Interaction Between Natural Convection and Radiation in a Square Cavity Heated from Below

INTERACTION BETWEEN NATURAL CONVECTION ANDRADIATION IN A SQUARE CAVITY HEATED FROM BELOW

E. H. Ridouane, M. Hasnaoui, and A. AmahmidLMFE, Department of Physics, Faculty of Sciences Semlalia,Marrakesh, Morocco

A. RajiLE, Department of Physics, Faculty of Sciences and Technics,Beni Mellal, Morocco

The radiation effect of gray surfaces on multiple steady-state solutions obtained in a square

enclosure filled with air has been investigated numerically, by a finite-difference procedure.

The bottom and top surfaces of the cavity are, respectively, heated and cooled at constant

temperatures, while its vertical walls are adiabatic. Parameters of the problem are the

Rayleigh number (103 � Ra � 2:5� 106) and the surface emissivity (0 � e� 1). The results

obtained show that surface radiation alters significantly the existence range of the solution

and the average heat transfer through the horizontal walls of the cavity. The parameter e is

found to reduce considerably the critical Rayleigh number characterizing the transition

toward the oscillatory convection.

INTRODUCTION

The problem of natural convection in enclosed spaces has received a great dealof attention due to its importance in many practical applications such as the designof solar collectors, the cooling of electronic equipment, and many other technolo-gical processes. In the literature, experimental, numerical, and analytical studiesdealing with natural convection in enclosures are so numerous as to render thesimple fact of thinking to enumerate them all an almost impossible task. Com-paratively, the works considering cavities with radiating walls are less numerous,although in practice the effect of radiation is not negligible, even when the confinedfluid is not participating in radiation.

The interaction of natural convection with thermal radiation induced by graysurfaces in a square cavity, differentially heated, and filled with air, was studiednumerically by Balaji and Venkateshan [1, 2] and by Akiyama and Chong [3].The results obtained by these authors show that surface radiation alters significantly

Received 22 November 2002; accepted 25 July 2003.

Address correspondence to M. Hasnaoui, Universite Cadi Ayyad, Faculte des Sciences Semlalia,

Departement de Physique, UFR de Thermique et Mecanique des Fluides, B.P. 2390, Marrakech,

Morocco. E-mail: [email protected]

Numerical Heat Transfer, Part A, 45: 289–311, 2004

Copyright # Taylor & Francis Inc.

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407780490250373

289

the temperature distribution inside the cavity and the flow patterns. The averageconvection Nusselt number variation with emissivity was found to be limited.However, the mean Nusselt number due to radiation was strongly dependent on theemissivity of the walls; it increases greatly when this parameter is increased. A usefulnumerical and experimental study was conducted by Ramesh et al. [4] in a squarecavity with differentially heated, isothermal vertical walls. High and low values ofemissivity were considered for the inner surfaces of the cavity. The main objective oftheir study was to answer whether it is possible, in practice, to realize adiabatichorizontal boundary conditions. It was found that this situation could be nearlyrealized, with a reasonable deviation, even with air as working fluid.

Purely experimental work using a differential interferometer has been con-ducted by Ramesh and Venkateshan [5] to study heat transfer by natural convectionand surface radiation in a square cavity, differentially heated, filled with air, andhaving highly emissive walls. A similar conclusion to those of the previous referencesis reported in this work concerning the emissivity effect on natural convection andoverall heat transfer through the active walls of the cavity. The same authors [6] havestudied experimentally, in a parallel work, the effect of surface radiation and par-tition resistance on natural-convection heat transfer in a square, fully partitionedcavity filled with air. The partition was located centrally inside the enclosure andparallel to the isothermal vertical walls. Special attention was paid to examining theeffect of the partition thermal resistance as well as the interaction between naturalconvection and surface radiation. In addition to the well-known effect of surfaceradiation on natural convection and radiation heat transfer, it was found that a

NOMENCLATURE

Fi; j view factor between segments i and j

g acceleration due to gravity, m=s2

Gi dimensionless radiosity of segment

i½¼ ð1� eiÞP

SjFijGj þ ei Ti

y þ 1� �4�

H0 cavity height, m

L0 cavity width, m

Nr conduction-radiation number

½¼ sT40H

0 =lðT 0H � T 0

CÞ�Nu average Nusselt number

¼ Nuconv þ Nuradð ÞPr Prandtl number ð¼ n=aÞQr dimensionless net radiation heat flux

Ra Rayleigh number

½¼ gbðT 0H � T 0

CÞH03=ðanÞ�t dimensionless time ð¼ t0a=H02 ÞT dimensionless temperature

½¼ ðT 0 � T 0CÞ=ðT 0

H � T 0CÞ�

T 00 reference temperature ½¼ ðT 0

H þ T 0CÞ=2�

(u, v) dimensionless velocities in x and y

directions ½¼ u0; v0ð ÞH 0=a�(x, y) dimensionless cartesian coordinates

½¼ x 0; y 0ð Þ=H 0�a thermal diffusivity, m2=s

b coefficient of thermal expansion, 1=K

e surface emissivity

y temperature ratio ½¼ T 00=ðT 0

H � T 0CÞ�

l conductivity of fluid, W=m K

n kinematic viscosity, m2=s

r fluid density, kg=m3

s Stefan-Boltzmann constant

(¼ 5.6761078W=m2K4)

O dimensionless vorticity ð¼ O0H02 =aÞC dimensionless stream function ð¼ C0=aÞCext extremum value of C (either Cmax

or CminÞ

Subscripts

C cold

cond conduction

conv convection

H hot

max maximum value

min minimum value

rad radiation

Superscript0 dimensional variables

290 E. H. RIDOUANE ET AL.

negative temperature gradient in the vertical direction occurs along the partitionheight. This gradient, which depends on the Rayleigh number and partition prop-erties, leads to an increase of the total Nusselt number when the partition resistanceparameter increases.

The objective of this article is to study the effect of surface radiation on mul-tiple steady-state solutions induced in a square, two-dimensional enclosure heatedfrom below. It was demonstrated in the past [7] that, in the absence of the radiationeffect, multiple steady-state solutions can be obtained in a rectangular cavity withpartial heating from below. A multiplicity of solutions is also possible in the case of asquare cavity heated entirely from below. Some useful information will be given hereconcerning the effect of wall emissivity on the range of existence of such solutionsand also on the critical values leading to the oscillatory regimes.

MATHEMATICAL FORMULATION AND GOVERNING EQUATIONS

The studied configuration, depicted in Figure 1, is a square cavity, heated frombelow and cooled from the top by maintaining the respective temperatures T 0

H andT 0C of these surfaces ðT 0

H > T 0CÞ constant. The cavity is filled with air (Pr¼ 0.71), and

its vertical walls are adiabatic. All the inner surfaces in contact with the studied fluidare assumed to be gray, diffuse emitters and reflectors of radiation. The governingequations for two-dimensional, incompressible, laminar flow of fluid, having con-stant properties and obeying the Boussinesq approximation, are the well-knownconservation of mass equation, the Navier-Stokes equations, and the energy equa-tion. These equations are written in dimensionless form using the vorticity ðOÞ andstream function ðCÞ formulation as follows:

Figure 1. Geometry of the physical system.

INTERACTION BETWEEN NATURAL CONVECTION AND RADIATION 291

qOqt

þ qðuOÞqx

þ qðvOÞqy

¼ Prq2Oqx2

þ q2Oqy2

� �� PrRa

qTqx

ð1Þ

qTqt

þ qðuTÞqx

þ qðvTÞqy

¼ q2Tqx2

þ q2Tqy2

ð2Þ

q2Cqx2

þ q2Cqy2

¼ �O ð3Þ

u ¼ � qCqy

v ¼ qCqx

ð4Þ

where

ðx; yÞ ¼ x0

H0 ;y0

H0

� �ðu; vÞ ¼ u0H0

a;v0H0

a

� �ðt;TÞ ¼ t0a

H02 ;T 0 � T 0

C

T 0H � T 0

C

� �

ðC;OÞ ¼ C0

a;O0H02

a

!ðRa;PrÞ ¼ gbðT 0

H � T 0CÞH03

an;na

" #9>>>>>=>>>>>;

ð5Þ

are the nondimensional variables.

Boundary Conditions

The boundary conditions associated to this problem are such that the velocitycomponents are zero on the rigid boundaries and the vorticity is evaluated, on theseboundaries, by the Woods formula [8]. For the stream function and temperature, theboundary conditions are

C ¼ 0 for all surfaces

T ¼ 1 for y ¼ 0

T ¼ 0 for y ¼ 1

qTqx

þNrQr ¼ 0 for adiabatic surfaces (vertical walls)

Here, Qr represents the dimensionless net radiative heat flux at the adiabatic walls.

The Radiosity Formulation

The nondimensional radiosity equation for the ith element on each wall isgiven by

Gi ¼ ð1� eiÞXSj

FijGj þ eiTi

yþ 1

� �4

ð6Þ


In the above equation, the summation over the surface element Sj is to be taken forall the elements of the boundary with which the element i can interact radiatively.The view factors between the elements are determined by Hottel’s [9] crossed stringmethod.

Since the view factors are only a function of the geometry, their calculation isperformed before the beginning of the main iteration procedure. The dimensionlessradiant heat flux Qr is then calculated by the expression

Qr ¼ ei 1þ Ti

y

� �4

�Xj

FijGj

" #ð7Þ

The total heat transfer across the horizontal walls is characterized by the normalizedNusselt number, Nu. The contribution of both natural convection and radiationshould be taken into account in the expression of Nu. The average natural-convection and radiation Nusselt numbers are, respectively, calculated as

Nuconv ¼Qconv

Qcond¼ 1

Qcond

Zwall

qTqy

��y¼0;1

dx ð8Þ

Nurad ¼ Qrad

Qcond¼ 1

Qcond

Zwall

NrQrjy¼0;1dx ð9Þ

where Qcond is the dimensionless heat transfer by pure conduction calculated by theexpression of Qconv in the pure conductive regime. The total average Nusselt numberis then obtained as

Nu ¼ Nuconv þNurad ð10Þ

METHOD OF SOLUTION AND VALIDATION OF THE NUMERICAL CODE

The vorticity and energy equations [Eqs. (1)–(2)], subject to the boundaryconditions, are solved by the alternating direction implicit (ADI) method. The firstand second derivatives were approximated by central differences and the time deri-vatives by a first-order forward difference. The vorticity and temperature equationsare written in conservative form for the nonlinear convective terms in order topreserve the property of conservation. The Poisson equation was solved by the pointsuccessive overrelaxation (PSOR) method. The effect of the grid size on the resultscorresponding to the solutions of S1, S2, and S3 types, obtained in this study (see thenext section), was studied using uniform grids of 51651, 81681, and 1016101. Itwas found that a grid of 51651 is sufficient to model accurately the heat transfer andfluid flow inside the cavity. Table 1 illustrates the effect of the grid size on the so-lution of S1 type, for different combinations of Ra and e corresponding to conditionsclose to those of oscillatory regimes. Hence, when the grid passes from 51651 to1016101, this table shows that the Nusselt numbers vary by less than 4.8% (in themost unfavorable case), while the variations in the case of cmin, characterizing theintensity of the main cell, do not exceed 0.6%. The differences in terms of cmax


(intensity of small secondary cells, located in the vicinity of the corners) reach 10% insome cases, but this difference has no significant effect on the flow structure and heattransfer. In the case of the solution of S3 type, the grid effect is presented in Table 2for Ra¼ 106 and e¼ 0 and 0.1. It is seen from this table that convective, radiative,and total Nusselt numbers induced by a grid of 51651 differ by less than 2.2% fromthose obtained by a grid of 1016101. In the case of cmax and cmin, characterizing,respectively, the intensities of positive and negative cells, the maximum differencesinduced do not exceed 4.5%. Similar tests were conducted also on the solution of S2type, which was obtained for lower values of Ra without observing significant effectof the grid (results not presented). The numerical code was validated first for thelimiting case of pure natural convection with the benchmark solution of De VahlDavis [10] with a uniform grid of 41641. The comparison performed in terms of theaverage Nusselt number and maximum stream function for different Ra showedgood agreement, the maximum differences being less than 2.69% and 0.84%, re-spectively. In the case of radiatively participating surfaces, the numerical code wasalso validated with the results obtained by Akiyama and Chong [3] by comparing theaverage convective Nusselt numbers calculated at the hot surface of a differentiallyheated enclosure. Comparisons have been made for Ra in the range 103 � Ra � 106.The maximum deviations observed were 0.62% and 2.3%, for e¼ 0 and e¼ 1, re-spectively. Other validation has also been performed with the results of Ramesh et al.[4], who considered the case of a differentially heated square cavity. The temperatureprofiles calculated with the present code along the adiabatic horizontal walls showedexcellent agreement with those presented in the latter reference. More details con-cerning the numerical code are given by Raji and Hasnaoui [11].

Table 1. Effect of the grid size on S1 for various Ra and e

Ra e Grid Cmax Cmin Nuconv Nuray Nutot

26106 0 51651 110.286 �12.137 7.617 0 7.617

81681 110.114 �12.530 7.299 0 7.299

1016101 109.651 �12.036 7.267 0 7.267

106 0.5 51651 6.612 �74.921 6.267 6.599 12.866

81681 6.165 �74.753 6.095 6.611 12.706

1016101 6.037 �74.658 6.051 6.623 12.673

46105 1 51651 3.827 �45.372 4.722 11.462 16.183

81681 3.567 �45.384 4.667 11.491 16.158

1016101 3.477 �45.390 4.658 11.506 16.164

Table 2. Effect of the grid size on S3 for Ra¼ 106 and e¼ 0 and 0.1

e Grid Cmax Cmin Nuconv Nuray Nutot

0 51651 38.725 �39.310 4.631 1.201 5.833

81681 38.937 �39.535 4.553 1.203 5.757

1016101 39.278 �39.306 4.540 1.214 5.754

0.1 51651 39.340 �38.677 4.628 1.200 5.828

81681 39.792 �38.655 4.545 1.194 5.740

1016101 41.196 �37.310 4.532 1.192 5.725


RESULTS AND DISCUSSION

In this study, the effect of the Rayleigh number ð103 � Ra � 2:5� 106Þ and theemissivity of the surfaces ð0 � e � 1Þ on the thermal and dynamic behavior of thefluid will be examined. The physical properties of air are evaluated at the referencetemperature T 0

0 ¼ 293:5KðT 0H ¼ 298:5K and T 0

C ¼ 288:5KÞ, and the temperatureratio was maintained constant at y ¼ T 0

0=ðT 0H � T 0

CÞ ¼ 29:35. All the enclosuresurfaces are considered to be gray with identical emissivities. Typical oscillatorysolutions will also be presented and the conditions of their occurrence will bespecified.

Multiplicity of Solutions in Pure Natural-Convection Regime

In the case of a cavity heated from below, it has been demonstrated in the past[7] that multiple steady-state solutions can be obtained for the case of pure naturalconvection ðe ¼ 0Þ. Special effort is devoted here to identifying multiple possiblesolutions and the ranges of Ra corresponding to their existence. The first solutionobtained is illustrated in Figure 2a for Ra¼ 105. The flow is monocellular andcounterclockwise rotating. This solution has been obtained with initial conditionscorresponding to the rest state. Note that the image of this solution through a verticalmirror is also a possible solution; it is clockwise rotating. Both solutions generate thesame mean heat transfer. This kind of solution will be stated as a solution of S1type. The second kind of solutions obtained here is illustrated in Figure 2b forRa¼ 46104; it is a bicellular ascending flow. Its determination necessitates adequateinitial perturbation of the C field. Note also that the image of this solution through ahorizontal mirror is a possible solution for the problem, characterized by a bicellulardescending flow. For the same reason evoked earlier, only the bicellular ascendingflow will be discussed here and it will be stated later as a solution of S2 type. Finally,the third kind of solution obtained is illustrated in Figure 2c for Ra¼ 1.76106. Theflow is bicellular, characterized by the presence of two horizontal cells, one (thenegative cell) above the other (the positive cell). Each cell is in contact with only oneactive wall and the heat transfer is from the hot wall to the lower cell, from the latterto the upper one (through the interface between the two cells), and finally from theupper cell to the cold surface. For this situation also, the image from a verticalmirror is a solution of the problem leading to the same mean heat transfer. Here-after, this kind of solution will be called a solution of S3 type. The obtaining of thissolution was somewhat special. In fact, when Ra exceeds the critical Rayleighnumber, RaC, characterizing the transition from steady-state solution of S1 typetoward the oscillatory one, the flow becomes unsteady and bicellular of S3 type evenif the initial conditions are those of a monocellular flow. Starting with Ra>RaC, theflow is bicellular and unsteady of S3 type. By decreasing Ra progressively, the flowfinishes by becoming steady from a given value of this parameter. The ranges of Rafor which the three solutions defined previously were obtained are summarized inTable 3. A close inspection of this table shows that the solution of S1 type has beenobtained for all the range of Ra explored, while the solutions of S2 and S3 types havebeen obtained for relatively small ranges of this parameter corresponding to weakand important effects of the natural convection, respectively.


Effect of Surface Emissivity

The effect of the emissivity of the walls will be discussed in this section. Thesurface radiation may alter significantly (depending on e and Ra) the flow structureinside the cavity. The natural-convection effect is reduced in general (diminution of

Table 3. Ranges of Ra corresponding to the different steady solutions

Solution type Corresponding range of Ra

S1 [103, 2.46106]

S2 [1.16104, 6.76104]

S3 [3.26105, 2.16106]

Figure 2. Steady-state streamlines and isotherms for e¼ 0: (a) Ra¼ 105 (S1); (b) Ra¼ 4� 104 (S2); (c)

Ra¼ 1:7� 106 (S3).


the flow intensity) due to the increase of the temperature of insulated walls under theradiation effect. This reduction of the convective effects is a natural result of theradiation contribution which tends to render the temperature uniform insidethe cavity. It is to be noted that the centro-symmetry of the steady-state solution ofS1 type, observed in the absence of the radiation effect, is maintained also in the caseof radiating surfaces. However, the radiation effect can lead to the destruction ofsymmetry in the case of bicellular flow (solutions of S2 and S3 types) when e exceedsa critical value which depends on Ra and the type of solution (the critical values of efor which the flow symmetry is destroyed are not given here). The asymmetryobserved is characterized by the augmentation of the size of one cell to the detrimentof the other, as e is increased.

It is well known that in Rayleigh–Benard problems, natural convection be-comes oscillatory when the Rayleigh number exceeds some critical value. The effectof the emissivity of the surfaces on such a critical value will be discussed in this study.In addition, it is to be noted that, when the transitions from S2 and S3 toward S1occur, they can occur in an oscillatory manner observed for short ranges of Ra. Thiskind of oscillatory convection is often observed as a prelude to the change of the flowstructure (passage from bicelluar to monocellular flow in the present study). Only thesolution of S1 type (monocellular flow) persists in the studied range of Ra until theappearance of the oscillatory convection. The effect of e on the transition of S1 to-ward the oscillatory regime is given in Table 4, where it is seen that it is precipitatedby increasing e. This result seems to be somewhat surprising since the increase of eleads to diminution of the natural-convection effect (the flow intensity decreases withincreasing e). However, a close examination of the flow structure shows that thesecondary flow, which is at the origin of oscillatory phenomena, is favored by theincrease of e. To illustrate the emissivity effect on the transition toward the oscil-latory convection, the value Ra¼ 1.96106 was selected since it leads to a steadyregime in pure natural-convection. The flow is monocellular clockwise with twosecondary negative cells located at opposite corners of the cavity. In addition, byincreasing e progressively (for the same value of Ra), the regime remains steady untila critical emissivity eCR’ 0.27 is reached. For values of e slightly higher than eCR, theconvection regime becomes unsteady and the oscillations are periodic in time. Thisbehavior is illustrated by presenting typical variations of Cext and NuH with time fora selected value of e (e¼ 0.3). Also, the corresponding streamlines and isotherms atselected instants during a flow cycle are presented to illustrate the changes undergoneby the flow structure as a result of the radiation effect. Hence, for e¼ 0.3, Figures 3aand 3b show the evolutions with time of Cmax and Cmin (Figure 3a) and NuH

Table 4. Critical values of Ra characterizing the transition of

S1 toward the oscillatory regime for various values of e

e RaC

0.0 2:5� 106

0.2 2:1� 106

0.5 1:3� 106

0.8 6:1� 105

1.0 4:1� 105


(Figure 3b). It is seen that the variations are periodic in time and their shapestestify to the large change in the flow structure during the flow cycle. In fact, the flowis dominated by a monocellular structure during a large range of time and thelife duration of the monocellular counterclockwise cell is almost twice thatcorresponding to the monocellular clockwise cell. In addition, the transition fromone monocellular flow to another occurs suddenly, generating peaks during the

Figure 3. Variations of Cext (a) and NuH (b) with time for Ra¼ 1.96106 and e¼ 0.3.


transition. Each transition is succeeded by the presence of two successive peakscharacterizing relative minimum and maximum in the evolution with time of NuH(Figure 3b). The streamlines and isotherms corresponding to selected instants inFigure 3a are presented in Figures 4a–4d. In Figure 4a the flow is mainly mono-cellular and characterized by the presence of a counterclockwise cell. Two clockwisesecondary cells are located in the upper right and lower left corners of the cavity. Theevolution of the cycle is favorable to the lower positive cell, which enters in com-petition with the upper negative one (Figure 4b). The growth of this cell (positiveone) continues until the disappearance of all the other cells (previously described),leading to a clockwise monocellular flow (Figure 4c) with two secondary negativecells located at the upper left and lower right corners of the cavity (not visible inFigure 4c). The lower negative cell will play a role similar to that played by thesecondary lower positive one of Figure 4a, leading to a bicellular flow in which themain negative cell is located in the lower part of the cavity (Figure 4d ). It is to benoted that, for the value of Ra (Ra¼ 1.96106) considered, when the value of e isprogressively increased from eCR¼ 0.27, the periodicity of the solution disappearsfor e in the range 0.38� e� 0.4. However, the periodicity appears again outside thisrange, with a large change in the flow structure as illustrated in Figures 5a and 5bwith e¼ 0.5. In fact, a close inspection of Figure 5a shows that the increase of e isfavorable to the negative cell and this to the detriment of the positive ones, which arereduced now to playing a secondary role during the flow cycle. At the end of the flowcycle, a sudden peak appears in the variations of Cmax during a very short period oftime in which the flow structure undergoes important modifications. The suddenvariation of Cmax is accompanied by an important increase of NuH (Figure 5b),reaching more than 25% when compared to its mean value. To illustrate the flowstructure, streamlines and isotherms are presented in Figures 6a–6d; they correspondto the instants indicated by the letters a–d in Figure 5a. The maximum number ofpoints (b–d) are selected during the main transition to put in evidence the trans-formations undergone by the flow structure. In fact, during a long period of time, theflow structure is similar to that presented in Figure 6a, which is itself similar to thestationary case obtained with e¼ 0. However, the transformations observed areimportant between instants corresponding to b and d in Figure 5a. The positive cellis more intense in Figure 6c than the negative one (60.5 for Cmax, compared with746.9 for Cmin). For the other situations, the negative cell is dominant. It should benoted that, for the value of Ra considered, the periodicity of the flow was destroyedin favor of a chaotic behavior from e¼ 0.7 (results not presented), the solution beingperiodic for e¼ 0.6.

The second kind of transition which will be discussed here is that corres-ponding to the passage from the solution of S2 type to the one of S1 type. In fact, itwas mentioned before that the solution S2 was obtained in a small range [Ra1, Ra2]of Ra. By decreasing Ra from Ra1, the solution S2 disappears in favor of a steadymonocellular flow. However, when this parameter was increased from Ra2, an un-steady convection, characterized by sustained periodic oscillations, was observed inthe range Ra2<Ra<Ra3. Further increase of Ra from Ra3 leads to a steadysolution of S1 type even if the initial conditions are favorable to S2. The values of Rai(i¼ 1, 2, and 3) are given in Table 5 for various e. It is seen from this table that theexistence range of this kind of solution decreases by increasing e while the transition


range corresponding to the unsteady convection presents different behavior. Toillustrate the oscillating behavior in the range Ra2<Ra < Ra3, the value e¼ 0.4was selected. Hence, for Ra¼ 56104, the variations of Cmax, Cmin, and NuH withtime are almost sinusoidal and the oscillations are characterized by small amplitudes

Figure 4. Streamlines and isotherms over one complete flow cycle for Ra¼ 1.96106 and e¼ 0.3.


(results not presented). However, by increasing Ra toward Ra3, the oscillations losetheir sinusoidal character and become more complicated following to the appearanceof secondary peaks in the variations of Cmax and Cmin (Figure 7a), while in thevariations of NuH (Figure 7b) the amplitude of the oscillations varies between twosuccessive peaks (separated in time by a half-period).

The third kind of transition is that between S3 and S1. It should be recalledthat the solution of S3 type is obtained only when e< eC (eC¼ 0.22, not to be



confused with eCR evoked previously). In addition, the range of Ra in which S3 isobtained depends on e. Three specific values of Ra (Ra4, Ra5, and Ra6, char-acterizing specific behaviors) have been identified. For Ra4<Ra<Ra5, an ini-tialization of the numerical code with a solution of S3 type leads to sustainedperiodic oscillations, while for Ra�Ra4, an initialization of the numerical codewith a solution of S3 type gives way to a steady solution of S1 type. The steadysolution of S3 type is obtained only in the range Ra5�Ra�Ra6, provided thevalue of e is lower than eC. Note finally that for Ra>Ra6, any initializationfavorable to S3 leads to sustained oscillatory behavior of S3 type. In Table 6, Rai(i¼ 4, 5, and 6) obtained for various e< eC are presented. It is seen from this tablethat Rai (i¼ 4 and 5) increase by increasing e while Ra6 undergoes an increasefollowed by a decrease with this parameter. To illustrate the periodic oscillationsobserved in the range [Ra4, Ra5], the selected values of e and Ra are 0.1 and56105, respectively. It is seen from Figures 8a and 8b that the oscillations areperiodic with time. The complex behavior of the oscillations is characterized by thepresence of four secondary peaks per period, and the amplitude of the oscillationsis relatively large. The streamlines during the flow cycle for this solution (notpresented) showed that the flow structure consists of two horizontal cells, in whicheach one is in contact with an active boundary and dominates during a half-periodof the flow cycle. For Ra slightly higher than Ra6, the oscillating behavior of thesolution is also illustrated with Ra¼ 26106 and e¼ 0.2. It is seen from Figures 9aand 9b that the oscillations of the solution are sinusoidal.

Heat Transfer

In the present section, attention will be focused on the heat transfer in order toquantify the contribution of natural convection and radiation to the overall Nusseltnumber [Eqs. (8)–(10)]. Hence, for the solution of S1 type, Figures 10a–10c show,respectively, the variations of Nuconv, Nurad, and Nu with Ra for various values of e.It is seen from Figure 10a that the radiation has a negative but limited effect onNuconv, since this quantity decreases by increasing e for a given Ra. The reduction ofthe convective effect is due to the radiation exchange between the inner walls of thecavity, which tends to make the temperature distribution in the system uniform andthereby to reduce the flow intensity. This tendency is more and more pronounced asthe value of e is increased. The effect of e on Nurad is, on the contrary, positive(Figure 10b), since the latter increases quickly with e (for a given Ra). Finally, the

Table 5. Critical values of Rayleigh number

e Ra1 Ra2 Ra3

0.0 1.16104 6.76104 7.26104

0.2 1.36104 5.26104 5.56104

0.4 1.36104 4.76104 5.66104

0.6 1.46104 4.56104 5.36104

0.8 1.46104 4.36104 4.96104

1.0 1.46104 46104 4.56104


Figure 6. Streamlines and isotherms over one complete flow cycle for Ra¼ 1.96106 and e¼ 0.5.

Table 6. Critical values of Rayleigh number

e Ra4 Ra5 Ra6

0.0 2.96105 3.2 �105 2.1�106

0.1 46105 5.8 �105 2.5�106

0.2 1.26106 1.24�106 1.7�106


quantification of the global contribution of the emissivity to the heat transfer isobtained by presenting the variations of Nu with Ra for various e (Figure 10c). It canbe seen from this figure that the weak reduction of the convection effect observedin Figure 10a is compensated largely by the contribution of the radiation. Such



observations have been formulated in the past by several authors [1–3]. In the case ofS2, similar conclusions can be formulated concerning the effect of e for each mode ofheat transfer (Figures 11a–11c). It should be mentioned that Nuconv of S2 is lowerthan that of S1, while the contrary is observed for Nurad in the range of Ra where

Figure 8. Variations of Cext (a) and NuH (b) with time for Ra¼ 56105 and e¼ 0.1.


both solutions exist. The effect of emissivity on heat transfer induced by S3 is pre-sented in Figures 12a–12c. The values considered for this parameter are 0.1 and 0.2,which are lower than eC¼ 0.22. Although these values of e are relatively weak, theeffects on Nurad (Figure 12b) and Nu (Figure 12c) are well visible. The maximumrelative difference observed in the case of Nu is about 37% for Ra¼ 1.56106.

Figure 9. Variations of Cext (a) and NuH (b) with time for Ra¼ 26106 and e¼ 0.2.


Figure 10. Variations with Ra of Nuconv (a), Nurad (b), and Nu (c) for various emissivities in the case of S1type solution.


Figure 11. Variations with Ra of Nuconv (a), Nurad (b), and Nu (c) for various emissivities in the case of S2type solution.


Figure 12. Variations with Ra of Nuconv (a), Nurad (b), and Nu (c) for various emissivities in the case of S3type solution (e< eC).


CONCLUSIONS

Natural convection, coupled with surface radiation, is studied numerically in asquare cavity, heated from below. The effects of the surface radiation on the multiplesolutions obtained in the pure natural-convection regime (e¼ 0) and also on thetransition toward the oscillatory regime are investigated. In the absence of radiation,three steady-state solutions have been obtained numerically using appropriate initialperturbations. Monocellular flow of S1 type has been obtained in the entire range ofRa explored. However, bicellular flows of types S2 and S3 have been obtained atrelatively low and high values of Ra, respectively. When surface radiation wasconsidered, it has been demonstrated that its effect on the existence ranges of thesesolutions depends strongly on e. Also, it has been found that there exists a criticalvalue of the emissivity, eC¼ 0.22, above which the solution of S3 type disappears. Inthe case of S1, the critical Rayleigh number characterizing the transition toward theoscillating convection is found to decrease notably by increasing e. The results ob-tained show also that it is possible to transit from a steady-state solution to a chaoticone by simply increasing e. Initial conditions favorable to S2=S3 lead to sustainedoscillatory behavior for Ra in the range [Ra2, Ra3]=[Ra4, Ra5]. Outside these rangesand outside the ranges corresponding to the existence of steady S2 and S3 solutions,the final state is that of steady S1 type solution (if Ra<RaC).

The conclusions concerning the effect of emissivity on heat transfer are globallysimilar to those previously reported in the literature. They are characterized by aweak effect of e on the convective Nusselt number (a slight decrease of Nuconv with eis observed), while the radiation and total Nusselt numbers increase monotonicallywith e, and their increase is more important for higher values of e and Ra. Thesolution of S1 type is found to be the most favorable to heat transfer through thehorizontal boundaries of the cavity.

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Interaction Between Natural Convection and Radiation in a Square Cavity Heated from Below

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