From temporal chaos towards spatial effects

Nuclear PhysicsB 0koc. Suppl.)2 (1987) 247-258 North-Holland,Amsterdam

247

FROM TEMPORAL CHAOS TOWARDS SPATIAL EFFECTS BERGE Service de Physique du Solide et de R~sonance Hagn~tique CEA Saclay, 91191 Gif-sur-Fvette Cedex, France

P.

The results presented in this lecture are those of experiments done at Saclay by M.Dubois and myself on Rayleigh-Benard (R.B.) convection I in high Prandtl number fluids (test fluid : silicone oil). The measurements are based on optical techniques. - The visualization of the whole structure (as well as that of the thermal oscillators) is achieved by "shadowgraphy" and Foucault (or Schlieren) images (knife-edge technique). - The dynamical measurements related to semi-local thermal gradients are based on a study of the current of photodiodes located at suitable places on the Foucault image ("semi-local" means that there is an integration along the path of the beam) s

The aim of this paper is to describe some features related to spatial effects in turbulence as opposed to chaos in "small boxes". I - CHAOS IN A SMALL BOX VERSUS TURBULENCE

IN LARGE CONTAINERS

Due to the restricted number of spatial modes allowed in a structure "locked" in a small box, it is well known that the system can be considered as a dynamical one (see the lecture by M.Dubois, this conference). Turbulence which sets in under such conditions very specific, namely,

is named "chaos" and its properties

are

- the convective structure remains well ordered even in the chaotic regime : spatial order is maintained, the streams being always present at the same location (spatial effects can be disregarded and one need consider only the temporal effects). - chaos arises (at relatively high values of the Rayleigh (Ra) number Ra > 200 Ra C , Ra C being the value for which convection sets in) from periodic or quasi-periodic regimes due to thermal oscillators located in the thermal boundary layers (that means that, near onset, many peaks coexist with the broad band spectrum characteristic of chaos) 3. - The fractal dimension of the corresponding

0920-5632]87[$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

attractor is small.

P Berge / From temporal chaos towards spatial effects

248

In connection with this last point, let us recall that, from a time one can calculate the correlation dimension v of the attractor 4.

series, However,

this dimension

robust

against

can be

change

considered as relevant only if its value is

of

the

various

reconstruction of the phase space. the case of chaos in a "small box ''~. The i0)

is

interaction rolls

situation found

completely between

(thermal)

Here,

involved

in

the

Such a robustness is generally found in

in larger

different.

parameters

containers the

oscillators

(aspect ratio larger than

turbulence

is

no

more

due to

: it is due to erratic motions of

boundaries in a disordered convective structure. This means that there

is no spatial order, since the streams can be located anywhere in the cell at any

time. Contrary to chaos, this turbulence can be named spatial turbulence

or phase turbulence to emphasize its spatial implication with continuous phase shifts of the rolls s . Phase turbulence sets in directly from steady convection (and at comparatively low values of Ra, Ra < i00 Rat). This means that, near onset, the spectrum is continuous and without peak. Lastly, the dimension

of

(equivalently,

the

corresponding

one says

that, in

attractor,

if

any,

is

very

large

contrast to the case of chaos, the system

has a large number of degrees of freedom). Note from now onwards that, though characteristic

of large

containers,

spatial

turbulence can

be

temporarily

present in small boxes. The

following table summarizes the main differences between temporal

chaos and spatial turbulence

(high Prandtl number fluids). Table I

Chaos in a small box aspect ratio r ~ 2

Turbulence in large containers F > i0

arises from periodic arises from steady regime Ra < i00 Ra regime Ra > 200 Ra C C

continuous spectrum continuous spectrum plus peaks spatial order kept low dimension

II - G E O M E T R I C A L

ASPECTS

OF PHASE

any spatial order high dimension

TURBULENCE

As mentioned above, the characteristic of spatial (or phase) turbulence is a perpetual and erratic motion of the streams, i.e. the spatial arrangement different

is continuously

changing; that is to say, an infinite number of

structural configurations

can be seen with time. Furthermore,

the

P. Berge / From temporal chaos towards spatial effects correlation

249

or ressemblance between successive structures is completely lost

after a certain lapse of time depending on the geometry and on the Ra number. In

other words, a structural arrangement at a given instant t has nothing in

common with an earlier arrangement at (t-T) provided x is sufficiently larger than a certain correlation or ressemblance characteristic time. In order to check experimentally that any configuration can be adopted by the streams, one can perform some kind of optical averaging of many different structural configurations visualized through shadowgraphy. To do so, one superimposes on the

same

photographic

instantaneous exposures

plate

configuration

many

exposures

(indeed,

the

each

time

corresponding

between

two

to

an

successive

has to be larger than the correlation time). When such a procedure

is applied, one observes an almost uniform "grey" aspect on the final picture. This shows that the bright (dark) zones corresponding to descending

S

FIGURE

1

Pictures of streams taken in phase turbulence. The size of the cell is dx2dx4.36d with d=1.25 cm. Silicone oil 3.10 -2 stokes (Ra=170Ra) . The c structures are "seen from above", dark lines corresponding to warm, uprising flows and bright lines to downgoing streams. The first three pictures correspond to structures at t, t+30 min, t+60 min. The fourth photograph S is the sum of 20 instantaneous pictures.

P. Berge / From temporal chaos towards spatial effects

250

(ascending) probability.

streams can he located anywhere in the cell with an equal In other words, in the case of phase turbulence, there is no

mean structure. This as averaged pictures

is illustrated in figure 1 where instantaneous as well are shown. Indeed in the relatively small aspect ratio

used for this experiment, III-

this phase turbulence is temporary

TRANSIENT SPATIAL TURBULENCE

As mentioned

above, even in a small box and before the setting in of

a definitive structure, there can turbulence like in large containers.

the

(see below).

exist

a

temporary

(transient)

phase

This transient phase turbulence generally follows a rapid starting of convection or the destabilization of a convective structure 7. It can

last from hours to weeks, this duration depending on the Ra number and on the size of the box. A very peculiar and striking effect is associated with this transient turbulence. Typically, the phase turbulence is interrupted from time to time by a quasi-locking of the structure in one of the many possible arrangements*. After this temporary locking, the phase turbulence reappears again until a new quasi-locking arises (in another configuration) etc... After many such successive sequences of phase turbulence and temporary lockings, a definitive locking occurs in one of the temporarily visited structure. In structures

other words, of low

an observer

symmetry (phase

spectacular slowing down structural arrangement.

and

would see turbulence)

temporary

rapidly evolving disordered with,

locking

on

from time a

more

to time, a symmetrical

In terms of (complex) phase space, one can say that there exists a certain number of attractors (the different possible structures) which are visited (pseudo-locked states) and, between two visits, the representative point wanders at random in the phase space. Finally, one of the visited attractors is choosen forever. This curious "hesitations" of the structure to choose a definitive arrangement is reminiscent of the very long transients existing in dynamical

systems T M

*

Recall

that

for

a

given

Ra

number

(Ra > 200 Ra c)

"equilibrium" structures can be locked in a small box".

many

different

P. Berge / From temporal chaos towards spatial effects One

can see

251

in figure 2, an example of such a phenomenon. The three

pictures above represent structures during sequences of phase turbulence; one can note their disorder and their low symmetry. Furthermore, these configurations are subjected to rapid changes : the memory of a given configuration is lost after a few minutes. On the contrary, one can see in the lower part of figure 2, six structures corresponding to temporary lockings. They are more symmetrical and, above all, a given structure may persist

(during its

temporary locking) without appreciable change during I0

minutes or more. In larger containers and for lower Ra numbers, the temporary lockings may last up to one hour.

FIGURE 2

Pictures of streams taken in transient turbulence with temporary lo~kings. Same experiment as in figure 1 except for the size of the cell dx2dx2.6d and Ra=290Ra . c The three pictures above correspond to phase turbulence with disordered and rapidly changing structures. The six pictures below correspond to more symmetrical arrangements related to remporary lockings.

P. Berge / From temporal chaos towards spatial effects

252

IV - P H A S E

TURBULENCE

As streams

VERSUS

seen above,

INTERMEDIATE

TURBULENCE

phase turbulence

is characterized

by the

associated with erratic motions of

absence of

a mean

structure and

sets in

directly from the steady state. On the contrary, chaos takes place within an ordered structure and originates from thermal oscillators. Let us describe now an intermediate case which we have found in relatively small aspect ratio cells.

For this

turbulence, common

mechanism,

turbulence: near

"intermediate turbulence",

related

to

an

there

both arise

erratic

are

two

motion

the mechanism is, like in phase of (some) streams. Due to this

similarities

directly from

between

the

two cases of

steady convection and their spectra -

onset - is continuous in both cases. However, as opposed to the case of

phase fixed

turbulence, mean

instantaneous structural

there

structure

exists for like

pictures taken

arrangements which

in

the "intermediate

chaos.

One

can

turbulence"

see

in

(I.T.) a

figure 3

three

in a sequence of I.T. These pictures show some differ by the configurations of some streams,

but, the main streams remain at the same place. The fourth picture is the sum of

some

structure

exposures; contrary

it to

reveals what

the

existence

happens

procedure does not reveal anything

with

phase

of

a well defined average turbulence

where such a

(see for comparison, figure i).

FIGURE

3

Pictures of streams taken in "intermediate turbulence" (size of the box equals dx2dx3.3d, Ra=170Ra ) C The first three pictures correspond to structures at different times, while the fourth comparison, fig.l

S

is

the

sum

of

i0

instantaneous pictures. See, for

P. Berge / From temporal chaos towards spatial effects Another striking As y e t m e n t i o n e d , be v e r y

appears between phase turbulence

due t o t h e v e r y l a r g e

in phase turbulence, to

difference

large.

the contrary,

and I . T .

number of d e g r e e s o f f r e e d o m i n v o l v e d

the dimension of the c o r r e s p o n d i n g On

253

careful

attractor

calculations

is expected

of the c o r r e l a t i o n

d i m e n s i o n y p e r f o r m e d i n t h e c a s e o f I . T . show t h a t t h e d i m e n s i o n i s l i k e l y t o be f i n i t e and c o m p a r a t i v e l y low. One can s e e i n f i g u r e 4 t h e r e s u l t s o f s u c h a c o m p u t a t i o n 11,5 as a function of t h e embedding t i m e . I n d e e d , when dimensions be

very

larger cautious

preliminary. the

than - let and

the

However, t h e

variations

of

us s a y - 5 a r e f o u n d , results

mentioned

robustness

t h e embedding

t i m e seems a good g u a r a n t y the attractor is finite s.

of

one knows t h a t

here

must be c o n s i d e r e d

the calculated

dimension,

we h a v e t o as

v a l u e s of v a g a i n s t

t h e t i m e d e l a y and t h e s a m p l i n g

a l l o w i n g us t o s a y t h a t ,

in I.T.

the dimension of

5

O

!

i

1000

2000

FIGURE

Correlation

dimension as

a function

S

4

of the

embedding time~ in the case of

"intermediate turbulence". The conditions are the same as for figure 3. Table ~ compares the turbulence shares (much

and p h a s e

properties less

extensive

of

both chaos

understood, study of

principal

turbulence.

this

at

As

one can

of chaos,

see "intermediate

(now, w e l l u n d e r s t o o d )

least

I.T.

characteristics

for

can l e a d

Pr ~ 1) us t o

intermediate turbulence"

and p h a s e t u r b u l e n c e

. Thus, we t h i n k t h a t

take a

step

established basis of deterministic chaos t o w a r d s more m y s t e r i o u s h a v i n g a much h i g h e r number o f d e g r e e s o f f r e e d o m .

V - TURBULENCE

A

IN A N A N N U L A R

natural idea

By

- achieved

increasing the

systems

CONTAINER

to create a system with a large number of degrees of

freedom is to couple many dynamical systems. Experimentally, principle

more

from t h e f i r m l y

this can be - in

with a one-dimensional chain of many convective rolls.

Ra number,

these rolls will oscillate and their coupling

will produce the required phenomena.

P. Berge / From temporal chaos towards spatial effects

254

Table

Chaos Mechanism

Transition

Intermediate turbulence

thermal oscillator

motion of (some) streams

from periodic or quasiperiodic regime

Spectrum Spatial structure

maintained

Dimension

motion of all the streams

directly from steady regime

peaks plus broad band

Phase turbulence

directly from steady regime

broad band

broad band

existence of a mean structure

totally disordered

finite

very large

low

The best system to do this is the convection in an annular container there is no boundary effect 12 By choosing a gap of the order of

because

magnitude

of

the

depth,

the

rolls

can

be

considered

as

mainly

two-dimensional. In order to be sure of a perfect homogeneity of the temperature along the

annulus, we chose to work between copper plates, the two rings confining

laterally more

the fluid

complicated,

being made but

we

of plexiglass.

adopted

an

optical

Indeed, the visualization is geometry

having the axial

symmetry of the system. A first

simplified scheme

conical mirror

radially

an incident

of the

whose axis

arrangement can

be seen in figure 5. A

is the same as that of the annulus spreads

(vertical) parallel

beam. This

radial and horizontal

beam crosses the convecting fluid and is collected by a second conical mirror with same axis as the first mirror. Thus, after having crossed the fluid, the beam

is

transformed

again

into

a

parallel

vertical beam which produces

shadowgraphic images of the streams on a horizontal plane. The

preliminary results presented here are related to a depth d=l cm

a gap of 1 cm and a mean radius of the annulus of 6.2 cm. The fluid is Silicone oil (2.10 -2 stokes viscosity). By increasing the Ra number from the ordered

state (a set of 50 identical rolls) the regime becomes turbulent for

Ra > 200 Ra . In this experiment, the striking fact is the coexistence of C spatially disordered (and turbulent) zones with ordered and laminar (almost steady)

ones. Figure

6 shows

such a

situation where

coexist with two patches of well ordered rolls.

two disordered zones

P. Berge / From temporal chaos towards spatial effects

WS

255

A

FIGURE 5

Schematic representation of th~ annular cell. A axis of the whole system R plexiglass rings F convecting fluid P copper plate W circulating thermostated water B incoming beam C I internal conical mirror C external collecting conical mirror 2 S plane of the screen where the shadowgraphic image is formed CS image of a cold stream WS image of a warm stream

The position and the size of the turbulent and laminar zones fluctuate with time, these changes occurring mainly through the displacement of the boundaries between turbulent and laminar zones (see figure 7).

observed

This behavior is very reminiscent of spatio-temporal intermittency in Kuramoto-Sivashinsky-like equation Is or in coupled mappings 14

However, the different features are complicated by the presence of phase defects in the arrangement of the rolls (abnormal size of a roll or of a wavelength, for example). Thus, many other experiments remain to be done in order to definitively interpret the observed phenomena.

P. Berge / From temporal chaos towards spatial effects

256

FIGURE 6

Shadowgraphic picture of the convection in an arrows indicate the direction of the streams.

annulus at

Ra=330 Ra . The C

ACKNOWLEDGEMENTS

I am greatly indebted to M.Dubois to whom most of results are due. I which to thank moreover C.Poitou for helpful assistance, Y.Pomeau, A.Pocheau and P.Manneville for stimulating discussions and B.Ozenda and M.Labouise for very nice experimental realizations.

REFERENCES

/1/ /2/ /3/ /4/

P.Bergb and H.Dubois, Contemporary Physics 25, 535 (1984). P.Berg~, Y.Pomeau and Ch.Vidal, Order within chaos (Wiley-Hermann 1986). M.Dubois, P.Bergb, Physics Letters, 76A, 53 (1980). P.Grassberger, I.Procaccia, Phys.Rev.Lett. 150, 346 (1983) and Physica 9D, 189 (1983).

P. Berge / From temporal chaos towards spatial effects

120 I

257

2&O

I

I

I

I

36O (°) Space

I

10 20 ~30 60 min

Time

®

r

0

20

~0

60

j

I

I

I

80 (0) I

spac e =

10

20 30 rain ~ime

® FIGURE 7

Evolution

of the turbulent

(hatched) and laminar zones as a function of time

(space is defined by the angle); r is the size of a roll. (A) typical annulus.

evolutions

(B) part of a turbulent one.

picture

mainly showing

due the

to

motion birth

of

of boundaries for the whole a

laminar zone L inside a

258

/5/

/6/ /7/ /8/ /9/ /i0/ /ii/ /12/ /13/ /14/

P. Berge / From temporal chaos towards spatial effects M.Dubois, P.Berg~, Physica Scripta, 33, 159 (1986). P.Atten, J.G.Caputo, B.Malraison, Y.Gagne, J.de M~canique, vol. special 1984, "Bifurcations et comportements chaotiques". P.Berg~ "Chaos and Order in Nature", Elmau 1981, ed. by H.Haken, (Springer-Verla~), p.14. P.Berg~, M.Dubois, Physics Letters, 93A, 365 (1983). M.Dubois, "Stability of thermodynamic system", Lecture notes in Physics, 164, 177 (1981). P.Manneville, Physics Letters 90A, n'7, 327 (1982). C.Grebogi, S.W.Mc Donald, E.Ott, d.A.Yorke, Physics Letters 99A, n'9, 415 (1983). B.Malraison, P.Atten, P.Berge and M.Dubois, J.Phys. Lettres 44, 897 (1983). A.Pocheau, V.Croquette, P.Le Gal and C.Poitou, Europhysics Letters, in print. H.Chat~ and P.Manneville, Phys.Rev.Lett. 58, 112 (1987). H.Chat~ and P.Manneville, C.R.Acad.Sci.Paris (s~ance du 19/1/87) in press.