OSCILLATING INSTABILITY IN ONE-DIMENSIONAL RAYLEIGH-BENARD CONVECTION

Transcription

1 OSCILLATING INSTABILITY IN ONE-DIMENSIONAL RAYLEIGH-BENARD CONVECTION F. Daviaud, P. Berge, M. Dubois To cite this version: F. Daviaud, P. Berge, M. Dubois. OSCILLATING INSTABILITY IN ONE-DIMENSIONAL RAYLEIGH-BENARD CONVECTION. Journal de Physique Colloques, 1989, 50 (C3), pp.c3-181-c < /jphyscol: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1989 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 JOURNAL DE PHYSIQUE Colloque C3, supplément au n 3, Tome 50, Mars 1989 C3-181 OSCILLATING INSTABILITY IN ONE-DIMENSIONAL RAYLEIGH-BENARD CONVECTION F. DAVIAUD, P. BERGE and M. DUBOIS Service de Physique du Solide et de Résonance Magnétique de Saclay, F Gif-sur-Yvette Cedex, France Résumé - Une oscillation collective périodique de la position des rouleaux est observée au cours d'expériences de convection de Rayleigh-Bénard en géométrie quasi-unidimensionnelle - c.a.d. quand l'une des dimensions horizontales est inférieure à l'épaisseur de la couche de fluide. Ces oscillations sont liées à la présence de très petites longueurs d'onde et conduisent directement à des comportements d'intermittences turbulentes spatio temporelles lorsque la géométrie est annulaire. Abstract - Collective periodic oscillations of the rolls' position are observed in experiments of Rayleigh-Benard convection, when one of the horizontal extensions is smaller than the depth of the fluid layer (quasi one dimensional geometry). The oscillations are related to the presence of very short wavelengths and can lead directly to turbulent spatio temporal intermittencies, when the convection is achieved in an annular geometry. INTRODUCTION The interest for one-dimensional systems has been growing up in the last few years, at least theoretically, in particular in the study of the transition to turbulence A/. By increasing progressively the length of these systems, the dynamics can evolve from regimes typical of dynamical systems /ll/, with a small number of temporal degrees of freedom to complex spatio-temporal behaviours. From a. theoretical point of view, models derived from a Kuramoto-Sivashinsky equation /2/ are used, whereas experimentally, Rayleigh-Benard convection is a system suitable to achieve a one dimensional chain of hydrodynamical rolls. Indeed, in this problem, the arrangement of the rolls in the horizontal plane has to be considered; in this sense, near threshold, a perfect structure with parallel rolls, can be entirely described by an unic horizontal space variable, X for example, whose axis is perpendicular to the rolls'axis. B"ut this is no longer true when, by increasing the Rayleigh number Ra (the control parameter of Rayleigh-Benard convection), a new bifurcation occurs at a given value Ra 2, for which a new set of rolls takes place, with their axis perpendicular to these of the former set /3/. Nevertheless, observations of convective patterns, developed in rectangular cells with different transverse aspect ratio F Y (T Y = L y /d, with d the depth of the fluid layer) have revealed that, in narrow cells, i.e. for r y <0.6, this bifurcation does not seem to occur, or if it does, it is at very high values of Ra; then, when time-dependence appears, the behaviour is practically the same all along the axis of the actual rolls and depends only on the distance along the longest dimension of the cell. In that sense, the convective structure acts as a one-dimensional chain of rolls. With this geometrical property, (r <0.6), new behaviours are observed, some of which are described in this paper. Article published online by EDP Sciences and available at

3 C3-182 JOURNAL DE PHYSIQUE A - COLLECTIVE OSCILLATING STATE IN A RECTANGULAR CELL. The experimental set-up can be described as follows: the fluid-% oil with viscosity *stokes at room temperature (Pr = 7.5) - fills a rectangular cell, with dimensions 180x2~5 ma, giving horizontal aspect ratios of 36 and 0.4. The cell is inserted between two horizontal copper plates, to assure good thermal conductivity of the horizontal boundaries. Their temperature is regulated by circulating water from thermostated baths, such that the fluid layer experiences a temperature difference AT, stable within 2 10-*C. The survey of the convective pattern and of its dynamics is performed through shadowgraphic images observed in a vertical plane. These images can be recorded on videotapes; moreover a part of them can be analyzed by a photodiode array of 256 pixels. using an optical device. This array is connected to a computer, allowing the intensity along a choosen horizontal line to be recorded at any time, or time series to be constructed with appropriate sampling time between each line record. Then the spatio-temporal evolution of the pattern can be followed quantitatively /4/. Increasing the Rayleigh number, a perfect structure takes place, formed by parallel rolls with axis perpendicular to the longer side of the cell. Though generally much shorter than Xc in such a narrow channel /5/ /6/ /7/ (Xc = 2d, the critical wavelength). the actual wavelength depends on the previous thermal history, but, if the fluid layer is submitted to quick AT increase, very short wavelengths are favoured. For Ra 2 lo6, the wavelength can be as short as 0.38 A, (or a/ac = 2.6, with a the wavenumber of the pattern). This wavelength remains stable up to very high Ra values (the highest explored value with the actual set-up was around 2.3 lo6). Along this whole Ra domain. the pattern is stationnary, except within a range where a collective oscillating state is observed. Increasing AT, oscillations appear for Ra= 1.17 lo6 and disappear for Ra = 1.7 lo6, whereas by decreasing AT, they are observed in the same domain, slightly shifted to lower Ra values by a small hysteretic phenomenon. Near their threshold, the oscillations are periodic; at higher Ra values, a biperiodic regime is observed before returning to stationnarity. The general features of these oscillations are shown in fig.1. The main streams (extrema of intensity on the shadowgraphic picture) move periodically around their mean position, with phase opposition between hot and cold streams motion. So when one considers a wavelength, one roll seems to increase periodically in size at the expense of the other and conversely. In the monoperiodic regime, the amplitude of the displacement 6Xo remains constant when the time runs, but depends on the position in the cell. At particular points, equally spaced, no motion is present; they define nodes of oscillations, with, in between, a maximum of the amplitude like in a stationnary wave (fig.2). Note that the nodes are not necessarily located at the main streams. The new length scale they introduce in the pattern is of order of 5 to 6 Xo, where A, is the actual wavelength. and can be incommensurate with Xo. When we speak of monoperiodic regime, it means that the fluid layer undergoes the same oscillatory mechanism, with the same frequency, whatever is the measurement point (except for the amplitude as described previously). Particularly impressive is the fact that, by performing cross correlation measurements. one can show that there exists a strict coherence all along the chain of the rolls: about a hundred rolls oscillate in perfect synchronism. When the regime becomes biperiodic, there is no,significant change in the global behaviour. except for the fact that the second frequency which appears in the dynamics (typically around f1/50 if fl is the frequency of the basic oscillations) is related to periodic displacement of the nodes of vibrations (previously fixed in the monoperiodic regime).

4 Fig. 1. Spatio-temporal evolution of the pattern in the rectangular cell. (monoperiodic regime. Ra = '). a) Intensity versus X of a part of the shadowgraphic images taken in the middle of the cell. Minima: uprising hot streams. Maxima: descending cold streams. The time lapse between each line record is 350ms. (fl = HZ) b) Positions of the intensity extrema versus time. (processed from the data of the fig. la). Fig.2. Amplitude of the displacement SX, = Xm-Xo of the streams versus X for the data shown in fig.1. The origin Xo of the positions is taken arbitrarily at the instant t=o. Xm is the maximum displacement for each stream. 0: hot streams; + Cold streams. In spite of the poor spatial resolution (one pixel corresponds to 0.17mm. i.e.-ao/20; &Xo max " 3 pixels), the periodicity of the spatial modulation of 6X, is clearly evidenced. B - OSCILLATIONS IN ANNULAR GEOMFTRY In order to study a one dimensional convective system with periodic boundary conditions. the Rayleigh-Benard convection is also achieved in an annular container filled with silicon oil of Prandtl number 22. As in the rectangular case, the walls of the cell are made of plexiglass and inserted between copper plates. Its internal diameter is 120 mm and its external diameter is 126 mm, giving a gap of 3 mm. With a depth of 10 mm. this cell has an aspect ratio rr=0.3 along the radius and rp=38.6 along the circumference.

5 C3-184 JOURNAL DE PHYSIQUE As in the rectangular cell, the shadowgraphic images of the convective structure give a picture of the pattern in the vertical plane, transformed into a circular image through a pair of conical mirrors /8/. In order to record the intensity of the pattern. an electromechanical system allows a photodiode to sweep the circumference at a given frequency. The photodiode signal is then digitized and stored in a microcomputer. and the spatiotemporal evolution of the pattern can be extracted from the resulting time series. The convective pattern consists, above the threshold, of a chain of rolls which presents a dispersion of the local wavelength far much greater than in the rectangular geometry. In fact, the wavelength is generally not homogeneous all along the circumference, ranging from Xc/3 to Xc/2, as can be seen in figure 3; nevertheless. like in the rectangular container, wavelengths shorter than Xc seem to be favoured. Moreover, this inhomogeneity depends on the Rayleigh number; when increasing it, one can observe, for given initial conditions, a widening of the range of the wavelengths present in the pattern. The same collective oscillation of the rolls axis is observed in the annular geomety, as the first time dependent regime. But, if this oscillating instability exhibits locally the same properties as in the rectangular cell, it strongly depends on the spatial properties of the pattern. Actually, some parts of the cell are oscillating and others not, the oscillating rolls being those with the shortest local wavelengths. This inhomogeneity of behaviour refers then to the dispersion of the local wavelengths, as can be seen in figure 4. and the oscillations look like a tendancy for the structure-and in particular for the shortest rolls- to adapt its local wavelength to a larger value. The oscillating regime is generally monoperiodic, with the same frequency inside an oscillating domain. But one can observe slightly different frequencies between two oscillating domains separated by stationary ones. The presence of nodes of oscillation (rolls that do not oscillate) is also evidenced inside an oscillating domain. as in the rectangular geometry, showing that they are not related to the presence of rigid lateral boundaries. When the Rayleigh number is increased, this collective instability remains, with an increasing amplitude of the oscillations, but other more complex regimes arise. such as spatial defects or spatio-temporal intermittencies. In fact, when the amplitude of oscillation is large enough, two rolls can merge and a turbulent regime can appear in a spatially. restricted domain while other regions remain organized, stationary or oscillating. C - CONCLUDING REMARKS. The previously described phenomenon of rolls' oscillations, with its particular properties, has been observed in different geometries, (rectangular and annular), and with different Pr number fluids. It seems to be a general phenomenon directly related to the presence of small wavelengths in narrow cells and two-dimensional motion. It looks similar to oscillations observed in Hele-shaw cells /5/, though in this case, the gap is much more smaller (ry<0.1) and the oscillations are chaotic very soon after their appearance. Note that the frequency of our reported oscillating behaviour ' *d2/~, when Pr = 7.5 and 'd2 /DT when Pr = 22 - are of the same order of magnitude as those reported in /5/. The observed behaviour, which points out the tendancy for the pattern to increase locally its wavelength, indicates that we are in presence of an instability in the plane [Ra. a]. Nevertheless, there are differences between the two geometries (annular and rectangular) due at first to the wavelength dispersion in the annular case. They can be summarized as follows. In the rectangular cell. the oscillations are present only in a defined window of Ra numbers, surrounded by a stationnary regime; the phenomenon is very pure and even when I the nodes move, they do so in a regular and periodic way. On the contrary, in the annular cell, the oscillating motion of the rolls is not organized spatially. even near threshold and it does not disappear, when increasing Ra. The behaviour becomes more complex, where the oscillations and the long waves, associated to their spatial amplitude modulation.

6 play an important role in the distabilization of the pattern leading to spatio-temporal intermittencies/lo/. Fig. 3: Distribution of the value of the local wavelength in the annular geometry at Ra = 3.8 lo6. Fig. 4. Existence domain of the oscillating behaviour in the annular geometry. The authors thank H.ChatE? and P.Manneville for constructive discussions and are indebted to R. Da Silva and A.Petrov for their efficient contribution. REFERENCES /1/ - H.Chat6, P.Mannevitle. Phys.Rev.Lett.. B. 112 (1987). /2/ - Y.Kuramoto, T.Tsuzuki, Prog.Theor.Phys., z, 356 (1976). /3/ - P.Berg4, N.Dubois, Contemporary Physics (1984). /4/ - H.Dubois, R. Da Sf Zva, F.Daviaud, P. Berg&, A. Petrov, Europhysics Letters 'submitted July 1988.

7 C3-186 JOURNAL DE PHYSIQUE /5/ - J.N.Koster, U.NiltZer, J. Fluid Mech., 125, 429 (1982). /6/ - O.KvenvoZd, Int. J. Heat Mass Transfer (1979). /7/ - D.Bensimon, Phys. Rev. A (1988). /8/ - P. Berg&, Nucl. Phys. B. (Proc.Supp1). 2, 247 (1987). /9/ - A. Pocheau, V.Croquette, P. Le Gal, C. Poitou, Europhysics Letters, 8, 915 (1987). /lo/ - S-Ciliberto, P. Bigazzi, Phys.Rev.Lett., a, 286 (1988). /11/ - Le Chaos. Thkorie et exp6riences. ed. by P. Berg&. Eyrolles, Paris. (1988)