numerical simulation of mixed convection within … · numerical simulation of mixed convection...

NUMERICAL SIMULATION OF MIXED CONVECTION WITHIN NANOFLUID-FILLEDCAVITIES WITH TWO ADJACENT MOVING WALLS

Mohammad Hemmat Esfe1, Ariyan Zare Ghadi2 and Mohammad Javad Noroozi11Department of Mechanical Engineering, Najaf Abad Branch, Islamic Azad University, Isfahan, Iran

2Department of Mechanical Engineering, Jouybar Branch, Islamic Azad University, Jouybar, IranE-mail: [email protected]

Received June 2012, Accepted August 2013No. 12-CSME-71, E.I.C. Accession 3391

ABSTRACTIn this study, nanofluid flow and heat transfer in a cavity with two moving lids are investigated. Governingequations are solved by finite volume approach using SIMPLE algorithm over a staggered gird system. Theresults show that when the moving lids have opposing effect, the streamlines contain two main vortices. Byincreasing the Richardson number, intensity of the vortex complying with buoyancy force increases, whileintensity of the other vortex decreases. When the moving lids have aiding effect, the streamlines containone the primary dominant vortex in which its strength increases with increase of the buoyancy force. In thiscase, rate of heat transfer is more than other cases.

Keywords: nanofluid; heat transfer; mix convection; double lid-driven cavity; numerical study.

SIMULATION NUMÉRIQUE DE LA CONVECTION MIXTE DANS DES NANO CAVITÉSREMPLIES DE LIQUIDE ET COMPORTANT DEUX PAROIS ADJACENTES EN MOUVEMENT

RÉSUMÉDans cette recherche, on s’intéresse à la circulation d’un nanofluide et au transfert de chaleur dans une cavitédont les deux couvercles sont mobiles. Pour résoudre les équations de contrôle, on se sert de la méthode desvolumes finis qui utilise l’algorithme SIMPLE dans un système alternatif. Ces résultats nous montrent deuxgrands mouvements rotatifs pour la circulation des fluides lorsque les couvercles mobiles se déplacent dansdes directions opposées. En augmentant le nombre de Richardson, l’intensité dans le mouvement, qui est enconformité avec la direction de la force propulsive, s’intensifie alors que l’intensité de l’autre est diminuée.Dans le cas où les couvercles mobiles se déplacent dans une même direction, la circulation est composéed’un mouvement rotatif principal dont l’intensité dépend de celle de la force propulsive. Ici, la chaleurtransférée est plus élevée que dans les autres cas.

Mots-clés : nanofluide ; transfert de chaleur ; convection mixte ; cavité double couvercle axée ; étude numé-rique.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4, 2013 1073

NOMENCLATURE

Cp specific heat at constant pressure (J/kg K)G gravitational acceleration (m/s2)H heat transfer coefficient (W/m2K)H cavity height (m)K thermal conductivity (W/m K)L cavity length (m)Nu local Nusselt number, hL/k fNum average Nusselt numberP pressure (Pa)P dimensionless pressure, pL2/ρn f α2

fPr Prandtl number, υ f /α fqw heat flux (W/m2)Ri Richardson number, (Ra/(Pr · Re2))T temperature (K)∆T Th−Tcu,ν velocity components in x,y direction (m/s)u0 characteristic velocityU,V dimensionless velocity, u/U0,v/U0x,y Cartesian coordinates (m)X ,Y dimensionless Cartesian coordinates, x/L,y/L

Greek symbolsα thermal diffusivity (m2/s)β thermal expansion coefficient (1/K)µ dynamic viscosity (kg/ms)ρ density (kg/m3)θ dimensionless temperature, (T −Tc)/(Th−Tc)ϕ solid volume fractionψ stream function

SubscriptsF pure fluidL differential lengthM meanN f nanofluid

1. INTRODUCTION

Analysis of mixed convective flow in a lid-driven cavity has some applications such as: materials processing[1], flow and heat transfer in solar ponds [2], crystal growing [3], float glass production [4], and coolingof electronic components, metal casting, food processing, galvanizing, and metal coating, among others.Fluid used in these applications is mostly water and ethylene glycol which has low heat transfer rate dueto its low thermal conductivity and does not meet the rising demand as an efficient heat transfer agent. Be-cause of a need to effective cooling and heating process, heat transfer enhancement in engineering is oneof the hottest topics in research. Employing nanofluids is an innovative method to augment heat transfer.Nanofluids are new type of heat transfer fluids containing a small quantity of nanosized particles (generallyless than 100 nm) that are uniformly and stably suspended in a liquid. These types of fluids with relativelyhigher thermal conductivities have received vast interest from researchers owing to their capability in im-provement of heat transfer. Many studies have been done on nanoparticles’ heat transfer inside cavity and

1074 Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4, 2013

effect of thermophysical properties such as thermal conductivity, dynamics viscosity and thermal expansioncoefficient.

In the following section, previous studies are reviewed. Khanafer et al. [5] investigated a 2-D enclosurefilled with nanoparticles. The nanofluid used in their study was water-copper. They observed that in aspecific Grashof number, increasing volume fraction of nanoparticles increases heat transfer rate. Anothernumerical study has been conducted by Jou et al. [6]. This study also confirms results of the previous study,which states that increasing the volume fraction of nanoparticles causes the heat transfer rate to increase.Oztop et al. [7] investigated the heat transfer in a cavity by adding different types of nanoparticles to thebase fluid. The results of this study show that increasing the volume fraction of nanoparticles increases theaverage Nusselt number. The above-mentioned studies are examples of studies that have been conducted toinvestigate natural convection inside cavities filled with nanofluids.

However, some studies are also done for the case of mixed convection inside cavities which occurs due tomoving lids as well as presence of buoyancy force. Muthtamilselvan et al. [8] studied a cavity with one mov-ing lid for different aspect ratios. Their findings include that the aspect ratio and nanoparticles concentrationhave great effect on heat transfer process and fluid flow. Talebi et al. [9] investigated combined convectionflow inside a lid-driven cavity filled with water-copper nanofluid. They observed that for a specific Reynoldsnumber, flow pattern and thermal behavior are completely dependent on the copper particles concentration.Abu-nada et al. [10] studied mixed convection flow inside a tilted lid-driven cavity. The results obtainedin their study demonstrate that increasing nanoparticles causes a considerable increase in the rate of heattransfer. Mahapatra et al. [11] investigated a cavity with two opposite moving lids. They studied effectof interaction of the lids for both cases of opposing or aiding movement of the lids. They also consideredeffect of radiation in their study. Hakan et al. [12] studied mixed convection flow inside a cavity with twoopposite moving lids in which they considered three different cases for movement of the lids. They inves-tigated effect of both opposing flow with the buoyancy force and the aiding flow with the buoyancy force.Oueslati et al. [13] studied aspect ratio in a 3-D cavity with two adjacent moving lids. Their results wereonly limited to forced convection. Recently, Sebdani et al. [14] have done a research on heat transfer insidea cavity when the two adjacent lids are moving. They chose the side walls to be the cold source and partof the lower wall to be the heat source. Their observations demonstrate that at a specific Reynolds numberand for high Rayleigh numbers, rate of heat transfer decreases by increasing nanoparticles. Furthermore,the results show that by increasing Rayleigh number, rate of decease in heat transfer also increases. Theyobtained the results based on variable viscosity and thermal conductivity models.

Previous studies confirm that lid-driven differentially heated cavities have various interesting applicationsin many fields. However, so far a study of cavities with two adjacent moving lids filled with nanofluid as theworking fluid in which the effect of direction of lids movement are also considered, has never been carriedout.

Depending on the problem, the interaction between forced convection and natural convection should beknown and thus in this study a mixed convection inside a cavity with two moving lids is investigated. Thenthe effect of parameters such as direction of movement of the lids and nanoparticles concentration in differentRichardson numbers, which includes forced convection-dominated and mixed convection-dominated on thestream pattern and thermal behavior, are analyzed.

2. MATHEMATICAL MODEL

Schematic diagram of the double lid-driven cavity for this study including boundary conditions and coordi-nate are shown in Fig. 1. The fluid is considered to be water–copper nanofluid. Size and form (shape) of theparticles are assumed to be uniform with a diameter equal to 47 nm.


Table 1. Thermophysical properties of water and copper.Property Water Coppercp 4179 383ρ 997.1 8954κ 0.6 400β 2.1×10−4 1.67×10−5

Particle diameter – 47 nm

Fig. 1. Schematic diagram of the double lid-driven cavity considered in the present study. Case 1: Upper wall movesto right and left wall moves to down. Case 2: Upper wall moves to left and left wall moves to up. Case 3: Upper wallmoves to right and left wall moves to up. Case 4: Upper wall moves to left and left wall moves to down.

The boundary conditions of problem are as follows: the right lid and the left moving lid are consideredto be insulated while the upper moving lid is kept at a low temperature. Also, heat sources are mountedon the lower lid. As shown in the figure, the left and the upper lids are moving with constant velocity andgravitation force is also acting downward.

The nanofluid is considered to be Newtonian and the flow is assumed to be laminar and incompressible.Also, the base fluid and the particles are in thermal equilibrium and no friction exists between them. Theonly body force acting on the system is buoyancy force which is included in momentum equation afterusing Boussinesq approximation. Thermophysical properties of water as the base fluid and properties of thecopper nanoparticles are tabulated in Table 1. The left and the upper lids are moving and four cases areconsidered for direction of their movement (as shown in Fig. 1).

For a steady, two-dimensional laminar and incompressible flow, the governing equations are

∂u∂x

+∂ν

∂y= 0, (1)

u∂u∂x

+ν∂u∂y

=− 1ρn f

∂ p∂x

+υn f ∇2u, (2)

u∂ν

∂x+ν

∂ν

∂y=− 1

ρn f

∂ p∂y

+υn f ∇2ν +

(ρβ )n f

ρn fg∆T, (3)


u∂T∂x

+ν∂T∂y

= αn f ∇2T. (4)

The following dimensionless parameters are introduced:

X =xL, Y =

yL, V =

ν

U0, U =

uU0

,

∆T = Th−Tc, θ =T −Tc

∆T, P =

pρn fU2

0. (5)

The Reynolds number and other parameters are

Re =ρ fU0L

µ f, Ri =

RaPr ·Re2 , Ra =

gB f ∆T L3

υ f α f, Pr =

υ f

α f. (6)

Equations (1–4) in dimensionless form are as follows:

∂U∂X

+∂V∂Y

= 0, (7)

U∂U∂X

+V∂U∂Y

=− ∂P∂X

+υnF

υ f

1Re·∇2U, (8)

U∂V∂X

+V∂V∂Y

=−∂P∂Y

+υn f

υ f·∇2V +Ri

βn f

β fθ , (9)

U∂θ

∂X+V

∂θ

∂Y=

αn f

α f∇

2θ . (10)

The general boundary conditions for all cases are as follows:{∂θ/∂X = 0, X = 0, 0 < Y < 1,

∂θ/∂X = 0, X = 1, 0 < Y < 1,

{U =V = 0, Y = 0, 0 < X < 1,

θ = 0, Y = 1, 0 < X < 1.

The special boundary conditions for each case are as follows:

Case 1 :

{U = 0, V =−1, X = 0, 0 < Y < 1

U = 1, V = 0, Y = 1, 0 < X < 1

Case 2 :

{U = 0, V = 1, X = 0, 0 < Y < 1

U =−1, V = 0, Y = 1, 0 < X < 1

Case 3 :

{U−0, V = 1, X = 0, 0 < Y < 1

U = 1, V = 0, Y = 1, 0 < X < 1

Case 4 :

{U = 0, V =−1, X = 0, 0 < Y < 1

U =−1, V = 0, Y = 1, 0 < X < 1.

The thermal diffusivity and effective density and other properties of the nanofluid are expressed by thefollowing relations:

αn f =kn f

(ρcp)n f, (11)


ρn f = ϕρs +(1−ϕ)ρ f . (12)

The heat capacitance and thermal expansion coefficient of the nanofluid can be given as

(ρcp)n f = ϕ(ρcp)s +(1−ϕ)(ρcp) f , (13)

(ρβ )n f = ϕ(ρβ )s +(1−ϕ)(ρβ ) f . (14)

The effective viscosity of the nanofluid which was proposed by Brinkman [15] is as follows:

µn f =µ f

(1−ϕ)2.5 . (15)

The effective thermal conductivity of the nanofluid is calculated by the Hamilton–Crosser model [16], whichis

kn f

k f=

ks +2k f −2ϕ(k f − ks)

ks +2k f +ϕ(k f − ks). (16)

The Nusselt number can be calculated asNu =

hLk f

, (17)

where heat transfer coefficient h is defined as

h =qw

Th−TC(18)

and the thermal conductivity may be expressed as

kn f =−qw

∂T/∂Y. (19)

The Nusselt number for hot wall can be written as

Nu =−(

kn f

k f

)(∂θ

∂x

). (20)

The average Nusselt number is calculated over hot surface by Eq. (18):

Num =1L

∫ L

0Nu dX . (21)

Also, for computing the stream function in the Cartesian coordinate system, we can use

u =∂ψ

∂yor ν =−∂ψ

∂x. (22)

3. NUMERICAL METHOD

The governing equations including continuity, momentum and energy equations associated with the bound-ary conditions are calculated numerically based on finite volume method and with staggered grid system.The SIMPLE algorithm introduced by Patankar [17] is implemented to solve coupled system of the govern-ing equations. The convection terms are approximated by a blend of central difference scheme and upwindscheme (hybrid-scheme) which results in a stable solution. Besides, a second-order central differencingscheme is served for the diffusion terms. The algebraic system arising from numerical discretization is


Table 2. Grid independence study for case 1 for a nanofluid with Ri = 1,ϕ = %2.Grid size Nuave11×11 2.505921×21 4.339931×31 4.5241×41 4.8351×51 5.0261×61 5.1171×71 5.230181×81 5.247991×91 5.2484101×101 5.2489

Table 3. Assessment of solutions for mixed convection in an enclosure.Re Ri Present study Ref. [19] Ref. [20] Ref. [21] Ref. [22] Ref. [23]1 100 1.00914 1.010134 1.00033 – – –100 0.01 2.0369 2.090837 2.03116 2.10 1.985 2.2400 0.000625 3.90 4.162057 4.02462 3.85 3.8785 4.01500 0.0004 4.5760 4.663689 4.52671 – – –1000 0.0001 6.3907 6.551615 6.48423 6.33 6.345 6.42

computed using Tridiagonal Matrix Algorithm (TDMA) [18]. The solution process is repeated until an ac-ceptable convergence criterion is reached. A FORTRAN computer code has been developed to solve theequations as described above.

Error =∑

Mj=1 ∑

Ni=1 |φn+1−φn|

∑Mj=1 ∑

Ni=1 |φn+1|

< 10−7.

Here, M and N refer to the number of grid points in x and y directions, respectively. N is the number ofiteration and Φ denotes any scalar quantity. To prove grid independence, numerical practice was executedfor nine different grid sizes, i.e. 21× 21,31× 31,41× 41,51× 51,61× 61,71× 71,81× 81,91× 91 and101× 101. Average Nusselt number for the bottom hot wall is attained for each mesh size as shown inTable 2. As can be seen, an 81×81 uniform grid size gives the accuracy needed in this work. All simulationsin this study have been performed using this grid sizes.

To validate our numerical approach, conditions of physical domain of present code was mimicked withthe conditions as invoked in [19–23]. Then the average Nusselt number for top heated moving lid extractedfrom our code was compared with the above-mentioned references, for different values of Reynolds andRichardson (Table 3). This comparison shows that our results are in high quality conformity with otherworks reported in the literature.

4. RESULTS AND DISCUSSION

In this study, fluid flow and heat transfer inside a 2-D cavity are investigated. The cavity contains twoadjacent moving lids. Four cases are considered for direction of the lids movement and the results areobtained based on the effect of direction of the lids movement on flow pattern and heat transfer mechanism.In this paper, the Richardson number is set to be between 0.001 and 10 and the Grashof number is consideredconstant and equal to 104. Nanoparticles concentration also varies in range of 0 to 0.06.


Figure 2 shows the results obtained for case 1 in the form of streamlines and isotherms. The solid linedenotes the pure fluid while the dashed line denotes the nanofluid with volume fraction of 0.06. For case 1,the upper moving lid moves to right and the left moving lid moves downward. These movements of thelids cause two primary vortices to be formed near the lids. According to the direction of the lids move-ment, the vortex which is formed by the force resulting from the movement of the upper lid is clockwiseand the vortex which is formed by the force resulting from movement of the left lid is counterclockwise.Richardson number in Figs. 2(a–b) is set to 0.001. This means that the force due to movement of the lidsdominates over the buoyancy force. Thus, the formed vortices are due to the shear forces and the domi-nant mechanism of heat transfer is forced convection. In addition, according to the figure, a small vortexis formed in the right corner of the cavity. However, another obvious point in the figure is the effect ofnanoparticles on the flow pattern. By increasing the nanoparticles, intensity of the vortices increases. Theisotherms for Richardson number of 0.001 are shown in the Fig. 2(b). The isotherms demonstrate that tem-perature gradient near the cold wall and particularly the hot wall is very high which shows high rate ofheat transfer in these areas. In fact, temperature distribution inside the cavity is highly dependent on theforced convection flow caused by movement of the lids. In Figs. 2(c–d), Richardson number is assumed tobe equal to unity. The same as the preceding case, streamlines contain two primary vortices being formednear the moving lids. A small vortex is also formed at lower right corner of the cavity. In this range ofRichardson number, the effect of buoyancy force increases in comparison to the previous case. Therefore,vortex formed by the shear force of the upper lid is reinforced by the buoyancy force, while the force createddue to the movement of the left lid decreases because its direction is opposite to the buoyancy force. Inthis figure, increase of strength of the vortex due to increase of nanoparticles is obvious. The heat transferin this case is more balanced in comparison to the previous case, thus temperature is distributed through-out the entire cavity. It can be understood from the figure that the temperature gradient on the lower wallis high which means there is a high heat transfer rate in this area. However, the temperature gradient isweaker compared to the previous case. Figures 2(e–f) shows the streamlines and the isotherms for Richard-son number equal to 10. In this range of Richardson number, the buoyancy force is the dominant forceacting on the system. Effect of the buoyancy force (which is the dominant force in this case) and the forcecaused by the upper lid (which is comply with the buoyancy effect) reinforce each other, thus a powerfulvortex is formed which occupies bulk of the cavity. Contrary to the upper vortex, strength of the left vortexdecreases in a way that it occupies a very small area near the left lid. The figure also demonstrates thatthe third vortex is completely disappeared. Using nanofluid instead of pure fluid increases strength of thevortices. The isotherms also mark a more uniform temperature distribution in comparison to the previouscase.

In case 2, the left lid moves upward and the direction of movement of the upper lid is leftward. This meansthat direction of movement of both lids is now opposed and as a result, forces caused by these movementsare opposite of each other.

In Figs. 3(a–b), Richardson number is low and the forced convection caused by movement of the lidsis dominant. Two primary vortices are formed inside the cavity and is caused by the shear forces of themoving lids. Besides, a small vortex is formed in lower section of the cavity. With increasing amount ofnanoparticles, flow pattern does not change and only strength of the vortices increases. The isotherms alsoshow a great gradient in the upper and the lower areas of the cavity which means there is high amount of heattransfer in the upper and lower sections. Figures 3(c–d) deal with mixed convection flow. Contrary to case1, force caused by movement of the left lid aids the buoyancy force, thus the left vortex is reinforced and theupper vortex is weakened. The isotherms also show temperature distributions throughout the entire cavity.In Fig. 3(e) since Richardson number is high enough, the buoyancy force is dominant and by augmenting theforce caused by movement of the left lid, the left vortex is reinforced and the upper vortex which moves in


Fig. 2. Stream lines and isotherms for case 1 at (a–b): Ri = 0.001, (c–d): Ri = 1, (e–f): Ri = 10.


Fig. 3. Stream lines and isotherms for case 2 at (a–b): Ri = 0.001, (c–d): Ri = 1, (e–f): Ri = 10.


opposite direction of the buoyancy force is weakened. The isotherms (Fig. 3f) are also completely dependenton the streamlines in a way that profile of temperature distributions inside the cavity is divided into twodistinct regions in which each region is located in an area of the formed vortices.

Figure 4 shows the results for case number 3. In this case, the left and upper lids move upward andrightward, respectively. This means that movements of the lids are in a way that support each other andagree with the buoyancy force. Thus in Fig. 4(a), a strong primary vortex is formed due to agreement of thetwo forces caused by the movement of the lids. Two small vortices are also formed in the lower section ofthe cavity. Figure 4(c) deals with mixed convection flow and thus a primary vortex caused by forced andnatural convection is formed. This figure also shows that small vortices disappear in this case. Temperaturedistribution inside the cavity is affected by both forced and natural convection. However, movement of thelids is in a way that reinforces the buoyancy force, thus a strong vortex is caused by the buoyancy force.Isotherms in this case show a uniform temperature distribution throughout the entire cavity.

In the last case, the upper lid moves leftward while the left lid moves downward. When forced convectionis dominant (Figs. 5a–b), a primary vortex is formed due to conformity of forces caused by the moving lids.Two small vortices are also formed in the right section of the cavity. In mixed convection flow (Figs. 5c–d), primary vortex is formed by the moving lids and the buoyancy force. In this case, small vortices aredisappeared and strength of the primary vortex increases. In Figs. 5(e–f), Richardson number is set to10 and the buoyancy force is dominant, thus a more powerful primary vortex than the previous cases isformed. Isotherms for case 4 for different flow regimes are also shown. It can be deduced from the figurethat increasing Richardson number decreases temperature gradient in the cold and hot walls, and for highRichardson numbers, a uniform temperature distribution forms inside the cavity.

The average Nusselt number diagrams for all the above discussed cases for different values of volumefraction and Richardson numbers of 0.001, 1 and 10 are shown in Fig. 6. The diagrams show that fordifferent Richardson number, increasing volume fraction of nanoparticles causes average Nusselt numberto increase in all cases. This can be a criterion to analyze rate of heat transfer in the system. Moreover,cases 4 and 3 have the maximum Nusselt number, respectively. This is because that forces caused by themoving lids amplify each other in these two cases. Cases 1 and 2 have lower Nusselt number in comparisonto cases 3 and 4, because the forces due to the moving lids are opposite in these cases. As shown in Figs. 2and 3, since the temperature gradient in case 1 is higher than that of case 2, the heat transfer rate is alsohigher for case 1. Figure 6 which is drawn for different values of Richardson number demonstrates thatincreasing Richardson number, which consequently increases the buoyancy force with respect to the shearforces, causes average Nusselt number and consequently heat transfer rate to decrease.

Figure 7 shows vertical velocity in mid-section of the cavity for all the four cases for volume fraction of0.02. In case 1 (Fig. 7a), there are velocity fluctuations in mid-section area of the cavity. In fact, this happensdue to presence of two oppositely-directed vortices. In the upper section of the cavity, velocity decreasesrapidly and then approaches to the lower section of the cavity with lesser gradient and finally tends to zeroat faster rate. For case 2 (Fig. 7b), in which the upper lid moves leftward and the left lid moves upward,there still exist some velocity fluctuations in mid-section of the cavity. First, velocity approaches to 0.5 athigh gradient and then decreases due to effect of the second vortex and finally tends to zero at high rate nearthe lower wall. For cases 3 and 4 (Figs. 7c–d), since effects of two moving lids reinforce each other andalso there is one primary vortex inside the entire cavity, velocity fluctuations are expected to be lower thanthe previous cases. In cases 3 and 4, quantity of velocity is expected to be maximum at the upper lid and todecrease linearly to zero at the lower lid. The results from the figures also confirm this trend.

Table 4 shows maximum stream function (ψmax) for cases 1 and 4. It is evident from the table that byadding the nanoparticles to the fluid, the strength of the vortices increases.


Fig. 4. Stream lines and Isotherms for case 3 at (a–b): Ri = 0.001, (c–d): Ri = 1, (e–f): Ri = 10.


Fig. 5. Stream lines and Isotherms for case 4 at (a–b): Ri = 0.001, (c–d): Ri = 1, (e–f): Ri = 10.


Fig. 6. Average for all cases Nu at (a): Ri = 0.001, (b): Ri = 1, (c): Ri = 10.


Fig. 7. U velocity profile in centerline of cavity at different Richardson numbers for (a): case 1, (b): case 2, (c): case 3,(d): case 4.

5. CONCLUSIONS

In this paper, mixed convection flow of water–copper nanofluid inside a cavity with two adjacent movinglids has been investigated. The following results have been obtained from an analysis of this study:

a. When movement of the lids are in a way that resulting forces augment each other, only one primaryvortex is formed and temperature lines distribute uniformly inside the cavity. Rate of heat transfer isalso higher than the condition when movement of the two lids counterbalances effect of each other.

b. When forces resulted from movement of the lids opposes each other, two primary vortices are formedinside the cavity. In this case, increasing Richardson number or in other word, increasing the buoyancyforce, causes the vortex complying with the buoyancy force to be reinforced and the other vortex isweakened.


Table 4. Vortices’ strength for cases 1 and 4 in different values of solid volume fraction.

c. By adding nanoparticles to the fluid, strength of the vortices increases in all cases. Also, by increasingvolume fraction of nanoparticles, average Nusselt number increases which also shows increase of heattransfer rate.

d. Velocity distribution in the mid-section of the cavity demonstrates that when effects of movement ofthe two lids agree with each other, velocity fluctuations are low in mid-section, while the effects areopposing with each other, there are more fluctuations in the mid-section. For all four cases, velocitygradient is high near the upper and lower walls.

The present theoretical effort is hoped to be helpful for the empirical works to examine combined con-vection heat transfer within a nanofluid-filled enclosure with more than one moving wall. Future work isrecommended to develop the current study for other nanofluids such as Al2O2-Water, TiO2-Water, ZnO-Water and comparing all findings with this paper. It would be also interesting to further study the fluidmotion and heat transfer inside a cavity with three or four moving walls for time-dependent flows.

REFERENCES

1. Jaluria, Y., “Fluid flow phenomena in materials processing – the 2000 Freeman Scholar Lecture”, Journal ofFluids Engineering, Vol. 123, No. 2, pp. 173–210, 2001.

2. Cha, C.K. and Jaluria, Y., “Recirculating mixed convection flow for energy extraction”, International. Journalof Heat and Mass Transfer, Vol. 27, No. 10, pp. 1801–1812, 1984.

3. Roy, B.N., Crystal Growth from Melts Applications to Growth of Groups 1 and 2 Crystals, Wiley, 1992.4. Pilkington, L.A.B., “Review lecture: the float glass process”, Proceedings of the Royal Society of London A,

Vol. 314, pp. 1–25, 1969.5. Khanafer, K., Vafai, K. and Lightstone, M., “Buoyancy driven heat transfer enhancement in a two-dimensional

enclosure utilizing nanofluids”, International Journal of Heat and Mass Transfer, Vol. 46, No. 19, pp. 3639–3653, 2003.

6. Jou, R.Y. and Tzeng, S.C., “Numerical research of nature convective heat transfer enhancement filled withnanofluids in rectangular enclosures”, International Communications in Heat and Mass Transfer, Vol. 33, No. 6,pp. 727–736, 2006.

7. Oztop, H.F. and Abu-Nada, E., “Numerical study of natural convection in partially heated rectangular enclosuresfilled with nanofluids”, International Journal of Heat and Fluid Flow, Vol. 29, No. 5, pp. 1326–1336, 2008.

8. Muthtamilselvan, M., Kandaswamy, P. and Lee, J., “Heat transfer enhancement of Copper-water nanofluidsin a lid-driven enclosure”, Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 6,pp. 1501–1510, 2010.

9. Talebi, F., Mahmoudi, A.H. and Shahi, M., “Numerical study of mixed convection flows in a square lid-drivencavity utilizing nanofluid”, International Communications in Heat and Mass Transfer, Vol. 37, No. 1, pp. 79–90,2010.

10. Oztop, H.F. and Abu-Nada, E., “Effects of inclination angle on natural convection in enclosures filled withCu-water nanofluid”, International Journal of Heat and Fluid Flow, Vol. 30, No. 4, pp. 669–678, 2009.


11. Mahapatra, S.K., Nanda, P. and Sarkar, A., “Interaction of mixed convection in two-sided lid driven differentiallyheated square enclosure with radiation in presence of participating medium”, Heat and Mass Transfer, Vol. 42,No. 8, pp. 739–757, 2006.

12. Oztop, H.F. and Dagtekin, I., “Mixed convection in two-sided lid-driven differentially heated square cavity”,International Journal of Heat and Mass Transfer, Vol. 47, No. 8–9, pp. 1761–1769, 2004.

13. Oueslati, F., Ben Beya, B. and Lili, T., “Aspect ratio effects on three-dimensional incompressible flow in a two-sided non-facing lid-driven parallelepiped cavity”, Comptes Rendus Mécanique, Vol. 339, No. 10, pp. 655–665,2011.

14. Sebdani, S., Mahmoodi, M. and Hashemi, S., “Effect of nanofluid variable properties on mixed convection in asquare cavity”, International Journal of Thermal Sciences, Vol. 52, pp. 112–126, 2012.

15. Brinkman, H.C., “The viscosity of concentrated suspensions and solutions”, Journal of Chemical Physics,Vol. 20, No. 4, pp. 571–581, 1952.

16. Hamilton, R.L. and Crosser, O.K, “Thermal conductivity of heterogeneous two component system”, Industrial& Engineering Chemistry Fundamentals, Vol. 1, No. 3, pp. 187–191, 1962.

17. Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Taylor and Fran-cis Group, 1980.

18. Versteeg, H.K. and Malalasekera, W., An Introduction to Computational Fluid Dynamics. The Finite VolumeMethod, John Wiley & Son Inc., 1995.

19. Abu-Nada, E. and Chamkha, Ali J., “Mixed convection flow in a lid-driven inclined square enclosure filled witha nanofluid”, European Journal of Mechanics B/Fluids, Vol. 29, No. 6, pp. 472–482, 2010.

20. Waheed, M.A., “Mixed convective heat transfer in rectangular enclosures driven by a continuously movinghorizontal plate”, International Journal of Heat and Mass Transfer, Vol. 52, No. 21–22, pp. 5055–5063, 2009.

21. Tiwari, R.K. and Das, M.K., “Heat transfer augmentation in a two-sided lid-driven differentially heated squarecavity utilizing nanofluids”, International Journal of Heat and Mass Transfer, Vol. 50, No. 9–10, pp. 2002–2018,2007.

22. Abdelkhalek, M.M., “Mixed convection in a square cavity by a perturbation technique”, Computational Materi-als Science, Vol. 42, No. 2, pp. 212–219, 2008.

23. Khanafer, K.M., Al-Amiri, A.M. and Pop, I, “Numerical simulation of unsteady mixed convection in a drivencavity, using an externally excited sliding lid”, European Journal of Mechanics B/Fluids, Vol. 26, No. 5, pp. 669–687, 2007.


numerical simulation of mixed convection within … · numerical simulation of mixed convection...

Documents