Two-Dimensional Steady Flow of a Power-Law Fluid Past a Square Cylinder in a Plane Channel: ...

Two-Dimensional Steady Flow of a Power-Law Fluid Past a SquareCylinder in a Plane Channel: Momentum and Heat-TransferCharacteristics

Abhishek K. Gupta,† Atul Sharma,‡ Rajendra P. Chhabra,*,† and Vinayak Eswaran‡

Departments of Chemical Engineering and Mechanical Engineering, Indian Institute of Technology,Kanpur, India 208016

The momentum and forced-convection heat-transfer characteristics for an incompressible andsteady flow of power-law liquids past a square cylinder in a plane channel have been analyzednumerically. The momentum and energy equations have been solved using a finite-differencemethod for a range of rheological and kinematic conditions as follows: power-law index, n )0.5-1.4, thereby covering both shear-thinning and shear-thickening behavior; Reynolds number,Re ) 5-40; and Peclet number, Pe ) 5-400. However, all computations have been performedfor one blockage ratio b/2h ) 1/8. Furthermore, the role of the type of thermal boundary condition,i.e., the constant heat flux and the isothermal surface, on the rate of heat transfer has alsobeen studied. Overall, when the drag coefficient and Nusselt number are normalized using thecorresponding Newtonian values, these ratios show only marginal additional dependences onthe flow behavior index. The shear thinning not only reduces the size of the wake region butalso delays the wake formation, and shear thickening shows the opposite effect on wakeformation, etc.

Introduction

The flow of fluids past bluff bodies, particularlycircular and square cylinders, represents an idealizationof many industrially important applications, and there-fore, it has received a great deal of attention over theyears. Most studies of this phenomenon have beenconcerned with the flow past a circular cylinder underfree-flow unconfined conditions, albeit limited resultson the extent of wall effects are also available. Thevoluminous literature relating to the flow of Newtonianfluids over a circular cylinder has been reviewed re-cently in two volumes.1,2 In contrast to the voluminousliterature on circular cylinders, the analogous case ofthe flow past a square cylinder has been investigatedmuch less extensively. Studies of such behavior havereceived impetus from two distinct but interrelatedobjectives. From a theoretical standpoint, such highlyidealized model configurations are convenient for thepurpose of elucidating fluid mechanical phenomenaincluding drag and wake characteristics, vortex shed-ding frequency, and detailed kinematics of the resultingflow field and their effects on properties such as the rateof heat transfer. On the other hand, this type ofinformation is frequently needed for the design ofstructures exposed to fluid flow (such as off-shorepipelines). Furthermore, because of changing processand climatic conditions, one also needs to calculate therate of heat transfer from such structures. Additionalapplications are found in polymer processing operationssuch as the formation of weld lines. Indeed, under

appropriate conditions of Reynolds number and blockageratio, a wide variety of phenomena has been observedeven with the flow of Newtonian fluids over a squarecylinder.3 It is now well-known that many materials(e.g., polymer solutions, melts, muds, emulsions, sus-pensions) encountered in chemical and processing ap-plications often exhibit complex rheological behavior,including shear thinning and shear thickening, vis-coelasticity, and yield stress.4 Despite their wide occur-rence, as far as is known to us, there has been no priorstudy of the flow of such non-Newtonian liquids pastbluff bodies, particularly square cylinders, and thepresent work aims to fill this gap in the existing lit-erature. It is, however, instructive and useful to brieflyrecount the corresponding body of knowledge availablefor the flow of Newtonian fluids, as this facilitates thesubsequent discussion of non-Newtonian liquids.

Previous Work

Because excellent accounts of the experimental andnumerical studies of the confined and unconfined,steady and unsteady flow of Newtonian fluids over asquare cylinder are available,1-3,5 only the salient pointsare repeated here. Okajima6 and Okajima et al.7 re-ported an extensive numerical and experimental studyfor an unconfined Newtonian fluid flow over a cylinderof rectangular cross section in the Reynolds numberrange from 100 to 2 × 104. The main thrust of their workwas to delineate the frequency of vortex shedding fromthe cylinder and to predict the corresponding Strouhalnumbers. The phenomenon of vortex shedding for asquare cylinder confined in a channel was also numeri-cally investigated by Mukhopadhyay et al.8 They re-ported that vortex shedding induces periodicity in theflow field. In particular, they investigated the effect ofthe blockage ratio and found the periodicity of flow tobe suppressed by the presence of confining boundaries.

* To whom correspondence should be addressed. Address:R. P. Chhabra, Department of Chemical Engineering, IndianInstitute of Technology, Kanpur, India 208016. Tel.: 0091-512-2597393. Fax: 0091-512-2590104. E-mail: [email protected].

† Department of Chemical Engineering.‡ Department of Mechanical Engineering.

5674 Ind. Eng. Chem. Res. 2003, 42, 5674-5686

10.1021/ie030368f CCC: $25.00 © 2003 American Chemical SocietyPublished on Web 10/04/2003

However, virtually no drag results were reported in anyof these studies. In a combined numerical and theoreti-cal study, Davis et al.9 reported a good match betweentheir predictions and experimental results for the cross-flow past a rectangular cylinder placed in a horizontalchannel with the Reynolds number of flow ranging from100 to 2000. More recently, Breuer et al.10 investigatedlow Reynolds number flow past a square cylinder placedin a channel with a parabolic inlet profile. They con-trasted the predictions of two numerical schemes,namely, the lattice-Boltzmann and the finite-volumemethods. Likewise, Valencia11 analyzed the unsteadylaminar flow pattern and convective heat transfer in achannel with a built-in square cylinder. He reportedsignificant enhancements in the rate of heat transferunder oscillatory separated flow conditions due to thecylinder being present in the channel. From a carefulscrutiny of the available literature for the flow ofNewtonian liquids past a square cylinder, two pointsseem to emerge clearly: First, it is generally believedthat steady two-dimensional flow occurs up to about theReynolds number of 50-55 in unconfined conditions,although steady flow is possible up to higher Reynoldsnumber when the blockage ratio is large; the firstmanifestation of the flow instability is through vortexshedding. Second, most of the aforementioned studiesattempt to elucidate the interplay between the blockageratio and the Reynolds number on one hand and thedetailed kinematics of flow (wake phenomena, vortexshedding, etc.) on the other, and indeed, only very scantresults are available on gross parameters of engineeringinterest such as the drag coefficient and Nusselt numberas functions of the relevant dimensionless parameterseven for Newtonian fluids, let alone for non-Newtonianliquids. For power-law fluids, as far as is known to us,there has been only limited activity relating to uncon-fined flow past a circular cylinder.12,13 Drag values asfunctions of the power-law index (0.7-1.2) and theReynolds number (5-40) were reported by D’Alessio andPascal.12 However, as the value of the Reynolds number(5-40) was progressively increased, they encounteredacute convergence difficulties for both shear-thinningand shear-thickening fluids even at a Reynolds numberof 20. Thus, for instance, for the maximum value of theReynolds number of 40 in their study, a fully convergedsolution could be obtained only for very weakly non-Newtonian fluids (0.95 e n e 1.1). On the other hand,Whitney and Rodin13 presented the drag results for theslow (creeping-flow) translation of spheres and of cyl-inders (of infinite and of finite aspect ratios) in uncon-fined power-law fluids (shear thinning). Their resultsare consistent with the previous extensive results forspheres and with the limited results for cylinders. It isalso appropriate to add here that the scant availableexperimental results of drag on nonspherical particles(including circular and square cross-section cylinders)in free-fall conditions in non-Newtonian fluids have beensummarized by several investigators.14-20 In most suchexperimental studies, the maximum aspect ratio (l/d)of the cylinders was typically on the order of 10-15, andtherefore, such models will constitute a very poorapproximation for infinitely long cylinders as assumedin the study of d’Alessio and Pascal.12

From the aforementioned account, it is thus safe toconclude that no prior theoretical results are availablefor the Poiseuille flow and the corresponding heattransfer in non-Newtonian liquids past a square cylin-

der situated symmetrically in a plane channel. Admit-tedly, the diversity of materials encountered in engi-neering practice (e.g., polymer melts and solutions,foams, emulsions, suspensions) display a wide varietyof rheological phenomena, including shear thinning,shear thickening, yield stress, and viscoelasticity, amongothers. However, it seems logical to begin with thesimplest and also the commonest type of non-Newtonianfeature, namely, the shear-thinning and shear-thicken-ing (dilatant) types of non-Newtonian behavior. Almostall polymeric melts (unfilled) and solutions displayshear-thinning behavior, whereas highly concentratedpastes and filled resins behave as shear-thickeningfluids over a range of shear rates.4 This initial modelcan, in turn, be used to build up the level of complexityin a gradual manner to incorporate the other non-Newtonian effects. In this work, the continuity, momen-tum, and energy equations have been solved numeri-cally for the unknown velocity, pressure, and temperaturefields for the steady Poiseuille flow of incompressiblepower-law liquids past a square cylinder placed in achannel. The results have, in turn, been processedfurther to deduce the values of the dimensionless dragcoefficient and Nusselt number as functions of theReynolds number, Peclet number, and power-law indexover wide ranges of conditions. Owing to the generallyhigh viscosity levels of non-Newtonian materials, coupledwith the fact that the steady flow past a square cylinderoccurs only up to about Re ) 50 in Newtonian fluids,the present study is limited to a maximum Reynoldsnumber of 40, which also ensures the two-dimensional-ity of the flow under these conditions.

Problem Statement and Governing Equations

Let us consider the steady incompressible two-dimensional flow of a power-law fluid past a squarecylinder (side length b) placed in a channel (width 2h),as shown schematically in Figure 1. The liquid entersthe channel with a fully developed velocity profile andat a constant temperature T∞. As noted earlier, becausethe maximum value of the Reynolds number consideredin this work is 40, the flow can be assumed to be two-dimensional and steady. Hence, no flow occurs in the zdirection, and no flow variable depends on the z coor-dinate, i.e., ∂X/∂z ) 0 for all variables X. Furthermore,the thermophysical properties (density, F; heat capacityCp; thermal conductivity, k; and power-law constants,m and n) are assumed to be independent of temperature.Under these conditions, the equations of continuity,momentum, and thermal energy (in the absence ofviscous dissipation) in their dimensionless form arewritten as follows

Figure 1. Schematics of the flow and the computational domain.

Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 5675

For a power-law fluid, the constitutive relation is givenby

where in Cartesian coordinates

The relevant components of the extra stress tensor arewritten as

In eqs 1-5, the velocities have been rendered dimen-sionless using the maximum (centerline) velocity Vo, alldistances in the x and y directions using the side of thesquare (b), the pressure using FVo

2, the extra stresscomponents using m(Vo/b)n, the viscosity using a refer-ence viscosity m(Vo/b)n-1, and the time using (b/Vo) asthe characteristic time. The temperature is made di-mensionless in two different ways depending on thethermal boundary condition specified at the surface ofthe solid cylinder. The two commonly used thermalboundary conditions are that of a uniform temperatureTw or that of a constant heat flux (qw) at the surface ofthe boundary CDD′C′ in Figure 1. In the case of theconstant temperature, the temperature difference (T ′- T∞) is scaled using (Tw - T∞) as the characteristictemperature difference. In the case of the constant heatflux boundary condition imposed on the surface of thesolid cylinder, (qwb/k) is used as the characteristictemperature difference. Furthermore, the two dimen-sionless groups, namely, the Reynolds (Re) and thePeclet number (Pe), appearing in eqs 2 and 3 are definedas follows

The advantage of using the Peclet number as opposedto the commonly employed Prandtl number is that thePeclet number is independent of the power-law con-stants, and this definition also thus coincides with thatcommonly used for Newtonian fluids.

After the extra stress components are substituted intothe momentum equations and the equations are rewrit-ten in their conservative forms, eqs 2a, 2b, and 3 become

It is worthwhile pointing out here that the viscousdissipation term has been neglected in the energyequations here, eqs 3 and 9. This term is usuallysignificant when either the shear rate is very high orthe fluid has a very high effective viscosity or both.Because the blockage ratio in the present case is 1/8 andthe maximum Reynolds number is 40, the maximumvalue of the shear rate close to the surface of thecylinder is not expected to be excessively high. Similarly,because the Reynolds number is not in the creepingrange, the effective viscosity of the fluid is also likelynot to be too high. On both these counts, the exclusionof the viscous dissipation term is probably a reasonableapproximation under such conditions.

The physically admissible and consistent boundaryconditions for this flow configuration (see Figure 1) areas outlined below.

At the inlet plane HH′

It is appropriate to add here that the velocity profilegiven by eq 10a is applicable under laminar flowconditions in the channel, i.e., for Reynolds numberbased on the width (2h) of the channel that are less than∼2000. Because the maximum value of the Reynoldsnumber (based on b) is only 40, this will translate intoa value of about 350 in terms of the channel Reynoldsnumber and somewhat higher in the annular region.Nevertheless, the Reynolds number is unlikely to reachsufficiently high values for the inertial effects to becomesignificant. It is thus reasonable to use this velocityprofile in the present study.

Continuity

∂Vx

∂x+

∂Vy

∂y) 0 (1)

Momentum

x component

∂Vx

∂t+ Vx

∂Vx

∂x+ Vy

∂Vx

∂y) - ∂p

∂x+ 1

Re(∂τxx

∂x+

∂τyx

∂y ) (2a)

y component

∂Vy

∂t+ Vx

∂Vy

∂x+ Vy

∂Vy

∂y) - ∂p

∂y+ 1

Re(∂τyx

∂x+

∂τyy

∂y ) (2b)

Thermal Energy

∂T∂t

+ Vx∂T∂x

+ Vy∂T∂y

) 1Pe(∂2T

∂x2+ ∂

2T∂y2) (3)

τij ) 2ηεij (4a)

η ) [12(∆:∆)]n-1/2(4b)

[12(∆:∆)] ) 2(∂Vx

∂x )2

+ 2(∂Vy

∂y )2

+ (∂Vx

∂y+

∂Vy

∂x )2

(4c)

τxx ) 2η∂Vx

∂x, τyy ) 2η

∂Vy

∂y, τxy ) τyx ) η(∂Vx

∂y+

∂Vy

∂x )(5)

Re )FVo

2-nbn

m(6)

Pe )FVobCp

k(7)

Momentum Equation

x momentum equation

∂Vx

∂t+ ∂

∂x(VxVx) + ∂

∂y(VxVy) )

- ∂p∂x

+ ηRe(∂2Vx

∂x2+

∂2Vx

∂y2 ) + 2Re(εxx

∂η∂x

+ εyx∂η∂y) (8a)

y momentum equation

∂Vy

∂t+ ∂

∂x(VxVy) + ∂

∂y(VyVy) )

- ∂p∂y

+ ηRe(∂2Vy

∂x2+

∂2Vy

∂y2 ) + 2Re(εyy

∂η∂y

+ εxy∂η∂x) (8b)

Thermal Energy Equation

∂T∂t

+ ∂

∂x(VxT) + ∂

∂y(VyT) ) 1

Pe(∂2T∂x2

+ ∂2T

∂y2) (9)

Vx ) 1 - (y/h)n+1/n, Vy ) 0, T ) 0 (10a)

5676 Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003

At the surface of the square cylinder (CDD′C′), theusual no-slip boundary condition is prescribed, i.e.

For the constant-temperature condition (Tw)

For the constant-heat-flux condition, at the surface ofthe cylinder

where n represents the directions normal to the surfaceof the cylinder.

At the confining boundaries HG and H′G′, the no-slipcondition is used, i.e.

At the exit plane GG′, there is no unique prescriptionavailable for outflow. The main concern here is that thecondition implemented at GG′ must not influence theflow upstream in any significant manner. This is bestaccomplished by using the so-called Orlanski condi-tion,21 written as follows for the momentum equationsin its dimensionless form

The corresponding form for the thermal energy equationis given by

where i ) x or y and Vconv ) 1 was used here. Thisparticular form of the boundary condition at the exitensures that the vortices can grow unhindered and exitthe flow domain without causing any appreciable dis-turbance to the flow upstream. It is appropriate to addhere that the use of this condition also reduces thenumber of iterations required per time step and allowsa shorter upstream computational domain as comparedto that required in the case of the Neumann boundarycondition.

Thus, eqs 1 and 8-10 provide the theoretical frame-work for describing the flow and heat-transfer phenom-ena between a power-law fluid and a square cylinderfor the flow conditions shown in Figure 1. Theseequations have been solved numerically using a finite-difference method for the unknown velocities (Vx, Vy),pressure (p), and temperature (T) over wide ranges ofphysical, rheological, and kinematic conditions. Once theflow domain has been mapped in terms of these vari-ables, the gross engineering parameters such as thedrag coefficient and Nusselt number can be evaluatedas described below.

It is convenient to introduce a dimensionless dragcoefficient (CD), defined as

where FD is the drag force experienced by the squarecylinder per unit length (in the z direction). This total

drag is made up of two components, namely, the form(pressure), CDP, and the friction (CDF) drag, which, inturn, are evaluated by integrating the pressure on thetwo faces of the cylinder CC′ and DD′ as follows

Similarly, the friction component acting in the x direc-tion stems from the shearing forces acting on the top(CD) and bottom (C′D′) faces of the cylinder, and thiscontribution is obtained by evaluating the followingintegral on these two faces

CD is then simply the sum of these two components.The Nusselt number, Nu, defined as hb/k is similarly

evaluated using the temperature field as follows: Forthe top (CD) and bottom (C′D′) faces of the cylinder

For the front (CC′) and rear (DD′) faces of the cylinder

where n represents the direction normal to the surfaceof the cylinder.

Equation 14 applies for the constant-temperaturecondition at the solid cylinder. Analogous expressionsfor the case of the constant heat flux can be obtainedby writing the heat balance at the surface of thecylinder. For instance, at each point on the surface ofthe cylinder, qw ) h(T ′-T∞), which, when made dimen-sionless using qwb/k as the characteristic temperaturedifference, yields the following expressions for the localNusselt number

Such local values were further averaged over eachface or over the whole cylinder to obtain the surfaceaveraged values of the Nusselt number and/or theoverall mean value of the Nusselt number for the wholecylinder, which is the quantity often required in processengineering design calculations. In addition to theaforementioned parameters, the flow domain was alsomapped in terms of the streamlines, isovorticity linesand isotherms, as these facilitate the visualization ofthe flow and temperature patterns, wake size, etc., toprovide further physical insights into the nature of theflow processes. For a fixed geometry, it can easily beshown that there are five dimensionless groups for thisproblem: Re, Pe, Nu, CD, and n. For a fixed geometry(blockage ratio and upstream and downstream dis-

Vx ) 0, Vy ) 0 (10b)

T ) 1 (10c)

∂T∂n

) -1 (10d)

Vx ) 0, Vy ) 0, T ) 0 (10e)

∂Vi

∂t+ Vconv

∂Vi

∂x) 0 (10f)

∂T∂t

+ Vconv∂T∂x

) 0 (10g)

CD )2FD

FVo2b

(11)

CDP ) 2(∫C ′

Cp dy - ∫D ′

Dp dy) (12)

CDF ) 2Re∫(η

∂Vx

∂y ) dx (13)

Nux ) - ∂T∂n

(14a)

Nuy ) - ∂T∂n

(14b)

For the top and the bottom faces of thesolid square cylinder

Nux ) 1TCD or C′D′(x)

(15a)

For the front and the rear faces of thesolid square cylinder

Nuy ) 1TCC′ or DD′(y)

(15b)


tances), this study aims to establish the functionaldependence of the drag coefficient CD on the Reynoldsnumber and the power-law index, as well as that of theNusselt number, Nu, on the Reynolds and Pecletnumbers and the power-law index.

Numerical Solution Procedure

The numerical solution procedure used here is anSMAC-type implicit method, as detailed in our previousstudies.22,23 Therefore, only the salient features of theprocedure are reported here. The SMAC-type implicitscheme was implemented on a staggered grid for thesolution of the continuity and momentum equations, i.e.,eqs 1, 8a, and 8b. The convective terms in eqs 8a and8b were discretized using the upwind scheme, whereasthe viscous terms were approximated using the stan-dard center-difference approach. The time-steppingsolution procedure entailed two steps: The first was apredictor step in which the unknown velocities werecalculated (predicted) using an assumed pressure field.Owing to the implicit discretization of the viscous terms,this step was iterative in nature. The second stepentailed the repeated correction of the pressure andvelocity fields until the equation of continuity wassatisfied. This was done by solving the Poisson equationfor the pressure correction iteratively within the pre-scribed limits together with appropriate boundary con-ditions. The use of the implicit scheme here helped avoidthe commonly encountered numerical instability, espe-cially at low Reynolds numbers as is the case in thisstudy. The time-stepping procedure was pseudotrans-ient; starting from arbitrary initial conditions, the timestepping of the momentum equation was continued untilthe steady-state solution was achieved. The steady-statevelocity field was then used to obtain the temperaturefield by applying an analogous time-stepping scheme tothe energy equation, eq 9, with an implicit method usingeither the upwind or the center-difference scheme forthe convective terms. Once again, the conduction termswere approximated by the center-difference approach.Starting with an arbitrary temperature field, the timestepping was continued until a steady-state temperaturefield was obtained.

The resulting pressure, velocity, and temperaturefields, in turn, formed the basis of the calculations ofthe derived variables affording a visualization of flowin terms of isovorticity, streamline, and isotherm pat-terns and the calculation of the gross engineeringparameters such as the drag coefficient and Nusseltnumber as functions of the pertinent variables.

Choice of Numerical Parameters

The accuracy and reliability of the numerical resultsis contingent upon an appropriate choice of the followingparameters defining the flow domain: upstream com-putational domain LU, downstream computational do-main LD, and grid size M × N. Naturally, theseparameters exert varying levels of influence on the flowand temperature fields, which, in turn, determine thevalues of the gross engineering parameters. Because noprior results are available for power-law liquids, ap-propriate values of the aforementioned parameters wereselected on the basis of the most severe case withNewtonian fluid behavior, i.e., Re ) 40 and Pr ) 10.Because the main idea here is to elucidate the non-Newtonian effects, and therefore the values of CDP, CDF

and Nu were calculated for a fixed value of b/2h ) 1/8,i.e., for a blockage ratio of 0.125. To establish anappropriate value of the upstream computational do-main, the value of LU was varied from 2 to 6, while thedownstream distance LD ) 12 was used in all ourcalculations. This experimentation clearly revealed thatthe resulting gain in the accuracy of CD and ⟨Nu⟩ wasonly marginal (less than approximately 1% in CD and0.36% in ⟨Nu⟩) at the expense of an exhortbitantincrease in CPU time when LU was increased beyond6. Similarly, limited calculations with n * 1 suggestedthe changes in the values of CD and ⟨Nu⟩ to be evensmaller than those quoted above. Thus, on the basis ofthese considerations, LU ) 6, LD ) 12, and b/2h ) 1/8were the values used in this study. Similar values havetypically also been used by others in the literature forthis flow problem.

Once these parameters had been fixed, attention wasturned to the choice of an appropriate grid. In fact, thisexploration was done for three values of n ) 0.5, 1, and1.4 for the maximum values of the Reynolds number,Re ) 40, and Prandtl number, Pr ) 10, i.e., Pe ) 400.Four grids 76 × 32, 114 × 48, 190 × 80, and 228 × 96were tested. Once again, weighing the marginal degreeof improvement in the values of drag coefficient andNusselt number obtained with the finest grid, i.e., 228× 96, against a disproportionately large increase in CPUtime, the 190 × 80 grid was regarded to be adequatefor the present results to be essentially grid-indepen-dent, especially for the values of n different from unity.Thus, to reiterate, all results reported in this studyrelate to the conditions of LU ) 6, LD ) 12, and b/2h )1/8 and were obtained using a 190 × 80 grid. It isworthwhile mentioning here that, owing to the two-dimensional and steady nature of the flow, the compu-tations were performed only for the upper half of thedomain, i.e., 0 e y e h.

Results and Discussion

Prior to undertaking the detailed presentation anddiscussion of the new results obtained in this work, itis necessary and desirable to validate the numericalsolution procedure used in this study, as this will alsohelp ascertain the level of uncertainty inherent in thenew results reported herein for power-law liquids.

(i) Validation of Numerical Solution Procedure.Validation of this procedure is generally accomplishedby benchmarking the numerical results against theavailable reliable numerical and/or analytical predic-tions for the analogous problem. In this work, the priornumerical results on drag due to Breuer et al.10 andSaha et al.5 for Newtonian liquids have been used tobenchmark the present results on drag. Saha et al.5numerically studied the two-dimensional steady flow ofan incompressible Newtonian fluid past a square cyl-inder, for a blockage ratio of 1/10 and upstream anddownstream distances of 5.5 and 33.5, respectively. ForRe ) 100, they reported the drag coefficient value of CD) 2.92. This flow (with LU ) 5.5, LD ) 33.5, and b/2h )0.1) was simulated in the present study using a uniformgrid (402 × 102) and using the first-order upwinddiscretization approximation for the convection termsin the momentum equation. The present value of thedrag coefficient, CD ) 2.941, is indeed in excellentagreement with that of Saha et al.;5 the differencebetween the two values is only 0.73%, which is not atall uncommon in this kind of work.


Similarly, Breuer et al.10 employed a nonuniformmesh and contrasted the performance of two numericalsolution procedures, namely, the finite-volume and thelattice-Boltzmann methods. For a blockage ratio of 1/8and upstream and downstream distances of 12 and 37,respectively, they reported CD ) 2.00 for Re ) 30. Thisvalue also compares favorably with that obtained in thepresent study, i.e., CD ) 2.06, resulting in a differenceof 3%. In fact, the two values differ by even smalleramounts at lower Reynolds number. Clearly, this smalldifference can safely be ascribed to the slightly differentvalues of LU and LD used, coupled with the differentnumerics employed in the two studies. It needs to beemphasized here that, because Breuer et al.10 varied theReynolds number from 0.5 to 300, the required down-stream computational domain increased with increasingReynolds number and, in fact, a value of LD ) 12 isbelieved to be quite adequate up to about Re ≈ 50.

Finally, Mukhopadhyay et al.8 studied the wakestructure around a square cylinder in a channel. Theyreported the flow field upstream of the cylinder to benearly unaltered by wake formation in the rear of thecylinder. They evaluated the shear stress distributionat the wall of the channel. For fully developed laminarflow, the local skin friction coefficient (Cf‚Re) on thechannel walls had a value of 12 in the absence of thecylinder, and clearly, this value increased as a result ofthe hydrodynamic resistance of the cylinder. For Re )50, the difference in the resulting peak values of theskin friction coefficient was on the order of 5% betweenthe present results and those of Mukhopadhyay et al.8Judging from these comparisons, it is perhaps fair tosay that the uniform grid and the first-order upwindscheme employed herein can yield reliable results at therelatively low Reynolds numbers used in this study.Furthermore, the values of drag coefficient presentedherein are believed to be accurate to within 3-4%.

Likewise, the accuracy of the solution procedureapplied to the thermal energy equation, eq 9, to obtainthe temperature field was validated using two bench-mark cases. In the first instance, the developing thermalboundary layer flow was considered in a two-dimen-sional channel without the square cylinder. For aReynolds number (based on the hydraulic diameter) of150, the computed temperature field was compared withthe corresponding analytical solution24 obtained usinga uniform grid of 400 × 200 for Pr ) 0.71 at variousaxial locations. The two values of the temperature werefound to be within (0.35% of each other, therebylending support to the correctness and accuracy of thenumerical solver for the unknown temperature. Thesecond benchmarking was done using unpublishedresults25 for free stream flow across a square cylinder,which were obtained using a hybrid grid, consisting ofa fine grid adjacent to the cylinder and a uniform coarsegrid away from the cylinder, to study heat transferbetween a Newtonian fluid (air) and a square cylinderwith a grid size of 323 × 264; a blockage ratio of 1/20 tosimulate free stream flow; and upstream and down-stream distances of 8.5 and 16.5, respectively. Thepresent values of the Nusselt number averaged over thefront, top, and rear faces of the cylinder and the overallmean values are compared with those of Sharma25 inTable 1 for the constant-temperature condition and inTable 2 for the constant-heat-flux condition imposed atthe cylinder surface. The present results were obtainedusing slightly different (but still adequate) values of the

upstream and downstream distances of 6 and 12,respectively; a blockage ratio of 1/15 with free-slipboundary condition at the transverse boundary; and auniform mesh of 190 × 120. An examination of theseresults shows that the two values differ by at most 1%from each other. Such minor differences are also notuncommon in such studies12 and can easily by attributedto the slightly different values of the blockage ratio andupstream and downstream lengths of the computationaldomain used in the present work. Conversely, one canargue that the heat-transfer characteristics are rela-tively insensitive to the blockage ratio and inlet and exitlengths under this range of conditions. This fact alsoprovides a justification for the choice of numericalparameters, including the blockage ratio, mesh, etc.,used in this work as described in the previous section.

Judging from the aforementioned benchmark com-parisons and our previous experience,22,23 it is perhapsreasonable to state that the new values of the total dragcoefficients and of the Nusselt number are reliable towithin 1-2%.

(ii) Drag Phenomena and Flow Field. The hydro-dynamic drag force exerted by the fluid on the squarecylinder is determined by the individual contributionsdue to the pressure and the shearing forces acting onthe object. As mentioned earlier, for a given geometry,this relationship can readily be expressed in terms ofthree dimensionless groups, namely, the drag coefficientCD, the Reynolds number Re, and the power-law indexn. Furthermore, the total drag coefficient comprises twocomponents, stemming from the pressure forces (CDP)and the shearing forces (CDF). To elucidate the role ofthe power-law index in an unambiguous manner, thedrag coefficient for power-law fluids was normalizedwith respect to the corresponding value for Newtonianfluids (n ) 1) at the same value of the Reynolds number.Figure 2 depicts the effects of the flow behavior indexn and the Reynolds number Re on the individual andtotal drag coefficients over the ranges 0.5 e n e 1.4 and5 e Re e 40. Note that the corresponding results for n) 1 are also shown as a horizontal line with the ordinatevalue of unity. Broadly speaking, shear-thinning be-havior (n < 1) is seen to increase the values of thepressure and total drag coefficients with reference to

Table 1. Comparison between Present Results andLiterature Values25 for the Constant-TemperatureCondition for Pr ) 0.70 (Air)

Re ref ⟨Nu⟩f ⟨Nu⟩t ⟨Nu⟩r ⟨Nu⟩

5 Sharma25 1.684 1.22 0.715 1.21present work 1.704 1.23 0.715 1.22




Table 2. Comparison of Present Results with Those ofSharma25 for the Constant-Heat-Flux Condition (Pr )0.7)

Re ref ⟨Nu⟩f ⟨Nu⟩t ⟨Nu⟩r ⟨Nu⟩






the corresponding values in a Newtonian fluid, andqualitatively, the opposite kind of behavior is observedfor shear-thickening liquids (n > 1), although the effectof n is seen to diminish with increasing value of theReynolds number in the range used in this work. Thisbehavior is qualitatively consistent with the analogous

results for spherical and spheroidal particles sediment-ing in power-law fluids.15,25 The effect of the flowbehavior index on the frictional component of the totaldrag is seen to be the opposite, i.e., shear thinningcauses a lowering of the frictional component, whereasshear-thickening behavior augments it. This effectbecomes accentuated at increasing value of the Reynoldsnumber. This trend is consistent with the notion that,as the Reynolds number of the flow is graduallyincreased, the general level of shearing increases,thereby resulting in lower values of the effective viscos-ity for a shear-thinning fluid and higher values of theeffective viscosity for dilatant fluids that then directlyinfluence the frictional component of the drag in thesame fashion, as seen in Figure 2.

Intuitively, as the value of the Reynolds number isprogressively increased, viscous forces make way forinertial forces, so one would expect the effect of the flowbehavior index to diminish with increasing value of theReynolds number. Indeed, this expectation is borne outby Figure 2c, wherein it is clearly seen that the ratioCDNN/CDN ranges from 1.15 to ∼0.88 as the value of nis gradually varied from 0.5 to 1.4. Some further insightscan be gained by examining the variation of the relativecontributions of the pressure and frictional componentsof the drag with the Reynolds number and the flowbehavior index. A detailed examination of these resultsreveals that, even at Re ) 5, the frictional contributionto the overall drag hovers around 30-45% as the valueof n is gradually increased from 0.5 to 1.4. The slightincrease in the value of CDF/CD is consistent with thefact that, in shear-thickening fluids (n > 1), the viscousforces would be more significant than in a shear-thinning fluid (n < 1). Furthermore, as the value of theReynolds number increases, the total drag coefficientincludes a progressively smaller contribution from fric-tion. For instance, at Re ) 40, the ratio CDF/CD rangesfrom 0.5 to 0.21 as the value of n is varied from 0.5 to1.4. Thus, as expected, overall, in this geometry, thepressure drag increasingly dominates as the value ofthe Reynolds number is gradually increased, even forshear-thinning and shear-thickening fluids. However,just as in the case of a sphere, non-Newtonian effectswould be expected to be more significant at low Reynoldsnumbers, and this expectation is borne out by theresults shown in Figure 2.

The intricate interactions between the various physi-cal and kinematic variables can also be seen throughthe detailed streamline and isovorticity contour plots.Because the main focus of the present study is tohighlight the role of the flow behavior index n, Figure3 shows the representative streamline and isovorticitycontour plots for a range of combinations of the valuesof n (0.5, 1, 1.4) and of the Reynolds number (5, 20, 40).An examination of this figure shows that, for a fixedvalue of Re, the flow is seen to be faster close to thecylinder in shear-thinning fluids (n < 1) than in aNewtonian medium (n ) 1) and, as expected, it is seento be impeded in shear-thickening fluids. This is a directconsequence of the dependence of the fluid viscosity onthe shear rate. However, moving away from the cylinderin the axial and lateral directions, this effect progres-sively becomes less prominent. On the other hand, asthe value of the Reynolds number is gradually in-creased, the wake region grows as the fluid behaviorundergoes a transition from shear-thinning to Newto-nian and, finally, to shear-thickening behavior. In other

Figure 2. Variation of (a) CDFNN/CDFN, (b) CDPNN/CDPN, and (c)CDNN/CDN with the Reynolds number and the power-lawindex.


words, the wake size is seen to be larger in a shear-thickening fluid than in a shear-thinning fluid. It needsto be emphasized that there is no inconsistency hereand that this counterintuitive result is a direct conse-quence of the scaling variables used in this study. Witha slight rearrangement of the definition of the Reynoldsnumber, it can readily be seen that the representativeshear rate is of order Vo/b. Clearly, although thisapproximation is appropriate close to the cylinder,elsewhere in the flow domain, the shear rate is likelyto be of order Vo/2h, i.e., 8 times smaller than Vo/b.Alternatively, one can argue that there is a layer of veryviscous fluid surrounding the square cylinder thatcauses the oncoming fluid stream to veer from itspath, which, in turn, results in a larger wake region inshear-thickening fluids than in Newtonian or shear-thinning fluids. In view of this result, within theframework of the present scaling scheme, the fluidviscosity is underestimated for shear-thinning fluids,and consequently, the Reynolds number is overesti-mated. This finding is also consistent with the corre-sponding results for a sphere falling in power-lawliquids, wherein it is seen that the velocity decays muchfaster in shear-thinning liquids than in Newtonianliquids.16 Such behavior is tantamount to a rapidincrease in the fluid viscosity slightly away from theobject, and presumably, it delays the formation of awake in pseudoplastic systems. One can construct asimilar argument to explain the presence of a largewake in dilatant systems (n > 1). However, for aconstant value of n, as the value of the Reynolds numberis increased, inertial forces outweigh the viscous forces.

Although it is difficult to pinpoint the precise value ofthe Reynolds number at which a wake will form, it isclear that the lower the value of the power-law index,the higher the value of the Reynolds number at whicha visible wake appears. For instance, at n ) 0.5, thereis no visible wake formed at Re ) 5, but a small wakeis present at Re ) 10.

As the Reynolds number is gradually increased, thewake region grows in size. This phenomenon is oftendescribed in terms of the so-called recirculation length(Lr), which is a measure of the distance from the rearsurface of the cylinder to the point of reattachmentalong the centerline of the wake. This quantity isestimated approximately from the streamline plot at theintersection of the Ψ ) 0 line and the channel centerlinei.e., the x axis. Figure 4 shows the variation of thedimensionless recirculation length with the Reynoldsnumber for a range of values of the power-law index.The present results are seen to be consistent with thelinear relationship observed for Newtonian fluid behav-ior reported in the literature.10 This linearity also seemsto apply reasonably well for shear-thickening fluids, butthe dependence is seen to be slightly weaker in shear-thinning fluids, which is, in a sense, consistent with thedelayed wake formation occurring in these systems, asmentioned above. Similarly, the increase in the valueof the recirculation length seen in Figure 4 is consistentwith the larger wake region shown in Figure 3 for shear-thickening media, and this is also borne out by theisovorticity contour plots shown in Figure 3.

(iii) Heat-Transfer Characteristics. Two addi-tional factors influence the heat-transfer aspects, namely,

Figure 3. Representative streamline (upper half) and isovorticity (lower half) plots for a range of values of the Reynolds number and thepower-law index.


the Peclet number and the type of thermal boundarycondition, i.e., a constant temperature on the surfaceof the cylinder (case I) or the constant-heat-flux condi-tion (case II). Owing to the underlying inherent differ-ences, the heat-transfer results corresponding to thesetwo conditions are presented and discussed separatelyin the following sections.

(a) Case I. As with the drag results presented in thepreceding section, the individual surface-averaged andoverall averaged values of the Nusselt number werenormalized using the corresponding Newtonian valueat the same values of the Reynolds and Peclet numbers.A distinct advantage of this form of representation liesin the fact that it allows for the most direct assessmentof the influence of the power-law index on the heat-transfer results.

Figures 5 (Re ) 5) and 6 (Re ) 40) show representa-tive variations of the normalized Nusselt numbers withthe flow behavior index (n) and the Peclet number (Pe).Qualitatively similar results are obtained for the othercases, and thus, these results are not shown here. Anexamination of Figures 5 and 6 clearly shows theinfluence of the flow behavior index on heat transfer tobe smaller than that on drag as seen in the previoussection. The maximum value of the ratio of ⟨Nu⟩NN/⟨Nu⟩Nis seen to be only about 1.21, thereby showing anenhancement of 21% in heat transfer from the topsurface of the cylinder in a shear-thinning fluid. In fact,the maximum enhancement in the overall heat-transfercoefficient is on the order of only 15% in shear-thinningfluids, and the corresponding reductions in shear-thick-ening media are even smaller, being on the order of only∼7-8%. Alternatively, one can argue that the effect ofthe power-law index is adequately embodied in themodified definition of the Reynolds number, and there-fore, the normalized Nusselt number, ⟨Nu*⟩, shows onlya weak additional dependence on the flow behaviorindex.

Turning our attention now to heat transfer from theindividual surfaces of the square cylinder, the effect ofn is seen to vary both qualitatively and quantitativelywith increasing values of the Reynolds and Pecletnumbers. For instance at low Re () 5) and Pe () 5), thevalue of ⟨Nu*⟩t for the top surface (CD in Figure 1) ofthe cylinder is most strongly influenced by the value of

the power-law index, and this ratio is greater than unityin shear-thinning fluids and smaller than unity inshear-thickening fluids. This is then followed by the

Figure 4. Dependence of the nondimensional recirculation lengthon the Reynolds number and the power-law index.

Figure 5. Effect of the power-law index (n) on the normalizedNusselt numbers corresponding to various faces of the squarecylinder for Re ) 5 (case I) for two values of the Peclet number.

Figure 6. Effect of the power-law index (n) on the normalizedNusselt numbers corresponding to various faces of the squarecylinder for Re ) 40 (case I) for two values of the Peclet number.


corresponding ratios of the Nusselt numbers for theoverall cylinder, for the front face and for the rear face,respectively. Under these conditions, because no wakeis present, the heat-transfer coefficient hardly variesalong this surface. At Re ) 5, as the value of the Pecletnumber is gradually increased, the values of ⟨Nu*⟩r areseen to be slightly above unity in shear-thinning mediaand slightly below unity in shear-thickening fluids. Thisdependence on n is seen to switch at Re ) 40, so thatthe effect of n is now reversed (Figure 6). This compo-nent is seen to show a reduction in heat transfer inshear-thinning liquids (because of the small wakeregion) and an enhancement in heat transfer in shear-thickening fluids, which is also consistent with thelarge wake region observed in dilatant media. Irrespec-tive of the value of the Reynolds number and/orPeclet number, the ratio ⟨Nu⟩NN/⟨Nu⟩N for the top facealways exceeds that for the front face of the cylinder.Further examination of the variation of the localNusselt number on the surface of the cylinder shows amonotonic increase in the value of the Nusselt numberas one traverses from point B to C, followed by adecrease along CD and another slight drop along DE.However, all of these variations appear to be self-canceling, thereby yielding values of the surface-aver-aged and overall averaged values of ⟨Nu⟩ that are onlymarginally lower or higher than the correspondingNewtonian values at the same values of the Reynoldsand Peclet numbers.

(b) Case II. Representative analogous results on thenormalized Nusselt number for this case are shown inFigures 7 and 8 for a range of values of the Reynoldsand Peclet numbers and the power-law index. Aninspection of these results shows qualitatively similartrends as seen in Figures 5 and 6 for case I, although

there are slight differences in the detailed variation ofthe Nusselt number along the surface of the cylinder.Once again, however, these differences appear to be self-compensating in determining the overall mean value ofthe Nusselt number.

Representative isotherm plots for case I (upper half)and II (lower half) are shown in Figure 9 for a range ofcombinations of Re, Pe, and n. For case I, the temper-ature field is seen to decay somewhat faster in shear-thinning fluids than in Newtonian media, therebysuggesting a thinner boundary layer in these fluids.However, the trend seems to be strongly dependent onthe values of the Reynolds and Peclet numbers. Quali-tatively, the opposite type of phenomenon is seen inshear-thickening fluids for case I. For case II also, theisotherms show a strong interplay between the valuesof the Reynolds and Peclet numbers and the value of n.For n ) 0.5, the temperature drop seems to be relativelyrapid, especially as the value of the Reynolds number/Peclet number is progressively increased. Although thiseffect is accentuated in shear-thinning media, it getssuppressed in shear-thickening fluids.

This section is concluded by reiterating the conclusionthat the drag coefficient and the Nusselt numbernormalized with respect to the corresponding Newto-nian values show only very weak additional depend-ences on the power-law index. In fact, if demonstratedfor other values of the blockage ratio, this fact offers aconvenient scheme for estimating the values of theNusselt number for shear-thinning and shear-thicken-ing fluids simply from a knowledge of the correspondingNewtonian values for this configuration. Shear-thinningbehavior results in a small degree of enhancement inheat transfer, whereas shear thickening has a deleteri-ous effect on it.

Figure 7. Effect of the power-law index (n) on the nor-malized Nusselt numbers corresponding to various faces of thesquare cylinder for Re ) 5 (case II) for two values of the Pecletnumber.

Figure 8. Effect of the power-law index (n) on the normalizedNusselt numbers corresponding to various faces of the squarecylinder for Re ) 40 (case II) for two values of the Peclet number.


Figure 9. Representative isotherm plots for case I (upper half) and case II (lower half): (a) Re ) 5, (b) Re ) 40.


Concluding Remarks

In this work, the governing equations describing thesteady two-dimensional flow over and heat transfer froma square cylinder immersed in a streaming power-lawliquid have been solved numerically to obtain detailedvelocity and temperature fields for a blockage ratio of1/8 and the following ranges of kinematic and physicalparameters: 0.5 e n e 1.4, 5 e Re e 40, 5 < Pe < 400.The role of the type of thermal boundary condition,i.e., constant heat flux and isothermal cylinder, on theoverall heat-transfer characteristics has also been stud-ied. Within the range of conditions investigated, thevalues of the normalized drag coefficient and Nusseltnumber (both individual and overall) hover aroundunity, with values slightly above unity in shear-thin-ning liquids and slightly below unity for dilatantfluids. The weak effect of the flow behavior index n ondrag is qualitatively similar to that well known for asphere.

The detailed flow patterns suggest relatively shorterwake regions in shear-thinning liquids and slightlylarger recirculation lengths in shear-thickening media.This counterintuitive trend can be ascribed to thescaling procedure used herein. The effect of the Reynoldsnumber on the flow patterns is qualitatively similar tothat seen for Newtonian fluids. Similarly, isotherm plotsreveal faster decay in the temperature field undercertain conditions (high Peclet numbers) in shear-thinning liquids, with the reverse behavior being ob-served in shear-thickening fluids. Unfortunately, nosuitable experimental results are available to refute/substantiate the theoretical predictions presented herein.

Nomenclature

b ) side of the square (m)CD ) drag coefficient (-)CDF ) friction drag coefficient (-)CDP ) pressure drag coefficient (-)Cp ) heat capacity of liquid (J/kg K)FD ) drag force per unit length of cylinder (N/m)h ) half-height of the computational domain (m)k ) thermal conductivity of liquid (W/m‚K)LD ) downstream distance (-)Lr ) recirculation length made dimensionless using b (-)LU ) upstream distance (-)m ) power-law consistency coefficient (Pa‚sn)n ) power-law index (-)Nu ) Nusselt number (-)⟨Nu⟩ ) average Nusselt number (-)p ) pressure (-)Pe ) Peclet number (-)Pr ) Prandtl number (-)t ) time (s)T ) dimensionless temperature (-)T′ ) fluid temperature (K)T∞ ) free stream fluid temperature (K)Vo ) centerline velocity (m/s)Vx ) x component of velocity (-)Vy ) y component of velocity (-)x y ) rectangular coordinates (m)

Greek Letters

εxx, εyy, εxy ) components of the rate of deformation tensor(s-1)

η ) power-law viscosity (-)F ) fluid density (kg/m3)

Subscripts

f ) front faceN ) NewtonianNN ) non-Newtonianr ) rear facet ) top face

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Received for review April 28, 2003Revised manuscript received August 21, 2003

Accepted September 3, 2003

IE030368F


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