Aiding-Buoyancy Mixed Convection from a Pair of Side-by-Side

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Aiding-Buoyancy Mixed Convection from a Pair of Side-by-Side Heated Circular Cylinders in Power-Law Fluids Alex Daniel and Amit Dhiman* Department of Chemical Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, India ABSTRACT: Mixed-convection flow and heat transfer of non-Newtonian power-law fluids over a pair of side-by-side circular cylinders in aiding buoyancy are investigated numerically. The calculations are performed in an unconfined domain for the following range of physical control parameters: Reynolds number (Re) = 1−40, Richardson number (Ri) = 0−1, and power-law index (n) = 0.2−1 for a transverse gap ratio of 1.5 at a constant Prandtl number (Pr) of 50. Extensive numerical results are presented, such as viscous, pressure, and total drag coefficients, average Nusselt number, and transition from a steady to a timeperiodic regime. The overall drag coefficient and its components decrease with the Reynolds number and increase with the Richardson number for all values of the power-law index considered. The average Nusselt number increases with an increase in both the Reynolds and Richardson numbers but decreases with an increase in the power-law index. It is observed that aiding buoyancy and shear-thinning tendency augment heat-transfer characteristics. The maximum enhancement in heat transfer is found to be approximately 85% for Re = 40, n = 0.2, and Ri = 1. However, the maximum enhancement in the value of the average Nusselt number for Ri = 0 and 0.5 is found to be approximately 70% and 81%, respectively. Finally, a simple heat-transfer correlation is obtained for the values of Re, n, and Ri covered in this study. flow configuration at low Reynolds numbers. Therefore, the buoyancy-induced free-stream non-Newtonian flow around a pair of side-by-side circular cylinders poses a challenging classical fluid mechanics and heat-transfer problem. Hence, the present study aims to investigate the effects of aiding buoyancy, non-Newtonian behavior, and interaction between the wakes around the cylinders at low Reynolds numbers.

1. INTRODUCTION Analysis of fluid flow and heat transfer around a bluff body like a cylinder at low Reynolds numbers has been a subject of intense research for several decades because of its fundamental significance as well as many related engineering applications. For instance, these flow situations are analogous to the one found in numerous technological applications such as the novel design of heat exchange systems, flow dividers in polymer processing, thermal treatment equipment of foodstuffs, and flow over different probes and sensors. Further, the buoyancy effect can significantly influence the flow field, thereby affecting heat-transfer characteristics. Buoyancy forces usually enhance the surface heat-transfer rate when the imposed flow is in the same direction of buoyancy force (i.e., aiding buoyancy), whereas they impede the same when they oppose the imposed flow (i.e., opposing buoyancy). When two or more of such cylinders are placed in proximity, the difficulty in predicting momentum and heat transfer around multicylinders is increased and interference effects are severe. Consequently, the wake behaves quite differently from that of an isolated single circular cylinder. The formation of cylinderlike structures, both alone and in groups (side-by-side cylinders, for instance), during the bluff-body flows and the associated heat-transfer characteristics encountered in applications like processing of substances of high molecular weight (polymer melts, emulsions, and suspensions) and multiphase fluid mixtures in food and pharmaceutical industries display various types of non-Newtonian flow behavior. Most non-Newtonian fluid systems display shear-thinning or shear-thickening characteristics under appropriate conditions of shear and/or their composition.1 This type of behavior is conveniently approximated by the simple power-law-type relationship between the stress tensor and the rate of deformation tensor. The exact form of the thermal and flow behavior interactions is highly dependent on the buoyancy effects in addition to the © 2013 American Chemical Society

2. PREVIOUS WORK It appears that Sparrow and Lee2 were the first to study the problem of mixed-convection boundary-layer Newtonian flow about a horizontal circular cylinder analytically for Re = 100− 3000, Pr = 0.7, and Ri = 0 to ∞. They reported that the heattransfer coefficients for mixed convection exceed those for either pure forced convection or pure natural convection under aiding conditions. Combined convection heat transfer from an isothermal circular cylinder placed with its axis horizontal and perpendicular to the free-stream direction was investigated by Badr3 for 1 < Re < 40, 0 ≤ Ri ≤ 5, and Pr = 0.7. The streamline and isotherm patterns were plotted, and the results were compared with previous experimental correlations. Patnaik et al.4 examined the flow past an isolated circular cylinder under the influence of aiding/opposing buoyancy (−1 ≤ Ri ≤ +1), Pr = 0.7, and Re = 20−200. The influences of buoyancy on the Nusselt number, wake structure, temporal lift, and drag forces were studied, and it was found that, at low Reynolds numbers, buoyancy opposing the flow could trigger vortex shedding and the flow past a cooled cylinder results in a wide wake because of greater entrainment from the ambient fluid while heating Received: Revised: Accepted: Published: 17294

August 22, 2013 November 7, 2013 November 12, 2013 November 12, 2013 dx.doi.org/10.1021/ie4027742 | Ind. Eng. Chem. Res. 2013, 52, 17294−17314

Industrial & Engineering Chemistry Research

Article

regime with aiding buoyancy for the ranges of conditions: Reynolds number (1 ≤ Re ≤ 40), power-law index (0.2 ≤ n ≤ 1.8), Richardson number (0 ≤ Ri ≤ 2), and Prandtl number (1 ≤ Pr ≤ 100). They reported both the drag coefficient and average Nusselt number augmented with the increasing aiding buoyancy, Reynolds number, and Prandtl number. An increase in the shear-thinning tendency enhances heat-transfer characteristics, whereas it is generally reduced in the case of shearthickening fluids. The buoyancy effects were found to be stronger in shear-thinning fluids and/or at low Reynolds numbers than in shear-thickening and/or at high Reynolds numbers. Soares et al.12 conducted numerical simulations of mixed-convection flow of power-law fluids over a cylinder in a horizontal configuration, with imposed flow oriented normal to the direction of gravity, for the ranges Re = 1−30, Pr = 1 and 100, Ri = 0−3, and n = 0.6−1.6. The drag coefficient decreased as a function of Re and n for all values of Ri. The effects of mixed convection on heat-transfer characteristics were more significant at lower values of the power-law index (shearthinning fluids), and such a decrease in n was found to increase the magnitude of the surface-averaged Nusselt number. Furthermore, Chandra and Chhabra13 investigated steady mixed-convection heat transfer from a heated semicircular cylinder immersed in power-law fluids with its curved surface facing upstream. The momentum and thermal energy equations were solved numerically over the ranges 1 ≤ Re ≤ 30, 0 ≤ Ri ≤ 2, 0.2 ≤ n ≤ 1.8, and 1 ≤ Pr ≤ 100. The drag coefficient has shown a monotonic increase as the Richardson or Reynolds number is increased. The average Nusselt number increased with an increase in the values of the Reynolds, Prandtl, and Richardson numbers. The shear-thinning behavior facilitated heat transfer, whereas shear thickening had an adverse influence on it. In addition to the above-discussed studies on a single cylinder under the impact of buoyancy, only a few studies are available on mixed-convection heat transfer and flow characteristics around two circular cylinders of equal diameter arranged in a side-by-side configuration. Song and Chang14 studied heat transfer and fluid flow in mixed convection for the double cylinders arranged transverse to the vertical air stream. They found that the Karman vortex street breaks down behind the double heated cylinders in a transient manner for a certain Richardson number range because of the buoyancy effect and vortex interaction, in contrast to the sudden breakdown applicable to a single heated cylinder. Other cylinder configurations with mixed convection were explored by Patnaik et al.15 like laminar mixed-convection heat transfer over an isolated circular cylinder and two cylinders in tandem, under the influence of aiding/opposing buoyancy (−1 ≤ Ri ≤ +1). Degeneration of a naturally occurring Karman vortex street into a twin eddy pattern was noticed for Re = 41−200 under buoyancy-aided convection. On the contrary, buoyancyopposed convection trigged vortex shedding even at a low Re range of 20−40, where only twin eddies were found in the natural wake. The influences of thermal buoyancy on the vortex shedding behind a heated circular cylinder in an infinite medium for cross-flow at low Reynolds numbers were investigated by Chatterjee and Mondal.16 They reported that the critical Ri for the onset of vortex shedding decreases and the Strouhal number (St) increases with Re. The effects of cross-buoyancy on vortex shedding behind circular and square cylinders for low Re are also discussed in ref 17. Chatterjee17 observed that the

narrows the same because of delayed separation. For the case of a heated cylinder under cross-flow conditions, it has been numerically found by Biswas and Sarkar5 that the hydrodynamic instabilities grow and the flow becomes periodically unsteady if the fluid is severely influenced by thermal buoyancy (1 ≤ Ri ≤ 2). The flow was steady in the absence of thermal buoyancy in the range of Reynolds number 10 ≤ Re ≤ 45 with Pr = 0.7. Thermal buoyancy brings about asymmetry in the wake and induces unsteadiness. Chang and Sa6 investigated laminar mixed convection about a heated circular cylinder of Newtonian fluid flow in the vertical direction for Re = 100 and 200, Pr = 0.73, and 0 ≤ Ri ≤ 1. They reported that excessive heating of the cylinder causes a singular decay of the dynamic vortices to the stationary twin vortices and ultimately to a vortex-free symmetric flow. Badr7 also studied the mixedconvection heat transfer from an isothermal horizontal cylinder for two cases: (i) the flow directed vertically upward (parallel flow) by varying Re = 5, 20, 40, and 60 and Gr = 0−7200 and (ii) the flow directed vertically downward (contra flow) for Re = 5, 20, and 40 and Gr = 0−3200 at Pr = 0.7. For parallel flow, an increase in the Grashof number resulted in a decrease in the wake length and increased heat transfer. Whereas for contraflow, with an increase in the Grashof number, the wake gets bigger and the Nusselt number is found to decrease because of a slowing down in the flow velocity near the cylinder surface, a further increase in the Grashof number results in an increase in the Nusselt number and the flow field becomes more dominated by natural convection. Merkin8 studied the problem of mixed convection from a horizontal cylinder in a stream flowing vertically upward in both the cases of a heated and a cooled cylinder. He found that heating the cylinder delays separation and, if the cylinder was warm enough, suppresses it completely. Cooling the cylinder brings the separation point nearer to the lower stagnation point, and for a sufficiently cold cylinder, there will not be a boundary layer on the cylinder. Gandikota et al.9 numerically investigated the effect of aiding/ opposing buoyancy on the laminar upward flow and heat transfer across a cylinder for Re = 50−150, −0.5 ≤ Ri ≤ +0.5, and Pr = 0.7 for both confined and unconfined cases. They attempted to understand the mechanism of buoyancy-induced breakdown of the Karman vortex street for increasing Reynolds number and to determine the critical Richardson number as a function of the Reynolds number. The critical Ri was found to increase with an increase in Re, and it was also observed that reducing the blockage increases the critical Ri at any Re. The Nusselt number was found to increase at a faster rate beyond the critical Ri, whereas it remained almost constant in the negative Ri ranges. The Nusselt number was also found to be larger for a higher blockage ratio, and heating caused a delay in the point of separation, as the higher blockage does. Similarly, in the vertically confined domain, Singh et al.10 determined the flow field and the temperature distribution around a heated/ cooled circular cylinder placed in an insulated vertical channel for Re = 100, −1 ≤ Ri ≤ +1, Pr = 0.7, and blockage ratio of 25%. They observed that the vortex shedding stopped completely at a critical Richardson number of 0.15, below which the shedding of vortices into the stream was quite prominent. For Ri < 0.15, wake broadening was observed, while for Ri > 0.15, separation delay and attached twin vortices were observed behind the cylinder. In the context of mixed convection around a circular cylinder subjected to vertically upward flow of power-law fluids, Srinivas et al.11 numerically investigated the steady mixed-convection 17295

dx.doi.org/10.1021/ie4027742 | Ind. Eng. Chem. Res. 2013, 52, 17294−17314


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critical Ri shows a declining tendency for both cylinders with an increase in Re. Subsequently, a numerical investigation of crossbuoyancy on vortex shedding behind a square cylinder for low Re for a blockage ratio of 0.05 was carried out by Chatterjee and Mondal.18 They found that the critical Ri for the onset of vortex shedding decreases and the corresponding critical St increases with an increase in Re for the range of parameters studied. While considering the effect of buoyancy on multiple square cylinders, the mixed-convection flow and heat-transfer characteristics around two isothermal square cylinders arranged in a tandem configuration in a confined vertical channel were simulated by Chatterjee19 for 1 ≤ Re ≤ 30. Mixed-convection heat transfer around five inline isothermal square cylinders periodically arranged within a vertical duct for both confined and unconfined cases was numerically investigated by Chatterjee and Raja20 at Re = 100. Niu and Zhu21 carried out a numerical study of three-dimensional flows around two identical square cylinders in staggered arrangements at Re = 250. Sarkar et al.22 investigated the flow and heat-transfer phenomena past two tandem square cylinders in a uniform upward flow at Re = 100 under aiding and opposing thermal buoyancy conditions. Experimentally, Chhabra et al.23 studied the terminal falling velocities of objects with different shapes in non-Newtonian media. They reported that the terminal falling velocity is influenced primarily by the shear-dependent viscosity. Thus, on the basis of the above discussion, it can be summarized that sufficient information is now available in the open literature on the non-Newtonian flow and mixedconvection heat-transfer characteristics around a single circular cylinder. However, in spite of having many engineering applications, the mixed-convection flow and heat transfer over a pair of side-by-side cylinders have not been investigated for non-Newtonian fluids to the same extent. Therefore, the present work attempts to investigate numerically the laminar flow and mixed-convection heat-transfer characteristics of power-law fluids around two side-by-side (center-to-center transverse distance of 1.5D) circular cylinders for a range of settings: Reynolds number (Re) = 1−40, power-law index (n) = 0.2−1, and Richardson number (Ri) = 0−1 for a Prandtl number of 50. Furthermore, the novelty of this work lies in both the results and the typical geometry studied with the different values of the physical parameters. This study will also enhance our process design capability of non-Newtonian fluids with the aiding-buoyancy and flow configuration studied, which has a great relevance from the industrial point of view, as given in the Introduction. The value of the Prandtl number on the order of 50 is very common in chemical, petroleum, and oil-related engineering applications.1,24 The dominancy of forced convection and/or subsidence of buoyancy effects with increasing Prandtl number is reported elsewhere in the literature.25−28 A constant Prandtl number of 50 is also employed by Bouaziz et al.29 for their simulations of flow and heat transfer in a plane channel with a built-in heated square cylinder. Thus, on the basis of the studies available in the literature,1,11−13,24−29 the present study is limited to a fixed value of the Prandtl number of 50 to delineate the effects of mixed convection and power-law fluid behavior. When the gap ratio (or the pitch ratio) is too small, the wake produced behind a pair of side-by-side cylinders acts like the one produced behind a single cylinder.30−32 However, at a gap ratio of 1.5D, the wakes behind the individual cylinders have their own identity, and at the same time, they try to interact

with each other. At higher gap ratios, again the interaction among wakes produced by the individual cylinders decreases and the wakes become separate entities behind independent bluff bodies. Because the aim of the present work is to study the effect when two cylinders are in close proximity of interaction, a gap ratio of 1.5D is found to be appropriate. Moreover, a pair of heated circular cylinders with a gap ratio of about 1.5D is frequently encountered in many specialized industrial applications such as polymeric resin/paint mixing, fabric dip-coating units, and others, where the non-Newtonian fluids are exposed to shearing force and heat transfer at close nip between cylindrical rolls. An especially remarkable point is that the flow properties at the pitch ratio of 1.5D are rather complex and strongly dependent on the Reynolds number, as reported by Kun et al. 33 and Chaitanya and Dhiman.24 However, investigations for different pitch ratios in aiding buoyancy may be carried out, but it is beyond the scope of the present study.

3. PROBLEM STATEMENT AND MATHEMATICAL FORMULATION 3.1. Problem Statement. The two-dimensional upward flow of power-law fluids over two side-by-side heated cylinders under the influence of mixed convection (aiding buoyancy) is schematically displayed along with boundary conditions in Figure 1. At the inlet, the flow is uniform and isothermal with velocity U∞ and temperature θ∞. The cylinders are considered

Figure 1. Schematic of mixed-convection upward flow past a pair of circular cylinders in a side-by-side arrangement with boundary conditions. 17296



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to be at a constant temperature of θw (>θ∞). The height and width of the computational domain in vertical and horizontal directions are represented by L (=Xu + Xd) and W, respectively. The circular obstacles are located symmetrically about the midplane at an upstream distance of Xu from the inlet to the center of the cylinders and at a downstream distance of Xd from the outlet to the cylinder centers. The distance between the centers of the cylinders is kept as 1.5D, where D is the diameter of each cylinder and also the nondimensionalizing length scale. 3.2. Governing Equations. The dimensionless governing equations for the problem under consideration with the Boussinesq approximation can be expressed in the following conservative forms.1,34

Vy Vx y p̅ x ; V= ; X= ; Y= ; P= ; φ U∞ D D U∞ ρU∞2 t ̅U∞ θ − θ∞ ; t= = D θw − θ∞

U=

τ=

In the present study, because of the relatively low value of the Reynolds number and weak buoyancy effects, the resulting shear rates are likely to be low and, therefore, it is reasonable to neglect the viscous dissipation effects in the thermal energy equation, as is supported by the limited results on viscous dissipation effects in an array of cylinders.35 Also, over the ranges of conditions spanned here, the gradients are not expected to be steep and, therefore, it is justified to neglect the viscous dissipation contribution in the energy equation; i.e., the Eckert number is assumed to be small (≪1), and this is considered to be a reasonable approximation in most situations.34−37 For incompressible fluids, the components of the stress tensor (τij) are related to the components of the rate of deformation tensor (εij) as follows:

Equation of continuity: ∂U ∂V + =0 ∂X ∂Y

η̅ τ̅ and η = m(U∞/D)n m(U∞/D)n − 1

(1)

X component of the momentum equation: ∂τyx ⎞ ∂(UV ) ∂(UU ) 1 ⎛ ∂τxx ∂P ∂U =− + + + + ⎜ ⎟ Re ⎝ ∂X ∂Y ⎠ ∂Y ∂X ∂X ∂t

τij̅ = 2η ̅ εij where (i , j) = (x , y)

(5)

(2)

The components of the rate of the strain tensor (εij), related to the velocity components and their derivatives, are given as

Y component of the momentum equation: ∂(VV ) ∂(UV ) ∂V + + ∂X ∂Y ∂t ∂τyy ⎞ 1 ⎛ ∂τxy ∂P + =− + ⎜ ⎟ + Riφ Re ⎝ ∂X ∂Y ⎠ ∂Y

εij =

(6)

For power-law fluids, the viscosity (η) is given by

(3)

⎛ I ⎞(n − 1)/2 η ̅ = m⎜ 2 ⎟ ⎝2⎠

where U and V are the dimensionless velocity components along the X and Y directions of a Cartesian coordinate system, respectively. P and t are the dimensionless pressure and time, respectively. If the density variation is not too large, it is sufficient and common to employ the well-known Boussinesq approximation to express its temperature dependence. In most analytical and numerical studies of natural/mixed convection flows, this approximation is customarily invoked to keep the level of complexity at a tractable level.11 A similar approach is employed in the current framework. For the present study, the temperatures of the surfaces of each cylinder and the free stream are taken as 300 and 298 K, respectively. Because the maximum temperature difference in the system is maintained small (2 K), it is therefore justified to neglect the variation in the thermophysical properties (such as thermal conductivity, specific heat capacity, consistency, power-law indices, etc.) of the fluid with temperature.

(7)

where m is the power-law consistency index, n is the power-law index of the fluid (n < 1, shear-thinning behavior; n = 1, Newtonian behavior; n > 1, shear-thickening behavior), and I2 is the second invariant of the rate of the strain tensor (εij), which is given as I2 = 2(εxx2 + εyy2 + εxy2 + εyx2). The components of the rate of the strain tensor are related to the velocity components in Cartesian coordinates as εxx =

∂Vy ⎞ ∂Vy ∂Vx 1 ⎛ ∂V ; εyy = and εxy = εyx = ⎜ x + ⎟ ∂x ⎠ ∂y 2 ⎝ ∂y ∂x

In general, the viscosity is a function of three invariants of the rate of the deformation tensor. However, the non-Newtonian model used for the present study is considered to be a function of the shear rate only and is related to the second invariant of the rate of the deformation tensor (I2). This is due to the fact that the first invariant (I1) is zero for an incompressible fluid. For viscometric flows (e.g., axial tube flow, axial and tangential annular flow, and flow in a film), the third invariant (I3) vanishes identically with that of I1. Also, it is commonly assumed that I3 is not very important in other flows.34 Furthermore, the available limited experimental evidence suggests the dependence of the viscosity on the third invariant to be much weaker than that on the second invariant and, hence, this approximation is used widely in the literature, especially in the absence of strong extensional effects.36,37

Equation of energy: ∂(Vφ) ∂(Uφ) ∂ 2φ ⎞ ∂φ 1 ⎛ ∂ 2φ = + + ⎟ ⎜ 2 + ∂Y Re × Pr ⎝ ∂X ∂X ∂t ∂Y 2 ⎠

∂Vj ⎤ 1 ⎡ ∂Vi + ⎢ ⎥ ∂i ⎦ 2 ⎣ ∂j

(4)

The dimensionless variables appearing in eqs 1−4 are defined as 17297



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Figure 2. Schematic of a nonuniform computational grid: (a) entire computational domain; (b) closer view around the cylinders.

all flow variables and is chosen based on the information documented in the literature.9−11,13,24

Generally, most of the non-Newtonian systems exhibit extensional or stretching flow effects under appropriate conditions, and one of the associated parameters is the Trouton ratio (defined as the ratio of the extensional viscosity to the corresponding shear viscosity38). Because for a nonNewtonian fluid the shear viscosity is a function of the shear rate and the elongational viscosity is a function of the rate of stretching, there exists a complexity while including nonNewtonian behavior in the Trouton ratio.1 Also, at low Weissenberg (or Deborah) number,39−41 nonlinear elastic effects due to normal stress differences or due to extensional effects are negligible.42 However, as the value of the Weissenberg number is slowly incremented, extensional effects in the wake region become increasingly more important, resulting in an increase in drag due to strain hardening. Furthermore, the model used here does have an extensional viscosity variation, but it is just that the Trouton ratio is fixed in order for such fluids to be the same as the Newtonian case. 3.3. Boundary Conditions. The physical boundary conditions for the present flow system can be written as follows (Figure 1). At the Inlet Boundary. The flow is uniform and isothermal and is flowing in the vertically upward direction: U = φ = 0, V = 1

∂φ ∂U ∂V = = =0 ∂Y ∂Y ∂Y

The numerical simulation starts with the initial conditions Vx = 0, Vy = 0, p̅ = 0, and θ = θ∞. Equations 1−7, in conjunction with the boundary (eqs 8−11) and initial conditions, are solved for the primitive variables, i.e., velocity (Vx and Vy), pressure (p), ̅ and temperature (θ) fields. At this stage, it is also useful to introduce the dimensionless physical parameters appearing in the present study: For power-law fluids, the Reynolds number (Re) and Prandtl number (Pr) are defined as Re = ρDnU∞2−n/m and Pr = (mCp/ k)(U∞/D)n−1, respectively. The Richardson number (Ri), which represents the importance of natural convection relative to that of forced convection, is defined as Ri = Gr/Re2 = gβΔTD/U∞2, where the Grashof number (Gr) is the ratio of the buoyancy to viscous forces acting on the fluid, and for power-law fluids, it is defined as Gr = (gβΔTρ2D3/m2)(U∞/ D)2(1−n). Unlike in the case of Newtonian fluids, Pr and Gr for a power-law fluid also depend upon the velocity and diameter of the cylinder, in addition to the thermophysical properties. However, the Richardson number (Ri) is independent of the power-law constants (m and n). The preceding definitions of Re, Pr, and Gr are based on the effective fluid viscosity, as given by m(U∞/D)(n−1). In the limit of the Newtonian fluid behavior (n = 1), these reduce to the corresponding definitions of Newtonian fluids. The total drag coefficient (CD), the sum of the friction and pressure components, is defined as CD = FD/(0.5ρU∞2D) = CDP + CDF = ∫ 10P dX + (2/Re)∫ 10(∂V/∂X) dY. The individual drag coefficients (CDP and CDF) are calculated using the definitions C DP = FDP/(0.5ρU ∞ 2 D) and C DF = FDF/(0.5ρU ∞ 2 D), respectively, where FDP and FDF are the pressure and frictional components of the drag force per unit length of the cylinder

(8)

At the Vertical Boundaries. The symmetry conditions simulating a frictionless wall and zero heat flux are imposed. ∂φ ∂V = U = 0 and =0 ∂X ∂X

(9)

On Circular Cylinders. A no-slip boundary condition is used with cylinders held at a constant temperature of θw. U = V = 0, φ = 1

(11)

(10)

At the Exit Boundary. It is located sufficiently far downstream from the cylinders with a zero diffusion flux for 17298



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Table 1. Grid Dependence Study for Re = 1 and 40, Pr = 50, Ri = 1, and n = 0.2, 0.4, and 1 Re = 1 total number of cells in the domain

minimum grid size (δ)

total drag coefficient (CD)

159000 155410 152280

0.005D 0.01D 0.02D

28.3960 28.4001 28.3922

0.01 0.01

159000 155410 152280

0.005D 0.01D 0.02D

23.5700 23.5730 23.5660

0.01 0.02

159000 155410 152280

0.005D 0.01D 0.02D

13.2564 13.2561 13.2553

0.002 0.01

% deviation

Re = 40

average Nusselt number (Nu) n = 0.2 4.3190 4.3229 4.3215 n = 0.4 2.9719 2.9735 2.9752 n = 1.0 2.2107 2.2110 2.2117

% deviation


% deviation

average Nusselt number (Nu)

% deviation

0.09 0.06

1.6897 1.6963 1.7212

0.39 1.86

28.5489 28.2744 27.8742

0.96 2.36

0.05 0.11

1.9449 1.9449 1.9450

0.003 0.005

22.5393 22.6423 22.9807

0.46 1.96

0.01 0.04

2.3552 2.3556 2.3558

0.02 0.03

15.2536 15.2762 15.3774

0.15 0.81

Table 2. Upstream Dependence Study (Xu) for Re = 1 and 40, Ri = 1, and Pr = 50 at Different Values of n Re = 1 total number of cells in the domain

upstream distance (Xu)


% deviation

130490 155410 164310

10D 20D 30D

28.2961 28.4001 28.3901

0.33 0.04

130490 155410 164310

10D 20D 30D

24.0238 23.5730 23.5418

2.05 0.13

130490 155410 164310

10D 20D 30D

14.0655 13.2561 13.1428

7.02 0.86

Re = 40


% deviation


% deviation


% deviation

3.34 0.08

1.7122 1.6963 1.6957

0.97 0.04

28.3679 28.2743 28.2707

0.34 0.01

2.39 0.15

1.9793 1.9449 1.9405

2.00 0.22

22.7901 22.6423 22.6235

0.74 0.08

2.88 0.36

2.4508 2.3556 2.3404

4.72 0.65

15.4133 15.2762 15.2538

1.05 0.15

momentum and energy equations. The semiimplicit method for the pressure-linked equations (SIMPLE) is used for solving the pressure−velocity decoupling. The second-order implicit timeintegration scheme is used, and the dimensionless time step (Δt) has been fixed at 0.01 because, for any smaller values of the time step (e.g., 0.005), the changes in the values of global characteristics are found to be negligible. The viscosity of the fluid is calculated by using constant density and non-Newtonian power-law viscosity models. The algebraic equations resulting after linearization are solved by using the Gauss−Seidel iterative method in conjunction with an algebraic multigrid solver (AMG). The AMG method greatly reduces the number of iterations and thus central processing unit time required to obtain a converged solution. Relative convergence criteria of 10−10 for the continuity and the x and y components of the velocity as well as 10−15 for the energy equation are employed in the steady regime. However, a residue of 10−20 is used in the case of the unsteady regime. To investigate the effect of the grid size on the physical parameters, three nonuniform grids of grid sizes of 152280, 155410, and 159000 cells are used, with the smallest grid sizes (δ) of 0.02D, 0.01D, and 0.005D clustered around the circular cylinders, respectively. The domain considered for these calculations refers to an upstream distance (Xu) = 20D, downstream distance (Xd) = 60D, and computational domain width (W) = 40D. The simulations are carried out for Re = 1 and 40, Pr = 50, and Ri = 1 for n = 0.2, 0.4, and 1 (Table 1).

due to pressure distributions and skin friction over the cylinder surface, respectively. The lift coefficient (CL) may be described as the ratio of the lift pressure to the dynamic pressure and is defined as CL = FL/(0.5ρU∞2D) = 2∫ 10P dY + (2/Re)∫ 10(∂U/ ∂Y) dX, where FL is the lift force per unit length of the cylinder. 3.4. Numerical Methodology. In this work, the grid generation and subsequent simulations are carried out by using computational fluid dynamics (CFD) solver Ansys Fluent 13.0. The computational grid generated here has both uniform and nonuniform distributions, with the mesh being fine near the cylinders. The grid distribution in the entire computational domain is shown in Figure 2a. The mesh around the cylinders is subdivided into two zones by a rectangular area, as shown in an expanded view near the cylinders (Figure 2b). The mesh around the cylinders varies from 0.01D to 0.03D for a distance of 0.25D in both the x and y directions. However, for the mesh away from the rectangular boundary (to the distance of 20D from the origin in both the x and y directions), a grid gradation from 0.03D to 0.4D is used. For the remaining portion of the domain in the y direction, a uniform grid spacing of 0.4D is used. The unstructured quadrilateral cells of nonuniform grid spacing are generated using the grid tool Ansys ICEM CFD. The two-dimensional, segregated laminar solver of Ansys Fluent has been employed to solve the incompressible flow equations on the nonstaggered grid arrangement. The second-order upwind scheme is used to discretize convective terms of 17299



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Table 3. Downstream Dependence Study (Xd) for Re = 1 and 40, Ri = 1, and Pr = 50 at Different Values of n Re = 1 total number of cells in the domain

downstream distance (Xd)


137610 155410 173210

40D 60D 80D

28.1997 28.4001 28.3809

0.64 0.07

137610 155410 173210

40D 60D 80D

23.5687 23.5730 23.5713

0.01 0.01

137610 155410 173210

40D 60D 80D

13.2555 13.2561 13.2561

0.005 0.00

% deviation

Re = 40


% deviation


% deviation


% deviation

5.30 0.15

1.6957 1.6963 1.6962

0.03 0.005

28.2710 28.2743 28.2739

0.01 0.002

0.01 0.01

1.9450 1.9449 1.9449

0.01 0.00

22.6430 22.6423 22.6423

0.003 0.00

0.002 0.00

2.3549 2.3556 2.3556

0.03 0.001

15.2753 15.2762 15.2762

0.01 0.00

Table 4. Computational Domain Width (W) Dependence Study for Re = 1 and 40, Ri = 1, and Pr = 50 at Different Values of n Re = 1 total number of cells in the domain

computational domain width (W)


125390 155410 183770

30D 40D 50D

28.3536 28.4001 28.3991

0.16 0.004

125390 155410 183770

30D 40D 50D

23.6470 23.5730 23.5667

0.34 0.03

125390 155410 183770

30D 40D 50D

13.5026 13.2561 13.1811

2.44 0.57

% deviation

Re = 40


% deviation


% deviation


% deviation

0.33 0.01

1.6952 1.6963 1.6980

0.17 0.10

28.2699 28.2743 28.2834

0.05 0.03

0.43 0.04

1.9486 1.9449 1.9450

0.19 0.01

22.6586 22.6423 22.6424

0.07 0.001

0.97 0.23

2.3729 2.3556 2.3508

0.94 0.21

15.3005 15.2762 15.2695

0.20 0.04

both the total drag coefficient and average Nusselt number are found to be negligible at Re = 40 for all n considered here. However, at Re = 1, the percentage relative changes in the total drag coefficient and average Nusselt number are found to be small for Xd = 40D, whereas for Xd = 60D, the corresponding changes are negligible compared to Xd = 80D. Furthermore, the downstream should have a longer length scale than the upstream because the disturbance in the flow in the upstream is less, but at the downstream, separation of the flow and vortex shedding from the cylinders occur.24 Therefore, the downstream distance of 60D is used in this study for better visualization of the flow and temperature characteristics. With the present case being that of unconfined cylinders, the vertical boundaries determine the width of the computational area. The influence of the width of the computational domain (W) has been investigated for the values of W = 30D (125390 cells), 40D (155410 cells), and 50D (183770 cells) for Re = 1 and 40, Ri = 1, Pr = 50, and n = 0.2, 0.4, and 1. The relative differences in the values of the total drag coefficient and average Nusselt number are given in Table 4. It can be observed here that W = 30D is not adequate because the results compared with W = 40D vary significantly. On the other hand, when W = 40D is compared with an even larger domain width (W) = 50D, only slight variation in the results is observed. Hence, a width of 40D is employed here for further calculations.

The percentage relative differences in the values of the total drag coefficient and average Nusselt number with respect to the finest grids are given in Table 1. Thus, it is notable from the table that the grid size of 155410 cells is adequate for the present flow system. The influence of the upstream distance (Xu) on the values of the physical output parameters is investigated for Xu = 10D (130490 cells), Xu = 20D (155410 cells), and Xu = 30D (164310 cells) for Re = 1 and 40, Ri = 1, and Pr = 50 for n = 0.2, 0.4, and 1 (Table 2). The relative percentage differences in the values of the total drag coefficient and average Nusselt number for the cases of shorter upstream distances (10D and 20D) are compared with those of the longest one (30D). For instance, at Re = 40, the relative deviations in the total drag coefficient and average Nusselt number for n = 0.2, 0.4, and 1 are found to be about 0.34%, 0.74%, and 1.05% for Xu = 10D and about 0.01%, 0.08%, and 0.15% for Xu = 20D compared the corresponding values at Xu = 30D. Thus, it can be observed from the table that the results vary considerably when the upstream distance is changed from 10D to 20D, but there is not much change in the results for Xu= 20D and 30D and, hence, an upstream distance of 20D (155410 cells) is used. The influence of the downstream distance (Xd) is examined for Xd = 40D (137610 cells), 60D (155410 cells), and 80D (173210 cells) for Re = 1 and 40, Ri = 1, and Pr = 50 for n = 0.2, 0.4, and 1 (Table 3). The relative percentage changes for 17300



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4. RESULTS AND DISCUSSION The two-dimensional unconfined flow and mixed-convection (i.e., with aiding buoyancy) heat transfer of non-Newtonian power-law fluids (n = 0.2, 0.4, 0.6, and 1) around a pair of sideby-side circular cylinders, covering shear-thinning (n < 1) and Newtonian (n = 1) behaviors, is simulated in the full computational domain for the Reynolds number (Re) = 1− 40, Richardson number (Ri) = 0−1, and Prandtl number (Pr) = 50 for a transverse gap ratio of 1.5. Values of n as low as 0.2− 0.3 are quite common for polymeric systems and particulate slurries.1 Validation of the present numerical results with literature values is reported in subsection 4.1. Further, variation of the flow and temperature patterns and dependence of the individual and overall drag coefficients and Nusselt number on Re, n, and Ri are discussed. Finally, the time histories of dimensionless output parameters (e.g., drag, lift, and Nusselt number) and the Strouhal number are presented and analyzed with Re, n, and Ri. 4.1. Benchmarking of Results. The benchmarking of the current framework is done by comparison with the only powerlaw results of Srinivas et al.11 for a single circular cylinder under the impact of aiding buoyancy in Tables 5 and 6. Mixed

and, hence, the results presented henceforth can be deemed trustworthy. This validates the present numerical solution procedure. Unfortunately, no experimental/numerical results are available in the open literature on the non-Newtonian flow and heat transfer around a pair of side-by-side circular cylinders in aiding buoyancy. 4.2. Flow Patterns. The flow patterns of non-Newtonian power-law fluids around a pair of side-by-side circular cylinders are characterized by analyzing the streamline contours (as shown in Figures 3 and 4 for Re = 1−40, n = 0.2−1, and Ri = 0−1). While considering Figures 3 and 4 of streamline contours, parts A, D, and G of Figure 4 are instantaneous snapshots of unsteady flow and the rest all are steady streamline contours. Also, the streamlines shown in these figures are at identical values of stream functions. Because the two cylinders are in very close proximity, coupled effects of the two cylinders are observed. However, the wake behind the two cylinders is found akin to that behind a single cylinder at low Reynolds numbers. The topology structure is symmetric with the geometrically symmetric line between two cylinders (or at x = 0) for the steady-flow cases. However, in contrast to a single cylinder in the unconfined domain, the front stagnation point of each cylinder in the present flow system is not on its own symmetric line through the cylinder center along the incoming flow direction. It is observed from the instantaneous streamline patterns that in the time-periodic regime the topological structure of the wake is asymmetric. Because the mixedconvection effect is increasing from small to large depending upon the value of the Richardson number, different flow behaviors around the two cylinders are distinguished by way of streamlines. Overall, in the present study, the flow patterns can be classified into four types, namely, (i) single bluff-body steady pattern (SBBSP), (ii) transitional pattern (TP), (iii) separated double-body steady pattern (SDBSP), and (iv) single bluffbody periodic pattern (SBBPP). The categorization of flow patterns here follows the nomenclature used by Chaitanya and Dhiman24 and Kun et al.33 For a better understanding, the flow patterns are discussed individually for the values of Ri at different Re and n values, and then finally the combined effects at different flow parameters are summarized. 4.2.1. Influence of Re and n on the Flow Patterns at Ri = 0. The topology structures of streamlines at Ri = 0 (i.e., forced convention) for different values of Re and n are shown in Figures 3A−C and 4A−C. Here, the flow is found to be steady and symmetric with the geometrically symmetric line between the two cylinders, for shear-thinning fluids (n < 1) for the entire range of Reynolds numbers covered. However, for n = 1, the flow is steady up to Re = 20 and thereafter it becomes timeperiodic. At Re = 1, the flow is uniform and no flow separation is observed, as shown in Figure 3A−C. A similar kind of trend at low Reynolds number is also reported by Srinivas et al.11 for the case of aiding buoyancy around a single cylinder for Re < 5. It can be seen that, as the value of power-law index (n) decreases (or as the shear-thinning tendency increases), the width of the closed or middle region (or residual bulge) between the individual cylinder separation lines tends to be decreased and the separation lines tend to get separated from one another at n = 0.2, as shown in Figure 3A−C. This is due to the thinning tendency of power-law fluids for the same amount of shear force near the cylinder surfaces. At Re = 5, similar characteristics are observed but with comparatively increased overall size (width and height) of the closed region between the

Table 5. Validation of the Present Results for the Drag Coefficient of a Single Cylinder with Srinivas et al.11 at Re = 1 and 40 for Ri = 1 at Different Values of n total drag coefficient (CD) Reynolds number (Re)

power-law index (n)

present work

Srinivas et al.11

1

1 0.6 1 0.2

15.2150 21.1544 2.1250 1.4198

15.1969 21.1702 2.1173 1.4428

40

Table 6. Validation of the Present Results for the Average Nusselt Number of a Single Cylinder with Srinivas et al.11 at Re = 1 and 40 for Ri = 1 at Different Values of n average Nusselt number (Nu) Reynolds number (Re)

power-law index (n)

present work

Srinivas et al.11

1

1 0.6 1 0.2

2.9978 3.4147 15.1479 27.8182

2.9976 3.4165 15.1594 27.4749

40

convection from an unconfined heated horizontal circular cylinder immersed in incompressible power-law fluids in a steady-flow regime with an aiding-buoyancy configuration is simulated with the same computational domain as that used for the present work. The results of extreme mixed-convection cases (i.e., at Ri = 1) for the range of parameters involved in the study are compared and illustrated in Tables 5 and 6 as follows. At Re = 40 for n = 1 and 0.2, the total drag coefficient differs by about 0.36% and 1.60%, respectively. However, at Re = 1, the drag values differ by about 0.12% for n = 1 and about 0.07% for n = 0.6 (Table 5). Similarly, the values of the average Nusselt number differ by about 0.08% for n = 1 and about 1.25% for n = 0.2 at Re = 40. With Re = 1, the average Nusselt number values differ only about 0.006% for n = 1 and about 0.05% for n = 0.6 (Table 6). Thus, the results obtained in the present study are in excellent agreement with the results available in the literature 17301



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Figure 3. Streamline contours around a pair of cylinders in side-by-side arrangement at Re = 1 for varying values of Ri and n [(A-I) SBBSP)].

transitional pattern is observed for n = 0.6. However, for Re = 40, the separated double-body steady pattern is found for all values of the power-law index (Figure 4B,C). From these figures, it is also observed that, as the power-law index decreases, the separated double-body size as well as elongation increases. The size of the corresponding wake patterns also were found to increase with an increase in the Reynolds number. As expected, early formation of the double-body steady pattern is observed as the shear-thinning tendency increases. Also, the double-body patterns are found to be in an antiphase for the entire range of investigations. 4.2.2. Influence of Re and n on the Flow Patterns at Ri = 0.5. The effects of Re and n on the streamlines for the current settings of study at Ri = 0.5 are shown in Figures 3D−F and 4D−F. For Ri = 0.5, the wake characteristics are observed to be similar in comparison with the corresponding cases at Ri = 0 for Newtonian fluids (n = 1) for Re ≤ 5, but the closed region between individual cylinder separation lines (residual bulge) is comparatively smaller in size. At Re = 10 and n = 1, a symmetric vortex pair in the downstream side is not observed in contrast to the pure forced-convection case (Ri = 0), while the single bluff-body periodic flow pattern (or time-periodic regime) is observed for Re ≥ 20 at n = 1 (Figure 4D, for instance). The transitional pattern is also observed for power-law fluids, except the separated double-body steady pattern encountered

separation lines (residual bulge) than those at Re = 1. Further, the tendency toward the formation of separate wakes (downstream of the cylinders) is observed with increasing Reynolds number. A symmetric vortex pair in the downstream side is observed for n = 1 and Re = 10; however, for other values of n at the same Reynolds number, no flow separation is observed, but the middle region (residual bulge) broadening tendency is found to increase in comparison to Re = 5. As Re is increased to 20 at n = 1, a vortex pair that is found to be elongated and pushed down toward downstream side is experienced. For Re ≥ 30 and n = 1, periodic unsteady-flow behavior begins, which can be envisaged by the instantaneous asymmetric streamlines and the vortex-shedding pattern (i.e., single bluff-body periodic pattern), as shown in Figure 4A. For shear-thinning fluids for Re < 20, the single bluff-body pattern with a closed region between the separation lines (or residual bulge) similar to the Newtonian case is observed. At Re = 20 with n = 0.2, the initiation of two small independent regions of wakes attached to each of the cylinders is observed (or here called the transitional pattern); however, no such independent small regions are observed for n > 0.2. This transitional pattern is found to be transforming into two independent regions of large recirculation zones, which are attached to the corresponding cylinders (called the separated double-body steady pattern) for Re = 30 for n ≤ 0.4, but the 17302



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Figure 4. Streamline contours around a pair of cylinders in side-by-side arrangement at Re = 40 for varying values of Ri and n [(A, D, G) SBBPP; (B, C, F) SDBSP; (E, H, I) TP].

at n = 0.2 for Re = 40 (Figure 4F). Here, the size of the closed region between the individual cylinder separation lines (residual bulge) decreases as the shear-thinning behavior increases. The reduction in the size of the residual bulge is also true when the value of Re is increasing for the fixed value of n. 4.2.3. Influence of Re and n on the Flow Patterns at Ri = 1. The flow characteristics here are observed to be similar to those at Ri = 0.5, but the respective patterns are found to be narrowed down in size (Figures 3G−I and 4G−I) for the range of physical parameters studied. Similar to Ri = 0.5, the time periodicity and single bluff-body periodic flow pattern are observed from Re = 20 onward for n = 1. For all other n values used in this study, the flow is found to be transitional at Re = 20. At Re = 40, cases with n = 0.4 and 0.2 resulted in the separated double-body and transitional patterns, respectively. Overall, the flow patterns for shear-thinning fluids (n < 1) are found as steady/transitional for the range of settings of 1 ≤ Re ≤ 40 and 0 ≤ Ri ≤ 1. The tendency for the change of the single bluff-body behavior through the transitional pattern to the separated double-body steady pattern increases with increasing Reynolds number and shear-thinning behavior for Ri = 0 (Figures 3C and 4C, for instance), whereas for Ri > 0, the effects are observed to be diminished and the above-mentioned change of the flow patterns is not encountered for the range of

conditions covered here. The closed region between the individual cylinder separation lines (i.e., residual bulge) is found to narrow in size with an increase in Ri. Also, the change from the double-body steady pattern to the single bluff-body steady pattern through the transitional pattern occurs with an increase in Ri, and these effects are more prominent at higher Re and lower n. This kind of behavior of the flow patterns with an increase in Ri at the rear of the cylinders is due to the dominance of the inertial force over the viscous force, resulting in the suppression of double-body formation and narrowing of the closed region because of the shifting of the point of separation toward the axis of symmetry. The shear thinning is caused by particles adopting a more flow-oriented arrangement, hence contributing to the above effects at low n values. Physically, the effect of an increase in Ri and a decrease in n causes the retention of steady laminar flow even at the higher Re considered here. A similar kind of observation in the case of a single cylinder under the influence of aiding buoyancy is also reported by Gandikota et al.9 Further, for Newtonian fluids (n = 1), the flow patterns are steady for all cases of Ri for Re ≤ 20, and for Re > 20, the flow becomes time-periodic. With increasing aiding buoyancy, the size of the wake region behind the cylinders is found to be reduced in both steady and (periodic) unsteady regimes. 17303



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Figure 5. Variation of the surface viscosity η(ϕ) with the angle measured from the front stagnation point of the cylinder ϕ for varying values of Re, Ri, and n.

4.3. Thermal Patterns. The heat-transfer characteristics are studied by using isotherm contours at different values of Re, n, and Ri, as shown in Figures 6 and 7. It is found that the isotherm profiles are analogues to the streamline patterns; therefore, the isotherm patterns can be classified in the same way as mentioned in subsection 4.2 by single bluff-body steady, a transitional, and separated double-body steady, and single bluff-body periodic patterns. Further, isotherms are found to be steady for shear-thinning fluids (n < 1) for the entire ranges of Re and Ri. However, for Newtonian fluids (n = 1), at Ri = 0, the isotherms are found to be steady up to Re = 20, and when Ri > 0, the isotherms become time-periodic for Re ≥ 20. In the steady regime, isotherm contours are found to be symmetric (about the channel centerline) like the streamlines.

To gain some more physical insight, the influence of dimensionless control parameters (Re, Ri, and n) on the viscosity in the vicinity of the cylinder is represented in Figure 5A−I. For instance, the results obtained for one of the cylinders (left cylinder in Figure 1) are shown in Figure 5, where ϕ and η(ϕ) are the angle measured from the front stagnation point of the cylinder and the surface viscosity at angle ϕ, respectively. For Newtonian fluids, the local viscosity over the cylinder is found to be constant and remains the same with changing shear strain rate. However, for shear-thinning fluids (n < 1), the viscosity changes with the rate of shear strain, becomes very large as the shear rate decreases, and tends to infinity when the shear rate is zero. Qualitatively, the maximum value of the viscosity can be seen at the point of separation for shearthinning fluids. 17304



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Figure 6. Isotherms around a pair of circular cylinders in a side-by-side configuration at Re = 1 for varying values of Ri and n.

It can be seen from Figures 6 and 7 that for all Ri and Re studied, the overall size of the isotherms is found to decrease with an increase in the shear-thinning behavior or with a decrease in the value of the power-law index (n). For instance, this effect can be clearly observed for Re = 1, 20, and 40 by comparing the cases of n = 0.4 and 0.2 in Figures 6 and 7 at a constant value of Ri. Also, the separation gap between isotherms behind cylinders (representing transitional and separated double-body steady patterns) is found to decrease as the value of the power-law index (n) decreases for Re > 10, but at lower values of Re (≤10), single bluff-body steady patterns (or merged isotherms) are observed, while for Newtonian fluids, for Re < 20, single bluff-body steady isotherm patterns are observed. Also, as expected, the crowding of the isotherms behind the cylinders is found to increase with an increase in Re. This is due to the dominance of the inertial forces over the viscous forces in the vicinity of the cylinders. Further, at Re = 20 for the pure forced-convection Newtonian case (i.e., Ri = 0), an eddy with symmetric nature inside the single bluff-body pattern is observed, whereas isotherms become time-periodic with the introduction of aiding buoyancy at this Re. For all other Newtonian cases (Re > 20), the unsteady nature of the isotherms is observed. Similar to the effect of the power-law index, increasing the value of Ri from 0 to 1 also favors reduction of the overall size of the isotherms and/or the gap between the isotherms of shear-thinning fluids and hence reduces the temperature distribution. For instance, when the temperature distribution behind the cylinders for Ri = 0 is compared with that for Ri = 1 at a constant value of n, the isotherm distribution narrows

which causes a higher temperature gradient and results in an increased heat-transfer rate with an increase in Ri. This effect is observed for all Re (Figures 6 and 7) and is found to be more predominant at higher Reynolds numbers for shear-thinning fluids. On the other hand, the asymmetry in the isotherm patterns (referred to as single bluff-body periodic patterns) with aiding buoyancy for Newtonian fluids is observed for Re ≥ 20 in this study (Figure 7A,D,G). The extent of asymmetry is characterized by the observed distortion and the induction of waviness in the isotherms, where the periodicity in the isotherm patterns is found to increase as Ri is increased. This waviness in the isotherm patterns can even be observed to persist like the vortex-shedding phenomena in flow patterns, while at Re = 20 in the absence of aiding buoyancy (i.e., the forced-convection case), the single bluff-body steady pattern of the isotherms with an eddy inside is observed. However, for Re < 20, the single bluff-body steady pattern of the isotherms is observed for Newtonian fluids and the crowding of the isotherms is found to increase with an increase in Ri (Figure 6, for instance). In general, the size of the plume produced from the two cylinders decays (or the crowding of the isotherms in the downstream direction) with an increase in both the shearthinning behavior and aiding buoyancy. Also, the temperature contours are denser around the front of the cylinders than its rear, indicating that the convective cooling of the cylinder surface is more intense in the upstream side. Thus, the heattransfer rate is maximum from the front of the cylinders and is found to be more significant at higher values of Re and Ri and lower values of n. Furthermore, no unsteadiness or wavy nature of the isotherm patterns is observed in the case of shear17305



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Figure 7. Isotherms around a pair of circular cylinders in a side-by-side configuration at Re = 40 for varying values of Ri and n.

thinning fluids under aiding buoyancy because of the fact that the values of the effective fluid viscosity decrease with increasing degree of the shear-thinning behavior, which increases the flow rate close to the surfaces of the cylinders, consequently reducing the effect of buoyancy forces on the isotherms. 4.4. Dependence of the Drag Coefficients on Re, n, and Ri. The frictional drag (CDF) and pressure drag (CDP) both contribute to the total hydrodynamic drag (CD) exerted by the fluid on the cylinders (i.e., CD = CDP + CDF). Variation of the coefficients of the individual drags (only for the steady region) and of the total drag shows a complex dependence on the dimensionless control parameters (Re, n, and Ri), as can be seen in Figures 8 and 9. The individual and total drag coefficients for both cylinders were found to be identical in magnitude for the range of settings studied. The flow and temperature fields in the vicinity of the cylinders are determined by a complex interplay among viscous, inertial, and buoyancy forces prevailing in the fluid. For a fixed value of the velocity, the inertial force does not change, but the viscous force can decrease or increase depending upon the value of the power-law index. Conversely, for a given fluid, with a gradual increase in the fluid velocity, the viscous term will diminish for a shear-thinning fluid (n < 1). Therefore, some of the nonmonotonic trends observed for drags in the preceding sections are due to such complex interactions. It is also worthwhile to add here that the level of shearing in the powerlaw fluids is likely to be more severe in the mixed-convection regime than in the forced-convection regime because of the additional contributions from the buoyancy-induced flow. The various trends of the individual and overall drag coefficients are

observed (as discussed below) for the different values of Re, Ri, and n. Parts A−C of Figure 8 present variation of the pressure drag coefficient (CDP) with Re and n for Ri = 0, 0.5, and 1. As expected, the value of CDP decreases with an increase in Re for all n and Ri. Variation in the values of CDP with n for Ri = 0 is found to be different from that of Ri = 0.5 and 1. For instance, at Ri = 0, the values of CDP decrease with increasing n (or as the shear-thinning tendency decreases) for 1 ≤ Re < 35 and thereafter (for 35 < Re ≤ 40) CDP increases with n. Therefore, a crossover among the lines of constant n values at Re ∼ 35 is observed, as can be seen in the magnified view of Figure 8A. A similar trend is also found to exist when the effect of aiding buoyancy is introduced (Ri > 0). However, the crossover point was found to be shifted to Re ∼ 30 at Ri = 0.5 and further to Re ∼ 25 in the case of Ri = 1. These observations are also consistent with the power-law fluid flow around side-by-side cylinders in a horizontal configuration at Ri = 0.24 At small values of the power-law index, the viscosity of the ambient fluid near the cylinders decreases. This results in an increase in the fluid velocity and causes a decrease in the pressure drop. This pressure drop is the reason for the increase in CDP for small values of n, even though the flow separation is relatively small at low Re. The viscous drag coefficient (CDF) is found to be higher for shear-thinning fluids than Newtonian fluids at low Re (∼< 5). From the definition of CDF, it is envisaged that an increased ycomponent velocity due to the shear-thinning nature and low Re values causes CDF to be at the higher side for shear-thinning fluids than Newtonian fluids, whereas an opposite trend is observed for Re > 5 for all values of Ri in the steady regime 17306



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Figure 8. Variation of the pressure drag coefficient (CDP), viscous drag coefficient (CDF), and CDP/CDF with Re, n, and Ri.

weaker dependence on Re compared to Newtonian fluids in this region. To present the dependence of CDF and CDP for the powerlaw fluid flow around side-by-side cylinders in aiding buoyancy, variation of the drag ratio (CDP/CDF) at different values of Re, n, and Ri is depicted in Figure 8G−I. For all cases of Ri studied, it can be envisaged that the contribution of the pressure drag coefficient (CDP) to the total drag coefficient (CD) is considerably higher in shear-thinning fluids than Newtonian fluids. Also, this contribution increases with increasing Re but decreases with an increase in Ri. Parts A−C of Figure 9 illustrate variation of the total drag coefficient (CD) with Re, n, and Ri along with the respective magnified views on the right-hand side, i.e., Figure 8A1−C1. Variations of the total drag coefficient (CD) are found to be similar to that of the viscous drag coefficient (CDF), but the contribution of the pressure drag coefficient (CDP) to the total

(Figure 8D−F). As a result, a crossover in the values of friction drag occurs at Re ∼ 5 and Re < 5 for Ri = 0 and Ri > 0, respectively. Therefore, a mixed dependence of the viscous drag coefficient (CDF) on Re, n, and Ri exists. It is also evident that the flow behavior index (n) exerts a stronger influence at low Reynolds numbers than at high Reynolds numbers. This is simply because of the fact that the role of viscosity diminishes with increasing Reynolds number. Similar to pressure drag, the value of CDF decreases with increasing Re for all Ri and n. Furthermore, before the crossover point, the slope of the CDF lines decreases with increasing power-law index (n) because the shear-thinning fluids exhibit a stronger dependence on Re in comparison to Newtonian fluids. The opposite trend is observed above the crossover point, where the value of CDF with an increase in Re increases with the power-law index (n), and this trend indicates that the shear-thinning fluids have a 17307



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Figure 9. Variation of the total drag coefficient (CD) with Re for n = 0.2−1 at Ri = 0−1 (A−C) and magnified views of the point of crossover (A1− C1).

over a pair of side-by-side circular cylinders in a horizontal configuration,24 the average Nusselt number and, hence, increased heat transfer from the increased contribution of convection with increasing Re is observed for fixed Ri and n. This higher heat transfer is due to the higher fluid velocity with an increase in the Reynolds number, which causes fast movement of the fluid medium and increased rate of advection, thereby leading to enhanced heat transfer from the surfaces of the two cylinders. As expected, heat transfer in the shear-

drag coefficient (CD) shifts the point of crossover to a high value of the Reynolds number. Therefore, the crossover in the values of CD for all fluids occurs at Re ∼ 20, 15, and 10 for Ri = 0, 0.5, and 1, respectively (Figure 9A1−C1). The drag force is also a measure of the expected pressure drop in the system. 4.5. Dependence of the Average Nusselt Number on Re, n, and Ri. Variation of the surface average Nusselt number (Nu) with Re, n, and Ri for a constant Prandtl number of 50 is depicted in Figure 10A−C. Similar to cross-flow heat transfer 17308



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transport rate in comparison to advection, which results a decrease in the Peclet number (Pe). In shear-thinning fluids, the particles adopt a more flow-oriented arrangement, and because of this, the relative importance of the diffusion transport rate increases. This explains the reason for an increase in the Nusselt number with shear thinning and aiding buoyancy. Also, it was found that the average Nusselt number increases with an increase in the buoyancy parameter (Ri) for any Re and n studied. However, the difference in the values of the average Nusselt number for the constant n lines increases with increasing Ri and more rapidly in shear-thinning fluids than in Newtonian fluids, indicating a stronger dependence of shearthinning fluids on Re compared to Ri. The maximum increment of heat transfer is observed for shear-thinning fluids at Ri = 1 for Re = 40. For instance, the percentage increase in the value of the average Nusselt number is found to be approximately 85% as the value of the power-law index changes from 1 to 0.2. On the other hand, the maximum enhancement in the value of the average Nusselt number is found to be approximately 81% (at Ri = 0.5) and approximately 70% (at Ri = 0). Furthermore, a heat-transfer correlation (eq 12) is introduced to establish a functional relationship among the Reynolds number, Richardson number, power-law index, and average Nusselt number from a total of 72 data points studied. This simple correlation can be utilized to calculate the intermediate values of the average Nusselt number for a range of values of various physical parameters investigated. Nu = 1.83Re 0.55n−0.38e 0.188Ri

(12)

Clearly, the average Nusselt number increases with increasing Reynolds and Richardson numbers but decreases with the power-law index. This correlation has an average deviation of about 3.81% and a maximum deviation of about 9.96% with the present computed results. Generally speaking, an increase in the Prandtl number results in an increase in the average Nusselt number, as reported by Moshkin and Sompong43 for Newtonian fluids around two rotating cylinders in a cross-flow arrangement. Likewise, the average Nusselt number increases with increasing Prandtl number for any Re and n for the flow across a single circular cylinder.1,44 Further, the heat-transfer results obtained for the problem under consideration are analogous to those reported for a single cylinder and, hence, the same could be attributed to variation of the heat-transfer characteristics with the Prandtl number. However, the specific flow and heat-transfer phenomena at varying values of the Prandtl number would be the scope for further research. Finally, in the present work, the definitions of the Reynolds and Prandtl numbers employ the free-stream velocity as the characteristic velocity scale, which does not account for the additional flow induced by buoyancy effects. It is thus proposed here to use the sum of the free-stream velocity and that induced by buoyancy effects as the characteristic velocity (Uch) of the aiding flow as follows:37,45

Figure 10. Variation of the surface average Nusselt number (Nu) with Re for n = 0.2−1 and Ri = 0−1.

thinning fluids (n < 1) is found to be considerably higher than that in the Newtonian fluids (n = 1) for the range of conditions covered here. This is due to the fact that, as the value of the power-law index decreases, for the same magnitude of inertial forces around cylinder surfaces, the effective viscosity decreases drastically and, hence, the heat-transfer rate increases. With an increase in the Richardson number (Ri), the potential energy of the molecules increases; this causes an increase in the diffusion

Uch = U∞ +

Dgβ ΔT

(13)

The corresponding scale for the effective shear rate accordingly is given by Uch/D, and this, in turn, leads to the following modified definitions of the Reynolds (Re*) and Prandtl (Pr*) numbers:37 Re* = Re(1 + Ri1/2)2−n and Pr* = Pr(1 + Ri1/2)n−1. The new definitions also offer the possibility of reconciling the results for forced convection (Ri = 0) and mixed convection 17309



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(Ri ≠ 0). Parts A and B of Figure 11 show representative plots in terms of the modified Reynolds number (Re*) for different

and Nusselt number) are obtained by taking the arithmetic average over 10 periodic cycles. The typical time histories of CD, CL, and Nu for steady and periodic unsteady regimes for extreme cases of Ri at different Re and n = 1 are shown in Figure 12A−D. For all simulations carried out here, the drag and lift coefficients are exhibiting only a single frequency in their respective oscillations, whereas a multiple frequency in the oscillations is a characteristic of three-dimensional flow. So, the two-dimensionality employed for simulations in the Reynolds number range explored in the present study is justified. It is evident from the figures that the frequency of the oscillation in the time history of the above-mentioned parameters (CD, CL, and Nu) is found to increase with an increase in the Reynolds and Richardson numbers. The periodicity is induced in the flow field as a result of the vortex-shedding phenomenon and is characterized by the Strouhal number (St). The Strouhal number (St) is a dimensionless parameter used to measure the frequency of vortex shedding in the fully periodic regime and is mathematically represented as St = f D/U∞. Variation of the Strouhal number (St) with respect to Ri and Re for n = 1 is explored in Figure 13. The temporal variation of the lift coefficient is used to calculate the frequency of vortex shedding (f). The Strouhal number becomes zero at the steady state because of the attachment of vortices symmetrically to the rear part of the cylinders about the midplane. On the other hand, in the timeperiodic regime, the Strouhal number is found to increase with an increase in Re. Similarly, the Strouhal number increases with aiding buoyancy (0 < Ri ≤ 1) as the heating increases the velocity in the wake region and causes the shear layer to be weakened. Also, the entrainment of the ambient fluid into the wake cavity is less easy under the impact of aiding buoyancy and results in a faster vortex shedding. A similar kind of phenomenon is also observed by Patnaik et al.4 for cross-flow over a single cylinder. The Strouhal numbers (St) at which the transition is occurring for Newtoinan and non-Newtonian fluids are also studied. For the maximum limit of aiding buoyancy studied here (i.e., at Ri = 1), the Strouhal numbers are found to be 0.2976, 0.0769, 0.0818, and 0.0734 for power-law indices of 0.2, 0.4, 0.6, and 1 at critical Reynolds numbers of 59, 49, 47, and 20, respecively. Here, the critical Reynolds number is the Reynolds number at which the transition occurs from a steady to a time-periodic regime for Newtonian and non-Newtonian fluids. 4.7. Transition from a Steady to a Time-Periodic Regime. Temporal variation of the lift coefficient (CL) is used to identify the steady to unsteady transition with an increase in the Reynolds number for Newtonian and non-Newtonian fluids at different values of Ri studied. For the steady cases, CL is found to be constant as the simulation time progresses, whereas for unsteady cases, an oscillation with constant amplitude in CL resulted. For a better understanding, the time history of the lift coefficient showing a steady to an unsteady transition with an increase in Re is shown in Figure 14A−D at the extreme values of the power-law index and Richardson number. The critical Reynolds number at different Richardson numbers is depicted in Figure 15. The critical Reynolds number is found to decrease with an increase in aiding buoyancy for Newtonian fluids. Even though the Reynods numbers of transitions are different, the slope of the transition line for the case of n = 0.6 is found to be similar to that of Newtonian fluids. As the shear-thinning tendency increases (or

Figure 11. Variation of the surface average Nusselt number (Nu) with modified Reynolds number (Re*) for n = 0.2−1 and Ri = 0.5 and 1.

values of the power-law index at Ri = 0.5 and 1. Obviously, this approach is found to yield a family of curves corresponding to the value of the modified Prandtl number (Pr*). For the forced-convection case, no changes are observed in the plots and are found to be the same as those in Figure 10A. As expected, it is observed that heat transfer is enhanced in shearthinning fluids compared to that in Newtonian media.37 An important observation that one can recognize here is that, owing to the nonlinearity of the governing equations for both free and forced-convection conditions, there is no theoretical justification for the addition of two velocities and, hence, for postulating the form of eq 13.37 On the other hand, this empirical approach does allow consolidation of the present results spanning wide ranges of Re, Ri, and n with acceptable levels of precision. This approach is also implicit in the works of Meissner et al., 45 Cameron et al., 46 and Wang and Kleinstreuer.47 4.6. Time History and Strouhal Number. For all timeperiodic cases, simulation is stopped and is said to have converged after 10 constant periodic cycles in CD, CL, and Nu. Further, the time-average values of the flow parameters (drag 17310



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Figure 12. Time history of the Nusselt number and total drag and lift coefficients for Newtonian fluids with Ri = 0 and 1 at Re = 20 and 40.

5. CONCLUSIONS The two-dimensional upward flow and heat transfer of powerlaw fluids across two cylinders in a side-by-side arrangement with aiding buoyancy are studied for values of the Reynolds number (1 ≤ Re ≤ 40), power-law index (0.2 ≤ n ≤ 1), and Richardson number (0 ≤ Ri ≤ 1) for a Prandtl number of 50. Some of the major findings of this work are as follows: (i) For all Ri, flow and heat transfer remain steady for shearthinning fluids (n < 1) for the entire range of Re; however, for Newtonian fluids, flow and heat transfer exhibit time-periodic behavior for Re ≥ 30 and Re ≥ 20 for Ri = 0 and Ri > 0, respectively. (ii) The crossover in the values of CD for all fluids occurs at Re ∼ 20, 15, and 10 for Ri = 0, 0.5, and 1 respectively. (iii) The average Nusselt number increases with an increase in both Re and Ri but decreases with an increase in n. The average Nusselt numbers are found to be higher for shearthinning fluids than Newtonian fluids. (iv) The maximum enhancement in heat transfer is found to be approximately 85% for Re = 40 and n = 0.2 at Ri = 1. However, the maximum enhancement in the value of the average Nusselt number is found to be approximately 81% (at Ri = 0.5) and approximately 70% (at Ri = 0). Broadly, a higher heating and shear-thinning tendency facilitates increased heat transfer. (v) The critical Reynolds numbers (i.e., transition of Newtonian and non-Newtonian fluids from a steady to a time-periodic regime) are determined.

Figure 13. Variation of the Strouhal number (St) with Re and Ri for n = 1.

with decreasing value of n), the slopes of the transtion lines are found to be reversed because of the highly nonlinear nature of non-Newtonian fluids. A crossover point is also observed among transition lines of non-Newtonian fluids below Ri = 0.5. At this crossover point, there exists a typical combination of Ri and Re at which a change of the non-Newtonian behavior will not affect the transition from steady to unsteady nature. 17311



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Figure 14. Time history of the lift coefficient (CL) showing a steady (LHS) to an unsteady (RHS) transition with an increase in Re for n = 0.2 at Ri = 0 and 1.

FDP

pressure drag force per unit length of the cylinder, N/ m FL lift force per unit length of the cylinder, N/m Gr Grashof number, (gβΔTρ2D3/m2)(U∞/D)2(1−n) acceleration due to gravity, m/s2 g h average convective heat-transfer coefficient, W/m2·K I1, I2, I3 first, second, and third invariants of the rate of the strain tensor, s−2 thermal conductivity of the fluid, W/m·K k length of the computational domain in the y direction, L m power-law consistency index, Pa·sn m n power-law index average Nusselt number, hD/k Nu P pressure Pr Prandtl number, (mCp/k)(U∞/D)n−1 Pr* modified Prandtl number Re Reynolds number, ρDnU∞2−n/m Re* modified Reynolds number Ri Richardson number, Gr/Re2 = gβΔTD/U∞2 St Strouhal number, f D/U∞ ΔT temperature difference t time Δt time step U∞ free-stream velocity of the fluid at the inlet, m/s Uch characteristic velocity, m/s component of the velocity in the x direction, m/s Vx component of the velocity in the y direction, m/s Vy W width of the computational domain in the x direction, m Xd downstream distance, m upstream distance, m Xu transverse coordinate, m x y streamwise coordinate, m

Figure 15. Re versus Ri at which there is transition in the Newtonian and non-Newtonian fluids from a steady to a time-periodic regime.

(vi) Finally, a simple heat-transfer correlation is established for the values of Re, n, and Ri covered here.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: dhimuamit@rediffmail.com or [email protected]. Tel.: +91-1332-285890 (office), +91-9410329605 (mobile). Notes

The authors declare no competing financial interest.

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NOMENCLATURE CD total drag coefficient, 2FD/ρU∞2D friction drag coefficient, 2FDF/ρU∞2D CDF CDP pressure drag coefficient, 2FDP/ρU∞2D lift coefficient, 2FL/ρU∞2D CL specific heat of the fluid, J/(kg·K) Cp D diameter of a cylinder, m f frequency of the oscillation, s−1 FD drag force per unit length of the cylinder, N/m FDF viscous drag force per unit length of the cylinder, N/ m

Greek Symbols

ε δ η̅ θ 17312

rate of the deformation tensor, s−1 smallest grid size, m power-law viscosity, Pa·s temperature, K dx.doi.org/10.1021/ie4027742 | Ind. Eng. Chem. Res. 2013, 52, 17294−17314

Industrial & Engineering Chemistry Research θw θ∞ ρ τ̅ β σ φ

Article

(19) Chatterjee, D. Mixed convetion heat transfer from tandem square cylinders in a vertical channel at low Reynolds numbers. Numer. Heat Transfer, Part A 2010, 58, 740−755. (20) Chatterjee, D.; Raja, M. Mixed convection heat transfer past inline square cylinders in a vertical duct. Therm. Sci. 2013, 17, 567−580. (21) Niu, J.; Zhu, Z. Numerical study of three-dimensional flows around two identical square cylinders in staggered arrangements. Phys. Fluids 2006, 18 (044106), 1−14. (22) Sarkar, S.; Dalal, A.; Biswas, G. Mixed convective heat transfer from two identical square cylinders in cross flow at Re = 100. Int. J. Heat Mass Transfer 2010, 53, 2628−2642. (23) Chhabra, R. P.; Rami, K.; Uhlherr, P. H. T. Drag on cylinders in shear thinning viscoelastic liquids. Chem. Eng. Sci. 2001, 56, 2221− 2227. (24) Chaitanya, N. S. K.; Dhiman, A. K. Non-Newtonian power-law flow and heat transfer across a pair of side-by-side circular cylinders. Int. J. Heat Mass Transfer 2012, 55, 5941−5958. (25) Villimpoc, V.; Cole, R.; Sukanek, P. C. Heat transfer in Newtonian liquids around a circular cylinder. Int. J. Heat Mass Transfer 1990, 33, 447−456. (26) Dhiman, A. K.; Anjaiah, N.; Chhabra, R. P.; Eswaran, V. Mixed convection from a heated square cylinder to Newtonian and power-law fluids. J. Fluids Eng. 2007, 129, 506−513. (27) Dhiman, A. K.; Chhabra, R. P.; Eswaran, V. Steady mixed convection across a confined square cylinder. Int. Comm. Heat Mass Transfer 2008, 35, 47−55. (28) Dhiman, A. K.; Sharma, N.; Kumar, S. Wall effects on the crossbuoyancy around a square cylinder in the steady regime. Braz. J. Chem. Eng. 2012, 29, 253−264. (29) Bouaziz, M.; Kessentini, S.; Turki, S. Numerical prediction of flow and heat transfer of power-law fluids in a plane channel with a built-in heated square cylinder. Int. J. Heat Mass Transfer 2010, 53, 5420−5429. (30) Sumner, D.; Wong, S. S. T.; Price, S. J.; Paidoussis, M. P. Fluid behaviour of side-by-side circular cylinders in steady cross-flow. J. Fluids Struct. 1999, 13, 309−338. (31) Shao, J.; Zhang, C. Numerical studies of flow past two side-byside circular cylinders, New Trends in Fluid Mechanics Research. Proceedings of the Fifth International Conference on Fluid Mechanics, Shanghai, 2007; Springer: Berlin, 2009; ISBN 978-3-540-75995-9. (32) Yoon, H. S.; Seo, J. H.; Kim, J. H. Laminar forced convection heat transfer around two rotating side-by-side circular cylinder. Int. J. Heat Mass Transfer 2010, 53, 4525−4535. (33) Kun, L.; Dong-Jun, M. A.; De-Jun, S.; Xie-Yuan, Y. Wake patterns of flow past a pair of circular cylinders in side-by-side arrangement at low Reynolds numbers. J. Hydrodyn. 2007, 19, 690− 697. (34) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 2000. (35) Chhabra, R. P.; Soares, A. A.; Ferreira, J. M.; Caramelo, L. Effects of viscous dissipation on heat transfer between an array of long circular cylinders and power-law fluids. Can. J. Chem. Eng. 2007, 85, 808−816. (36) Tanner, R. I. Engineering Rheology, 2nd ed.; Oxford University Press: London, 2000. (37) Nirmalkar, N.; Chhabra, R. P. Mixed convection from a heated sphere in power-law fluids. Chem. Eng. Sci. 2013, 89, 49−71. (38) Hawared, S. J.; Sharma, V.; Butts, C. P.; McKinley, G. H.; Rahatekar, S. S. Shear and extensional rheology of cellulose/ionic liquid solutions. Biomacromolecules 2012, 13, 1688−1699. (39) McKinley, G. H. Steady and transient motion of spherical particles in viscoelastic liquids, Transport Processes in Bubbles. Drops and Particles, 2nd ed.; Taylor & Francis: New York, 2002. (40) Kim, N. J.; Pipe, C. J.; Ahn, K. H.; Lee, S. J.; McKinley, G. H. Capillary breakup extensional rheometry of a wormlike micellar solution. Korea−Australia Rheol. J. 2010, 22, 31−41. (41) Pipe, C. J.; McKinley, G. H. Microfluidic rheometry. Mech. Res. Comm. 2009, 36, 110−120.

temperature at the surface of the cylinder, K temperature of the fluid at the inlet, K density of the fluid, kg/m3 stress tensor, Pa coefficient of volume expansion, K−1 total stress tensor, Pa dimensionless temperature

Superscripts

- dimensional quantity

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REFERENCES

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Aiding-Buoyancy Mixed Convection from a Pair of Side-by-Side

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