OSCILLATING INSTABILITY IN ONE-DIMENSIONAL RAYLEIGH-BENARD CONVECTION

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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-

C3-186. �10.1051/jphyscol:1989327�. �jpa-00229468�

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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-91191 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 Ra2, 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 FY (TY = Ly/d, with d the depth of the fluid layer) have revealed that, in narrow cells, i.e. for ry<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 http://dx.doi.org/10.1051/jphyscol:1989327

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C3-182 JOURNAL DE PHYSIQUE

A

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COLLECTIVE OSCILLATING STATE I N A RECTANGULAR CELL.

The e x p e r i m e n t a l set-up can b e d e s c r i b e d a s follows: t h e f l u i d - % o i l with v i s c o s i t y 0.65 1 0 - * s t o k e s a t room temperature ( P r = 7.5)

-

f i l l s a r e c t a n g u l a r c e l l , with dimensions 1 8 0 x 2 ~ 5 ma, g i v i n g h o r i z o n t a l a s p e c t r a t i o s o f 36 and 0 . 4 . The c e l l i s i n s e r t e d between two h o r i z o n t a l copper p l a t e s , t o a s s u r e good thermal c o n d u c t i v i t y o f t h e h o r i z o n t a l boundaries. T h e i r temperature i s r e g u l a t e d by c i r c u l a t i n g w a t e r from t h e r m o s t a t e d b a t h s , such t h a t t h e f l u i d l a y e r e x p e r i e n c e s a temperature d i f f e r e n c e AT, s t a b l e w i t h i n 2 10-*C.

The s u r v e y of t h e c o n v e c t i v e p a t t e r n and o f its dynamics is performed through shadowgraphic images observed i n a v e r t i c a l plane. These images can b e recorded on v i d e o t a p e s ; moreover a p a r t o f them can be analyzed by a photodiode a r r a y of 256 p i x e l s . u s i n g an o p t i c a l d e v i c e . T h i s a r r a y is connected t o a computer, a l l o w i n g t h e i n t e n s i t y a l o n g a choosen h o r i z o n t a l l i n e t o be recorded a t any t i m e , o r time s e r i e s t o be c o n s t r u c t e d with a p p r o p r i a t e sampling time between e a c h l i n e r e c o r d . Then t h e s p a t i o - t e m p o r a l e v o l u t i o n of t h e p a t t e r n can be followed q u a n t i t a t i v e l y / 4 / .

I n c r e a s i n g t h e Rayleigh number, a p e r f e c t s t r u c t u r e t a k e s p l a c e , formed by p a r a l l e l r o l l s with a x i s p e r p e n d i c u l a r t o t h e l o n g e r s i d e of t h e c e l l . Though g e n e r a l l y much s h o r t e r than Xc i n s u c h a narrow channel /5/ / 6 / /7/ (Xc = 2d, t h e c r i t i c a l w a v e l e n g t h ) . t h e a c t u a l wavelength depends on t h e p r e v i o u s thermal h i s t o r y , b u t , i f t h e f l u i d l a y e r is submitted t o q u i c k AT i n c r e a s e , very s h o r t wavelengths a r e favoured. For Ra 2 l o 6 , t h e wavelength can be a s s h o r t as 0.38 A, ( o r a/ac = 2 . 6 , with a t h e wavenumber of t h e p a t t e r n ) . This wavelength remains s t a b l e up t o v e r y h i g h Ra v a l u e s ( t h e h i g h e s t e x p l o r e d v a l u e with t h e a c t u a l s e t - u p was around 2 . 3 l o 6 ) . Along t h i s whole Ra domain. t h e p a t t e r n i s s t a t i o n n a r y , e x c e p t w i t h i n a range where a c o l l e c t i v e o s c i l l a t i n g s t a t e is o b s e r v e d . I n c r e a s i n g AT, o s c i l l a t i o n s appear f o r Ra= 1.17

lo6

and d i s a p p e a r f o r Ra = 1.7 l o 6 , whereas by d e c r e a s i n g AT, t h e y a r e observed i n t h e same domain, s l i g h t l y s h i f t e d t o lower Ra v a l u e s by a small h y s t e r e t i c phenomenon. Near t h e i r t h r e s h o l d , t h e o s c i l l a t i o n s a r e p e r i o d i c ; a t higher Ra v a l u e s , a b i p e r i o d i c regime is observed b e f o r e r e t u r n i n g t o s t a t i o n n a r i t y .

The g e n e r a l f e a t u r e s o f t h e s e o s c i l l a t i o n s a r e shown i n f i g . 1 . The main s t r e a m s (extrema of i n t e n s i t y on t h e shadowgraphic p i c t u r e ) move p e r i o d i c a l l y around t h e i r mean p o s i t i o n , with phase o p p o s i t i o n between h o t and c o l d streams motion. So when one c o n s i d e r s a wavelength, one r o l l seems t o i n c r e a s e p e r i o d i c a l l y i n s i z e a t t h e expense of t h e o t h e r and c o n v e r s e l y . I n t h e monoperiodic regime, t h e amplitude of t h e d i s p l a c e m e n t 6 X o remains c o n s t a n t when t h e time r u n s , b u t depends on t h e p o s i t i o n i n t h e c e l l . A t p a r t i c u l a r p o i n t s , e q u a l l y spaced, no motion i s p r e s e n t ; they d e f i n e nodes of o s c i l l a t i o n s , with, i n between, a maximum of t h e amplitude l i k e i n a s t a t i o n n a r y wave ( f i g . 2 ) . Note t h a t the nodes a r e n o t n e c e s s a r i l y l o c a t e d a t t h e main s t r e a m s . The new l e n g t h s c a l e t h e y i n t r o d u c e i n t h e p a t t e r n is of o r d e r o f 5 t o 6 Xo, where A, is t h e a c t u a l wavelength. and can be incommensurate with Xo.

When w e s p e a k of monoperiodic regime, i t means t h a t t h e f l u i d l a y e r undergoes t h e same o s c i l l a t o r y mechanism, with t h e same frequency, whatever i s t h e measurement p o i n t (except f o r t h e amplitude a s d e s c r i b e d p r e v i o u s l y ) . P a r t i c u l a r l y i m p r e s s i v e i s t h e f a c t t h a t , by performing c r o s s c o r r e l a t i o n measurements. one can show t h a t t h e r e e x i s t s a s t r i c t coherence a l l along t h e chain of t h e r o l l s : about a hundred r o l l s o s c i l l a t e i n p e r f e c t synchronism. When t h e regime becomes b i p e r i o d i c , t h e r e i s n o , s i g n i f i c a n t change i n t h e g l o b a l behaviour. e x c e p t f o r t h e f a c t t h a t t h e second frequency which a p p e a r s i n t h e dynamics ( t y p i c a l l y around f1/50 i f f l i s t h e frequency o f t h e b a s i c o s c i l l a t i o n s ) is r e l a t e d t o p e r i o d i c displacement o f t h e nodes o f v i b r a t i o n s ( p r e v i o u s l y f i x e d i n t h e monoperiodic regime).

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Ra = 1.27 10').

a) I n t e n s i t y v e r s u s X of a p a r t o f t h e shadowgraphic images taken i n t h e middle of t h e c e l l . Minima: u p r i s i n g h o t streams. Maxima: descending cold streams. The t i m e lapse between each l i n e record i s 350ms. ( f l = 0.275 HZ)

b) P o s i t i o n s of t h e i n t e n s i t y extrema v e r s u s time. (processed from t h e d a t a o f t h e f i g . l a ) .

Fig .2.

Amplitude of t h e displacement SX, = Xm-Xo of t h e streams versus X f o r t h e d a t a shown i n f i g . 1 . The o r i g i n Xo of the positions is taken a r b i t r a r i l y a t t h e i n s t a n t t = O . Xm i s the maximum displacement f o r each stream. 0: h o t streams; + Cold streams. I n s p i t e o f t h e poor s p a t i a l r e s o l u t i o n (one p i x e l corresponds t o 0.17mm. i.e.-Ao/20; &Xo max

"

3 p i x e l s ) , the p e r i o d i c i t y of t h e s p a t i a l modulation of 6X, is c l e a r l y evidenced.

B

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OSCILLATIONS I N ANNULAR GEOMFTRY

I n o r d e r t o study a one dimensional convective system with p e r i o d i c boundary conditions.

t h e Rayleigh-Benard convection i s a l s o achieved i n an annular c o n t a i n e r f i l l e d with s i l i c o n o i l of P r a n d t l number 22. A s i n t h e r e c t a n g u l a r c a s e , t h e w a l l s of t h e c e l l a r e made of p l e x i g l a s s and i n s e r t e d between copper p l a t e s . Its i n t e r n a l diameter i s 120 mm and i t s e x t e r n a l diameter is 126 mm, g i v i n g a gap of 3 mm. With a depth of 10 mm. t h i s c e l l has an a s p e c t r a t i o rR=0.3 along t h e r a d i u s and

rp=38.6

along t h e circumference.

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A s i n t h e r e c t a n g u l a r c e l l , t h e shadowgraphic images o f t h e c o n v e c t i v e s t r u c t u r e g i v e a p i c t u r e of t h e p a t t e r n i n t h e v e r t i c a l p l a n e , transformed i n t o a c i r c u l a r image through a p a i r of c o n i c a l m i r r o r s /8/. I n o r d e r t o r e c o r d t h e i n t e n s i t y o f t h e p a t t e r n . an e l e c t r o m e c h a n i c a l system a l l o w s a photodiode t o sweep t h e c i r c u m f e r e n c e a t a given frequency. The photodiode s i g n a l i s t h e n d i g i t i z e d and s t o r e d i n a microcomputer. and t h e s p a t i o t e m p o r a l e v o l u t i o n of t h e p a t t e r n c a n b e e x t r a c t e d from t h e r e s u l t i n g time s e r i e s . The c o n v e c t i v e p a t t e r n c o n s i s t s , above t h e t h r e s h o l d , o f a c h a i n o f r o l l s which p r e s e n t s a d i s p e r s i o n o f t h e l o c a l wavelength f a r much g r e a t e r t h a n i n t h e r e c t a n g u l a r geometry. I n f a c t , t h e wavelength i s g e n e r a l l y n o t homogeneous a l l a l o n g t h e c i r c u m f e r e n c e , ranging from Xc/3 t o Xc/2, a s can b e s e e n i n f i g u r e 3; n e v e r t h e l e s s . l i k e i n t h e r e c t a n g u l a r c o n t a i n e r , wavelengths s h o r t e r t h a n Xc seem t o b e favoured. Moreover, t h i s inhomogeneity depends on t h e Rayleigh number; when i n c r e a s i n g i t , one c a n o b s e r v e , f o r g i v e n i n i t i a l c o n d i t i o n s , a widening of t h e range of t h e wavelengths p r e s e n t i n t h e p a t t e r n .

The same c o l l e c t i v e o s c i l l a t i o n o f t h e r o l l s a x i s i s observed i n t h e a n n u l a r geomety, a s t h e f i r s t time dependent regime. B u t , i f t h i s o s c i l l a t i n g i n s t a b i l i t y e x h i b i t s l o c a l l y t h e same p r o p e r t i e s a s i n t h e r e c t a n g u l a r c e l l , i t s t r o n g l y depends on t h e s p a t i a l p r o p e r t i e s of t h e p a t t e r n . A c t u a l l y , some p a r t s o f t h e c e l l a r e o s c i l l a t i n g and o t h e r s n o t , t h e o s c i l l a t i n g r o l l s b e i n g t h o s e w i t h t h e s h o r t e s t l o c a l wavelengths. T h i s inhomogeneity of behaviour r e f e r s t h e n t o t h e d i s p e r s i o n o f t h e l o c a l wavelengths, a s can b e s e e n i n f i g u r e 4. and t h e o s c i l l a t i o n s look l i k e a tendancy f o r t h e s t r u c t u r e - a n d i n p a r t i c u l a r f o r t h e s h o r t e s t r o l l s - t o a d a p t i t s l o c a l wavelength t o a l a r g e r v a l u e .

The o s c i l l a t i n g regime is g e n e r a l l y monoperiodic, w i t h t h e same frequency i n s i d e an o s c i l l a t i n g domain. But one can o b s e r v e s l i g h t l y d i f f e r e n t f r e q u e n c i e s between two o s c i l l a t i n g domains s e p a r a t e d by s t a t i o n a r y ones. The p r e s e n c e o f nodes o f o s c i l l a t i o n ( r o l l s t h a t do n o t o s c i l l a t e ) i s a l s o evidenced i n s i d e an o s c i l l a t i n g domain. a s i n t h e r e c t a n g u l a r geometry, showing t h a t t h e y a r e n o t r e l a t e d t o t h e p r e s e n c e of r i g i d l a t e r a l boundaries.

When t h e Rayleigh number i s i n c r e a s e d , t h i s c o l l e c t i v e i n s t a b i l i t y remains, with an i n c r e a s i n g amplitude of t h e o s c i l l a t i o n s , b u t o t h e r more complex regimes a r i s e . such a s s p a t i a l d e f e c t s o r s p a t i o - t e m p o r a l i n t e r m i t t e n c i e s . I n f a c t , when t h e amplitude of o s c i l l a t i o n i s l a r g e enough, two r o l l s can merge and a t u r b u l e n t regime c a n appear i n a s p a t i a l l y . r e s t r i c t e d domain w h i l e o t h e r r e g i o n s remain o r g a n i z e d , s t a t i o n a r y o r o s c i l l a t i n g .

C

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CONCLUDING REMARKS.

The p r e v i o u s l y d e s c r i b e d phenomenon o f r o l l s ' o s c i l l a t i o n s , w i t h its p a r t i c u l a r p r o p e r t i e s , h a s been observed i n d i f f e r e n t g e o m e t r i e s , ( r e c t a n g u l a r and a n n u l a r ) , and with d i f f e r e n t P r number f l u i d s . I t seems t o b e a g e n e r a l phenomenon d i r e c t l y r e l a t e d t o t h e presence o f s m a l l wavelengths i n narrow c e l l s and two-dimensional motion. It l o o k s s i m i l a r t o o s c i l l a t i o n s observed i n Hele-shaw c e l l s /5/, though i n t h i s c a s e , t h e gap is much more s m a l l e r ( r Y < 0 . 1 ) and t h e o s c i l l a t i o n s a r e c h a o t i c very soon a f t e r t h e i r appearance. Note t h a t t h e frequency of o u r r e p o r t e d o s c i l l a t i n g behaviour ^'-

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1 0 - * d 2 / ~ , when P r = 7.5 and 0 . 6 10-'d2 /DT when P r = 22

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a r e o f t h e same o r d e r o f magnitude a s those r e p o r t e d i n /5/.

The observed b e h a v i o u r , which p o i n t s o u t t h e tendancy f o r t h e p a t t e r n t o i n c r e a s e l o c a l l y its wavelength, i n d i c a t e s t h a t we a r e i n presence o f an i n s t a b i l i t y i n t h e p l a n e [Ra. a].

N e v e r t h e l e s s , t h e r e a r e d i f f e r e n c e s between t h e two geometries ( a n n u l a r and r e c t a n g u l a r ) due a t f i r s t t o t h e wavelength d i s p e r s i o n i n t h e a n n u l a r c a s e . They can b e summarized a s follows. I n t h e r e c t a n g u l a r c e l l . t h e o s c i l l a t i o n s a r e p r e s e n t o n l y i n a d e f i n e d window of Ra numbers, surrounded by a s t a t i o n n a r y regime; t h e phenomenon i s v e r y p u r e and even when

I

t h e nodes move, t h e y do s o i n a r e g u l a r and p e r i o d i c way. On t h e c o n t r a r y , i n t h e a n n u l a r c e l l , t h e o s c i l l a t i n g motion o f t h e r o l l s is n o t organized s p a t i a l l y . even n e a r t h r e s h o l d and i t does n o t d i s a p p e a r , when i n c r e a s i n g Ra. The behaviour becomes more complex, where t h e o s c i l l a t i o n s and t h e l o n g waves, a s s o c i a t e d t o t h e i r s p a t i a l amplitude modulation.

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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.

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