CONVECTION HEAT TRANSFER AROUND A SINGLE ROW OF CYLINDERS

Computational Thermal Sciences, 6 (6): 477–492 (2014)

CONVECTION HEAT TRANSFER AROUND A SINGLEROW OF CYLINDERS

A. L. F. Lima E Silva∗ & S. M. M. Lima E Silva

Federal University of Itajuba - UNIFEI, Institute of Mechanical Engineering - IEM, Prof. JoseRodrigues Seabra Campus, 1303 BPS Av., Pinheirinho District, 37500-903, Box-Postal 50, Ita-juba, Minas Gerais, Brazil

∗Address all correspondence to A. L. F. Lima E Silva E-mail: [email protected]

The convection heat transfer around circular cylinders has been studied to be applied in heat exchangers, heaters, elec-tronic circuits, and other thermal equipment. In these applications, it is necessary to analyze the influence of theheated cylinders in the fluid flow and vice versa. This paper presents a study about the Newtonian incompressibletwo-dimensional crossflow fluid around isothermal cylinders using the immersed boundary method (IBM) with the vir-tual physical model (VPM). The validation of the IBM–VPM homemade dynamic simulation code was performed fortwo particular cases. In case 1, the heat transfer effect in the flow around a group of three cylinders was studied in forcedand mixed convection. In case 2, different numbers of cylinders were arranged in a single-row configuration with differ-ent distances between them. Navier-Stokes and energy equations were discretized using the finite difference method forspace and a second-order Runge–Kutta method for time, and solved in a two-dimensional confined domain. Streamlines,isotherms, drag and lift coefficients, and Strouhal and Nusselt number are presented. The results confirmed the greatinfluence of the distances between the cylinders over the aerodynamic coefficients and over the Nusselt number and theability of the present numerical code to simulate flows over heated structures formed by any kind of immersed bodies.

KEY WORDS: numerical simulation, convection heat transfer, immersed boundary method, bank of cylin-ders

1. INTRODUCTION

The heat transfer phenomenon is presented in severalkinds of flows like external or confined flows. In addition,an immersed body can generate the formation and detach-ment of vortices, changing the flow dynamics. The classi-cal Von Karmam vortex street has been extensively stud-ied. These phenomena are simultaneously present, for ex-ample, in the anemometry field, where the sensor can af-fect the flow and interfere in the results. To minimize thisinfluence, the project of this instrument is assembled withthe previous knowledge of the aerodynamics and heattransfer coefficients. This class of flows is also presentin heat exchangers, risers of petroleum, cooling of elec-tronic components, and others. For this reason many ex-perimental, numerical, and theoretical studies have beenaccomplished.

A vast number of numerical papers may be found inliterature for the benchmark problem of isothermal flowswith a single cylinder. Some papers are based on classi-cal methodologies and have the main objective of gener-ating data to better understand the flow dynamics. Oth-ers proposed new numerical methodologies with the aimof improving the quality of the results, the efficiency,and the speed to obtain the results. Some classical pa-pers may be cited, like the work of Williamson (1996),who presented a bibliographical revision on flows aroundcircular cylinders, Saiki and Biringen (1996) used the im-mersed boundary method (IBM) to simulate steady androtating cylinders, and the work of Meneghini and Bear-man (1997) and Meneghini et al. (2001) showed the re-sults of the simulations with oscillating and side-by-sidecylinders and the configuration in tandem. Sumner et al.(2005) presented experimental results of the aerodynam-

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478 A.L.F. Lima E Silva & S.M.M. Lima E Silva

NOMENCLATURE

Cd drag coefficient Sh longitudinal distances betweenCd time-averaged drag coefficient cylinders centersd diameter Sv transversal distances betweenDij distribution function cylinders centers∆s distance between two Lagrangian points St Strouhal numberF Lagrangian vector force t nondimensional timef Eulerian vector force (IBM) U free stream velocity, m/sg distances between cylinders V vector velocityGr Grashof number x vector of Eulerian pointsLx length of the domain xk vector of Lagrangian pointsLy width of the domainn number of cylindersNu local Nusselt number Greek SymbolsNu average Nusselt numberPr Prandtl number,ν/α α thermal diffusivity, m2/sQ Lagrangian term of energy equation θ nondimensional temperatureq Eulerian term of energy equation (IBM) µ dynamic viscosity of fluid, Pa sRe Reynolds number,ρUd/µ ν kinematic viscosity of fluid, m2/sRi Richardson number ρ density, kg/m3

ics forces and vortex shedding frequency for two circu-lar cylinders with equal diameter disposed diagonally, forReynolds numbers of 3.2× 104 to 7.4× 104. Due to itsgreat practical applications, this class of flows is still be-ing studied. Lima E Silva (2002) and Lima E Silva et al.(2003) proposed the physical virtual model (VPM), whichis based on the momentum equations applied to a fluidparticle located over the fluid–solid interface. The valida-tions were carried out for a single cylinder and the heattransfer phenomenon was not considered.

Some papers deal with the flow around a bank ofgeometries like cylinders. Cylinderlike structures canbe found both alone and in groups, in heat exchang-ers, offshore structures, cables, and others. Ishigai andNishikawa (1975) presented an experiment for a single ar-ray of cylinder with different gap sizes between the cylin-ders. The authors observed two types of flows. In the first,the vortex shedding from two adjacent cylinders is 180◦

out of phase, and in the second behavior, the cylinderscloser to the wall shed vortices in phase whereas cylindersfarther from this side shed vortices out of phase. Kangand Su (2012) simulated using the incompressible lat-tice Boltzmann method. They analyzed flows around two,

three, and four equal circular cylinders in order to studythe physics of fluid–structure interaction of multicylindersin a crossflow and prove the accuracy of the methodology.The authors presented pressure distributions, time histo-ries of the drag and lift coefficients, and the power spec-trum of the fluctuating lift coefficient. Huang et al. (2006)used the commercial code Fluent to study flows arounda bank of cylinders in order to understand the dynamicsof the forming section of a paper machine. The authorssimulated laminar flow around two rows of unequal-sizedcylinders at low Reynolds number and velocity profiles.Lima E Silva et al. (2007) presented a study of the flowdynamics around two circular cylinders in tandem, sideby side, inV configuration and a composition of differ-ent geometries using the IBM-VPM. The results of dragand lift coefficients and Strouhal number were presentedwith the vorticity and pressure fields. The authors high-lighted the advantages of the IBM in using a Cartesiangrid to simulate flows over any geometric shape and theease of calculating the force coefficients for an isolatedbody or the global value for the configuration.

When the buoyancy effect is present, the vortex shed-ding becomes more complicated. Patnaik et al. (1999b)

Computational Thermal Sciences

Convection Heat Transfer around a Single Row of Cylinders 479

simulated flows over single cylinders with the buoyancyeffect. The aerodynamic coefficients, the vortex sheddingfrequency, and the mean Nusselt number as functions ofReynolds and Richardson were presented. Also in thework of Patnaik et al. (1999a), the authors reported theirresults of flows with single cylinders and cylinders in tan-dem, for Richardson numbers from−1 to 1. They pre-sented Strouhal number, and the drag and lift coefficientsas a function of Richardson number, vorticity, and tem-perature fields. Many other papers concerning forced andmixed convection with cylinders have been published bySharma and Eswaran (2005a,b). Some studies based onthe immersed boundary method with heat transfer havebeen outstanding in literature, since they take advantageof the Cartesian grid generation to simulate complex ge-ometries, as in Giacomello et al. (2006); Pacheco et al.(2005) and Pacheco-Vega et al. (2006). Studies concern-ing multiple bodies with heat transfer are relatively scarsein literature.

In the present work, the numerical investigations werecarried out to simulate flows over a group of heated cylin-ders to analyze their flow interaction and the thermal ef-fects. The flow is assumed to be two-dimensional andlaminar and the fluid is Newtonian and incompressible.The simulations were carried out in forced and mixedconvection in order to compare the effect of buoyancywhich is orthogonal to the flow direction. Drag and liftcoefficients, and Strouhal and Nusselt numbers were com-puted from the velocity, pressure, and temperature fields.In case 1 the cylinders were arranged in a triangular con-figuration in a confined channel for Re = 200, since thesame configuration can exist in very compact heat ex-changers. The vertical and horizontal distances betweenthe cylinders were varied, and the influence on the flowdynamics was analyzed. In case 2, a single-row simula-tion was done with the objective of analyzing the influ-ence of the gap between the cylinders and their distancesto the walls. The purpose of these studies was to clarifythe heat transfer and flow characteristics in both simulatedcases. These studies also contributed to the validation ofthe virtual physical model applied for the analysis of theconvection heat transfer phenomenon and the improve-ment of the homemade code that can be used for furtherstudies.

2. METHODOLOGY

Navier–Stokes equations for nonisothermal fluids and theimmersed boundary method with the Physical Virtual

Model (VPM) equations are written here considering thefollowing assumptions:

• The fluid is Newtonian and the flow is of an incom-pressible fluid.

• The immersed body (i.e., the cylinders) is describedby the VPM instead of by the direct imposition of theboundary conditions of velocity and temperature.

• The effect of the density variation due to temperaturechanges is considered only in the gravitational force(Boussinesq hypothesis).

• The energy generation term is neglected, becauseneither the effects of internal heat are considered (forexample, absorption or emission radiation) nor thehumidity which could be responsible for the latentheat exchange.

• There are no Coriolis force nor rotation effects of thecoordinate system.

The equations of mass conservation, momentum, andenergy (temperature) can be written in a nondimensionalform as

∇ · V = 0, (1)

∂V

∂t+∇ · (V V )=−1

ρ∇P+

1

Re∇2V +Riθ+f (2)

∂θ

∂t+∇ · (V θ) =

(1

Re Pr

)∇2θ+ q, (3)

where V , P , and θ are the nondimensional veloc-ity fields, pressure, and temperature, respectively. Therelevant nondimensional parameters are Reynolds (Re),Prandtl (Pr), and Richardson (Ri = Gr/Re2). The dis-cretized forms of the transport equations, Eqs. (1), (2),and (3) are derivated by using the finite difference methodwith second-order Runge-Kutta for the momentum equa-tions and the second-order Adams Basforth for the tem-perature equation. The fractional step method presentedby Armfield and Street (1999) is applied as a pressure cor-rection.

Any kind of immersed body is represented by a La-grangian grid that can be mobile or deformable. A La-grangian point coordinatexk is represented in Fig. 1 to-gether with a Eulerian point.

The Eulerian forcef added to the momentum equa-tions [immersed boundary method, Lima E Silva (2002)],is given by

f(x, t) =∑

F (xk, t)Dij(xk)(x− xk)∆s2. (4)

Volume 6, Number 6, 2014


FIG. 1: Eulerian and Lagrangian grid representation(Lima E Silva, 2002)

Dij(xk) is the interpolation function,xk = (Xk, Yk)are the Lagrangian points, over the interface,∆s isthe distance between two Lagrangian points andx =(Xi, Yj) are the Eulerian points.F (xk, t) is the La-grangian force density, modeled by the VPM presentedbelow. f(x, t) is the Eulerian force, which is differentfrom zero only over the interface. Equation (4) representsthe fluid–solid interaction.

Analogously,q has the function to model the tempera-ture of the cylinder and is given by

q(x, t) =∑

Q(xk, t)Dij(xk)(x− xk)∆s2, (5)

whereQ is calculated based on the temperature valuesinterpolated from the Eulerian grid. In the present work,the interpolations of the variables from and to the Euleriangrid are done with the distribution function suggested byPeskin and McQueen (1994) as

Dij(xk) =g[(Xk −Xi)/h][(Yk − Yj)/h]

h2, (6)

g(r) =

g1(r) if | r |< 1,1/2− f1(2− | r |) if 1 <| r |< 2,0 if | r |> 2,

(7)

where

g1(r) =3− 2 | r | +

√1 + 4 | r | −4 | r |28

, (8)

r is equal to(Xk − Xi)/h, or (Yk − Yj)/h andh is theEulerian grid length. The VPM proposed by Lima E Silva

et al. (2003) is used to calculate the Lagrangian force,F (xk, t):

F (xk, t) =∂V

∂t(xk, t) + (V ·∇)V (xk, t)

− 1

Re∇2V +

1

ρ∇p(xk, t). (9)

The terms on the right-hand side of Eq. (9) are obtainedfrom the Eulerian field interpolation. The same procedureis done forQ, calculated as

Q(xk, t) =∂θ

∂t(xk, t) +∇ · [V (xk, t)θ(xk, t)]

− 1

Re Pr∇2θ(xk, t). (10)

After calculatingQ(xk, t) over the Lagrangian grid,the Eulerian variableq(x, t), as in Eq. (5), is obtained tobe used in Eq. (3) for the calculation of the new temper-ature field. The distribution and interpolation proceduresare detailed in Lima E Silva (2002) and Lima E Silva et al.(2003).

3. NUMERICAL RESULTS

Two different configurations were studied with a numberof geometries (n) varying from 3 to 10. In case 1, thedynamic and the thermal field were analyzed in a triangu-lar configuration in forced and mixed convection. In case2, a bank of cylinders disposed side by side was simu-lated only in forced convection. For both configurations,the no-slip condition was applied to the channel walls.The results of drag and lift coefficients and Nusselt andStrouhal numbers are presented for both cases.

Before the presentation of both configurations, the nextsection shows the results with a test case of a single heatedcylinder immersed in an infinite channel. This case wasperformed for the numerical validation of the methodol-ogy and for the discussions about the numerical parame-ters and grid resolution chosen.

3.1 Test Case – A Single Heated Cylinder

A benchmark test frequently used for comparison is theflow over a single cylinder. This geometry appears indifferent practical problems. In order to validade theIBM/VPM code, the present simulations with a singleheated cylinder were compared with the literature. Thedomain has a length ofLx = 55d and width ofLy = 30dto reproduce the free boundary condition. The cylinder



was placed atx = 16d andy = 15d in order to minimizethe influence of the boundaries; the mesh resolution forthese simulation was 318×164, which was based on pre-vious studies presented by Lima E Silva et al. (2003) andLima E Silva et al. (2007).

Table 1 shows the average values of drag coefficientsand Nusselt numbers for Reynolds 100 using differentnumbers of cells inside the cylinder. For all the simula-tions presented here, the cylinder hasd = 1 and at least30 cells inside its surface. The values of drag varied lessthan 0.2% of the coaster case to the more refined case.The variation of the Nusselt number was higher when thenumber of cells inside the cylinder increased from 20 to30, but after this value the Nusselt variation was less than2%.

The tests were carried out for Reynolds numbers be-tween 47≤ Re≤ 250 and a constant Prandtl number of0.71. The nondimensional temperature of the cylinder isθ = 1, and the fluid is atθ = 0 at the beginning of thesimulation. The results of drag coefficient, and Strouhaland Nusselt numbers are presented. The local Nusseltnumber (Nu) is defined as−∂θ/∂n, wheren is the cylin-der surface normal direction. The average Nusselt number(Nu) was calculated by averaging the local Nusselt num-ber over the surface of the cylinder and averaging overtime. The Strouhal number defined as St =fd/U∞ wasobtained by using the fast Fourier transform of the lift co-efficient and represents the nondimensional frequency inwhich the vortex shedding occurs.

Table 2 presents the drag coefficient and Strouhal num-ber. Good results were obtained for the range of simulatedReynolds numbers with differences at less than 4%.

The local Nusselt number results are plotted inFig. 2(a) as a function of the angle over the cylinder sur-face (θ) for Reynolds number 200. The patterns are sim-ilar for all authors, with the largest Nusselt number lo-cated at the front stagnation point of the cylinder and theminimum value occurring at about 130◦, near the flow

TABLE 1: Average drag coefficient and average Nusseltnumber for a single heated cylinder and Reynolds 100 fordifferent mesh refinement

Cells inside Cd Nu Total of cells20 1.3693 4.2382 35,640

30 1.3665 4.6067 52,152

40 1.3664 4.6563 67,116

50 1.3664 4.7286 84,316

TABLE 2: Average drag coefficient and Strouhal numberfor a single heated cylinder

Author Reynolds Cd St47 1.46 0.12

80 1.40 0.15

Present 100 1.37 0.16

work 200 1.35 0.19

250 1.34 0.21

Braza et al. (1986) 100 1.36 0.16

200 1.40 0.20

Ding et al. (2007) 100 1.356 0.166

200 1.348 0.196

Liu et al. (1998) 100 1.35 0.164

200 1.31 0.192

separation point. Then the Nusselt number increases con-tinuously to the rear stagnation point. The present resultunderpredicts the Nusselt number on the front portion ofthe surface compared with the results of Momose and Ki-moto (1999), but compares better with the results of Pat-naik et al. (1999b) and Zang et al. (2008).

The average Nusselt number results as a function ofReynolds, obtained with the present study, are plotted inFig. 2(b). The results are also compared with the follow-ing correlations available in the literature:Zhuauskas (1972) correlation:

Nu = 0.51Re0.5 (11)

Knudsen and Katz (1958):

Nu = 0.683Re0.466Pr1/3 (12)

Churchill and Bernstein (1977) correlation:

Nu=0.3+0.62Re1/2Pr1/3

[1+(0.4/Pr)2/3]1/4

[1+

(Re

282000

)5/8]4/5(13)

It is interesting to observe that the time-averagedNusselt number increased approximately 10% when theReynolds number increased from 80 to 100, for the au-thors presented in Fig. 2. The increase of Nusselt numberwas by 28% when compared to the values obtained forReynolds values of 100 and 200. The time-averaged Nus-selt number increases as the Reynolds number increasesdue to the higher flow velocity and the presence of thevortex shedding, which increases the heat transfer fromthe cylinder surface.



(a) (b)

FIG. 2: Local Nusselt number for Re = 200(a) and time-averaged Nusselt number as a function of Reynolds number(b)

3.2 Case 1 – Triangular Configuration

In the triangular configuration, the cylinders are consid-ered to be at uniform temperature (θ = 1) and the channelwalls are at the same inlet fluid temperature (θ = 0). Fig-ure 3 shows the two-dimensional domain and the bound-ary conditions for this configuration. For these simula-tions the following values were chosen: length and widthof the domainLx = 6d andLy = 20d, horizontal andvertical distances between the cylinders,Sh = 2.5d, 3dandSv = 1.25d, 1.5d whered is the cylinder’s diameter.The channel was simulated on the horizontal position, butall the figures are presented in a vertical disposition forbetter visualization. The channel has solid walls on leftand right faces and the inner flow has a uniform profile. Anonuniform grid of 274× 171 points was used in a uni-form region near the cylinders, as shown in closer viewin Fig. 3. The simulation was carried out for Reynolds at200, Prandtl at 0.7, and for the cases of forced convectionRi = 0.0 and mixed convection Ri = 2.0 and 5.0.

Figure 4(a) shows the isotherms forSh = 2.5d andSv = 1.25d with Ri = 0 (or null Grashof number) (left),Ri = 2 (middle), and Ri = 5 (right). The buoyancy effectcan change the flow pattern, as shown in the two first fig-ures. For the small domain the symmetry is lost even forRi = 0. Therefore the nonsymmetric flow patterns on thefront stagnation point of cylinder 1 exist, and there is a re-

FIG. 3: Computational domain and boundary conditionsfor case 1



(a)Sh = 2.5d andSv = 1.25d

(b) Sh = 3d andSv = 1.5d

FIG. 4: Instantaneous isotherms for the triangular configuration with Re = 200 and three different Ri values:(a)Sh =2.5d andSv = 1.25d, (b) Sh = 3d andSv = 1.5d



versal that includes two vortices along the downstream lo-cation near the right channel wall, as can be seen in Fig. 5with the streamlines. For the bigger domain 4(b) the flowis symmetric (Ri = 0).

Other simulations with forced convection and differ-ent values ofSh andSv were also performed, and the re-sults showed the high influence of these parameters on thetemperature fields and aerodynamic coefficients. It was

(a)Sh = 2.5d andSv = 1.25d

(b) Sh = 3d andSv = 1.5d

FIG. 5: Instantaneous streamlines for the triangular configuration with Re = 200 and three different Ri values:(a) Sh = 2.5d andSv = 1.25d, (b) Sh = 3d andSv = 1.5d



observed that as the distance between the cylinders in-creases, the vortex shedding starts earlier and the flowsymmetry is lost. For small values ofSh andSv (Sh = 2andSv = 2), the vortices behind cylinders 2 and 3 are big-ger and the flow develops as though the cylinders were asingle structure. There is a great influence of the vorticesof cylinders 1 on cylinders 2 and 3 whenSv is increased.The coefficients are also influenced by the flow dynam-ics. For a comparison between Figs. 6 and 7, the ampli-tude variation of the drag coefficient increased in cylinder3 and decreased in cylinders 2 and 3 in the mixed convec-tion cases.

Table 3 presents the results of the drag and liftcoefficients, and Strouhal and Nusselt numbers for eachcylinder in each simulation. In the simulation withforced convection (Ri = 0), the flow is symmetric when

the distance between the cylinders is higher and eachcylinder has its own vortex street. Cylinders 2 and3 present the same values of the drag and lift coef-ficients, and Strouhal and Nusselt numbers, as expected.For the other two simulations with mixed convection, theflow is not symmetric on the front stagnation point, be-cause the direction of the buoyancy effect is orthogonalto the flow direction. The Nusselt number is higher formixed convection than for forced convection. It is ob-served that the buoyancy effect makes the flow patternsbehind the cylinders towards the left wall. The drag coef-ficient values for cylinder 1 are smaller than for cylin-ders 2 and 3 because they are closer to the wall andhave obvious vortex shedding. The exception is when theflow is reversal for Ri = 5 and the drag coefficient dec-reases.

(a) Ri = 0 (b) Ri = 2

(c) Ri = 5

FIG. 6: Temporal evolution of drag coefficient forSh = 2.5d andSv = 1.25d



(a) Ri = 0 (b) Ri = 2

(c) Ri = 5

FIG. 7: Temporal evolution of drag coefficient forSh = 3d andSv = 1.5d

TABLE 3: Drag and lift coefficients, and Strouhal and Nusselt numbers for the three cylinders

Sh = 1.25 andSv = 2.5Cylinder Ri Cd Cl St Nu

10 2.6 10−4 0.3 8.32 2.5 0.01 0.4 8.35 6.1 5.0 0.5 9.9

20 2.7 0.3 0.3 8.92 4.2 2.9 0.4 10.15 8.3 6.6 0.5 11.4

30 2.7 −0.3 0.3 6.12 1.9 1.5 0.3 8.35 1.5 5.4 0.5 7.4

Sh = 1.5 andSv = 3.0Cylinder Ri Cd Cl St Nu

10 2.4 10−6 0.3 5.62 2.3 0.3 0.4 7.15 5.8 5.4 0.5 10.3

20 2.9 0.3 0.3 6.22 4.5 2.3 0.4 8.45 7.3 5.7 0.5 11.5

30 2.9 −0.3 0.3 6.22 1.9 1.6 0.3 7.15 1.9 4.9 0.5 7.1



3.3 Case 2 – Single Row of Cylinders in ForcedConvection

This case consists of a bank of cylinders disposed side byside and confined in a two-dimensional channel, as pre-sented in Fig. 8. The domain isLx = 21d, Ly = 40d.The simulations were carried out by varying the numberof cylinders, the distances between them (g), and the dis-tances to the wall (h). For this configuration, only theforced convection was considered. It is expected that amore uniform flow distribution results in a higher overallNusset number and an improved efficiency, for example,if a heat transfer exchanger is being studied.

For a single row of cylinders of equal diameter, differ-ent regimes were observed when the gap between cylin-ders and their distances to the wall were changed. The

FIG. 8: Computational domain for case 2

Reynolds number was 150 for all simulations in case 2and the grid was also refined near the cylinders. Figures 9and 10 show the vorticity fields for some instants of time.The interaction between the vortex streets of the cylin-ders increases when the distances decrease. For values ofg ≤ 1.5d the flow between the cylinders is strongly de-viated from the free-stream direction. The pressure up-stream of the cylinders increases and and the velocity ishigher near the walls. For these simulations, the range ofthe lateral distance to the wall was0.4d ≤ h ≤ 6.25d.The distance to the wall also contributed to the flow pat-tern. The two inner cylinders tend to shed out-of-phasevortices, whereas other cylinders shed in-phase vortices,as observed when the gap between the lateral wall andcylinders ish ≥ 1d. The two central cylinders are gen-erally in a separate group, showing the biased flow pat-tern seen in the flow around two side-by-side cylinders,as observed by Huang et al. (2006). Our simulations alsoshowed that small values of the gap to the lateral wall(h < 1d) may trigger the flow of the last cylinder to flipfrom in-phase to antiphase vortex shedding with its group.When the number of cylinder was increased to (n ≥ 8)the number of cylinders in a group, shedding vortices inphase varied. The coupling within each cylinder in a clus-ter is so strong that all wakes therein move in-phase. Fig-ure 11 shows the temperature field for three simulationswith nine cylinders where the cylinders shedding vortexin-phase are highlighted.

Table 4 shows the drag coefficient and Nusselt numberfor the cylinder near the right wall, wheren is the numberof geometries,g the distance between the centers of thetwo adjacent cylinders, andh the distance of the first andlast cylinder’s centers to the wall. It is possible to note theinfluence of the distance of the cylinders and the gap to thewall with these values. The drag coefficient decreases asg increases for all numbers of cylinders. For the same dis-tance between two cylinders (same value ofg), the dragof the last cylinder on the right increases as the numberof geometries increases because the value ofh is smallerand the influence of the wall is bigger. The average dragcoefficient is higher for the configuration with the highestnumber of cylinders. The central cylinder tends to havethe highest drag coefficient, but this difference decreaseswhenh becomes smaller.

Nusselt number presents the same behavior for all sim-ulations, except forn = 10 andg = 1.2. In this case,the group of cylinders is similar to a single body. Nus-selt number increases when the gap between the cylindersdecreases to 1.5. Below this value (g ≤ 1.2), the Nus-selt value decreases for all cylinders. This behavior is ex-



(a) g = 1.5d (b) g = 2d

(c) g = 2.5d (d) g = 3d

FIG. 9: Vorticity field for six cylinders (n = 6) in a single row with different values ofg and Re = 150



(a) g = 1.5d (b) g = 2d

(c) g = 2.1d (d) g = 2.4d

FIG. 10: Vorticity field for nine cylinders (n = 9) in a single row with different values ofg and Re = 150



(a) g = 2d (b) g = 2.1d (c) g = 2.4d

FIG. 11: Temperature field for nine cylinders in a single row with different values ofg and Re = 150. Highlightedcylinders are in phase

pected because the convective heat transfer coefficient de-creases when the bank forms a single set.

Figure 12 shows the Nusselt number for a centralcylinder as a function of gapg. It is possible to observe amaximum value close tog = 1.5 for all simulations. Thisis probably the optimum configuration to maximize theheat transfer when using the described domain.

4. CONCLUSIONS

The immersed boundary method with the virtual phys-ical model was used to simulate incompressible two-dimensional flows on a bank of heated cylinders. In or-der to validate the IBM-VPM code, a test case of a sin-gle heated cylinder was first presented. The numerical re-sults of two configurations of a group of cylinders werechosen. The time variations of the aerodynamic coeffi-cients and the Nusselt number are easily obtained withthis methodology because the relevant parameters are ob-

tained directly from the Eulerian field, even when a com-plex geometry is presented. It is very simple to calculatedrag and lift coefficients, for example, over any kind ofgeometry when using this methodology. The wake struc-ture and the interaction between the cylinders were stud-ied for different spacings between the cylinders. For thetriangular configuration, cylinders 2 and 3 are arrangeddownstream of cylinder 1. For the small gap (Sv = 2d andSh = 1.25d), the vortices downstream of the cylindershave a strong interaction with each other and cannot beseparately distinguished. By increasing longitudinal andtransversal distances, the drag coefficient decreases forcylinder 1 but increases for cylinders 2 and 3. This be-havior was noted for Ri = 0 and Ri = 2. For Ri = 5 thedrag tends to increase for three cylinders due to the rever-sal flow. The general behavior for the Nusselt number isto decrease when the cylinders are farther apart.

In mixed convection, the flow interactions between thecircular cylinders and the channel walls are quite differ-



TABLE 4: Drag coefficient and Nusselt number for thelast cylinder on the right

n g/d h/d Cd Nu6 2.0 5.0 2.7 4.76 2.5 3.75 2.3 4.56 3.0 2.5 2.1 4.46 3.5 1.25 2.1 4.46 3.7 0.75 2.1 4.4

7 1.2 6.4 2.9 3.77 2.0 4.0 3.0 4.77 2.5 2.5 2.5 4.67 3.0 1.0 2.4 4.67 3.2 0.4 2.3 4.4

8 1.5 4.75 3.6 5.18 2.0 3.0 3.1 4.88 2.5 1.25 2.7 4.78 2.7 0.55 2.7 4.7

9 1.2 5.2 4.2 4.29 1.5 4.0 3.7 5.29 2.0 2.0 3.3 4.99 2.1 1.6 3.3 4.99 2.2 1.2 3.2 4.8

10 1.2 4.6 5.0 4.410 1.5 3.25 4.4 5.410 1.7 2.35 4.1 5.110 2.0 1.0 3.9 5.0

FIG. 12: Average Nusselt number as a function of gapfor all simulations (Re = 150) of case 2

ent from those in forced convection. Therefore, the av-erage time, Nusselt number, and the drag and lift coeffi-cients are also different. The flow dynamics and isotherms

are symmetric on the front stagnation point only in forcedconvection. In mixed convection, the effect of buoyancygenerates an inclined flow and some vortices along theleft wall of the channel. In general the drag coefficientand Nusselt number increase for mixed convection exceptfor cylinder 3 and Ri = 5. In this case, a big recirculationbubble forms next to the wall, maybe due to the proximitywith the channel inlet.

For the simulations with a row of cylinders in a forcedconvection flow, the results showed the great influence ofthe walls as the number of cylinders increased. Varyingthe gap between the lateral wall and cylinders can changethe flow from out-of-phase to in-phase vortex sheddingand it may be observed that groups with different numbersof cylinders are formed.

The developed code based on the immersed bound-ary method with the virtual physical model was able toreplicate the overall flow characteristics. A good agree-ment with the aerodynamic coefficients and nondimen-sional numbers was presented.

ACKNOWLEDGMENTS

The authors would like to acknowledge financial sup-port from the Minas Gerais State Research Foundation(FAPEMIG), National Council for Scientific and Techno-logical Development (CNPq), and the Federal Universityof Itajuba-MG (UNIFEI).

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CONVECTION HEAT TRANSFER AROUND A SINGLE ROW OF CYLINDERS

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