC/DPE/SPEA, CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France F. Daviaud DSM/DRECAM/SPEC, CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France ͑Received 25 April 1996; accepted 16 January 1997͒

FIG. 3 .

 3 FIG.3. Radial velocity profile at three different radii r; hϭ14.6 mm and ⌬Tϭ2.9 K.

FIG. 1 .

 1 FIG. 1. Schematic drawing of the experimental apparatus.

Phys. Fluids 9 (5), May 1997 1070-6631/97/9(5)/1473/3/$10.00 © 1997 American Institute of Physics Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 132.166.49.33 On: Thu, 29 Sep

ACKNOWLEDGMENTS

We wish to thank N. Mukolobwiez, J. F. Mercier, and C. Normand for stimulating discussions, and M. Labouise and B. Ozenda for their technical assistance.