Surface deflection in Rayleigh-Benard convection

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Surface deflection in Rayleigh-Benard convection

Javier Jiménez-Fernandez, Javier Garcia-Sanz

To cite this version:

Javier Jiménez-Fernandez, Javier Garcia-Sanz. Surface deflection in Rayleigh-Benard convection.

Journal de Physique, 1989, 50 (5), pp.521-527. �10.1051/jphys:01989005005052100�. �jpa-00210934�

Short Communication

Surface deflection in Rayleigh-Benard ^convection

Javier Jiménez-Fernandez(1) and Javier Garcia-Sanz(2)

(1)Dpto. ^{de Física E.T.S.} Arquitectura Av. Juan de Herrera s/n, ²⁸⁰⁴⁰ Madrid, Spain

(2)Dpto. ^{de Física} Fundamental, Universidad Nacional de Educación a Distancia, Apdo.

60141, ²⁸⁰⁸⁰ Madrid, Spain

(Reçu ^{le 19} juillet 1988, révisé le 22 décembre 1988, accepté ^le 4 janvier 1989)

Résumé.2014 Nous avons analysé ^la déformation d’une surface supérieure ^{libre dans une}

couche fluide chauffée uniformément _par dessous. Nous avons obtenu une expression ^ana- lytique pour l’amplitude ^{de la} déformation en fonction des paramètres caractéristiques

du fluide et de l’épaisseur ^{de la} couche, lorsque ^{le mouvement convectif est} provoqué

par la force d’Archimède, c’est-à-dire, ^{dans le cas} des couches profondes.

Abstract.2014 The problem of the deformation of a free _upper surface in a convective layer subjected ^{to an adverse} temperature gradient ^is analyzed. ^An analytical expression ^for

the amplitude of the surface deflection in terms of the fluid layer parameters ^{has been} obtained when buoyancy force is the responsible mechanism for instability (thick layers).

Classification

Physics ^Abstracts 47-25Q

1. Introduction.

Convective motions in a fluid layer open to the atmosphere produce ^{a deforma-}

tion of the _upper free surface. In the past ^interest has been focused in the study ^of

the influence of surface deformation on the critical conditions of instability ^{in both :} buoyancy ^and thermocapillary convection [1-4]. ^{More recently, some} experimental ^re-

sults on the pattern selection for different geometries ^{and the} stability ^{of rolls versus} hexagons ^{have been} published [5-7]. ^{In addition to stability} considerations an impor-

tant question arises in convective surface problems : ^the shape ^{and the} amplitude

of the surface deflection. In fact, ^the shape of the deformed surface provides ^{an ob-}

servational criterion to indentify ^the physical ^mechanism responsible for convective motion : density ^{or surface tension} gradients. Experimental ^results [8] confirm this fact and provide ^{also the} dependence of the surface profile ^{on the} layer depth ^{and the}

distance to the convective threshold for thin layers. Some numerical results have been

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005005052100

522

obtained [9] ^{from a linear} analysis ^{in order to find the} regions where the relief is con- cave or convex and the influence of the fluid parameters ^on the deflection. However,

the results are only qualitative because the amplitude ^{can not} be determined from a

linear analysis.

In this note, ^the problem ^{is revisited} following closely the method developed by Joseph [10] for this and other free surface problems [11-12]. ^An analytical expression

for the amplitude of the surface deflection in terms of the fluid layer parameters ^has been obtained. We consider the convective motion in a cell bounded by symmetry surfaces and boundary conditions which leads to simple boundary-value problems.

More complex boundary conditions including ^the dependence of surface tension on the temperature (thermocapillary convection) will be consider in a forthcoming paper.

2. Analysis ^{and results.}

Consider an infinite horizontal layer heated from below and a cartesian coordinate system (x,y,z) ^{with its z-axis} pointing upwards. ^The upper free surface is defined by

a function : z ⁼ h(x, y ; ,) ^{where e is a} perturbation parameter : the Nusselt number

discrepancy : e2= ^{Nu - 1. The bottom surface at z = 0 is} horizontal and is held at a

fixed temperature To + AT. We assume that the horizontal boundaries are surfaces of symmetry ^between adjacent convective cells (rolls ^or hexagons) insulating for heat and

mass transfer. Thus, ^{the fluid} layer ^{is confined in} domains of horizontal cross-section E so that the mean height ^{of a cell is} given by :

In the rest state E ⁼ 0, ^h = h ; and the solution of the Boussinesq equations ^{for the} temperature ^{is :}

where To ^{is the} temperature ^{at z = h.} Following ^reference [12] ^{we shall as-}

sume that this condition for the temperature ^{holds at} any point of the de- formed surface and introduce the invertible mapping between the perturbed ^do- mains Ve; = ((x, y, z) , (x, y) E E, 0 z h) and the domain of the rest state :

which is analytic ^{in the} parameter E ^{and maps} boundary points ⁱⁿ boundary points.

We introduce the functions :

where 8(x, y,z ; e), H(x, y,z ; -) ^and b(x, y ; e) ^{are the} perturbations ^{from the rest state}

solution of the temperature, pressure and deflection respectively, p is the reference

density, ^aT the volumetric expansion coefficients ^the conductivity, and g ^{the ac-}

celeration due to gravity, ^{and take as} normalizing condition the value of the Nusselt

number at z ⁼ 0 :

...::.

where the overbar denote horinzontal averaging. Then, ^{if the} perturbation quantities 0,H, 6, ^the velocity ^u and the function A are expanded ^{as power series in the small} parameter E ^the mapping (3) ^{leads at} successive orders to boundary ^value problems

for the partial derivatives of the perturbations ^with respect ^{to e in} the undeformed domain Vo. Then, the solution into V£ may be obtained by inversion of the mapping.

(For detailes, ^{see Ref.} [12]).

At first order the boundary ^value problem ^{is :}

in Vo, ^{with the} boundary conditions :

where _nr is the outward normal to the side boundary E, V2=ex :x ^{ax +} ey â ^y ay ^{is the}

porjected gradient ^{is the} viscosity, a the surface tension coefficient and li§1 ^the

stress tensor components. At the bottom surface at z ⁼ 0 : w(l) ⁼ 4>(1) ^{= 0. In}

addition we have for a rigid ^{surface : u = v = 0 and for a free surface :} S(1) ⁼ g (1) ^{= 0.}

Except for the normal stress condition (8b), ^{the above boundary value} problem

is the standard Rayleigh-Benard problem [13]. ^Note that the condition assumed for the temperature ^at the free surface provides ^a considerable simplification. In virtue of

JOURNAL DE PHYSIQUE T. 50, N- 5, ^{MARS 1989} 28

524

condition (9) ^{a solution} may be obtained on the basis of a separation of the variables.

Therefore we write :

where f(X, Y) ^{is a} "plan form" defined as a solution of :

and a is a real number. System (5-7) ^with (8a) ^{and one of the two} possible boundary

conditions at Z ⁼ 0 lead to the well known Rayleigh-Benard problem ^for W(Z). ^The eigenfunction fV(Z) belongs ^{to a} given eigenvalue is determined to whithin a constant

A, ^the amplitude ^{of the} z-component ^{of the} velocity. Then, the normal stress condition

(8b) gives ^the following expression ^for b(l) (X, Y)

where D ⁼ d/dz ^{and Wo is the} eigenfunction ^{for A = 1. The} amplitude ^{A is obtained}

from the normalizing ^condition (4) ^{which at} lower order is :

the horinzontal averaging of the heat equation ^at second order gives :

(note ^{that w(l) = 0} Vl) ^{and after some} calculations the above expression may be reduced to :

-- -

1

and by integration ^of (14) ^{it arises}

where (.) denote the vertical mean value. On the other hand, the second-order deriva- tive 0(2) (h) may be related to first order quantities ^{from the} boundary condition : 0

= 0 at Z = h :

Finally, inserting (11), (16) ^into (15) :

where 03A60 may be determined from (7) ^{in terms of Wo}

J2 depends ^{on the cell} geometry ; ^we have obtained the value 0.5 for rolls and

5 1- 3~3) ^{for hexagons using the} Christopherson’s ^solution [13].

18- ( 8-r )

The simplest ^case corresponds ^{to a} free surface at Z ⁼ 0 and a two dimensional motion for which : W (1) ⁼ A sin xi ^{Z cos} aX, f3 = h ^and

is the characteristic equation. Insertion of this solution for W 1> in (11) and (17) ^leads

to the following expressions for the deflection and velocity amplitudes

The minimum critical value of A 0>2-4 ^{(the Rayleigh number) is} obtained when

ah= 03C0 and then :

From (22) ^the following upper bound for the amplitude ^{A is obtained :}

which _may be used as an approximative value of A. In fact, ^this expresion ^show simply ^{that the} amplitude ^{of the} velocity ^is proportional to : h where T = h2 is the

T K

characteristic ^time of heat diffusion.

526

On the other hand, ^{we note} that A decreases as 1/h and therefore K decreases as

1/h2. Figure ¹ illustrate this variation of the amplitude K ^versus h, ^{for a} silicone oil : p ⁼ 0.968, v =n= ^{1.00, K = 0.001095, u = 11.36 in C.G.S. units). In a previous work}

[9] ^the qualitative behavior of K with the depth h ^{has been} analyzed. ^{It was shown}

that K increases with h for thin layers, ^{reaches a} maximum and then decreases. A dual behavior which _may be explained ^{from a} physical point ^{of view} by ^the competition

between surface tension and buoyancy ^{forces. In our} analysis ^we have assumed that convection is driven only by buoyancy forces which is the dominant mechanism in thick layers, ^{and we obtain in} accord with [9] that the deflection must be a decreasing

function of the depth h.

Fig. ^{1.- The} amplitudes ^{of the} deflection h’ and the z-component ^{of the} velocity A ^{as functions}

of the mean layer depth h, ^{for a silicone oil.}

Expression (22) ^and (23) ^{provide the} quantitative dependence of surface deflection

on h and the fluid parameters : ~, P, 03C3, K. A result which _may be tested by straighfor-

ward experimental ^{work. The} analysis given here will be enlarged ^{in order to include} thermocapillary convection in a forthcoming ^paper.

Acknowledgements.

J. Jiménez-Fernandez wishes to acknowledge ^the support ^{of the} "Direcci6n Gen-

eral de Investigaciôn Cientifica _y Técnica (DGICYT)" ^{of the} Spanish Government.

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