Two-Dimensional Steady Poiseuille Flow of Power-Law Fluids Across a Circular Cylinder ... ·...

Two-Dimensional Steady Poiseuille Flow of Power-Law Fluids Across a CircularCylinder in a Plane Confined Channel: Wall Effects and Drag Coefficients

Ram Prakash Bharti and R. P. Chhabra*

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

V. Eswaran

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

The Poiseuille flow of incompressible power-law fluids past a circular cylinder placed midway between twoparallel plates has been investigated numerically by solving the continuity and momentum equations usingFLUENT (Version 6.2). Extensive results highlighting the roles of the Reynolds number (Re), the power-lawindex (n), and the blockage ratio (â) on the global and detailed flow characteristics have been presentedover wide ranges of conditions (1e Ree 40, 0.2e n e 1.9, and 1.1e â e 4). For a fixed value of theblockage ratio, the drag coefficient increases as the shear-thickening (n > 1) tendency of the fluid increases,whereas shear-thinning (n < 1) fluid behavior shows the opposite dependence. At smallRe, this effect isobserved to be very strong and it gradually diminishes asReincreases. The effect ofRediminishes forn > 1with a decrease inâ, whereas the dependence becomes stronger forn < 1. Individual drag coefficientsalso show qualitatively similar dependence onRe, n, and â. In addition, the streamline and pressureprofiles have also been presented to provide further physical insights into the detailed kinematics ofthe flow. The wake size is observed to increase as the flow behavior index (n) decreases. Because ofwall effects, the flow separation seems to be delayed in shear-thickening fluids, whereas the opposite trendwas observed in shear-thinning fluids. While the pressure profiles are observed to be similar to that for anunconfined flow, the flattening of the pressure curve in the rear portion of the cylinder suggestssluggish pressure recovery due to wall effects. In contrast to an unconfined flow, the front stagnationpressure coefficient values can be negative in magnitude, under certain conditions. The dependence of thepressure coefficient on the flow behavior index intensifies in shear-thickening fluids with a decrease in theblockage ratio.

1. Introduction

Because of its fundamental and pragmatic significance,considerable research efforts have been devoted to the study ofcrossflow of fluids past cylinders of circular and noncircularcross sections. Typical examples include the flow in tubularand pin heat exchangers, in the resin-transfer molding (RTM)process of manufacturing fiber-reinforced composites, in filtra-tion screens and aerosol filters, etc. Consequently, a voluminousbody of knowledge is now available on the transverse flow ofNewtonian fluids over a circular cylinder in an unconfined flowconfiguration (e.g., see, Zdravkovich1,2), although much less isknown about the effect of confining walls on the flowphenomenon, even for Newtonian fluids. The severity ofconfinement is characterized by defining a blockage ratio (â orλ). For a channel of heightH and circular cylinder of diameterD, the blockage ratio is defined as follows:

or

Zdravkovich2 classified the blockage effects for the Newtonianfluid flow over a circular cylinder as follows:

(1) Forλ < 0.1, the wall effects are small and may be ignored.(2) For 0.1e λ e 0.6, the blockage modifies the flow and

the correction is necessary.(3) Forλ > 0.6, the blockage radically alters the flow around

the cylinder and the correction of data is unjustified.This rather crude classification is applicable for all flow

regimes except in the two-dimensional laminar flow. At verylow Reynolds number (Re) values in the two-dimensionallaminar flow regime, the wall effects are known to be important,even for λ < 0.001, because of the slow decay of the flow(velocity) field. All in all, a reasonable body of information isnow available on the crossflow of Newtonian fluids over acircular cylinder.

On the other hand, it is readily acknowledged that manysubstances encountered in industrial practice (pulp andpaper, food, polymer and process engineering applications)display shear-thinning and/or shear-thickening behavior.3

Because of their high viscosity levels, non-Newtoniansubstances are generally processed under laminar flowconditions. Admittedly, many non-Newtonian fluids (notably,polymeric systems) display viscoelastic behavior; theavailable scant literature both for the creeping flow past asingle cylinder and over a periodic array of cylinders seemsto suggest the viscoelastic effects to be minor in this flowconfiguration.4 Furthermore, the fluid relaxation time oftendecreases with shear rate. Thus, the relaxation time willalso decrease asRe increases and, hence, a suitably definedDeborah number would be small. Therefore, the viscoelasticeffects are not expected to be significant in this case.

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

â ) HD

(1a)

λ ) DH

) 1â

(1b)

3820 Ind. Eng. Chem. Res.2007,46, 3820-3840

10.1021/ie070166+ CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 05/02/2007

Similarly, the currently available numerical simulations examinethe role of viscoelasticity in the absence of shear-dependentviscosity and predict very little change in the value of dragcoefficient and, thus, it seems that the steady flow resistance isdetermined primarily by the viscous properties of the fluid.4

Therefore, it seems reasonable to begin with the flow of purelyviscous power-law type fluids as long as the power-lawconstants are evaluated in the shear-rate range that is appropriatefor the flow over a cylinder and the level of complexity can bebuilt up gradually to accommodate the other non-Newtoniancharacteristics.

To our knowledge, there has been no prior study on thecrossflow of power-law fluids past a circular cylinder confinedin a channel. This work intends to fill this gap in the literature.At the outset, it is instructive; however, to briefly recount theavailable limited work on the flow of power-law fluids past anunconfined circular cylinder to facilitate the subsequent pre-sentation of the new results for the flow of power-law fluidspast a confined circular cylinder.

2. Previous Work

The problem of Poiseuille flow (see Figure 1 for theschematics) of Newtonian fluids past a cylinder placed sym-metrically between two fixed parallel walls was studied analyti-cally by Faxen5 for λ e 0.5. Using the method of images toobtain the series expansion, he presented the following expres-sion for drag:

where

and

For small values ofλ, Faxen’s solution is quite accurate. Someother attempts have also been made to solve the Oseen’slinearized equations.6,7 For λ < 0.2, the numerical resultsshow good agreement with the limited experimental results ofWhite.8

On the other hand, the extent of wall effects on dragcoefficient in an uniform flow past a cylinder placed sym-metrically between two plane walls (moving with the uniformstream) has been investigated numerically by Huang and Feng9

for 0.1e Ree 10 and 1.25e â e 500, by Ben Richou et al.10

for 0.01e λ e 0.6 at very low Reynolds numbers (10-4 e Ree 1) and by Chakraborty et al.11 in the range 1.54e â e 20and 0.1e Re e 200, respectively. They also presented anequation for the drag coefficient as a function of the blockageratio â and the Reynolds numberRe. Using a stream function-vorticity formulation, in conjunction with the finite elementmethod, Anagnostopoulos et al.12 presented results on theinfluence of blockage ratio (λ ) 0.05, 0.15, and 0.25) on thesteady and unsteady wake characteristics at a fixed value ofthe Reynolds number (Re ) 106). Recently, the influence ofwall blockage (0.01e λ e 0.6) on the drag force at very lowReynolds numbers (10-4 e Re e 1) has been investigated inPoiseuille flow by Ben Richou et al.13 They reported a goodmatch with the predictions of Faxen and with their experimentalresults.

Aside from these experimental and numerical studies, theeffects of blockage on the flow and heat transfer have also beeninvestigated analytically using the standard boundary layerapproximation.14,15 Vaitiekunas et al.14 investigated the effectof channel blockage on the dimensionless shear stress, thelocation of Umax (which is the point of boundary layerdetachment), and the local heat-transfer coefficient. Using theexperimental results available in the literature,16 they ap-proximated the velocity distribution outside the boundary layerby the modified Hiemenz polynomial, as a function of the wallblockage. Recently, Khan et al.15 have presented a functionaldependence of the drag and the Nusselt number (Nu) on theReynolds number and blockage ratio using the modified vonKarman-Pohlhausen method, which uses the fourth-order veloc-ity profile inside the boundary layer. Outside the boundary layer,they obtained the potential flow velocity using the method ofimages. They assumed a third-order temperature profile to solvethe corresponding thermal energy equation for the two com-monly used thermal boundary conditions.

Thus, in summary, only very limited information is availableon the role of confining walls, even for the flow of Newtonianfluids past a circular cylinder. On the other hand, reliable resultsare now available on the power-law fluid flow and the associatedconvection past an unconfined circular cylinder (e.g., see refs17-29). Overall, the currently available numerical results goup toRe) 40 and power-law indexes in the range of 0.2e ne 2. The results obtained using different numerics and domains,etc., show excellent correspondence, thereby confirming theirreliability and accuracy.

However, to the best of our knowledge, there has been noprior study elucidating the role of confining (planar) walls on

Figure 1. Schematic representation of a channel confined (Poiseuille) flowover a circular cylinder.

CD )FD(λ)

µUavg) 6π

f(λ) + g(λ)(2)

f(λ) ) A0 - (1 + 0.5λ2 + A4λ4 + A6λ

6 + A8λ8) ln(λ)

g(λ) ) B2λ2 + B4λ

4 + B6λ6 + B8λ

8

A0 ) -0.9156892732

A4 ) 0.05464866

A6 ) -0.26462967

A8 ) 0.792986

B2 ) 1.26653975

B4 ) -0.91804

B6 ) 1.87710

B8 ) -4.66549

Ind. Eng. Chem. Res., Vol. 46, No. 11, 20073821

the flow of power-law fluids past a circular cylinder. The presentwork is concerned with the investigation of the two-dimensional,incompressible, Poiseuille flow of power-law fluids past acircular cylinder confined in a channel over wide ranges ofReynolds number (1e Ree 40), power-law index (0.2e n e1.9), and blockage ratio (â ) 4, 2, 1.6, and 1.1).

3. Problem Statement and Governing Equations

The channel confined flow is simulated here by consideringthe two-dimensional steady, incompressible, Poiseuille flow(fully developed velocity profile) of power-law fluids over aninfinitely long circular cylinder (of diameterD) placed sym-metrically between two parallel plane walls (blockage ratio,â

) H/D), as shown schematically in Figure 1. The length andheight of the computational domain are defined in terms of theaxial and lateral dimensions (L and H, respectively). Thecylinder is placed at a distanceLu (upstream length), from theinlet to the center of the cylinder and at a distanceLd

(downstream length), from the center of the cylinder to theoutflow boundary. (Note:L ) Lu + Ld.)

The fully developed velocity profile for the laminar flow ofpower-law fluids flow in a channel (of heightH) is given as

whereUmax is the maximum velocity, which is related to thearea-averaged velocity (Uavg) as

The governing equations for this flow problem are given asfollows:

whereF is the density,u the velocity (Ux andUy componentsin Cartesian coordinates),f the body force, andσ the stresstensor. The stress tensorσ, which is the sum of the isotropicpressure (p) and the deviatoric stress tensor (τ) is given by

The rheological equation of state for incompressible fluidsis 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,n the power-lawindex of the fluid (n < 1 corresponds to a shear-thinning fluid;n ) 1 corresponds to a Newtonian fluid; andn > 1 correspondsto a shear-thickening fluid), andI2 the second invariant of therate of strain tensor (ε), which is given by

Figure 2. Schematics representation of a two-dimensional nonuniformcomputational grid (a) in the full domain and (b) in the region close to thecylinder.

Table 1. Effect of Downstream Length (Ld) on the DragCoefficientsa

n ) 0.2 n ) 1 n ) 1.9

Ld/D CDP CDF CD CDP CDF CD CDP CDF CD

Re) 1, â ) 10040 0.19 -1.48 -0.31 -0.05 -0.05 -0.05 -0.08 -0.09 -0.0960 -1.35 1.71 -0.44 -0.04 -0.03 -0.03 -0.07 -0.08 -0.0780 -0.21 0.32 -0.06 -0.03 -0.03 -0.03 -0.07 -0.08 -0.08

Re) 40,â ) 10040 -0.05 -0.07 -0.05 0.03 0.01 0.03-0.10 -0.10 -0.1060 0.04 0.01 0.03-0.10 -0.09 -0.1080 -0.21 -0.04 -0.19 0.04 0.01 0.03-0.10 -0.09 -0.09

Re) 1, â ) 1.140 -0.01 -0.01 -0.01 <0.01 <0.01 <0.01 <0.01 <0.01 <0.0160 -0.04 -0.05 -0.04 <0.01 <0.01 <0.01 <0.01 <0.01 <0.0180 -0.10 -0.07 -0.10 <0.01 <0.01 <0.01 <0.01 <0.01 <0.01

Re) 40,â ) 1.140 <0.01 <0.01 <0.01 <0.01 <0.01 <0.01 <0.01 <0.01 <0.0160 <0.01 <0.01 <0.01 <0.01 <0.01 <0.01 <0.01 <0.01 <0.0180 <0.01 -0.02 <0.01 <0.01 -0.03 <0.01 <0.01 <0.01 <0.01

a Values shown represent the percent relative change in the dragcoefficient values, with respect to the drag values atLd ) 20D.

Table 2. Grid Independence Studya

n ) 0.2 n ) 1 n ) 1.9

grid CDP CDF CD CDP CDF CD CDP CDF CD

Re) 1G2 0.07 -2.10 -0.07 0.32 0.28 0.32 0.59 0.83 0.60G3 -0.20 -3.63 -0.42 0.43 0.33 0.43 0.79 1.12 0.81

Re) 40,â ) 100G2 0.05 -1.50 -0.05 0.24 0.40 0.25 0.59 0.83 0.60G3 -0.15 -2.80 -0.35 0.33 0.50 0.34 0.79 1.12 0.81

a Values shown represent the percent relative change in the dragcoefficient values, with respect to the drag values at Grid G1.

Table 3. Comparison of the Present Total Drag Values with theAvailable Literature Values for a Channel Confined Flow Across aCircular Cylinder

Present Total Drag Value

source â ) 2 â ) 4 â ) 10

Re) 0.01present work 132.26 31.933 13.358Faxen5,a 138.48 31.943 13.359Ben Richou et al.13,b 133.23 13.745

Re) 1present work 17.105Ben Richou et al.13,b 17.336

a Analytical approximation.b Numerical value.

Up(y,n) ) Umax[1 - (|1 - 2yH |)(n+1)/n] (for 0 e y e H) (3)

Umax ) (2n + 1n + 1 )Uavg (4)

Continuity equation: ∇‚u ) 0 (5a)

Momentum equation: F(u‚∇u - f) - ∇‚σ ) 0 (5b)

σ ) -pI + τ (6)

τ ) 2ηε(u) (7)

ε(u) )(∇u) + (∇u)T

2(8)

η ) m(I2

2)(n-1)/2

(9)

I2 ) 2(εxx2 + εyy

2 + εxy2 + εyx

2) (10)

3822 Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007

and the components of the rate of strain tensor are related tothe velocity components in Cartesian coordinates as follows:

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

(1) At the inlet (x) 0) boundary: The flow is assumed tobe fully developed. The following conditions are imposed atthe inlet:

(2) At the bottom (y) 0), top (y) H) boundaries and onthe surface of the cylinder: The standard no-slip condition isused, i.e.,

(3) At the exit (x) L) boundary: The default outflowboundary condition option in FLUENT (a zero-diffusion fluxfor all flow variables) was used in this work. Physically, thischoice implies that the conditions of the outflow plane areextrapolated from within the domain and, as such, have anegligible impact on the upstream flow conditions. The ex-trapolation procedure used by FLUENT updates the outflowvelocity and pressure in a manner that is consistent with the

fully developed flow assumption, when there is no areachange at the outflow boundary. It is important to note herethat gradients in the cross-stream direction may still existat the outflow boundary. Also, the use of this condition obviatesthe need to prescribe any boundary condition for pressure. Thisis similar to the homogeneous Neumann condition, given by

The computations have been performed in the full compu-tational domain (Figure 1), to ensure that the flow is symmetricabout the mid-plane over the range ofRe values used in thiswork. The numerical solution of governing equations (eqs5-11), in conjunction with the previously noted boundaryconditions (eq 12), yields the primitive variables, i.e., thevelocity (Ux andUy) and pressure (p) fields. These, in turn, areused to deduce the local and global momentum characteristics,as outlined below. At this point, it is appropriate to introducesome definitions:

• TheReynolds 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 angle,θ, andp∞ is thefree stream pressure at the exit boundary.

Table 4. Dependence of the Pressure Drag Coefficient (CDP) on the Reynolds Number (Re), the Power-Law Index (n), and the BlockageRatio (â)

Pressure Drag Coefficient,CDP

Re n) 0.2 n ) 0.4 n ) 0.6 n ) 0.8 n ) 1 n ) 1.2 n ) 1.4 n ) 1.6 n ) 1.8 n ) 1.9

Blockage Ratioâ ) 41 15.9185 14.4961 14.4013 15.0159 16.1496 17.7733 19.8533 22.4439 25.5924 27.42355 3.8031 3.5032 3.4276 3.4843 3.6440 3.9104 4.2784 4.7584 5.3570 5.7109

10 2.2765 2.1387 2.1200 2.1551 2.2271 2.3407 2.4970 2.7067 2.9763 3.139620 1.4776 1.4393 1.4539 1.4950 1.5467 1.6114 1.6883 1.7838 1.9014 1.972930 1.2108 1.2151 1.2303 1.2687 1.3142 1.3668 1.4247 1.4920 1.5705 1.616940 1.0986 1.1024 1.1160 1.1510 1.1925 1.2391 1.2885 1.3435 1.4054 1.4412

Blockage Ratioâ ) 21 23.8923 29.6039 39.7198 56.0063 81.6320 122.2101 186.4858 289.1772 453.9137 571.43105 5.7232 6.5696 8.3748 11.4913 16.5257 24.5808 37.3946 57.9046 90.8324 114.3311

10 3.5642 3.9414 4.7270 6.1467 8.5484 12.4941 18.8431 29.0569 45.4917 57.229020 2.5394 2.7238 3.0778 3.7023 4.7700 6.6119 9.6869 14.7225 22.8880 28.736430 2.2556 2.3317 2.5676 2.9698 3.6311 4.7680 6.7252 10.0117 15.4045 19.283540 2.1401 2.1349 2.3128 2.6148 3.1003 3.9064 5.3014 7.7003 11.6957 14.5858

Blockage Ratioâ ) 1.61 33.1534 48.1044 75.7211 125.6794 216.2630 383.0144 692.2424 1272.0347 2366.1372 3242.34475 8.0385 10.4919 15.6550 25.4453 43.4436 76.7223 138.5241 254.4554 473.2601 648.4952

10 5.3138 6.3497 8.5910 13.1905 22.0099 38.5425 69.3775 127.3021 236.6788 324.287620 4.2036 4.5705 5.5320 7.4808 11.5578 19.6157 39.0134 63.7946 118.4338 162.221330 3.8970 3.9977 4.6250 5.8292 8.2943 13.4445 23.5050 42.6813 79.0566 108.230040 3.7889 3.7172 4.1705 5.0667 6.7921 10.4566 17.8636 32.1630 59.3938 81.2558

Blockage Ratioâ ) 1.11 151.4033 393.1404 1170.6764 3725.3817 12336.2970 42062.6970 146236.5200 516268.2800 1842719.2000 3496383.40005 NCa 103.2389 239.7179 746.9738 2467.9545 8412.8099 24247.4200 103253.7200 368543.8800 699276.7100

10 NCa 81.5961 132.3580 376.5562 1235.0564 4206.8156 14623.8780 51626.9420 184271.9800 349638.400020 NCa NCa 93.5961 195.5396 619.7177 2104.2196 7312.2672 25813.6190 92136.0690 174819.270030 NCa NCa NCa 141.4003 415.6870 1403.7127 4875.2026 17209.2380 61424.1280 116546.240040 NCa NCa NCa 118.1024 314.6362 1053.7346 3656.7735 12907.0910 46068.1740 87409.7330

a Solution not converged.

εxx )∂Ux

∂x(11a)

εyy )∂Uy

∂y(11b)

εxy ) εyx ) 12(∂Ux

∂y+

∂Uy

∂x ) (11c)

Ux ) Up(y,n) and Uy ) 0 (12a)

Ux ) 0 and Uy ) 0 (12b)

∂Ux

∂x) 0 and

∂Uy

∂x) 0 (12c)

Re)FDnUavg

2-n

m(13)

Cp ) static pressuredynamic pressure

)p(θ) - p∞

(1/2)FUmax2

(14)


• The total drag coefficient(CD), which is the sum of thefriction and pressure components, is defined as

whereFD is the total drag force on the cylinder per unit length.The individual drag coefficients,CDP andCDF, are calculated

using the following definitions:

whereFDP is the pressure component of the total drag forceandS is the surface area, and

whereFDF is the frictional component of the total drag forceandRe* is the Reynolds number (based onUmax), which is givenas

The unit vector,ns, normal to the surface of the cylinder, isgiven 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 theaforementioned equations, (D/2) and Umax are used as thecharacteristic length and velocity, respectively.

4. Numerical Solution Procedure

This numerical study has been performed using FLUENT(Version 6.2). The unstructured “quadrilateral” cells of non-uniform grid spacing were generated using the commercial gridgenerator GAMBIT. The two-dimensional (2-D), steady, lami-nar, segregated solver module of FLUENT was used to solvethe incompressible flow on the collocated grid arrangement. TheQUICK scheme 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. The constant densityand “non-Newtonian power-law” viscosity models were used.The fully developed (Poiseuille) velocity profile has beenincorporated in the FLUENT via user-defined functions (UDFs).

Figure 3. Effect of the Reynolds number (Re), the power-law index (n), and the blockage ratio (â) on the normalized pressure drag coefficient (CDPN ).

CD )FD

12FUmax

2D) CDF + CDP (15)

CDP )FDP

12FUmax

2D) ∫S

Cpnx dS (16)

CDF )FD

(1/2)FUmax2D

) 2n+1

Re* ∫S(τ,ns) dS) 2n+1

Re* ∫S(τxxnx +

τxyny) dS (17)

Re* )FDnUmax

2-n

m) Re( n + 1

2n + 1)2-n(18)

ns )xex + yey

xx2 + y2) nxex + nyey (19)

τij ) η(∂Ui

∂j+

∂Uj

∂i ) ) (I2

2)(n-1)/2(∂Ui

∂j+

∂Uj

∂i ) (20)


FLUENT solves the system of algebraic equations using theGauss-Siedel (G-S) point-by-point iterative method inconjunction with the algebraic multi-grid (AMG) method solver.The use of AMG scheme can greatly reduce the number ofiterations and, thus, CPU time required to obtain a con-verged solution, particularly when the model contains alarge number of control volumes. Relative convergencecriteria of 10-10 for the continuity andx- andy-components ofthe velocity were prescribed. In addition, the values of thedrag coefficients were also monitored and only when thesevalues had stabilized to four significant digits were thesevalues finally accepted. However, for certain combinations ofthe values ofn, Re, andâ, it was not possible to achieve theaforementioned level of convergence. Thus, for instance, forâ) 1.1, the lower the value of the power-law index (n), the lowerthe Re value up to which fully converged results could beobtained. Similar convergence problems for small values of thepower-law index have also been reported in the literature byothers.17,27-31

5. Choice of Numerical Parameters

Needless to say, the reliability and accuracy of the numericalresults is contingent upon a prudent choice of numericalparameters, namely, the values ofLu, Ld, and the grid itself. Asnoted earlier, in a channel confined flow, the flow is boundedby the solid walls in the lateral direction and, therefore, theboundary layers also grow on these confining walls. The

confining walls may act as a stabilizing or destabilizing factor,depending on the extent of the wall effects (â). Also, the flowbehavior does not seem to be monotonic, with respect to thewall confinement, even for Newtonian fluids.9-13

5.1. Computational Grid Details. Because of the no-slipcondition prescribed both on the channel walls and on thesurface of the cylinder, steep velocity gradients will occur inthese regions. Therefore, one requires fine grids near theseboundaries to capture the flow field adequately.

The schematics of the 2-D, nonuniform unstructured com-putational grid structure used here are shown in Figure 2 inthe full domain (Figure 2a) and in the region close to the cylinder(Figure 2b) for a blockage ratio of 1.6. The computations havebeen performed on the full domain for four values of theblockage ratio (â ) 4, 2, 1.6, and 1.1), using a similar gridstructure. The unstructured “quadrilateral” grid has been gener-ated by dividing the computational domain into subdomains asfollows:

• In the first subdomain (of diameter 5D upstream/downstream) close to the cylinder, a circular grid of fine uniformspacing (δc) is used.

• Along the length of the channel, a fine grid (spacingδc) isstretched to a coarser grid in the both upstream (successive ratio(SR) of 1.035) and downstream (SR) 1.015) sides and thecoarse grid of uniform size (δe) in the outer regions, i.e., in theregion close to the inlet and outlet boundaries).

• In the lateral direction (i.e., along they-direction), theuniform grid (with a spacing ofδc ) 0.011) has been generated

Figure 4. Dependence of the normalized pressure drag coefficient (CDP,N) on the Reynolds number (Re), the power-law index (n), and the blockageratio (â).


in the central region covering the height of the circular cylinder.This grid has been stretched (SR) 0.95) to the fine grid (δw)in the outer region with the fine grid of uniform spacing nearthe boundary walls.

• A uniform grid distribution (grid points,Nc ) 400) on thesurface of the cylinder was used.

The following grid specifications have been used to studythe effects of domain and grid sizes on the flow parameters.

Figure 5. Influence of the Reynolds number (Re), the power-law index (n), and the blockage ratio (â) on the normalized friction drag coefficient (CDFN ).

Table 5. Dependence of the Friction Drag Coefficient (CDF) on the Reynolds Number (Re), the Power-Law Index (n), and the Blockage Ratio(â)

Friction Drag Coefficient,CDF

Re n ) 0.2 n ) 0.4 n ) 0.6 n ) 0.8 n ) 1 n ) 1.2 n ) 1.4 n ) 1.6 n ) 1.8 n ) 1.9

Blockage Ratioâ ) 41 5.9635 7.2444 8.7048 10.3914 12.3861 14.8176 17.7772 21.4069 25.8608 28.45295 1.1771 1.4833 1.8239 2.2014 2.6260 3.1238 3.7153 4.4341 5.3115 5.8275

10 0.5595 0.7517 0.9657 1.1967 1.4462 1.7232 2.0356 2.4006 2.8358 3.089320 0.2472 0.3703 0.5170 0.6754 0.8437 1.0236 1.2159 1.4248 1.6590 1.789130 0.1466 0.2450 0.3608 0.4892 0.6271 0.7745 0.9299 1.0947 1.2717 1.367540 0.1037 0.1842 0.2806 0.3908 0.5109 0.6402 0.7764 0.9198 1.0711 1.1515

Blockage Ratioâ ) 21 7.0022 10.2560 15.3153 23.3029 35.9811 56.3685 89.2216 142.5608 229.3903 291.91595 1.3247 2.0187 3.0643 4.6798 7.2214 11.2973 17.8656 28.5294 45.8912 58.3904

10 0.6107 0.9880 1.5371 2.3620 3.6421 5.6826 8.9634 14.2904 22.9661 29.214620 0.2652 0.4870 0.7882 1.2210 1.8711 2.8947 4.5333 7.1909 11.5209 14.641130 0.1634 0.3280 0.5455 0.8511 1.2947 1.9793 3.0699 4.8377 7.7183 9.795140 0.1154 0.2483 0.4241 0.6674 1.0118 1.5297 2.3463 3.6686 5.8244 7.3792

Blockage Ratioâ ) 1.61 7.6095 12.5834 21.2777 37.0263 65.8331 119.3332 219.2990 407.8174 765.1082 1051.91605 1.4026 2.4477 4.2335 7.4073 13.1774 23.8788 43.8704 81.5719 153.0276 210.3888

10 0.6108 1.1699 2.0918 3.7035 6.6020 11.9564 21.9508 40.7986 76.5233 105.202520 0.2750 0.5810 1.0513 1.8613 3.3218 6.0059 12.3093 20.4227 32.2800 52.616930 0.1730 0.3936 0.7266 1.2697 2.2388 4.0299 7.3617 13.6386 25.5389 35.094640 0.1240 0.2998 0.5656 0.9845 1.7080 3.0478 5.5457 10.2512 19.1727 26.3374

Blockage Ratioâ ) 1.11 10.3509 28.1214 79.8897 242.3704 772.2247 2554.8483 8674.8112 30055.7840 105666.7000 199194.57005 NCa 4.4692 15.6655 48.3956 154.4273 510.9678 1734.9635 6011.1585 21133.3420 39838.9080

10 NCa 2.2437 7.2192 24.0697 77.1856 255.4810 867.4843 3005.5823 10566.6730 19919.457020 NCa NCa 3.5940 11.7501 38.5339 127.7337 433.7470 1502.7970 5283.3412 9959.733130 NCa NCa NCa 7.6690 25.6222 85.1468 289.1693 1001.8710 3522.2329 6639.826740 NCa NCa NCa 5.7585 19.1483 63.8499 216.8810 751.4095 2641.6795 4979.8740

a Solution not converged.


The grid spacings areδw ) 0.01, δe ) 0.0303 (in thedownstream), andδe ) 0.0323 (in the upstream), respectively.

The aforementioned three grids have been used here todemonstrate that the results are grid-independent.

5.2. Domain Independence Study.Using the nonuniformgrid (G2), the domain independence study was performed tochoose the appropriate values of the upstream length (Lu) anddownstream length (Ld) for the range of blockage ratios (â)covered in this work. These dimensions (Lu andLd) are variedsimultaneously to find a domain size that is large enough toobtain the results free from the domain effects and yet is notprohibitively demanding, in terms of the computing require-ments.

5.2.1. Effect of Downstream Length (Ld). The effect ofdownstream length (Ld) on the drag coefficient values has beenstudied by keeping the upstream length,Lu ) 10D, at theextreme values of the Reynolds numbers (Re) 1 and 40) andfor three values of the power-law index (n ) 0.2, 1, and 1.9).The length (Ld/D) was varied as 20, 40, 60, and 80, respectively.The effect of downstream length has been analyzed at two values

of the blockage ratio covering the maximum possible confine-ment (â ) 1.1) and an almost-unconfined flow (â ) 100). Table1 shows the effect ofLd/D on the values of the individual andtotal drag coefficients (CDP, CDF, CDP). The values reported inthis table denote the percentage relative change in the valuesof the drag coefficients, with respect to the downstream lengthof Ld ) 20D. The increase in the value of (Ld/D) above 40 showsa very small change in the results. However, the computationaltime (using a computer processing unit (CPU), Intel PentiumIV, 3.2 GHz, 1 GB RAM) required to simulate the flow increasesby more than 2-fold with an increase in the downstream lengthfrom 40 and 60 and from 60 to 80, respectively. Furthermore,the choice ofLd/D ) 40 is also consistent with that used in theliterature.32 Bearing in mind all these factors, the downstreamlengthLd/D ) 40 is believed to be adequate to obtain the resultsthat are essentially free from the downstream (or outflow)boundary effects.

5.2.2. Effect of Upstream Length (Lu). Having fixed thedownstream length (Ld/D ) 40), the corresponding effect ofupstream length (Lu) on the drag coefficients has been studiedfor two values ofLu/D, namely, 10 and 15 at a blockage ratioof 1.1. It was found that an increase in the upstream lengthfrom 10 to 15 had an insignificant effect (change in the valuesof the order of<0.01%) on the drag values. Thus, keeping inmind the significant increase in CPU time, the upstream lengthLu/D ) 10 is believed to be adequate to obtain the resultspresented in this study.

5.3. Grid Independence Study.Having fixed the size ofdomain (Lu/D ) 10 andLd/D ) 40), the grid independence study

Figure 6. Dependence of the normalized friction drag coefficient (CDF,N) on the Reynolds number (Re), the power-law index (n), and the blockage ratio (â).

Grid 1 (G1): δc ) 160

Grid 2 (G2): δc ) 1100

Grid 3 (G3): δc ) 1160


has been performed using the three nonuniform unstructuredgrids (G1, G2, and G3) for the two extreme values of theReynolds number (Re) 1 and 40) and for three values of thepower-law index (n ) 0.2, 1, and 1.9) at a blockage ratio ofâ) 1.1. Grids G1, G2, and G3 forâ ) 1.1 have 27 960, 54 600,and 106 560 quadrilateral cells, respectively. Table 2 shows therelative change (with respect to grid G1) in the drag coefficientvalues. Admittedly, the friction drag coefficient shows a changeof ∼1%-2%, with respect to grid G2 in an extremely shear-thinning fluid; however, the total drag coefficient varies by∼0.5%. This is not at all surprising, given the fact that thefriction drag coefficient contributes very little to the total dragcoefficient in this case. Thus, grid G2 seems to be sufficientlyrefined to obtain results that are free from grid effects. Also,the values ofδc ) 1/100 was reported to be adequate to capturethe boundary layer phenomenon for unconfined flow of power-law fluids across elliptical and circular cylinders.17,19,20

Thus, keeping in mind a marginal change in the results atthe expense of a rather significant increase in the computingtime, all results reported in this work are based on the followingchoices of the numerical parameters:Lu/D ) 10 andLd/D )40 and grid G2 (consisting of 183 940, 94 740, 76 900, and54 600 quadrilateral cells forâ ) 4, 2, 1.6, and 1.1, respec-tively). To add further weight to our claim for the accuracy ofthe results, the numerical results obtained herein have beencompared with the literature values in the next section.

6. Results and Discussion

In this work, steady flow computations have been performedusing the full domain for the following range of conditions:

Reynolds number,Re ) 1, 5, 10, 20, 30, and 40; the power-law index,Re) 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, and 1.9thereby covering both shear-thinning (n < 1) and shear-thickening (n > 1) fluid behaviors and for four values of theblockage ratio,â ) 4, 2, 1.6, and 1.1. Because extensive resultsfor a power-law flow past an unconfined circular cylinder havebeen reported recently17,20over wide ranges of conditions (0.01e Ree 40, and 0.2e n e 2), these are not duplicated here.However, prior to presenting the new results, it is appropriateto validate the solution procedure, to ascertain the accuracy andreliability of the results presented herein for the confined flowcase.

6.1. Validation of Results.Because an extensive validationof the FLUENT solver for the flow of Newtonian and power-law fluids over a circular cylinder has been reported else-where,11,17,19,20,33only additional comparisons for the confinedflow of Newtonian fluids are reported herein. Table 3 shows acomparison between the present results for three values of theblockage ratio (â ) 10, 4, and 2) with the literature values. Anexcellent correspondence can be seen between the present workand the literature results. This inspires confidence in the choiceof grid and numerical parameters used in this work.

Bearing in mind all these factors, together with our previousexperience, the present results are believed to be reliable towithin (2%-3%.

6.2. Drag Phenomena.Dimensional considerations suggestthe individual and total drag coefficients to be functions ofthe Reynolds number (Re), flow behavior index (n), andthe blockage ratio (â). The ensuing sections explore thisrelationship.

Figure 7. Influence of the Reynolds number (Re), the power-law index (n), and the blockage ratio (â) on the total drag coefficient (CD).


6.2.1. Pressure Drag Coefficient (CDP). The dependence ofthe pressure drag coefficient (CDP) on the Reynolds numberRe,power-law index, and blockage ratio (â) is shown in Table 4.For a fixed Re value, the pressure drag coefficient (CDP)increases as the fluid behavior changes from Newtonian (n )1) to shear-thickening (n > 1) for all values of the blockageratio (â); however, an opposite dependence is seen when thefluid behavior changes from Newtonian to shear-thinning (n <1). For a fixed value of the flow behavior index (n), the valueof the pressure drag coefficient increases as theRe valuedecreases. Furthermore, the influence of the power-law indexis more prominent at lowRevalues, which gradually diminisheswith increasingRe value. With the decreasing value of theblockage ratio (â) (i.e., an increase in the wall confinement),the CDP value increases as theRevalue increases. At smallâvalues, the value of the power-law index (n) shows a stronginfluence on theCDP value. The effect ofn on CDP also isreduced asâ increases. For an unconfined flow, the velocitygradients progressively diminish away from the surface of thecylinder. On the other hand, the velocity gradients initiallydecrease and increase again in the annular region for the flowover a confined cylinder. For a Newtonian liquid, this resultsin additional dissipation and, therefore, the drag on the obstacle(sphere/cylinder) increases, in comparison to that for anunconfined cylinder under otherwise identical conditions (i.e.,Re). However, the increase in drag for a circular cylinder isknown to be non-monotonic, even in Newtonian fluids.1,2 Thisbehavior gets accentuated in power-law fluids, because of theirshear-dependent viscosity. Broadly, the inertial forces scale asUref

2, whereas viscous forces scale asUrefn (at least in the low-

Re regime). Evidently, the pressure forces will display anintermediate scaling on velocity, depending on theRevalues,i.e., whether the flow is inertia-dominated or viscosity-

dominated. Therefore, the minimum inCDP observed atRe) 1and forâ ) 4 for shear-thinning fluids (n < 1) is probably amanifestation of the two competing mechanics, namely, thereduction of the effective viscosity in the annular region (forn< 1) and the additional dissipation due to the sharpening of thevelocity gradients at the wall. In other words, atâ ) 4, theadditional dissipation due to walls is a minor contribution. Onthe other hand, for smallerâ values, the increase in the dragdue to the walls outweighs the reduction in drag due to shear-thinning behavior. Similarly, for shear-thickening fluids (n >1) and for all values ofâ, both phenomena (enhanced shearingand wall effects) go hand in hand andCDP shows a monotonicdependence onn, Re, andâ.

To elucidate the role of wall blockage (â) on CDP in anunambiguous manner, theCDP value has been normalized usingthe corresponding unconfined value (CDP

N ) CDP,â/CDP,∞) at thesameReandn values. Figure 3 shows the influence ofRe, n,andâ on the normalized pressure drag coefficient (CDP

N ). Thenormalized values are larger than unity over the range ofconditions, except forn < 0.4 andâ ) 4. Irrespective of theâandn values, theCDP value shows a very strong wall effect inshear-thickening fluids. At large values of the blockage ratio,the normalized pressure drag coefficient (CDP

N ) shows a verystrong effect ofRein shear-thickening fluids, especially at lowRevalues, which diminishes with increasingRe. On the otherhand, while the normalized drag coefficient (CDP

N ) values showa dependence of the power-law indexn in shear-thickening typesof fluids (n > 1), which is qualitatively similar for theRerange,an opposite and quite interesting dependence onReis observedin shear-thinning fluids (n < 1). The normalized values (CDP

N )decrease as theRevalue increases for the range of power-lawindex of 0.6e n e1, and these increase as theRevalues increase

Figure 8. Influence of the Reynolds number (Re), the power-law index (n), and the blockage ratio (â) on the normalized total drag coefficient (CDN).


for n e 0.4. A crossover in the normalized values can beclearly be observed atn ) 0.4 (Figure 3) with the decreasingvalue of the power-law index (n). This flip-flop behaviorstems from two possible effects: first, it clearly indicatesthat the role of wall effects is different for power-law andNewtonian liquids otherwise under identical conditions (âand Re). Second, all else being equal, as the velocity (orvelocity gradients) is progressively increased, the effectiveviscosity of a shear-thinning fluid (n < 1) progressivelydecreases asn decreases, whereas no such mechanics existsin Newtonian fluids. The decrease in viscosity is particularlysteep forn e ∼0.4 and, hence, the switchover occurs near thispoint.

Similarly, to delineate the role of power-law rheology onCDP

in an unambiguous manner, theCDP value in power-law fluidshas been normalized using the corresponding Newtonian value(CDP,N ) CDP,n/CDP,n)1) at the sameReandâ values. Figure 4shows the influence ofRe, n, andâ on the normalized pressuredrag coefficient (CDP,N). It is observed that the normalized values(CDP,N) increase as the power-law index (n) increases andCDP,N

> 1 in shear-thickening (1< n e 1.9) fluids for allReandâvalues. At â ) 4, the normalized pressure drag coefficient(CDP,N) with the power-law index (n < 0.6) shows a verydifferent behavior and, once again, this is due to the twocompeting mechanisms mentioned in the preceding sections..This is because the effects of the wall blockage (â) diminish astheâ values increase. It has been shown elsewhere17,19that, foran unconfined cylinderCDP,N > 1 in shear-thinning (n < 1)fluids andCDP,N < 1 for shear-thickening (n > 1) fluids forsmall Revalues. Therefore, as the value ofâ is progressively

increased, one would expect to approach the unconfined flowresults. However, the dependence of drag coefficients onâ ishighly nonlinear and such an extrapolation is fraught withuncertainty. Furthermore, theRevalue has a strong influenceat high â values. Forâ < 1.6, as the values ofCDP,N mergeinto a single line, thereby showing a negligible influence ofReon theCDP,N for shear-thickening (n > 1) fluids. Also, the rateof change of the normalized pressure drag coefficient (CDP,N)with the power-law indexn increases asâ increases. On theother hand, shear-thinning (n < 1) fluids show the opposite typeof dependence. The values ofCDP,N decreases and are observedto be<1 for all values ofReandâ and for power-law index inthe range of 0.4e n e 1. For n < 0.6, the value of thenormalized pressure drag coefficient increases asn decreasesat the largest blockage ratio (â ) 4). The complexity of thedependence ofCDP,N onReandn increases asâ increases. Thus,the CDP value is determined by a complex interplay betweenthe values ofRe, n, andâ.

6.2.2. Friction Drag Coefficient (CDF). Table 5 shows theinfluence of the Reynolds number (Re), power-law index (n),and the blockage ratio (â) on the friction drag coefficient (CDF).For fixed values ofRe and â, the value of the viscous dragcoefficient (CDF) increases asn increases in both shear-thinning(n < 1) and shear-thickening (n > 1) fluids for all values ofâ.At high â values, the variation of the viscous drag coefficient(CDF) with n was determined to be highly nonlinear at highRevalues on the semilogarithmic scale, which diminishes as theRevalue decreases. The degree of nonlinearity also increaseswith decreasingâ, even at highRevalues. For fixed values ofthe flow behavior index (n) and the blockage ratio (â), the

Figure 9. Dependence of the normalized total drag coefficient (CD,N) on the Reynolds number (Re), the power-law index (n), and the blockage ratio (â).


friction drag coefficient decreases with increasingRevalue. Forfixed values ofReandn, the value ofCDF increases as the walleffects increase (i.e., a decrease in the value ofâ). The influenceof the power-law rheology is observed to be more prominentat low â values. These trends are clearly due to the nonlinearnature of the viscosity over the surface of the cylinder17 and/oron the confining walls.

The normalized friction drag coefficientCDFN (which is

defined asCDFN ) CDF,â/CDF,∞) is shown in Figure 5. Qualita-

tively similar wall effects on the friction drag coefficient (CDFN )

are observed here, relative to that observed in Figure 3 for thepressure drag coefficient (CDP

N ). However, theCDFN values are

observed to be almost an order of magnitude smaller than thoseof CDP

N , which clearly show that the pressure forces dominatethis flow.

Finally, the normalized friction drag coefficient (CDF,N )CDF,n/CDF,n)1) is shown in Figure 6. It can clearly be seen thatCDF,N > 1 for shear-thickening (n > 1) fluids and opposite trendis observed in shear-thinning (n < 1) fluids. For n > 1, thenormalized value (CDF,N) increases with increasingn and withdecreasingâ for all values ofReandâ. A decrease in the valueof â shows a rather negligible influence ofReonCDF,N in shear-thickening (n > 1) fluids, under otherwise identical conditions.The increasing value ofRe reduces the normalized values inshear-thinning (n < 1) fluids. An increase in the value ofâalso modulates the effect ofRe. The streamline profiles (seeFigures 11-13 presented later in this paper) clearly show thereduced propensity for the flow separation with decreasingReand increasingn (i.e., change in the fluid behavior from shear-thinning to shear-thickening). At sufficiently small value of the

blockage ratio (â < 1.6), as the fluid behavior changes fromshear-thinning (n < 1) to shear-thickening (n > 1), no flowseparation was observed at highn values, even atRe) 40 (e.g.,see Figure 13, presented later in this work) for the range ofRevalues. Because of this fact, although there is a significantchange in the drag values (see Tables 4 and 5), the effects ofnon-Newtonianness on the drag values shown in Figures 4 and6 are quite similar. It can be concluded from Figures 5 and 6that the power-law rheology, in conjunction with the wall effects(â), has a stronger influence on the friction component of thedrag coefficient than that in unconfined (â ) ∞) flow over therange of conditions covered herein. This is obviously due tothe additional friction at the confining walls, which alters theflow field near the cylinder. A linear relationship between thefriction drag coefficient and power-law index is observed onthe semilogarithmic scale, which changes to nonlinear with anincrease in the blockage ratio (â) and a decrease in the power-law index (n) at high values ofâ. Note that the viscosity of ashear-thinning (n < 1) fluid becomes very large as the shearrate decreases and, hence, it approaches infinity as the shearrate approaches small values (zero) far away from the cylinderin the axial direction, whereas the effective viscosity of the fluidwill be minimum near the cylinder and the wall in the lateraldirection, as the velocity gradients are maximum in theseregions. On the other hand, even if one uses the other fluidmodels (such as Ellis or Carreau), the value of the viscositywill approach the so-called zero-shear viscosity far away fromthe cylinder. Of course, the shear-thickening fluid will exhibitthe maximum viscosity near the cylinder and the wall.

Figure 10. Effect of the Reynolds number (Re), the power-law index (n), and the blockage ratio (â) on the drag ratio (CDP/CDF).


Finally, there is a sort of consistency in the trends seen inFigures 4 and 6. For shear-thickening fluids (n > 1), both inertialand viscous forces scale asUref

2 andUref1-1.9, and, in the extreme

case ofn ) 1.9, both scale almost identically, thereby diminish-

ing the role of the power-law index (n). The decrease in thepressure and friction components with the increasing Reynoldsnumber (Re) is balanced by the increase due to the additionalresistance of walls. This is clearly observed in Figures 4d and

Figure 11. Streamline profiles atâ ) 4: (a) Re) 1, (b) Re) 10, (c)Re) 20, and (d)Re) 40.


6d. For shear-thinning fluids (n < 1), there are opposingmechanisms that result in varying levels of reduction or enhance-ment in drag, with reference to the corresponding Newtonianvalue, depending on the values ofn, Re, andâ.

6.2.3. Total Drag Coefficient (CD). The total drag coefficient,CD (which is defined asCD ) CDP + CDF) is plotted in Figure7 for a range of values ofRe, n, and â studied herein. Herealso, the dependence is observed to be similar to that of the

Figure 12. Streamline profiles atâ ) 2: (a) Re) 1, (b) Re) 10, (c)Re) 20, and (d)Re) 40.


pressure drag coefficient (CDP) and the friction drag coefficient(CDF) shown in Figures 3 and 5, respectively.

The normalized total drag coefficient (CD) using the corre-sponding unconfined value (CD

N ) CD,â/CD,∞) at the sameRe

and n values is shown in Figure 8 to highlight the role ofblockage otherwise under identical conditions. The wall effectson the total drag coefficient (CD

N) are qualitatively similar tothat of the pressure and friction drag coefficients (CDP

N and

Figure 13. Streamline profiles atâ ) 1.6: (a)Re) 1, (b) Re) 10, (c)Re) 20, and (d)Re) 40.


CDFN , respectively), observed in Figures 3 and 5, respectively.

In all cases, the normalized drag coefficients are greater thanunity, except for atâ ) 4 andn e ∼0.4. This behavior, again,is believed to be the flow being almost, but not quite,unconfined, as well as being in the limits of the end of thecreeping flow and, hence, due to different scaling with respectto the velocity, as mentioned previously.

Figure 9 depicts the variation of normalized total dragcoefficient, i.e., total drag coefficient normalized using thecorresponding Newtonian drag value (CD,N ) CD,n/CD,n)1) underotherwise identical conditions. Once again, the dependence isobserved to be similar to that of normalized pressure and frictioncoefficients (CDP,N and CDF,N) shown in Figures 4 and 6,respectively.

6.2.4. Drag Ratio,CDP/CDF. Finally, to elucidate the role ofthe power-law rheology and wall effects on the relativecontributions of the individual drag components, Figure 10shows the variation of the drag ratio (CDP/CDF) with the flowbehavior index (n), the Reynolds number (Re), and the blockageratio (â). The ratio is observed to be (CDP/CDF) > 1 for therange of conditions covered herein, except forâ ) 4 andn e1.8 atRe) 40, which suggests that this flow is dominated by

the pressure forces rather than the viscous forces. For shear-thickening (n > 1) fluid behavior, the drag ratio is seen to beindependent ofReat lowerâ values. The decreasing blockageratio shows a stronger dependence of the drag ratio in shear-thinning (n < 1) fluids, whereas only a weak dependence canbe observed in shear-thickening (n > 1) fluids. As expected,the wall effects become less prominent with an increase inâvalue, eventually approaching the unconfined flow limit. Thus,the drag ratio (CDP/CDF) < 1 for â ) 4 andn g 1.8 atRe) 40,indicates the switch over in the importance of these forces, whichis similar to that observed in the unconfined flow of power-law fluids across a circular cylinder.17,20

6.3. Detailed Flow Characteristics.Evidently, the depen-dence of the individual and total drag coefficients observed inthe preceding section should also be reflected in similardifferences in the detailed kinematics of the flow. This aspectis explored in this section. In particular, the streamline profilesand variation of the pressure coefficient over the surface of thecylinder, and their values at the stagnation points are analyzedto gain some further insights into the nature of the flow.

6.3.1. Streamline Patterns.Representative streamline profilesare shown in Figures 11-13 for blockage ratios (â) of 4, 2,

Figure 14. Distribution of pressure coefficient (Cp) over the surface of the cylinder for the range of Reynolds number (Re) and the power-law index (n) ata blockage ratio ofâ ) 4: (a)Re) 1, (b) Re) 10, (c)Re) 20, and (d)Re) 40. The anglesθ ) 0° and 180° correspond to the front and rear stagnationpoints, respectively.


and 1.6, for four values of the Reynolds number (Re) 1, 10,20, and 40) and for five values of the power-law index (n )0.2, 0.6, 1, 1.4, and 1.8), respectively. These figures show theflow to be symmetric for the range of conditions investigatedin this work. However, the spreading of the streamlines closeto the cylinder with a decreasing value of the wall blockage(â) shows the acceleration in the flow (e.g., see Figures 11a-13a). For a fixed value ofâ, as theRe value is graduallyincreased, the size of the wake grows and the point of separation(θs) moves forward on the surface of the cylinder for both shear-thinning (n < 1) and shear-thickening (n > 1) fluids. This effectis observed to be even stronger at low values ofâ (e.g., seeFigure 13a). On the other hand, no flow separation wasobserved, even atRe) 40 for the smallest blockage (â ) 1.6)at n ) 1.8, whereas the largest wake can be observed atn )0.2. For a fixed value ofRe, the wake size grows with decreasingn, i.e., as the fluid behavior changes from shear-thickening (n> 1) to Newtonian (n ) 1), and finally, to shear-thinning (n <1) for the range ofâ values studied herein. However, this trendis contrary to the streamline patterns for an unconfined (â )∞) flow past a circular cylinder.17,20 At high Re values, anincrease in the wake size shows the acceleration of the flowwith the increasing wall effect (i.e., a decrease inâ, in shear-

thinning (n < 1) fluids), the opposite behavior is observed inshear-thickening (n > 1) fluids. For fixed values ofn and â,the size of wake increases with increasingRe. However, asâvaries, its behavior is dependent on the flow behavior index ofthe fluid (n). The increase in the wake size withRe is muchfaster in shear-thinning (n < 1) fluids that that in shear-thickening fluids, especially at low values of the blockage ratio.Thus, in summary, the wake size increases asReincreases and/or â values in shear-thinning (n < 1) fluids decrease. On theother hand, the opposite type of tendency is observed in shear-thickening (n > 1) fluids.

The wall effects, in conjunction with the power-law rheology,clearly influence the flow behavior in a complex manner. Thesetrends are linked to the varying levels of the pressure recovery,depending on the values ofâ, n, andRe. One of the importantissues in the flow past bluff bodies is the recovery of thepressure. The variation of the pressure coefficient (Cp) over thesurface of the cylinder, relative toRe, n, andâ, is discussed inthe next section.

6.3.2. Surface Pressure Coefficient (Cp). Typical variationof the pressure coefficient (Cp) over the surface of the cylinderwith Reynolds number (Re) and the power-law index (n) isshown in Figures 14-16 for blockage ratios (â) of 4, 2, and

Figure 15. Distribution of pressure coefficient (Cp) over the surface of the cylinder for the range of Reynolds number (Re) and the power-law index (n) atâ ) 2: (a) Re) 1, (b) Re) 10, (c)Re) 20, and (d)Re) 40.


1.6, respectively. The value of the pressure coefficient decreasesfrom its maximum value at the front stagnation point (θ ) 0)along the surface of the cylinder toward the rear of the cylinder,followed by an increase due to the recirculation of fluid (if any)in the rear side of the cylinder. A minimum value of the pressurecoefficient (Cp,min) is observed to occur at the rear stagnationpoint (θ ) π) and/or at the point of separation (θ ) θs). Forfixed values ofn andâ, the point ofCp,min shifts forward (fromrear to the front side) with a gradual increase in theRevalue asthe point of separation continues to move forward. However, itis almost independent of the flow behavior index at higherâvalues. These trends are also similar to that observed in thecase of an unconfined circular cylinder.17,20 The wall effectsalso lead to somewhat sluggish pressure recovery, as the pressurecurve in the rear part of the cylinder flattens out with decreasingâ values for all values ofRe and n. At high values of theblockage ratio (â ) 4), the Cp value over the surface of thecylinder is higher for shear-thinning (n < 1) fluids than thatfor Newtonian (n ) 1) and lower for shear-thickening (n > 1)fluids at low values ofRe in the upstream side of the cylinder

(e.g., see Figure 14a), which switches over in the downstreamside, akin to that for the unconfined case. However, at highvalues of the Reynolds number (Re), the variation of pressurecoefficient shows a complex behavior with respect to the power-law index (e.g., see Figures 14b-14d). The value ofCp is alwaysobserved to be higher in shear-thinning fluids than that inNewtonian fluids and lower for shear-thickening fluids over theentire surface of the cylinder at lowâ values (e.g., see Figures15 and 16). An increase in the wall effects and/or a decrease inRe, or both, show a very sharp drop in the pressure over thesurface of the cylinder. This dependence is stronger in shear-thickening (n > 1) fluids than that in Newtonian and shear-thinning (n e 1) fluids. The lowest value of the pressurecoefficient (Cp) can be observed in shear-thickening fluids atsmallest blockage (â ) 1.6). The fact that no separation hasbeen observed at lowReandâ values and shear-thickening (n> 1) fluids is possibly due to the very small pressure values inthe downstream side of the cylinder.

Furthermore, the dependence ofCp at the front (θ ) 0 or2π) and rear (θ ) π) stagnation points and at the midpoints

Figure 16. Distribution of pressure coefficient (Cp) over the surface of the cylinder for the range of Reynolds number (Re) and the power-law index (n) atâ ) 1.6: (a) Re ) 1, (b) Re ) 10, (c) Re ) 20, and (d)Re ) 40. The upper and lower part of the figures represent the shear-thinning (n e 1) andshear-thickening (n g 1) behaviors, respectively.


(top: θ ) π/2; bottom: θ ) 3π/2) on the Reynolds number(Re), the flow behavior index (n) and blockage ratio (â) is shownin Figure 17, plotted in terms of the values ofCpRe. The top,middle, and bottom rows in this figure represent [Cp(0)Re], [Cp-(π/2)Re and Cp(3π/2)Re], and Cp(π)Re, respectively. Similarvalues of the midpoint pressure coefficients, i.e.,Cp(π/2)ReandCp(3π/2)Re, show the flow to be symmetric over the range ofconditions embraced by this study. For fixed values of the

blockage ratio (â) and flow behavior index (n), the value of thefront stagnation pressure coefficient increases as theRevalueincreases. At a low Reynolds number (Re) 1) and a high valueof the blockage ratio (â ) 4), the front stagnation pressurecoefficient increases as the fluid behavior changes from shear-thinning to Newtonian and finally to shear-thickening. Similartrends are also observed with increasingRein shear-thickening(n > 1) fluids; however, shear-thinning (n < 1) fluids show

Figure 17. Dependence of the front stagnation pressure coefficient [Cp(0) or Cp(2π)] (top row), midpoint pressure coefficient [Cp(π/2) andCp(3π/2)](middle row), and rear stagnation pressure coefficient [Cp(τ)] (bottom row) on the Reynolds number (Re), the power-law index (n), and the blockage ratio(â).


the opposite type of dependence. However, the rate of changeof Cp(0)Rewith the power-law index (n) at highRe is small.The front stagnation pressure coefficient always decreases withn and Re at small values of the blockage ratio (â e 2). Forfixed values ofReandn, the increasing value ofâ reduces thevalue ofCp(0). This effect is more prominent at small valuesof â. The negative values ofCp(0) were also observed withdecreasingReandâ. No such negative values were observedfor an unconfined flow over a cylinder;17,20therefore, this effectmust be ascribed to the additional dissipation due to wall effects.

Both midpoint and rear stagnation point pressure coefficients(Cp(π/2) andCp(π)) are negative in magnitude for the range ofconditions studied herein. Qualitatively similar dependences ofthe pressure coefficient at the midpoint and at the rear stagnationpoint onRe, n, andâ are observed in Figure 17. Atâ ) 4, themagnitudes of the both decrease with increasingRein the rangeof 0.2e n e 1.9. The change in the fluid behavior from shear-thinning to Newtonian increases the values ofCp(π/2) andCp-(π), these values decrease with increasingn (for n > 1). Forsmall values of the blockage ratio (â e 2), these values showstronger dependence on the power-law index forn > 1. In shear-thinning (n < 1) fluids, these values are influenced onlymarginally byn. The influence ofRe on Cp(π/2) andCp(π)diminishes with increasingReand/or decreasingâ, or both.

In summary, the detailed kinematics of the flow and the grossengineering parameters are influenced in an intricate mannerby the value ofRe, n, and the wall effects (â). This interplay isfurther accentuated by the fact that, even at lowRe, the viscousterm in the momentum equations is highly nonlinear for power-law fluids. As theRevalue is increased, the flow is governedby the two nonlinear terms, namely, inertial and viscous, whichscale differently with velocity. For instance, as noted earlier,the viscous forces will approximately scale asUref

n, whereasthe inertial forces approximately scale asUref

n. Here,Uref is thecharacteristic scaling velocity of the fluid for the fully developedflow at the inlet. Thus, keeping everything else fixed, thedecreasing value of the power-law index (n) suggests diminish-ing importance of the viscous effects for shear-thinning (n <1) fluids, while the inertial terms will still scale as beingproportional toUref

2. On the other hand, viscous effects are likelyto grow with the increasing value of the power-law indexn fora shear-thickening (n > 1) fluid. For the extreme case ofn ≈2, the viscous term will also approximately scale asUref

2,identical to the inertial term. These nonlinear interactions, inconjunction with the wall effects (due to the modified velocityfield and the sharpening of the gradients in the annular region),exert a strong influence on the overall flow kinematics in thiscase. It is believed that these different types of dependencieson the flow behavior index and velocity are also responsiblefor the non-monotonic behavior of flow kinematics and dragphenomenon. Finally, the highly nonlinear form of the momen-tum equations also poses enormous challenges in choosingappropriate numerics and numerical parameters, even in thesteady-flow regime for such flows.

7. Concluding Remarks

Extensive results on the individual and total drag coefficients,surface pressure coefficients, and their values at stagnation pointsand streamline patterns have been presented to provide physicalinsights into the hydrodynamics of the confined power-law fluidflow over a cylinder over the following ranges of conditions:1 e Ree 40, 0.2e n e 1.9, and four values of the blockageratio (â ) 4, 2, 1.6, and 1.1). Irrespective of theâ value, anenhancement in the drag coefficient was observed with an

increase in the shear-thickening (n > 1) tendency of the fluid,whereas the shear-thinning (n < 1) fluid shows an oppositedependence. This effect is rather strong at smallRe, and itgradually diminishes with increasingRe. The effect of Rediminishes in shear-thickening (n > 1) fluids with increasingwall effects (decrease inâ), whereas the dependence becomesstronger for shear-thinning (n < 1) fluids. The individual dragcoefficients also show similar dependence onRe, n, andâ. Thisflow is dominated by the pressure forces rather than the viscousforces, as indicated by the values of the drag ratio (CDP/CDF >1). This phenomenon is much stronger in shear-thinning fluidsthan that in Newtonian or shear-thickening fluids. The flowpatterns show that the flow is symmetric about the mid-planeof the channel for the range of conditions studied herein, exceptfor n ) 0.2,Reg 5, n ) 0.4,Reg 20, andn ) 0.6,Reg 30at blockage ratio ofâ ) 1.1. The confining walls seem to thedelay the flow separation in shear-thickening fluids, whereasthe opposite trend was observed forn < 1. Thus, depending onthe type of fluid behavior, the confining walls may stabilize ordestabilize the flow. The wall effects result in a somewhatsluggish pressure recovery in the rear of the cylinder. However,the maximum value of the pressure coefficient (Cp) occurred atthe front stagnation point for the range of conditions investigatedherein; the maximum values, however, decrease with decreasingâ. The dependence of the pressure coefficient onn intensifiesin shear-thickening (n < 1) fluids with a decrease inâ.

Nomenclature

CD ) total drag coefficientCD,∞ ) total drag coefficient for an unconfined (â ) ∞) cylinderCD

N ) normalized total drag coefficient;CDN ) CD,â/CD,∞

CD,N ) normalized total drag coefficient;CD,N ) CD,n/CD,n)1

CDF ) frictional component of the drag coefficientCDF,∞ ) friction drag coefficient for an unconfined (â ) ∞)

cylinderCDF

N ) normalized friction drag coefficient;CDFN ) CDF,â/CDF,∞

CDF,N ) normalized friction drag coefficient;CDF,N ) CDF,n/CDF,n)1

CDP ) pressure component of the total drag coefficientCDP,∞ ) pressure drag coefficient for an unconfined (â ) ∞)

cylinderCDP

N ) normalized pressure drag coefficient;CDPN ) CDP,â/

CDP,∞CDP,N ) normalized pressure drag coefficient;CDP,N ) CDP,n/

CDP,n)1

Cp ) pressure coefficientCp(0) ) pressure coefficient at the front stagnation point (θ )

0)Cp(π) ) pressure coefficient at the rear stagnation point (θ )

π)Cp(θ) ) pressure coefficient at an angleθ on the surface of the

cylinderD ) diameter of the cylinder (m)ex, ey ) unit vectorsFD ) drag force per unit length of the cylinder (N/m)FDF ) frictional component of the drag force per unit length of

the cylinder (N/m)FDP ) pressure component of the drag force per unit length of

the cylinder (N/m)H ) height of the computational domain (m)I2 ) second invariant of the rate of the strain tensor (s-2)m ) power-law consistency index (Pa sn)n ) power-law flow behavior index


nx ) x-component of the direction vectorny ) y-component of the direction vectorns ) direction vector normal to the surface of the cylinderp ) pressure (Pa)p(θ) ) surface pressure at angleθ (Pa)Re) Reynolds numberUp ) fully developed velocity of the fluid (m/s)Uavg ) average velocity of the fluid at the inlet (m/s)Umax ) maximum velocity of the fluid at the inlet (m/s)Ux ) x-component of the velocity (m/s)Uy ) y-component of the velocity (m/s)x, y ) streamwise and transverse coordinates (m)

Greek Symbols

â ) blockage ratio;â ) H/Dη ) viscosity (Pa s)θ ) angular displacement from the front stagnation (θ ) 0)

(degrees)λ ) blockage ratio;λ ) D/HF ) density of the fluid (kg/m3)τ ) shear stress (Pa)τxx,τxy ) x- andy-components of the shear stress (Pa)

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ReceiVed for reView January 29, 2007ReVised manuscript receiVed March 21, 2007

AcceptedMarch 27, 2007

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