steady flow of power law fluids over a pair of cylinders in tandem arrangement · fluids over a...

Steady Flow of Power Law Fluids over a Pair of Cylinders in TandemArrangement

Rahul C. Patil, Ram P. Bharti,† and Raj P. Chhabra*

Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India

The steady flow of incompressible power-law fluids over a pair of cylinders in tandem arrangement has beenstudied numerically. The field equations have been solved using a finite volume method based solver (FLUENT6.2). In particular, the effects of the power-law index (0.4e n e 1.8), Reynolds number (1e Ree 40), andthe gap ratio between the two cylinders (2e G e 10) on the local and global flow characteristics such asstreamline profiles, center line velocity, surface pressure coefficient, and individual and total drag coefficients,etc. have been studied in detail. The wake interference in conjunction with the power-law rheology exerts astrong influence on the flow dynamics at high Reynolds numbers, even in the steady-flow regime, whereasat low Reynolds numbers, the flow is influenced by the rheological behavior of the fluid. The increasingdegree of shear-thinning behavior delays the flow separation, whereas early separation is seen in Newtonianand in shear-thickening fluids. The pressure coefficient distribution on the surface of the cylinders shows anintricate dependence on the power-law index, Reynolds number, and gap ratio. The shear-thinning behaviorexerts stronger effects on the drag characteristics than those seen in Newtonian and shear-thickening fluids.Both upstream and downstream cylinders show smaller values of the individual and total drag coefficientsthan those for a single circular cylinder under otherwise identical conditions.

1. Introduction

Over the years, considerable research efforts have beendevoted to the study of the cross-flow of Newtonian and non-Newtonian fluids over cylinders of circular and non-circularcross-sections. Indeed, depending upon the value of the Rey-nolds number, this flow displays a wide variety of flow regimesover a relatively small range of Reynolds numbers, even inNewtonian fluids.1,2 The fluid-flow interactions for multiple bluffbodies are very important as well as complex, yet only limitedinformation is available in the literature, even for Newtonianfluids.1-4 The hydrodynamic forces and flow configurations aremajor criteria for the design of structures in many engineeringapplications, that is, tubes in tubular and in pin-type heatexchangers, which are used extensively in the cooling ofelectronic-components and in food and polymer processingapplications. Further applications are found in the resin transfermolding process of manufacturing fiber reinforced composites,in filtration screens and aerosol filters, etc. Also, extra bodiesare often used in flow-control strategies to modify, or to probe,the original wake. A pair of two cylinders in various arrange-ments is the simplest case of multiple bluff bodies. A thoroughunderstanding of the flow dynamics of such an idealized caseis germane to understanding the flow around complex arrange-ments, such as in the shell of tubular heat exchangers, in pin-type heat exchangers, and in tubular membrane modules, etc.Evidently, an improved understanding of the hydrodynamicsof the flow (such as the dead-zones or the zones of maximumshear, wake size, etc.) will lead to improved design methodsand ensure homogeneous product quality for temperature-sensitive materials (such as foodstuffs). In such cluster con-figurations, the types of interference encountered are varied suchas theproximity interference, wake interference, anda combina-

tion of both. Among many possible arrangements in which twocircular cylinders can be positioned relative to the cross-flow,tandem arrangement has been studied extensively. This con-figuration is suited to studywake interference, wherein the wakeof the upstream cylinder may extend up to the downstreamcylinder.4 This work is concerned with the flow of power-lawfluids over a pair of cylinders in tandem arrangement.

It is probably reasonable to say that a voluminous body ofinformation is now available on the various aspects of flowphenomena associated with the transverse flow of Newtonianfluids over a single cylinder (e.g., refs 1-7). On the other hand,many substances of multiphase nature and/or of high molecularweight encountered in industrial practices (pulp and papersuspensions, food, polymer melts and solutions, etc.) displayshear-thinning and/or shear-thickening behavior.8 Owing to theirhigh viscosity levels, these materials are generally processedin the laminar flow conditions. Therefore, it seems reasonableto begin with the analysis of purely viscous power-law typefluids, and the level of complexity can be built up gradually toaccommodate other non-Newtonian characteristics such as yieldstress, visco-elasticity, etc. As far as known to us, there hasbeen no prior study on the steady cross-flow of incompressiblepower-law liquids over a pair of cylinders in tandem arrange-ment. This constitutes the main objective of this work. At theoutset, it is desirable, however, to briefly recount the availablelimited work on the flow of Newtonian fluids over a pair ofcylinders and of power-law fluids past a single cylinder tofacilitate the subsequent presentation of the new results for thepair of cylinders in tandem configuration.

2. Previous Work

A pair of cylinders can geometrically be arranged inside-by-side, or tandem(or aligned), or staggeredconfigurations withrespect to the direction of the approaching flow. It is useful tonote here that both types of arrangements are encountered onthe shell-side of tubular heat exchangers and membrane separa-

* To whom correspondence should be addressed. Tel:+91-512-259 7393. Fax:+91-512-259 0104. E-mail: [email protected].

† Present Address: Department of Chemical and BiomolecularEngineering, The University of Melbourne, Parkville 3010, Melbourne,Victoria, Australia.

1660 Ind. Eng. Chem. Res.2008,47, 1660-1683

10.1021/ie070854t CCC: $40.75 © 2008 American Chemical SocietyPublished on Web 02/02/2008

tion units. Over the years, many previous investigations havealready revealed complex flow behavior, different flow patterns,and wake interferences depending upon the relative positioningof the two cylinders, even in Newtonian fluids. A detailedclassification of the different types of flow interference regimeshas been proposed in the literature (e.g., refs 4, 9-13, etc.).For instance, Igarashi10,11identified six different possible typesof interferences for a pair of cylinders in tandem arrangement.Some of these features have been studied at high Reynoldsnumbers among others by Okajima,14 Arie et al.,15 and Xu andZhou16 and have been recently summarized.17

Among the numerical studies on the flow around twocylinders, Stansby and Slaouti18 used the inviscid discrete-vortexmethod to investigate the flow around two side-by-side circularcylinders at high Reynolds numbers. Their results are in linewith the available experimental studies. Subsequently, they19

used a random vortex method to model the 2D flow aroundtwo (tandem and side-by-side) circular cylinders atRe) 200.Similar flow configurations have been studied by Mittal et al.20

by using a stabilized finite element method atRe ) 100 and1000. In a recent bi-dimensional numerical analysis (using bothlinear stability analysis and DNS), Mizushina and Suehiro21

investigated the 2D wake instability of two tandem circularcylinders for the gap ratio (G) ranging from 2 to 7 and theReynolds number in the range of 60 to 100. They reported thebifurcation diagram to become complex in the vicinity of theinstability threshold. Subsequently, Huang et al.22 simulated the2D unsteady flow of water in aligned and staggered arrays ofcylinders atRe) 150 usingFLUENT. Recently, using the mesh-free least-square-based finite difference (MLSFD) method, Dinget al.23 have simulated the flow around two cylinders in side-by-side and tandem arrangements for two values of the ReynoldsnumberRe) 100 and 200. They characterized the flow fieldby presenting the instantaneous streamlines and vorticitycontours and in terms of global engineering parameters such asthe Strouhal number, the mean values, and the amplitudes ofdrag and lift coefficients. These results are also consistent withthe previous experimental findings3 in the various flow regimesand with interactive vortex shedding. More recently, Juncu24

has numerically solved the transformed vorticity-stream functionformulation of the Navier-Stokes equations using a finitedifference method for the steady Newtonian flow around twocylinders (of different diameters,D1 and D2) in tandemarrangement (gap ratio,G* ) 2L1/D1, whereL1 andL2 are thedistances of the upstream and downstream cylinders from theorigin of the coordinate systems). He presented extensive resultsshowing the effects of the Reynolds number (based onD1), Re) 1 to 30, of the diameter ratio,D1/D2 ) 0.5, 1, and 2 on thepressure coefficient, vorticity, and the individual and total dragcoefficients for the two-cylinder system.

Among the experimental studies, the evolution of the flowinterference behind the two side-by-side cylinders was inves-tigated using flow-visualization methods by Bearman andWadcock.25 Similarly, Williamson26 experimentally studied thisproblem in the Reynolds number range of 50 to 200. Heobserved synchronized behavior, that is, anti-phase and in-phasefeatures of the wakes for certain values of the gap ratio.Recently, Tasaka et al.27 have confirmed experimentally theexistence of two (slow and fast) modes of vortex shedding fortwo circular cylinders in tandem. Their results confirm the 2Dnumerical predictions21 that resist finite size effects or eventual3D instabilities. For specific gap ratios, they detected thesubcritical and saddle node bifurcations that lead to hystereticexchanges between the two modes of vortex shedding. As a

result of the finite length of the cylinders, they observed slightlydifferent ranges and transition values of the Reynolds numberand gap ratios from those based on the numerical results forinfinitely long cylinders (e.g., ref 21, etc.).

In contrast, the corresponding limited information for the flowof power-law fluids over a single cylinder has been summarizedin recent studies.28-42 The limits of the cessation of the creepingflow regime and of the transition from the 2D steady symmetricflow to the asymmetric flow regimes have been delineated onlyrecently for power-law fluids.38 This study shown that shear-thickening fluid behavior can advance the formation of asym-metric wakes to lower values of the Reynolds number than thatin Newtonian liquids. All in all, reliable results are now availablefor the flow of power-law fluids over a cylinder in the 2D steadysymmetric flow regime embracing the range of conditions as:Ree 40, 0.2e n e 2. Similarly, some information36,37 is alsoavailable for the flow past a cylinder confined in a planarchannel. Aside from these results based on the solution of thecomplete governing equations, there have been some studiesbased on the boundary layer flow approximation (e.g., refs 40and 42 and references therein).

In summary, as far as known to us, there has been no priorstudy dealing with the steady flow of power-law fluids overtwo tandem cylinders in an unconfined and cross-flow config-uration. This work is concerned with the 2D steady andunconfined cross-flow of an incompressible power-law fluidover two tandem cylinders over the following ranges ofconditions: Reynolds number (1e Ree 40), power-law index(0.4 e n e 1.8), and gap ratio (2e G e 10).

3. Problem Statement and Governing Equations

Consider the 2D, steady, cross-flow of an incompressiblepower-law liquid streaming with a uniform velocity (Uo) overa pair of infinitely long circular cylinders (of equal diameter,D) in tandem arrangement (gap ratio,G ) L/D, whereL is thecenter-to-center distance), as shown in part a of Figure 1. Theunconfined flow condition is simulated here by enclosing the

Figure 1. Schematic representation of (a) the flow over a pair of cylindersin tandem arrangement and (b) an approximation of an unconfined flow(uniform velocity,U0) configuration.

Ind. Eng. Chem. Res., Vol. 47, No. 5, 20081661

two circular cylinders in a circular outer boundary (of diameterDo), as shown schematically in part b of Figure 1. The diameterof the outer circular boundaryDo is taken to be sufficientlylarge to minimize the boundary effects.

The continuity and momentum equations for this flow in theircompact forms are written as follows,

whereF, U, f, andσ are the density, velocity, body force, andthe stress tensor, respectively. The stress tensor, sum of theisotropic pressure (p), and the deviatoric stress tensor (τ), isgiven by

For incompressible fluids, the extra stress tensor is given by

whereε(U), the components of the rate of strain tensor, are givenby

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

wherem is the power-law consistency index andn is the power-law index of the fluid (n < 1, shear-thinning;n ) 1, Newtonian;andn > 1, shear-thickening) andI2 is the second invariant ofthe rate of strain tensor (ε). The components of the rate of thestrain tensor are related to the velocity components and theirderivatives and are available in standard text books (e.g., ref43).

The physically realistic boundary conditions for this flowconfiguration may be written as follows:

• At the inlet boundary: The uniform flow condition isimposed at the inlet.

• On the surface of the cylinders: The standardno-slipconditionis used.

• At the exit boundary: The default outflow boundary conditionoption in FLUENT (a zero diffusion flux for all of the flowvariables) was used in this work. This choice implies that theconditions of the outflow plane are extrapolated from withinthe domain and as such have negligible influence on theupstream flow conditions. The extrapolation procedure used byFLUENT updates the outflow velocity and the pressure in amanner that is consistent with the fully developed flowassumption, when there is no area change at the outflowboundary. However, the gradients in the cross-stream directionmay still exist at the outflow boundary. Also, the use of thiscondition obviates the need to prescribe a boundary conditionfor pressure. This is similar to the homogeneous Neumanncondition, that is,

The numerical computations have been carried out in the fullcomputational domain, that is, without assuming midplanesymmetry (Figure 1). The numerical solution of the governingequations (eqs 1-2) in conjunction with the above-notedboundary conditions (eqs 7-9) maps the flow domain in termsof the primitive variables, that is, velocity (Ux and Uy) andpressure (p) fields. These, in turn, are used to deduce the localand global momentum characteristics as outlined below anddetailed elsewhere.34,36-40 However, at this stage, it is useful tointroduce some dimensionless parameters.

• The Reynolds number (Re) for power-law fluids is definedas follows

•Surface pressure coefficient(Cp) is defined as follows

wherep(θ) is the surface pressure at an angleθ andpo is thefree stream pressure at the exit boundary.

•The total drag coefficient(CD), the sum of the friction andpressure components, is defined as

whereFD is total drag force on the cylinder per unit length.The individual drag coefficients,CDP and CDF, are calculatedusing the following definitions:

whereFDP is the pressure component of the drag force andS isthe surface area.

whereFDF is the frictional component of the drag force andns

(unit vector normal to the surface of the cylinder) is given as

whereex andey are thex andy components of the unit vector,respectively, andτ, the dimensionless shear stress, is expressedas

where η and I2 are the dimensionless viscosity and secondinvariant of the rate of strain tensor, respectively. In the aboveequations, the radius of the cylinder (D/2) and the uniform inletvelocity (Uo) are used as the characteristic length and charac-teristic velocity, respectively.

•Continuity equation: ∇.U ) 0 (1)

•Momentum equation: F(U.∇U - f) - ∇.σ ) 0 (2)

σ ) -pI + τ (3)

τ ) 2ηε(U) (4)

ε(U) )(∇U) + (∇U)T

2(5)

η ) m(2I2)n-1

2 whereI2 ) (εxx2 + 2εxy

2 + εyy2 ) (6)

Ux ) Uo andUy ) 0 (7)

Ux ) 0 andUy ) 0 (8)

∂Ux

∂x) 0 and

∂Uy

∂x) 0 (9)

Re)FDnUo

2 - n

m(10)

Cp ) Static pressureDynamic pressure

)p(θ) - po

(1/2)FUo2

(11)

CD )FD

(1/2)FUo2D

) CDP + CDF (12)

CDP )FDP

(1/2)FUo2D

) ∫SCpnxdS (13)

CDF )FDF

(1/2)FUo2D

) 2n + 1

Re ∫S(τ.ns)dS)

2n + 1

Re ∫S(τxx.nx + τxy.ny)dS (14)

ns )xex + yey

xx2 + y2) nxex + nyey (15)

τij ) η(∂Ui

∂j+

∂Uj

∂i ) ) (I2

2)(n - 1)/2 (∂Ui

∂j+

∂Uj

∂i ) (16)

1662 Ind. Eng. Chem. Res., Vol. 47, No. 5, 2008

Dimensional analysis of the field equations and the boundaryconditions suggests the local and global flow characteristics tobe functions of the Reynolds number (Re), power-law index(n), and the gap ratio (G). This functional relationship is exploredin this work.

4. Numerical Solution Procedure

Because detailed descriptions of the numerical solutionprocedure are available elsewhere,36-40 only the salient featuresare recapitulated here. In this study, the field equations havebeen solved usingFLUENT (version 6.2). The unstructuredquadrilateral cells of non-uniform grid spacing were generatedusing the commercial grid tool GAMBIT. The 2D, steady,laminar, segregated solver was used to solve the incompressibleflow on the collocated grid arrangement. Thesecond orderupwindscheme has been used to discretize the convective termsin the momentum equations. The semi-implicit method for thepressure linked equations (SIMPLE) scheme was used forsolving the pressure-velocity decoupling. Theconstant densityand non-Newtonian power-lawviscosity modules were used.FLUENT solves the system of algebraic equations using theGauss-Siedel point-by-point iterative method in conjunction

with the algebraic multi-grid (AMG) method solver. The useof the AMG scheme can greatly reduce the number of iterationsand thus the CPU time required to obtain a converged solution,particularly when the model contains a large number of controlvolumes. Relative convergence criteria of 10-10 for the continu-ity and x andy components of the velocity were prescribed inthis work.

5. Choice of Computational Domain and Grid Size

Needless to say, the reliability and accuracy of the numericalresults are contingent upon a prudent choice of the numericalparameters, namely, the optimal domain and grid sizes. In thisstudy, the domain is characterized by the diameter (Do) of thefaraway cylindrical envelope of the fluid. An excessively largevalue ofDo will warrant enormous computational resources anda small value will unduly influence the results, and hence ajudicious choice ofDo is vital to the accuracy of the results.Similarly, an optimal grid size should meet two conflictingrequirements, namely, it should be fine enough to capture theflow field yet it should not be excessively resources intensive.The effects of these parameters (Do and grid size) on the dragcoefficient values for the power-law fluid flow past a singlecylinder have been explored extensively recently,38,40 only theadditional results showing the influence of these parameters onthe drag coefficients for the two-cylinder system are presentedhere, thereby ensuring the present results to be free from theseartifacts.

Several values of (Do/D) ranging from 300 to 1300 have beenused in this study to examine the role of domain size on thepresent numerical results. Table 1 shows the effect of domainsize (Do/D) on the individual and total drag coefficient valuesfor G ) 2, for three values of the power-law index (n ) 0.4, 1,and 1.8) and for the extreme values of the Reynolds number(Re ) 1, 30 and/or 40) considered in this work. The domainindependence study has been carried out with gridM1 detailedin Table 2. It shows a very small change (<0.05%) in dragvalues with an increase in domain size from 1100 to 1200 andfrom 1200 to 1300 at Reynolds numberRe) 1. Furthermore,the change in domain size from 300 to 400 and 400 to 500 forRe) 40 (n ) 0.4 and 1) and forRe) 30 (n ) 1.8) is foundto a yield very small change in the drag values (<0.18%). Itneeds to be emphasized here that the extremely small changesseen in the values of the drag coefficients are accompanied bya 2- to 3-fold increase in CPU time in the extreme condition.

Table 1. Effect of Domain Sizes (with GridM1) on the Individualand Total Drag Coefficients for Two Circular Cylinders for G ) 2

upstream cylinder downstream cylinder

n Do/D CDP,1 CDF,1 CD,1 CDP,2 CDF,2 CD,2

Re) 11100 3.7932 4.0007 7.7939 2.0270 2.4785 4.5055

1 1200 3.7916 3.9990 7.7906 2.0263 2.4776 4.50391300 3.7901 3.9975 7.7876 2.0257 2.4768 4.50251100 9.6271 6.5864 16.2135 8.1161 6.0858 14.2019

0.4 1200 9.6265 6.5858 16.2123 8.1155 6.0852 14.20061300 9.6264 6.5858 16.2122 8.1155 6.0852 14.20071100 2.2792 2.8058 5.0850 0.5188 0.9590 1.4778

1.8 1200 2.2783 2.8046 5.0829 0.5186 0.9585 1.47711300 2.2776 2.8039 5.0815 0.5185 0.9583 1.4768

Re) 40300 0.9387 0.5104 1.4491 0.03250 0.1196 0.1521

1 400 0.9378 0.5100 1.4478 0.03254 0.1195 0.1520500 0.9372 0.5097 1.4469 0.03257 0.1195 0.1521300 0.9151 0.2462 1.1613 0.1042 0.1227 0.2269

0.4 400 0.9145 0.2462 1.1607 0.1044 0.1227 0.2271500 0.9142 0.2462 1.1604 0.1043 0.1227 0.2270

Re) 30300 0.9471 0.8273 1.7744 0.003899 0.1118 0.1157

1.8 400 0.9457 0.8261 1.7718 0.003887 0.1116 0.1155500 0.9450 0.8255 1.7705 0.003880 0.1115 0.1154

Table 2. Grid Specifications (a) Details of Grids Used in Grid Independence Study, (b) Number of Grid Cells Used in Final Computations forDifferent Values of Gap Ratio (G) with the M4 Grid

(a)

number of cells in the computational domain (Ncells)

grid δ/Da NbDo/D ) 400,

G ) 2Do/D ) 1200,

G ) 2Do/D ) 200,

G ) ∞Do/D ) 1200,

G ) ∞

M1 0.010 200 175 852 213 852 160 202 221 818M2 0.010 400 240 208 278 208 192 708 254 524M3 0.005 400 279 990 317 990 221 302 290 862M4 0.004 400 302 388 340 388 245 050 309 050M5 0.003 400 318 388 356 388 268 702 332 702

(b)

gap ratio,G Ncells for Do/D ) 400 Ncells for Do/D ) 12002 302 388 340 3883 430 210 468 2104 379 146 417 1466 383 096 421 09610 352 386 390 386∞ 245 050* 309 050

a δ/D: grid spacing in the proximity of the cylinders; *Do/D ) 200. b N: number of grid points on the surface of the cylinder.


These results also reinforce two points: first, owing to the slowdecay of the velocity field, a much larger domain is needed atlow Reynolds numbers than that at high Reynolds numbers.Second, all else being equal, it appears that the flow field decaysmuch faster in power-law fluids (n < 1) than that in Newtonianfluids, and it is therefore possible to work with somewhat shorterdomains for power-law fluids. This is a distinct advantage, atleast at low Reynolds numbers. Thus, keeping in mind thesetwo conflicting requirements, based on the above study and ourprevious experience,34-42 the domain sizes (Do/D) of 400 and

1200 are believed to be adequate in the Reynolds number rangeof 1 e Ree 5 and 5< Ree 40, respectively, for gap ratio 2e G e 10 over the power-law index range (0.4e n e 1.8)considered here, to obtain the results that are essentially freefrom domain effects.

Having fixed the domain size, the grid independence studyhas been carried out for five non-uniform unstructured grids(M1, M2, M3, M4, andM5) for the extreme values of the Reynoldsnumber (Re) 1, 30 and/or 40), for three values of the power-law index (n ) 0.4, 1, and 1.8) and for the gap ratioG ) 2.The grid details are shown in Table 2. The influence of thegrid size on the individual and total drag coefficients for thetwo-cylinder case is shown in Table 3. It can be clearly seenfrom these results that, in moving from gridM4 to M5, the dragcoefficient values show a very small change, the maximumchange being 0.45, 0.17, and 0.11% atRe) 1 for n ) 0.4, 1,and 1.8, respectively. The corresponding changes atRe) 40are seen to be 0.62 and 0.6% forn ) 0.4 and 1, respectively,2.48% atRe ) 30 andn ) 1.8. In view of these negligiblechanges (accompanied by an enormous increase in the compu-tational time), the gridM4 is believed to be sufficiently refinedto resolve the momentum transfer phenomena with acceptablelevels of accuracy within the range of conditions of interest here.The number of grid cells with gridM4 for different values ofthe gap ratio (G) is also shown in Table 2. Finally, to add furtherweight to our claim for the accuracy of the results, the numericalresults obtained herein have been compared with the literaturevalues in the next section in the limiting case of Newtonianfluid behavior.

6. Results and Discussion

In this work, the 2D steady-flow computations have beencarried out for the following values of the dimensionlessparameters: Reynolds number,Re ) 1, 2, 5, 10, 20, and 30and/or 40; the power-law index,n ) 0.4, 0.6, 1, 1.4, and 1.8,thereby covering both shear-thinning (n < 1) and shear-thickening (n > 1) fluids, and for five values of the gap ratio,G ) 2, 3, 4, 6, and 10. Because the flow of power-law fluidsover a single cylinder is known38 to become asymmetric forRe) 40 atn ) 1.8, the new results for this power-law index arerestricted toRe e 30. Without assuming the flow symmetry

Table 3. Effect of the Grid Size on the Individual and Total DragCoefficients for Two Circular Cylinders in Tandem Arrangement (G) 2)

Upstream Cylinder Downstream Cylinder

n grid CDP,1 CDF,1 CD,1 CDP,2 CDF,2 CD,2

Re) 1M1 3.7916 3.9990 7.7906 2.0263 2.4776 4.5039M2 3.7899 3.9988 7.7887 2.0251 2.4776 4.5027

1 M3 3.8207 3.9687 7.7894 2.0441 2.4589 4.5029M4 3.8272 3.9621 7.7893 2.0482 2.4548 4.5030M5 3.8333 3.9561 7.7894 2.0521 2.4511 4.5032M1 9.6271 6.5864 16.2135 8.1161 6.0858 14.2019M2 9.6341 6.5748 16.2089 8.1250 6.0757 14.2006

0.4 M3 9.7583 6.4523 16.2106 8.2405 5.9612 14.2017M4 9.7862 6.4253 16.2095 8.2672 5.9344 14.2016M5 9.8139 6.3958 16.2097 8.2928 5.9088 14.2016M1 2.2783 2.8046 5.0829 0.5186 0.9585 1.4771M2 2.2753 2.8059 5.0812 0.5176 0.9591 1.4767

1.8 M3 2.2857 2.7959 5.0815 0.5210 0.9557 1.4766M4 2.2879 2.7945 5.0824 0.5220 0.9557 1.4777M5 2.2893 2.7922 5.0815 0.5224 0.9546 1.4770

Re) 40M1 0.9378 0.5100 1.4478 0.03254 0.1195 0.1520M2 0.9378 0.5099 1.4477 0.03223 0.1193 0.1515

1 M3 0.9414 0.5063 1.4477 0.03321 0.1185 0.1517M4 0.9422 0.5055 1.4477 0.03341 0.1183 0.1517M5 0.9429 0.5047 1.4476 0.03362 0.1181 0.1517M1 0.9145 0.2462 1.1607 0.1044 0.1227 0.2271M2 0.9153 0.2458 1.1611 0.1035 0.1222 0.2257

0.4 M3 0.9185 0.2418 1.1603 0.1057 0.1202 0.2259M4 0.9194 0.2409 1.1603 0.1063 0.1197 0.2260M5 0.9202 0.2400 1.1601 0.1069 0.1192 0.2261

Re) 30M1 0.9457 0.8261 1.7718 0.003887 0.1116 0.1155M2 0.9454 0.8267 1.7721 0.003301 0.1113 0.1146

1.8 M3 0.9482 0.8238 1.7720 0.003728 0.1110 0.1147M4 0.9488 0.8233 1.7721 0.003823 0.1109 0.1147M5 0.9493 0.8228 1.7721 0.003920 0.1108 0.1147

Table 4. Comparison of the Newtonian (n ) 1) Flow Results for a Single Cylinder (G ) ∞) and the Two Cylinders in Tandem Arrangement.(CD: Average Drag Coefficient; CL: Total Lift Coefficient; St: Strouhal Number)

Upstream Cylinder Downstream Cylinder

Re source CD,1 (CL,1 St1 CD,2 (CL,2 St2

G ) 2.5present results 1.275 0.0 -0.0792 0.0

100 Mittal et al.20 1.271 0.0 -0.0750 0.0Ding et al.23 1.163 0.0 -0.0895 0.0

G ) 5.5present results 1.446( 0.015 0.413 0.169 0.952( 0.164 1.772 0.169

100 Mittal et al.20 1.433( 0.015 0.403 0.168 0.952( 0.164 1.741 0.168Ding et al.23 1.329( 0.013 0.330 0.160 0.858( 0.125 1.554 0.160

G ) 2CDP,1 CDF,1 CD,1 CDP,2 CDF,2 CD,2

1 Present Results 3.8272 3.9621 7.7893 2.0482 2.4548 4.5030Juncu24 3.9150 3.9900 7.9050 2.1000 2.482 4.5820






about the midplane, the results have been obtained using thefull computational domain (Figure 1). However, prior topresenting the new results, it is appropriate to validate thesolution procedure to ascertain the accuracy and reliability ofthe results presented herein.

6.1. Validation of Results.Because extensive validation forthe case of a single cylinder is dealt with in detail elsewhere,34-41

the present results for the flow of Newtonian fluids over twocylinders in tandem arrangement are compared in Table 4 withthe available literature values. Because only limited results areavailable for the flow over the two-cylinder system in the rangeof conditions studied herein, limited time-dependent computa-tions have also been carried out atRe) 100 for the purpose ofvalidation only (Table 4). An excellent correspondence can beseen in this table. The orders of the deviations seen in Table 4are not at all uncommon in such numerical studies due to theinherent differences stemming from different flow schematics,grid sizes, and solution methodologies, etc. For instance, the

rectangular domain used by Mittal et al.20 employsLu ) 5D,Ld ) 16D andH ) 16D, whereLu, Ld, andH are the upstreamand downstream lengths and height of the computationaldomain, respectively. The present results for this case (shownin Table 4) are also based on this domain size. On the otherhand, Ding et al.23 also used a rectangular domain, but withLd

) 24D and H ) 25D, respectively. Juncu24 has used theorthogonal curvilinear transformed stream function-vorticityformulation to solve this problem. Notwithstanding theseinherent differences in these studies (including the use of finiteelement,20 finite difference,24 and MLSFD23 methods), thecorrespondence seen in Table 4 is regarded to be satisfactoryand acceptable.

On the basis of these comparisons together with our previousexperience and coupled with the fact that the numericalpredictions for power-law fluids tend to be less accurate, thepresent results for a pair of cylinders are believed to be reliableto within (2-3% .

Figure 2. Streamline profiles for different values of the Reynolds number (Re) and power-law index (n) at a gap ratio ofG ) 2. The bracketed quantitiesrepresent (ψmin, ψmax, Ns), whereψmin and ψmax are the minimum and maximum values of the dimensionless stream function andNs is the number ofstreamlines.

Figure 3. Streamline profiles for different values of the Reynolds number (Re) and power-law index (n) at a gap ratio ofG ) 4. The bracketed quantitiesrepresents (ψmin, ψmax, Ns), whereψmin and ψmax are the minimum and maximum values of the dimensionless stream function andNs is the number ofstreamlines.


6.2. Detailed Flow Kinematics.Some physical insights intothe nature of the flow can be gained by examining the streamlinepatterns, center line velocity, and surface pressure profiles.

6.2.1. Streamline Profiles.Representative plots showing thedependence of the streamline patterns in the vicinity of thecylinders on the Reynolds number (Re), power-law index (n),and gap ratio (G) are presented in Figures 2 and 3. For fixedvalues of the power-law index (n) and gap ratio (G), the size ofthe recirculation zone grows and the point of the separationmoves forward on the surface of the cylinders for both shear-thinning (n < 1) and shear-thickening (n > 1) fluids as theReynolds number (Re) is progressively increased. For fixedvalues of the Reynolds number (Re) and gap ratio (G), as thefluid behavior changes from Newtonian (n ) 1) to shear-thickening (n > 1), the recirculation region grows in size. The

shear-thinning (n < 1) fluid behavior shows complex depen-dence of the wake size for all values ofReandG. For smallvalues of the Reynolds number (Re e 5), no separation wasobserved for any value of the power-law index (n) and gap ratio(G). Similarly, irrespective of the value of the gap ratio, no flowseparation occurred in highly shear-thinning fluids (n ) 0.4)for Reynolds numbers up to 20. In shear-thickening (n > 1)fluids, at low values of the gap ratio (G e 4), the wakeinterference phenomena can clearly be seen in Figures 2 and 3,where the wake of the upstream cylinder is being suppresseddue to the downstream cylinder being too close to it. For shear-thinning fluids at low Reynolds numbers, the viscosity becomesvery large as the shear rate decreases, and hence it tends toinfinity, far away from the cylinder when shear rate is zero. Onthe other hand, for shear-thickening fluids (n > 1), the inertial

Figure 4. Velocity profiles at the horizontal center line (y ) 0) for different values of the power-law index (n) at a gap ratio ofG ) 2 for (a)Re) 1, (b)Re) 10, and (c)Re) 40. The locationsXR1, XF2, andXR2 are the rear stagnation point (θ ) π) of the first cylinder and the front (θ ) 0) and rear (θ )π) stagnation points of the second cylinder, respectively.


effects dominate, even when far away from the cylinders. Thus,the gap ratio shows stronger dependence on the wake interfer-ence in shear-thickening fluids and/or in high Reynolds numberflows. It is, however, appropriate to add here that, in Newtonianfluids, the flow separates at aboutRe ) 5 for a singlecylinder,38,44 but depending upon the gap between the twocylinders, the presence of the downstream cylinder can causethe flow to separate at much smaller values of the Reynoldsnumber, as observed in experiments with a viscous glycerinesolution forG ) 2 andRe) 0.01.45 However, no separationwas observed in the present study, even atRe) 1.

6.2.2. Center-Line Velocity Profile.Figures 4-6 show thevariation of the dimensionlessx-component of the velocity(Ux

/ ) Ux/Uo) along the center line (y ) 0) with the Reynolds

number (Re), flow behavior index (n), and gap ratio (G). Theleft side of the figures shows the variation of the velocity alongthe center line connecting the two cylinders, that is,XR1, fromthe rear stagnation point (θ ) π) of the upstream cylinder toXF2, the front stagnation point (θ ) 0) of the downstreamcylinder. In the right-side figures, the corresponding velocityvariation fromXR2, the rear stagnation point of the downstreamcylinder to the downstream center line. At a fixed location (x,0),the center line velocity increases with the increasing value ofthe power-law index (n) for all values of the Reynolds number(Re) and gap ratio (G). At low values of the gap ratio (G ) 2),the dimensionless velocity (Ux

/) in the region between the twocylinders is seen to be small (and negative suggesting reverseflow) and it decreases as one moves away from the cylinders.

Figure 5. Velocity profiles at the horizontal center line (y ) 0) for different values of the power-law index (n) at a gap ratio ofG ) 4 for (a)Re) 1, (b)Re) 10, and (c)Re) 40.


The negative values imply flow separation, but the values aretoo small to be captured in streamline patterns. Because of suchsmall values, the streamline patterns in Figures 2 and 3 showno streamlines in this region. The increasing value of the gapratio (G) shows quite an interesting dependence of the velocityon the Reynolds number and the power-law index in both shear-thinning and shear-thickening fluids. The magnitude of thevelocity is seen to be maximum at the midpoint (G/2 distanceforward from the upstream cylinder) of the cylinders at lowvalues of the gap ratio and low values of the Reynolds number( parts a.1-c.1 of Figure 4, part a.1 of Figure 5, and part a.1 ofFigure 6). This point is seen to move forward with the increasingvalue of the gap ratio (G). The velocity (Ux

/) profiles along thecenter line of the downstream cylinder show qualitatively similar

dependence on the power-law index (n), Reynolds numbers (Re),and the gap ratio (G). The dimensionless velocity (Ux

/) in therear of the downstream cylinder is seen to be continuouslyincreasing along the center line for all values of the power-lawindex; the increasing values of Reynolds number show adecrease (from zero to negative) followed by an increase in thevalues.

6.2.3. Pressure Coefficient (CP) on the Surface of Cylin-ders.Representative results showing the variation of the pressurecoefficient (CP) on the surface of the upstream and downstreamcylinders are plotted in Figures 7-9 for a range of values ofthe power-law index (n), for three values of the Reynoldsnumber (Re) 1, 10, and 40), and for a range of values ofG.For fixed values of the Reynolds number, the power-law index,

Figure 6. Velocity profiles at the horizontal center line (y ) 0) for different values of the power-law index (n) at a gap ratio ofG ) 10 for (a)Re) 1, (b)Re) 10, and (c)Re) 40.


and the gap ratio, the variation of the pressure coefficient overthe upstream cylinder shows a behavior qualitatively similar tothat seen for a single cylinder.34,40 Because of varying extentsof interferences, the values for the downstream cylinder showquite a complex dependence on the dimensionless parameters.The CP values are always seen to be higher for the upstreamcylinder than those for the downstream cylinder, under otherwiseidentical conditions. For both upstream and downstream cyl-inders, the pressure profiles show symmetric patterns aroundthe midplane, thereby suggesting the flow to be symmetric overthe ranges of conditions studied herein.

For fixed values of the Reynolds number (Re), power-lawindex (n), and gap ratio (G), the pressure coefficient (CP) forthe upstream cylinder is seen to decrease from its maximumvalue at the front stagnation point (θ ) 0) along the surfacetoward the rear of the cylinder. In the absence of flow separation,the decrease in the pressure coefficient continues all the wayup to the rear stagnation point (θ ) π), where it reaches theminimum value (CP,min). On the other hand, the minimum valueof the pressure coefficient (CP,min) occurs at the point ofseparation,θ ) θs(< π), and beyondθ > θs, it increases up toθ ) π. For a fixed value of the flow behavior index (n), the

Figure 7. Variation of the pressure coefficient over the surface of the upstream (left) and downstream (right) cylinders for different values of power-lawindex (n), for a gap ratio ofG ) 2 and (a)Re) 1, (b) Re) 10, and (c)Re) 40. θ ) 0 andθ ) 180 correspond to the front and rear stagnation points,respectively.


point of minimum pressure coefficient (CP,min) on the surfaceof the upstream cylinder shifts toward the front stagnation point(θ ) 0) with a gradual increase in the Reynolds number (Re).The pressure coefficient over the surface of both cylinders ishigher for shear-thinning (n < 1) fluids than that for Newtonian(n ) 1) and lower for shear-thickening (n > 1) fluids in theupstream side of the cylinders; the behavior is seen to switchover in the downstream side of the cylinders. At small valuesof the gap ratio (G), the values of the pressure coefficient atthe front and rear stagnation points, that is,CP(0) andCP(π),are seen to be strongly influenced by the value of the power-law index (n) at low Reynolds numbers (Re).

6.3. Macroscopic Characteristics.In this section, the rolesof the flow behavior index (n), the Reynolds number (Re), andthe gap ratio (G) on the individual and total drag coefficientsare discussed.

6.3.1. Pressure Drag Coefficient (CDP). Table 5 shows theinfluence of the Reynolds number (Re), power-law index (n),and gap ratio (G) on the pressure drag coefficient for theupstream (CDP,1) and downstream (CDP,2) cylinders, respectively.Also, included in table are the corresponding values for a singlecylinder (G ) ∞). For fixed values of the Reynolds number(Re) and gap ratio (G), the pressure drag coefficient (CDP)increases with the decreasing value of the power-law index (n)

Figure 8. Variation of the pressure coefficient over the surface of the upstream (left) and downstream (right) cylinders for different values of power-lawindex (n), for a gap ratio ofG ) 4 and (a)Re) 1, (b) Re) 10, and (c)Re) 40.


for both cylinders. The shear-thinning (n < 1) always yields ahigher value of the pressure drag coefficient (CDP) than thecorresponding Newtonian (n ) 1) values; an opposite trend isseen for shear-thickening (n > 1) fluids. Table 5 also shows astronger effect of the power-law index (n) on the pressure dragcoefficient (CDP) in shear-thinning (n < 1) fluids than that inshear-thickening (n > 1) fluids. For fixed values ofRe, n, andG, the upstream cylinder always shows a higher value of thepressure drag coefficient (CDP) than that for the downstreamcylinder, that is,CDP,1 > CDP,2. The highly shear-thinning fluid(n ) 0.4), largest gap ratio (G ) 10), and small value of theReynolds number (Re) 1) show the smallest value of the ratio(CDP,1/CDP,2), which increases with the increasing value of the

power-law index (n) and of the Reynolds number (Re). Theincrease in the gap ratio (G) exerts an opposite influence onthe pressure drag for two cylinders. The pressure drag coef-ficients for both cylinders are always smaller than that for asingle cylinder (CDP > CDP,1 > CDP,2), which is clearly due tothe interference between the flow fields brought about by theproximity of the two cylinders. For instance, forG ) 2, theratio of the pressure drag coefficients for the two cylinders(CDP,1/CDP,2) increases from 1.18 to 4.38 as the power-law index(n) was increased from 0.4 to 1.8 for a Reynolds number (Re)of 1 and from 8.65 to 1323 as the power-law index (n) wasincreased from 0.4 to 1.4 for a Reynolds number of 40,respectively. On the other hand, the change in the gap ratio (G)

Figure 9. Variation of the pressure coefficient over the surface of the upstream (left) and downstream (right) cylinders for different values of power-lawindex (n), for a gap ratio ofG ) 10 and (a)Re) 1, (b) Re) 10, and (c)Re) 40.


from 2 to 10 decreases this ratio (CDP,1/CDP,2) from 1.18 to 1.05and 4.38 to 2.65 atn ) 0.4 andn ) 1.8, respectively, forRe)1. The corresponding decrease forRe) 40 was seen to be from8.65 to 1.43 and 1323 to 3.59 atn ) 0.4 and 1.4, respectively.

It can also be seen from Table 5 that the influence of the dimen-sionless parameters on the pressure drag coefficient is strongerfor the downstream cylinder than that for the upstream cylinder.These trends clearly suggest strong wake interference on thebehavior of the downstream cylinder. At small values ofG, thepressuredragvaluesfor thetwocylindersarequiteclose,asnowakeis formed under these conditions. On the other hand, the influenceof power-law index (n) is seen to be stronger on the pressure dragcoefficient of the downstream cylinder (CDP,2) than that for theupstream cylinder (CDP,1) at such low Reynolds numbers. How-ever, the effect is higher in shear-thinning (n < 1) fluids thanthat in Newtonian/shear-thickening fluids and for the downstreamcylinder than for the upstream cylinder. For instance, forG )2 as the Reynolds number (Re) is increased from 1 to 20, theCDP,1 values decrease by factors of 8.15, 3.46, and 2.26 atn )0.2, 1, and 1.8, respectively, and the corresponding changes in thevalues for the downstream cylinder (CDP,2) are 25, 13.17, and

11.97, respectively. The pressure drag coefficients for the up-stream cylinder show an opposite dependence on the power-lawindex (n) (i.e., decrease inCDP,1 with decreasing values ofn) athigh values of the Reynolds numbers (Re) 40) and small gapratios (G ) 2 and 3). It should also be noted that at high valuesof the Reynolds number (Re) and the power-law index (n), thevalues of the pressure drag coefficient for the second cylinder areseen to be very close to zero, for example,CDP,2) 0.0007 atRe) 40 andn ) 1.4. It is probably due to the fact that the trailingcylinder is completely within the wake of the front cylinder.

To elucidate the role of the power-law index (n) on the dragcharacteristics, the drag coefficient values have been normalizedusing the corresponding Newtonian values, under otherwiseidentical conditions, defined as follows:

Figure 10 shows the normalized pressure drag coefficients,and both CDP,1

N and CDP,2N are >1 in shear-thinning fluids,

whereas they are<1 in shear-thickening fluids. As the fluid

Table 5. Dependence of Pressure Drag Coefficient (CDP) on the Reynolds Number (Re), Power-law Index (n), and Gap Ratio (G)a

n ) 0.4 n ) 0.6 n ) 1 n ) 1.4 n ) 1.8

G CDP,1 CDP,2 CDP,1 CDP,2 CDP,1 CDP,2 CDP,1 CDP,2 CDP,1 CDP,2

Re) 12 9.7862 8.2672 6.4787 5.0369 3.8272 2.0482 2.8120 0.9804 2.2879 0.52203 10.7250 9.3050 7.0358 5.5985 4.0668 2.2714 2.9180 1.1034 2.3294 0.97824 11.4360 10.1209 7.4403 6.0111 4.2313 2.4297 2.9824 1.1954 2.3442 0.90846 12.3901 11.3232 8.0041 6.6130 4.4463 2.6630 3.0594 1.3392 2.3623 0.783410 13.3811 12.7207 8.6429 7.3689 4.6741 2.9749 3.1444 1.5317 2.4037 0.9084∞ 14.9888 9.9526 5.2088 3.4441 2.6031

Re) 22 5.3500 3.9164 3.9033 2.5279 2.6880 1.1662 2.1404 0.6031 1.8287 0.33923 5.8052 4.4740 4.2029 2.8675 2.8158 1.3394 2.1871 0.7166 1.8371 0.68924 6.1426 4.9181 4.4174 3.1207 2.9005 1.4658 2.2134 0.8013 1.8368 0.67036 6.5751 5.5932 4.7033 3.4950 3.0092 1.6517 2.2513 0.9238 1.8502 0.572710 6.9961 6.3891 4.9962 3.9722 3.1267 1.8920 2.3077 1.0741 1.8850 0.6703∞ 7.6831 5.4821 3.3971 2.4912 2.0193

Re) 52 2.6671 1.4645 2.2201 1.0430 1.7785 0.5590 1.5426 0.3177 1.3919 0.19093 2.8324 1.7397 2.3363 1.2417 1.8213 0.6897 1.5488 0.4140 1.3846 0.42364 2.9526 1.9692 2.4197 1.3967 1.8533 0.7857 1.5575 0.4812 1.3860 0.45206 3.0990 2.3251 2.5274 1.6306 1.9014 0.9240 1.5799 0.5730 1.3987 0.372810 3.2342 2.7512 2.6335 1.9285 1.9604 1.0988 1.6160 0.6878 1.4222 0.4520∞ 3.4459 2.7930 2.0869 1.7189 1.5059

Re) 102 1.7240 0.7089 1.5599 0.5364 1.3697 0.3146 1.2519 0.1882 1.1692 0.11333 1.7915 0.8917 1.6067 0.6853 1.3824 0.4209 1.2508 0.2664 1.1645 0.27564 1.8461 1.0530 1.6446 0.8036 1.3976 0.4996 1.2560 0.3210 1.1661 0.33736 1.9185 1.3007 1.7008 0.9847 1.4265 0.6145 1.2708 0.3992 1.1737 0.263910 1.9884 1.5992 1.7604 1.2128 1.4663 0.7616 1.2969 0.5015 1.1904 0.3373∞ 2.0827 1.8458 1.5502 1.3737 1.2580

Re) 202 1.2011 0.3305 1.1627 0.2587 1.1046 0.1555 1.0553 0.0891 1.0135 0.04363 1.2150 0.4687 1.1680 0.3762 1.1017 0.2401 1.0500 0.1506 1.0082 0.16214 1.2370 0.5889 1.1825 0.4711 1.1058 0.3059 1.0496 0.1970 1.0057 0.12296 1.2780 0.7794 1.2136 0.6197 1.1208 0.4062 1.0556 0.2675 1.0064 0.173610 1.3220 1.0063 1.2529 0.8066 1.1482 0.5386 1.0725 0.3643 1.0152 0.2451∞ 1.3720 1.3050 1.2063 1.1293 1.0668

Re) 40 (*Re) 30)2 0.9194 0.1063 0.9345 0.0770 0.9422 0.0334 0.9265 0.0007 0.9488* 0.0038*3 0.9032 0.2052 0.9199 0.1653 0.9285 0.0980 0.9186 0.0478 0.9407* 0.0423*4 0.9014 0.3041 0.9156 0.2438 0.9220 0.1541 0.9108 0.0890 0.9350* 0.0745*6 0.9177 0.4664 0.9247 0.3756 0.9219 0.2466 0.9055 0.1560 0.9308* 0.1249*10 0.9511 0.6649 0.9522 0.5501 0.9376 0.3776 0.9113 0.2538 0.9339* 0.1972*∞ 0.9846 0.9906 0.9826 09562 0.9785*

a CDP,1: Pressure drag coefficient for an upstream cylinder;CDP,2: pressure drag coefficient for a downstream cylinder.G ) ∞: Single cylinder.

φN )

φ(n)

φ(n ) 1)where,φ ) CDP,1,CDF,1CD,1, CDP,2,CDF,2CD,2


behavior changes from the Newtonian (n ) 1) to shear-thinningfluids (n < 1), the normalized values are seen to increase withthe decreasing value of the power-law index (n) and Reynoldsnumber (Re). For all values of the Reynolds number (Re) andgap ratio (G), the downstream cylinder shows a somewhatstronger influence of the flow behavior index (n) on the pressuredrag coefficients (CDP,2

N ) than the upstream cylinder. The gapratio (G) is seen to have only a weak effect on the normalizedpressure drag coefficient for both cylinders. In shear-thickeningfluids (n > 1), the normalized values (CDP,1

N ) are seen to beinfluenced by the power-law index (n) in a manner opposite tothat seen in the shear-thinning (n < 1) fluids. Furthermore, forall values of the gap ratio (G), CDP,2

N is seen to be stronglyinfluenced by the power-law rheology (n) but it is almostindependent of the low Reynolds number (Re).

Similarly, to delineate the role of the gap ratio,G, in anunambiguous manner, it is useful to define the normalized dragcoefficient as follows:

The normalized values of the pressure drag coefficient forboth cylinders are always found to be less than one (Figure11), which implies strong interference between the two cylindersover the range of the gap ratio (2e G e 10). For the upstreamcylinder, the normalized values are seen to increase with theincreasing value of the power-law index (n) at small values ofthe gap ratio (G) and the Reynolds number (Re); complex

Figure 10. Dependence of the normalized pressure drag coefficient for an upstream (CDP,1N) and for the downstream (CDP,2

N) cylinders on the Reynoldsnumber (Re) and power-law index (n) for gap ratios of (a)G ) 2, (b) G ) 4, and (c)G ) 10.

φSC ) φ

φ∞where,φ ) CDP,1,CDF,1CD,1, CDP,2,CDF,2CD,2

φ∞ ) CDP,CDF CD (for single cylinder)


dependence is seen at highRe and G. The low values of thegap ratio (G) and the power-lax index (n < 1) exert a strongeffect on the normalized values (CDP,1

SC ). This is due to the wakeinterference at a low gap ratio (G), which becomes moreprominent with increasing values of the Reynolds number (Re).On the other hand, at low Reynolds numbers (Re), the closeproximity of the two cylinders causes the formation of a stagnantzone in between the two cylinders, as can be seen from thestreamline patterns (Figures 2 and 3). As a result of this stagnantregion, the drag values for the upstream cylinder are seen to bevery small compared to the single cylinder values, whereas theyare quite comparable with the increasing values of the Reynoldsnumbers and/or in shear-thickening fluids. Increasing the valueof the gap ratio (G) reduces the wake interference and the valuesfor single cylinders and the upstream cylinders seem to be

approached. The normalized pressure drag coefficient valuesfor the downstream cylinder (CDP,2

SC , right) show a completelyopposite dependence on the dimensionless parameters to thatfor the upstream cylinder (CDP,1

SC , left). Over the range of theconditions, the normalized values are seen to decrease with theincreasing value of the power-law index (n), increasing valueof the Reynolds number (Re) and the decreasing value of thegap ratio (G). Once again, the values are seen to be stronglydependent on the power-law index (n) at low values of Reynoldsnumbers (Re) and gap ratio (G).

6.3.2. Friction Drag Coefficient (CDF). Table 6 shows theeffects of the Reynolds number (Re), power-law index (n), andgap ratio (G) on the friction drag coefficient (CDF). For the sakeof completeness, the corresponding values for a single cylinder

Figure 11. Dependence of the normalized pressure drag coefficient for an upstream (CDP,1SC) and for the downstream (CDP,2

SC) cylinders on the Reynoldsnumber (Re) and power-law index (n) for gap ratios of (a)G ) 2, (b) G ) 4, and (c)G ) 10.


(G ) ∞) are also included in this table. The friction dragcoefficient values for both cylinders decrease with an increasein the Reynolds number (Re) and decrease in the gap ratio (G).For fixed values of the gap ratio (G), the friction drag coefficient(CDF) increases with the decreasing value of the power-law index(n) for the upstream cylinder at low Reynolds numbers (Re)1 and 5) and for downstream cylinder over the entire range ofconditions studied herein. Here also, the flow behavior index(n) exerts a stronger influence at low Reynolds numbers thanat high Reynolds numbers. At high Reynolds numbers (5e Ree 40), the friction drag coefficient for the upstream cylinder(CDF,1) decreases with the decreasing values ofn for all valuesof G. For the downstream cylinder,CDF,2 reaches a minimumvalue in a highly shear-thickening fluid,n ) 1.4, at the highestvalue of the Reynolds number,Re ) 40 and for the smallestvalue ofG ) 2. The friction drag coefficient values for bothcylinders over the range 2e G e 10 are always smaller thanthose for a single cylinder. As expected, the smaller the gapG,the larger the deviations from the single cylinder values. Forfixed values of the dimensionless parameters (Re, n, andG),the upstream cylinder always shows a higher value of the frictiondrag coefficient (CDF) than the downstream cylinder. The

increase in the gap ratio (G) shows an opposite dependence onthe friction drag for two cylinders. For instance, the ratio (CDF,1/CDF,2) at Re) 1 increases from 1.08 to 2.92 and from 1.04 to2.45 as the power-law index (n) is increased from 0.4 to 1.8for the gap ratio,G ) 2 and 10, respectively. The correspondingchanges forRe) 40 are seen to be from 2.01 to 6.28 and from1.12 to 3.13 as the power-law index (n) is increased from 0.4to 1.4.

Figure 12 shows the variation of the normalized friction dragcoefficientsCDF,1

N andCDF,2N . Over the range of conditions, the

variation of the normalized friction drag coefficient for theupstream cylinder shows a mirror image (Figure 12) in shear-thinning and shear-thickening fluids, respectively. For all valuesof gap ratio (G), the dependence ofCDF,1

N on n is seen to switchover atRe) 10. For Re< 10, the normalized values for theupstream cylinder (CDF,1

N ) decrease as the fluid behavior changesfrom highly shear-thinning (n ) 0.4) to Newtonian (n ) 1) andfinally to highly shear-thickening (n ) 1.8). At Re) 10, thesevalues are seen to be almost independent of the power-law index(n) for all values ofG. A further increase in the value of theReynolds number (Re > 10) shows a completely reverse

Table 6. The Dependence of the Friction Drag Coefficient (CDF) on the Reynolds Number (Re), Power-Law Index (n), and Gap Ratio (G)a

n ) 0.4 n ) 0.6 n ) 1 n ) 1.4 n ) 1.8

G CDF,1 CDF,2 CDF,1 CDF,2 CDF,1 CDF,2 CDF,1 CDF,2 CDF,1 CDF,2

Re) 12 6.4233 5.9344 5.3976 4.5785 3.9621 2.4548 3.2293 1.4545 2.7945 0.95573 6.7521 6.2776 5.6272 4.7591 4.1107 2.5079 3.3301 1.4811 2.8577 0.97824 7.0102 6.5686 5.8401 4.9492 4.2404 2.5867 3.4054 1.5302 2.8977 1.02226 7.3205 6.9448 6.1522 5.2532 4.4325 2.7495 3.5005 1.6394 2.9400 1.107910 7.6474 7.3701 6.5109 5.6473 4.6496 3.0145 3.6025 1.8183 2.9980 0.9084∞ 8.3448 7.3424 5.1661 3.9437 3.2510

Re) 22 3.3557 2.8978 3.1007 2.3482 2.6677 1.4391 2.3750 0.9381 2.1711 0.65703 3.5204 3.0806 3.2323 2.4605 2.7523 1.4910 2.4283 0.9754 2.2008 0.68924 3.6444 3.2392 3.3462 2.5777 2.8217 1.5580 2.4655 1.0253 2.2182 0.73126 3.7889 3.4509 3.5020 2.7676 2.9185 1.6879 2.5142 1.1188 2.2440 0.797810 3.9325 3.6857 3.6649 3.0170 3.0257 1.8861 2.5780 1.2551 2.2889 0.6703∞ 4.2190 3.9609 3.2731 2.7796 2.4550

Re) 52 1.4918 1.1300 1.5685 0.9959 1.6074 0.7206 1.5924 0.5273 1.5621 0.39833 1.5543 1.2165 1.6197 1.0610 1.6381 0.7649 1.6078 0.5627 1.5674 0.42364 1.5992 1.2968 1.6620 1.1296 1.6637 0.8149 1.6208 0.6007 1.5736 0.45076 1.6486 1.4076 1.7163 1.2423 1.7021 0.9049 1.6446 0.6657 1.5891 0.495110 1.6936 1.5233 1.7702 1.3886 1.7490 1.0366 1.6811 0.7630 1.6171 0.4520∞ 1.7665 1.8536 1.8502 1.7840 1.7133

Re) 102 0.8282 0.5616 0.9505 0.5258 1.1014 0.4246 1.1815 0.3346 1.2239 0.26383 0.8534 0.6122 0.9703 0.5703 1.1110 0.4588 1.1838 0.3586 1.2221 0.27564 0.8724 0.6629 0.9876 0.6169 1.1216 0.4957 1.1891 0.3856 1.2243 0.29346 0.8938 0.7331 1.0118 0.6943 1.1412 0.5625 1.2024 0.4357 1.2325 0.329110 0.9130 0.8042 1.0370 0.7929 1.1682 0.6614 1.2256 0.5154 1.2502 0.3373∞ 0.9376 1.0726 1.2252 1.2936 1.3208

Re) 202 0.4549 0.2749 0.5703 0.2710 0.7485 0.2383 0.8708 0.1980 0.9538 0.15963 0.4609 0.3056 0.5734 0.3007 0.7473 0.2608 0.8667 0.2101 0.9483 0.16214 0.4670 0.3364 0.5788 0.3314 0.7497 0.2864 0.8665 0.2286 0.9460 0.17276 0.4761 0.3816 0.5892 0.3843 0.7579 0.3357 0.8709 0.2674 0.9466 0.200810 0.4849 0.4261 0.6018 0.4512 0.7729 0.4116 0.8836 0.3337 0.9547 0.2544∞ 0.4936 0.6178 0.8047 0.9262 1.0020

Re) 40 (*Re) 30)2 0.2409 0.1197 0.3365 0.1249 0.5055 0.1183 0.6423 0.1023 0.8233* 0.1109*3 0.2383 0.1381 0.3330 0.1432 0.4995 0.1304 0.6315 0.1063 0.8159* 0.1107*4 0.2387 0.1583 0.3323 0.1627 0.4968 0.1467 0.6267 0.1171 0.8111* 0.1179*6 0.2414 0.1896 0.3346 0.1994 0.4968 0.1821 0.6235 0.1451 0.8075* 0.1408*10 0.2458 0.2191 0.3409 0.2482 0.5034 0.2427 0.6270 0.2003 0.8101* 0.1897*∞ 0.2493 0.3490 0.5223 0.6544 0.8475*

a CDF,1: Friction drag coefficient for an upstream cylinder;CDF,2: Friction drag coefficient for a downstream cylinder.G ) ∞: Single cylinder.


dependence for the upstream cylinder (CDF,1N ) with the increas-

ing value of the power-law index (n). The normalized frictiondrag coefficient for the downstream cylinder (CDF,2

N ) shows adependence on the power-law rheology (n), which is qualita-tively similar to that seen for the normalized pressure dragcoefficient in Figure 10. Here, also the downstream cylindershows a stronger dependence of the normalized values on thepower-law index (n) than that for the upstream cylinder.

Figure 13 depicts the dependence of the normalized frictiondrag coefficient for the upstream (CDF,1

SC , left) and the down-

stream (CDF,2SC , right) cylinders onRe, n,and G. The depend-

ences seen in this Figure are qualitatively similar to that for thenormalized pressure drag coefficient as that seen in Figure 11.

6.3.3. Total Drag Coefficient (CD). The dependence of thetotal drag coefficients,CD () CDP + CDF), on the Reynoldsnumber (Re), the power-law index (n), and the gap ratio (G) isshown in Figure 14. The total drag coefficient (CD) shows adependence on the dimensionless parameters (Re, n, G) similarto the friction drag coefficient (CDF) for both upstream anddownstream cylinders. The total drag coefficient (CD) values

Figure 12. Dependence of the normalized friction drag coefficient for an upstream (CDF,1N) and for the downstream (CDF,2

N) cylinders on the Reynoldsnumber (Re) and power-law index (n) for gap ratios of (a)G ) 2, (b) G ) 4, and (c)G ) 10.


for both cylinders decrease with the decreasing value of thegap ratio (G). For fixed values of the gap ratio (G), the totaldrag coefficient (CD) increases with the decreasing value of thepower-law index (n) for the upstream cylinder at low Reynoldsnumbers (Re) 1 and 5) and for the downstream cylinder. Athigh values of the Reynolds numbers (5e Ree 40), the totaldrag coefficients for the upstream cylinder (CD,1) show anopposite dependence on the power-law index (n) for all valuesof the gap ratio (G).

The influence of the power-law rheology (n) on the total dragcoefficient for an upstream (CD,1

N , left) and downstream

(CD,2N , right) cylinders for the range of dimensionless param-

eters is shown in Figure 15. The dependence of the normalizedtotal drag coefficient on the power-law index (n) is seen to bequalitatively similar to that seen for normalized pressure andfriction drag coefficients, seen in Figures 10 and 12, respectively.The normalized total drag coefficient for the two cylinders alsoshows a dependence (Figure 16) onRe, n, and G, which isqualitatively similar to that of the normalized pressure andfriction drag coefficients seen in Figures 11 and 13.

Broadly, shear-thinning (n < 1) fluid behavior always yieldshigher values of the drag coefficient (CD) than the corresponding

Figure 13. Dependence of the normalized friction drag coefficient for an upstream (CDF,1SC) and for the downstream (CDF,2

SC) cylinders on the Reynoldsnumber (Re) and power-law index (n) for gap ratios of (a)G ) 2, (b) G ) 4, and (c)G ) 10.


Newtonian (n ) 1) values; an opposite trend is seen for theshear-thickening (n > 1) fluids. These figures also show astronger effect of power-law index (n) on the total dragcoefficient (CD) in shear-thinning (n < 1) fluids than that inshear-thickening (n > 1) fluids.

6.3.4. Relative Contributions ofCDP and CDF . To elucidatethe role of the gap ratio (G) on the relative contributions of theindividual drag components, a drag ratio (CDR)CDP/CDF) hasbeen plotted in Figure 17 for a range of values ofRe, n, andG.For fixed values of the Reynolds number (Re) and of the power-law index (n), the drag ratio (CDR) for both cylinders increaseswith the increasing value of gap ratio (G), thereby suggestingno or little interaction between the two cylinders. The change

in fluid behavior from Newtonian (n ) 1) to shear-thinning (n) 0.4) enhances the relative contribution of the pressurecomponent over the range of conditions, whereas the drag ratio(CDR) reduces with the increasing value of the power-law indexfor shear-thickening fluids (n > 1). In shear-thickening fluids,the drag ratio of the second (downstream) cylinder (CDR,2) showsa very strong dependence on the power-law index (n), especiallyat high Reynolds numbers.

At low Reynolds numbers (Re) 1), the gap ratio (G) is seento have a very small effect on the drag ratio for the upstreamcylinder (CDR,1) in shear-thickening fluids (n > 1). With theincreasing value of the Reynolds number (Re), theCDR,1 valuesare seen to be almost independent of the gap ratio (G). These

Figure 14. Dependence of the total drag coefficients for an upstream (CD,1) and for the downstream (CD,2) cylinders on the power-law index (n) and thegap ratio (G) for Reynolds numbers of (a)Re) 1, (b) Re) 2, (c) Re) 5, (d) Re) 10, (e)Re) 20, and (f)Re) 40 (0:4e n e 1:4) and 30 (n ) 1:8).


values are, however, higher for the upstream cylinder than thatfor the downstream cylinder, under otherwise identical condi-tions. For instance, atRe ) 1 as the gap ratio (G) increasesfrom 2 to 10,CDR,1 changes from 1.52 to 1.75, from 0.97 to1.00, and from 0.82 to 0.80 for power-law indexesn ) 0.4, 1,and 1.8, respectively. The corresponding changes atRe) 40are almost nonexistent over the same ranges of conditions. Thechange in the drag ratio for the downstream cylinder (CDR,2)with the gap ratio is rather strong, for example, the value changesfrom 1.39 to 1.73, from 0.83 to 0.99, and from 0.55 to 0.74

with an increase in the gap ratio (G) from 2 to 10 atRe) 1 forn ) 0.4, 1, and 1.8, respectively. The corresponding changesat Re ) 40 and forn ) 0.4 and 1 are from 0.89 to 3.04 andfrom 0.28 to 1.56, and atRe) 30 andn ) 1.8 is 0.034 to 1.04.This clearly shows that the relative contribution of the twocomponents is strongly dependent on the proximity of the twocylinders in shear-thinning fluids (n < 1) for the upstreamcylinder, whereas this dependence is weak in shear-thickening(n > 1) fluids. On the other hand, the downstream cylindershows a stronger dependence on both the gap ratio (G) and the

Figure 15. Dependence of the normalized total drag coefficient for an upstream (CD,1N) and for the downstream (CD,2

N) cylinders on the Reynolds number(Re) and the power-law index (n) for gap ratios of (a)G ) 2, (b) G ) 4, and (c)G ) 10.


flow behavior index (n), which accentuates with the increasingvalue of the Reynolds number (Re).

In summary, the flow characteristics of power-law fluids pasta pair of cylinders in a tandem arrangement are seen to beinfluenced in an intricate manner by the value of the Reynoldsnumber (Re), the power-law index (n), and the gap ratio (G).At high values of the Reynolds numbers, the wake interferencesare more prominent when the gap ratio (G) is very small. Onthe other hand, when the cylinders are placed far from each

other (G f ∞), no wake interference will occur and the twocylinders would behave in a non-interfering manner. Thisinterplay is further accentuated by the fact that, even at lowReynolds numbers, the viscous terms in the momentum equa-tions are highly nonlinear for power-law fluids. As the Reynoldsnumber is increased, the flow is governed by two nonlinearterms, namely, inertial and viscous, which scale differently withvelocity. For instance, the viscous forces approximately scaleas ∼Uo

n, whereas the inertial forces scale as∼Uo2. Thus,

Figure 16. Dependence of the normalized total drag coefficient for an upstream (CD,1SC) and for the downstream (CD,2

SC) cylinders on the Reynolds number(Re) and power-law index (n) for gap ratios of (a)G ) 2, (b) G ) 4, and (c)G ) 10.


keeping everything else fixed, the decreasing value of the power-law index (n) suggests diminishing importance of the viscouseffects for shear-thinning (n < 1) fluids, whereas the inertialterms will still scale as∝Uo

2. On the other hand, viscous effectsare likely to grow with the increasing value of the power-lawindex (n) for a shear-thickening (n > 1) fluid. For the extremecase ofn ) 1.8, the viscous terms will also scale as∼Uo

1.8,almost identical to the inertial term. These nonlinear interactionsin conjunction with the distance between the two cylinders exerta strong influence on the momentum transfer characteristics.Even though when the cylinders are placed closed to each other

(small values ofG) there is no wake interference at lowReynolds numbers, the power-law rheology exerts a stronginfluence on the flow field and drag phenomena. It is believedthat these different kinds of dependencies on the flow behaviorindex and velocity are also responsible for the non-monotonicbehavior of kinematics of the flow as seen in this work.

7. Concluding Remarks

Extensive numerical results on the momentum characteristicsof the steady unconfined flow of power-law fluids over a pair

Figure 17. Dependence of the drag ratio for an upstream (CDR,1) and for the downstream (CDR,2) cylinders on the Reynolds number (Re) and power-lawindex (n) for gap ratios of (a)G ) 2, (b) G ) 3, (c) G ) 4, (d) G ) 6, and (e)G ) 10.


of cylinders in tandem arrangement have been studied over wideranges of conditions as: 1e Ree 40, 0.4e n e 1.8, and forfive values of the gap ratio (G ) 2, 3, 4, 6, and 10). The effectsof the dimensionless parameters (Re, n, G) on the detailedkinematics of the flow (streamline patterns, velocity, pressureprofiles) and the drag phenomena (individual and total dragcoefficients) are presented in detail. The wake interference inconjunction with the power-law rheology is seen to be a stronginfluence at high Reynolds numbers, whereas at low Reynoldsnumbers the flow is altered by the fluid behavior. The flowseparation is seen to be delayed in shear-thinning fluids ascompared to that in Newtonian and in shear-thickening fluids.The pressure coefficient distribution over the surface of thecylinders shows a complex dependence on the power-law index,Reynolds number, and gap ratio. The shear-thinning fluids showstronger effects on the drag characteristics than in Newtonianand in shear-thickening fluids. The drag coefficient values showa dependence on the dimensionless parameters, which isqualitatively similar to that of a single cylinder. Both upstreamand downstream cylinders show smaller values of the individualand total drag coefficients than those for a single circularcylinder when these interact with each other.

Notations

CD ) Total drag coefficient, dimensionlessCD

N ) Normalized total drag coefficient using the correspond-ing Newtonian value, [)

CD(Non-Newtonian)/CD(Newtonian)], dimensionlessCD

SC ) Normalized total drag coefficient using the correspond-ing value for single cylinder

value, dimensionlessCDF ) Frictional component of the drag coefficient, dimension-

lessCDF

N ) Normalized friction drag coefficient using the corre-sponding Newtonian value,

[) CDF(Non-Newtonian)/CDF(Newtonian)], dimensionlessCDF

SC ) Normalized friction drag coefficient using the corre-sponding value for single

cylinder value, dimensionlessCDP ) Pressure component of the drag coefficient, dimension-

lessCDP

N ) Normalized pressure drag coefficient using the corre-sponding Newtonian value,

dimensionlessCDP

SC ) Normalized pressure drag coefficient using the corre-sponding value for single

cylinder value, dimensionlessCDR ) Drag ratio, dimensionlessCP ) Pressure coefficient, dimensionlessCP(0) ) Pressure coefficient at the front stagnation (θ ) 0) point,

dimensionlessCP(π) ) Pressure coefficient at the rear stagnation (θ ) π) point,

dimensionlessD ) Diameter of the cylinders, mDo ) Outer boundary of the computational domain, mFD ) Drag force per unit length of the cylinder, N/mFDF ) Frictional component of the drag force per unit length

of the cylinder, N/mFDP ) Pressure component of the drag force per unit length of

the cylinder, N/mG ) Gap ratio between the two cylinders, dimensionlessI2 ) Second invariant of the rate of the strain tensor, s-2

L ) Center-to-center distance between the cylinders, m

m ) Power-law consistency index, Pa.sn

n ) Power-law flow behavior index, dimensionlessp ) Pressure, PaRe) Reynolds number, dimensionlessUo ) Uniform inlet velocity of the fluid, m/sUx,Uy ) x andy components of the velocity, m/sx,y ) Stream-wise and transverse coordinates, m

Greek Symbols

η ) Viscosity, Pa.sθ ) Angular displacement from the front stagnation (θ ) 0),

degreesF ) Density of the fluid, kg/m3

Ψ ) Stream function, dimensionlessτ ) Shear stress, Paτxx,τxy ) x andy components of the shear stress, Pa

Subscripts

1 ) Upstream cylinder2 ) Downstream cylindero ) Free stream

Superscripts

N ) Newtonian valueSC ) Single circular cylinder

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ReceiVed for reView June 22, 2007ReVised manuscript receiVed October 31, 2007

AcceptedNovember 8, 2007

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