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Numerical Heat Transfer, Part A:Applications: An International Journal ofComputation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unht20

Mixed Convection Heat Transfer fromTandem Square Cylinders in a VerticalChannel at Low Reynolds NumbersDipankar Chatterjee aa Simulation and Modeling Laboratory, Central MechanicalEngineering Research Institute (Council of Scientific and IndustrialResearch), Durgapur, IndiaPublished online: 19 Nov 2010.

To cite this article: Dipankar Chatterjee (2010) Mixed Convection Heat Transfer from TandemSquare Cylinders in a Vertical Channel at Low Reynolds Numbers, Numerical Heat Transfer, PartA: Applications: An International Journal of Computation and Methodology, 58:9, 740-755, DOI:10.1080/10407782.2010.516703

To link to this article: http://dx.doi.org/10.1080/10407782.2010.516703

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MIXED CONVECTION HEAT TRANSFER FROMTANDEM SQUARE CYLINDERS IN A VERTICALCHANNEL AT LOW REYNOLDS NUMBERS

Dipankar ChatterjeeSimulation and Modeling Laboratory, Central Mechanical EngineeringResearch Institute (Council of Scientific and Industrial Research),Durgapur, India

The fluid flow and heat transfer characteristics around two isothermal square cylinders

arranged in a tandem configuration with respect to the incoming flow within an insulated

vertical channel at low Reynolds number range (1�Re� 30) are estimated in this article.

Spacing between the cylinders (S) is fixed at four widths of the cylinder dimension (d) and,

the blockage parameter (B) is set to 0.25. The buoyancy-aided/opposed convection is

examined for the Richardson number (Ri) ranges from �1 to 1 with a fixed Prandtl

number (Pr) of 0.7. The transient numerical simulation for this two-dimensional, incom-

pressible, laminar flow and heat transfer problem is carried out by a finite volume code

based on the PISO algorithm in a collocated grid system. The results suggest that the flow

remains steady for the entire range of parameters chosen in this study. The representative

streamlines, vorticity, and isotherm patterns are presented to interpret the flow and thermal

transport visualization. Additionally, the time average drag coefficient (CD) as well as time

and surface average Nusselt number (Nu) for the upstream and downstream cylinders

are determined to elucidate the effects of Re and Ri on flow and heat transfer phenomena.

INTRODUCTION

Analysis of hydrodynamics and thermal transport phenomena behind bluffobstacles such as circular=square cylinders at low Reynolds numbers (Re) has beena subject of considerable interest to the research community for several decades. Thetransport processes occurring behind the bluff bodies have tremendous engineeringimportance, since these processes often arise in numerous technological applicationssuch as heat exchangers, solar extraction systems, cooling towers, oil and gas pipe-lines, electronic cooling, and so on. When flow passes over multiple bodies, wakesform behind the bodies and a complex flow structure originates as a consequenceof the mutual interactions among them. Accordingly, the flow structure obtainedbehind multiple bluff bodies is significantly different from that obtained behind asingle obstacle. Furthermore, the wake interactions strongly depend on the configur-ation and the spacing of the bodies to the incoming flow as well as on the geometrical

Received 31 May 2010; accepted 31 July 2010.

Address correspondence to Dipankar Chatterjee, Simulation and Modeling Laboratory, Central

Mechanical Engineering Research Institute (Council of Scientific and Industrial Research), Durgapur –

713209, India. E-mail: d_chatterjee@cmeri.res.in

Numerical Heat Transfer, Part A, 58: 740–755, 2010

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2010.516703

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shapes (for example circular or square cross section) of the bodies. Consequently,the in-line, staggered or tandem arrangements of the obstacles offer different flowpatterns. It should be mentioned that these arrangements of the obstacles are oftenfollowed in compact heat exchangers. Again, the flow patterns and the wake struc-tures for the case of flow over square cylinders are considerably different from thatover circular cylinders because of the fact that unlike the circular cylinders the squarecylinders tend to fix the separation point, causing differences in the critical regimes.Furthermore, the separation mechanisms depending on the shedding frequencies andthe aerodynamic forces also differ significantly for these two geometries.

The flow becomes even more complicated when the wakes are further influ-enced by heat transfer. It should be recognized that for low to moderate Re flows,the buoyancy effect can significantly complicate the flow field, thereby affecting heattransfer characteristics. When the flow velocity is not very high but the temperaturedifference between the body and the fluid is significantly high, the heat transfer beha-vior is strongly influenced by thermal buoyancy. Thermal buoyancy plays a role ofparamount importance on estimating the wake behavior, since the wake structure isperturbed due to the superimposed thermal buoyancy resulting in hydrodynamicinstabilities even at small Re. The buoyancy parameter Ri¼Gr=Re2, popularlyknown as the Richardson number (with Gr being the Grashof number), providesa measure of the strength of free convection over the forced convection. The flowand thermal fields are completely dominated by free convection when Ri is suffi-ciently high (Ri>> 1). Forced convection characterizes the transport phenomenafor Ri<< 1 and both free and forced convections (i.e., mixed convection) are equallyimportant for Ri� 1. Buoyancy forces usually enhance the surface heat transfer ratewhen they aid the forced flow, whereas they impede the same when they opposethe forced flow. For the aiding case (flow past a heated body) the forced flow is in

NOMENCLATURE

B blockage parameter

CD drag coefficient

d cylinder size

g acceleration due to gravity

Gr Grashof number

h local heat transfer coefficient

H width of computational domain

k thermal conductivity of fluid

L length of computational domain

Ld downstream length

Lu upstream length

Nu local Nusselt number

n normal direction

p dimensionless pressure

Pr Prandtl number

Re Reynolds number

Ri Richardson number

S spacing between cylinders

t dimensionless time

T temperature

T1 free stream temperature

TW cylinder temperature

u1 free stream velocity

u dimensionless axial velocity

v dimensionless normal velocity

x dimensionless axial coordinate

y dimensionless normal coordinate

a thermal diffusivity of fluid

b coefficient of volume expansion

g kinematic viscosity of fluid

h dimensionless temperature

q density of fluid

Subscripts

W cylinder surface

1 free stream

Superscripts

� dimensional quantity

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the same direction as the buoyancy force, whereas for the opposing case (flow past acooled body) it is in the opposite direction.

Studies involving flow and heat transfer over a single bluff body or multiplebluff bodies (especially cylinders with circular or square cross sections) have been asubject of intense research in the past. Accordingly, a huge amount of pertinentexperimental and numerical works are available in the literatures which are notexplicitly mentioned here for the sake of brevity. In general, it is observed froma detailed survey that the studies related to the buoyancy-aided=opposed flowand heat transfer over multiple cylinders, especially with the square cross sectionare less frequent in the literature compared to the studies involving the circularcylinders. In the context of laminar mixed convection heat transfer aroundhorizontal cylinders in a vertical channel, Oosthuizen and Madan [1] studied exper-imentally the unsteady mixed convection for the range Re¼ 100�300. Merkin [2]reported that heating the circular cylinder delays separation and finally, the bound-ary layer does not separate at all. Jain and Lohar [3] found an increase in sheddingfrequency with a corresponding increase in the cylinder temperature. Farouk andGuceri [4] investigated numerically the laminar natural and mixed convection heattransfer around a heated circular cylinder placed within adiabatic channel walls.They also studied the effects of varying the ratio of width across the walls to cyl-inder diameter in the steady flow regime. Badr [5, 6] studied the laminar combinedconvection heat transfer from an isothermal horizontal circular cylinder for thetwo cases when the forced flow is directed either vertically upward (parallel flow)or vertically downward (contra flow) through the solution of the full vorticitytransport equation together with the stream function and energy equations. Thebuoyancy-aided (0�Ri� 4) steady convection heat transfer at low Re (¼20, 40,and 60) from a horizontal circular cylinder situated in a vertical adiabatic ducthas been studied numerically by Ho et al. [7] for the blockage parameter ofB¼ 0, 0.1667, 0.25, and 0.5. They observed a significant enhancement of the pureforced convection heat transfer due to the blockage effect as a result of placing thehorizontal cylinder in the vertical duct. Chang and Sa [8] investigated the phenom-enon of vortex shedding from a heated=cooled circular cylinder in the mixed con-vection regime and predicted the degeneration of purely periodic flows into asteady vortex pattern at a critical Richardson number of 0.15. Nakabe et al. [9]studied the effect of buoyancy on the channel-confined flow across a heated=cooledcooled circular cylinder with a parabolic inlet velocity profile via a finite differencemethod. The aiding, opposing, and cross buoyancy cases were considered atRe¼ 80 and 120 for �1�Ri� 1.6 and B¼ 0.15 and 0.3. The authors found threemain results: first, at constant Re, the value of Ri at which the vortex sheddingdegenerates decreases with increasing blockage ratio; second, at constant blockageratio, the value of Ri at which vortex shedding degenerates increases with Re; andthird, at constant Ri, the value of Re at which vortex shedding starts goes onincreasing with the increasing blockage ratio. By exploiting a finite elementmethod, Singh et al. [10] determined the flow field and temperature distributionaround a heated=cooled circular cylinder placed in an insulated vertical channel,with parabolic inlet velocity profile at Re¼ 100, �1�Ri� 1 and B¼ 0.25.They observed that the vortex shedding stopped completely at a critical Ri of0.15, below which the shedding of vortices into the stream was quite prominent.

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Patnaik et al. [11] studied numerically the influence of aiding and opposingbuoyancy on the flow past an isolated circular cylinder. They observed that atlow Reynolds numbers range (e.g., Re¼ 20–40), buoyancy opposing the flow couldtrigger vortex shedding. Saha [12] investigated the natural convection phenomenapast a square cylinder placed centrally within a vertical parallel plate channelnumerically using the marker and cell (MAC) method. The flow was foundunstable above the critical Grashof number (¼3� 104). The drag coefficient wasfound to decrease while the Strouhal and surface-averaged Nusselt numberswere seen to increase with Grashof number. The effect of buoyancy on the flowstructure and heat transfer characteristics of an isolated square cylinder in upwardcross flow was investigated numerically by Sharma and Eswaran [13] for Re¼ 100and Pr¼ 0.7. Like the circular cylinder case, the degeneration of the Karman vor-tex street was also observed to occur at a critical Ri of 0.15. In another article,Sharma and Eswaran [14] studied the effect of channel-confinement of variousdegrees (B¼ 0.1, 0.3, and 0.5) on the upward flow and heat transfer around a hea-ted=cooled square cylinder by considering the effect of aiding=opposing buoyancyat �1�Ri� 1, for Re¼ 100 and Pr¼ 0.7. They observed that with an increasein the blockage parameter, the minimum heating (critical Ri) required for thesuppression of vortex shedding decreases up to a certain blockage parameter(¼0.3), but thereafter increases. Singh et al. [15] performed a comprehensiveschlieren-interferometric study for the wakes behind heated circular and squarecylinders placed in a vertical test cell. A detailed dynamical characteristic of vorti-cal structures was reported in their study. The problem of the laminar upwardmixed convection heat transfer for thermally developing air flow in the entranceregion of a vertical circular cylinder under buoyancy effect and wall heat fluxboundary condition has been numerically investigated by Hussein and Yasin [16]through an implicit finite difference method and the Gauss elimination technique.The investigation covers Reynolds number range from 400 to 1,600 and the heatflux from 70W=m2 to 400W=m2. The results revealed that the secondary flow cre-ated by natural convection have a significant effect on the heat transfer process ashas also been shown by [17]. Ameziani et al. [18] have numerically investigated theproblem of natural convection in a vertical opened porous cylinder. The type offlows (with and without recirculation) depended on the filtration number, theaspect ratio and the Biot number. Mixed convection over an in-line bundle ofcylinders has been studied numerically by Gowda et al. [19, 20] for �1�Ri� 1.In a subsequent effort, the same authors [21] reported numerical results for the heattransfer and fluid flow over a row of in-line cylinders placed between two parallelplates. The effect of Ri on the flow and heat transfer was investigated in the workfor �5�Ri� 5 with Re¼ 50, 100 and Pr¼ 0.7. Lacroix and Carrier [22] presenteda numerical study of mixed convection heat transfer around two vertically sepa-rated horizontal cylinders within confining adiabatic walls. They investigated theeffect of the cylinder spacing, the distance between the plates, and Ri on the flowand heat transfer. Jue et al. [23] predicted numerically the heat transfer of transientmixed convective flow around three heated cylinders arranged in an isoscelesright-angled triangle between two vertical parallel plates. In a recent article,Gandikota et al. [24] studied the effect of thermal buoyancy on the upward flowand heat transfer characteristics around a heated=cooled circular cylinder for the

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range of parameters �0.5�Ri� 0.5, 50�Re� 150 and the blockage parameters ofB¼ 0.02 and 0.25. They observed that the vortex shedding frequencies (Strouhalnumbers) increase with increased heating and suddenly reduce to zero at the criticalRi. The critical Ri is again found to increase with Re for a particular blockageratio and the higher the blockage ratio, the smaller the critical Ri.

From the above critical evaluation of the pertinent available literature in thesubject area, it is obvious that although there are some results for the flow and heattransfer around circular cylinders placed in a vertical channel, the square counterpartis surprisingly less for the same configuration. Additionally, there is no reported workon the buoyancy-aided=opposed mixed convection heat transfer over tandem squarecylinders in a vertical configuration as frequently encountered in various chemicalengineering applications. Accordingly, the aim of the present work is to numericallyinvestigate the mixed convection heat transfer over two heated=cooled cylinders withsquare cross section arranged in a tandem fashion to the incoming flow in a verticaltwo-dimensional channel. The study will be conducted for the blockage parameter ofB¼ d=H¼ 0.25 in the Reynolds and Richardson numbers ranges of 1�Re� 30 and�1�Ri� 1 keeping the Prandtl number constant (Pr¼ 0.7).

DESCRIPTION OF THE PROBLEM, GOVERNING EQUATIONS,AND BOUNDARY CONDITIONS

The geometry of the problem considered in this study along with the coordi-nate system used is shown schematically in Figure 1 Two fixed identical squarecylinders with sides d are placed within a vertical channel in a tandem arrangementand are exposed to a uniform upward free stream with velocity u1 and temperatureT1. The cylinders are heated or cooled to a temperature TW and the channel wallsare considered adiabatic. The blockage ratio (B¼ d=H, where H is the width of thecomputational domain) is fixed at 0.25. The upstream and downstream distances ofthe computational domain are chosen as Lu¼ 8 d and Ld¼ 15d, respectively, and thespacing between the cylinders is set to S¼ 4d (refer to Figure 1). These values arechosen so as to reduce the effect of the inlet and outlet boundary conditions onthe flow patterns in the vicinity of the cylinders. Furthermore, these choices are alsoconsistent with the other contemporary studies available in the literature [8].

The dimensionless governing equations for this two-dimensional, laminar,incompressible flow with constant thermophysical properties along with Boussinesqapproximation and negligible viscous dissipation can be expressed in the followingconservative forms.

Continuity

quqx

þ qvqy

¼ 0 ð1Þ

Momentum

quqt

þ q uuð Þqx

þ q uvð Þqy

¼ � qpqx

þ 1

Re

q2uqx2

þ q2uqy2

!ð2aÞ

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qvqt

þ q uvð Þqx

þ q vvð Þqy

¼ � qpqy

þ 1

Re

q2vqx2

þ q2vqy2

!þRi h ð2bÞ

Energy

qhqt

þ q uhð Þqx

þ q vhð Þqy

¼ 1

RePr

q2hqx2

þ q2hqy2

!ð3Þ

where u, v are the dimensionless velocity components along x and y directions of aCartesian coordinate system, respectively, p and t are the dimensionless pressure andtime, Re (¼u1d=g) is the Reynolds number based on the cylinder dimension, Ri(¼Gr=Re2) is the Richardson number, Gr (¼gb(TW�T1)d3=g2) is the Grashofnumber with g and b are the gravitational acceleration and volumetric expansioncoefficient, h is the dimensionless temperature, and Pr¼g=a is the Prandtl number.The fluid properties are described by the density q, kinematic viscosity g, and

Figure 1. Schematic diagram of the computational domain.

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thermal diffusivity a. The dimensionless variables are defined as

u ¼ u

u1; v ¼ v

u1; x ¼ x

d; y ¼ y

d; p ¼ p

qu21; h ¼ T � T1

TW � T1; t ¼ u1t

d

ð4Þ

The boundary conditions of interest in this study are as follows. At the inlet (faceMP in Figure 1) a uniform flow of constant temperature is prescribed.

u ¼ 0; v ¼ 1; h ¼ 0 ð5Þ

It should be mentioned that a parabolic velocity profile at the inlet would be moreappropriate to ensure a fully developed flow before the upstream cylinder. Neverthe-less, the upstream length proves to be sufficient to obtain a fully developed velocityprofile before the flow reaches the upstream cylinder.

At the outlet (face NO in Figure 1) an outflow boundary condition is given, i.e.,

quqy

¼ qvqy

¼ qhqy

¼ 0 ð6Þ

Although this boundary condition is strictly valid only when the flow is fullydeveloped, its use in other flow conditions is also permissible for computational con-venience provided that the outlet boundary is located sufficiently far downstreamfrom the region of interest [6].

The channel walls (MN and OP in Figure 1) and the cylinder surfaces havebeen given the no-slip velocity boundary conditions, and the confining walls arealso assumed to be thermally insulated. Mathematically these can be expressed asChannel walls:

u ¼ v ¼ 0;qhqx

¼ 0 ð7Þ

Cylinder surfaces:

u ¼ v ¼ 0; h ¼ 1 ð8Þ

The flow is assumed to start impulsively from rest.The drag force acting on the cylinders includes the pressure and viscous drags

and can be obtained as

CD ¼ 2

Z 1

0

p dxþ 2

Re

Z 1

0

qvqx

dy ð9Þ

The heat transfer between the cylinders and the surrounding fluid is calculated by theNusselt number. The local Nusselt number based on the cylinder dimension is given by

Nu ¼ hd

k¼ � qh

qnð10Þ

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where h is the local heat transfer coefficient, k is the thermal conductivity of the fluid,and n is the direction normal to the cylinder surface. Surface average heat transfer ateach face of the cylinder is obtained by integrating the local Nusselt number along thecylinder face. The time average Nusselt number is computed by integrating the localvalue over a large time period.

METHOD OF SOLUTION

The conservation equations subjected to the aforementioned boundaryconditions are solved using a finite volume based method according to the PISOalgorithm in a collocated grid system. A second order upwind scheme for discretizingthe convective terms and a central difference scheme for the diffusive terms of themomentum and energy equations are used. The time discretization is carried outby a second order accurate fully implicit Adams-Bashforth scheme. The conditionsnecessary to prevent numerical oscillations are determined from the Courant-Friedrichs-Lewy (CFL) and the grid Fourier criteria. The final time step size is takenas the minimum of the two criteria mentioned above. Furthermore, the time stepsize is varied from 0.01 to 0.1 to determine an optimum value that results in lesscomputational time, but produces sufficiently accurate results. A dimensionlesstime step size of 0.05 is finally used in the computation satisfying all of theabove restrictions. A body-force-weighted pressure interpolation technique is usedto interpolate the face pressure from the cell center value. The discretized governingequations are finally solved by an algebric multigrid solver developed earlier for suchkind of flows [24–27]. The convergence criteria for the inner (time step) iterationsare set as 10�6 for all the discretized governing equations.

A nonuniform grid distribution having a close clustering of grid points in theregions of large gradients and coarser grids in the regions of low gradients is used.A grid size of 0.01 units clustered around the cylinders over a dimensionless dis-tance of 1 unit is used in the present simulation. A comprehensive grid sensitivitystudy is carried out to select the most economical mesh sizes for the problem underconsideration. Three nonuniform mesh sizes (50� 240, 80� 270, and 110� 300) areused to demonstrate the grid independence study for the highest values of thedimensionless parameters such as Re¼ 30 and Ri¼ 1. The percentage change inthe values of time average CD and time and surface average Nu for the coarsestgrid (50� 240) with respect to the finest grid (110� 300) are found to be 1.82%and 0.68%, respectively, while the same for the final two sets of grids are only0.32% and 0.3%. Accordingly, 80� 270 mesh size is preferred keeping in viewthe accuracy of the results and computational convenience in the simulations.All of the computations are carried out in an Intel Core 2 DuoTM (2.26GHz)workstation.

The accuracy of the present numerical code is tested extensively in our earlierworks [24–27] for the drag coefficient, dimensionless recirculation length, andStrouhal and Nusselt numbers for the fluid flow and forced convection heat transfer.Hence, no further attempt is done here to validate the present results since there areno experimental or numerical results reported in the literature for the physical prob-lem configuration, ranges of Reynolds and Richardson numbers and the boundaryconditions considered in the present study.

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RESULTS AND DISCUSSION

The objective of the present study is to investigate the influence ofaiding=opposing buoyancy on the overall flow patterns and the heat transfer char-acteristics for the upward flow past the tandem square cylinders at low Reynoldsnumbers. Furthermore, understanding of the variations of some important globalflow and heat transfer parameters such as drag coefficient and Nusselt numberunder these conditions are also an important agenda. The flow exhibits 2-D andsteady nature for the entire range of conditions considered in the present study.The 2-D behavior of the flow is unquestionable at the Reynolds number consideredhere, whereas the steady character has been confirmed by noticing that the dragcoefficient and transverse velocity component monitored at a location 3d down-stream of the upper cylinder are not changing appreciably with time (notshown for the purpose of brevity). With a dimensionless time step size of 0.05,the transient simulations for all the cases are performed over 5,000 time iterations(corresponding to a dimensionless time instant of 250) and results are obtained andpresented for visual appreciation.

The overall flow patterns and the heat transfer characteristics are presentedin Figures 2–4 in the form of streamline, vorticity contours, and isotherm plots atsome representative, Re and for different Ri. The negative Ri signifies that thebuoyancy force is in a direction opposite to that of the flow. At very low Re(e.g., Re¼ 5), for all Ri values, the streamlines are found to completely stick tothe walls of the cylinders and there is no flow separation (refer to Figure 2a).

Figure 2. Streamline plots at (a) Re¼ 5 and (b) Re¼ 30 for different Ri.

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Figure 4. Isotherm plots at (a) Re¼ 5 and (b) Re¼ 30 for different Ri.

Figure 3. Vorticity contours at (a) Re¼ 5 and (b) Re¼ 30 for different Ri.

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The vorticity contours under these conditions, as shown in Figure 3a, exhibit nosignificant longitudinal spread owing to the small inertia effect of the fluid parti-cles. There is no wake formation irrespective of the Ri at very low Re since theviscous force dominates over the inertia as well as the buoyant forces at smallerRe. The vorticity field is found to be high close to the sharp edges of the cylindersdue to the corner effects. As Re increases, the streamlines are no longer fullyattached to the walls of the cylinders and separation occurs which leads to the for-mation of wakes behind the cylinders, as depicted in Figure 2b. A pair of steadysymmetric vortices is found to form behind the cylinders at higher Re. Thesevortices show significant streamwise spread for both the cylinders, as depictedin Figure 3b, owing to the enhanced inertia effect at higher Re, which is in con-trast to the phenomena at low Re. It is also interesting to note from Figure 2bthat in the negative Richardson regimes (i.e., opposed buoyancy) at the vicinityof the cylinder, the inertia force is opposed by the buoyancy as well as the viscousforces causing an early separation. Consequently, the wakes become broaderalong the streamwise direction which makes the recirculation lengths behind thecylinders larger for negative Ri compared to that for positive Ri where the inertiaand buoyancy forces are in the same direction. The isotherms are having dif-fusion-type like profiles at low Re where there is no flow separation, whereas, theyspread more along the streamwise direction at higher Re due to higher inertiaeffect. It is worth noting from Figure 4a that the effect of aiding=opposing buoy-ancy on heat transfer at very small Re is not quiet prominent since the convectiveheat transfer is substantially small. However, at higher Re (refer to Figure 4b) dueto the counteracting effect of convective and thermal buoyancy for negative Ri,the thermal plume becomes wider along the transverse direction, which againbecomes narrower for positive Ri as a result of cumulative effect of convectionand buoyancy driven heat transfer. The isotherms are found to be crowded onthe front surfaces of the cylinders (preferably for Ri� 0) indicating higher heattransfer rates on the front surfaces as compared to the other surfaces of thecylinders and this crowding of isotherms increase with increase in Re, as expected.Furthermore, at very small Re, the isotherms are crowded more on the upstreamcylinder (with respect to the incoming flow) than the downstream one, whereas itreverses at higher Re. This establishes that the heat transfer at very small Re ismore from the upstream cylinder than the downstream one and the oppositeis true at higher Re.

To represent the effect of Re and Ri on the mean flow and heat transferquantities Figures 5 and 6 are plotted, which depict the variation of time averagedrag coefficient (CD) and time and surface average cylinder Nusselt numbers(Nu) with Re for different Ri. The CD decreases with increasing Re due to theformation of recirculation zones behind the cylinders at higher Re and attainsan almost constant value thereafter. The pressure and viscous forces reduce con-siderably as a result of flow separation in the recirculation zones at higher Rewhich is responsible for the corresponding reduction in drag. It is to be notedfrom Figure 5 that at very small Re, the Richardson numbers have almost noeffect on CD, whereas at comparatively higher Re, CD increases with Ri parti-cularly for the upstream cylinder. This is attributable once again by recallingthat at negative Ri there is a larger recirculation zone which reduces gradually

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with a subsequent increase in Ri. From Figure 6, it is evident that the time andsurface average Nu for all the cylinders increases with Re and Ri as expected.Furthermore, the downstream cylinder is always having higher Nu valuescompared to the upstream one as depicted in Figures 6a and 6b. This is quitenatural, since the heat flux is convected to the downstream by the fluid. Thevariations of time average local Nu on the cylinder surfaces are represented inFigures 7a and 7b for different Ri and at Re¼ 30. The heat transfer rate isthe highest on the front surface of the downstream cylinder for all Ri, whereas

Figure 5. Variation of time average CD with Re for (a) upstream and (b) downstream cylinders at

different Ri.

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it is so for Ri� 0 in case of upstream cylinder. A symmetric distribution of Nu isobserved on the front and rear surfaces with respect to the mid-longitudinalplane through the cylinders. The corresponding Nu is minimum at the midpointand maximum close to the corners. Since the heat transfer rate is closely relatedto the flow field, the local heat transfer rate is minimum where the velocity mag-nitudes are relatively small. On the vertical side faces of the cylinders, heat trans-fer decreases along the flow direction as the heat flux in the upstream isconvected to the downstream by the fluid.

Figure 6. Variation of time and surface average Nu with Re for (a) upstream and (b) downstream cylinders

at different Ri.

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CONCLUSION

Fluid flow and heat transfer from two heated=cooled square cylinders keptin a tandem fashion within a vertical confining space (blockage ratio, B¼ 0.25)in the two-dimensional steady laminar flow regimes are investigated in thiswork. The effects of aiding=opposing buoyancy on the flow and heat transferare extensively studied for the Reynolds and Richardson numbers ranges of1�Re� 30 and �1�Ri� 1 keeping a fixed Prandtl value (Pr¼ 0.7). A largerrecirculation zone is produced behind both the cylinders at relatively higherReynolds number under the buoyancy opposed conditions. However, this is

Figure 7. The time average local Nu on each surface of (a) upstream and (b) downstream cylinders at

different Ri.

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observed to reduce for the buoyancy-aided conditions. Accordingly, the heattransfer from the cylinders increases with increased heating which is evidencedfrom the increase of Nu with Ri. In general it is observed that CD decreasesand Nu increases with Re and the front surfaces of the cylinders are experiencingmore heat transfer compared to the other surfaces. Furthermore, it is observedthat the heat transfer for the downstream cylinder is always higher compared tothe upstream one.

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