Study of unsteady convection through simultaneous velocity and interferometric measurements

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Study of unsteady convection through simultaneous

velocity and interferometric measurements

P. Bergé, M. Dubois

To cite this version:

P. Bergé, M. Dubois. Study of unsteady convection through simultaneous velocity and

interfer-ometric measurements.

Journal de Physique Lettres, Edp sciences, 1979, 40 (19), pp.505-509.

�10.1051/jphyslet:019790040019050500�. �jpa-00231675�

Study

of

_unsteady

convection

_through

simultaneous

_velocity

and

interferometric

measurements

P.

_Bergé

and M. Dubois

Service de _Physique du Solide et de Résonance magnétique, DPh-G/PSRM, CEN, Saclay, B.P. n° _{2, 91190 Gif-sur-Yvette,} France

(Reçu le 2 _juillet 1979, accepté le 20 aout 1979)

Résumé. 2014 Nous

présentons un _système

_optique

très

_simple,

qui permet de mesurer simultanément la vitesse au

sein d’un fluide en convection et la carte des

_isogradients

de température. La simultanéité de ces deux mesures

permet une _approche

_beaucoup

_{plus complète}

des

_phénomènes

_convectifs,

_{particulièrement}

de ceux

_dépendants

du _temps. Nous avons

_appliqué

cette méthode à l’instabilité de

_{Rayleigh-Bénard}

et avons _pu mettre ainsi en

évidence le mécanisme

_{physique, responsable}

du comportement oscillant, observé dans une

_petite

boite. Abstract. 2014 A

very

simple optical

device is

proposed,

which enables the

_velocity

in a convective fluid, and the map of the temperature

isogradients

to be

_{simultaneously}

measured. The

_simultaneity

of these two measurements

is an _{important step} in the

_{understanding}

of _{unsteady phenomena.} This fact is illustrated _by the study of the time

oscillatory

behaviour of the motion observed in a fluid confined in a small _box, and submitted to the

Rayleigh-Bénard

_instability.

LE

JOURNAL DE

_{PHYSIQUE - LETTRES}

Classification

Physics Abstracts

07.60L - 44.90 -

47.25Q - 47.80

The usefulness of local

_{optical velocity}

measure-ments for the

_{understanding}

of

_{Rayleigh-Benard}

convection has been

_widely

demonstrated

_[1,

2,

3].

Nevertheless,

up to now, this

powerful technique

has

only provided

local information on

velocity

behaviour at a

_given

time. In contrast with this

_technique,

a full _map of the

_temperature

_{perturbations}

can be

obtained

_through

interferometric measurements

_[4],

and, then,

usefully

combined with

_velocity

measure-ments. An alternative differential interferometric

method,

using

Wollaston

_prisms,

has been

pro-posed [5,

6]

and

_widely

used in the case of

Rayleigh-Benard convection

_[7,

_8].

In this

letter,

we _present a much

_{simpler optical}

_system

which

_provides

dif-ferential

_{interferogrammes ;}

moreover, the simul-taneous use of this interferometric

technique

with

laser

_{Doppler velocimetry}

on the same

_sample,

is a very

powerful

method;

this

point

is illustrated in

the

_study

of

_unsteady

convective

motion,

from which

new information on the mechanism of the

oscillatory

regime

in

_high

Prandtl number fluids is obtained.

Background.

-

Rayleigh-Benard

convection is the

problem

of a

_simple,

dilatable

fluid,

confined between

two

_{highly conducting rigid}

horizontal

_plates

and submitted to a thermal

gradient

åT/d

where AT is

the

_temperature

difference

_applied

between the

_plates

and d is the

_depth

of the fluid

_layer.

If this

_gradient,

parallel

to the

_gravity

_axis,

is

_higher

than a certain

threshold,

regular

convective motions set in. A

_good

stability

parameter is the

_Rayleigh

number Ra

where g

is the

_{gravitational}

acceleration and a, v

and

_DT

are

respectively

the volume

expansion

coef-ficient,

the kinematic

_viscosity

and the thermal

diffusivity

of the fluid.

If Ra is lower than a critical value

_Rac,

the fluid

remains at rest. On the

_contrary,

if Ra >

_Rac,

the

system is unstable and convective

_velocity

is

_present

and

_{spatially organized}

in rolls. It was

_recently

found

_[3,

9,

10]

that,

by increasing

the

_Rayleigh

number,

the

_following

states _appear

_{successively :}

1)

Steady

two dimensional rolls at

_Ra~.

2)

Steady

three dimensional rolls at Rall.

3) Monoperiodic

time oscillation of the

_velocity

at

Raosc.

4) Biperiodic

oscillations of the

_velocity

at

_Rabip.

5)

First _appearance of chaotic behaviour of the

velocity

i.e. turbulence at

_RaT.

In

_fact,

the existence and the threshold of some of these

_steps

in this cascade towards turbulence

_depend

on the

_geometry

of the box

_confining

the

fluid,

and also on the actual structure

_{[10, 16].}

In a

small

L-506 JOURNAL DE PHYSIQUE - LETTRES

box,

with an

_aspect

ratio

(ratio

between the horizontal

extension to the

_depth

of the

_layer)

F =

2,

all the sequences described above may be found in

high

Prandtl number fluid

₍₁₎

_[10]

_(Pr

=

v/DT).

The

experiments reported

here use this

geometry.

Experimental

conditions. - The convective fluid

is silicone oil with

_{viscosity 11}

= 0.11 stoke at 20

~~,

DT

= 7.64 x 10-4

cm2 s - 1

and P~ ~ 130. It is con-tained in a

_rectangular

_plexiglass

frame

_{(fig. 1);}

the dimensions of the fluid

_layer

are d = 1.25

cm,

Lx =

2.50 cm,

_Ly

= 1.50 _cm. The

_top

and the bottom of the cell consist of massive copper

plates

thermally

regulated

to within

10’~C.

In the

_experiment

described

here,

we have in fact two identical

_cells,

filled with the same oil and inserted between the same _copper

_plates,

so that their axes Jazz are

per-pendiular ;

in this way we were able to observe the

dif-ferential

_{interferogrammes along}

the two

_principal

axes X’ X and Y’ Y of the convective motion.

The laser

_Doppler

velocimeter is of the conventional

type

and has

_already

been described

_[13].

The

instan-taneous

_Doppler

_{frequency fd}

is

_directly

measured with an automatic _{counter ;} its time variations can

be Fourier

_analysed

with an FFT

_{analyser working}

(1) And also in low Prandtl _{convection, except} for the second step [11, 12].

Fig. 1. - Scheme of the

experimental cell used : d = 1.25 _cm,

L. = 2.5 cm and _L, = 1.5 cm. At the _bottom, scheme of the

velo-city field at the _height Z = _{0.2 d The curved horizontal full lines}

represent the lines _{where Vz} = _0.

at a _{very low}

_frequency.

In the _{geometry used} in these

experiments,

the

_{frequency fd corresponds}

to the

vertical

_velocity

_component

_P~

measured at the

mid-height

of the cell

_(Z

=

d/2).

The differential interferometer is _very

_{simple (fig.}

_2).

!

Fig. 2. - Scheme of the

optical set up I and II are the two narrow beams of the _{Doppler velocity} meter. P is the point where the _velocity is measured. PMT is the photomultiplier which receives the _light scattered from P. G is the _{glass plate} which recombines _pairs of _rays such as

A 5 mW He-Ne laser beam is

_expanded,

crosses the

convective fluid and is then reflected

_by

both faces of a

glass plate, plane

and

parallel

to ~. At each

_point

in the field after the

_{glass plate,}

two _{rays 1} and 2

interfere after

_having

crossed the fluid under

_study

along

two

_{nearby paths separated by}

8e. The inter-ference order is

_proportional

to the difference between

the mean refractive

_indexes,

and therefore to the difference between the mean

_temperatures

of the two

_paths.

_By

_imaging,

we then obtain the map of

the absolute value of the

temperature

derivative in the 8e direction. In

_practice,

we choose 8e to be

parallel

to X’ X or Z’ Z. be can

_easily

be

_adjusted

through

the width e of the

_glass

_plate

_{(here, e}

= 2

mm)

and the

_angle

9.

Furthermore,

the presence of the

glass plate,

near the exit of the

cell,

does not

_perturb

in _{any way} the laser

_Doppler

_velocimeter,

which works

_{simultaneously}

at the same

_place.

The

simpli-city

of this

_set-up

leads to a _very

appealing

cost-efficiency

ratio.

Convective structure. - Some

_examples

of the results are shown in

_figure

3.

_They

show the

_velocity

field

and the

_isogradient

lines of the convective motion established in the

_previously

described cell for

Ra

143

_{(2) .}

The motion is

_stationary

and the

Ra

=

( )

convective structure can be described

_by

the

super-position

of two rolls

_along

the X’ X direction and two

rolls

_along

the Y’ Y direction

_[the

_dependence

Vz

=

f (Y)

_at X =

Lx/2

is

quite

similar _to that of

Vz

=

f (X )

shown in the

figure],

perturbed by

boun-dary

effects. The

_isogradient

_lines,

_projected

on the

[Z, X] plane,

confirm

_{qualitatively}

the

_velocimetry

results,

although

the interference order is related to

the

_{integration along}

the Y’ Y

direction,

and it is

difficult,

in the case of three dimensional

motions,

to calculate the local

temperature

perturbation.

However,

the hot

_region

at the bottom of the

_layer,

which is characteristic of an

ascending

motion

_[14],

can be

noticed,

particularly

in the horizontal

iso-gradients picture. [The

same kind of

_picture

is observed

along

the Y’ Y

_direction.]

Oscillatory

regime.

- The main interest of this

temperature

visualisation is to allow the

instan-taneous determination of the whole structure in

unsteady

motion,

while

velocimetry

gives

more

quanti-tative,

but local information.

For a

_given

_fluid,

the threshold of the time

_dependent

phenomena

depends

on the dimensions of the

layer

and in a small box

(r

=

2),

time

independent

convec-tion _may be observed _up to more than _many hundred times the critical onset, with a fluid

having

a Prandtl

number of

130;

this threshold also

_{strongly depends}

on the actual convective structure.

Fig. 3. -

Dependence of the _{V. component} with X measured at

the _mid-height of the cell (Z = d/2) and _{at Y} =

Ly/2.

Ra/Rac = 143.

In I and II, corresponding isogradient lines : I - horizontal

iso-gradient lines with _(8e)., = 1.53 mm. II - vertical isogradient

lines with _(be)z = 0.65 mm.

The

_simplest

_unsteady

_state, which _may be

_obtained,

corresponds

to the

_periodic

time

_dependence

of the

velocity ; although

this has been observed in _many different

_systems

_[9,

_11,

_{15, 16,}

_17],

_up to now, no

definitive mechanism has been

_brought

to

_light.

So let us

analyse

the

_physical

data obtained in this

oscillatory regime

through

simultaneous

_velocity

and interferometric measurements, with

_Ra/Rac

= 280.

The time

_dependence

of the

_velocity

_component

_Vz,

measured at z =

d/2,

in the maximum

ascending

L-508 JOURNAL DE _{PHYSIQUE -} LETTRES

Fig. 4. -

a) Time dependence of the _{Vz component} measured at

the maximum _{ascending flow ;} Z = _d/2 and f ~

L,/2.

Ra/ Rae = _280.

b) Photographs of the most _{representative} vertical _isogradient

lines

_[(be,)

= 0.65

mm] taken _{simultaneously} with the velocity

measurements at A, B... and so on. _Practically a _{complete cycle}

is _{representated} from A to A’. _A’, B’ and C’ _correspond to the

same phase as _A, B and C. The _respective times _{(tA being} taken as

the _origin) of the _{images are : tB} = 6 s ; tc = ₁₄

s ; tD = _{20 s ;}

the = 23 s ; tF = 26 s ; tA, = _{30 s ;}

tB, = _{36 s;}

tc’ = 44 s. _Here, the hot _{droplet originates} in a _right lower comer (see A, B) and is advected at C ; D, E and F show 3 _steps in the advection _by the roll.

can be maintained for several weeks and is

charac-terized

_by

a

velocity

_{power spectrum} with a _very

narrow

_{frequency peak (here,}

_fo

= 33 x

10-3

S-l)

together

with its harmonics

_nfo.

_{Simultaneously,}

the

images

of the

_isogradient

lines are filmed

(16 mm

camera)

at the rate of 1

_image

_per second.

Synchro-nisation marks

_give

the

_{correspondance}

between each

image

and the measured

_V

_amplitude.

In

_figure

4b,

we _present some selected

images

corresponding

to

the more

_{representative}

_steps

of the

oscillatory cycle

(projected

on an XZ

_plane).

Observations and

_proposed

mechanism. - The observed

_results,

as shown in

_figure

_4, can be

sum-marized as follows : there are

_{schematically}

two

main

_phases

in a

complete cycle.

The first one

cor-responds

to the

_growing

of a hot

_droplet.

The second

stage

corresponds

to the advection of this

_droplet,

together

with the increase in

_{velocity amplitude.}

During

its

_advection,

the

_droplet

relaxes

_thermally,

and then

_disappears

with a reduction in the

velocity

field

_amplitude.

In order to

_gain

a

_{good understanding}

of the

observed

behaviour,

it is _necessary to examine the known

_velocity

_field,

_together

with the

_isogradient

images.

The hot

_droplet,

which _grows on the bottom

plate,

and in a comer of the small

_box,

is

_initially,

in a

_region

of _very low

_{velocity (see}

_{fig. 1),}

where

the heat flux is not _yet well advected.

_During

its

growth,

the

_droplet

has no

_perceptible

horizontal

motion;

after a

_while,

it reaches its final extension b

(here,

b ~

_{2 mm).}

At the end of its

_growth,

the

_droplet

becomes unstable and reaches a

_region

where the horizontal

velocity

is

_high

_(Vx

is maximum at Z = 0.2

d) [1].

So,

it is

_completely

advected. The

hot,

less

dense,

droplet obviously

acts as an additional force in the

convective

_ascending

motion and the

_velocity

field - and thus the vertical mass flux - is

highly

enhanced;

moreover, the passage of the

droplet

in

the

_ascending

motion

_gives

rise to a deformation

in the convective structure. This last

_point

can

_easily

be seen on the « F »

_image

and is checked

_{by velocity}

measurements at different

_points

in the structure.

During

its

_advection,

the

_droplet

relaxes

thermally

and a new

_cycle

follows.

This mechanism is

_{principally governed by}

a

local thermal

_singularity,

itself controlled

_by

the

thermal

_diffusivity

of the

_fluid;

note that there is

good

agreement

between the

_growth

time of the

_droplet

(~ 15 s.),

the half of the

_{oscillating period,}

and the

thermal

_diffusivity

time

_b2/4

_DT.

To some extent,

this mechanism can be

compared

to that

_proposed

by

Howards

_[18],

which is related to an

_instability

in the

_sublayer

and leads to the law

_[19]

also

_obeyed

in small boxes. In each

_periodic

situation,

observed here at

_{RalRa,, -}

_280,

we

_always

observed

only

one

droplet,

in

_spite

of the presence of four

(equivalent

at first

_sight)

comers.

Nevertheless,

in

different

_{experiments performed}

with the same small

When an

_ascending

motion is _present in the centre

of the cell

_(Fig.

_1),

a hot

_droplet

_may be formed in

any bottom comer ; when a

_descending

motion is

present

in the centre of the cell

_(opposite

situation

to

that

in

_{figure 1),}

a cold

_droplet

_may be formed in any

top

comer. The increased

_{degeneracy originates}

from a

_{slight dissymmetry}

in the structure, the unstable corner

_being

that of the

_largest

roll

_(the

actual lowest

_wavenumber).

In contrast, we notice

that in a

_{perfectly symmetrical}

structure

_(rolls

of the

same

size),

we did not observe either

_droplets,

or

oscillations,

until Ra N 420

_Rac.

Furthermore,

in

_large

_boxes,

we observed the same

law to be

_obeyed

₍₁₎

_{’Losc =}

_{f (Ra),}

as in small ones.

In

_large

boxes,

the _presence of structure defects

-

(local

distorsions of the

_{wavelength leading}

to

large regions

where Y is _very small

_[20])

can

probably

play locally

the same role as the boundaries of the

small box considered in this paper; so we believe

the mechanism to be the same as in small boxes.

In

_conclusion,

the

_simultaneity

of

_velocity

and

interferometric measurements

_(performed

in a

_simple

manner)

has allowed the

_physical

mechanism

res-ponsible

for

_periodic

behaviour in convective

_high

Prandtl number fluid to be

_understood;

from our

observations,

it now seems clear that this behaviour

is

_{governed by}

the

_growth

of a local

_temperature

heterogeneity.

Acknowledgments.

- We wish

to thank G. Zalczer and Y. Pomeau for

_helpful

discussions and B. Ozenda and M. Labouise for their technical assistance

contributions.

References

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