We report experimental observations of traveling waves in a pure fluid with a
free surface situated in a long container submitted to a horizontal temperature
gradient perpendicular to its large extension. Above a critical value of the
gradient and depending on the height of liquid $ h, $ a source of waves is
created in the container for small value of $ h, $ while the system exhibits
stationary patterns for larger values of $ h. $ The spatio-temporal properties
of the waves are studied and compared to theoretical predictions. Email
contact: daviaud@amoco.saclay.cea.frComment: Saclay-S93/017 Email: daviaud@amoco.saclay.cea.f

Daviaud, F.

Vince, J. M.

English

arXiv

URL: http://www-spht.cea.fr/articles/S93/017 http://fr.arxiv.org/abs/cond-mat/9303049International audienceWe report experimental observations of traveling waves in a pure fluid with a free surface situated in a long container submitted to a horizontal temperature gradient perpendicular to its large extension. Above a critical value of the gradient and depending on the height of liquid $ h, $ a source of waves is created in the container for small value of $ h, $ while the system exhibits stationary patterns for larger values of $ h. $ The spatio-temporal properties of the waves are studied and compared to theoretical predictions. ----- Ce court article résume les premiers résultats acquis dans cette nouvelle configuration longitudinale: comme dans les travaux précédents des mêmes auteurs, nous sommes dans le cas d'un canal horizontal rempli d'un liquide. Cette fois l'instabilité est provoquée par un gradient horizontal de température. La phénoménologie observée rappelle fortement le cas où le chauffage se fait à l'aide d'un fil conducteur placé sous la surface: un système d'ondes propagatives apparaît dès le seuil de l'instabilité, lequel subit ensuite des bifurcations l'amenant à une dynamique spatiotemporelle non-triviale. En dehors de la nouveauté indiscutable de ces résultats, c'est bien l'analogie avec le cas du fil chaud qui me paraît la plus prometteuse: elle montre la généralité de ces phénomènes, et permet d'espérer une clarification des véritables mécanismes à l'origine de l'instabilité dans un cas comme dans l'autre. Ce travail mérite d'être publié et poursuivi. \vskip 20mm \hfill{Huges Chaté

Daviaud, François

Vince, J.M.

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Traveling waves in a fluid layer subjected to a horizontal temperature gradient

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Traveling waves in a fluid layer subjected to a horizontaltemperature gradientFranc¸ois Daviaud, J.M. VinceTo cite this version:Franc¸ois Daviaud, J.M. Vince. Traveling waves in a fluid layer subjected to a horizontal temper-ature gradient. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, AmericanPhysical Society, 1993, 48, pp.4432-4436. <10.1103/PhysRevE.48.4432>. <hal-00164453>HAL Id: hal-00164453https://hal.archives-ouvertes.fr/hal-00164453Submitted on 20 Jul 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Traveling Waves in a Fluid LayerSubmitted to a Horizontal Temperature GradientF. Daviaud and J.M. VinceService de Physique de l’Etat Condense´, CEA - Centre d’Etudes de SaclayF-91191 Gif-sur-Yvette Cedex, FranceWe report experimental observations of traveling waves in a pure fluid witha free surface situated in a long container submitted to a horizontal temperaturegradient perpendicular to its large extension. Above a critical value of the gradi-ent and depending on the height of liquid h, a source of waves is created in thecontainer for small value of h, while the system exhibits stationary patterns forlarger values of h. The spatio-temporal properties of the waves are studied andcompared to theoretical predictions.1While the nature of turbulence in fluids is still today an open question, theevolution towards turbulent states has received much attention in the last decadeand important results have been obtained in weakly confined systems. In thisframe, the study of one-or two- dimensional (1D or 2D) systems in numericalmodels and in experiments has shown that a periodic cellular state may destabi-lize when a control parameter is varied and becomes turbulent through differentscenarios. A transition to turbulence via spatio-temporal intermittency has beenobserved in models of phase equations [1] and coupled map lattices [2] as well asin 1D Rayleigh-Be´nard convection [3], in the Printer’s instability [4] or in capillaryripples [5]. A similar regime is also observed in circular [6] or plane Couette [7]flows. Spatio-temporal chaos can also appear by an increase of the number ofdefects as it is seen in Ginzburg Landau equations [8] or in convection in binaryfluid mixtures [9]. Much work has been devoted to this last system in which thefirst state observed consists of traveling waves which arise from a subcritical Hopfbifurcation. This system displays a rich variety of dynamical states, such as con-fined states or defects and spatio-temporal chaos near onset in 1D [10] and 2Dsystems [11]. More recently, 1D propagative patterns have also been observed ina new experiment involving a hot wire below the free surface of a liquid. Thisdynamic state is qualitatively consistent with theories based on Ginzburg-Landauequations [12]. However, the physical mechanisms responsible for this instabilityremain unknown as well as the exact role of defects.In this letter, we report new experimental results concerning different dynam-ical regimes, including traveling waves, obtained in a very simple configuration. Apure fluid with a free surface is situated in a long and narrow container and is sub-mitted to a horizontal temperature gradient perpendicular to its large extension.Many experiments have been devoted to this problem, but always in containerswith a small extension perpendicular to the gradient [13]. In this case, the prop-agating modes were never observed, whereas they can develop in our geometry.This problem has also received increasing attention in the past decade, becauseof microgravity facilities, in methods of float-zone crystal-growth with cylindricalgeometries [14]. It was theoretically studied, in the case where gravity is ignored[15,16,17]. Our system thus appears to be a good tool to study both the mech-anisms which drive this instability and the dynamics of nonlinear 1D travelingwaves.The apparatus consists of a rectangular container 20 cm long and 1 cm large(see Fig.1). The long vertical walls are made of copper and can be thermallyregulated by circulating water. The container is closed by two small plexiglasswalls. The lower boundary consists of a glass plate to allow visual observationwhile a plexiglass plate is inserted a few millimiters above the surface of the fluid2to avoid evaporation problems. The container is filled with silicon oil (viscosityν = 0.0065 Stokes) of Prandtl number P = 7 up to a height h that is measuredwith a precision of 0.05 mm. The horizontal temperature gradient ∆T is imposedbetween the two copper walls and is measured using thermocouples. This gradientis regulated with a stability of 10−2 K.The patterns are visualized by shadowgraphic imaging. A parallel light beamcrosses vertically the container from top to bottom and forms a horizontal pictureon a screen, both due to surface deformations at the interface oil-air and to tem-perature gradients in the fluid (see Fig.2). The spatio-temporal evolution of thestructures is recorded using a video camera and the images are digitized along aline of 512 pixels perpendicular to the gradient.The two parameters which control our experimental system are h, the heightof Si oil in the cell, and ∆T , the horizontal temperature difference between the twowalls. We have observed that, for each h value, when ∆T is increased from zero toa threshold value ∆Tc, a longitudinal (parallel to the gradient) pattern appears.But depending on the height of liquid, two different regimes take place. When∆T ≥ ∆Tc, for small h values (from 0 to 2.8 mm), the system exhibits longitudinalwaves propagating perpendicularly to the gradient (see Fig.2) while, for larger hvalues (from 2.8 mm to 10 mm), stationary “rolls”with axis parallel to the gradientare observed. Figure 3 shows the dependence of ∆Tc on h for these two kinds ofstructures. One can notice that the value of height h = 2.8 mm for which the twodomains meet each other corresponds to a ratio between the Marangoni and theRayleigh number W = Ma/Ra ' 1 with Ma = −(∂σ/∂T )∆Th/ρ0κν and Ra =gα∆Th3/κν, where g is the gravitational acceleration, α the thermal expansioncoefficient, κ the thermal diffusivity, ρ0 the density of the fluid, ν the kinematicviscosity and σ the surface tension. The traveling waves domain correspondsto W > 1, which means that surface tension effects are dominant, while in thestationary rolls domain (W < 1), buoyancy is preponderant.When studying the regime of propagative waves in more details, one canobserve that the structures are not exactly perpendicular to the large side of thecontainer but exhibit an angle ψ ' 80◦. This angle does not seem to depend on∆T or on h. When ∆T is increased above ∆Tc, a source of waves is created witha dimension which depends on ε = (∆T − ∆Tc)/∆Tc. The size of the source isvery large near the threshold and decreases in length as ε is increased.Space-time diagrams have been performed to study the dynamic of the waves(see Fig. 4). With such diagrams, the wavelength and the period of these waves caneasily be measured. First of all, spatial and temporal Fourier spectra reveal thatthe system of waves has a unique wavelength and is monoperiodic at threshold.For h values between 0 and 2.5 mm, the period at threshold Tc increases with h (cf.3Figure 5). In the same range of h, the λc/h variation versus h is no-monotonous,with a maximum for h = 1.4 mm. These two behaviours can be compared to thoseobtained when a hot wire is used to destabilize a fluid layer with a free surface[12]. The physical context is somewhat different but the behaviour of the two setsof curves is very similar, allowing us to think that the physical mechanism whichdrives the instability in the hot wire experiment could have the same origin as inour configuration.We have also performed some measurements for ∆T larger than ∆Tc at dif-ferent h values. The curves giving the period T and the phase velocity vϕ versus εare respectively displayed in Fig.6a and Fig.6b. The figures show that T decreasesand vϕ increases when ε is increased. One can notice that, for small h values,the values of T and vϕ are clearly different from these obtained for the middlerange of h values. This fact shows how this phenomenon is sensitive to the surfacetension effects which are more important as h is small. Finally, when ε is furtherincreased, the system of traveling waves exhibit phase instabilities leading to spacetime dislocations.For the stationary rolls, the experiment shows that they are strictly perpen-dicular to the large side of the cell, contrary to traveling waves. The wavelengthassociated to these rolls appears to be proportional to the height of liquid h. Then,when ε is increased, the rolls begin to oscillate under the form of an optical mode,and, for larger ε, the pattern is destabilized and gives place to spatio-temporalchaos. These dynamical regimes are reminiscent of what is observed in Rayleigh-Benard convection in narrow gap geometries [3].Our observations raise several interesting issues concerning both the hydrody-namical and the phenomenological aspects. First of all, thermocapillarity appearsto be the main physical mechanism driving the instability observed in our ex-periment for small values of h, while buoyancy is preponderant for larger valuesof h. It is known that, due to thermocapillary effects, fluid motion can developwhen a temperature gradient is imposed on a thin fluid layer with a free surface[16]. Although we have not yet recorded velocity profiles, by seeding the flow withparticles and illuminating it with a laser sheet, we have seen that the horizontalvelocity profile reveals a shear flow near the surface and a return flow at the bot-tom, the convection being in the form of a monocellular cell (cf. [13]). Smith andDavis have shown that this dynamic state can destabilize under the form of twotypes of thermoconvective instabilities: stationary longitudinal rolls and travelinghydrothermal waves [15]. Some phenomena, such as temperature oscillations, hadbeen previously observed in floating zones experiments [14], but, to our knowledge,our observations provide the first clear evidence of these instabilities. However,4our experimental results share some differences with the results obtained by Smithand Davis. These authors consider the destabilization of two basic flow profiles:the linear flow state (LFS) and the return flow state (RFS). Stationary rolls arenot stable in RFS but only in LFS. Moreover, oblique traveling waves are notpredicted in fluids with P > 1 in LFS while they do exist in RFS, but with anangle ψ ' 20◦ instead of 80◦ as observed in the experiments.These differences could be explained by the absence of gravity in their model.More recently, the stability of convective motion in a differentially heated cavityhas been studied, in the presence of gravity [18,19,20]. However, certainly dueto different boundary conditions, our experimental observations differ from theirresults. A more realistic model including both surface tension and gravity togetherwith appropriate boundary conditions is at the moment studied and the first resultsare promising [21].As for the phenomenological aspect, our configuration reveals to be a goodsystem for exploring nonlinear 1D traveling waves and their destabilization. Thespatio-temporal properties of the waves can be studied in the frame of two coupledamplitude equations, that could be derived from the physical model [21]. As amatter of fact, they have general properties that do not depend on the exactexperimental system. In our experiement, the presence of a source stabilizes theright- and left-going waves with a definite wavelength and near threshold, thewidth of the source increases when decreasing ε, as it is observed in the hot wireexperiment [12] and in recent numerical simulations [22]. Our experiment alsoshows similarities with some dynamical regimes (such as confined states with astable sink of rolls) oberved in binary convection in an annulus [9,10]. We areperforming experiments in this direction, using an annular configuration with aradial temperature gradient.We wish to thank M.Dubois, J.Lega and C.Normand for stimulating discus-sions and M.Labouise, P. Hede and B. Ozenda for their technical assistance.5References[1] H.Chate´ and P.Manneville Phys.Rev.Lett. 58, 112 (1987)[2] K.Kaneko Prog.Theor.Phys. 74, 1033 (1985)H.Chate´ and P.Manneville, Physica D 32, 409 (1988)[3] S.Ciliberto and P.Bigazzi, Phys.Rev.Lett. 60, 286 (1988)F.Daviaud, M.Dubois and P.Berge´, Europhys.Lett. 9, 441 (1989)[4] M.Rabaud, S.Michalland and Y.Couder, Phys.Rev.Lett. 64, 184 (1990)[5] N.B.Tufillaro, R.Ramshankar and J.P.Gollub, Phys.Rev.Lett. 62, 422 (1989)[6] C.D.Andereck, S.S.Liu and H.L.Swinney, J.Fluid Mech. 164, 155 (1986)[7] F.Daviaud, J.Hegseth and P.Berge´, Phys.Rev.Lett. 69, 2511 (1992)[8] P.Coullet and J.Lega, Europhys.Lett. 7, 511 (1988)[9] see e.g. D.Bensimon, P.Kolodner, C.M.Surko, H.Williams and V.Croquette,J. Fluid Mech. 217, 441 (1990).[10] P.Kolodner, J.A.Glazier and H.Williams, Phys.Rev.Lett. 65, 1579 (1990)P.Kolodner, Phys.Rev. A 46, 6431 (1992)[11] V.Steinberg, E.Moses and J.Fineberg, Nucl.Phys. B2, 109 (1987)[12] J.M. Vince and M.Dubois, Europhys.Lett. 20, 505 (1992)[13] see e.g. D.Villers and J.K.Platten, J. Fluid Mech. 234, 487 (1992).[14] F.Preissier, D.Schwabe and A.Scharmann, J. Fluid Mech. 126, 545 (1983)J.J.Xu and S.H.Davis, Phys.Fluids 27, 1102 (1984)[15] M.K.Smith and S.H.Davis, J. Fluid Mech. 132, 119 (1983).[16] S.H.Davis, Ann.Rev.Fluid Mech. 19, 403 (1987).[17] M.K.Smith, J. Fluid Mech. 194, 391 (1988).[18] P.Laure and B.Roux, J. Cristal Growth 97, 226 (1989)[19] B.M.Carpenter and G.M.Homsy, J. Fluid Mech. 207, 121 (1989).[20] G.Z. Gershuni, P.Laure, V.M.Myznikov, B.Roux, E.M.Zhukhovitsky, preprintsubmitted to Europ.J.Mech. B (Fluids)[21] C.Normand and J.Lega, private communication[22] F.Plaza, J.M.Vince and M.Dubois, private communication6Figure CaptionsFigure 1: Schematic drawing of the experimental apparatus.Figure 2: Shadowgraphic image of a traveling wave pattern in an horizontalplane. The top (respectively the bottom) of the picture corresponds to the hot(repectively the cold) plate.Figure 3: Critical temperature difference ∆Tc vs. the height of liquid h.Figure 4: Spatio-temporal evolution showing a left- and a right going wave emit-ted from a source and corresponding to the pattern shown in Figure 2. The totalacquisition time is 25 sec and the spatial extension is 20 cm.Figure 5: Evolution of the period Tc with the height of liquid h.Figure 6: Evolution of the period T (a) and of the velocity vϕ (b) of the waveswith the reduced temperature difference ε.7

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J. M. Vince

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