Mixed Convection Heat Transfer from a Cylinder in Power-Law Fluids

Ind. Eng. Chem. Res. 2009, 48, 9735–9754

9735

Mixed Convection Heat Transfer from a Cylinder in Power-Law Fluids: Effect of Aiding Buoyancy Avadhani T. Srinivas, Ram P. Bharti,† and Rajendra P. Chhabra* Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, U.P., India

Mixed convection heat transfer from an isothermally heated horizontal cylinder immersed in incompressible power-law fluids is considered here in the steady flow regime when both the imposed flow and the buoyancy induced motion are in the same direction, i.e., the so-called buoyancy aiding configuration. Within the framework of the Boussinesq approximation, the suitable forms of the momentum and thermal energy equations for the power-law fluid model have been solved numerically using the finite volume based FLUENT (version 6.3) solver for the following ranges of conditions: the buoyancy parameter (Richardson number, 0 e Ri e 2), power-law index (0.2 e n e 1.8), Reynolds number (1 e Re e 40), and Prandtl number (1 e Pr e 100). In particular, the effects of these dimensionless parameters on the detailed local and global kinematics of flow and heat transfer characteristics such as streamline, vorticity, and pressure profiles, individual and total drag coefficients, and local and average Nusselt numbers have been presented. The wake size shows trends which are qualitatively similar to that seen in the pure forced convection (Ri ) 0) regime, though it decreases with increasing Richardson number (Ri) and/or Prandtl number (Pr). The pressure coefficient decreases with the increasing values of Reynolds number (Re) and Prandtl number (Pr) and with decreasing Richardson number (Ri). It is, however, seen to be relatively insensitive to the value of power-law index (n). Both drag coefficients and average Nusselt number are augmented with the increasing buoyancy effects, Reynolds (Re) and, Prandtl (Pr) numbers. An increase in the shear-thinning tendency of the fluid enhances the drag and heat transfer, whereas both of these are generally reduced in shear-thickening fluids. Similarly, the fluid behavior also modulates the role of Richardson number (Ri); namely, the pressure forces dominate over the viscous forces in shear-thinning fluids and vice versa in shear-thickening fluids. Therefore, the buoyancy effects are found to be stronger in shear-thinning fluids and/or at low Reynolds numbers than that in shear-thickening and/or at high Reynolds numbers. 1. Introduction Owing to their wide ranging applications, considerable research efforts have been devoted toward exploring the rates of heat and momentum transfer from cylinders immersed in Newtonian fluids for well over 100 years now.1,2 Suffice it to say that adequate information has accrued over the years on most aspects including global characteristics (like drag, lift and Nusselt number, for instance), wake phenomena (like wake length, angle of separation, type of wake, etc.), and vortex shedding phenomena, etc. for Newtonian fluids. On the other hand, many substances of multiphase nature and/or of high molecular weight encountered in industrial practice (pulp and paper suspensions, food, polymer melts and solutions, etc.) also display varying levels of shear-thinning and/or shear-thickening behavior, viscoelasticity, yield stress, etc.3 Owing to their high viscosity levels, these materials are generally processed in the laminar flow conditions. Furthermore, the currently available viscoelastic simulations examine the role of viscoelasticity in the absence of the shear-dependent viscosity in the limit of zero Reynolds number (creeping flow), and under these conditions, the effect of fluid elasticity is predicted to be small. On the other hand, few experimental studies available on this flow employ experimental fluids which show both shear-dependent viscosity and viscoelasticity.4,5 Therefore, it is neither justified to compare these experiments with numerical predictions nor * To whom correspondence should be addressed. Tel.: +91-512259 7393. Fax: +91-512-259 0104. E-mail: [email protected]. † Present address: Department of Chemical & Biomolecular Engineering, Melbourne School of Engineering, The University of Melbourne, Parkville 3010, Victoria, Australia.

it is obvious whether the significant differences from the Newtonian kinematics seen in the aforementioned experiments are due to shear-dependent viscosity or due to viscoelasticity or due to a combination of both.4-6 Therefore, it seems reasonable to begin with the analysis of purely viscous powerlaw type fluids and the level of complexity can be then built up gradually to accommodate other non-Newtonian characteristics such as yield stress, viscoelasticity, etc. As far as is known to us, there has been no prior study on heat transfer in the steady mixed convection regime from a cylinder submerged in power-law fluids when the buoyancy is superimposed on the imposed flow thereby resulting in the case of so-called aiding buoyancy. This constitutes the main objective of this work. At the outset, it is desirable, however, to briefly recount the available limited pertinent work to facilitate the subsequent presentation of the new results. 2. Previous Work As noted earlier, the bulk of the information available on momentum and heat transfer from a cylinder submerged in Newtonian fluids has been thoroughly reviewed by Zdravkovich,1,2 Ahmad,7 and Bharti et al.8 The corresponding limited information for the flow of and heat transfer from powerlaw fluids over a single cylinder has been summarized in recent studies.9-28 On the basis of the aforementioned studies, the current state of the knowledge can be summarized as follows: drag, wake, and pure forced convection (Ri ) 0) heat transfer characteristics have been studied extensively in the steady flow regime (Reynolds number not exceeding 40) over wide ranges of the power-law index (0.2 e n e 2) and Prandtl number (Pr)

10.1021/ie801892m CCC: $40.75  2009 American Chemical Society Published on Web 04/07/2009

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Ind. Eng. Chem. Res., Vol. 48, No. 21, 2009

ranging from 1 to 1000. The critical values denoting the onset of flow separation and the cessation of the steady flow regime for an unconfined cylinder are also available.19 Recently, vortex dynamics and drag characteristics of power-law fluids across a two-dimensional circular cylinder in unconfined flow configuration has also been studied28 over wide ranges of the powerlaw index (0.4 e n e 1.8) and Reynolds number (40 e Re e 140). Similarly, extensive experimental results on the onset of vortex shedding and pressure profiles around a cylinder submerged in shear-thinning polymer solutions exhibiting varying levels of viscoelasticity are also available.29-31 It is perhaps fair to say that flow characteristics have been investigated much more thoroughly than the corresponding heat/mass transfer characteristics. Broadly speaking, all else being equal, shearthinning behavior facilitates convective heat/mass transfer and shear-thickening impedes it. This trend is also borne out by the limited results for a cylinder confined in a planar slit18 and for two cylinders in tandem configuration submerged in powerlaw fluids.25 Most of the research on the convective heat transfer from a cylinder is focused on either pure forced convection or pure natural convection, even in Newtonian fluids.7,8 At high Reynolds numbers, heat transfer occurs mainly by forced convection, but the contribution of natural convection gradually increases as the Reynolds number of the flow is progressively reduced. In most practical applications, natural convection effects are always present as density gradients are setup in the ambient medium due to the prevailing temperature gradients in the continuous medium. The importance of mixed convection is gauged by the value of the Richardson number (Ri ) Gr/Re2). In the two extreme cases, the heat transfer occurs primarily by forced (Ri f 0) and free (Ri f ∞) convection, respectively. The mixed convection is believed to occur in between these two limits. Further complications arise depending upon the orientation of the cylinder with respect to the direction of streaming flow. For instance, aiding buoyancy appears when an imposed (vertically upward) flow approaches a horizontal cylinder (at a temperature higher than that of the fluid) and hence the buoyancy force also acts in the upward direction; whereas the opposing buoyancy situation appears when the direction of the imposed flow and that induced by buoyancy are opposite and parallel to each other. Additional complications arise based on whether the cylinder is being heated (i.e., cylinder surface temperature is less than the streaming fluid temperature, Tw < T0) or cooled (Tw > T0). Aiding buoyancy arises in the heat transfer from bluff body to fluid (Ri > 0), and (Ri < 0) represents the opposing buoyancy flow situation. Similarly, the rate of heat transfer is also influenced in the cross buoyancy flow configuration where the buoyancy induced velocity is oriented normal to the imposed flow. Obviously, the cross buoyancy flow configuration shows a greater degree of asymmetry in velocity field than that in aiding or opposing buoyancy case. This work is, however, concerned with the aiding buoyancy (Ri > 0) flow, where both the imposed velocity and the buoyancy induced velocity are in upward direction over a heated horizontal cylinder. Among others, the role of buoyancy on convective heat transfer in cross-flow at low Reynolds and Grashof numbers was investigated experimentally by Collis and Williams,32 who derived a rough criterion for the onset of buoyancy effects as Re ) 1.85Gr0.35. On the basis of the standard boundary layer analysis for one-dimensional flow, Kreith33 concluded that natural convection effects can not be ignored for Ri > 1. Fand

and Keswani34 experimentally investigated the rate of heat transfer in the mixed convection regime from a horizontal cylinder to water in the cross-flow configuration. On the basis of the value of the Richardson number (Ri), they identified four regimes of heat transfer as the following: In the first zone characterized by (Ri e 0.5), heat transfer occurs by forced convection; in the second zone (0.5 e Ri e 2), the main heat transfer mechanism is still forced convection, but natural convection contributes up to about 10%. In the third zone (2 e Ri e 40), both natural and forced convection effects are of the same order of magnitude and the heat transfer process is unsteady (4 e Ri e 40) when the forced flow is horizontal and heat transfer coefficient oscillates with a random period between the two extreme values. Finally, in the fourth zone (Ri g 40), natural convection is the main mode of heat transfer. In summary, when Ri , 1, buoyancy is unimportant in the flow and, for Ri .1, the buoyancy effect can not be neglected (i.e., there is insufficient kinetic energy to homogenize the fluid). Subsequently, Sparrow and Lee35 obtained a similarity solution for the aiding flow mixed convection heat transfer from a horizontal cylinder by expanding velocity and temperature profiles using power-law type expressions of the distance from the lowest point of the cylinder. The local Nusselt number distribution was only obtained in the region upstream of the point of separation. Sarma and Sukhatme36 experimentally investigated the influence of free convection and free stream turbulence on the local heat transfer from a cylinder to air for wide ranges of parameters (50 e Re e 7000 and 0.02 e Gr/Re2.5 e 32). Badr37,38 solved the combined heat transfer from an isothermal cylinder with its axis horizontal and perpendicular to the free stream direction and with free stream parallel and opposite to the buoyancy flow for 1 e Re e 40 and 0 e Ri e 5. Subsequently, Badr39 also assessed the effect of flow direction from aiding flow to opposing flow for air. Wong and Chen40 studied the mixed convection heat transfer characteristics from an isothermal cylinder by solving the full Navier-Stokes and energy equations using the finite element method. In an experimental study, Villimpoc et al.41 used the holographic interferometery method to study heat transfer in a series of Newtonian liquids (water and corn syrup) with the temperature difference of 0.1 °C (negligible variation in physical properties) covering the range of parameters as 0.002 e Re e 64, 5.5 e Pr e 2.7 × 104, and 0.001 e Ri e 4.4. They reported that forced convection dominates at high Prandtl numbers even up to Ri ) 4.4. This is presumably so in part due to the thinning of the thermal boundary layer at high Prandtl numbers. The phenomenon of vortex shedding from a heated/cooled circular cylinder in mixed, natural, and forced convection regimes was investigated numerically by Chang and Sa42 over a wide range of Grashof numbers. Subsequently, Ahmad and Qureshi43,44 solved the laminar mixed convection from a heated horizontal cylinder in cross-flow by using finite difference method for a wide range of Reynolds and Grashof numbers, but for a single value of the Prandtl number (Pr) for air. The influence of buoyancy on Nusselt number, wake characteristics, temporal lift, and drag forces over heated/cooled cylinders was numerically investigated by Patnaik et al.45 This work has recently been extended to mixed convection from a cylinder in powerlaw fluids when the imposed velocity is normal to the direction of the velocity induced by the buoyancy.27 Depending upon the values of the Richardson number (Ri), Prandtl number (Pr), Reynolds number (Re), and the power-law index (n), they reported the contribution of free convection to be of the order of 10-15%. More importantly cross-flow buoyancy effects


induce asymmetry in and/or alter the streamlines and isotherm contours which might adversely influence the homogeneity of a temperature and shear sensitive product such as that encountered in thermal processing of foodstuffs. Other than this, preliminary results on the nature of mixed convection in Newtonian and power-law fluids are also available for cylinders of square cross-section.46-49 In summary, it is thus fair to conclude that only scant studies on mixed convection from a cylinder in the steady flow regime (0.01 e Re e 40) are available, even for Newtonian fluids (most of which relate to air and water). Likewise, there has been only one study27 dealing with mixed convection in power-law fluids from a cylinder in the two-dimensional (2D) flow regime, when the imposed flow is oriented normal to the direction of gravity. Indeed no prior work is available on the mixed convection in the aiding-buoyancy situation and this study endeavors to fill this gap in the current literature. In particular, the governing equations have been solved numerically for the following ranges of conditions: Reynolds number (1 e Re e 40), power-law index (0.2 e n e 1.8), Prandtl number (1 e Pr e100), and Richardson number (0 e Ri e 2). 3. Physical Model and Governing Equations The physical problem considered here is the forced (vertically upward) flow over a heated horizontal cylinder maintained at a constant temperature. The buoyancy force arises from the variation of the fluid density with temperature in the vicinity of the cylinder. At very high Reynolds numbers (Re) of the imposed flow, the buoyancy effects are often neglected (forced convection). The other extreme is when there is no imposed flow (Re ) 0) and the heat transfer is purely by natural convection. In the intermediate range of conditions, however, both free and forced convections contribute to the overall heat transfer in varying proportions depending upon the value of the Richardson number (Ri). In this work, the buoyancy force is considered to be acting parallel (vertically upward) to the imposed flow direction (vertically upward) thereby resulting in the so-called aiding-buoyancy mixed conVection flow. Generally speaking, the thermophysical properties of the fluid, notably viscosity and density, are temperature dependent. The extent of their variation with temperature varies from one fluid to another. 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 as F ) F0[1 β(T - T0)], where β is the coefficient of volumetric expansion at a constant pressure and F0 is the density at a reference temperature (taken here as the temperature of the ambient fluid) T0. 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. Over the range of Reynolds numbers being considered here, the flow is known to be steady and two-dimensional for Newtonian fluids7,38,43,44 and it is assumed to be so for powerlaw fluids also, for no information is available in this regard for power-law media. Therefore, let us consider the twodimensional (2D), steady flow of an incompressible power-law liquid (at temperature T0) streaming vertically upward with a uniform velocity (U0) over a long circular cylinder (of diameter, D), as shown in Figure 1a. The cylinder is placed horizontally in an unconfined fluid medium. The surface of the cylinder is maintained at a constant wall temperature, Tw (>T0). The viscous dissipation effects are assumed to be negligible and the thermophysical properties (heat capacity, thermal conductivity, viscosity) of the liquid are assumed to be independent of

9737

Figure 1. Schematic representation of (a) (upward) flow over an unconfined cylinder maintained at constant wall temperature of Tw and (b) an approximation of an unconfined flow (uniform velocity, U0, and temperature T0) configuration.

temperature, except for the density appearing in the body force term in the momentum equation. These two assumptions restrict the applicability of the present results to the situations where the temperature difference between the fluid and the cylinder is not too large and for moderate viscosity and/or shearing levels so that the viscous dissipation effects are negligible. The temperature difference ∆T () Tw - T0) is maintained small so that it is justified to neglect the variation of viscosity with temperature. An unconfined flow condition is simulated here by enclosing the cylinder (of diameter, D) in a large concentric cylindrical envelope of fluid (of diameter D0), as shown schematically in Figure 1b. The value of D0 is taken to be sufficiently large to minimize the boundary effects (Figure 1b). The continuity, momentum, and thermal energy equations for this flow are written as follows: • Continuity equation ∂Uy ∂Ux + )0 ∂x ∂y

(1)

• x-Component of momentum equation

(

F Ux

)

(

∂τyx ∂Ux ∂Ux ∂τxx ∂p + Uy + + )∂x ∂y ∂x ∂x ∂y

)

(2a)

• y-Component of momentum equation

(

F Ux

)

(

)

∂τyy ∂Uy ∂Uy ∂τxy ∂p + Uy + + )+ Fgy ∂x ∂y ∂y ∂x ∂y (2b)

Since the gravity is acting in the negative y-direction (Figure 1a), the y-component of the gravity (gy) is taken as (-g). By using the Boussinesq approximation for density (F) in the body force term, eq 2b is rewritten as

(

F Ux

)

∂Uy ∂Uy ∂p + Uy + )∂x ∂y ∂y ∂τyy ∂τxy + + F0gβ(T - T0) (2c) ∂x ∂y

(

)

• Thermal energy equation: The thermal energy equation, in terms of the constant transport properties and in the absence of viscous dissipation, can be written as

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(

Fcp Ux

(

∂T ∂T ∂2T ∂2T + Uy )k 2 + 2 ∂x ∂y ∂x ∂y

)

)

(3)

In the above equations, F, F0, Ux, Uy, T, T0, k, and cp are the fluid density at temperatures T and T0 respectively, x- and y-components of the velocity, temperature, free-stream fluid temperature, thermal conductivity, and specific heat of the fluid, respectively. For incompressible fluids, the components of the extra stress tensor (τij) are related to the components of the rate of deformation tensor (εij) as follows: τij ) 2ηεij where (i, j) ) (x, y)

(4)

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

[

∂Uj 1 ∂Ui + εij ) 2 ∂j ∂i

]

(5)

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

Ux ) 0

Ux ) 0 Uy ) U0 and T ) T0

(7)

• On the surface of the cylinder: The standard no-slip condition and constant temperature are used as follows: Ux ) 0 Uy ) 0 and T ) Tw

(8)

• At the exit boundary: The rear half of the enclosing fluid envelope constitutes the outlet of the flow. On this surface, the default outflow boundary condition option in FLUENT (a zero diffusion flux for all flow variables) was used in this work. This choice implies that the conditions of the outflow plane are extrapolated from within the domain and as such have negligible influence on the upstream flow conditions. The extrapolation procedure used by FLUENT updates the outflow velocity and the pressure in a manner that is consistent with the fully developed flow assumption, when there is no area change at the outflow boundary. However, the gradients in the cross-stream direction may still exist at the outflow boundary. Also, the use of this condition obviates the need to prescribe a boundary condition for pressure. This is similar to the homogeneous Neumann condition, i.e., ∂Ux )0 ∂y

∂Uy ) 0 and ∂y

∂T )0 ∂y

(9)

• At the plane of symmetry (x ) 0): The symmetric flow condition is imposed.

∂T )0 ∂x

(10)

Evidently, the mixed convection flow governing equations are a set of coupled equations. The numerical solution of the governing equations (eqs 1-3) in conjunction with the abovenoted boundary conditions (eqs 7-10) maps the flow domain in terms of the primitive variables, i.e., velocity (Ux and Uy), temperature (T), and pressure (p) fields. These, in turn, are used to deduce the local and global momentum and heat transfer characteristics as outlined below and detailed elsewhere.8,11,17,18 However, at this stage, it is useful to introduce some dimensionless parameters. • The Reynolds number (Re) and Prandtl number (Pr) for powerlaw fluids are defined as follows Re )

( )

cpm U0 FDnU02-n ; Pr ) m k D

n-1

(11)

• The Richardson number (Ri), which represents the importance of natural convection relative to that of the forced convection, is defined as follows

η ) m(I2 /2)(n-1)/2 where I2 ) 2(εxx2 + εxy2 + εyx2 + εyy2) (6) where m is the power-law consistency index, n is the powerlaw index of the fluid, and I2 is the second invariant of the rate of strain tensor.50 Evidently, n < 1 indicates shear-thinning behavior, n > 1, shear-thickening, and n ) 1, the standard Newtonian fluid behavior. The physically realistic boundary conditions for this flow configuration are written as follows: • At the inlet boundary: The front half of the fluid envelope constitutes the inlet boundary where the uniform flow condition is imposed, i.e.,

∂Uy ) 0 and ∂x

Ri )

gβ(Tw - T0)D Gr ) 2 Re U02

(12)

where, the Grashof number (Gr), the ratio of buoyancy to viscous forces acting on the fluid, for power-law fluids is defined as follows

[( ) ]

Gr ) gβ(Tw - T0)D3

F U0 m D

(1-n) 2

(13)

Note that, unlike in the case of Newtonian fluids, the Prandtl (Pr) and Grashof (Gr) numbers 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, n). It is useful to add here that in the limit of free convection, i.e., Re ) 0 and Ri f ∞, there will be a characteristic velocity induced by the density gradient and it is, thus, still possible to define a Reynolds number (Re) based on this velocity; obviously, the Reynolds number will not be zero. However, the definitions employed here are used extensively in the heat transfer literature and have gained wide acceptance to describe the results of mixed-convection heat transfer. • The surface pressure coefficient (Cp) is defined as follows Cp )

p(θ) - p0 static pressure ) dynamic pressure (1/2)FU02

(14)

where p(θ) is the surface pressure at an angle θ and p0 is the free stream pressure at the exit boundary. • The total drag coefficient (CD), sum of the friction and pressure components, is defined as CD )

FD (1/2)FU02D

) CDP + CDF

(15)

where FD is the drag force exerted by the fluid on the cylinder per unit length. The individual drag coefficients, CDP and CDF, are calculated using the following definitions:


CDP )

CDF ) )

FDF 2

∫ (τ S

yx·nx

)

2

2n+1 Re

∫C n S

(1/2)FU0 D

)

(1/2)FU0 D 2n+1 Re

FDP

p y

dS

(16a)

∫ (τ·n ) dS s

S

+ τyy·ny) dS

(16b)

where FDP and FDF are the pressure and frictional components of the drag force per unit length of cylinder and S is the surface area. The ns (unit vector normal to the surface of the cylinder) is given as ns )

xex + yey

) nxex + nyey

√x2 + y2

(17)

where ex and ey are the x- and y-components of the unit vector, respectively, and τ, the dimensionless shear stress, is expressed as τij ) η

(

) () (

∂Uj ∂Ui I2 + ) ∂j ∂i 2

(n-1)/2

∂Uj ∂Ui + ∂j ∂i

)

(18)

where η and I2 are the dimensionless viscosity and second invariant of the rate of strain tensor, respectively. In the above equations, radius of the cylinder (R ) D/2) and the uniform inlet velocity (U0) are used as the characteristic length and velocity scales, respectively. • The local Nusselt number, Nu(θ), on the surface of the cylinder is evaluated using the temperature field as follows: Nu(θ) )

∂T hD )k ∂ns

(19)

Such local values are further averaged over the surface of the cylinder to obtain the surface aVeraged (or overall mean) Nusselt number as follows: Nu )

1 2π

∫

2π

0

Nu(θ) dθ

(20)

The average Nusselt number (or dimensionless heat transfer coefficient) can be used in process engineering design calculations to estimate the rate of heat transfer from an isothermal cylinder, or the temperature of the cylinder for a given heat flux. Dimensional analysis of the field equations and the boundary conditions suggests the local and global flow and heat transfer characteristics to be functions of Reynolds number (Re), powerlaw index (n), Prandtl number (Pr), and Richardson number (Ri). This functional relationship is explored in this work. 4. Numerical Solution Procedure Since detailed descriptions of the numerical solution procedure are available elsewhere,17-21,23-26 only the salient features are recapitulated here. In this study, the field equations have been solved using FLUENT (version 6.3). The unstructured “quadrilateral” cells of nonuniform grid spacing were generated using the commercial grid tool GAMBIT. The two-dimensional, steady, laminar, segregated solver was used to solve the incompressible flow on the collocated grid arrangement. The second-order upwind scheme has been used to discretize the convective terms in the momentum and thermal energy equations. The semi-implicit method for the pressure linked

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equations (SIMPLE) scheme was used for solving the pressurevelocity decoupling. The constant density and non-Newtonian power-law viscosity models were used. FLUENT solves the system of algebraic equations using the Gauss-Siedel (GS) point-by-point iterative method in conjunction with the algebraic multigrid (AMG) method solver. The use of the AMG scheme can greatly reduce the number of iterations and, thus, the CPU time required to obtain a converged solution, particularly when the model equation contains a large number of control volumes. Relative convergence criteria of 10-10 for the continuity and xand y-components of the velocity and of 10-15 for the thermal energy equation were prescribed in this work. Over the range of conditions covered here, no difficulty was encountered in meeting the prescribed convergence criterion, albeit the CPU time increased several fold for small values of power-law index (n) and/or large values of the Richardson number (Ri). 5. Choice of Computational Parameters In this study, the effects of Reynolds number (Re), powerlaw index (n), Richardson number (Ri), and Prandtl number (Pr) on the momentum and heat transfer characteristics are investigated in the following ranges of conditionssReynolds number (Re ) 1, 5, 10, 40), power-law index (n ) 0.2, 0.6, 1, 1.4, 1.8)sthereby covering both shear-thinning (n < 1) and shearthickening (n > 1) fluid characteristics. The Richardson number values are chosen as Ri ) 0, 0.5, 1, 1.5, 2. The Ri ) 0 case corresponds to forced convection, whereas at Ri ) 1, the characteristics velocity induced by density variation is of the same order as the imposed flow velocity. The case of Ri ) 2 corresponds to the situation in which strong free convection effects are expected. Besides, due to the relatively smaller value of the coefficient of thermal expansion for liquids, it is generally not possible to achieve as high values of the Richardson number (Ri) in liquids as that in gases. Finally, the Prandtl number values are chosen as Pr ) 1, 10, 20, 50, and 100 so that the contribution of free convection can be assessed over the whole range of Prandtl numbers (Pr) used in this work. It is appropriate to add here that it is not at all uncommon to encounter industrial fluids possessing the value of Prandtl number (Pr) as large as 100, or even higher. Besides, in numerical studies, the maximum value of the Prandtl number (Pr) is also restricted by the fact that very fine grids are required near the cylinder owing to the progressive thinning of the thermal boundary layer with increasing Prandtl number (Pr). However, a 100-fold variation in the value of Prandtl number (Pr) covered in this work should provide an adequate guide for delineating the scaling of the Nusselt number (Nu) with Prandtl number (Pr). Needless to say, the reliability and accuracy of the numerical results is contingent upon a prudent choice of the numerical parameters, namely, optimal domain and grid sizes. In this study, the domain is characterized by the diameter (Do) of the outer cylindrical envelope of the fluid. An excessively large value of Do will warrant enormous computational resources and a small value will unduly influence the results; hence, a prudent choice is vital to the accuracy of the results. Similarly, an optimal grid size should meet two conflicting requirements, namely, it should be fine enough to capture the flow field yet it should not be exorbitantly resource intensive. The effects of these parameters (Do and grid size) on the drag coefficient and Nusselt number values for the power-law fluid flow past a cylinder have been explored extensively elsewhere;15,16,19,21,24,25 only the additional results showing the influence of buoyancy parameter on the mixed convection characteristics are presented here thereby ensuring the present results to be free from these artifacts.

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Table 1. Influence of Domain Size (with Grid G1 and M1 for Re ) 1 and Re g 5) on Mixed Convection Flow Characteristics for Pr ) 100 Ri ) 0

Ri ) 2

D0/D

Re(n)

CDP

CDF

CD

Nu

Re(n)

CDP

CDF

CD

Nu

1100 1200 1300 1100 1200 1300 1100 1200 1300 300 400 500 300 400 500 300 400 500 300 400 500 300 400 500 300 400 500

1 (0.2)

20.3724 19.5202 19.5398 5.1363 5.1356 5.1344 2.5796 2.5792 2.5788 4.2602 4.2579 4.2780 2.0693 2.0674 2.0667 1.4987 1.4967 1.4963 0.9917 0.9912 0.9914 0.9729 0.9731 0.9731 0.9139 0.9134 0.9133

6.7700 7.2516 7.4074 5.1928 5.1921 5.1906 3.2595 3.2590 3.2586 1.5091 1.5099 1.4910 1.8672 1.8657 1.8651 1.7232 1.7211 1.7206 0.1474 0.1472 0.1473 0.5250 0.5247 0.5247 0.7481 0.7477 0.7476

27.1424 26.7718 26.9472 10.3291 10.3278 10.3251 5.8391 5.8382 5.8374 5.7693 5.7679 5.7679 3.9365 3.9332 3.9318 3.2219 3.2178 3.2169 1.1391 1.1384 1.1387 1.4987 1.4978 1.4978 1.6620 1.6610 1.6609

4.9446 5.6551 5.4934 3.3453 3.3452 3.3448 3.1100 3.1103 3.1102 11.2969 11.3065 11.0492 6.6071 6.6062 6.6056 5.8194 5.8185 5.8183 29.7471 29.8030 29.7504 17.9168 17.9168 17.8778 15.6635 15.6617 15.6101

1 (0.6)

13.0374 12.9939 12.8892 8.4712 8.4768 8.4816 5.5579 5.5614 5.5644 3.2796 3.2846 3.2885 2.7105 2.7142 2.7175 1.1618 1.1622 1.1634 1.4260 1.4272 1.4281 1.4201 1.4211 1.4220

9.7665 9.5185 9.3761 8.2979 8.3033 8.3079 6.7640 6.7689 6.7727 2.8914 2.8954 2.8984 3.0513 3.0554 3.0590 0.2892 0.2893 0.2896 0.8820 0.8825 0.8828 1.2487 1.2495 1.2502

22.8039 22.5124 22.2652 16.7690 16.7801 16.7896 12.3219 12.3303 12.3371 6.1710 6.1800 6.1869 5.7618 5.7697 5.7765 1.4510 1.4514 1.4530 2.3079 2.3097 2.3109 2.6688 2.6706 2.6722

4.5392 4.4094 4.2868 3.8278 3.8286 3.8300 3.5711 3.5715 3.5719 7.6219 7.6254 7.6281 6.7627 6.7642 6.7655 39.6514 39.6352 39.6091 19.9606 19.9643 19.9670 16.1547 16.1562 16.1571

1 (1.0) 1 (1.8) 5 (0.2) 5 (1.0) 5 (1.8) 40 (0.2) 40 (1.0) 40 (1.8)

Table 2. Details of the Grids Used in the Domain and Grid Independence Studies grid spacing (δ/D)

grid points region Bb on the half-cylinder grid region Aa (stretched from) region Cc surface (Nc) Ncellsd for Re < 5 G1 G2 G3 G4

0.020 0.010 0.005 0.010

0.020 f 0.5 0.010 f 0.5 0.005 f 0.5 0.010 f 0.5

M1 M2 M3 M4

0.010 0.005 0.002 0.010

0.010 f 0.5 0.005 f 0.5 0.002 f 0.5 0.010 f 0.5

0.5

200 200 200 300

488400 494400 505200 741600

200 200 200 300

134200 145200 185200 210000

for Re g 5 0.5

a Uniform fine grid, circular region of inner diameter D (cylinder diameter) and outer diameter 2D. b Nonuniform grid, circular region of inner diameter 2D and outer diameter 10D. c Uniform coarse grid, circular region of inner diameter 10D and outer diameter Do. d Total number of grid cells for a domain size of D0/D ) 1200 for Re < 5 and that of D0/D ) 300 for Re g 5.

At low Reynolds numbers (Re < 5), the hydrodynamic boundary layer is known to be thick (hence the velocity field decays slowly) and thus a large computational domain is required19,21,24,25 as compared to that at high Reynolds numbers (Re > 5). Several values of (D0/D) ranging from 300 to 1200 have been used in the domain independence test. The study is carried out for three values of the Reynolds number (Re ) 1, 5, 40), three values of the power-law index (n ) 0.2, 1, 1.8), extreme values of the Richardson number (Ri ) 0, 2), and the highest value of the Prandtl number (Pr ) 100) used in this work. Table 1 shows the effect of the domain size (D0/D) on the values of the drag coefficients (CDP, CDF, CD) and average Nusselt number (Nu). The domain independence study has been carried out with grids G1 and M1 detailed in Table 2. At Re ) 40, in moving from a domain size of 300 to 400 and from 400 to 500, the changes in drag coefficients values are seen to be 1) fluids. For fixed values of power-law index (n) and Reynolds (Re) and Prandtl (Pr) numbers, the pressure coefficient (Cp) increases with an increasing value of the Richardson number (Ri). As expected, the pressure coefficient (Cp) values at the front (and rear) stagnation point are strongly enhanced (reduced) by the shear-

thinning (n < 1) nature of the flow; however, an opposite influence is seen in shear-thickening (n > 1) fluids. As the Reynolds number (Re) increases, the front stagnation pressure coefficient, Cp(0) decreases whereas the rear stagnation pressure coefficient Cp(π) increases for all values of the Richardson (Ri) and Prandtl numbers (Pr) and flow behavior index (n). The front stagnation pressure coefficient, Cp(0) increases with increasing Prandtl number (Pr), whereas the rear stagnation pressure coefficient, Cp(π) decreases. Broadly, the value of pressure coefficient (Cp) on the surface of the cylinder is higher in shear-thinning (n < 1) fluids than that for Newtonian (n ) 1) and lower for shear-thickening (n > 1) fluids, otherwise under identical conditions. Irrespective of the type of fluid behavior, the value of pressure coefficient at the front and rear stagnation points are seen to be more strongly influenced by the value of the Richardson number (Ri) at low Reynolds numbers than that at higher values of Reynolds numbers. This

9744


Figure 4. Influence of Richardson (Ri) number on the streamline (left half of the figures) and isovorticity (right half of the figures) contours for power-law fluid (n ) 1.8) flow at Re ) 40 for different values of Prandtl number (a) Pr ) 1, (b) Pr ) 10, (c) Pr ) 20, (d) Pr ) 50, and (e) Pr ) 100.

is simply due to the fact that the role of free convection diminishes with increasing Reynolds number (Re). The variation is seen to be rather significant in shear-thinning (n < 1) liquids which are probably more conducive to free convection currents than their counterparts, shear-thickening (n > 1) fluids. 6.2.2. Macroscopic Characteristics. The changes brought about by different governing parameters (Re, Pr, n, Ri) in the local flow characteristics seen in the previous section also influence the behavior at the macroscopic level. In this section, the role of flow behavior index (n), Reynolds number (Re), Prandtl number (Pr), and Richardson number (Ri) on the individual and total drag coefficients is elucidated. (a) Pressure Drag Coefficient (CDP). Table 5 shows the influence of the dimensionless parameters (Re, n, Pr, Ri) on the pressure drag coefficient (CDP). At Re ) 40, for a fixed value of the Richardson number (Ri), the pressure drag

coefficient (CDP) increases as the value of power-law index (n) is gradually increased. The shear-thinning (n < 1) behavior of the fluid always yields a higher value of the pressure drag coefficient (CDP) than the corresponding Newtonian (n ) 1) value (for Re e 40), whereas an opposite trend is seen for the shear-thickening (n > 1) behavior of the fluid; both these observations are directly linked to the nature of the surface pressure (Cp) profiles presented in the preceding section. The influence of the flow behavior index (n) on the pressure drag coefficient (CDP) is observed to be stronger in shear-thinning (n < 1) fluids than that in the shear-thickening (n > 1) fluids. For fixed values of the Richardson number (Ri), power-law index (n), and Prandtl number (Pr), the value of the pressure drag coefficient (CDP) increases with decreasing value of Reynolds number (Re). The increasing buoyancy effect (Ri) always increases pressure drag coefficient (CDP), though the


9745

Figure 5. Variation of pressure coefficient (Cp) over the surface of the cylinder for a range of values of the Prandtl number (Pr ) 1, 10, 20, 50, 100), for extreme values of Richardson number (Ri ) 0, 2), and for Reynolds numbers (a) Re ) 1, (b) Re ) 5, and (c) Re ) 40 for Newtonian (n ) 1) fluids.

effect is particularly striking at low Prandtl numbers. Due to the increased distortion of flow thereby resulting in steeper velocity gradients (and hence stresses), the pressure drag coefficient (CDP) is always larger in mixed convection (Ri > 0) than that for forced convection (Ri ) 0). This effect increases as the Reynolds number (Re) is progressively reduced thereby weakening the role of forced convection. For instance, for a fixed value of the Prandtl number (Pr ) 100) in forced convection flow, as the Reynolds number (Re) is increased from 1 to 40, the pressure drag coefficient (CDP) decreases from 9.77 to 0.98, from 5.14 to 0.97, and from 3.41 to 0.95 at n ) 0.6, 1, and 1.4, respectively. The corresponding changes for Richardson number of Ri ) 2 are from 12.99 to 1.38, from 8.47 to 1.42, and from 6.57 to 1.43 at n ) 0.6, 1, and 1.4, respectively. On the other hand, for a fixed Prandtl number (Pr ) 100), an increase in the Richardson number (Ri) from 0 to 2 enhances the pressure

drag coefficient (CDP) values at Re ) 1 from 9.78 to 12.99, from 5.14 to 8.48, and from 3.41 to 6.57 at n ) 0.6, 1, and 1.4, respectively. The corresponding changes at Re ) 40, as expected, were seen to be much smaller (Table 5) owing to weaker buoyancy currents. This shows that the pressure drag coefficient (CDP) increases with the increasing Richardson number (Ri) for all values of Reynolds number (Re), powerlaw index (n), and Prandtl number (Pr). Table 5 also shows that the influence of the dimensionless parameters (Re, n, Pr) on the pressure drag coefficient (CDP) is accentuated in mixed convection (Ri > 0) when compared to that in forced convection (Ri ) 0). Similarly, the influence of power-law index (n) on the pressure drag coefficient (CDP) is seen to be stronger at low Reynolds numbers and the effect is stronger in shear-thinning (n < 1) fluids than that in Newtonian (n ) 1) and shear-thickening (n > 1) fluids. For instance, as the value of the Reynolds number (Re) is increased from 1 to

9746


Figure 6. Variation of pressure coefficient (Cp) over the surface of the cylinder for a range of values of the Prandtl number (Pr ) 1, 10, 20, 50, 100), for extreme values of Reynolds number (Re ) 1, 40), and for Power-law index (a) n ) 0.6, (b) n ) 1, and (c) n ) 1.4 for Richardson number, Ri ) 2.

40 at a Richardson number Ri ) 2, pressure drag coefficient (CDP) increases by factors of 11.18, 5.94, and 4.59 at n ) 0.6, 1, and 1.4, respectively. It decreases from 1.66 to 1.16, 2.28 to 1.43, and 2.48 to 1.42 at n ) 0.2, 1, and 1.4, respectively, as the Prandtl number (Pr) is increased from 1 to 100 for Reynolds number, Re ) 40, and Richardson number, Ri ) 2. Clearly, the role of Prandtl number (Pr) is seen to be less significant than that of the Reynolds number (Re) and/or Richardson number (Ri). (b) Friction Drag Coefficient (CDF). Table 6 shows the dependence of the friction drag coefficient (CDF) on the Reynolds number (Re), power-law index (n), Prandtl number (Pr), and Richardson number (Ri). For fixed values of power-law index (n) and Prandtl number (Pr), the friction drag coefficient (CDF) decreases with increasing Reynolds number (Re) and/or with increasing value of Richardson number (Ri). For fixed values of the Richardson number (Ri), it increases with increasing power-law index (n) over the entire range of conditions

considered herein. At high Reynolds numbers (Re ) 40), the friction drag coefficient (CDF) decreases with the decreasing value of power-law index (n) for all values of Richardson number (Ri). Similar to the pressure drag coefficient (CDP), here also, the flow behavior index (n) exerts a stronger influence at low Reynolds numbers than that at high Reynolds numbers. This is simply due to the fact that the role of viscosity diminishes with the increasing Reynolds number (Re). Over the range of conditions, the friction drag coefficient (CDF) values for mixed convection (Ri > 0) are always larger than those for forced convection (Ri ) 0) due to the increased distortion of stream lines near the cylinder. The greater the Richardson number (Ri), the larger the deviation in the friction drag coefficient (CDF) from the corresponding value in the pure forced convection regime. The friction drag coefficient shows an opposite dependence on the Prandtl number (Pr), i.e., it decreases with a gradual increase in the Prandtl number (Pr), irrespective of the type of fluid behavior.


9747

Table 5. Dependence of Pressure Drag Coefficient (CDP) on Reynolds Number (Re), Power-Law Index (n), Richardson Number (Ri), and Prandtl Number (Pr) Re ) 40 Pr

Re ) 10

Re ) 5

Re ) 1

Ri n ) 0.2 n ) 0.6 n ) 1.0 n ) 1.4 n ) 1.8 n ) 0.6 n ) 1.0 n ) 1.4 n ) 1.8 n ) 0.6 n ) 1.0 n ) 1.4 n ) 1.8 n ) 0.6

100 0 1 2 50 0 1 2 20 0 1 2 10 0 1 2 1 0 1 2

0.9917 1.1673 1.1618 0.9917 1.1950 1.1910 0.9917 1.2325 1.2500 0.9917 1.2686 1.3134 0.9917 1.4845 1.6689

0.9829 1.2603 1.3756 0.9829 1.3000 1.4322 0.9829 1.3617 1.5205 0.9829 1.4161 1.6005 0.9829 1.6680 2.0089

0.9738 1.2757 1.4260 0.9738 1.3281 1.5063 0.9738 1.4102 1.6318 0.9738 1.4826 1.7436 0.9738 1.8109 2.2800

0.9469 1.2620 1.4304 0.9469 1.3201 1.5224 0.9469 1.4120 1.7970 0.9469 1.4938 1.7968 0.9469 1.8666 2.4155

0.9139 1.2404 1.4201 0.9139 1.3017 1.5193 0.9139 1.3984 1.6756 0.9139 1.4847 1.8153 0.9139 1.8784 2.4764

1.8205 2.2947 2.6612 1.8205 2.3968 2.7713 1.8205 2.5576 3.0179 1.8205 2.6990 3.2288 1.8205 3.3452 4.1527

1.5287 2.0365 2.3783 1.5287 2.1466 2.5567 1.5287 2.3223 2.8392 1.5287 2.4810 3.0932 1.5287 3.2444 4.3217

1.3547 1.8713 2.2234 1.3547 1.9827 2.4091 1.3547 2.1611 2.7054 1.3547 2.3231 2.9744 1.3547 3.1366 4.3359

1.2408 1.7591 2.1165 1.2408 1.8697 2.3053 1.2408 2.0465 2.6072 1.2408 2.2072 2.8824 1.2408 3.0315 4.3031

2.7615 3.4177 3.7685 2.7615 3.5800 4.1383 2.7615 3.7582 4.5586 2.7615 4.0728 4.9065 2.7615 5.0659 6.3721

2.0693 2.7728 3.7685 2.0693 2.9415 3.5541 2.0693 3.2156 3.9945 2.0693 3.4672 4.3952 2.0693 4.7059 6.3597

1.7082 2.4193 2.9305 1.7082 2.5843 3.2063 1.7082 2.8527 3.6529 1.7082 3.1009 4.0650 1.7082 4.3988 6.2225

1.4987 2.2003 2.7105 1.4987 2.3596 2.9844 1.4987 2.6186 3.4298 1.4987 2.8591 3.8438 1.4987 4.1624 6.0869

9.7777 11.6489 12.9938 9.7777 12.1761 13.8273 9.7777 13.0928 15.1703 9.7777 13.8560 16.3500 9.7777 17.3092 21.4941

n ) 1.0

n ) 1.4

n ) 1.8

5.1350 3.4059 2.5791 7.0655 5.2349 4.2894 8.4768 6.5704 5.5614 5.1350 3.4059 2.5791 7.6298 5.7411 4.7467 9.3696 7.4000 6.3391 5.1350 3.4059 2.5791 8.5784 6.6061 5.5332 10.8511 8.8074 7.6778 5.1350 3.4059 2.5791 9.4935 7.4489 6.3100 12.2436 10.1752 9.0014 5.1350 3.4059 2.5791 14.2828 12.2927 11.0069 19.4596 17.4911 16.8765

Table 6. Dependence of Friction Drag Coefficient (CDF) on Reynolds Number (Re), Power-Law Index (n), Richardson Number (Ri), and Prandtl Number (Pr) Re ) 40

Re ) 10

Re ) 5

Re ) 1

Pr

Ri n ) 0.2 n ) 0.6 n ) 1.0 n ) 1.4 n ) 1.8 n ) 0.6 n ) 1.0 n ) 1.4 n ) 1.8 n ) 0.6 n ) 1.0 n ) 1.4 n ) 1.8 n ) 0.6 n ) 1.0 n ) 1.4 n ) 1.8

100

0 1 2 0 1 2 0 1 2 0 1 2 0 1 2

50

20

10

1

0.1474 0.2467 0.2892 0.1474 0.2478 0.2935 0.1474 0.2468 0.2967 0.1474 0.2443 0.2958 0.1474 0.2421 0.2779

0.3532 0.5205 0.6170 0.3532 0.5448 0.6558 0.3532 0.5789 0.7103 0.3532 0.6048 0.7522 0.3532 0.6859 0.8843

0.5250 0.7460 0.8820 0.5250 0.7892 0.9516 0.5250 0.8568 1.0608 0.5250 0.9153 1.1561 0.5250 1.1478 1.5426

0.6551 0.9242 1.0893 0.6551 0.9823 1.1835 0.6551 1.0772 1.3375 0.6551 1.1637 1.4789 0.6551 1.5479 2.1281

0.7481 1.0612 1.2487 0.7481 1.1313 1.3622 0.7481 1.2483 1.5521 0.7481 1.3576 1.7313 0.7481 1.8757 2.6151

1.0846 1.3741 1.5788 1.0846 1.4359 1.6763 1.0846 1.5302 1.8221 1.0846 1.6098 1.9400 1.0846 1.8452 2.4078

1.2293 1.6346 1.9224 1.2293 1.7342 2.0841 1.2293 1.8957 2.3445 1.2293 2.0422 2.5799 1.2293 2.7017 3.6375

1.2925 1.7813 2.1220 1.2925 1.9070 2.3289 1.2925 2.1165 2.6742 1.2925 2.3131 2.9996 1.2925 3.2740 4.6073

1.3158 1.8686 2.2445 1.3158 2.0128 2.4835 1.3158 2.2569 2.8906 1.3158 2.4902 3.2834 1.3158 3.6904 5.3511

1.8793 2.2887 2.6532 1.8793 2.3892 2.7441 1.8793 2.5623 2.9982 1.8793 2.6876 3.2083 1.8793 3.2734 4.0702

1.8672 2.4576 2.8914 1.8672 2.6164 3.1487 1.8672 2.8788 3.5692 1.8672 3.1216 3.9555 1.8672 4.2715 5.7701

1.7969 2.4998 3.0051 1.7969 2.6929 3.3241 1.7969 3.0194 3.8631 1.7969 3.3304 4.3777 1.7969 4.9160 7.0114

1.7232 2.5032 3.0513 1.7232 2.7180 3.4104 1.7232 3.0857 4.0291 1.7232 3.4416 4.6333 1.7232 5.3504 7.9209

7.4087 8.6454 9.5193 7.4087 8.9941 10.0709 7.4087 9.6198 10.9798 7.4087 10.1337 11.8218 7.4087 12.5303 15.5855

5.1912 6.9994 8.3034 5.1912 7.5671 9.1979 5.1912 8.5325 10.7010 5.1912 9.4692 12.1259 5.1912 14.3120 19.4121

3.9578 3.2590 5.9473 5.3044 7.3743 6.7689 3.9578 3.2590 6.5635 5.9298 8.3820 7.8199 3.9578 3.2590 7.6376 7.0306 10.1308 9.6789 3.9578 3.2590 8.6947 8.1325 11.8497 11.5496 3.9578 3.2590 14.6426 14.6893 21.4258 22.6256

Table 7. Dependence of Total Drag Coefficient (CD) on Reynolds Number (Re), Power-Law Index (n), Richardson Number (Ri), and Prandtl Number (Pr) Re ) 40 Pr

Re ) 10

Re ) 5

Re ) 1


100 0 1 2 50 0 1 2 20 0 1 2 10 0 1 2 1 0 1 2

1.1391 1.4141 1.4510 1.1391 1.4428 1.8485 1.1391 1.4794 1.5467 1.1391 1.5129 1.6092 1.1391 1.7266 1.9467

1.1361 1.7809 1.9926 1.1361 1.8448 2.0880 1.1361 1.9405 2.2308 1.1361 2.0210 2.5527 1.1361 2.3539 2.8932

1.4987 2.0217 2.3079 1.4987 2.1173 2.4579 1.4987 2.2670 2.6926 1.4987 2.3979 2.8996 1.4987 2.9586 3.8225

1.6020 2.1863 2.5197 1.6020 2.3024 2.7059 1.6020 2.4892 3.0048 1.6020 2.6575 3.2759 1.6020 3.4145 4.5436

1.6620 2.3016 2.6688 1.6620 2.4330 2.8815 1.6620 2.6467 3.2277 1.6620 2.8423 3.5466 1.6620 3.7542 5.0915

2.0951 3.6688 4.1900 2.0951 3.8327 4.4477 2.0951 4.0877 4.8400 2.0951 4.3087 5.1712 2.0951 5.1904 6.5605

2.7579 3.6711 4.3006 2.7579 3.8808 4.6408 2.7579 4.2180 5.1837 2.7579 4.5232 5.6731 2.7579 5.9462 7.9592

2.6472 3.6526 4.3454 2.6472 3.8898 4.7380 2.6472 4.2776 5.3796 2.6472 4.6362 5.9739 2.6472 6.4106 8.9432

(c) Total Drag Coefficient (CD). The dependence of the total drag coefficient (CD ) CDP + CDF) on the Reynolds number (Re), power-law index (n), Richardson number, and Prandtl number (Pr) is shown in Table 7 and Figure 7. For a fixed value of Prandtl number (Pr), the total drag coefficient (CD) shows a complex dependence on the dimensionless parameters (Re, n, Ri). In forced convection (Ri ) 0), the drag coefficient (CD) increases with the decreasing value of power-law index (n) at low Reynolds number (Re). As the Reynolds number (Re) is gradually increased, the drag coefficient (CD) shows the opposite type of variation (Ri ) 0 in Figure 7b and c) with power-law index (n), albeit the change is negligibly small.11,15 But increasing values of the Richardson number (Ri) enhance this effect, as can be clearly seen in Figure 7c. For fixed values of the Richardson number (Ri), the total drag coefficient (CD)

2.5566 3.6277 4.3611 2.5566 3.8826 4.7888 2.5566 4.3034 5.4978 2.5566 4.6973 6.1658 2.5566 6.7219 9.6542

4.6408 3.9365 3.5051 3.2219 5.7063 5.2304 4.9192 4.7035 6.4217 6.1710 5.9355 5.7618 4.6408 3.9365 3.5051 3.2219 5.9692 5.5579 5.2773 5.0776 6.8824 6.7028 6.5304 6.3948 4.6408 3.9365 3.5051 3.2219 6.3205 6.0944 5.8721 5.7042 7.5568 7.5637 7.5161 7.4589 4.6408 3.9365 3.5051 3.2219 6.7604 6.5888 6.4312 6.3007 8.1148 8.3508 8.4427 8.4771 4.6408 3.9365 3.5051 3.2219 8.3393 8.9774 9.3148 9.5128 10.4422 12.1298 13.2339 14.0079

17.1864 20.2944 22.5131 17.1864 21.1702 23.8983 17.1864 22.7125 26.1501 17.1864 23.9897 28.1718 17.1864 29.8395 37.0797

10.3262 14.0649 16.7803 10.3262 15.1969 18.5675 10.3262 17.1107 21.5521 10.3262 18.9627 24.3695 10.3262 28.5949 38.8716

7.3636 11.1821 13.9447 7.3636 12.3046 15.7820 7.3636 14.2437 18.9382 7.3636 16.1436 22.0249 7.3636 26.9353 39.3669

5.8381 9.5939 12.3303 5.8381 10.6765 14.1591 5.8381 12.5639 17.3567 5.8381 14.4426 20.5510 5.8381 25.6963 39.5021

increases with the decreasing value of the Prandtl number (Pr). The influence of Prandtl number (Pr) is highly pronounced at low Reynolds numbers than that at higher values of Reynolds numbers due to the thinning of the momentum boundary layer. Some further insights regarding the role of buoyancy can be gained by examining the relative contributions of the individual drag components by defining a drag ratio CDR () CDP/CDF) for a range of values of the dimensionless parameters (Re, n, Pr, Ri). The main trends are summarized here. Irrespective of the values of Prandtl number (Pr), Richardson number (Ri), and power-law index (n), the drag ratio (CDR) increases with increasing Reynolds number (Re). For instance, the ratio CDR increases almost by a factor of 2 for power-law index of n ) 0.6, 1, and 1.4 in the forced convection regime (Ri ) 0) at the highest value of the Prandtl number, Pr ) 100, as the Reynolds

9748


Figure 7. Dependence of the total drag coefficient (CD) on the power-law index (n), Prandtl number (Pr), and Reynolds number (a) Re ) 1 (0.6 e n e 1.8), (b) Re ) 5 (0.6 e n e 1.8), and (c) Re ) 40 (0.2 e n e 1.8) at two extreme values of the Richardson number (Ri ) 0, 2).

number (Re) is progressively increased from 1 to 40. The corresponding increase is only on the order of 60% in shearthinning (n < 1) and Newtonian (n ) 1) fluids and only about 25% at n ) 1.4 thereby showing the strong interplay between the shear-dependent viscosity, buoyancy effects and the Reynolds number (Re). At low Reynolds numbers (Re ) 1), the Richardson number (Ri) is seen to have a very small effect on the drag ratio for all values of power-law index (n) thereby suggesting that the both components are influenced to the same extent and in the same way. On the other hand, with increasing values of the Reynolds number (Re), this ratio is seen to be strongly dependent on the value of the Richardson number (Ri) at high Prandtl numbers (Pr). The drag ratio (CDR) shows a very strong dependence on the power-law index (n), especially at high Reynolds numbers. This is simply due to the different scaling of the viscous and inertial forces with the characteristic flow velocity.

Similarly, in order to elucidate the role of power-law index (n) on the drag characteristics, the drag coefficient values have been normalized using the corresponding Newtonian values, under otherwise identical conditions, defined as follows XN )

X(n) X(n ) 1)

where X ) CDP, CDF, CD, Nu

(21)

Once again, the key trends are summarized here. With reference to the Newtonian fluids (n ) 1), at low Reynolds number (Re e 10), the pressure drag coefficient is augmented as (CDPN) > 1 in shear-thinning (n < 1) fluids whereas (CDPN) < 1 thereby suggesting a reduction in shear-thickening (n > 1) fluids. However, the reverse behavior is seen at Re ) 40 which arises from the different scaling of the viscous and inertial forces with velocity. These trends are also seen for all values of Richardson number (Ri) considered here. Finally, the effect of the Prandtl


9749

Figure 8. Influence of the buoyancy parameter (Ri ) 0, 2) and Prandtl number (Pr ) 1, 100) on the isotherm patterns in the vicinity of an isothermal cylinder for a range of power-law index (a) n ) 0.2, (b) n ) 0.6, (c) n ) 1, (d) n ) 1.4, and (e) n ) 1.8 at a Reynolds number of Re ) 40.

number (Pr) is seen to be more marked in shear-thickening (n > 1) fluids at high Reynolds number while the reverse is true in shear-thinning (n < 1) fluids. This is mainly due to the relative thicknesses of the momentum and thermal boundary layers. On the other hand, while the normalized frictional component of drag not only exhibits similar trends as seen above, a switchover occurs in its dependence on power-law index at about Re ∼ 5. Furthermore, in this case, the effect of Prandtl number (Pr) is seen to be stronger at low Reynolds numbers than that at high Reynolds number. Finally, the total drag shows trends which are linear combinations of the individual behaviours seen above. (d) Effect of Buoyancy Parameter (Ri). Finally, it is useful to delineate the effect of the buoyancy parameter (Ri) on the drag coefficient. This is easily done by normalizing the value

of the drag coefficient using the corresponding forced convection values (Ri ) 0), under otherwise identical conditions, as follows: XM )

X(Ri) X(Ri ) 0)

where X ) CDP, CDF, CD,Nu

(22)

The main findings can be summarized as follows: Irrespective of the values of Prandtl number (Pr), Reynolds number (Re), and Richardson number (Ri), the normalized pressure component of drag always increases as the value of the power-law index (n) increases, i.e., as the fluid behavior transits from shearthinning (n < 1) to shear-thickening (n > 1) via the Newtonian (n ) 1). The increase is particularly steep at low Prandtl numbers and in shear-thickening fluids. The normalized friction drag component shows qualitatively similar behavior.

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Figure 9. Influence of the buoyancy parameter (Ri ) 0, 2) and Prandtl number (Pr ) 1, 100) on the isotherm patterns in the vicinity of an isothermal cylinder for a range of power-law index (a) 0.2, (b) 0.6, (c) 1, and (d) 1.4 at a Reynolds number of Re ) 1. Table 8. Dependence of Average Nusselt Number (Nu) on Reynolds Number (Re), Power-Law Index (n), Richardson Number (Ri), and Prandtl Number (Pr) Re ) 40

Re ) 10

Re ) 5

Re ) 1

Pr


100

0 1 2 0 1 2 0 1 2 0 1 2 0 1 2

50

20

10

1

29.7471 35.5427 39.6514 22.7238 27.4749 30.6812 15.8176 19.4048 21.6065 11.9925 14.8671 16.4786 4.7335 6.2531 6.5131

21.0687 23.3187 25.2120 16.4838 17.7201 20.3600 11.9303 13.9673 15.2868 9.3081 11.1489 12.2377 4.0552 5.1103 5.6069

17.9168 18.7893 19.9606 14.0449 15.1594 16.1785 10.2412 11.4214 12.2613 8.0605 9.2160 9.9329 3.6526 4.4812 4.8582

16.4578 16.6562 17.5374 12.8824 13.4519 14.2145 9.3937 10.1555 10.7848 7.4086 8.2156 8.7567 3.3990 4.0829 4.3826

15.6635 15.4288 16.1547 12.2496 12.4628 13.0865 8.9164 9.4128 9.9259 7.0309 7.6201 8.0624 3.2305 3.8203 4.0712

10.4262 8.8876 11.6441 9.9054 12.4133 10.4462 8.2686 7.1127 9.3846 8.0263 10.0641 8.5056 6.0850 5.3085 7.0599 6.0952 7.6204 6.4979 4.8230 4.2603 5.6890 4.9587 6.1618 5.3071 2.2103 2.0580 2.8512 2.5099 2.9999 2.7085

8.1741 9.0872 9.5277 6.5284 7.3744 7.7619 4.9051 5.6154 5.9398 3.9608 4.5822 4.8635 1.9709 2.3671 2.5320

In summary, all else being equal, the individual and the total drag coefficients increase with Richardson number (Ri). This can safely be ascribed to the higher stresses due to the aiding buoyancy in this case. 6.3. Heat Transfer Characteristics. The solution of thermal energy equation along with the equation of continuity and motion yields the thermal field, which, in turn is processed to obtain the isothermal (constant temperature) contours, local, and surface averaged Nusselt numbers. (a) Isotherm Patterns. The role of the pertinent dimensionless parameters (Re, n, Pr, Ri) on the temperature contours for a range of conditions is shown in Figures 8 and 9. For a fixed value of the Richardson number (Ri), the crowding of isotherms in the upstream direction increases with the increasing Reynolds (Re) and/or Prandtl (Pr) numbers. Thus, the resulting temperature gradients and, hence, the heat transfer rate increase with Reynolds (Re) and/or Prandtl (Pr) numbers, albeit the dependence on the Reynolds number (Re) is somewhat stronger than that on the Prandtl number (Pr). The influence of the powerlaw index (n) on isotherm patterns is seen to be more pronounced at high Reynolds (Re) and/or Prandtl (Pr) numbers.

7.8956 8.6242 9.0067 6.2744 7.0009 7.3349 4.6805 5.3342 5.6123 3.7826 4.3569 4.5974 1.9147 2.2736 2.4145

7.6429 8.3630 8.546 6.0634 6.7484 7.1767 4.4673 5.1252 5.4968 3.5469 4.1244 4.4485 1.6524 2.0511 2.2347

6.6071 7.2463 7.6219 5.2910 5.8887 6.2261 3.9553 4.4960 4.7838 3.1810 3.6773 3.9291 1.5619 1.9115 2.0610

6.1077 6.7489 7.0662 4.9196 5.4962 5.7782 3.7093 4.2104 4.4498 3.0049 3.4558 3.6650 1.5196 1.8131 1.9567

5.8194 6.4802 6.782 4.6997 5.2804 5.5286 3.5624 4.0493 4.2575 2.8998 3.3270 3.5085 1.4951 1.7796 1.8887

3.9275 4.2197 4.4097 3.1265 3.4165 3.5958 2.3197 2.6088 2.7661 1.8562 2.1313 2.2819 0.9173 1.1319 1.2340

3.3449 3.6458 3.8286 2.6970 2.9976 3.1662 2.0410 2.3337 2.4820 1.6621 1.9445 2.0760 0.8767 1.0998 1.1816

3.1752 3.4784 3.6429 2.5765 2.8704 3.0201 1.9688 2.2467 2.3768 1.6165 1.8797 1.9951 0.8760 1.0806 1.1522

3.1103 3.4210 3.5715 2.5340 2.8253 2.9603 1.9476 2.2136 2.3295 1.6066 1.8538 1.9562 0.8819 1.0713 1.1348

As the fluid behavior changes from shear-thickening (n > 1) to shear-thinning (n < 1), an increase in the compactness of the isotherms results in an overall increase in the temperature gradients. This is because the thermal boundary layer is known to be thinner in shear-thinning (n < 1) fluids than that in Newtonian (n ) 1) fluids under otherwise identical conditions. Figures 8 and 9 also show that with increasing Richardson number (Ri), the clustering of the isotherms becomes dense close to the cylinder surface which indicates, once again, higher temperature gradients and accordingly higher heat transfer rates. (b) Local Nusselt Number, Nu(θ). Figures 10 and 11 depict the representative variation of the local Nusselt number on the surface of the cylinder at two extreme values of Richardson number (Ri ) 0, 2), three representative values of the Reynolds number (Re ) 1, 5, 40), and for three values of the power-law index (n ) 0.6, 1, 1.4) for a range of values of Prandtl number (Pr). It is seen that for fixed values of Re, n, and Pr, the local Nusselt number values are higher in the mixed convection (Ri ) 2) regime than that for forced convection (Ri ) 0), under otherwise identical conditions. This suggests the sharpening of the temperature gradients brought about by the buoyancy-


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Figure 10. Variation of local Nusselt number over the surface of the cylinder at extreme values of Richardson number (Ri ) 0, 2) for a range of values of the Prandtl number (Pr) at Reynolds numbers (a) Re ) 1, (b) Re ) 5, and (c) Re ) 40 for Newtonian (n ) 1) fluids.

induced flow. These figures also show that the nature of variation of the local Nusselt number in the mixed convection (Ri > 0) is qualitatively similar to that in the forced convection (Ri ) 0) heat transfer, e.g., the maximum and minimum values of the local Nusselt number seem to occur at locations other than θ ) 0° and θ ) 180° (measured from the front stagnation point). Two distinct peaks are seen under these conditions. All other features of the dependence of the local Nusselt number in the mixed convection regime are qualitatively similar to that seen in the forced convection case,14-18,20,24,25 especially, the local values increase with the increasing values of Reynolds number (Re), Richardson number (Ri), and Prandtl number (Pr). The maximum and minimum values of the local Nusselt number occur close to the front (θ ) 0°) and rear (θ ) 180°) stagnation points, respectively. The curves also show a downstream shift in the position of the minimum values of the local Nusselt number with increasing values of Reynolds (Re) and Prandtl (Pr) numbers. In shear-thinning (n < 1) fluids, an upstream

(downstream to the front stagnation point) shift in the position of the maximum value and a downward shift (upstream to the rear stagnation point) in the location of the minimum value of the local Nusselt number is seen with increasing Prandtl number (Pr). The power-law index (n) appears to influence the maximum values more than the minimum values of the local Nusselt number in both forced and mixed convections. Irrespective of the type of fluid behavior, the influence of the powerlaw index (n) on the maximum value of local Nusselt number increases with an increase in the Richardson number (Ri). When the imposed flow and that caused by the buoyancy are in the same direction (aiding buoyancy), the thermal boundary layer is further thinned and hence there is an increase in the local value of the heat transfer coefficient. (c) Average Nusselt Number (Nu). Table 8 shows the functional dependence of the average Nusselt number (Nu) on the Reynolds number (Re), power-law index (n), Richardson number (Ri), and Prandtl number (Pr). For a fixed value of the

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Figure 11. Variation of local Nusselt number over the surface of the cylinder at extreme values of Reynolds number (Re ) 1, 40) for a range of values of Prandtl number (Pr) and for power-law index of (a) n ) 0.6, (b) n ) 1, and (c) n ) 1.4 at a Richardson number of Ri ) 2.

Richardson number (Ri), the functional dependence of the average Nusselt number (Nu) in the mixed convection is qualitatively similar to that in the forced convection.14-18,20,24,25 For instance, the average Nusselt number (Nu) shows an increase with Reynolds number (Re) and/or Prandtl number (Pr) and/or both. For fixed values of the parameters (Ri, Re, Pr), the average Nusselt number increases as the fluid behavior changes from Newtonian (n ) 1) to shear-thinning (n < 1); however, the opposite effect is seen in shear-thickening (n > 1) fluids. For fixed values of Reynolds number (Re), flow behavior index (n), and Richardson number (Ri), an increasing value of the Prandtl number (Pr) also enhances the rate of heat transfer, irrespective of the type of the fluid behavior. The average Nusselt number (Nu) values in the mixed convection are always higher than those in the forced convection. For a fixed Reynolds number (Re), the average Nusselt number (Nu) is seen to increase with increasing Richardson number (Ri). It is also seen that the influence of Richardson number (Ri) on the average Nusselt

number (Nu) is more prominent at low Reynolds number than that for high Reynolds numbers. For instance, for fixed Pr ) 100, Re ) 1, and n ) 0.6, an increase in the Richardson number (Ri) from 0 to 2 enhances the average Nusselt number (Nu) by 12.27% whereas the corresponding enhancements are 33% at n ) 0.2 and 20% at n ) 0.6 and Re ) 40. Further examination of heat transfer results reveals that all else being equal, shear-thinning (n < 1) fluid behavior can enhance heat transfer by up to 40-50% whereas shearthickening (n > 1) has an adverse effect and one can observe reduction of about 10% in heat transfer with respect to the corresponding behavior in Newtonian fluids. With the aiding buoyancy, the enhancement can be as much as ∼100% at a Richardson number of Ri ) 2 and power-law index of n ) 0.2. On the other hand, comparing the results of forced convection with that for mixed convection suggests increase in the value of Nusselt number (Nu) on the order of 45% depending upon the values of the pertinent parameters. Broadly, the degree of


enhancement shows a positive dependence on Richardson number (Ri), Reynolds number (Re), and inverse dependence on power-law index (n) and Prandtl number (Pr). Finally before leaving this section, it is useful to emphasize here that both the flow and temperature fields in the vicinity of the cylinder are determined by a complex interplay between the viscous, inertial, pressure, and buoyancy forces prevailing in the fluid. These forces in turn show different scaling with respect to the power-law index, velocity, and the characteristic linear dimension. For instance, the viscous forces scale as ∼U0n whereas the inertial forces scale as U02. Now, for a fixed value of velocity, the inertial force does not change, but the viscous force can decrease or increase depending upon the value of the power-law index (n). Conversely, for a given fluid, with a gradual increase in the fluid velocity, the viscous term will diminish for a shear-thinning (n < 1) fluid whereas it will grow in a shear-thickening (n > 1) fluid. Therefore, some of the nonmonotonic trends seen in the preceding sections are due to such complex interactions. These interactions are further accentuated in mixed convection due to the coupled nature of the momentum and thermal energy equations. 7. Concluding Remarks In this work, the momentum and heat transfer characteristics of the steady flow of power-law fluids across a cylinder in the buoyancy aiding mixed convection have been studied numerically over wide ranges of conditions as follows: Reynolds number (1 e Re e 40), Prandtl number (1 e Pr e 100), Richardson number (0 e Ri e 2), and power-law index (0.2 e n e 1.8). The effect of the dimensionless parameters (Re, Pr, n, Ri) on the detailed kinematics and global aspects of the flow and heat transfer characteristics are elucidated in detail. For a range of values of power-law index (n) and Reynolds number, the wake size shows qualitatively the same trend as that seen in the forced convection regime, whereas it decreases with increasing Richardson number (Ri). The surface pressure coefficient decreases with the increasing Reynolds and Prandtl numbers, and it is influenced only weakly by the power-law index; it increases with increasing Richardson number. Increasing values of the Prandtl number lower the value of drag coefficients. The mixed convection heat transfer characteristics are qualitatively similar to the forced convection in power-law fluids. The local and average Nusselt numbers increase with increasing values of Reynolds, Prandtl, and Richardson numbers. Broadly, the rate of heat transfer increases with the increasing Reynolds, Richardson, and Prandtl numbers and with the decreasing value of the power-law index. The dependence is strongest on the Reynolds number, however. It is possible to augment the rate of heat transfer by up to about 40-45% by imposing aiding buoyancy flow under appropriate combinations of the power-law index, Prandtl number, Reynolds number, and Richardson number in a given application. Notations cp ) specific heat of the fluid, J/(kg K) CD ) total drag coefficient, dimensionless CDM ) normalized total drag coefficient using the corresponding forced convection value [) CD(mixed convection)/CD(forced convection)], dimensionless CDN ) normalized total drag coefficient using the corresponding Newtonian value [) CD(non-Newtonian)/CD(Newtonian)], dimensionless CDF ) frictional component of the drag coefficient, dimensionless

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) normalized friction drag coefficient using the corresponding forced convection value [) CDF(mixed convection)/CDF(forced convection)], dimensionless CDFN ) normalized friction drag coefficient using the corresponding Newtonian value [) CDF(non-Newtonian)/CDF(Newtonian)], dimensionless CDP ) pressure component of the drag coefficient, dimensionless CDPM ) normalized pressure drag coefficient using the corresponding forced convection value [) CDP(mixed convection)/CDP(forced convection)], dimensionless CDPN ) normalized pressure drag coefficient using the corresponding Newtonian value [) CDP(non-Newtonian)/CDP(Newtonian)], dimensionless CDR ) drag ratio, [) CDP/CDF], dimensionless Cp ) pressure coefficient, dimensionless Cp(0) ) pressure coefficient at the front stagnation (θ ) 0) point, dimensionless Cp(π) ) pressure coefficient at the rear stagnation (θ ) π) point, dimensionless D ) diameter of the cylinder, m D0 ) outer boundary of the computational domain, m FD ) drag force per unit length of the cylinder, N/m FDF ) frictional component of the drag force per unit length of the cylinder, N/m FDP ) pressure component of the drag force per unit length of the cylinder, N/m Gr ) Grashof number, dimensionless h ) local convective heat transfer coefficient, W/(m2 K) I2 ) second invariant of the rate of the strain tensor, s-2 k ) thermal conductivity of the fluid, W/(m K) m ) power-law consistency index, Pa sn n ) power-law flow behavior index, dimensionless Nu ) surface averaged Nusselt number, dimensionles NuM ) normalized average Nusselt number using the corresponding forced convection value, [) Nu(mixed convection)/Nu(forced convection)], dimensionless NuN ) normalized average Nusselt number using the corresponding Newtonian value, [) Nu(non-Newtonian)/Nu(Newtonian)], dimensionless Nu(θ) ) local Nusselt number over the surface of the cylinder, dimensionless p ) pressure, Pa Pr ) Prandtl number, dimensionless Re ) Reynolds number, dimensionless Ri ) Richardson number () Gr/Re2), dimensionless T ) temperature, K T0 ) temperature of the fluid at the inlet, Tw ) temperature of the surface of the cylinder, K U0 ) uniform inlet velocity of the fluid, m/s Ux, Uy ) x- and y-components of the velocity, m/s x, y ) Transverse and streamwise coordinates, m Greek Symbols η ) viscosity, Pa s θ ) angular displacement from the front stagnation (θ ) 0), deg ω ) vorticity, dimensionless F ) density of the fluid, kg/m3 τ ) extra stress tensor, Pa τxx, τxy ) x- and y-component of the shear stress, Pa CDFM

Note Added after ASAP Publication: The version of this paper that was published on the Web April 7, 2009 had errors in eq 16b. The correct version of this paper was reposted to the Web April 10, 2009.

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ReceiVed for reView December 9, 2008 ReVised manuscript receiVed March 9, 2009 Accepted March 11, 2009 IE801892M

Mixed Convection Heat Transfer from a Cylinder in Power-Law Fluids

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