On multiple steady states for natural convection (low Prandtl number fluid) within porous square...

International Journal of Heat and Mass Transfer 59 (2013) 230–246

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International Journal of Heat and Mass Transfer

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On multiple steady states for natural convection (low Prandtl number fluid)within porous square enclosures: Effect of nonuniformity of wall temperatures

Madhuchhanda Bhattacharya a, Tanmay Basak b,⇑a C2-5-4C, Delhi Avenue, Indian Institute of Technology Madras Campus, Chennai 600 036, Indiab Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

a r t i c l e i n f o

Article history:Received 11 June 2012Received in revised form 5 November 2012Accepted 14 November 2012Available online 8 January 2013

Keywords:Natural convectionMultiple steady statesBifurcationSymmetry breaking bifurcation

0017-9310/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.11

⇑ Corresponding author.E-mail address: [email protected] (T. Basak).

a b s t r a c t

A Brinkman extended Darcy model has been used to study the effect of spatial nonuniformity of walltemperatures on the multiplicity of steady convective flows within a square porous enclosure saturatedwith low Prandtl fluid (Pr = 0.026). The convection is assumed to be driven by the hotter bottom wall inconjunction with colder side walls, where bottom-side wall junctions maintain continuity of temperatureto represent a more realistic situation. This configuration enforces nonuniformity on either bottom wallor side walls or both bottom and side wall temperatures. The degree of nonuniformity of wall tempera-tures are varied parametrically in terms of thermal aspect ratio (H) to simulate various possible scenariosof nonuniform bottom/side wall temperatures and steady solution branches are traced in the parameterspace of H to investigate the variation of multiplicity of the flows. A perturbation technique has beenused to initiate the steady solution branches, which are then traced by numerical continuation scheme.A penalty finite element approximation in conjunction with Newton–Raphson method and parametercontinuation scheme has been used to construct the bifurcation diagrams of various steady branches.This work reveals three steady symmetric branches and four steady asymmetric branches with exhibitionof symmetry breaking bifurcation. This work also shows that the nonuniformity of wall temperaturesplay a crucial role for the existence of multiple solutions as well as the multiplicity of convective flows.

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1. Introduction

Buoyancy driven natural convection within fluid saturated por-ous media appear in almost all fields especially in the fields of geo-thermal, chemical, fiber and granular insulation, food processing,etc., and hence have remained an active area of research for morethan five decades. Over the years, various fundamental and engi-neering aspects of buoyancy driven flows within porous mediahave been studied both experimentally and theoretically, whichcan be found in few books entirely dedicated on ‘‘transport/flowthough porous media’’ and the references therein [1–3]. Most ofthe theoretical work on natural convection within porous mediaoriginated from the pioneering work of Lapwood [4] on the natureof convective rolls within a horizontal porous layer subjected touniform vertical temperature gradient. Subsequently, a significanteffort have been diverted to study the changes in flow structuresand associated heat transfer that may arise due to non-uniformityof thermal gradients encountered in practical situations of con-fined square/rectangular enclosures [5–23]. These works analyzedthe flow structures under a variety of nonuniform temperature/

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heating profiles applied on either bottom wall or side wallsor both bottom and side walls. Most of these studies consideredcontinuous linear or sinusoidal functions [5–20], while a few alsoconsidered discrete temperature/heating patterns [21–23]. Simul-taneously, a significant effort have also been devoted to developa rigorous mathematical model for accurate description of the flow[24–28,30,31,31–33] and efficient solution techniques for solvingthe governing equations [34–40].

Earlier works have also revealed a rich spectrum of bifurcationphenomenon leading to multiple steady or oscillatory flow struc-tures during buoyancy driven convection within fluid saturatedporous enclosures [41–47]. These investigations are primarilybased on a rectangular porous enclosure with buoyancy opposingtemperature gradients being imposed on isothermally maintainedbottom and top walls and studied the evolution of variousbranches of multiple solutions (oscillatory as well as stationary)as intensity of imposed temperature difference (in terms of Ray-leigh number or Ra) increases beyond a certain limit (critical Ray-leigh number of Rac). Effect of the geometry of the enclosure oncritical Rayleigh number and multiplicity of the solutions have alsobeen investigated in details. However, neither of the previous workinvestigated the effect of nonuniformity of wall temperatures,which may indeed be present in reality, on the multiplicity of

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bifurcation patterns of the solutions. Isothermally hot/cold wallsare mere ideal representation of a more complex physical systemespecially when the temperature difference is applied on two adja-cent walls. In many practical situations, convection is driven by hotbottom wall with cold side walls. Since temperature is continuousat the junction of cold and hot walls, temperature can never be iso-thermal in reality for such situations. This motivated the presentwork to study the multiplicity of free convective flows arising fromthe nonuniformity of wall temperatures.

In this work, we consider a more realistic situation of a fluid sat-urated square porous enclosure, whose bottom wall is hotter thanside walls with continuity of temperature maintained at the boththe bottom corners. The side walls are assumed to be at same tem-perature profiles to impose symmetry, which enables to illustratethe presence of symmetry breaking bifurcation in such systems.This specific configuration requires either bottom wall or sidewalls or both bottom and side walls to be at nonisothermal condi-tions and allows to simulate various possibilities of nonuniformi-ties of bottom/side wall temperatures that may exist in realsituations. Various possible degree of nonuniformities of bottom/side wall temperatures are parameterized by defining thermal as-pect ratio (H), which is the ratio of nonisothermality of side wallsto the total nonisothermality imposed in the bottom and sidewalls. This parameterization allows to simulate the entire spec-trum of possible situations starting from isothermally cold sidewalls with nonisothermal bottom wall to isothermal bottom wallwith nonisothermally cold side walls and study the variation ofmultiplicity of convective flows by systematically varying H from0 to 1. Steady solution branches are traced in the parameter spaceof H to illustrate the effect of nonisothermality of wall tempera-tures on the multiplicity of convective flows in porous media (withDarcy number of 0.001) at Ra = 106 (convective regime) and for alow Prandtl fluid (Pr = 0.026). We have used a perturbation tech-nique to initiate different steady branches, which reveals a widespectrum of multiple solutions as well as symmetry breaking bifur-cation arising from the nonuniformity of wall temperatures. In allthe cases, the top wall is assumed to be thermally insulated. Wehave used a Brinkman extended Darcy model to describe the flowwithin the porous media, which is shown to represent accuratedescription of physical system over a wide range of parameters[24–31].

2. Problem formulation

Consider steady buoyancy driven convection within a fluid sat-urated porous medium confined in a square enclosure of dimen-sion L as illustrated schematically in Fig. 1. The third dimension

Fig. 1. Schematic description of the physical system.

of the enclosure is assumed to be large enough such that thetwo-dimensional approximation of the flow field and temperaturevariations is valid. The walls of the enclosure are impermeable tomass flow, whereas only the top wall of the enclosure is thermallyinsulated with other walls being perfect conductor of heat. Thetemperature of the bottom wall maintains a sinusoidal distributionfrom colder Th at the corners to hotter TH at the center given by:

Tðx; y ¼ 0Þ ¼ Th þ ðTH � ThÞ sinðpx=LÞ : TH P Th; ð1aÞ

where x, y is the Cartesian coordinate with origin at the left bottomcorner of the enclosure (refer to Fig. 1). Simultaneously, both theside walls of the enclosure maintain following linearly decreasingtemperature profile from Th at the bottom to Tc at the top ensuringthe continuity of temperature at both the bottom corners({x = y = 0}, {x = L,y = 0}):

Tðx ¼ 0; yÞ ¼ Tðx ¼ L; yÞ ¼ Th � ðTh � TcÞy=L : Tc 6 Th: ð1bÞ

It may be noted that although chosen geometry and boundary con-ditions of the present problem are symmetric with respect to thevertical center line (x = L/2), we consider the problem over the en-tire regime (0 6 x 6 L, 0 6 y 6 L) in order to allow possible asym-metric solutions. The symmetry of the problem is chosenintentionally since we want to investigate the possibility of exis-tence of asymmetric solutions and symmetry breaking bifurcationof the present symmetric problem.

The thermal boundary conditions considered here is the generalrepresentation of a variety of classical buoyancy driven convectionwithin a square enclosure involving either isothermal or distributedbottom and side wall temperatures. In our formulation, the extent ofnonuniformity of the bottom wall temperature is controlled byDTb = TH � Th, while DTs = Th � Tc controls the deviation from iso-thermal cold side walls. Thus, various combinations of isothermaland/or distributed wall temperatures can be achieved by varyingDTb and DTs in order to attain desired intensity of nonuniformityin the bottom and side wall temperature distributions. In order toparametrize the effect of varying degree of nonuniformity of sideand bottom wall temperatures, we define thermal aspect ratio (H) as

H ¼ Th � Tc

TH � Tc; 0 6 H 6 1; ð2Þ

where H is the ratio of the nonuniformity of side wall temperatures(DTs) to the overall temperature gradient imposed within thedomain (DTmax = TH � Tc � DTb + DTs). Equivalently, 1 �H signifiesthe relative intensity of the bottom wall temperature nonunifor-mity (DTb) compared to DTmax. According to the definition, H = 0corresponds to isothermally cold side walls (T = Tc) with sinusoidal-ly distributed bottom wall temperature from cold Tc at the cornersto hot TH at the center. At the other extreme, H = 1 representsisothermally hot bottom wall (T = TH) with linearly distributed sidewall temperatures from hot TH at the bottom to cold Tc at the top.Any intermediate value of H resembles the case of nonuniformtemperature distribution at both bottom and side walls withintermediate nonuniformities (DTs = HDTmax, DTb = (1 �H)DTmax)between the two limiting cases of H = 0 and 1.

We assume that the thermal and physical properties of theporous medium are invariant of temperature except fluiddensity (qf), which varies linearly with temperature given byqf = q0[1 � b(T � Tc)]. Here, q0 is the density of fluid at T = Tc andb is the volume expansion coefficient. We also assume that thebuoyancy driven flow within the porous media can be modeledby Boussinesque and local thermal equilibrium approximations.With further assumption of incompressible and Newtonian fluid,governing mass, momentum and energy balance equations at stea-dy state can be written as [24–31]

232 M. Bhattacharya, T. Basak / International Journal of Heat and Mass Transfer 59 (2013) 230–246

r� � u ¼ 0; ð3aÞ

q0u � r�u ¼ �r�pþ lfr�2u�

�lf

Kuþ q0gbðT � TcÞ; ð3bÞ

�q0Cpfu � r�T ¼ keffr�2T: ð3cÞ

In the above equations, r⁄ is the gradient and r⁄2 is the Laplacianoperators in Cartesian coordinate system {x,y}. In the above equa-tions, u and p are the intrinsically averaged velocity vector and dy-namic pressure of fluid, respectively, T is temperature, g is theacceleration vector due to gravity ([0 g]), lf and Cpf

are fluid viscos-ity and specific heat, respectively, and keff is the effective thermalconductivity of the porous medium. The velocity vector satisfiesno-slip boundary conditions and thermal boundary conditions arealready mentioned at the beginning of this section.

In terms of the following dimensionless variables

X ¼ xL; Y ¼ y

L; a ¼ keff

�q0Cpf

; U ¼ uLa;

h ¼ T � Tc

TH � Tc; P ¼ pL2

q0a2 ; ð4Þ

the dimensionless form of Eq. (3) and the associated boundary con-ditions can be written as

rU ¼ 0; ð5aÞ

U � rU ¼ �rP þ Prr2U� PrDa

Uþ RaPrhbe; ð5bÞ

U � rh ¼ r2h; ð5cÞat X ¼ 0 and X ¼ 1; fU ¼ 0; h ¼ Hð1� YÞ; ð5dÞat Y ¼ 0; fU ¼ 0; h ¼ Hþ ð1�HÞ sinðpXÞ; ð5eÞ

at Y ¼ 1; U ¼ 0;@h@Y¼ 0:

�ð5fÞ

In Eq. (5) be ¼ ½01�T is the unit vector,r andr2 are the gradient andLaplacian operators, respectively in the dimensionless coordinatesystem {x, y} and Pr = lf/q0a, Da = K/�L2 and Ra = q0gb (TH� Tc)L3/lfaare the dimensionless Prandtl, Darcy and Rayleigh numbers,respectively.

It may be noted that Eq. (5) has four parameters, namely H, Pr,Da and Ra, which govern the buoyancy driven convective flow andassociated heat transfer characteristics within the above definedfluid saturated porous media. However, this work concentratesonly on effect of thermal aspect ratio (H) for a low Prandtl numberfluid (Pr = 0.026) and investigates all the possible steady convec-tive flow patterns over the entire range of H = 0 to 1. For this pur-pose, the parametric analysis is performed in the parameter spaceof H by freezing other parameters at Da = 0.001 and Ra = 106. Itmay be pointed out that the selection of the above parametersare arbitrary and are only to illustrate the effect of H on the possi-ble convective flow patterns for a low Prandtl number fluid. Never-theless, Ra = 106 is selected to ensure convection dominant heattransfer, while Da = 0.001 represent a moderately permeable por-ous media.

3. Computational procedure

The solutions of Eq. (5) for a fixed set of parameters are com-puted using Galerkin penalty finite element method, where theweak variational form of Eq. (5) is treated as a constrained optimi-zation problem with continuity equation being the constraint [48].In this formulation, Eqs. (5a) and (5b) are reformulated to a singleequation by eliminating pressure as P = �cr � U, where c is thepenalty parameter. The combined form of Eqs. (5a) and (5b) canbe written as

U � rU ¼ crðr � UÞ þ Prr2U� PrDa

Uþ RaPrhbe: ð6Þ

Thus, Galerkin penalty finite element method reduces the numberof unknowns of the original equations (Eq. (5)) and solves Eq. (6)along with Eqs. (5c)–(5f) for velocity vector and temperature underthe constraint of continuity equation. This is achieved by assigninga large value for c (generally 104–108) [48], which ensures thedivergence free property of U. Here, we have used c = 107 for allour calculations [18,48].

Upon elimination of pressure, the state variables (U and h) areexpanded in terms of finite element basis set /kf gN

k¼1 such that

U �XN

k¼1

Uk/kðX;YÞ; h �XN

k¼1

hk/kðX;YÞ; ð7Þ

where N is the total number of nodes within the computational do-main. Here, we have used bi-quadratic basis set, which correspondsto biquadratic finite element space discretization. Using Eq. (7), theweak formulation of Galerkin finite element converts Eqs. (6), (5c)–(5f) to the following discrete vector–matrix form

RðfUk; hkgNk¼1Þ ¼ ½cK1 þ K2�

fUkgNk¼1

fhkgNk¼1

" #� F ¼ 0; ð8Þ

where K1 and K2 contain the Galerkin finite element projections ofthe terms of Eqs. (6) and (5c) with and without penalty parameter,respectively and F is Galerkin finite element projection of theboundary terms. The above set of nonlinear equations is solvedusing Newton–Raphson method after evaluating the Galerkin finiteelement projections by appropriate Gaussian quadratures for K1, K2

and F (three point Gaussian quadrature for K2 and F and two pointGaussian quadrature for K1) [18]. Here, we have used 20 � 20biquadratic elements corresponding to 41 � 41 nodes within thedomain for all the simulations, which satisfies convergence test aswill be discussed in the following section.

The solution procedure as outline above is applied to paramet-rically trace the trajectory of possible steady solution branches inthe parameter space of H. A steady branch is initiated fromH = 1 by solving Eq. (8) using Newton–Raphson method startingfrom appropriate initial guess. Once a branch is initiated, it is thentraced while solving Eq. (8) successively at each H along the trajec-tory using the solution of the previous point as initial guess. It maybe pointed out that the convergence of Newton–Raphson dependson the initial guess and consequently, initiation at H = 1 convergesto a particular branch based on the proximity of the initial guess tothat branch. Thus, the initial guess used for the initiation of abranch is extremely critical for this analysis in order to exploreall the possible solutions. The simplest initial guess for initiatinga branch at H = 1 may be either h = 0 or h = 1 "{x, y} without priorknowledge of solution patterns. However, these initial guesses maynot reveal all possible branches especially the possible asymmetricmodes. Thus, we adopt the following procedure to generate initialguess for initiation of possible branches. We introduce perturba-tions in the bottom and/or side wall temperature distributionssuch that the perturbed temperature profiles are physically mean-ingful. The perturbation may be either symmetric or asymmetricaround the vertical center line (x = 0.5). It is important to note thatasymmetric perturbations are essential to initiate asymmetricmodes of the solutions. The solutions obtained with the perturbedboundary conditions are then successively used as initial guesswhile slowly reducing the perturbations and approaching the de-sired bottom and/or side wall temperature distributions given byEqs. (5d) and (5e) with H = 1. Starting from the perturbed bound-ary conditions, Newton–Raphson method converges to varioussteady solutions at H = 1 based on the nature of perturbations.Here, we have explored various symmetric and asymmetric pertur-


bations which revealed seven steady solution branches as will bediscussed in the following section.

4. Results and discussion

In order to illustrate the bifurcation diagram of various steadybranches, we define average Nusselt numbers at bottom (Nub), left(Nul) and right (Nur) walls of the enclosure as

Nub ¼Z 1

0

@h@Y

� �Y¼0

dX�� ; Nul ¼

Z 1

0

@h@X

� �X¼0

dY�� ;

Nur ¼Z 1

0

@h@X

� �X¼1

dY�� ; ð9Þ

and determine them along the trajectory of various solutionbranches in the parameter space of thermal aspect ratio (H). It fol-lows from Eq. (9) that the Nusselt numbers, which represent theabsolute value of average heat flux at the respective walls, mustsatisfy the heat balance within the enclosure given byNub ¼ Nul þ Nur . This relation, which is obtained by integratingthe energy balance equation (Eq. (5c)) over the computational do-main, ensures the accuracy of the numerical solutions. The maxi-mum error in the heat balance was observed to be within 2–3%in all our numerical simulations justifying the accuracy of thesolutions.

It may be noted that the difference between Nur and Nul is zerofor symmetric solutions, while non-zero values of Nur � Nul corre-spond to asymmetric solutions. Accordingly, the differenceNur � Nul, which remains zero along the trajectories of symmetricbranches, may either retain its sign (positive or negative) or under-go reversal of sign (positive to negative or vice versa) along the tra-jectory of asymmetric solutions. Thus, Nur � Nul is an indication ofthe evolution of the symmetric/asymmetric nature of steady solu-tions along various branches especially in case of symmetry break-ing bifurcation [49] and has been used to illustrate the bifurcationdiagram as shown in Fig. 2. Fig. 2 illustrates the entire bifurcationdiagram in terms of the variations of Nub (top panel) and Nur � Nul

(bottom panel) along the trajectories of various steady branches inthe parameter space of H. Here, we have obtained seven steadybranches among which three branches (1(s)–3(s)) are associated

Nu b

(a)

Nu

− N

ur

l

(b)

1.2

1.4

1.6

1.8

2

2.2

1 (s) 2 (s)

5 (as)

4 (as)

6 (as)

3 (s)

7 (as)

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

5 (as)

5 (as)′

4 (as)

4 (as)′

6 (as)

6 (as)′

7 (as)

7 (as)′

Thermal aspect ratio, Θ0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

246

0 0.41 0.82

1(s)

Fig. 2. Bifurcation diagram of various branches in terms of variations of Nub

(subplot a) and Nur � Nul (subplot b) in parameter space of thermal aspect ratio H.

with symmetric solutions and rest of the four (4(as)–7(as))branches correspond to asymmetric flow field and temperaturedistributions. In Fig. 2, the solution branches are sequentially num-bered as 1,2,3, . . . ,7 with (s) and (as) following the number repre-senting symmetric and asymmetric nature of the solutions,respectively. In addition to satisfactory heat balance by all thesolutions as mentioned above, convergence of 20 � 20 biquadraticfinite elements (41 � 41 nodes) is also shown in Table 1 for H = 1.In Table 1, bottom and side wall Nusselt numbers corresponding to20 � 20 biquadratic elements are compared with those corre-sponding to 28 � 28 biquadratic elements (57 � 57 nodes) for allthe seven solution branches at H = 1. It may be noted that the dif-ferences between various Nusselt numbers with refined mesh oc-cur in second decimal place for all the cases showing theadequacy of 41 � 41 nodes.

Fig. 2 shows that all the solution branches except ‘‘1(s)’’ are con-fined within the range of 0.8305 6H 6 1. The symmetric branch‘‘1(s)’’ spans the entire range of thermal aspect ratio (0 6H 6 1as shown in the inset) with all its turning points appearing forH > 0.855. Hence, only one steady symmetric solution correspond-ing to branch ‘‘1(s)’’ exists for H 6 0.8305, while multiple solutionsinvolving symmetric as well as asymmetric flow and temperaturedistributions emerge for H P 0.8305. For each asymmetric solu-tion, there exists its refection counterpart since the problem issymmetric by definition. It may be noted that the flow field andtemperature profiles corresponding to the reflection counterpartof an asymmetric solution can be obtained by the transformationx ? 1 � x. Thus, the Nusselt number at the left wall (Nul) of anasymmetric solution becomes the Nusselt number at the right wall(Nur) for its refection counterpart and vice versa. However, theNusselt number at the bottom wall (Nub) remains same for the pairof asymmetric solution and its counterpart. Thus, each asymmetricbranch of Nub in Fig. 2(a) is associated with two branches ofNur � Nul having same magnitude but opposite sign correspondingto asymmetric solution and its counterpart. This can be seen fromFig. 2(b), where the trajectories of Nur � Nul corresponding to thepair of asymmetric branches and their counterparts are markedwith and without prime, respectively (e.g. ‘‘4(as)’’ and ‘‘40(as)’’). Itmay be noted that the choice of the asymmetric solution and itscounterpart (i.e. ‘‘ 4(as)’’ and ‘‘40(as)’’) is interchangeable. Fig. 2(b)clearly illustrates the evolution of degree of asymmetry along thebranches especially for the case of symmetry breaking bifurcationof branches ‘‘6(as)’’ and ‘‘7(as)’’, where a asymmetric solutioneither emerges from or merges to a symmetric branch (‘‘2(s)’’ inthis case).

The complete bifurcation diagram presented in Fig. 2 exempli-fies the role of thermal aspect ratio (H) on the multiplicity of stea-dy flow field and temperature distribution within the porousmedia defined in Section 2. It is interesting to note that there existsa critical thermal aspect ratio (Hcr = 0.8305) which demarcates theregion of unique and multiple solutions. Multiple solutions appearfor H P Hcr, where there exist significant nonuniformities in theside wall temperatures. Multiple solutions disappear as the non-uniformities of side wall temperatures decrease below those corre-sponding to Hcr. This implies that the temperature of side wallshave to maintain certain degree of nonuniformity based on otherparameters (such as Rayleigh and Darcy numbers) in order to ini-tiate multiple steady solutions. It may be further noted that newsymmetric as well as asymmetric solution branches appear asthe temperature gradients of the side walls increase beyond Hcr.Not only the new solution branches but also their multiple turningpoints dramatically increase the number of multiple solutions forH > 0.95. As many as 25 possible solutions are observed atH = 0.97, while H = 1 is associated with 17 possible steady solu-tions. Thus the occurrence of multiple solutions is greatly facili-tated by the nonuniformity in the side wall temperatures with

Table 1Bottom and side wall Nusselt numbers for various solution branches at H = 1 with20 � 20 and 28 � 28 biquadratic finite elements.

20 � 20 28 � 28

Nub Nur Nul Nub Nur Nul

Branch ‘‘1(s)’’ 2.0399 1.0157 1.0157 2.0262 0.9999 0.9999

Branch ‘‘2(s)’’ 1.6939 0.8197 0.8197 1.6894 0.8152 0.81521.2692 0.6477 0.6477 1.2529 0.6227 0.6227

Branch ‘‘3(s)’’ 1.9432 0.9704 0.9704 1.9058 0.9429 0.94291.0991 0.5599 0.5599 1.0890 0.5413 0.5413

Branch ‘‘4(as)’’ 1.6119 0.8732 0.7057 1.6116 0.8528 0.69401.2503 0.5507 0.6810 1.2366 0.5338 0.6642

Branch ‘‘5(as)’’ 1.5850 0.9072 0.7018 1.5706 0.8775 0.68971.6168 0.9088 0.7111 1.5735 0.8709 0.6843

Branch ‘‘6(as)’’ 1.3298 0.8381 0.499 1.3318 0.8278 0.4879

Branch ‘‘7(as)’’ 1.2153 0.5662 0.6333 1.1991 0.5538 0.6100


uniformly hot bottom wall since the temperature gradient of bot-tom wall is proportional to 1 �H.

In the following section we will discuss about individualbranches and associated circulation and temperature patternsalong the trajectories. The circulations within the enclosure arevisualized in terms of stream function (w) defined as @w

@Y ¼ Ux and� @w

@X ¼ Uy, where Ux and Uy denote x and y components of the veloc-ity vectors, respectively. According to this definition, positive valueof w corresponds to anticlockwise circulation and negative value ofw is associated with clockwise circulation. For each H, streamfunction is evaluated from the computed velocity vector U by solv-ing the following Poisson equation

@2w

@X2 þ@2w

@Y2 ¼@Ux

@Y� @Uy

@Xð10Þ

along with the boundary condition of w = 0 at the walls.

4.1. Branch ‘‘1(s)’’

Fig. 3 plots the trajectories of various Nusselt numbers corre-sponding to branch ‘‘1(s)’’ (Nub in subplot (a) and Nur or equiva-lently Nul in subplot (b)) in the parameter space of H. Since Nur

and Nul are same for symmetric solutions, the heat balance condi-tion for branch ‘‘1(s)’’ reduces to Nub ¼ 2Nur . This condition is wellsatisfied along the entire branch as can be seen from Fig. 3(a) and(b), where maximum error in the heat balance was observed to be2.2%. It may be noted that first turning point of branch ‘‘1(s)’’ ap-pears around H = 0.888 while approaching from H = 0. Subse-quently, branch 1(s) takes a second turn at H � 0.855 andcontinues to H = 1 thereafter. This results in the formation of afolded S shape with region of multiplicity confined between

Nu b

(a)

1

2

3

4

5

6 A

B

CD,E F

G

H I

0 0.2 0.4 0.6 0.8 1

Branch 1(s)

Thermal aspect ratio, Θ

1.49

1.495

1.5

DE

0.79 0.795 0.8

Fig. 3. Variations of Nub (subplot a) and Nur (subplot b) with

0.855 < H < 0.888, where each H corresponds to three symmetricsolutions. Between H = 0 and first turning point, branch ‘‘1(s)’’ con-tinues smoothly except a minor stretched S shape formation with nofolding around H = 0.7937 (shown in the inset).

Evolution of flow patterns and temperature distributions asthermal aspect ratio approaches 1 from 0 along branch ‘‘1(s)’’ isshown in Fig. 4, which shows the streamline and temperature con-tours at various locations along branch ‘‘1(s)’’ indicated by alpha-bets ‘‘A’’–‘‘I’’ in Fig. 3. At each location of Fig. 4, unfilled contoursat left show streamlines, while filled contours at right show thetemperature distribution within the enclosure. In case of stream-lines, darker and lighter shaded contours represent positive andnegative stream functions, respectively and indicate anticlockwiseand clockwise circulations, respectively. On the other hand, varia-tions of temperature within the enclosure from h = 0 to 1 are rep-resented by dark to bright shadings, respectively. The sameconventions will be used in all other illustrations of stream func-tion and temperature contours. Since the magnitude of streamfunctions signify the strength of circulations, maximum and mini-mum of w denoted by wmax and wmin, respectively correspondingto various locations are reported in all the illustrations. Accordingto the present definition of stream function, wmax represent theintensity of strongest anticlockwise circulations, while wmin is ameasure of the maximum strength of clockwise circulations. Abso-lute magnitudes of maximum and minimum stream functions aresame for symmetric flows (jwjmax = jwjmin), while one of themdominates over the other in case of asymmetric flows.

Entire side walls being uniformly cold at T = Tc, hot bottom wallsolely drives the convective flow at H = 0. This leads to the forma-tion of two strong primary circulations (marked P in the figure)stretching over most of the enclosure and rotating in the oppositedirections as can be seen from location ‘‘A’’ in Fig. 4. The primarycirculations emanate from the hottest center (T = TH) of the bottomwall and rotate anticlockwise in the left half and clockwise in theright half of the enclosure. It may be observed from location ’’A’’in Fig. 4 that a pair of counter rotating secondary circulations (de-noted S1 and S2) are formed one above another near both the bot-tom corners of the enclosure due to high intensity of primarycirculations. Among the two secondary circulations, the one atthe bottom (S1) is attached and co-rotates with the correspondingprimary circulation while being stronger than the other secondarycirculation S2. This flow pattern is typically observed in enclosureswith hot bottom wall and cold side walls. As thermal aspect ratioincreases, temperature at both the bottom corners increase in pro-portion to H (h(x = {0,1},y = 0) = H) with an associated redistribu-tion of side and bottom wall temperatures according to Eqs. (5d)and (5e), respectively. As a result, colder portion of side walls moveupward with an associated strengthening of both the secondarycirculations (S1 and S2) as thermal aspect ratio increases alongbranch ‘‘1(s)’’ (refer to locations ‘‘B’’, ‘‘C’’, etc. of Fig. 4). Growth ofsecondary circulations (S1 and S2) towards vertical center line not

Nu

or

Nu

rl

(b)

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1


A

B

C D,E F

G

H I

Branch 1(s)

thermal aspect ratio H along symmetric branch ‘‘1(s)’’.

A (Θ = 0)( ψ = ψ = 10.61)⏐ ⏐ ⏐ ⏐max min

S2

S2S1 S1P

B (Θ = 0.4)( ψ = ψ = 8.63)⏐ ⏐ ⏐ ⏐max min

S2

S2S1 S1P

C (Θ = 0.7)( ψ = ψ = 7.16)⏐ ⏐ ⏐ ⏐max min

S2

S2

S1

S1S3

P

D (Θ = 0.792)( ψ = ψ = 6.72)⏐ ⏐ ⏐ ⏐max min

S2

S2

S1

S1S3

P

E (Θ = 0.796)( ψ = ψ = 6.69)⏐ ⏐ ⏐ ⏐max min

S2

S2

S1

S1S3

P

F (Θ = 0.87)( ψ = ψ = 6.72)⏐ ⏐ ⏐ ⏐max min

S4

S4S2 S2

S1

S1S3

P

G (Θ = 0.87)( ψ = ψ = 6.64)⏐ ⏐ ⏐ ⏐max min

S4

S4S2 S2

S1

S1S3

P

H (Θ = 0.87)( ψ = ψ = 6.60)⏐ ⏐ ⏐ ⏐max min

S4

S2 S2

S1

S1S3

P

I (Θ = 1)( ψ = ψ = 6.59)⏐ ⏐ ⏐ ⏐max min

S4

S2 S2

S1

S1S3

P

Fig. 4. Flow (unfilled contours in left panel) and temperature (filled contours in right panel) profiles within the porous enclosure at various locations on branch ‘‘1(s)’’. Darkerand lighter streamlines in left panel indicate anticlockwise and clockwise circulations, respectively, while and darker to brighter shadings in right panel indicate colder tohotter zones, respectively.


only weaken primary circulations but also pushes them towardsthe upper region of the enclosure creating a new subcell (S3) withinS1–P union near the center of the bottom wall at H = 0.7 (location‘‘C’’ in Fig. 3). Upon further increase of H to ‘‘E’’, all the secondarycirculations intensify with the circulation S2 trying to impinge be-tween the two subcells (S1 and S3) of primary circulation. This is re-flected in the Nusselt numbers as a stretched S shape formationaround H = 0.7937. Impingement and subsequent growth of S2 to-wards bottom wall during the stretch ‘‘E’’ to ‘‘F’’ lead to the detach-ment of secondary circulation S1 from its union with parentprimary circulation and secondary circulation S3. During thisstretch, a subcell S4 is also formed within the extension of circula-tion S2 towards corresponding side wall. Secondary circulations S2

further expand towards corresponding bottom corner as branch‘‘1(s)’’ undergoes turning point bifurcation impelling S1 to shrinkand S4 to weaken. This trend is followed during rest of the tracefrom ‘‘F’’ to ‘‘I’’, which results in splitting and subsequent disap-pearance of S1 from the corners as H travels to location ‘‘H’’. Be-yond location ‘‘H’’, flow pattern remains almost unchanged withsecondary circulations at the bottom corners (S2) dominating theflow in the lower region and primary circulations controlling theflow in the upper region of the enclosure. It may be noted that sub-cell S3 remains attached to its parent cell P during the entire stretchfrom ‘‘C’’ to ‘‘I’’ while undergoing resizing along the foldings ofbranch ‘‘1(s)’’.

In accordance with the flow transition from ‘‘A’’ to ‘‘I’’, temper-ature profiles within the enclosure also get redistributed as can beseen from Fig. 4. Temperature distribution is dominated by ther-mal boundary layers from hot and cold walls at location ‘‘A’’, whereprimary circulations control heat flow within the entire domain. Atthis location, hot region penetrates from the center of bottom wall,while side walls impose colder boundary layer in the surroundings.As H increases, more portion near bottom corners and lower sidewalls remain at higher temperature causing wider as well as higherpenetration of hot region pushing colder thermal boundary layerupward. It may be noted that isotherms in the lower region ofthe enclosure start bending along the secondary circulations asthey strengthen enough with advancement of H. This results in

stratification of isotherms near bottom wall at sufficiently highH, where multiple secondary circulations control the flow in thelower region of the enclosure. Within stratified region, isothermsbend along multiple secondary cells leading to wavy patterns withmaxima and minima appearing at the junction of two cells.

4.2. Branch ‘‘2(s)’’

Variations of bottom (Nub in subplot (a)) and side (Nur=Nul insubplot (b)) Nusselt numbers along the trajectory of branch‘‘2(s)’’ in the parameter space of H are shown in Fig. 5. The maxi-mum error in the heat balance was observed to be 3% along thisbranch. It may be observed that unlike branch ‘‘1(s)’’, branch‘‘2(s)’’ is confined within a very short span of high thermal aspectratio given by 0.9528 < H < 1. Originating from location ‘‘A’’(H = 1), branch ‘‘2(s)’’ undergoes a folding at H � 0.9528 andtraces back to H = 1 corresponding to location ‘‘D’’ in Fig. 5. Conse-quently, the region of multiplicity of this branch is spanned over itsentire range of existence, where each H is associated with twosteady symmetric solutions. It may be noted that Nusselt numbersvary much significantly along the upper part (‘‘A’’ to ‘‘B’’) comparedto the lower part of the trajectory (‘‘C0’’ to ‘‘D’’), where Nub and Nur

or Nul remain almost constant. This can be attributed to the reten-tion of similar flow and temperature patterns along the lower partof branch ‘‘2(s)’’ as shown in Fig. 6. Fig. 6 shows the flow patternsand corresponding temperature distributions at various locations(‘‘A’’ to ‘‘D’’) along the entire trajectory of branch ‘‘2(s)’’. The sameconventions as in Fig. 4 have been used to illustrate the streamfunction and temperature contours in Fig. 6.

As observed in Fig. 4, significant temperature gradient in lowerpart of side walls along with hot bottom wall at high thermal as-pect ratio lead to a series of counter rotating secondary circulationsin the bottom of the enclosure. Simultaneously, colder upper sidewalls drive the formation of two counter rotating primary circula-tions spreading over the upper region of the enclosure either iso-lated or connected to some of the co-rotating secondarycirculations depending on the solution branch and the locationtherein. Correspondingly, temperature distributions also show

Nu b

(a)

Nu

or

Nu

rl

(b)

1.3

1.4

1.5

1.6

1.7

Branch 2 (s)

A

B

C D0.95 0.96 0.97 0.98 0.99 1


0.65

0.7

0.75

0.8

0.95 0.96 0.97 0.98 0.99 1

A

B

C

F


Branch 2 (s)

Fig. 5. Variations of Nub (subplot a) and Nur (subplot b) with thermal aspect ratio H along symmetric branch ‘‘2(s)’’.

A (Θ = 1)( ψ = ψ = 6.96)⏐ ⏐ ⏐ ⏐max min

S3

S3S1S2 S2

B (Θ = 0.97)( ψ = ψ = 7.13)⏐ ⏐ ⏐ ⏐max min

S3

S3S1S2 S2

C (Θ = 0.97)( ψ = ψ = 6.92)⏐ ⏐ ⏐ ⏐max min

S3

S3S1S2 S2

D (Θ = 1)( ψ = ψ = 6.76)⏐ ⏐ ⏐ ⏐max min

S3

S3S1S2 S2

Fig. 6. Flow (unfilled contours in left panel) and temperature (filled contours inright panel) profiles within the porous enclosure at various locations on branch‘‘2(s)’’. Darker and lighter streamlines in left panel indicate anticlockwise andclockwise circulations, respectively, while and darker to brighter shadings in rightpanel indicate colder to hotter zones, respectively.


two different regime within the upper and lower part of the enclo-sure. In the upper part, boundary layer is formed from the side walls,where cold regions try to penetrate inside the enclosure. On theother hand, the temperature near the bottom wall get stratified withwavy contours due to formation of secondary circulations. The fre-quency and amplitude of oscillations of the isotherms depend onthe number and strength of the secondary circulations. The abovementioned flow and temperature characteristics are observed inall the locations on branch ‘‘2(s)’’ as well as for all the locations ofbranch ‘‘1(s)’’ with high thermal aspect ration (see Figs. 4 and 6).However, strength and extent of individual circulation as well asthe connectivity between various cells depend on specific branchand the location therein. For example, at location ‘‘A’’ of branch‘‘2(s)’’ corresponding to H = 1, secondary circulations near the cen-ter of the bottom wall (denoted by S1) dominate the flow in lower re-gion of the enclosure in contrast to the location ‘‘I’’ of branch ‘‘1(s)’’shown in Fig. 4, where secondary circulations at the bottom cornerspredominate over other secondary cells. Also the connectivity be-tween primary and secondary circulations get reversed from loca-tion ‘‘I’’ of branch ‘‘1(s)’’ to location ‘‘A’’ of branch ‘‘2(s)’’. Primarycirculations remain attached to the corresponding secondary cellnear center of bottom wall in former location, while secondary circu-lations at bottom corners (marked S2 in Fig. 6) turn out to be subcellof corresponding primary circulation at later location.

At location ‘‘A’’ of branch ‘‘2(s)’’, predominant secondary circu-lations S1 occupy most of the bottom region of the enclosure andcompress the weaker circulations S2 towards the correspondingside wall. Other than S1 and S2, there also exist a much weakercounter rotating circulation (S3) at the junction between each pri-

mary circulation and its corresponding subcell. As H varies alongthe upper part of the trajectory (‘‘A’’ to ‘‘B’’), both weaker second-ary circulations, namely S2 and S3, expand with an associatedshrinking of S1 towards vertical center line. It may be noted thatalthough growth of S3 impinges the connectivity between primarycirculation and its subcell, they remain to be connected along theentire trajectory while experiencing significant resizing of S2. Afterthe folding at H � 0.9528, the flow and temperature patterns re-main almost unaffected except the fact S3 starts expanding alongthe side walls instead of its growth towards vertical center line.Subsequently, S3 extends till the bottom wall pushing S2 slightlytowards center plane (x = 0.5) with an associated resizing of S1. Itmay be noted that flow in the lower region of the enclosurechanges from predominance of single circulation (S1) to dominanceof multiple circulations as H travels from location ‘‘A’’ to ‘‘D’’.Accordingly, waviness of stratified isotherms smoothen from loca-tion ‘‘A’’ to location ‘‘D’’ in order to follow redistributed multiplesecondary cells as may be observed from Fig. 6.

4.3. Branch ‘‘3(s)’’

Bifurcation diagram of branch ‘‘3(s)’’ in terms of the variationsof Nusselt numbers with thermal aspect ratio is shown in Fig. 7,where maximum error in heat balance was determined to be1.9%. It may be noted that the qualitative bifurcation features ofthis branch is similar to that of the previous branch except therange of existence. Undergoing a turning point bifurcation ath � 0.863, branch ‘‘3(s)’’ exists over a wider spectrum of high ther-mal aspect ratio given by 0.863 [ H 6 1. Also, the variations ofNusselt numbers are of comparable magnitudes along upper andlower trajectories of branch ‘‘3(s)’’. Correspondingly, the differencebetween the Nusselt numbers corresponding to the two solutionsat H = 1 (locations ‘‘A’’ and ‘‘F’’) is much higher compared to thatof branch ‘‘2(s)’’. The associated transition of flow patterns andtemperature distributions from location ‘‘A’’ to ‘‘F’’ along the trajec-tory of branch ‘‘3(s)’’ are shown in Fig. 8.

At H = 1 corresponding to location ‘‘A’’ on branch ‘‘3(s)’’, sec-ondary circulations near each bottom corner, which will be re-ferred to as S1 in the following text, dominate convective flow inthe lower region of the enclosure. Each circulation of S1 has a veryweak subcell S01 (marked only at left half in Fig. 8 for clarity of pre-sentation) within its extension towards corresponding side wall.There also exists another counter rotating secondary cell belowS01. It may be noted that circulations of S1 are inclined towardsthe center and tend to impinge between the corresponding pri-mary circulation and its secondary subcell near the bottom wall,which will be denoted by S2 for ease of illustration. The impinge-ment and subsequent growth of S1 along path ‘‘A’’ to ‘‘B’’ not onlydetach the secondary circulations S2 from their correspondingprimary circulation but also drive them towards the bottom wall.

A (Θ = 1)( ψ = ψ = 6.33)⏐ ⏐ ⏐ ⏐max min

S1

S1′

S2 S1 S1′

B (Θ = 0.864)( ψ = ψ = 5.64)⏐ ⏐ ⏐ ⏐max min

S3S3S1

S1′

S2 S1

C (Θ = 0.9)( ψ = ψ = 5.59)⏐ ⏐ ⏐ ⏐max min

S3

S3

S1

S2 S1

D (Θ = 0.93)( ψ = ψ = 5.80)⏐ ⏐ ⏐ ⏐max min

S3

S3

S1

S1

E (Θ = 0.95)( ψ = ψ = 5.81)⏐ ⏐ ⏐ ⏐max min

S3

S3

S1

S1

F (Θ = 1)( ψ = ψ = 5.95)⏐ ⏐ ⏐ ⏐max min

S3

S3

S1

S1

Fig. 8. Flow (unfilled contours in left panel) and temperature (filled contours in right panel) profiles within the porous enclosure at various locations on branch ‘‘3(s)’’. Darkerand lighter streamlines in left panel indicate anticlockwise and clockwise circulations, respectively, while and darker to brighter shadings in right panel indicate colder tohotter zones, respectively.

Nu b

(a)

Nu

or

Nu

rl

(b)

1

1.2

1.4

1.6

1.8

2 A

B

CD E F

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1Thermal aspect ratio, Θ

Branch 3 (s)

0.5

0.6

0.7

0.8

0.9

1 A

B

CD E F

0.86 0.88 0.9 0.92 0.94 0.96 0.98 1Thermal aspect ratio, Θ

Branch 3 (s)

Fig. 7. Variations of Nub (subplot a) and Nur (subplot b) with thermal aspect ratio H along symmetric branch ‘‘3(s)’’.


As a result, secondary circulations S2 spread along the bottom wall asbranch ‘‘3(s)’’ undergoes folding (refer to location ‘‘B’’). Simulta-neously, a pair of secondary circulations (denoted by S3) rotatingin the same direction as S2 originate and grow at each bottom cornerof the enclosure, which act as an attractor for circulation S2. Thegrowth of S3 along with the spreading of S2 towards the bottom cor-ner force the secondary circulations S1 to dislocate and move awayfrom the bottom wall with an associated shrinking of the primarycirculations. Subsequently, the secondary circulations S2 and S3

merge together at each half of the enclosure as H varies along ‘‘B’’to ‘‘C’’. It may be noted that the upward movement of S1 from ‘‘B’’to C’’ results in disappearance of weaker cells (S01 and the ones belowthem). With further increase of H from ‘‘C’’ to ‘‘D’’, the secondary cir-culations S2 weaken and disappear strengthening S3 at each bottomcorners of the enclosure. This is associated with an weakening andrealignment of secondary circulations S1 to be inclined towardsthe center of the bottom wall (see location ‘‘E’’). Subsequently, S1

gradually extends towards bottom wall from both the sides of S3

along with the formation of co-rotating subcells and associated relo-cation of S3 away from the side walls as H approached 1 along thelower trajectory of branch ‘‘3(s)’’ (‘‘E’’ to ‘‘F’’). However, it may benoted that the flow patterns do not alter the temperature distribu-tion significantly along this part of the trajectory (‘‘D’’ to ‘‘F’’), whichis reflected in Fig. 7 as almost constant Nusselt numbers during thatstretch. On the other hand, temperature within stratified regionundergoes significant change from highly wavy to almost flat pat-terns along the path ‘‘A’’ to ‘‘C’’ due to transformation of the singlelayer of secondary flows into two layers of weaker circulations nearbottom wall of the enclosure. As a result, Nusselt numbers also dropsignificantly along path ‘‘A’’ -‘‘C’’ compared to rest of the trajectory asmay be seen from Fig. 9.

4.4. Branch ‘‘4(as)’’

Fig. 9 illustrates the bifurcation pattern of pair of asymmetricbranch ‘‘4(as)’’ and its counterpart ‘‘40(as)’’ via trajectories of bot-tom and side wall Nusselt numbers in subplot a and b, respectivelyand that of the difference between side wall Nusselt numbersðNur � NulÞ in subplot c. In Fig. 9b and c, the trajectories shownin darker and lighter shades refer to branch ‘‘4(as)’’ and its counter-part ‘‘40(as)’’, respectively. It may be noted that trajectory of Nur forbranch ‘‘40(as)’’ also represents the trajectory of Nul for branch‘‘4(as)’’ since left and right wall Nusselt numbers swap amongthemselves for any asymmetric solution pair. Hence, the discussionon these pair of branches will be based on branch ‘‘4(as)’’, whichcan be easily interpreted for its counterpart branch ‘‘40(as)’’ withthe one-to-one mapping hðx; yÞj4ðasÞ ¼ hð1� x; yÞj40 ðasÞ andwðx; yÞj4ðasÞ ¼ �wð1� x; yÞj40 ðasÞ.

Originating from location ‘‘A’’ (H = 1), Nur corresponding tobranch ‘‘4(as)’’ remains greater than Nul up to H � 0.909, at whichNur � Nul undergoes a transformation of sign from positive to neg-ative as may be observed from Fig. 9. The difference between sidewall Nusselt numbers (Nur � Nul) retains its sign along the rest ofthe branch, while it continues to H = 0.8305 and then traces backto H = 1 (location ‘‘F’’) undergoing three turning point bifurcationwithin 0.8305 [ H [ 0.8322. This makes ‘‘4(as)’’ to be the longestspanned solution branch except ‘‘1(s) with maximum 2% error inthe heat balance along the entire trajectory. Also, branch ‘‘4(as)’’and ‘‘40(as)’’ are the only asymmetric solution pair that exist forH < 0.895. It is interesting to note that the trajectory of right wallNusselt number for branch ‘‘4(as)’’ resembles that of Nub withNur jA � Nur jF being of same order of magnitude as NubjA � NubjF.On the other hand, left wall Nusselt number remains of similar or-

A (Θ = 1)(ψ = 9.29, ψ = −6.61)max min

P2

S1

P1

S2 S5

B (Θ = 0.85)(ψ = 10.89, ψ = −6.65)max min

P2

S1

P1

S2S4 S5

C (Θ = 0.835)(ψ = 10.04, ψ = −6.15)max min

P2

S1

P1

S2S4 S5

D (Θ = 0.88)(ψ = 8.93, ψ = −5.88)max min

P2

S1

P1

S2S3S4 S5

E (Θ = 0.94)(ψ = 8.40, ψ = −5.75)max min

P2

S1

P1

S2S3S4 S5

F (Θ = 1)(ψ = 7.99, ψ = −5.39)max min

P2

S1

P1

S2S3S4 S5

S6

Fig. 10(a). Flow (unfilled contours in left panel) and temperature (filled contours in right panel) profiles within the porous enclosure at various locations on branch ‘‘4(as)’’.Darker and lighter streamlines in left panel indicate anticlockwise and clockwise circulations, respectively, while and darker to brighter shadings in right panel indicate colderto hotter zones, respectively.

Nu b

(a)

Nur(b) Nu − Nur l(c)

1.2

1.5

1.8

2.1

2.4

A

B

C

D

EF

0.84 0.88 0.92 0.96 1

Thermal aspect ration, Θ

Branch 4 (as) − 4′(as)

0.5

0.7

0.9

1.1

A

F

B

C

D

E

A′

B′

C′

D′E′

F′

0.84 0.88 0.92 0.96 1Thermal aspect ratio, Θ

4 (as)

4′(as)

−0.2

−0.1

0

0.1

0.2 A

BC

D E

F

A′

B′C′

D′ E′

F′

4 (as)

4′(as)

0.84 0.88 0.92 0.96 1


Fig. 9. Variations of Nub (subplot a), Nur (subplot b) and Nur � Nul (subplot c) with thermal aspect ratio H along asymmetric branch ‘‘4(as)’’ and its counterpart ‘‘40(as)’’.


der of magnitude at locations A and F as may be observed fromFig. 9.

Transition of flow and temperature patterns from location ‘‘A’’to ‘‘F’’ along branch ‘‘4(as)’’ are shown in Fig. 10(a). The counterpartmirror reflective flow and temperature profiles corresponding tobranch ‘‘40(as)’’ (marked as ‘‘A0’’–‘‘F0’’) are shown in Fig. 10(b).Asymmetric flow mainly originates from preferential growth ofone of the two primary circulations observed in symmetric flows.This results in distortion of isotherms towards a particular sidewall in accordance with the flow profile as shown in Figs. 10(a)and 10(b). In Fig. 10(a), the isotherms are twisted towards left walldue to strengthening of the primary cell in left half of the enclo-sure. In accordance with the mirror reflective similarities between‘‘40(as)’’ and ‘‘4(as)’’, the isotherms in Fig. 10(b) are inclined to-wards right wall associated with the growth of primary circulationin right half of the enclosure. However, isotherms in all the casescharacterize high thermal aspect ratio temperature distributionswith boundary layer formation near upper side walls and stratifi-cation near bottom wall.

Preferential growth of primary circulation occur diagonally atlocation ‘‘A’’ (H = 1), where stronger primary circulation (P1)extending up to bottom wall separates other weaker circulationsat the two diagonally opposite sides of the enclosure. Here, weakerprimary circulation P2 along with its subcell S2 spans diagonallyover a wide portion of the enclosure, while other side of the diag-onal is mostly occupied by P1 except the presence of an isolatedsecondary cell S1 at the corner. It may be noted that union of P2–S2 is constantly impinged by a weaker counter rotating circulation,which originates from side wall and retains its presence along theentire trajectory. During initial trace of ‘‘A’’ to ‘‘B’’, stronger primarycirculation P1 intensifies further and expands along top and bottomwall towards P1 and S1, respectively driving them to their corre-sponding corners. As a result, weaker primary circulation P2 re-tracts along top wall, while its co-circulating secondary cell S2

grows as H varies from ‘‘A’’ to ‘‘B’’. It may be noted that P2 andS2 continue their respective trend along the entire branch, wheresecondary cell S2 intensifies and weaker primary circulation (P2)gradually looses its intensity while shrinking upward. However a

A′ (Θ = 1)(ψ = 6.61, ψ = −9.29)max min

P2

S1P1S2

S5

B′ (Θ = 0.85)(ψ = 6.65, ψ = −10.89)max min

P2

S1 S4P1S2

S5

C′ (Θ = 0.835)(ψ = 6.15, ψ = −10.04)max min

P2

S1P1S2

S5S4

D′ (Θ = 0.88)(ψ = 5.88, ψ = −8.93)max min

P2

S1P1S6S2

S5S4

E′ (Θ = 0.94)(ψ = 5.75, ψ = −8.40)max min

P2

S1P1S6S2

S5S4

F′ (Θ = 1)(ψ = 5.39, ψ = −7.99)max min

P2

S1P1S6S2

S5S4S6

Fig. 10(b). Flow (unfilled contours in left panel) and temperature (filled contours in right panel) profiles within the porous enclosure at various locations on branch ‘‘40(as)’’.Darker and lighter streamlines in left panel indicate anticlockwise and clockwise circulations, respectively, while and darker to brighter shadings in right panel indicate colderto hotter zones, respectively.


reversal of growth pattern is observed for P1 and S1 as H passesthrough the turning points. During this period (‘‘B’’ to ‘‘C’’), secondarycell S1 revives its growth, while primary circulation P1 retracts alongthe bottom wall. Secondary circulation S1 continues its growth alongthe rest of the trajectory and subsequently merge with S2–P2 duo asH approaches ‘‘E’’. This compels primary circulation P1 to move awayfrom the bottom wall leaving its secondary co-rotating subcell nearthe center of bottom wall, which was formed during the trace ‘‘C’’to ‘‘D’’. Finally, S1–S2–P2 union entirely encircle primary circulationP1 along with the presence of two new secondary cells below themat each bottom corner of the enclosure. These cells being formedduring the foldings of branch ‘‘4(as) grow along with S1 and S2

through the entire bottom part of the trajectory.It may be noted that turning point bifurcations initiate the growth

of secondary flows in the lower region of the enclosure. Prior to thefoldings of branch ‘‘4(as)’’ or ‘‘40(as)’’, primary circulation P1 convect amajor portion of heat from the bottom wall. This leads to thinner strat-ified layers of isotherms near bottom wall at location ‘‘A’’, which under-go further thinning due to expansion of P1 along bottom wall as Hprogresses towards first turning point. Upon foldings, stratified layersof isotherms starts thickening as multiple secondary cells acquire pre-dominance in the lower region of the enclosure along lower part ofthe trajectory. This leads to much thicker stratification at location‘‘F’’ compared to location ‘‘A’’ although both the locations corre-spond to H = 1. Accordingly, inclination of isotherms towards P1

also get gradually restricted to upper region of the enclosure asmay be observed from the temperature profiles of location ‘‘A’’ to‘‘F’’ or ‘‘A0’’ to ‘‘F0’’ in Figs. 10(a) and 10(b).

4.5. Branch ‘‘5(as)’’

Bifurcation diagrams of second longest asymmetric branch‘‘5(as)’’ and its counterpart ‘‘50(as)’’ are shown in Fig. 11 using sameconventions as in Fig. 9 (i.e. darker/lighter trajectories for branch‘‘5(as)/‘‘50(as)’’). In Fig. 11, the trajectories of Nub, Nur andNur � Nul are shown in subplots (a), (b) and (c), respectively, wheremaximum error in the heat balance was observed to be 1.5%. Sim-ilar to previous branch, the discussion for this branch will be basedon ‘‘5(as)’’, which can be appropriately interpreted for ‘‘50(as)’’. Itmay be noted that similar to branch ‘‘4(as)’’, branch ‘‘5(as)’’ alsooriginates from and traces back to H = 1 but undergoes a singleturning point bifurcation at H � 0.895. Also, unlike previousbranch this branch retains the sign of Nur � Nul along its entiretrace. It is interesting to note that in this case also one of the sidewall Nusselt number (Nul for branch ‘‘5(as)’’ or Nur for branch‘‘50(as)) follows similar trajectory as bottom wall Nusselt number

(Nub), while the other one follows a different pattern forming loopsnear the turning point. Nevertheless, upper and lower trajectoriesof all the three Nusselt numbers remain close to each other exceptnear the turning point suggesting similar temperature patterns atcorresponding locations as shown in Figs. 12(a) and 12(b).

Fig. 12(a) shows streamlines and isotherms at various locationson branch ‘‘5(as)’’, while their corresponding mirror reflectivecounterparts along branch ‘‘50(as)’’ are shown in Fig. 12(b). Unlikeprevious branch (‘‘4(as)’’–‘‘40(as) pair), convection in the lower re-gion is always controlled by multiple secondary circulations in thiscase. As a result, unevenly intensified primary circulations andassociated preferential bending of isotherms along stronger circu-lation are restricted in the upper region of the enclosure during theentire trail of branch ‘‘5(as)’’ or ‘‘50(as)’’ from ‘‘A’’ to ‘‘F’’. It may benoted that there exists a distinct demarcating plane between pri-mary and secondary circulation controlled regions, which remainalmost unchanged along the entire trajectory. As a result, iso-therms undergo similar extent of stratification from location ‘‘A’’to ‘‘F’’ as may be observed from Figs. 12(a) and 12(b). Also, iso-therms at corresponding locations on upper and lower trace ofthe branch (e.g. ‘‘A’’ and ‘‘F’’) follow similar patterns near bottomwall, while differing around and above the demarcation zone be-tween primary and secondary circulations. This is due to the factthat although individual circulation more or less resembles eachother at corresponding locations on the branch, the connectivitybetween them undergo severe relocation as H traces from location‘‘A’’ to ‘‘F’’. At location ‘‘A’’, dominant primary circulation P1,remaining attached with its subcell S1 at diagonally opposite bot-tom corner, isolates the weaker primary circulation (P2) and thepair of interlinked secondary cells (S21 and S22 ) at other two diago-nally opposite corners. As H traces back to location ‘‘F’’, weakerprimary circulation P2 establishes connection with S21 –S22 duo iso-lating P1 and S1 at diagonally opposite corners.

It may be interesting to note that number of convective cells re-mains same in most part of branch ‘‘5(as)’’ or ‘‘50(as)’’ except a smalltrace of ‘‘B’’ to ‘‘D’’ around the turning point, where a new secondarycirculation S4 originates and subsequently dies down at the bottomcorner below S21 . During its initial trace to the turning point (‘‘B’’–‘‘C’’), circulation S4 grows from the bottom corner and dislocatesS21 upward via merging with the secondary cell S3 present at thejunction of S21 –S22 duo. This is followed by splitting and subsequentshrinking of S4 towards the bottom corner as S21 retraces back to thebottom wall upon folding of branch ‘‘5(as)’’ or ‘‘50(as)’’. However, S21

retains its connectivity with its co-cell S22 along the entire trajectoryas may be observed from Figs. 12(a) and 12(b). Subsequently, S4

vanishes while S3 rejuvenates at the junction of S21 –S22 duo as H

Fig. 11. Variations of Nub (subplot a), Nur (subplot b) and Nur � Nul (subplot c) with thermal aspect ratio H along asymmetric branch ‘‘5(as)’’ and its counterpart ‘‘50(as)’’

A (Θ = 1)(ψ = 7.10, ψ = −5.13)max min

P1

P2

S1S3

S21

S22

B (Θ = 0.914)(ψ = 7.25, ψ = −4.92)max min

S4

P1

P2

S1S3

S21

S22

C (Θ = 0.9)(ψ = 7.08, ψ = −4.99)max min

S4

P1

P2

S1S3

S21

S22

D (Θ = 0.9)(ψ = 7.11, ψ = −5.08)max min

S4

P1

P2

S1S3

S21

S22

E (Θ = 0.96)(ψ = 7.22, ψ = −5.26)max min

P1

P2

S1S3

S21

S22

F (Θ = 1)(ψ = 7.26, ψ = −5.43)max min

P1

P2

S1S3

S21

S22

Fig. 12(a). Flow (unfilled contours in left panel) and temperature (filled contours in right panel) profiles within the porous enclosure at various locations on branch ‘‘5(as)’’Darker and lighter streamlines in left panel indicate anticlockwise and clockwise circulations, respectively, while and darker to brighter shadings in right panel indicate colderto hotter zones, respectively.

A′ (Θ = 1)(ψ = 5.13, ψ = −7.10)max min

P2

P1

S1

S3S22S21

B′ (Θ = 0.914)(ψ = 4.92, ψ = −7.25)max min

P2

P1

S1

S3S22S21

S4

C′ (Θ = 0.9)(ψ = 4.99, ψ = −7.08)max min

P2

P1

S1

S3S22S21

S4

D′ (Θ = 0.9)(ψ = 5.08, ψ = −7.11)max min

P2

P1

S1

S3S22S21

S4

E′ (Θ = 0.96)(ψ = 5.26, ψ = −7.22)max min

P2

P1

S1

S3S22S21

F′ (Θ = 1)(ψ = 5.43, ψ = −7.26)max min

P2

P1

S1

S3S22S21




proceeds towards ‘‘F’’. Folding of branch ‘‘5(as)’’ or ‘‘50(as)’’ also re-sults in preferential growth of S22 along the upper trace (‘‘D’’ to‘‘F’’) of branch ‘‘5(as)’’ or ‘‘50(as) leading to its impingement at P1–S1 junction and their subsequent detachment via merging of S22 withP2 at location ‘‘F’’. During the entire trajectory, secondary cell S1 alsoremains linked to its minor subcell within its extension towards cor-responding side wall (right for branch ‘‘5(as)’’ or left for branch‘‘50(as)) demarcating the region of weaker primary circulation P2. Itmay be noted that the connectivity between S1 and its subcellstrengthen as H travels from ‘‘A’’ to ‘‘F’’.

4.6. Branch ‘‘6(as)’’

Unlike previous asymmetric branches, branch ‘‘6(as)’’ exhibitssymmetry breaking bifurcation with symmetric branch ‘‘2(s)’’ asshown in Fig. 13, which plots the trajectories of Nub, Nur andNur � Nul in subplots (a), (b) and (c), respectively using the sameconventions as in previous figures. Fig. 13 also shows relevant partof branch ‘‘2(s)’’ for illustration purpose. It may be noted that sym-metry breaking bifurcation can be clearly visualized fromFig. 13(b), where trajectories of right and left wall Nusselt numbersstarting from different values corresponding to locations ‘‘A’’ and‘‘A0’’, respectively coincide and merge with that of branch ‘‘2(s)’’at location ‘‘F’’ or ‘‘F0’’ (H = 0.96159). Accordingly, the differencebetween the side wall Nusselt numbers gradually disappear asbranch ‘‘6(as)’’ merges with symmetric branch ‘‘2(s)’’ after twosubsequent foldings at H � 0.9085 and 0.987, respectively. Itmay be noted that maximum error in the heat balance was foundto be 1.7% in this branch.

Similar to branch ‘‘5(as)’’, right wall Nusselt number of branch‘‘6(as)’’ remains to be greater than Nul and correspondingly the dif-ference between the side wall Nusselt numbers retains its signalong the entire trajectory as can be seen from subplots (b) and(c). In this case, side wall Nusselt numbers as well as bottom wallNusselt number approach the trajectory of branch ‘‘2(s)’’ from be-low. It may be noted that initial trace of branch ‘‘6(as)’’ from loca-tion ‘‘A’’ to the first turning point corresponds to maximumasymmetry as reflected in the magnitude of Nur � Nul among allother branches. Beyond the first turning, the difference betweenthe side wall Nusselt numbers (Nur � Nul) reduces as asymmetric

Nu b

(a)

Nur(b)

1.2

1.3

1.4

B

C

D

0.9 0.92 0.94Thermal asp

Branch 6 (a

0.5

0.6

0.7

0.8

A

B

C

D

EF,F′

A′B′

C′

D′ E′

6 (as)

6′(as)

2 (s)

0.9 0.92 0.94 0.96 0.98 1


Fig. 13. Variations of Nub (subplot a), Nur (subplot b) and Nur � Nul (subplot c) with th

flow and temperature distributions gradually transform intosymmetric ones and converge with branch ‘‘2(s)’’ at H = 0.96159as shown in Fig. 14(a). The mirror reflective profiles along branch‘‘60(as)’’ are shown in Fig. 14(b).

It may be noted from the isotherms of Figs. 14(a) and 14(b) thatalthough the extent of stratification remains almost constant at allthe locations, its pattern changes as H travels around branch‘‘6(as)’’. Isotherms within the stratified zone continue to be nearlystraight till a single convective cell dominates the flow in the lowerregion of the enclosure. This persists up to the first turning point ofbranch ‘‘6(as)’’ (up to location ‘‘C’’), beyond which multiple cellsstarts appearing in the lower region of the enclosure inflictingundulations in the isotherms near bottom wall (see location ‘‘D’’).Gradually, multiple circulations gain dominance in the lower regionof the enclosure causing further intensification of the wavy patternof isotherms as branch ‘‘6(as)’’ completes its trail and merges tobranch ‘‘2(s)’’ at location ‘‘F’’. Simultaneously, flow and temperaturepatterns within the enclosure transform from highly asymmetric tosymmetric arrangement as may be seen from Fig. 14(a) or 14(b).

Branch ‘‘6(as)’’ starts with a highly asymmetric assembly of cir-culations at location ‘‘A’’, where dominant primary circulation P1

diagonally spreads across the enclosure and isolates weaker circu-lations at other two diagonally opposite corners. The circulation atthe upper corner will be referred to as P2, while the one at bottomwill be denoted as S1 in the following text. It may be noted that atlocation ‘‘A’’, majority of the fluid in the upper portion is circulatedby circulation P1, while S1 controls the circulation near bottom wallby horizontally stretching over most of the width except corners.Accordingly, isotherms remain almost parallel to bottom wallwithin stratified lower region and preferentially bend along theflow of P1 in the upper region of the enclosure excluding a smallarea surrounding P2. Preferential inclination of isotherms graduallydisappears as P2 strengthen while P1 retracts from the bottom cor-ner in order to evolve into the mirror image of each other at loca-tion ‘‘F’’ upon realignment of both the circulations along thevertical center line. As circulation P1 retracts, its subcell S2 initiatesand starts growing adjacent to the bottom corner along withspreading of another circulation S3 from the corner below it. As aresult, subcell S2 remaining attached to its parent circulation P1

shrinks upward upon first folding of branch ‘‘6(as)’’ (location ‘‘C’’)

Nu − Nur l(c)

A

E

F

0.96 0.98 1ect ratio, Θ

2 (s)

s) − 6′(as)

−0.4

−0.2

0

0.2

0.4 A

BC

DEF,F′

A′B′C′

D′E′


6 (as)

6′(as)

2 (s)

0.9 0.92 0.94 0.96 0.98 1

ermal aspect ratio H along asymmetric branch ‘‘6(as)’’ and its counterpart ‘‘60(as)’’

A (Θ = 1)(ψ = 7.51, ψ = −4.42)max min

P1

P2

S1S6

P21

B (Θ = 0.93)(ψ = 7.40, ψ = −4.82)max min

S2

P1

P2

S1S6 P21

C (Θ = 0.91)(ψ = 7.09, ψ = −5.03)max min

S2

P1

P2

S1S6 S3

D (Θ = 0.94)(ψ = 7.11, ψ = −5.36)max min

S4

S5

S2

P1

P2

S1S6 S3

E (Θ = 0.985)(ψ = 7.33, ψ = −5.75)max min

S4

S5

S2

P1

P2

S1S6 S3

F (Θ = 0.962)(ψ = 7.20, ψ = −7.07)max min

S4

S5

S2

P1

P2

S1S6 S3


A′ (Θ = 1)(ψ = 4.42, ψ = −7.51)max min

P1

P2

S1S6

P21

B′ (Θ = 0.93)(ψ = 4.82, ψ = −7.40)max min

S2

P1

P2

S1 S6

P21

C′ (Θ = 0.91)(ψ = 5.03, ψ = −7.09)max min

S2

P1

P2

S1 S6S3

D′ (Θ = 0.94)(ψ = 5.36, ψ = −7.11)max min

S4 S5

S2

P1

P2

S1 S6S3

E′ (Θ = 0.985)(ψ = 5.75, ψ = −7.33)max min

S4 S5

S2

P1

P2

S1 S6S3

F′ (Θ = 0.962)(ψ = 7.07, ψ = −7.20)max min

S4 S5

S2

P1

P2

S1 S6S3



but then extends around S3 to get reattached with bottom wallconfining S3 in the corner (see location ‘‘D’’). However, circulationS3 continues strengthening along branch ‘‘6(as)’’ forcing S2 to gen-erate a sub-circulation (S4) within its extension near bottom wall,which starts spreading towards S1. During this trail from location‘‘A’’ to ‘‘D’’, subcell S2 also realigns itself along the side wall byinfringing the bonding between P2 and its short living subcell P21

and subsequently encroaching the area upon disappearance ofP21 . Meanwhile, circulation S1 also develops a subcell (S5) near cor-responding side wall due to intensification of a secondary circula-tion S6 at the corresponding bottom corner along the entire trailfrom ‘‘A’’ to ‘‘D’’. It may be noted that intensification of S6 as wellas spreading of S4 force S1 to realign and extend upwards imping-ing the junctions of S2–S4 duo as well as P1–S2 union. Consequently,P1–S2 bonding ruptures and S1 merges with P2 (see location ‘‘D’’).Henceforth flow pattern remains to be similar till the second turn-ing point except preferential enhancement of some of the second-ary circulations adjacent to bottom wall and associated resizing ofothers. It is interesting to note that preferential growth of second-ary circulations beyond location ‘‘D’’ of branch ‘‘6(as)’’ transformsthe highly nonuniform flow structure near bottom wall to a nearuniform one with significant flow enhancement at location ‘‘E’’.This is manifested as the sudden enhancement of bottom Nusseltnumber near location ‘‘E’’ as may be observed from Fig. 13. Subse-quently, S3 and S6 merge with the corresponding primary circula-

tions (P2 and P1, respectively) via splitting of S1–S5 and S2–S4

bonding as branch ‘‘6(as)’’ merge with branch ‘‘2(s)’’ (see location‘‘F’’). During this trace, various circulations resize accordingly in or-der to mirror replicate S2–S3–S4 assembly in S5–S6–S1 or vice versa.

4.7. Branch ‘‘7(as)’’

Fig. 15 illustrates bifurcation characteristics of last as well asshortest spanned solution branch ‘‘7(as)’’ and its mirror reflectivecounterpart ‘‘70(as)’’ via darker and lighter shaded trajectories,respectively following the conventions used for other asymmetricbranches. It may be noted that maximum error in heat balancewas observed to be 1.3% along branch ‘‘7(as)’’, which exists onlywithin 0.96 [ H 6 1. In Fig. 15, variations of bottom and side wallNusselt numbers are shown in subplots (a) and (b), respectively,while subplots (c) traces the difference between side wall Nusseltnumbers from location ‘‘A’’ or ‘‘A0’’ to ‘‘F’’ or ‘‘F0’’ along the trajecto-ries of two counterpart branches. It is evident from the trajectoriesof Fig. 15 that branch ‘‘7(as)’’ also exhibits symmetry breakingbifurcation and merges with symmetric branch ‘‘2(s)’’ after goingthrough three turning point bifurcations within a short span of0.96 [ H 6 0.9807. Originating from H = 1 corresponding to loca-tion ‘‘A’’, branch ‘‘7(as)’’ takes it first turn at H � 0.96 and then un-dergo two subsequent foldings at H � 0.9807 and 0.9687 beforemerging to the lower trajectory of branch ‘‘2(s)’’ at location ‘‘F’’

Nu b

(a)

Nur(b) Nu − Nur l(c)

1.2

1.25

1.3

1.35

A

B

C

DE

F

Branch 7 (as) − 7′(as)

2 (s)

0.95 0.96 0.97 0.98 0.99 1


0.55

0.6

0.65

0.7

A

B

C

DE

F/F′D′ A′B′

C′

E′

7 (as)

7′(as)

2 (s)

0.95 0.96 0.97 0.98 0.99 1


−0.08

−0.04

0

0.04

0.08

A

B

C

D

E

F

A′

B′

C′

D′E′

F′

7 (as)

7′(as)

2 (s)

0.95 0.96 0.97 0.98 0.99 1Thermal aspect ratio, Θ

Fig. 15. Variations of Nub (subplot a), Nur (subplot b) and Nur � Nul (subplot c) with thermal aspect ratio H along asymmetric branch ‘‘7(as)’’ and its counterpart ‘‘70(as)’’.


(‘‘F0’’) corresponding to H = 0.978. It may be noted that the follow-ing discussion will be based on branch ‘‘7(as)’’, which can be easilyinterpreted for its counterpart based on the relation between themas discussed before.

It is interesting to note that Nul of branch ‘‘7(as)’’ (lighter trajec-tory in subplot b) exhibits much smaller variations (varies between0.633 and 0.659) compared to Nur . Consequently, Nur (darker tra-jectory in subplot b) follows the variation pattern of Nub due tothe constraint Nub ¼ Nur þ Nul and accordingly the two trajectoriesreplicate each other as may be observed from subplots (a) and (b).In this case, right wall Nusselt number of branch ‘‘7(as)’’ lies abovethat of left wall for a very short period of 0.96 6H 6 0.9718 start-ing form first turning point and remains to be lower than Nul forrest of the trajectories. Correspondingly, the difference betweenright and left wall Nusselt numbers undergoes its first transitionof sign at H = 0.96 followed by its second transition atH = 0.9718 to flip back to its sign during initial trace of ‘‘A’’ to‘‘B’’. However, the difference between side wall Nusselt numbersremains low along the entire trajectory compared to other asym-metric branches. Also, variations of bottom as well as side wallNusselt numbers along branch ‘‘7(as)’’ are much smaller comparedto other branches as may be observed from Figs. 15 and 2. Thus,temperature distributions are expected to remain similar near bot-tom and side walls while evolving from asymmetric to symmetricpatterns along branch ‘‘7(as)’’ as may be observed from Figs. 16(a)and 16(b).

Fig. 16(a) shows streamlines and isotherms at various locationson branch ‘‘7(as)’’ and corresponding mirror reflective counterpartsalong branch ‘‘70(as) are shown in Fig. 16(b). It may be noted thatflow pattern at location ‘‘A’’ (‘‘A0’’) is very similar to that at location‘‘F’’ (‘‘F0’’) of branch ‘‘4(as)’’ (‘‘40(as)’’), where dominant primary cir-culation P1 is surrounded by a union of three secondary circula-tions denoted from top as S1, S2 and S3, respectively. Other thanthese main circulations, there exist several weaker convective cellsat each bottom corner (S4 and S5) as well as at both the junction ofS1–S2 and S2–S3 denoted by S1�2 and S2�3, respectively. Also, circu-lation S4 enforces a minor subcell S6 within the extension of S3 to-wards the corresponding side wall. Existence of counter rotatingcirculations at the junction of two interlinked cells is the typical

characteristics of multi-celled circulations as also observed in pre-vious flow profiles. Intensification of these circulations at the junc-tion between two interlinked cells with the variation of thermalaspect ratio leads to weakening and subsequent rupture of therespective bonding creating two separate cells. On the other hand,severe wakening of the circulation at any junction may lead to col-lapse of the associated interlinked two cells into a single one asthermal aspect ratio varies along a branch. Similar trends are alsoobserved in several parts of branch ‘‘7(as)’’ or ‘‘70(as)’’ as may beseen from Figs. 16(a) and 16(b). For example, during the progressof H from location ‘‘A’’ to location ‘‘B’’, subcell S6 disappears dueto weakening of S4 at the bottom corner. Simultaneously, S2�3

intensifies from location ‘‘A’’ to location ‘‘B’’ leading to its mergerwith the primary circulation P1 via rupture of S2–S3 bonding. Circu-lation S2�3 continues its growth beyond the first turning point tilllocation ‘‘C’’. Also, circulation S4 revives its growth upon comple-tion of first turning point with reappearance and subsequentstrengthening of subcell S6 as may be observed from the streamlinecontours of location ‘‘C’’. During this stretch of B to C, circulation S5

moves up followed by its merger with S1�2 as secondary cell S2

shrinks and aligns towards the corner. This leads to more uniformcell structures and consequent flow enhancement near bottomwall from location ‘‘B’’ to location ‘‘C’’. Correspondingly, Fig. 15shows a sudden increase of Nub during that stretch. However, cir-culation S2�3 undergoes severe weakening along path ‘‘C’’ to ‘‘D’’,while other circulations near bottom wall continue intensifying.As a result, nonuniformity of flow structure near bottom wall in-creases along path ‘‘C’’ to ‘‘D’’, which causes a sudden decrease ofNub as reflected in Fig. 15. Growth of S2 along path ‘‘C’’ to ‘‘D’’ alsocauses S5 to retrace back to the corner upon detachment from S1�2.Beyond location ‘‘D’’, circulation S3 starts shrinking and S2�3 growsto become replication of each other as branch ‘‘7(as)’’ completes itstwo foldings to merge with branch ‘‘2(s)’’. During this trace, circu-lations S4 and S2 also undergo resizing to become mirror image ofeach other at location ‘‘F’’. Similarly, circulations S6 and S1�2 be-come mirror reflection of each other at location ‘‘F’’ upon theirgrowth along path ‘‘D’’ to ‘‘F’’. It may be noted that intensificationof S4 during second folding of branch ‘‘7(as)’’ or ‘‘70(as)’’ also leadsto its merger with primary circulation P1 and detached S6 from its

A (Θ = 1)(ψ = 7.89, ψ = −5.26)max min

P1

S1

S3

S4

S2 S1−2

S6

S2−3 S5

B (Θ = 0.962)(ψ = 7.96, ψ = −5.27)max min

P1

S1

S3

S4 S2S1−2 S2−3 S5

C (Θ = 0.97)(ψ = 7.63, ψ = −5.75)max min

P1

S1

S3

S4S2 S1−2S2−3

S6

D (Θ = 0.978)(ψ = 7.3, ψ = −6.2)max min

P1

S1

S3

S4S2

S1−2

S5S2−3

S6

E (Θ = 0.97)(ψ = 7.13, ψ = −6.65)max min

P1

S1

S3

S4S2 S1−2

S5S2−3

S6

F (Θ = 0.978)(ψ = 6.87, ψ = −6.87)max min

P1

S1

S3

S4S2 S1−2S2−3

S6


A′ (Θ = 1)(ψ = 5.26, ψ = −7.89)max min

P1

S1

S3 S6 S4

S2

S1−2

S2−3S5

B′ (Θ = 0.962)(ψ = 5.27, ψ = −7.96)max min

P1

S1

S3 S4

S2

S1−2

S2−3

S5

C′ (Θ = 0.97)(ψ = 5.75, ψ = −7.63)max min

P1

S1

S3 S4

S2

S1−2

S2−3 S6

D′ (Θ = 0.978)(ψ = 6.20, ψ = −7.30)max min

P1

S1

S3 S4

S2

S1−2

S2−3S5 S6

E′ (Θ = 0.97)(ψ = 6.65, ψ = −7.13)max min

P1

S1

S3 S4

S2

S1−2

S2−3S5 S6

F′ (Θ = 0.978)(ψ = 6.87, ψ = −6.87)max min

P1

S1

S3 S4

S2

S1−2

S2−3 S6



parent cell S3 (see location ‘‘E’’). This linkage between P1 and S4

evolve into the replication of S1–S2 bonding as H proceeds towards‘‘F’’. During this entire trail, primary circulation P1 shrinks, whilesecondary circulation S1 expands along the vertical center line inorder to replicate each other at location ‘‘F’’. Needless to say, trans-formation from highly nonuniform flow structure at location ‘‘D’’ touniform flow structure at location ‘‘F’’ strengthen the flow nearbottom wall. Accordingly Nub also increases beyond the secondturn of branch ‘‘7(as)’’ as may be observed from Fig. 15.

It may be noted that although secondary circulations undergo aseries of merging and splitting, their region of dominance remainalmost unchanged along the trajectory of branch ‘‘7(as)’’ or‘‘70(as)’’. Consequently, isotherms exhibit similar extent of stratifi-cation at all the locations during the transformation from asym-metric to symmetric pattern. However, stratified isothermsbecome more wavy as H travels from location ‘‘A’’ to ‘‘F’’. This isdue to evolution of secondary cells near bottom wall into a moreevenly distributed structure as may be observed from Figs. 16(a)and 16(b). Also, isotherms in the upper region of the enclosuretransform into symmetric configuration along the vertical centerline as P1 and S1 transform into mirror image of each other.

5. Conclusive remarks

This work presents a detailed analysis on the possibility ofmultiple steady flow structures and corresponding temperature

patterns during natural convection within a low Prandtl number(Pr = 0.026) fluid saturated square porous enclosure with insulatedtop wall and other walls at various imposed temperature distribu-tions. Temperature distributions of bottom (Tb = Th + DTbsinpx/L)and side (Ts = Th � DTsy/L) walls are specifically selected not onlyto ensure continuity at both the bottom corners but also to main-tain geometric symmetry of the problem with respect to verticalcenter line in order to study the presence of asymmetric solutionsfor symmetrically defined problems. Existence of multiple steadysolutions involving symmetric as well as asymmetric structuresare examined for varying relative magnitude of DTs or DTb with re-spect to the total imposed temperature gradient within the compu-tational domain (DTs + DTb) and the results are presented in termsof thermal aspect ratio H defined as 0 6H = DTs/(DTb + DTs) 6 1.All the simulations are performed for fixed Darcy (Da = 0.001)and Rayleigh (Ra = 106) numbers, while multiplicities of steadyconvective flows for various steady branches have been traced inthe parameter space of H.

It has been shown that imposed thermal gradient of side wallsplay a critical role for the occurrence of multiple steady flow struc-tures, multiple solutions starts appearing at H > Hcr � 0.8305 cor-responding to high degree of thermal nonuniformity in the sidewalls compared to that of bottom wall. Thus, a high degree of ther-mal nonuniformity has to be maintained at the side walls in orderto observe multiplicity of steady convective flows. Within therange of multiplicity (Hcr < H 6 1), there exist seven steady


branches involving symmetric (‘‘1(s)’’–‘‘3(s)’’) as well as asymmet-ric (‘‘4(as)’’–‘‘7(as)’’) solution patterns. It may be noted that foreach asymmetric branch, there exists its mirror reflective counter-part since the problem is chosen to be symmetric around verticalcenter line. Symmetry of the problem is specifically selected in or-der to show existence of symmetry breaking bifurcation exhibitedby asymmetric branches ‘‘6(as)’’ and ‘‘7(as) even for this simpleclassical problem of natural convection. Starting with highly asym-metric pattern at H = 1, these two branches gradually transforminto symmetrically patterned solution branch ‘‘2(s)’’ and ultimatelymerge with it upon a series of folding via turning point bifurca-tions. Other solution branches are isolated and except symmetricbranch ‘‘1(s)’’ return to their point of origin (H = 1) via turningpoint bifurcations restricting their presence within the range ofmultiplicity (Hcr < H 6 1). On the other hand symmetric branch‘‘1(s)’’ continues from H = 1 to H = 0 although exhibiting all itsfoldings for H > Hcr. It may be noted that branch ‘‘1(s)’’ is the mainsolution branch which exists over the entire range of 0 6H 6 1,while all other branches appear around it for H > Hcr. Turningpoint bifurcations of each of these solution branches lead to multi-ple steady flows within Hcr < H 6 1, where multiplicity of solu-tions vary with H. There exists 17 solutions at H = 1, whereas 25solutions were found for H = 0.97.

The new contribution of this work is to unfold all possible sevensteady solution branches by perturbation technique and trace indi-vidual branches in the parameter space of H. It is interesting tonote that the simple problem of natural convection within a poroussquare enclosure can lead to such rich bifurcation patterns basedon imposed thermal boundary conditions. In absence of such a rig-orous and systematic analysis, most of the solution branches willbe unexplored and most of the simulations will converge to onlymain solution branch ‘‘1(s)’’. It is also important to note that solv-ing the problem in half domain by exploiting symmetry of the de-fined problem will miss all the asymmetric branches and thus leadto the wrong conclusion about solution spectrum. The variety offlow structures around each branch also reveal complexity of suchproblem and emphasize the need for special attention to reveal allthe possible solution patterns for various thermal aspect ratio (H).Tracing of the entire spectrum of flow structures and correspond-ing temperature patterns around the branches also reveal widevariations of Nusselt numbers, which is of direct importance toengineering applications.

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