Rayleigh–Marangoni horizontal convection of low Prandtl number fluidsE. Bucchignani and D. Mansutti Citation: Physics of Fluids 16, 3269 (2004); doi: 10.1063/1.1772381 View online: http://dx.doi.org/10.1063/1.1772381 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/16/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermal-solutal capillary-buoyancy flow of a low Prandtl number binary mixture with a −1 capillary ratio in anannular pool Phys. Fluids 28, 084102 (2016); 10.1063/1.4959211 Assessment of the role of axial vorticity in the formation of particle accumulation structures in supercriticalMarangoni and hybrid thermocapillary-rotation-driven flows Phys. Fluids 25, 012101 (2013); 10.1063/1.4769754 Marangoni convection and heat transfer in thin liquid films on heated walls with topography: Experiments andnumerical study Phys. Fluids 17, 062106 (2005); 10.1063/1.1936933 Transient Marangoni convection in hanging evaporating drops Phys. Fluids 16, 3738 (2004); 10.1063/1.1772380 Transitional regimes of low-Prandtl thermal convection in a cylindrical cell Phys. Fluids 9, 1287 (1997); 10.1063/1.869244

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Rayleigh–Marangoni horizontal convection of low Prandtl number fluidsE. BucchignaniCentro Italiano Ricerche Aerospaziali, Via Maiorise, 81043 Capua CE, Italy

D. Mansuttia)

Istituto per le Applicazioni del Calcolo (IAC/CNR), Viale del Policlinico, 137, 00161 Roma, Italy

(Received 10 September 2003; accepted 21 May 2004; published online 22 July 2004)

In this paper we describe the results of the numerical study of the Boussinesq–Navier–Stokesequations for the convection flow in an open top three-dimensional parallelepipedic cavitys43131d driven by a horizontal temperature difference at two opposite vertical walls in presence ofthermocapillary effects. Two fluids have been considered with Prandtl number Pr=0.015 and Pr=1, at fixed Rayleigh number, Ra=150; simulations for several values of the Marangoni number(Ma) have been developed in order to detect fully three-dimensional flow configurations. From theanalysis of the numerical flows we reach the following conclusions: the increase of the Prandtlnumber enhances the coupling of the heat and mass transfer phenomena, moreover while at Pr=0.015 a fully three-dimensional behavior is observed at Maù200, at Pr=1 the flow configurationappears clearly three dimensional just at Ma=30; the increase of the Marangoni number induces theappearance of secondary counter-rotating vortices around to the primary one. The values of theNusselt number at the main stream perpendicular middle plane are provided. ©2004 AmericanInstitute of Physics. [DOI: 10.1063/1.1772381]

I. INTRODUCTION

By horizontal convection flow it is meant the convectivemotion of a fluid induced by a temperature difference along ahorizontal direction due to buoyancy effects. The case of ashallow horizontal convective cell is adopted in theliterature1 as a model for the Hadley circulation cell, one ofthe persistent flow structures established within the earth at-mosphere near the equator and due to horizontal temperaturedifference. Actually Hadley-type circulations are found inseveral environmental situations such as within geophysicaland meteorological fluid flows driven by temperature differ-ences.

Horizontal convection is relevant also to several indus-trial applications such as unidirectional crystal growth frommelt (e.g., horizontal Bridgman technique), glass manufac-turing, welding material processing and others. In some ofthese examples the liquid is enclosed in a three-dimensional(3D) box bounded by rigid and impermeable walls but thetop free surface which is exposed to a gaseous phase. Inthese conditions, the temperature induced variation of thesurface tension at the top produces thermocapillary effectsthat combine with the buoyancy force and influence the con-vection flow characteristics. Actually very complex flow pat-terns and thermal fields might result as it can be inferredfrom experiments.2 Obviously under microgravity condi-tions, the so-called thermocapillary convection becomesmore critical; especially in view of applications in this con-text (e.g., artificial crystal growth in microgravity condi-tions), the interaction between buoyancy and thermocapillary

convection needs to be explicitly analyzed and understood.Most of the existing numerical and analytical works3,4

deal with two-dimensional domains or semi-infinite layers,while there is a lack of numerical solution for what concernsthree-dimensional limited domains. This is due to the highcomputational resources required by the classical algorithmsused for this kind of problem. The recent advent of cheappowerful processors, together with the use of faster implicitcodes, has made possible the execution of true transientsimulations even on a PC in reasonable CPU time.

In this paper, we deal with flows in a top free parallel-epipedic cavity filled with a fluid undergoing thermocapillaryeffects at the ambient interface. A horizontal temperaturegradient is applied by differentially heating two opposite ver-tical walls.

Theoretical studies have revealed the existence of stableplane parallel flows in infinite horizontal layers(Poiseuille–Couette flow).5 In a limited domain this approximation canbe valid only in the core region, far from the vertical walls.The numerical simulations performed in two-dimensionalrectangular cavities, for fluids of Prandtl numbers rangingbetween Pr=0.05 and 50(Zebib et al.6), highlighted that atlow Prandtl numbers a recirculation roll develops near thecold wall, while at higher Prandtl numbers the roll developsnear the hot wall. In the case of Pr=0.015, Ben Hadid andRoux7,8 observed that for low values of the Reynolds–Marangoni number, the fluid configuration reaches the paral-lel flow state in the central region of the cavity. In addition tothis central region, they found an upwind region in which theflow is accelerated and a downwind region in which the flowis decelerated. As the Marangoni number is increased, a re-circulation roll is reported near the cold wall, whereas nearthe hot wall a boundary layer regime is observed. They also

a)Author to whom correspondence should be addressed. Telephone: 39–06–8847–0264; fax: 39–06–440–4306. Electronic mail:mansutti@iac.rm.cnr.it

PHYSICS OF FLUIDS VOLUME 16, NUMBER 9 SEPTEMBER 2004

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found that the flow field is almost independent of the tem-perature field. Considering a fluid with a higher Prandtl num-ber, Villers and Platten4 observed the appearance of a recir-culation roll near the hot wall. This was confirmedexperimentally using acetonesPr=4d. They also found thatthe number of convective cells changes with the aspect ratioand proposed that the length of a cell is approximately twotimes its thickness that is not too far from the critical wave-length predicted by Smith and Davis,5 who indicated a valueof 2.5. The influence of the Prandtl number on the location ofthe recirculation has recently been investigated by Mercierand Normand.3 They assumed that in the core region thevelocity has only the horizontal component, with a parabolicprofile. Near the walls this flow is distorted and becomes twodimensional. This deformation is analyzed as a perturbationin the fluid, superimposed to the basic flow. The disturbancesare characterized by a complex wave number that gives in-formation on the configuration of the flow. For large or smallPrandtl numbers there is a strong asymmetry between the hotand cold wall. For intermediate values of the Prandtl numberthere is a range of Reynolds number for which the flow iscenter symmetric with eddies near the hot and the cold cor-ners.

Mancho and Herrero9 studied the linear stability of abasic flow for a fluid with infinite Prandtl number, assumingno boundary conditions in thez direction, which is consid-ered infinite. They found a three-dimensional(3D) structureconsisting of stationary longitudinal rolls. The instability isnot homogeneous in thex direction, but it grows on the hotside.

The effect of various thermal boundary conditions on thelinear stability of surface-tension-driven flow in an un-bounded liquid layer was analyzed by Priede and Gerbeth.10

Gershuniet al.11 studied the stability of the convective mo-tions by means of small perturbations in a long horizontalcavity. They considered either rigid or free upper boundary, awide range of Prandtl numbers(0.001–50) and analyzed theeffect of three perturbing mechanisms(hydrodynamic, heli-cal wave and Rayleigh mode).

By linear stability analysis, Smith and Davis5 confirmedthat, besides the multicellular state, the basic flow may alsolose its stability due to hydrothermal waves(HW). This mo-tion is enhanced by a temperature disturbance wave thatpropagates in a direction that depends on the Prandtl number.Schwabeet al.12 developed experiments with thermocapil-lary flows in presence of negligible buoyancy effects: a thinlayer of fluid at Prandtl number 17 was considered and, forgrowing Marangoni numbers63102øMaø33103d, thetransitions to steady multicellular flow and, then, to time-dependent travelling waves were detected.

HW were also observed experimentally by Daviaud andVince13 in a rectangular cavity containing silicon oilsPr=8.4d, with a temperature gradient applied across the smallerdimension.

Experiments have been conducted by Riley and Neitzel14

to investigate the HW instability in a cavity filled with asilicon oil sPr=14d. The transition from steady unicellularflow to HW flow was observed in a thin layersheight/width=1/30d; experiments with deep layers do not

exhibit this kind of instability, but rather a transition tosteady multicellular flow and then to oscillatory flow.

Pelacho and Burguete15 performed an experiment on afluid layer of Prandtl number 10.3, observing oblique HWcharacterized by a wave number of 2.2, similar to that foundin theoretical works, but with a propagation angle different,probably due to the geometry of the cell. Burgueteet al.16

observed experimentally in a fluid layer of the same Prandtlnumber HW corresponding to bulk temperature oscillations,exhibiting different behaviors depending on the value of theheight.

Three-dimensional simulations were performed by Mun-drane and Zebib17 considering a fluid with Pr=8.40. Theirsolutions displayed two- or three-dimensional behavior whenthe Marangoni number was, respectively, below and abovesome critical value: at Ma=2.933103 the numerical flow istwo dimensional and characterized by a vortex naturally at-tracted by the hot wall. At Ma=1.953105 the numericalflow appears fully three dimensional with twin corkscrewstructures superimposed on the expected recirculation pat-tern.

The mechanisms of instability have been analyzed ex-perimentally by Braunsfurth and Homsy18 in a parallelepi-pedic domain with equal length and width, filled with ac-etone sPr=4.4d. In this system, just at small Marangoninumbers, they observed a transition from steady two-dimensional flow to steady three-dimensional flow. They alsotraced the aspect ratio dependence of the critical Marangoninumber and found a strong decrease of the critical Ma as theaspect ratio increases. They also investigated the stability ofthe three-dimensional flow at large Marangoni numbers andobserved a transition to oscillatory behavior.

In this paper we extend the numerical results obtained byMundrane and Zebib and contribute to the understanding ofthe mechanisms of transition from steady two-dimensionalflow to steady three-dimensional flow. The cavity here stud-ied has dimensions 43131; the Rayleigh number has beenkept constantly equal to Ra=150, while the Marangoni num-ber has been varied through a sufficiently large range. Be-sides, in order to understand the influence of the Prandtlnumber on the flow configuration, two distinct values havebeen selected, Pr=0.015 and 1.

We solved the vorticity–velocity formulation of the time-dependent Navier–Stokes equations for a nonisothermal in-compressible fluid. A true-transient procedure based on a lin-earized fully implicit finite difference second order schemehas been adopted; the Bi-CGSTAB algorithm with a ILUpreconditioner has been used to solve the linear systems. Thenumerical solutions are visualized by means of LIC(LineIntegral Convolution), which is a vector field representationfor the global realistic visualization of a surface flow.19 Thisnumerical procedure was used by the authors20 and is hereadjusted in order to include the thermocapillary effects.

This paper is organized as follows: in the next sectionwe describe the mathematical formulation, in Sec. III webriefly explain the numerical method for solution; the nu-merical results are described and discussed in Sec. IV andfinally conclusions are drawn in Sec. V.

3270 Phys. Fluids, Vol. 16, No. 9, September 2004 E. Bucchignani and D. Mansutti

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II. MATHEMATICAL FORMULATION

An open parallelepipedic cavity whose dimensions areLx=4, Lz=1 andH=1 (H is the vertical dimension) is con-sidered(Fig. 1). A driving temperature difference in thexdirection is imposed by prescribing the temperature at thetwo solid walls, while adiabatic conditions are assumed atthe four remaining boundaries. There is a flat free surfacelocated aty=H across which no mass transport takes place;the fluid above the surface is assumed to be a gas of negli-gible viscosity and conductivity and therefore it will not in-fluence the flow and temperature fields in the liquid. Forliquids of interest, surface tensionssd decreases with in-creasing temperaturesud and the following linear function isconsidered an adequate approximation to this relation:

s = st − gsu − utd, g =]s

]u.

Through the tangential stress balance at the interface,surface-temperature gradients generate an interfacial shearstress which drives the surface flow in the direction oppositeto that of the temperature gradient.

The fluid is governed by the Navier–Stokes equations,assuming the Boussinesq approximation to be valid. Theyare formulated in terms of vorticityv, velocity u=su,v ,wd,and temperatureu,

]v

]t+ ¹ 3 sv 3 ud =

Pr

Ma¹2v −

Ra Pr

Ma2 ¹ 3 Sug

uguD ,

s1d

¹2u = − ¹ 3 v, s2d

]u

]t+ su · ¹ du =

1

Ma¹2u. s3d

The nondimensional parameters Ra, Pr, and Ma are de-fined as

Ra =gbDTH3

kn, Pr =

n

k, Ma =

gDTH

mk

in which g is the gravitational acceleration,b is the coeffi-cient of thermal expansion,H is the height of the box,DT isthe temperature difference between hot and cold walls,k isthe thermal diffusivity,n is the kinematic viscosity, andm isthe dynamic viscosity. The nondimensional scheme is based

on a reference velocityu* = gDT/m and a reference timet* = Hm /gDT. This scheme is particularly advantageous, asit avoids the appearance of nondimensional parameters in theexpressions of the boundary conditions on the free surface.

Heat flux, from hot to cold wall, has been evaluatedusing the proper definition of the mean Nusselt number on asectionS,

Nu =1

SE

S

suu − ¹ ud ·n dS.

It represents the ratio between the total heat flux versusthe heat flux related to the purely diffusive case.

Boundary conditions: The formulation used here allowsa very simple way of applying the boundary conditions:

(i) The boundary conditions associated to Eq.(1) are ob-tained by the vorticity definition written on the bound-ary. Due to the time discretization scheme used, thisboundary condition is updated and enforced at eachcurrent time step. This procedure contributes to thecorrect coupling between the kinematic and the dy-namic parts of the problem.

(ii ) Dirichlet boundary conditions are associated to theelliptic velocity equation(2). In particular null veloc-ity field is assigned at the solid walls; at the top freesurface, that is assumed to be flat, it holds

v = 0,]u

]y= −

]u

]x,

]w

]y= −

]u

]z,

where the last two conditions are obtained by enforc-ing the second dynamics law. We observe that thesecond condition above introduces a jump discontinu-ity at x=0 andx=4. Actually at those Cartesian planesthe velocity componentu is null due to the presenceof a rigid wall; consequently]u/]y is null. On thecontrary, at the free surface,]u /]x is a measure of thethermocapillary effects and is not at all vanishing.However this inconsistency of the formulation has noeffect within the numerical solution procedure hereadopted because this is based on a finite differencediscretization and the discontinuity edge is left out ofthe computational domain as if regularized boundaryvalues were imposed.

(iii ) The boundary conditions for the energy equation areeasily derived from the definitions and are of Dirichletor Neumann type in those portions of the boundarywhere, respectively, the value of temperature or itsnormal derivative are known,

u = Tc, x= 0,u = Th, x= Lx,

]u

]n= 0,

y= 0, y= H,

z= 0, z= Lz.

III. SOLUTION PROCEDURE

The governing equations(1)–(3) together with theproper boundary conditions are discretized by using finite

FIG. 1. The computational domain.

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difference approximations. Spatial derivatives are discretizedon a uniform mesh through central second-order differenceswhile time derivatives are discretized through three-pointsecond-order backward differences.

By the chosen finite difference schemes, the discretemodel ensures maximum accuracy by a staggered variablelocation. At this purpose the MAC stencil, originally built forthe su ,ud formulation by Harlow and Welch,21 has beenadapted to the presentsu ,v ,ud formulation. In general wheneach velocity component is evaluated at the middle of thefaces of the computational cell which are orthogonal, themass conservation law at the discrete level can be satisfiedup round-off error. In a similar way, by evaluating the vor-ticity components at the midpoint of the edges of the com-putational cell which are parallel, the natural property of so-lenoidality of the vorticity can be met at the discrete level upto round off error. Furthermore in our model staggering ofthe variables allows to discretize the Eq.(2) without anyneed of variable averaging. On the contrary, in discretizingEq. (1), averaging is necessary for the treatment of the ad-vective term¹3 sv3ud, where the productv3u is hereaveraged in the whole(Guj and Stella22). In this way theresulting discrete equation is consistent with the implicitproperty of solenoidality of the vorticity field expressed indiscrete form. The computation of the numerical solutionshas been performed by a time-dependent algorithm. A truetransient procedure requires particular care when used to-gether with the vorticity–velocity formulation. In fact thecontinuity equation is imposed only in an implicit way bydropping the term¹ ·u in Eqs. (1)–(3) so that mass conser-vation and definition of vorticity could be violated if goodcoupling among the full set of the equations is not ensured.Then the time integration has been executed by means of animplicit numerical scheme that has been afterwards linear-ized through the “frozen coefficients” technique.23

In this way at each time step our original system ofpartial differential equations gives rise to a large linear sys-tem of equations of the type

Ax = b,

wherex is the unknown vector andb is the known vector.The coefficient matrixA has a very sparse structure. Thesolution of this linear system via a direct method is obvi-ously not recommended due to the size of the problem andan iterative procedure has been adopted, a variant of precon-ditioned conjugate gradient. This approach has the advantageto allow a great flexibility in writing the discretized form ofnumerical model. We have used the Bi-CGSTABalgorithm,23,24because of its numerical stability and speed ofconvergence.

Although, from a theoretical point of view, iterativemethods can be used without preconditioning the linear sys-tems of equations, the use of a preconditioning technique is,in many practical applications, essential to fulfill the conver-gence and stability requirements of the iterative procedureitself. The aim of the preconditioner is to convert the originallinear system to an equivalent but better-conditioned system.This consists of finding a real matrixC such that the newlinear system

C−1Ax = C−1b

has(by construction) better convergence and stability char-acteristics than the original system. It is obvious that thematrix C must be chosen carefully so thatC−1 should beclose to the inverse ofA and easy to be inverted in order tocompress the computational cost. As a matter of fact the ILUfactorization is one of the most widely used preconditioners:C is defined as the productLU of a lower sLd and an uppersUd triangular matrix generated by a variant of the Croutfactorization algorithm where only the elements ofA that areoriginally nonzero are factorized and stored. In this way thesparsity structure ofA is completely preserved.

The testing of the described numerical model has beenextensively discussed by Stella and Bucchignani.23 The nu-merical inaccuracies usually reported in meeting mass con-servation and vorticity definition had the the same order ofmagnitude as the round-off error at each time step.

Here the numerical solutions have been visualized bymeans of LIC(Line Integral Convolution), which is a vectorfield representation for the global realistic visualization of asurface flow.19 In contrast to tracing single streamlines, theLIC technique is based on the tracing of a high number ofrelatively short streamlines, that are actually convoluted withan initially random distribution of black/white points on thetarget surface(seeds). Resulting LIC images appear verysimilar to some typical flow visualizations(oil flow patterns),thus well suited for numerical–experimental comparison.However, it is worth observing that LIC allows only qualita-tive comparisons, therefore numerical tables are also in-cluded. The criterion used in order to assess the steadiness ofthe flow is based on the numerical evaluation of the timederivative ofv averaged on the whole flow field: when thisvalue is smaller than a fixed quantitys10−9d the flow is con-sidered steady.

IV. NUMERICAL RESULTS

The numerical model described above was adopted tosolve several thermal convection flow problems in parallel-epipedic domains and the computational code was repeatedlyvalidated.20

In the study of the horizontal thermal convection flow ina shallow three-dimensional cavity, the space mesh sensitiv-ity analysis lead the authors to the conclusion that a meshwith a spatial resolution of 0.05 in the vertical direction issufficient to obtain accurate results.20 For the buoyancy andthermocapillary case treated in this paper, which relates to acavity dimensioned 43131 and a flow at fixed Rayleighnumber Ra=150, we developed a space mesh refinementanalysis for a fluid at Pr=0.015 and Ma=30. The previousestimate is still confirmed as it can be checked in Table Iwhere we find the list of the values ofumax, vmax, wmax, andNux, respectively, the maximum of the modulus of each ve-locity component and the Nusselt number at the middle planenormal to thex axis, obtained at different space meshes. Thenumerical model resulted to be nearly second-order accuratein space. Finally for our numerical tests we adopted thespace mesh 41321321 that seems sufficient to guaranteethe description of the whole flow structures.

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A similar mesh sensitivity was already performed by theauthors25 for a different Prandtl numbersPr=23d, obtainingthe same performances: this reasonably leads to the conclu-sion that the mesh sensitivity is not significantly dependenton the value of Pr.

It is worthwhile to mention that, by time and space meshsensitivity analysis developed in a two-dimensional problem,it has been noted that this numerical model appears “nearly”second order accurate both in time and in space.23 Actuallythe linearization procedure applied to the convective termswithin the discretized equations slightly lowers the degree ofaccuracy of the overall discrete operator with respect to thesingle discrete schemes used.

The parallel flow solution valid for an infinite layeryields a simple cubic polynomial expression for the horizon-tal velocity profile usyd taking into account both the ther-mocapillary and the buoyancy effects, while a simpler para-bolic profile is obtained when thermocapillary acts alone.These expressions are very useful when one is interested inthe basic convective states, for low values of the controlparameters; however, for limited domains they are valid onlyin the core region; besides, for increasing values of the con-trol parameters the basic flow is more and more distorted, aswill be shown in the following.

A. Case A: Pr=0.015

The simulations presented in this section are performedusing a fluid with Prandtl number equal to 0.015, as in thecase of molten gallium. The nondimensional values of tem-

perature at cold and hot walls are, respectively,Tc=0 andTh=4. We remind that the Rayleigh number has been setequal to 150 for all the simulations, while the Marangoninumber has been varied in the range 30–300, in order tounderstand the influence of the increasing thermocapillaryeffects on the final flow configuration. The first simulationhas been executed at Ma=30. Figure 2 shows the transienthistory of thex component of velocitysud: after a a timeinterval of 160 nondimensional units, the flow is steady. Suc-cessively, further simulations have been performed at Ma=100,200,300. For all these values, steady solutions wereobtained. Table II reports the maximum values of the threevelocity components and the average Nusselt number in thexdirection as a function of Ma. In Fig. 3 the flow evolution forthe selected values of Ma can be seen. In this figure, the LICrepresentations in thez-normal plane atz=0.5 are reported.At the lowest value, a concentrated vortex forms near thecold wall, superimposed on the base parallel flow, which canbe clearly observed in the central part of the domain. Thestrength of the vortex increases with Ma. The existence of arecirculation roll near the cold wall confirms the numerical

TABLE I. Mesh sensitivity analysis: Pr=0.015, Ra=150, Ma=30.

25315315 41321321 61331331

umax 15.316310−2 15.330310−2 15.337310−2

vmax 10.889310−2 10.880310−2 10.876310−2

wmax 4.346310−2 4.338310−2 4.334310−2

Nux 1.0842 1.0935 1.0982

TABLE II. Pr=0.015. Maximum values of the three velocity componentsand Nusselt number in the middle plane normal to thex axis as a function ofMa.

Ma umax vmax wmax Nux

30 15.330310−2 10.880310−2 4.338310−2 1.0935

100 8.379310−2 6.510310−2 2.165310−2 1.1561

200 6.459310−2 4.815310−2 1.455310−2 1.2336

300 5.351310−2 4.231310−2 1.214310−2 1.2722

FIG. 2. Pr=0.015, Ra=150, Ma=30. Time history of thex component ofvelocity u at the point 0.8, 0.5, 0.4.

FIG. 3. Pr=0.015, Ra=150, LIC representation at the planez=0.5 for Ma=30 (a), 200 (b), 300 (c).

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findings3,6 about the influence of Pr on the location of therecirculation eddies. At Ma=200 a counter-rotating cell oc-curs in the right-lower part of the layer. And finally, at Ma=300 a distorted configuration is observed, due to the move-ment of the new vortex towards the middle of the section.The corresponding isothermal lines are shown in Fig. 4.There are only slight differences among them, and this im-plies that they are not affected by the flow field, as alreadyshown by Ben Hadid and Roux.7 This is typical of low-Prfluids, which are characterized by a weak coupling betweenflow and heat transfer, as confirmed by the weak increase ofNu when Ma is increased(Table II). The coupling betweenflow and heat transfer is stronger for high Pr. The tempera-ture variation from cold to hot wall appears monotonic withrespect tox, very close to a temperature distribution derivedexclusively from buoyant forces. This means that the ther-mocapillary effects do not contribute significantly to the ther-mal field. The surface temperature gradient is about constant(linear temperature distribution) with a small increase nearthe cold wall, related to the increase ofu near this wall. It isinteresting to make a comparison with the numerical resultsof Ben Hadid and Roux7,8 which, even if referred to a two-dimensional domain, provide a good reference solution, es-pecially for low values of Ma. Their flow configurationsagree qualitatively with our solutions, even if they consid-ered a case without buoyancy effects7 or with higher valuesof buoyancy.8 In Fig. 5 we compare the profile of thexcomponent of velocity atx=2 andz=0.5 sMa=30d with theanalytical solution valid for infinite domains.14 The quadraticprofile of Smith and Davis5 is modified by the addition of acubic term due to the buoyancy. Comparison shows a good

agreement among numerical and analytical data. The smalldifferences are due to the walls of the domain. It can be seenthat u is relatively small in the lower 80% of the domain,while being quite large near the free surface. The large ther-mocapillary driven velocity at the free surface is clearly thedominant flow in thex direction. Continuity would dictatethat velocities iny must be sufficiently large to support thissurface flow, which is consistent with buoyancy effects in thevicinity of the isothermal walls.

As shown by Mundrane and Zebib,17 the effects due tothe increasing value of Ma are not only limited to raising ofvortices, superimposed to the base flow, but cause a transi-

FIG. 4. Pr=0.015, Ra=150. Isothermal lines at the planez=0.5 for Ma=30 (a), 200 (b), 300 (c).

FIG. 5. Pr=0.015, Ra=150, Ma=30. Velocity component alongx as a func-tion of y at the fixed positionx=2 andz=0.5.

FIG. 6. Pr=0.015, Ra=150, Ma=200. LIC representation at the planez=0.15 (a) andz=0.85 (b).

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tion to fully three-dimensional configurations. Figure 6shows the LIC representation in two differentz-normalplanes, respectively, atz=0.15 andz=0.85, for Ma=200.These sections, together with the one atz=0.5, show thatthere is a significant variation alongz and, for this value ofMa, the flow is three dimensional. In the midplane, there ismore activity near the free surface and the central circulationis larger and less vigorous. In the two planes far from themidplane, the central circulation moves towards the free sur-face and becomes more vigorous. In this case, it is conve-nient to perform three-dimensional plots, for a detailed de-scription of the flow; Fig. 7 shows thex component ofvelocity u as a function ofy andz at the two fixed positionsx=0.4 (near the cold wall) andx=2. It is not surprising thatmaximum plotted velocity is located at the free surface,where the thermocapillary effect is present. This is the domi-nant flow influence in thex direction. Near the cold wall(butthe same is near the hot wall), the most vigorous activity inxdirection is limited to a small direction at the free surface. Ananalysis of the velocity distribution shows that the flow isdirected towards the hot wall, except for a small region near

the free surface. Roughly two orders of magnitude separatethe respective maxima. Atx=0.4, three-dimensional varia-tion is small and is limited to the region in the immediatevicinity of the free surface. More significant variation ofualongz is registered forx=2; this means that the region thatdisplays the greatest degree of three-dimensional behavior isin the middle of thex direction, in the top half of the calcu-lation domain.

Three-dimensional effects are more significantly ob-served at Ma=300. Figure 8 shows the LIC representation intwo differentz-normal planes, respectively, atz=0.15 andz=0.85. These sections, together with the one atz=0.5, showa strong variation alongz. In fact, in the middle section, themain vortex(near the cold wall) has the center near the freesurface, while it moves downward in the other sections.

Figure 9 contains the LIC representation at the planesx=1, 2, 3. From this figure, the effects due to the rigidboundary conditions in thez direction are clearly high-lighted, as well as the different behavior with respect to thesolutions of Mundrane and Zebib,17 who used periodicboundary conditions. Near the hot wall[Fig. 9(c)], even ifthe buoyant forces contribute to the vertical motion, there areregions where the velocity vector is not directed as the posi-tive y direction. A vertical symmetric split in the flow field isvisible, but it is more evident farther from the hot wall[Fig.9(b)]: here two counter-rotating cells can be seen, due to thelateral no-slip surfaces. The two cells are also visible nearthe cold wall[Fig. 9(a)], but they are relatively high in thecomputational domain. This flow pattern, characterized by atwin corkscrew structure, is similar to the analogous foundby Mundrane and Zebib, so near the cold wall the effects of

FIG. 7. Pr=0.015, Ra=150, Ma=200. Velocity component alongx as afunction of y andz at the fixed positionx=0.4 (a) andx=2 (b).

FIG. 8. Pr=0.015, Ra=150, Ma=300. LIC representation at the planez=0.15 (a) andz=0.85 (b).

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the lateral walls for this low value of Pr are less pronounced.Other interesting flow configurations are represented in Fig.10, where the LIC at the planesy=0.3, 0.5, 0.8 are shown.The vertical symmetric split is visible along all thex exten-sion of the domain. Near the hot wall two symmetriccounter-rotating vortices are visible: their size increases to-wards the free surface[Fig. 10(c)]. Otherwise the twin cork-screw structure observed near the cold wall does not producerecirculations in the horizontal planes: in fact the flow struc-ture is almost everywhere stratified, with the exception of asymmetric recirculation in the low middle region of the cav-ity [Fig. 10(a)].

At Ma.300 a transition to unsteady regime is observed,however this important topic is beyond the purposes of thepresent paper.

B. Case B: Pr=1

In this section, the simulations performed using a fluidwith Prandtl number equal to 1 are presented. Also in thiscase Ra has been fixed at 150, while Ma has been varied inthe range 30–5000. For all these values, steady solutionswere obtained. Table III reports the maximum values of thethree velocity components and the average Nusselt numberin the x direction as a function of Ma. The LIC representa-tions in thez-normal plane atz=0.5 are reported in Fig. 11for Ma=30, 500, 1000 and in Fig. 12 for Ma=2000, 5000.The representation at Ma=30 looks like the one obtained atPr=0.015, with a large circulation in the counterclockwise

TABLE III. Pr=1. Maximum values of the three velocity components andNusselt number in the middle plane normal to thex axis as a function of Ma.

Ma umax vmax wmax Nux

30 0.2535 9.403310−2 1.534310−2 1.2870

100 0.1717 8.744310−2 2.776310−2 1.9216

200 0.1407 8.056310−2 2.883310−2 2.8153

500 0.1199 6.688310−2 2.921310−2 4.9582

1000 0.0922 5.202310−2 2.453310−2 7.2357

2000 0.0734 3.668310−2 1.656310−2 9.6567

5000 0.0481 2.334310−2 1.218310−2 12.118

FIG. 9. Pr=0.015, Ra=150, Ma=300. LIC representation at the planex=1 (a), x=2 (b), x=3 (c).

FIG. 10. Pr=0.015, Ra=150, Ma=300. LIC representation at the planesy=0.3 (a), 0.5 (b), 0.8 (c).

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direction. For Ma=100, 200, 500 a concentrated vortex be-comes visible near the cold wall, but in the flow structure atMa=1000 a significant difference is highlighted: the pres-ence of a second vortex near the hot wall. Then, at Ma=2000 the new vortex is increasing its size, while the otherone is reducing its dimension. And finally, Ma=5000, theflow configuration is dominated by a unique large cell local-ized at the center of the cavity. The corresponding isothermallines are shown in Fig. 13 for Ma=30, 100, 200, 500 and inFig. 14 for Ma=1000, 2000, 5000. In this case we observethat for Ma less or equal 200 the temperature variation ap-pears monotonic with respect tox, and it is only at Ma=500 that thermocapillary effects give a contribution to the

thermal field and the temperature variation is no more mono-tonic. Temperature gradients are largest near the hot and coldwalls, while the gradient near the midplane is relativelysmaller. For the highest values of Ma, the thermal fields areless regular and in fact wide regions characterized by a tem-perature gradient with opposite sign are observed. Finally, atMa=5000, the thermal gradients are all concentrated near thehot and cold walls, while in a large central region(about90% of the domain), the temperature remains nearly con-stant.

The differences with respect to the behavior observed atPr=0.015 are deeper, because indeed at Ma=30 the flow hasa three dimensional configuration. In fact, in the middle sec-tion z=0.5 the vortex is localized near the cold wall, but inthe other sections the vortex moves towards the hot wall.Figure 15 shows the LIC representation in two differentz-normal planes(z=0.15 and 0.85) at Ma=30. These repre-sentations are quite different from the one atz=0.5. In fact inthese two sections, the central vortex is localized near the hotwall. This suggests the conclusion that, even for this smallvalue of Ma, the flow is already three dimensional.

Figure 16 shows the LIC representation in thex-normalplane atx=1, 2, 3; in this case the twin corkscrew structuresimilar to the one of Mundrane and Zebib is observed near

FIG. 11. Pr=1, Ra=150. LIC representation at the planez=0.5 for Ma=30 (a), 500 (b), 1000(c).

FIG. 12. Pr=1, Ra=150. LIC representation at the planez=0.5 for Ma=2000(a), 5000(b).

FIG. 13. Pr=1, Ra=150. Isothermal lines at the planez=0.5 for Ma=30(a),100 (b), 200 (c), 500 (d).

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the hot wall[Fig. 16(c)]. In the middle of the cavity instead[Fig. 16(b)], two couples of symmetric counter-rotating cellsare visible. They are approximately equally sized and occupyall the section, so the flow energy is evenly distributedthrough the cutting plane. And finally, near the cold wall

[Fig. 16(a)], a wide region where the velocity vector is di-rected as the negativey direction is observed. The verticalsymmetry is preserved, but the centers of the lower cellsmoved towards the lateral vertical walls. In Fig. 17 the LICat the planesy=0.5, 0.8 are shown: in these representationsthe flow appears stratified alongx and the structure does notchange with the value ofy. Only at y=0.8 [Fig. 17(b)] twosmall symmetric recirculation zones are visible near the hotwall.

V. CONCLUSIONS

In the literature, in the context of horizontal convection,most of the existing numerical and analytical works3,4 deal

FIG. 14. Pr=1, Ra=150. Isothermal lines at the planez=0.5 for Ma=1000(a), 2000(b), 5000(c).

FIG. 15. Pr=1, Ra=150, Ma=30. LIC representation at the planez=0.15(a) andz=0.85 (b).

FIG. 16. Pr=1, Ra=150, Ma=30. LIC representation at the planex=1 (a),x=2 (b), x=3 (c).

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with two-dimensional domains or semi-infinite layers, whilethere is a lack of numerical solutions in three-dimensionallimited domains. This is due to the high computational re-sources required by the classical algorithms used for thiskind of problem. The recent advent of cheap powerful pro-cessors, together with the use of implicit codes, has madepossible the execution of true transient simulations even on aPC and within a reasonable CPU time.

Most of the theoretical studies have revealed the exis-tence of stable plane parallel flows in infinite horizontal lay-ers(Poiseuille–Couette flow).5 However, in a limited domainthis approximation can be valid only in the core region, farfrom the vertical walls, as the side walls can strongly distortthe plane parallel flows. The numerical simulations per-formed in two-dimensional rectangular cavities, for fluids ofPrandtl numbers ranging between Pr=0.05 and 50(Zebib etal.6), highlighted that for low Prandtl numbers a recirculationroll develops near the cold wall, while for higher Prandtlnumbers the roll develops near the hot wall.

Three-dimensional simulations were performed by Mun-drane and Zebib17 considering a fluid characterized by Pr=8.40. Their solutions have displayed two- or three-dimensional behavior when the Marangoni number was be-low and above some critical value. They observed at Ma=2.933103 a two-dimensional flow characterized by a vor-tex attracted, as expected, to the hot wall. On the other side,they found a flow field at Ma=1.953105 contained fullythree-dimensional involvement and displayed twin cork-screw structures superimposed on the expected recirculationpattern.

In this work our aim has been the extension of the nu-merical results obtained by Mundrane and Zebib in order tobetter clarify the mechanisms of transition from steady two-dimensional flow to steady three-dimensional flow. Here wehave described the results of the numerical study of theRayleigh–Marangoni horizontal convection flow in an opentop 3D parallelepipedic cavitys43131d. We have consid-ered two fluids, for Pr=0.015 and Pr=1, at fixed Rayleighnumber, Ra=150, and have developed simulations for sev-

eral values of the Marangoni number in order to detect fullythree-dimensional flow configurations. From the analysis ofthe numerical flows we reached the following conclusions:the increase of the Prandtl number enhances the coupling ofthe heat and mass transfer phenomena, moreover while atPr=0.015 a fully three-dimensional behavior is observed atMaù200, at Pr=1 the flow configuration appears clearlythree dimensional just at Ma=30; the increase of the Ma-rangoni number induces the appearance of secondarycounter-rotating vortices close to the primary one. The valuesof the Nusselt number at the main stream perpendicularmiddle plane have been provided.

ACKNOWLEDGMENTS

Besides the financial support of I.A.C. to which one ofthe authors is affiliated, the authors acknowledge the supportby A.S.I. (Agenzia Spaziale Italiana), project “Interazione fragocce e fronti di solidificazione unidirezionali in presenza dimoti alla Marangoni,” 2001.

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