Instabilities produced by hot-wire heating below a free surface with emphasis on relationships between temporal and spatial dynamics

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Instabilities produced by hot-wire heating below a free surface with emphasis on relationships between

temporal and spatial dynamics

E. Ringuet, S. Meunier-Guttin-Cluzel, C. Rozé, G. Gouesbet

To cite this version:

E. Ringuet, S. Meunier-Guttin-Cluzel, C. Rozé, G. Gouesbet. Instabilities produced by hot-wire heat-

ing below a free surface with emphasis on relationships between temporal and spatial dynamics. Jour-

nal de Physique II, EDP Sciences, 1994, 4 (8), pp.1243-1260. �10.1051/jp2:1994197�. �jpa-00248040�

Classification Phj,vics Abstiacts 05.45

Instabilities produced by hot-wire heating below

_a

free surface

with emphasis

_on

relationships between temporal and spatial dynamics

E.

Ringuet,

S.

Meunier-Guttin-Cluzel,

C. Rozd and G. Gouesbet

L.E.S.P. (*), CORIA, Rouen INSA, B.P. 08, 76131 Mont Saint

Aignan

Cedex, France

(Receiied 22 DeieJ1ibei /993, revised /6 April /994, accepted /6 May /994)

Abstract. Bifurcations up to the appearance of temporal chaos _are observed when hot-wire heats

below _a free surface. New obwrvations _are reported, and _some

emphasis

is laid _on the

relationships between

temporal dynamics

produced by signal~ averaged along the hot-wire and

spatial

behaviour of _waves generated above it.

1. Introduction.

There is _a

growing

interest in the

study

of instabilities

generated by

local

heating

below a free surface,

leading

_to the observation of many nonlinear

phenomena including developed

chaos and intermittencies.

Heating by

a laser

produces

the so-called

optical

heartbeats _or thermal lens oscillations

which,

beside

allowing

fundamental

investigations

in the framework of nonlinear

dynamics (Refs. [1-31

and references

therein)

also leads to

applications

for concentration

measurements of chemical

species [41.

In another

procedure

to which this paper is

devoted,

heating

is achieved

by

means of _a hot-wire. This allows _a better control of the distance between the

heating

location and the free surface.

Temporal signals

recorded _at the ends of the

hot-wire,

thus

averaging phenomena occurring along

the wire, exhibited various kinds of behaviour

including

type II

intermittency (Refs. [5-71

and references

therein).

On the other hand,

spatial phenomena

are involved in hot-wire

experiments

under the form of _a train of _waves

propagating

_on the free surface,

parallel

to the wire,

allowing

_an

experimental study

^of lD

spatio-temporal

chaos

[8-101.

In

particular,

Vince and Dubois [81 ^{examined the}

dynamics

above _a

long (L

₌ 60 _cm hot-wire

showing

that the train of _waves may exhibit several defects

(sources

and

sinks)

which however

eventually

^relax to a

simpler

situation with 0 _or defect.

No defect is observed when the distance between the wire and the surface is

large.

In such _a case,

spatio-temporal disorganization

of the train of _waves may be observed.

Furthermore,

in this _case, the _waves have _a

significant spatial

extension

perpendicular

to the wire and may be

(*) U.R.A. CNRS 230.

interacting

with confinement walls. In the

present

paper, hot-wire

experiments

_are carried _out with _a smaller wire

(6 cm), allowing

_a better

study

of

temporal signals averaging along

^the wire and

spatial phenomena,

^without any lateral confinement effect. Also, _{a very}

simple

model

enables _us to

predict

under which conditions

nonperiodic phenomena

_may ^be

expected

in the

case when _no

spatial

defect _occurs.

The paper is

organized

_as follows. Section 2

briefly

describes the

experimental

_set-up and the

primary instability

which is characterized _{as a}

supercritical Hopf

bifurcation. Section 3 introduces _a model

relating spatial

_{wave patterns} and

temporal signals

in the absence of any

spatial

defect.

Experiments

are

reported

to demonstrate the

pertinence

of the model. Section 4 presents a zoo of

temporal

behaviours

examplifying

the richness of the

system.

In the _case of

quasiperiodicity, temporal

behaviour may

easily

be related to the

spatial organization

of the

wave

pattern.

^{Section 5 is devoted} to the coexistence of attractors. In sections 4 and

5,

the observed behaviours

correspond

to the existence of _one

spatial

defect. In all these sections, the relations between

temporal signals

and

spatial

_{patterns are} discussed whenever

possible.

Section 6 is _a conclusion.

2.

Experimental

set-up and the

primary instability.

2.I EXPERIMENTAL _SET-uP. The

experimental

_set-up is similar _to the _one described

by

Rozd et al.

[61,

but with two modifications _:

(I)

the wire

length

is 6 _cm instead of 3 and

(it)

it is

current controlled instead of

being

temperature controlled.

The fluid under

study

is _a silicon oil

(Rhodorsil

47V

lo)

with _a Prandtl number

equal

to 130.

The oil is contained in _a tank (17 x12 _x ID

cm3)

whose walls _are

kept

at _a constant

temperature

(To

_m 25

°C) by using

thermostated _water. This tank is located in _a thermostated box which

keeps

the ambient temperature at To ^{within 0.05 K. The fluid is heated}

by

_a

platinum

wire with _a diameter

equal

to 20 _~Lm. Its resistance _can be written _as R

IT)

=

Ruji

_{+ a}

IT To)] (')

where

Ro

~ 18 Q is the wire resistance at the reference temperature

Tu,

_a ^is the temperature coefficient of the

platinum

(a 3.9 _x

10~~

_K~

')

_and T is the

spatial

average of the wire temperature. ^{The ratio} of the wire

length

L over the wire diameter is

large enough

to ensure that the temperature

profile

is

essentially

constant

along

the wire in the

steady

state, in such _a way that

edge

effects _are assumed _to be

negligible [I1, 121.

The wire is soldered at the top ^{of two thin}

strips

and is

tight horizontally (Fig.

Ii below the free surface of the fluid.

Horizontality

is

adjusted by making

the wire and its

image (formed by

reflexion _on the free

surface) parallel

when the wire is very close to the surface. The distance between the surface and the wire is taken _to be _zero when the wire and its

image

_are blended.

The vertical

position

^of the wire is

adjusted by

_means of _a micrometric

pedestal

described in reference

[131.

The wire is

supplied

^with a constant current

(I)

which is controlled

by using

_a micro- computer. ^{The Joule effect}

resulting

^{from the} current induces the

heating

of both the wire and the fluid

surrounding

the wire.

By using

a second

micro-computer,

the wire

voltage

is recorded

with _a

sampling frequency

^which may be

adjusted

between 20 and 400 Hz. Time series of 160 000 consecutive values may be recorded.

The wire

voltage

U is related to the

spatially

_average temperature T

by

Ohm's law U

=

Ro[I

+ a IT

To

II (2)

The

dynamics depends

_on two control parameters

ii)

the distance d between the wire and the free surface and

(it)

^the

intensity

I

supplied

to the wire.

~ SUPPORTS ~

~fi

Fig, I. -Schematic representation of the supports ^{of the wire.}

2.2 THE _PRIMARY INSTABILITY.- For _a fixed distance d and for _a low temperature

T, the system ^is

steady.

The free surface is deformed,

producing

_a crest or a

trough depending

on whether d is

large

_or small. A double convection roll is established

along

the wire (see for instance numerical simulations in Ref.

[141).

When T is increased, the system _may ^undertake a

supercritical Hopf

bifurcation

[61, leading

to

oscillatory

behaviour, if d is _{not too}

large.

Waves then appear on the free

surface, propagating parallel

to the wire. Some characteristics of these

phenomena

_may be understood, and _even

quantitatively predicted, by

_a

simple

model

relying

on the existence of two characteristic times

(I)

_a heat transport ^time

corresponding

to the time

required

for the heat transport ^{between the wire and the free surface and}

(it)

a

Marangoni

time

corresponding

to the

disruption

of temperature

gradients

at the surface

[lsl.

For

measuring

the critical temperature _T~

(critical intensity I~)

at which the system becomes unstable, the

intensity

is

gradually

increased for _a fixed d _up to the appearance of

oscillatory

behaviour. The _onset of

oscillatory

motion may be detected

(I) optically, by illuminating

^the surface with _an

expanded

laser beam and

observing

the reflected

light

on a screen or

(it) electrically, by examining

the wire

voltage

time series which becomes

periodic

at the onset

[61.

The critical

intensity I~

is

displayed

t,eisus d in

figure

2a, the

points

below and above the

curve

corresponding

_to

steady

and

oscillatory

behaviour,

respectively.

Periods

Fj

^' at the _onset

versus d _are

displayed

in

figure

2b.

Increasing

d from _zero,

periods

_are first small

(0.

_s for

330 (a) 4 (b)

$i

_{280 QS3}

E

~

~'

230

t2

3

_

180

130

0 2 3 4 0 2 3 4

d

(mm)

d

(mm)

Fig.

~. The _onset of

oscillatory

motion _occurs for different values I~ ^{of the}

intensity depending

on the

distance d (a), with _a certain period _T~ I/F~ (b).

d

= 0.2

mm),

then increase in _a

non-regular

_{way up} to 4.5 _s and

eventually

^{decrease. These} results _are similar to those

previously

described

[6,

8,

131.

The critical

wavelength

^{of the}

propagating

waves is about ID times the distance d

[6, 81. Relying

_on the

experimental

determination of the

disruption

relation of these _waves, it is claimed that

they

_are neither pure

capillary

nor pure

gravity

waves

[61.

Their exact nature is therefore still unknown. Let _us however mention that _a connection with

hydrothermal

surface-wave

instability

disturbances

governed by

the

Kuramoto-Sivashinsky equation

has been

recently

discussed

[161.

When the-

horizontality

of the wire is

perfectly

well

adjusted,

there is no

privileged

direction of

propagation.

3.

Spatial

wave

pattern

and

temporal signals

: model and

experiments.

In the

steady

_state, the temperature ^is constant

along

the wire and _can

readily

be evaluated

by measuring

U

(Eq. (2)).

When _waves propagate

along

the wire, it is assumed that the local

temperature at location.r

along

the wire and at time t,

T(.r,

t is slaved _to the local _state of the waves, I.e.

T(x, t) ₌

f (x

vt )

(3)

qhere

v is the _wave

velocity.

This

expression

is allowed

by

the fact that _waves propagate without any deformation. Also,

equation (3)

_assumes the absence of any

spatial

defect in the

wave train. When the

phenomena

_are

periodic,

the unknown function

f

_may be

expanded

as a

Fourier series

reading

T(>.,

t)

=

f a~

cos

~ _"~

(x

_vt

)j

^{+ b~, sin ~ "~ ix vt}

)j ⁽⁴⁾

,,

A

Whirl

_{A' is} the

spatial period (wavelength)

of the _wave, and the _{constant term} has been dismissed.

The

spatial

_average of the temperature

along

the

length

L of the wire then reads

j ^L

T(t

=

T(,i,

t At

(5)

L

o

which becomes _:

T(t)

-

i,,f loco,

^CDs

~

i~ _{vii Csn}

_sin ^~

i~

_vi

(6)

where

C[~,

=

~"

sin

(2

^{arn ~}

+'

(l

^cos

(2

^{arn ~}

⁽⁷⁾

n A _n A

~~~ ~'

^~°~

^~

~~

'~~~ _~

"~ ~~~

which also

gives

^{the form} of the time

signal U(t)

which is

proportional

to

f(t),

with the

constant term dismissed.

If the number of _waves L/A

along

the wire is _an

integer

_p, then

C[~,

=

Cl',~

₌ ⁰ and

f(t)

_{is also zero (or}

_actually

_{a constant).} On the other hand, if L/A

= ~p

+1/2),

then the coefficients _{are zero} if _n is _even. In such _{a case,} the spectrum ^{of the}

signal

will not exhibit

peaks

at

frequencies

2

Fo,

4

Fo..

where

Fo

is the fundamental

frequency.

Averaging (T)~

_{over a} time

period

of the

signal T(t

), we obtain the total power of the

signal

P

(L/A

^which reads

The function

f(x

vi )

being unknown,

it is

tentatively

assumed that

only

_aj is different from _zero in relation

(4), leading

to

P

=

2

a(

^{~ ~}

sin~ I

lo)

2 wL A

This result is

displayed

_as A

=

,,I

versus AIL in

figure

3. In the framework of this

simple

model, the dimensionless ratio AIL appears to be _a

pertinent

_parameter to

organize

the results

in the absence of any

spatial

defect. Let _us remark that the control parameters are

I and d

(Sect. 2.I).

For

I, given, depends

_on d _as demonstrated in references

[6, 81.

Therefore,

parameters

might

be taken _as the control parameter ^I and the model parameter

(AIL).

The control parameter ^I would

modify

the function

f(,i

vi of

equation (3).

This

function

being

assumed to be unknown in the above

simple

model, I is _no

longer

_a relevant parameter in this restricted framework.

1.o

O.8

~'~

PI

~

~ ~ ~~

P2

0.2

O.0

O,10 O.25 0.40 O.55 O.70 0.85 1.00

~ /

L Fig. 3. A

= f(A ) ~imple model.

Experimentally,

the total power of the

signal

is extracted

by computing

the power spectrum and

summing

_power

spectral

densities _over all

frequencies.

For _a fixed current I, the

wavelength

A is

adjusted by changing

the distance d,

decreasing

when d decreases. In reference

[61,

the

wavelength

A has been measured

by using

a laser reflection

technique,

the reflected

image being

studied

by

a

photodiode

_array. In the present paper, we measured the

propagation

time t~ ^{of a wave over the}

length

L of the

wire, yielding

the _wave

velocity

v

=

L/t~

^{and the} characteristic

frequency

^F

yielding

the

period

T,

leading

to A

=

LT/t~.

^An

example

of

experimental

results is

displayed

as A _versus d, for1

= 250 mA, in

figure

4,

1.0

O.8

Pi

P2 O.6

~

O.4

O.2

-- p~

0,O

0.4 0.9 IA 1.9 2A 2.9 3A

d

(mm)

Fig. 4. -A f(d) _: experimental results for /

=

250 mA.

exhibiting

the

pertinence

of the model

(Fig. 3).

In

particular,

for the

right bump

in which

d evolves from 3.4 _to 1.4 _mm, AIL evolves from about _to 0.5. For the _next

bump,

AIL evolves from 0.5 to 1/3.

During

these

experiments,

the direction of

propagation

of the

waves remained

unchanged.

This situation _can be

sketchily represented by

the

following

diagram

----.

Figure

5

displays

a time

signal

and its power spectrum ^{for L/A} ₌ ^3.5 ± 0. I,

illustrating

the

simplicity

of the

dynamics

when L/A is of the form ~p + 1/2 j, _p

being

an

integer. Although

all harmonics _are present, ^the

amplitude

of the

peak

2

Fo,

for

instance,

is much smaller than the

4100

~

$J ^~~~

~

2100

0 2 3 4

Time

(s)

~

°

Fo

~

-50 3 Fo

~' 2 Fo

CQ _loo

j~~

-150

0 2 4 6 8 10

Frequency (Hz)

Fig.

5. Time

signal

and power spectrum density (PSD) obtained for L/A

= 3.5 _± 0. (d

= 0.74 _mm

and / 280 mA). U is given in

arbitrary

unit.

amplitude

of the

peak

³

Fo.

This situation

corresponding

to AIL

1 0?9 is indicated

by

the label P in

figures

3 and 4. The

signal

_power

being large,

^the

Signal/Noise

Ratio

(SNR)

is very

large

too.

Conversely, figure

6

displays

_a time

signal

and its power spectrum ^{for L/A} ₌ ^2.0 ± 0. I, which is very different from

(p

+ 1/2), _p

integer (label

P2 in

Figs.

3 and

4).

The

signal

^is more

complicated,

_even harmonics _are

bigger

and the SNR is

significantly

deteriorated.

8200

'F#

~ 74fl~

6600

0 3 6 9

Time

(s)

o

$~

~° 2 Fo

e ^-50 3 Fo

iii

-100

©-

-150

0 2 3 4

Frequency (Hz)

Fig.

6. Time

signal

and power ~pectrum

density

(PSD) obtained for L/A 2.0 _± 0. (d

= 1.45 _mm

and / 250 mA).

Actually,

the states in which the total power is _a minimum _are

special

_states.

They

mark the transition between two

bumps,

_one for which L/A reads N _{+ r,} 0

< I <

I,

N

integer

~ 0, and

another _one for which L/A reads N _{+ r +} I.

According

to the aforementioned idealized

simple

model, the _power minima _are 0

(Fig.

3).

Actually they

are not,

probably partially

due to the fact that such minima _are very difficult _to detect

experimentally (Fig.

4). In the absence of any defect, _a

periodic

behaviour is observed in the domains

corresponding

to the

bumps

and _non-

periodic

behaviour is observed _near the minima. On the other hand, in the presence of _a defect,

non-periodic

behaviour may ^be observed _up to (A IL about

equal

^{0.5, I-e- when there} are at least two waves on the wire. With _{two waves} and _no defect

separating

the _{two waves,}

only periodic

behaviour has been observed.

As _an

example

of

non-periodic

behaviour, let _us consider the time series and its associated power spectrum in

figure

7 for L/A

=

3.0±0.1

(labelP3

in

Figs.

3 and 4). The power

spectrum ^{has been evaluated}

by averaging

10 spectra ^{built from 8192 data}

points.

This _state has been characterized _as

being

chaotic, in the spectrum, ^{the chaos leads to} a broadband

contribution _at

frequencies

smaller than the fundamental

frequency (Fo).

There, with _waves

propagating

in the _same direction, the observed

simple periodic

behaviour

corresponds

to a case when there _are ~p + 1/2) individual _waves

along

the wire, _p

being

an

integer,

while,

conversely,

_more

complex

behaviour may be

expected

when there is _an

integer

number of

waves

along

the wire. The

physical

reason

why

it is _so remains to be identified.

In order _to discuss the relation between

temporal signals

^and

spatial

patterns more

extensively,

the free surface is filmed

by

means of _a thermal _camera,

allowing

_us to record the time evolution of the temperature field. To match the small

sampling frequency

of the _camera (25

images

_per

second), experiments

must be carried out in a small

frequency regime

such _as

1o2oo

I

9400 (a)

8600

0 50 100 150 200

Time

(s)

o

~

_~~

F° 2Fo

~ (b)

u~ -80

©- -120

0 2

Frequency (Hz)

Fig.

7.- Chaotic behaviour observed for _a minimum of the total power for L/A =3.0± o-I

(d 1.0 _mm and / 260 mA).

for d

=

3.5 _mm, 1

=

260 mA. In this _case for which the state is

periodic,

the temperature ^of a

point

^located on the surface, _on the vertical above the wire, has been recorded

during

_a few

periods ^leading

to the temperature

profile

of _an individual _wave

displayed

in

figure

8. This temperature

profile

_{owns a strong}

similarity

with the surface deformation

profile published

in reference

[6], figure

15. The

profile

in

figure

8 may be viewed _{as an} oscillation made of _two steps :

(I)

_an increase of the temperature ^for x/A

ranging

from _to

0.4,

due to the arrival of heat from the

heating

_source followed

by (it)

_a decrease due _to the

Marangoni disruption

of the

temperature

gradients.

is

~s

~

~ lo

~

$

5

0

0.0 0.2 0.4 0.6 0.8 1.0

x

/ I

Fig.

8.

Spatial

temperature

profile

of the _waves observed when the wire _is far from the surface (d

=

3.5 mm). T is

given

in

arbitrary

unit.

The

question

then arises _to know to which extent the information

gained by measuring

the

average temperature

T(t)

from

U(t)

is

equivalent

to that

gained by measuring

a local

temperature T(>.o,

t). In some cases at least, the _answer is

positive. First,

this is the _case for the trivial situation of the basic stable _state in which both

f

_and

T(xo), Vxo,

^do not

depend

on t.

Second,

this is also the _case for _a

periodic

state in which both

I(t

_and

_{T(xo, t)}

are

periodic

in

time, with, furthermore,

the _same

period.

Next, we consider the _case of _a chaotic

signal

in which the

spatial

pattern ^exhibits one sink where

counter-propagating

_waves encounter and

disappear, according

to the

diagram

: _----. It then appears that the

temporal

chaos of the time

signal

is correlated with _a

spatial

chaotic behaviour of the sink whose

position

is observed to be

chaotically wandering.

In

figure

9 this statement is illustrated

by comparing

several time

signals (I)

the average temperature

T(t) (a) (it)

_a local temperature ⁱⁿ a small domain of the surface in which the sink motion is confined

(b)

and

(iii)

_a local temperature ^{recorded far} away from the sink

(c).

The strong

similarity

between

figures

9a and

9b, showing

six

synchronized

groups of oscillations

(separated by

^vertical

lines), visually

illustrates the correlation between the chaotic average temperature

T(t)

and the chaotic

spatial

^motion of the sink. On the other hand, the

comparison

between

figures

9b and 9c indicates that the

spatial

^{chaotic behaviour}

associated with the sink is indeed

strongly

localized. As _a

whole,

the system ^remains

strongly organized

in space and the information of the sink

wandering

is then transferred _to the average

temperature ⁱⁿ an

easily

_way.

Therefore, in

spite

of the apparent

complexity

of the system ^which is open and involves

propagating

_waves, the

spatial

auto-confinement of defects allows _{us to} characterize states with _a small number of

degrees

of freedom

by using

the tools of nonlinear

dynamics.

4. A _zoo of

temporal

behaviours.

All the _states

analyzed

in this section _are

generated by

the presence of _a

single

sink

point,

I-e-

the

simple

model discussed in section 3 is _no

longer

relevant. The process of

production

of the sink will be discussed in section 5.

Yet,

_even with _a

single

sink

point,

_many different kinds of

behaviour may appear,

simple

or

complex. Typical

transient times needed to reach the

(a)

I I I I

I ^' I

I I

I

~ ^l

I

$

(b)

(c)

0 20 40 60 80

Time

(s)

Fig.

9.

-Comparison

between

averaged

along the wire and local temperatures at the free surface (d ₌ 0.9 and / 280 mA).

described

asymptotic

states _are short, at worst a few seconds. The obtained _{states are} very stable.

They

_may be observed without detectable modifications

during

runs

lasting

several hours.

Reproducibility

is very

good

when the ratio AA/Ad is small in

figure

4, in which AA is the modification of A induced

by

the variation Ad of d. In

particular,

this is the _case for the first _two

bumps (starting

from the

rightj

down _to d

i mm.

4.

QUASIPERIODICITY.

The sink

point

_separates two domains in which _waves propagate ⁱⁿ

opposite

directions. Wave

frequencies

in each domain _are found _to be

slightly

different. Two

frequencies

are therefore present ^{in the} system,

leading

to

quasiperiodicity

if these

frequencies

are incommensurable. The difference in

frequencies

also

implies

that the sink

point, I-e-,

the

point

where the _waves meet, must move.

A wire

voltage

^{time series} is

displayed

in

figure

lo, its power spectrum

(Fig.

17b) exhibits

two incommensurable

frequencies Fj

=

2.osl Hz and

F~

=

2.199 Hz

(AF,

₌ ± o.oo3

Hz).

The attractor is _a 2-torus which may ^be reconstructed in _a

3D-phase

_space

using

the

time-delay

method. The intersection of the reconstructed torus with _a Poincard section is

displayed

in

figure

I.

Frequency incommensurability implies

that all the surface of the torus is visited.

6500

£

~

55i0

4500

o lo 20

Time

(s)

Fig. ^lo.

Quasiperiodic

behaviour of U(t) (d

= 0.62 _mm and /

=

260 mA).

~~~~y#ti,~"~~W #,k~j;-

~~~j~~

/ ~''h£.

fi&

~~~~

~)

..~,

~~

'w~ #

ki /~' ',

'"j _j ^-'

i'j~j.,":'

Fig.

ii- Poincard section of the 2-torus.

Let

P,

be the successive

points

in the Poincard section of

figure

I I and B be the

barycenter

of the _set of

points,

^I ₌ 0,

,

N. The

phase angle

_~b, of

points P,,

I

= I,

.,

N is defined _as the

angle (Po BP, ).

^{From the}

~b,'s,

taken in the range

(0,

2w and then normalized in the range

(o, I),

_we ^obtain the first return map ~b,

~ ⁼

f(~b,

)

displayed

in

figure

12. After

fitting

the data

by

parts,

by using

a least squares method associated with _a SVD

(Singular

Value

1-o

0.8

~~,,'

'$

_0.6

fl ^,,'

,

~

0A

,"'

0.2

~~,,'

0.0

"'

0.0 0.2 0.4 0.6 0.8 1.0

Phase(I)

Fig.

12. -First _return map for the

phase angle

defined in the Poincard section.

Decomposition) algorithm,

the

slope

of the curve may be evaluated _at each

point

~b,. The

Lyapunov

exponent ^{of the} map can be

computed

^afterward

by using [1711

j ^N

A

=

p jj

in

f'(~b,)) (i1)

,-,

This

Lyapunov

exponent, ^{which is} also the greatest

Lyapunov

exponent ^{of the time} series, is then found to be A

=

0.063 _± 0.005,

indicating

a nonchaotic behaviour.

4.2 CHAOTIC _PHASE INTERMITTENCY. The

temporal

evolution is

displayed

in

figure 13,

its

careful examination shows that the groups of oscillations _are of different

lengths.

The spectrum exhibits two

frequencies

F

=

1.024 Hz and

F~

=

1.106 Hz _{near a}

frequency locking

of ratio

Fj/F~=

I. However in contrast with the

previous

case where _two incommensurable

frequencies

led to

quasiperiodicity,

_we are here faced with chaotic

phase intermittency.

The

only

other

experimental

observation of

phase intermittency

we are aware of has been obtained in

Rayleigh-Bdnard

convection

by Bergd

and Dubois

[18].

To illustrate this

phenomenon, figure

14

displays

_a few

points

in _a Poincard section of _a 3D- time

delay

reconstruction. The

points

are numbered in

chronological

order. There is _a

regular

drift from

points

to 10 followed

by

a

slowing

down for

points

to 14,

produced by

the

proximity

of the

frequency locking,

and then

by

a

large change

from

point14

to 15. This behaviour is characteristic of the

phase intermittency phenomenon.

1800

# 1300

5

800

300

0 15 30 45 60

Tbne

(s)

Fig.

^{13. Chaotic} time series of

phase intermittency.

The

longest

_group of oscillations _is in the middle

(d 1.04 _mm and /

=

290 mA ).

7

9

s

11

13 14

Fig.

14. Illustration of the interrnittent variation of the phase angle in the Poincard section.

The first return map ~b,

~ ⁼

f(~b, ),

is shown in

figure 15, exhibiting

two channels

nearly

tangent to the bisector, associated with the

intermittency.

This map is _more

complicated

than in the

previous

case

(Fig. 12)

with the result that the map

Lyapunov

exponent ^{is difficult} to

extract

by using

relation (I I

). Therefore,

the greatest

Lyapunov

exponent ^{has been evaluated}

by studying

the

temporal

^evolution in _a reconstructed

phase

_space,

according

to an

algorithm developed by

Rosenstein _et al.

[19], leading

to A

m 0.20 s~

'.

_This

positive

value indicates _a chaotic behaviour. The

probability

distribution of the duration of _one apparent ^{rotation in the} Poincard section is

displayed

in

figure

16. The presence of chaos is also confirmed

by

the time series power

spectrum (Fig, 17a)

which exhibits _a

noisy background,

to be

compared

with the power spectrum ^{of the}

previously

discussed

quasiperiodic signal

in

figure17b.

4.3 TYPE-I PSEUDO-INTERMITTENCY.

Type-) intermittency

^is

generated by

the loss of

stability

^of a limit

cycle

^when _a real

eigenvalue

of the linearized Poincard map crosses the unit circle _at (+ ^and leads to time

signals

^made of laminar

phases interrupted by

chaotic

bursts

[201.

^This kind of behaviour has been

previously

observed in hot-wire

experiments [5-

1.0

~,

o g

~

_~

j~ ,,'''

o.5

'i

o.6

/ ~~"~'

E

~~

# ^" ,'

ff

fl

,

,'' a

^o.3

~

_0A

,w

f~

_~

~'

~P

,,''

^0.2

0.2 ,

,,''

l''

^~

0.0 ^{' '"} o_0

0.0 0.2 0A 0.6 0.8 1.0 9 II 13 15 17

Phase(I)

Lengdl (s)

Fig, 15. Fig. 16.

Fig,

15. First return map far the phase angle.

Fig-

16. Relative

probability

distribution of the time

required

for _one apparent ^{rotation in the Poincard} section.

0

~

_~~

~'~~~

~~ ~

~~~~ /~

~

(a)

u~ -80

©- -120

0.0 0.5 1.0 1-S 2.0 2.5

~

~

~/

^{"~~' F' ~ ~ F2 2 Fl ~}

^~/

²

CQ ~~

2~ (b)

iii

-100

©-

-150

0 2 3 4 5

Frequency (Hz)

Fig.

17.

Comparison

between _two power spectra : (al chaotic

phase intermittency,

(b)

quasiperiodi- city.

211.

In this section, _we present a case

sharing

_{many common}

points

^with

type-I intermittency

stt.icto _sensu except ^{for the} presence of

hysteresis (discussed

in the _next

section),

hence the

name of

type-I pseudo-interrnittency.

By varying

the distance

d,

_we observed _a bifurcation from _a

periodic

state to the attractor

displayed

in

figure18.

The dark

region

is reminiscent of the old limit

cycle

which lost its

stability.

The first _retum map based _on the successive maxima of the

signal

exhibits _a channel

near the bisector _as in theoretical models

(Fig, 19).

The

probability

distribution P

(L) giving

the

probability

P of

observing

a laminar

phase

of

length

L is shown in

figure

20. The maximal

probability

is for L

m

40 _s. For

larger

L's, the

probability

decreases fast. Moreover, there is _a second

significant peak

of

probability

^{for L} _m ¹⁰ s. Such characteristics _are in fair agreement with theoretical results

[22].

1250 1250

'~

_l150

( ^fl

850

1050 ,'

450 950

450 850 1250 950 1050 l150 1250

u(t) Max(I)

Fig. 18. Fig. 19.

Fig.18.

-Reconstruction of _a 2D-attractor of

type-I pseudo

intermittency (d=0.6mm and / =285mA).

Fig. 19. First return map for the maximum of the time series.

0.2

£~s

if

~$ 0.1

~$

~'~

0 20 40 60 80

Length (s)

Fig. ^20. Relative probability distribution of the

length

of laminar phases.

Figure

21

displays

the power spectrum ^evaluated

by averaging

29 spectra ^{built from} 4 096 data

points.

The

broadening

of the distribution around 40 _s induces _a

spectral

domain _at

small

frequencies

where _we may write PSD _cc F~" The exponent a is

equal

to 2.6 which is

quite

far from theoretical results where _a

m

[231.

^Our

experimental

value cannot be

compared

with other

experimental

values, due _to the lack of

experimental intermittency

spectra in the literature.

0

~

Fo

~i

_{m0 ',}

,

CQ -80

,, j~~

-120

-20 ~

~4°

20

Log10(F) (dB)

Fig.

21.

Average

_power spectrum

density

in

logarithmic

scales.

5. Coexistence of attractors.

Another classical feature of nonlinear systems ^{is the} coexistence of

multiple

attractors, each attractor

being

surrounded

by

its _own basin of attraction. This behaviour _occurs in hot-wire

experiments,

I-e- for _a

given

_parameter vector

(d,

I

),

the state of the system ^is not

necessarily unique.

Examples

are

provided by

two

experiments performed

with _{a constant} current1

=

295 mA.

Each

experiment

starts with the wire located _at d~, ^{far from the surface. Then} d is decreased step

by

_step down to

d~. Finally,

d is increased from

d~

back to

d~.

^{At each} step, ^{the time series}

fit

_{is recorded and the power spectrum is}

_computed.

_{The free surface}

_spatial

_{state is also}

observed, to

possibly emphasize

the relation between

temporal

behaviour T(t) and

spatial

behaviour of the _waves.

5. FIRST EXPERIMENT. Here, d~ ₌ ^3.0 mm and

d~

= 0.56 mm. Three different _cases may be

distinguished.

(I)

For d

ranging

from 3.0 down _to 0.94 _mm, all individual _waves propagate ⁱⁿ

single

direction, _say from the

right

to the left,

according

to : _----.

ii)

For d

decreasing

further down to 0.56 _mm, and also for d

increasing

back from 0.56 up to 1.46 _mm, there is _a sink

point

on the left part ^{of the}

wire, according

to the

diagram

(iii)

irection, but now rom

the

left to ^the right : ----.

All _{signals are} periodic, except near the _minima

of

which

not

given).

Therefore,

it

_{is observed} that the resenceof a _sink does _notnecessarily

a

omplicated behaviour.

In

measured on both sides of

the sink were

identical

and

the sink

location _{was constant.}

5.2 SECOND EXPERIMENT.

Here,

_d~

=

2.5 _mm and

d~

=

0.76 _mm. The square root of the total power (A i>eisiis d is

displayed

in

figure

?2.

Again,

three domains _can be

distinguished.

1-o

0.8

1

0.6

/

~f _i1

0A

' '

0.2

(1

1

0.0

0.5 1.0 1-S 2.0 2.5

d

(mm)

Fig. 22.- Evolution of A

,~

_{with d}

_showing

_{three different domains for the second}

experiment.

(I) For d

ranging

from 2.5 down _to 0.84 _mm

(dashed line),

the _waves propagate ^{from the left}

to the

right (----)

and

signals

are

periodic.

(ii)

For d

decreasing

further down to 0.76 _mm and then

increasing

back up to 1.24 _mm

(white circles),

there is _a sink in the

right

_part of the wire

(----)

and

signals

are not

periodic,

_as

presently

discussed.

(iii) Finally,

for d

increasing again

_up to 2.5 _mm

(black circles),

_{waves propagate}

again

from the left to the

right (----)

and

signals

_are

periodic.

The appearance and

disappearance

^{of the sink therefore exhibit} a strong

hysteresis.

The

states of the system ⁱⁿ

phases (it),

I.e, in the presence of the sink, _are rather

complex

as

illustrated in

figure

23 where attractors are reconstructed for d

=

0.84

(a),

0.88

(b),

0.92

(c),

0.96

(d),

1.00

(e)

and I.16 _mm

(fl.

Poincard sections for each _{attractor are} shown in

figure

24.

The evolution of the _attractors from

(a)

to

(fl,

and

particularly

^from

(a)

to

(dj,

shows _a

gradual simplification

of the structure

corresponding

to the

unfolding

of the

top-right

^dark zone and to the fact that the central part ^{of the} attractor becomes less and less visited.

(a) ''.~ _(b)

..

(c)

..

_. =t

'

~ l

J

r£ . _

"

.

'

,

~z

>"

(d)

' ~ ,'

~

..-, '

~

~,

- ^- '

'

,_ -

, ."

," "

Fig. 23. Reconstructed attractors for different values of d.

~

*

~'~

*~

_~~~~

___~

~~4

^~"

^~»

^+,

.+

~

.

f ($~ ^'~

f

i~

~

+

,,

.

'

^r

+

I

f

^~

+

,

+

,_~

~~

', _~~

^'

(~)

~'~~t

^{~ *}

(b) ^"*~$~

~+

(~) b.,

~/~

~

y~

I

j _{~* ~.**'~} f

,~

', ".: _i~'*~,~ (

~

i

~".

_ ^.

i; ',.~ ~'[~~

.~ ^'

~

~)r..~ ~i<.. 'j _~;

~

~

~ ,

)

(d) (e) _"**w~i

^*

(f~

~

Fig.

24. Poincard sections of the attractors

displayed

in

figure

23.

Poincard sections show that the attractors _are

essentially

two-tori with _a

probability

of presence

evolving

to a more discontinuous

repartition

from

(a)

to

(fl.

In

particular,

in

(fl,

all data

points

_are contained in

eight

disconnected boxes.

In

la),

we are faced with

phase intermittency,

the _arrow

pointing

to a dark _zone where the

phase

evolves

slowly.

The

disappearance

of

phase intermittency

_may then be followed up ^to

(fl

where the presence of

eight

distinct domains

corresponds

to a

frequency locking

_near the

rational 7/8

= 0.875.

Indeed,

the power spectrum ^{then shows} two basic

frequencies F,

and

F~

in the ratio

F,/F~

=

0.88.

One also has to remark that the appearance and

disappearance

of the sink _occur when the total power is close _{to a} minimum. On both sides of the

minimum,

the number of individual

waves is

changed by

a value of I.

Both

experiments

indeed show the presence of

hysteresis

connected with the coexistence of attractors. When the wire is far

enough

from the surface

(d

_m 2 _mm

),

_we

always

found _a

single periodic

state. On the other

hand,

when the wire is closer _to the

surface, multiple

attractors may be present. ^For

example,

for d

=

1.0 _{mm, we} may observe

(ij

_a

periodic

state with all the

waves

propagating

in the _same direction

(it)

_a

periodic

state with _a sink

point

_or

(iii)

_a

quasiperiodic

state with _a sink

point.

6. Conclusion.

New

experimental

results _on instabilities

produced

when hot-wire

heating

_a fluid below the

free surface have been

presented.

^In

spite

of the interaction of many

phenomena (heat

conduction and convection,

Marangoni

effect, _wave

propagationj,

^the system can be reduced to _a

dynamical

system with a small number of

degrees

of freedom. Some

emphasis

has been

laid _on the relations between

temporal averaged temperature signals

^and wave

spatial

_patterns

along

the wire. In the present case where

only

one defect _{at most} is present ^{in the train of}

waves, it has been demonstrated that there is _a strong

relationship

between

temporal

and

spatial

behaviour.

Many

different kinds of behaviour may be observed and

analyzed

in _a

precise

_way, thanks _to the very

good Signal/Noise

Ratio achieved,

including quasiperiodicity

or

type-I

^and

type-]]

intermittencies. A

simple

model enables _us to

predict

in which situations

complex

behaviour may be

expected. Interesting

future

investigations

would include the

study

of hot-wire forced

experiments

in which the

supplying

current I would be modulated.

Experiments along

_{a very}

long

hot-wire allow the

experimental study

of

spatio-temporal

chaos.

Also, the unsolved issue to understand the _{exact nature} of the _waves is warranted _to be of utmost interest.

References

Ill Gouesbet G. and Lefort E.,

Dynamical

states and bifurcations of _a thermal lens using

spectral analysis.

Phys. Rein. A 37 (1988) 4903-4915.

[2] Gouesbet G., Pashinin P. P., Rastopov S. F. and

Sukhodolsky

A. T.~

Oscillatory instability

of liquids drops-lens by continuous laser heating, Sov. Phy~. Lebedei> hi.~titute Rap. (USA II

(1991) 30-2 Translation of Sb. Kiatk. Sorb.~h(.h. Fir. AN SSSR. Fiz. Inst. P-N- Lebedeva, Russia II (1991) 33-5.

[31 Meunier-Guttin-Cluzel S., Maheu B. and Gouesbet G.. Combined

approaches

and characterizations of experimental chaotic attractors in thermal lensing, Physica D 58 (1992) 423~440.

[41 Enokida Y. and Suzuki A., _«

Application

of laser-induced thermal lens oscillation _to concentration control in solvent _extraction processes ».

Proceeding<

of the International

Symposium

Instabilities in

multiphase

flows, Rouen~ May I lth-14th, 1992. G. Gouesber, A. Berlemont, Eds. (Plenum

Publishing

Corporation~ 1993).