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The Effect of Aiding/Opposing Buoyancyon Two-Dimensional Laminar Flow Acrossa Circular CylinderGurunath Gandikota a , Sakir Amiroudine b , Dipankar Chatterjee c &Gautam Biswas da LPMI, Arts et Métiers Paris Tech, Angers, Franceb Laboratoire TREFLE UMR CNRS, Talence, Francec Simulation & Modeling Laboratory, Central Mechanical EngineeringResearch Institute, (Council of Scientific & Industrial Research),Durgapur, Indiad Indian Institute of Technology Kanpur, Kanpur, India

Version of record first published: 09 Sep 2010.

To cite this article: Gurunath Gandikota, Sakir Amiroudine, Dipankar Chatterjee & Gautam Biswas(2010): The Effect of Aiding/Opposing Buoyancy on Two-Dimensional Laminar Flow Across a CircularCylinder, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation andMethodology, 58:5, 385-402

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THE EFFECT OF AIDING/OPPOSING BUOYANCY ONTWO-DIMENSIONAL LAMINAR FLOW ACROSSA CIRCULAR CYLINDER

Gurunath Gandikota1, Sakir Amiroudine2,Dipankar Chatterjee3, and Gautam Biswas41LPMI, Arts et Metiers Paris Tech, Angers, France2Laboratoire TREFLE UMR CNRS, Talence, France3Simulation & Modeling Laboratory, Central Mechanical EngineeringResearch Institute, (Council of Scientific & Industrial Research),Durgapur, India4Indian Institute of Technology Kanpur, Kanpur, India

The effect of thermal buoyancy on the upward flow and heat transfer characteristics around

a heated/cooled circular cylinder is studied. A two-dimensional finite-volume model is

deployed for the analysis. The influence of aiding/opposing buoyancy is studied for the

range of parameters �0.5�Ri� 0.5, 50�Re� 150, and the blockage ratios of B¼ 0.02

and 0.25. The flow shows unsteady periodic nature in the chosen range of Reynolds numbers

for the forced convective cases (Ri¼ 0), and the vortex shedding stops completely at some

critical values of Richardson numbers.

INTRODUCTION

Analysis of fluid flow and heat transfer around bluff obstacles such ascircular=square cylinders at low Reynolds number has been a subject of intenseresearch for several decades. The transport processes have tremendous engineeringimportance since these are often found crucial in numerous technological applica-tions, such as heat exchangers, solar heating systems, natural circulation boilers,nuclear reactors, dry cooling towers, electronic cooling, etc. When flow passes overa bluff obstacle, wakes form behind it as a result of flow separation and formation ofrecirculation zones. The flow structure strongly depends on the shape and size of thebody, the inflow and outflow conditions, and the blockage parameter. The flowbecomes more complicated when the wake is further influenced by heat transfer. Itshould be recognized that for low to moderate Re flows, the buoyancy effect cansignificantly influence the flow field, thereby affecting heat transfer characteristics.The wake behavior is significantly different for the case of cross flow around ahorizontal cylinder in comparison to the flow in a vertical configuration under the

Received 11 December 2009; accepted 26 May 2010.

Address correspondence to Sakir Amiroudine, Laboratoire TREFLE UMR CNRS, 8508

Esplanade des Arts et Metiers, 33405 Talence Cedex, France. E-mail: sakir.amiroudine@ensam.eu

Numerical Heat Transfer, Part A, 58: 385–402, 2010

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2010.505167

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influence of buoyancy. Buoyancy forces usually enhance the surface heat transferrate when they aid the forced flow; whereas, they impede the same when they opposethe forced flow. For the aiding situation (flow past a heated body), the forced flow isin the same direction as the buoyancy force; whereas, for the opposing situation(flow past a cooled body), it is in the opposite direction.

In the context of laminar mixed convection heat transfer around circular=square cylinder subjected to vertically upward=downward flow, Oosthuizen andMadan [1] studied experimentally the unsteady mixed convection for Re¼ 100–300. Merkin [2] pointed out that heating the circular cylinder delays separationand, finally, the boundary layer does not separate at all. Jain and Lohar [3] reportedan increase in shedding frequency with an increase in cylinder temperature. Faroukand Guceri [4] investigated numerically the laminar natural and mixed convectionheat transfer around a heated circular cylinder placed within adiabatic channel wallsin the steady flow regime. Badr [5, 6] studied the laminar mixed convection heattransfer from an isothermal horizontal circular cylinder for the two cases whenthe forced flow is directed either vertically upward (parallel flow) or downward(contra flow). The buoyancy-aided (0�Ri� 4) steady convection heat transfer atlow Re (¼20, 40, and 60) from a horizontal circular cylinder in a vertical adiabaticduct has been studied numerically by Ho et al. [7] for the blockage parameters 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. However, there wasappreciable degradation of the buoyancy-aided enhancement in the heat transferrate. Chang and Sa [8] investigated the phenomenon of vortex shedding from a hea-ted=cooled circular cylinder in the mixed convection regime, and predicted thedegeneration of purely periodic flows into a steady vortex pattern at a criticalRichardson number of 0.15. Nakabe et al. [9] studied the effect of buoyancy onthe channel-confined (channel walls maintained at ambient temperature) flow acrossa heated=cooled circular cylinder with parabolic inlet velocity profile by using afinite-difference method. Three flow configurations were considered: (1) aiding

NOMENCLATURE

B blockage ratio

d cylinder size, m

f vortex shedding frequency, Hz

g gravitational acceleration, m=s2

Gr Grashof number

H width of computational domain, m

Lu upstream length, m

Ld downstream length, m

Nu local Nusselt number

P dimensionless pressure

Pe Peclet number

Pr Prandtl number

Re Reynolds number

Ri Richardson number

St Strouhal number

T temperature, K

U dimensionless axial velocity

V dimensionless normal velocity

X dimensionless axial coordinate

Y dimensionless normal coordinate

a thermal diffusivity, m2=s

/ polar angle, �

n kinematic viscosity, m2=s

h dimensionless temperature

q density, kg=m3

s dimensionless time

Subscripts

av average

cr critical

W cylinder wall

1 free stream

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buoyancy at Re¼ 80 and 120, 0�Ri� 1.6, and B¼ 0.15 and 0.3; (2) opposingbuoyancy at Re¼ 50, �1�Ri� 0, and B¼ 0.15, and (3) cross-stream buoyancy atRe¼ 80 and B¼ 0.3. For the aiding buoyancy case, the authors found three mainresults. First, at constant Re, the value of Ri at which the vortex shedding degener-ates (into twin vortices) decreases with increasing B. Second, at constant B, the valueof Ri at which vortex shedding degenerates increases with Re. Third, at constant Ri,the value of Re at which vortex shedding starts increases with increasing B. For theopposing buoyancy case, the flow remained unsteady periodic without vortex sup-pression. By exploiting a finite-element method, Singh et al. [10] determined the flowfield and temperature distribution around a heated=cooled circular cylinder placed inan 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 com-pletely at a critical Richardson number of 0.15, below which the shedding of vorticesinto the stream was quite prominent. Flow past an isolated circular cylinder isnumerically simulated, under the influence of aiding and opposing buoyancy byPatnaik et al. [11]. They found that at low Reynolds numbers (e.g., Re¼ 20–40),buoyancy opposing the flow could trigger vortex shedding. Saha [12] investigatedthe natural convection phenomena past a square cylinder placed centrally within avertical parallel plate channel numerically using the MAC method. The flow wasfound unstable above the critical Grashof number (¼3� 104). The drag coefficientwas found to decrease, while the Strouhal and surface-averaged Nusselt numberswere seen to increase with the Grashof number. The effect of buoyancy on the flowstructure and heat transfer characteristics of an isolated square cylinder in upwardflow was investigated numerically by Sharma and Eswaran [13] for Re¼ 100 andPr¼ 0.7. Like the circular cylinder case, the degeneration of the Karman vortexstreet was also observed to occur at a critical Richardson number of 0.15 from theirstudy for the square cylinder. In another article, Sharma and Eswaran [14] studiedthe effect of channel-confinement of various degrees (B¼ 0.1, 0.3, and 0.5) on theflow and heat transfer characteristics around a heated=cooled square cylinder, byconsidering the effect of aiding=opposing buoyancy at �1�Ri� 1, for Re¼ 100and Pr¼ 0.7. They observed that with an increase in the blockage parameter, theminimum heating (critical Ri) required for the suppression of vortex sheddingdecreases up to a certain blockage parameter (¼0.3), but thereafter increases. Singhet al. [15] performed a comprehensive schlieren-interferometric study for the wakesbehind heated circular and square cylinders placed in a vertical test cell. A detaileddynamical characteristic of vortical structures was reported in their study. The prob-lem of the laminar upward mixed convection heat transfer for thermally developingair flow in the entrance region of a vertical circular cylinder under buoyancy effectand wall heat flux boundary condition, has been numerically investigated by Husseinand Yasin [16] through an implicit finite-difference method and the Gauss elimin-ation technique. The investigation covers Reynolds number range from 400 to1600, and the heat flux from 70W=m2 to 400W=m2. The results revealed that thesecondary flow created by natural convection have a significant effect on the heattransfer process, as has also been shown by reference [17]. Ameziani et al. [18] havenumerically investigated the problem of natural convection in a vertical openedporous cylinder. The type of flows (with and without recirculation) depended onthe filtration number (Ra), the aspect ratio (A), and the Biot number (Bi).

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The aim of the present study is to investigate numerically the effect of aiding=opposing buoyancy on the two-dimensional laminar upward flow and heat transferacross a circular cylinder for various Reynolds (Re¼ 50–150) and Richardson(�0.5�Ri� 0.5) numbers, and two different blockage parameters: B¼ 0.25 that cor-responds to the channel confined case, and 0.02 which is close to the flow in an infinitemedium [19]. The study attempts to understand the mechanism of buoyancy-inducedbreakdown of Karman Vortex Street for increasing Re under channel confinedand unconfined flow situations, and to determine the critical Richardson number asa function of the Reynolds number for these two blockage ratios.

PROBLEM DESCRIPTION

The geometry of the problem considered in this study along with the coordi-nate system used, is shown schematically in Figures 1a and 1b. The system of interestconsists of a fixed two-dimensional circular cylinder of diameter d heated or cooledto a constant temperature TW and exposed to an upward stream of temperature T1.Figures 1a and 1b, respectively, depict the configurations for the channel unconfined(B¼ d=H¼ 0.02, where H is the width of the computational domain) and confined(B¼ 0.25) flows. Two adiabatic vertical walls of finite length are placed at a distanceof H=2 on either side of the center of the cylinder for the channel confined flow case;whereas, artificial confining boundaries are considered for the unconfined flow case.The center of the cylinder is placed on the vertical axis at a downstream distance ofLu¼ 8d from the entry plane. The lengths of the computational domains are fixed asL¼LuþLd¼ 24d for B¼ 0.25 and L¼ 34d for B¼ 0.02. These values are chosen soas to reduce the effect of the inlet and outlet boundary conditions on the flow

Figure 1. Schematic diagram of the computational domain. (a) B¼ 0.02 and (b) B¼ 0.25.

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patterns in the vicinity of the obstacle. Furthermore, these choices are also consistentwith the other contemporary studies available in the literature [10, 14].

Governing Equations

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

qUqX

þ qVqY

¼ 0 ð1Þ

Momentum

qUqs

þUqUqX

þ VqUqY

¼ � qPqX

þ 1

Re

q2UqX 2

þ q2UqY 2

!ð2aÞ

qVqs

þUqVqX

þ VqVqY

¼ � qPqY

þ 1

Re

q2VqX 2

þ q2VqY 2

!þRi h ð2bÞ

Energy

qhqs

þUqhqX

þ VqhqY

¼ 1

Pe

q2hqX 2

þ q2hqY 2

!ð3Þ

where U, V are the dimensionless velocity components along X and Y directions of aCartesian coordinate system, respectively; P and s are the dimensionless pressure andtime; Re (¼v1d=n) is the Reynolds number based on the cylinder diameter with v1being the average velocity at the inlet; Ri (¼Gr=Re2) is the Richardson number; Gr(¼gb(TW�T1)d3=n2) is the Grashof number with g and b being the gravitationalacceleration and volumetric expansion coefficient; h is the dimensionless tempera-ture; and Pe (¼RePr) is the Peclet number with Pr (¼n=a) being the Prandtlnumber. The fluid properties are described by the density q, kinematic viscosity n,and thermal diffusivity a. The dimensionless variables are defined as follows.

U ¼ u

v1V ¼ v

v1X ¼ x

dY ¼ y

dP ¼ p

qv21h ¼ T � T1

TW � T1s ¼ t v1

dð4Þ

The corresponding dimensional quantities are denoted by u, v, x, y, p, T, and t,respectively.

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Boundary Conditions

A parabolic velocity profile with the average velocity vav¼ v1 is considered atthe inlet for the channel confined flow; whereas, a uniform free stream with velocity,v1 is assumed for the unconfined flow. The vertical channel walls are assumedinsulated, and far-field boundary conditions are imposed on the artificial confiningboundaries. The cylinder temperature TW is governed by the magnitude of theRichardson number. A no-slip boundary condition is imposed on both the cylinderand channel walls. The exit boundary is located sufficiently far downstream from theregion of interest; hence, an outflow boundary condition is proposed at the outlet.Mathematically, one can write the following.

At the inlet

U ¼ h ¼ 0; V ¼ 1:5ð1� X 2=4Þ for B ¼ 0:251 for B ¼ 0:02

�ð5Þ

At the outlet

qUqY

¼ qVqY

¼ qhqY

¼ 0 ð6Þ

At the vertical boundaries

U ¼ V ¼ qhqX

¼ 0 for B ¼ 0:25

U ¼ h ¼ 0; V ¼ 1 for B ¼ 0:02ð7Þ

At the cylinder surface

U ¼ V ¼ 0; h ¼ 1 ð8Þ

The flow is assumed to start impulsively from rest.The Strouhal number, which characterizes the periodicity in a flow field, is

defined as St¼ f d=v1, where f is the vortex shedding frequency. The heat transferbetween the cylinder and the surrounding fluid is calculated by the Nusselt number.The local Nusselt number based on the cylinder dimension is given by

Nu ¼ hd

k¼ � qh

qnð9Þ

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 transferis obtained by integrating the local Nusselt number along the cylinder face.The time-average Nusselt number is computed by integrating the local value overa large time period.

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METHOD OF SOLUTION

The conservation equations subjected to the aforementioned boundaryconditions are solved using a finite-volume-based method according to theSIMPLEC [20] algorithm in a collocated grid system. A second order upwind schemefor discretizing the convective terms, and a central difference scheme for the diffusiveterms of the momentum and energy equations are used. The time discretization iscarried out by a second order accurate fully implicit Adams-Bashforth scheme.The conditions necessary to prevent numerical oscillations are determined fromthe Courant-Friedrichs-Lewy (CFL) and the grid Fourier number criteria. The finaltime step size is usually taken as the minimum of the two criteria mentioned above.Furthermore, the time step size has been varied from 0.01 to 0.1 to determine anoptimum value that results in less computational time, but produces sufficientlyaccurate results. A dimensionless time step size of 0.05 is finally used in the computa-tions satisfying all of the above restrictions. A body-force-weighted pressure interp-olation technique is used to interpolate the face pressure from the cell centervalue. The discretized governing equations are finally solved by an algebric multigridsolver developed by the same research group earlier for such flows [21]. The conver-gence criteria for the inner (time step) iterations are set as 10�6 for all discretizedgoverning equations.

A nonuniform grid distribution having a close clustering of grid points inthe vicinity of the walls (i.e., the regions with higher gradients of flow and thermalvariables), have been used in the present computation. Figure 2a shows the griddistribution in the entire computational domain; whereas, an expanded view near

Figure 2. Representative grid distribution for B¼ 0.25: (a) entire computational domain and (b) closer

view around the cylinder.

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the obstacle is depicted in Figure 2b. A comprehensive grid sensitivity study iscarried out to select the most economical mesh sizes for the two blockage ratios.For B¼ 0.25, 110� 312, 124� 352, and 140� 398 mesh sizes and for B¼ 0.02,212� 312, 236� 348, and 252� 370 cells along the X and Y directions, respectively,were attempted. The Strouhal number and the time- and surface-average cylinderNusselt number were computed at Re¼ 100 and Ri¼ 0 for all grids mentioned

Figure 3. Isotherm plots at Re¼ 100, Ri¼ 0, and B¼ 0.25: (a) Singh et al. [10] and (b) present

computation.

Figure 4. Comparison of Strouhal number as a function of Ri with the experimental results of Singh et al.

[13] for Re¼ 94 and 110, B¼ 1.55%.

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above, and a maximum variation of �2% has been observed. Finally, 110� 312 and212� 312 grids were selected as the most computationally economical onefor B¼ 0.25 and 0.02, respectively. All computations were carried out in an IntelCore 2 DuoTM (2.26GHz) workstation.

The results due to present simulation are compared with the available numeri-cal [10] and experimental [15] results in the literature. Figure 3 depicts the compari-son in terms of the isotherm profiles for the case of Re¼ 100, Ri¼ 0, and B¼ 0.25.Figure 4 presents the Strouhal number variation with Ri at Re¼ 94 and 110 for ablockage parameter of 1.55%. An excellent matching is observed between the presentcomputation and the corresponding numerical and experimental results.

RESULTS AND DISCUSSION

Air (Pr¼ 0.7) is considered the working fluid for the present study. Simulationsare carried out for blockage ratios of 0.02 and 0.25 at Re¼ 50–150 in steps of 25, andRi¼�0.5 to 0.5 with a dimensionless time step size of 0.05 and results are presentedat a dynamic steady state condition (5,000 time iterations that correspond to a

Figure 5. Streamlines for different Richardson numbers for B¼ 0.25 and Re¼ 100.

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dimensionless time instant of 250). The negative Richardson number signifies thatthe buoyancy force is in a direction opposite to that of the flow. Representativestreamline and isotherm patterns under the aiding=opposing buoyancy conditionsare shown in Figures 5–8 for B¼ 0.25 and 0.02 at Re¼ 100. The streamline and iso-therm patterns show some intriguing aspects of vortex dynamics under the influenceof aiding=opposing buoyancy flows. In the negative Richardson regimes (i.e., oppos-ing buoyancy) at the vicinity of the cylinder, the inertia force is opposed by the buoy-ancy as well as the viscous forces. As a consequence, the point of separation movestowards the forward stagnation point of the cylinder, causing an early separation.This also makes the wakes broader, and a periodic vortex shedding can be observed.The vorticity contours (not shown) are observed to have a wavering motion whichincreases with increased cooling of the cylinder (Ri< 0). It should be mentioned thatin the entire regime of negative Richardson numbers for both cases of channel con-fined and unconfined flows, the vortex shedding phenomena are quite prominentwhich is justified from the respective streamline plots. The shedding phenomena con-tinue to characterize the flow even at Ri¼ 0 (i.e., for pure forced convection), and 0.1for Re¼ 100. However, the flow structure changes considerably at some critical

Figure 6. Streamlines for different Richardson numbers for B¼ 0.02 and Re¼ 100.

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value of Ri (Ricr), where the vortex shedding is observed to stop completely. Underthis aiding buoyancy condition, in the close proximity to the cylinder, the inertiaforce is added with the viscous force, resulting in a separation delay and thepoint of separation shifts towards the trailing edge. A steady symmetric twin vortexstructure is observed at Ri� 0.15 for B¼ 0.25 and at Ri� 0.18 for B¼ 0.02, whichestablishes that the flow becomes completely steady at these Ri values. Accordingly,Ricr¼ 0.15 and 0.18 (streamlines and isotherms are not shown for both cases) areconsidered as the critical values corresponding to B¼ 0.25 and 0.02 at Re¼ 100.For unconfined flow, the absence of side wall effect results in a further separationdelay, due to which the critical Ri becomes slightly higher compared to the channelconfined flow. This can further be explained by the fact that the side walls (for thechannel confined case) try to confine the streamlines near the cylinder surface, whichimparts additional stability to the vortex street and as a result, less heating isrequired to stabilize the flow. The enhanced stability at higher blockage is againdue to the suppression of the propagation of disturbances, causing wake instability

Figure 7. Isotherms for different Richardson numbers for B¼ 0.25 and Re¼ 100.

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by the channel walls. The above observations are in excellent agreement with theearlier studies of Chang and Sa [8] and Singh et al. [10].

The isotherm profiles are the reflection of the physical phenomena observedfrom the analysis of the streamline patterns. The waviness of the isotherms down-stream of the cylinder under the opposing buoyancy conditions is a clear indicationof the periodic nature of the flow and thermal fields. This waviness can even beobserved to persist like the vortex shedding phenomena up to the critical Ri,above which the plume spread becomes narrower and the waviness subsequentlydisappears. The isotherm profiles show an almost symmetric nature (about thechannel centerline), like the streamlines for Ri�Ricr.

To further establish the above observations regarding the suppression ofvortices at the critical Ri for different Re, the dimensionless frequency of vortexshedding (Strouhal number) is plotted against Ri for the two blockage parametersin Figures 9a and 9b. The Strouhal number (St) is obtained from the fast Fouriertransform (FFT) of the transverse velocity component signal at the location 3ddownstream of the cylinder on the channel centerline. The Strouhal number is foundto decrease with increased cooling of the cylinder below the fluid temperature (i.e.,with decreasing Ri) for all Re. This can be attributed to the fact that with anincreased cooling of the cylinder, the fluid in the vicinity of the cylinder becomesdenser and, hence, the vortices stay attached to the cylinder for long periods.Up to the critical values of Ri, St increases steadily; however, for Ri higher than

Figure 8. Isotherms for different Richardson numbers for B¼ 0.02 and Re¼ 100.

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the critical values, the frequency becomes suddenly zero, i.e., the shedding hasstopped completely as a result of the breakdown of the Karman vortex street.It should be mentioned that for higher blockage ratio, the shedding frequency isobserved to be more compared to that for the lower one, particularly for lowervalues of Ri, as a result of the effect of the confining walls which is consistent withearlier studies reported in the literature [22].

The critical Ri is not a universal property for the degeneration of the vortexshedding mechanism, and is a function of the flow Reynolds number [15].Figure 10 depicts the dependence of the critical Richardson number on the Reynoldsnumber for the two blockage ratios. The critical Ri is found to increase with Re at

Figure 9. Strouhal versus Richardson numbers at (a) B¼ 0.25 and (b) 0.02.

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the constant blockage ratio. At low Reynolds number, the vorticity pumped into thewake from the boundary layers on the cylinder can be diffused away from the shearlayers merely by viscous diffusion [23]. However, as the flow Reynolds numberincreases viscous diffusion alone cannot keep up with the increased vorticityproduction in the upstream boundary layers, and vortices break away at regularintervals, constituting vortex shedding. Hence, the flow destabilizes at higher Reand, accordingly, more heating is required to weaken the breakdown of the Karmanvortex street. Figure 10 further reveals that the critical Ri is always higher for thelower blockage ratio at a particular Re. This is due to the increase in the stabilityof the wake for the channel confined case (higher blockage), which requires lessheating for stabilization of the flow.

To represent the enhancement=suppression of the average heat transferfrom the cylinder subjected to thermal buoyancy and the two blockage ratios, thetime- and surface-average cylinder Nusselt number against Ri for various Re andB is plotted in Figure 11. The Nusselt number is found to increase with Re as usualfor both blockage ratios. However, the variation of Nu with Ri has some interestingfeatures. For example, below the critical Ri, increasing magnitude of Ri serves toincrease Nu in general but with a lower gradient; whereas, above the critical Riit increases rapidly. This is because of the sudden stopping of shedding at thecritical Ri, the vortices become smaller in size resulting in a reduction of the poorheat transfer zones and, consequently, the heat transfer is enhanced. Below thecritical Ri and particularly in the range of negative Ri, due to the longer residencetime of the larger vortices behind the cylinder the heat transfer is comparatively less.It is also observed from Figure 11 that the higher blockage (B¼ 0.25) yields higherNu since the higher blockage increases the shedding frequency and, hence, thevortices are shed at a faster rate, resulting in an enhanced heat transfer rate.

The variation of time-average local Nusselt number on the cylinder surfaceagainst the polar angle / is shown in Figures 12a and 12b for Re¼ 100 and at

Figure 10. Variation of critical Richardson number with Reynolds number.

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different Ri and B. The locations /¼ 0� and 180� correspond to the front and rearstagnation points with respect to the incoming fluid. The local Nusselt numberincreases initially along the surface of the cylinder due to the fluid acceleration upto /¼ 40� for B¼ 0.25, and /¼ 30� for B¼ 0.02 at Ri¼ 0. Thereafter, it reducesabruptly as a result of boundary layer development on the cylinder. At the pointof separation (/¼ 140� and 145� for B¼ 0.25 and 0.02, respectively, at Ri¼ 0), theNusselt number becomes minimum. It has been observed from the figure that thepoint of separation is delayed due to heating (e.g., Ri¼ 0.5); whereas, it is rushed

Figure 11. Time- and surface-average Nusselt number versus Richardson numbers for (a) B¼ 0.25

and (b) 0.02.

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due to cooling (e.g., Ri¼�0.5). After the point of separation, a further rise in polarangle witnesses a rapid increase in heat transfer for the case of negative Ri; whereas,there is no significant increase in heat transfer for positive Ri. This is attributable tothe phenomena of suppression of vortex shedding above the critical Ri due to whichthe vortices remain attached with the cylinder at higher positive Ri, resulting in areduced heat transfer rate.

CONCLUSION

Numerical investigation using a finite-volume method has been performed inthis work to demonstrate the influence of thermal buoyancy on the degeneration

Figure 12. Variation of time-average local Nusselt number on the cylinder surface for (a) B¼ 0.25 and (b)

0.02.

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of vortex shedding phenomena behind a heated=cooled circular cylinder fixed in avertical channel, and subjected to an upward flow of an incompressible viscous fluid.Results are obtained for the cases of confined and near-unconfined configurationsfor different Reynolds numbers in the laminar two-dimensional regime. The criticalRichardson number is found to increase with Reynolds number in the chosen rangefor both the channel confined and unconfined cases, and it also has been observedthat reducing the blockage increases the critical Ri at any Re. The Nusselt numberincreases at a faster rate beyond the critical Ri, whereas it remains almost constantin the negative Ri ranges. The Nusselt number is also found to be larger for higherblockage. Heating causes a delay in the point of separation, as the higher blockagedoes the same. Finally, the results are found in good agreement with the availablenumerical and experimental results.

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