Two-Dimensional Steady Flow of a Power-Law Fluid Past a Square

5674

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

Two-Dimensional Steady Flow of a Power-Law Fluid Past a Square Cylinder in a Plane Channel: Momentum and Heat-Transfer Characteristics Abhishek K. Gupta,† Atul Sharma,‡ Rajendra P. Chhabra,*,† and Vinayak Eswaran‡ Departments of Chemical Engineering and Mechanical Engineering, Indian Institute of Technology, Kanpur, India 208016

The momentum and forced-convection heat-transfer characteristics for an incompressible and steady flow of power-law liquids past a square cylinder in a plane channel have been analyzed numerically. The momentum and energy equations have been solved using a finite-difference method for a range of rheological and kinematic conditions as follows: power-law index, n ) 0.5-1.4, thereby covering both shear-thinning and shear-thickening behavior; Reynolds number, Re ) 5-40; and Peclet number, Pe ) 5-400. However, all computations have been performed for one blockage ratio b/2h ) 1/8. Furthermore, the role of the type of thermal boundary condition, i.e., the constant heat flux and the isothermal surface, on the rate of heat transfer has also been studied. Overall, when the drag coefficient and Nusselt number are normalized using the corresponding Newtonian values, these ratios show only marginal additional dependences on the flow behavior index. The shear thinning not only reduces the size of the wake region but also delays the wake formation, and shear thickening shows the opposite effect on wake formation, etc. Introduction The flow of fluids past bluff bodies, particularly circular and square cylinders, represents an idealization of many industrially important applications, and therefore, it has received a great deal of attention over the years. Most studies of this phenomenon have been concerned with the flow past a circular cylinder under free-flow unconfined conditions, albeit limited results on the extent of wall effects are also available. The voluminous literature relating to the flow of Newtonian fluids over a circular cylinder has been reviewed recently in two volumes.1,2 In contrast to the voluminous literature on circular cylinders, the analogous case of the flow past a square cylinder has been investigated much less extensively. Studies of such behavior have received impetus from two distinct but interrelated objectives. From a theoretical standpoint, such highly idealized model configurations are convenient for the purpose of elucidating fluid mechanical phenomena including drag and wake characteristics, vortex shedding frequency, and detailed kinematics of the resulting flow field and their effects on properties such as the rate of heat transfer. On the other hand, this type of information is frequently needed for the design of structures exposed to fluid flow (such as off-shore pipelines). Furthermore, because of changing process and climatic conditions, one also needs to calculate the rate of heat transfer from such structures. Additional applications are found in polymer processing operations such as the formation of weld lines. Indeed, under * To whom correspondence should be addressed. Address: R. P. Chhabra, Department of Chemical Engineering, Indian Institute of Technology, Kanpur, India 208016. Tel.: 0091512-2597393. Fax: 0091-512-2590104. E-mail: chhabra@ iitk.ac.in. † Department of Chemical Engineering. ‡ Department of Mechanical Engineering.

appropriate conditions of Reynolds number and blockage ratio, a wide variety of phenomena has been observed even with the flow of Newtonian fluids over a square cylinder.3 It is now well-known that many materials (e.g., polymer solutions, melts, muds, emulsions, suspensions) encountered in chemical and processing applications often exhibit complex rheological behavior, including shear thinning and shear thickening, viscoelasticity, and yield stress.4 Despite their wide occurrence, as far as is known to us, there has been no prior study of the flow of such non-Newtonian liquids past bluff bodies, particularly square cylinders, and the present work aims to fill this gap in the existing literature. It is, however, instructive and useful to briefly recount the corresponding body of knowledge available for the flow of Newtonian fluids, as this facilitates the subsequent discussion of non-Newtonian liquids. Previous Work Because excellent accounts of the experimental and numerical studies of the confined and unconfined, steady and unsteady flow of Newtonian fluids over a square cylinder are available,1-3,5 only the salient points are repeated here. Okajima6 and Okajima et al.7 reported an extensive numerical and experimental study for an unconfined Newtonian fluid flow over a cylinder of rectangular cross section in the Reynolds number range from 100 to 2 × 104. The main thrust of their work was to delineate the frequency of vortex shedding from the cylinder and to predict the corresponding Strouhal numbers. The phenomenon of vortex shedding for a square cylinder confined in a channel was also numerically investigated by Mukhopadhyay et al.8 They reported that vortex shedding induces periodicity in the flow field. In particular, they investigated the effect of the blockage ratio and found the periodicity of flow to be suppressed by the presence of confining boundaries.

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

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

However, virtually no drag results were reported in any of these studies. In a combined numerical and theoretical study, Davis et al.9 reported a good match between their predictions and experimental results for the crossflow past a rectangular cylinder placed in a horizontal channel with the Reynolds number of flow ranging from 100 to 2000. More recently, Breuer et al.10 investigated low Reynolds number flow past a square cylinder placed in a channel with a parabolic inlet profile. They contrasted the predictions of two numerical schemes, namely, the lattice-Boltzmann and the finite-volume methods. Likewise, Valencia11 analyzed the unsteady laminar flow pattern and convective heat transfer in a channel with a built-in square cylinder. He reported significant enhancements in the rate of heat transfer under oscillatory separated flow conditions due to the cylinder being present in the channel. From a careful scrutiny of the available literature for the flow of Newtonian liquids past a square cylinder, two points seem to emerge clearly: First, it is generally believed that steady two-dimensional flow occurs up to about the Reynolds number of 50-55 in unconfined conditions, although steady flow is possible up to higher Reynolds number when the blockage ratio is large; the first manifestation of the flow instability is through vortex shedding. Second, most of the aforementioned studies attempt to elucidate the interplay between the blockage ratio and the Reynolds number on one hand and the detailed kinematics of flow (wake phenomena, vortex shedding, etc.) on the other, and indeed, only very scant results are available on gross parameters of engineering interest such as the drag coefficient and Nusselt number as functions of the relevant dimensionless parameters even for Newtonian fluids, let alone for non-Newtonian liquids. For power-law fluids, as far as is known to us, there has been only limited activity relating to unconfined flow past a circular cylinder.12,13 Drag values as functions of the power-law index (0.7-1.2) and the Reynolds number (5-40) were reported by D’Alessio and Pascal.12 However, as the value of the Reynolds number (5-40) was progressively increased, they encountered acute convergence difficulties for both shear-thinning and shear-thickening fluids even at a Reynolds number of 20. Thus, for instance, for the maximum value of the Reynolds number of 40 in their study, a fully converged solution could be obtained only for very weakly nonNewtonian fluids (0.95 e n e 1.1). On the other hand, Whitney and Rodin13 presented the drag results for the slow (creeping-flow) translation of spheres and of cylinders (of infinite and of finite aspect ratios) in unconfined power-law fluids (shear thinning). Their results are consistent with the previous extensive results for spheres and with the limited results for cylinders. It is also appropriate to add here that the scant available experimental results of drag on nonspherical particles (including circular and square cross-section cylinders) in free-fall conditions in non-Newtonian fluids have been summarized by several investigators.14-20 In most such experimental studies, the maximum aspect ratio (l/d) of the cylinders was typically on the order of 10-15, and therefore, such models will constitute a very poor approximation for infinitely long cylinders as assumed in the study of d’Alessio and Pascal.12 From the aforementioned account, it is thus safe to conclude that no prior theoretical results are available for the Poiseuille flow and the corresponding heat transfer in non-Newtonian liquids past a square cylin-

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

der situated symmetrically in a plane channel. Admittedly, the diversity of materials encountered in engineering practice (e.g., polymer melts and solutions, foams, emulsions, suspensions) display a wide variety of rheological phenomena, including shear thinning, shear thickening, yield stress, and viscoelasticity, among others. However, it seems logical to begin with the simplest and also the commonest type of non-Newtonian feature, namely, the shear-thinning and shear-thickening (dilatant) types of non-Newtonian behavior. Almost all polymeric melts (unfilled) and solutions display shear-thinning behavior, whereas highly concentrated pastes and filled resins behave as shear-thickening fluids over a range of shear rates.4 This initial model can, in turn, be used to build up the level of complexity in a gradual manner to incorporate the other nonNewtonian effects. In this work, the continuity, momentum, and energy equations have been solved numerically for the unknown velocity, pressure, and temperature fields for the steady Poiseuille flow of incompressible power-law liquids past a square cylinder placed in a channel. The results have, in turn, been processed further to deduce the values of the dimensionless drag coefficient and Nusselt number as functions of the Reynolds number, Peclet number, and power-law index over wide ranges of conditions. Owing to the generally high viscosity levels of non-Newtonian materials, coupled with the fact that the steady flow past a square cylinder occurs only up to about Re ) 50 in Newtonian fluids, the present study is limited to a maximum Reynolds number of 40, which also ensures the two-dimensionality of the flow under these conditions. Problem Statement and Governing Equations Let us consider the steady incompressible twodimensional flow of a power-law fluid past a square cylinder (side length b) placed in a channel (width 2h), as shown schematically in Figure 1. The liquid enters the channel with a fully developed velocity profile and at a constant temperature T∞. As noted earlier, because the maximum value of the Reynolds number considered in this work is 40, the flow can be assumed to be twodimensional and steady. Hence, no flow occurs in the z direction, and no flow variable depends on the z coordinate, i.e., ∂X/∂z ) 0 for all variables X. Furthermore, the thermophysical properties (density, F; heat capacity Cp; thermal conductivity, k; and power-law constants, m and n) are assumed to be independent of temperature. Under these conditions, the equations of continuity, momentum, and thermal energy (in the absence of viscous dissipation) in their dimensionless form are written as follows

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

Continuity

Pe )

∂Vx ∂Vy + )0 ∂x ∂y

(1)

Momentum x component

(

)

(2a)

(

)

(2b)

∂Vx ∂Vx ∂Vx ∂p 1 ∂τxx ∂τyx + Vx + Vy )+ + ∂t ∂x ∂y ∂x Re ∂x ∂y y component

∂Vy ∂Vy ∂Vy ∂p 1 ∂τyx ∂τyy + Vx + Vy )+ + ∂t ∂x ∂y ∂y Re ∂x ∂y Thermal Energy

(

)

∂T ∂T ∂T 1 ∂2T ∂2T + 2 + Vx + Vy ) ∂t ∂x ∂y Pe ∂x2 ∂y

(3)

For a power-law fluid, the constitutive relation is given by

τij ) 2ηij

(4a)

[21(∆:∆)]

(4b)

η)

n-1/2

where in Cartesian coordinates

[

2

+2

∂Vy ∂y

2

+

)

∂Vx ∂Vy + ∂y ∂x

2

(4c)

The relevant components of the extra stress tensor are written as

τxx ) 2η

(

)

∂Vx ∂Vy ∂Vx ∂Vy , τyy ) 2η , τxy ) τyx ) η + ∂x ∂y ∂y ∂x (5)

In eqs 1-5, the velocities have been rendered dimensionless using the maximum (centerline) velocity Vo, all distances in the x and y directions using the side of the square (b), the pressure using FVo2, the extra stress components using m(Vo/b)n, the viscosity using a reference viscosity m(Vo/b)n-1, and the time using (b/Vo) as the characteristic time. The temperature is made dimensionless in two different ways depending on the thermal boundary condition specified at the surface of the solid cylinder. The two commonly used thermal boundary conditions are that of a uniform temperature Tw or that of a constant heat flux (qw) at the surface of the boundary CDD′C′ in Figure 1. In the case of the constant temperature, the temperature difference (T ′ - T∞) is scaled using (Tw - T∞) as the characteristic temperature difference. In the case of the constant heat flux boundary condition imposed on the surface of the solid cylinder, (qwb/k) is used as the characteristic temperature difference. Furthermore, the two dimensionless groups, namely, the Reynolds (Re) and the Peclet number (Pe), appearing in eqs 2 and 3 are defined as follows

Re )

FVo2-nbn m

(6)

(7)

The advantage of using the Peclet number as opposed to the commonly employed Prandtl number is that the Peclet number is independent of the power-law constants, and this definition also thus coincides with that commonly used for Newtonian fluids. After the extra stress components are substituted into the momentum equations and the equations are rewritten in their conservative forms, eqs 2a, 2b, and 3 become

Momentum Equation x momentum equation ∂Vx ∂ ∂ + (VxVx) + (VxVy) ) ∂t ∂x ∂y 2 2 2 ∂η ∂η η ∂ Vx ∂ Vx ∂p + + + yx + xx (8a) 2 ∂x Re ∂x2 Re ∂x ∂y ∂y

(

) (

)

y momentum equation ∂Vy ∂ ∂ + (VxVy) + (VyVy) ) ∂t ∂x ∂y 2 2 2 ∂η ∂η ∂p η ∂ Vy ∂ Vy + + yy (8b) + + xy 2 ∂y Re ∂x2 Re ∂y ∂x ∂y

(

) (

Thermal Energy Equation

] ( ) ( ) (

∂Vx 1 (∆:∆) ) 2 2 ∂x

FVobCp k

)

(

)

∂T ∂ ∂ 1 ∂2T ∂2T + (VxT) + (VyT) ) + 2 ∂t ∂x ∂y Pe ∂x2 ∂y

(9)

It is worthwhile pointing out here that the viscous dissipation term has been neglected in the energy equations here, eqs 3 and 9. This term is usually significant when either the shear rate is very high or the fluid has a very high effective viscosity or both. Because the blockage ratio in the present case is 1/8 and the maximum Reynolds number is 40, the maximum value of the shear rate close to the surface of the cylinder is not expected to be excessively high. Similarly, because the Reynolds number is not in the creeping range, the effective viscosity of the fluid is also likely not to be too high. On both these counts, the exclusion of the viscous dissipation term is probably a reasonable approximation under such conditions. The physically admissible and consistent boundary conditions for this flow configuration (see Figure 1) are as outlined below. At the inlet plane HH′

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

(10a)

It is appropriate to add here that the velocity profile given by eq 10a is applicable under laminar flow conditions in the channel, i.e., for Reynolds number based on the width (2h) of the channel that are less than ∼2000. Because the maximum value of the Reynolds number (based on b) is only 40, this will translate into a value of about 350 in terms of the channel Reynolds number and somewhat higher in the annular region. Nevertheless, the Reynolds number is unlikely to reach sufficiently high values for the inertial effects to become significant. It is thus reasonable to use this velocity profile in the present study.


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

Vx ) 0, Vy ) 0

(10b)

For the constant-temperature condition (Tw)

T)1

(10c)

(10d)

where n represents the directions normal to the surface of the cylinder. At the confining boundaries HG and H′G′, the no-slip condition is used, i.e.

Vx ) 0, Vy ) 0, T ) 0

(10e)

At the exit plane GG′, there is no unique prescription available for outflow. The main concern here is that the condition implemented at GG′ must not influence the flow upstream in any significant manner. This is best accomplished by using the so-called Orlanski condition,21 written as follows for the momentum equations in its dimensionless form

∂Vi ∂Vi + Vconv )0 ∂t ∂x

(10f)

The corresponding form for the thermal energy equation is given by

∂T ∂T + Vconv )0 ∂t ∂x

(10g)

where i ) x or y and Vconv ) 1 was used here. This particular form of the boundary condition at the exit ensures that the vortices can grow unhindered and exit the flow domain without causing any appreciable disturbance to the flow upstream. It is appropriate to add here that the use of this condition also reduces the number of iterations required per time step and allows a shorter upstream computational domain as compared to that required in the case of the Neumann boundary condition. Thus, eqs 1 and 8-10 provide the theoretical framework for describing the flow and heat-transfer phenomena between a power-law fluid and a square cylinder for the flow conditions shown in Figure 1. These equations have been solved numerically using a finitedifference method for the unknown velocities (Vx, Vy), pressure (p), and temperature (T) over wide ranges of physical, rheological, and kinematic conditions. Once the flow domain has been mapped in terms of these variables, the gross engineering parameters such as the drag coefficient and Nusselt number can be evaluated as described below. It is convenient to introduce a dimensionless drag coefficient (CD), defined as

CD )

2FD FVo2b

∫CC′p dy - ∫DD′p dy)

CDP ) 2(

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

∂T ) -1 ∂n

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

(11)

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

(12)

Similarly, the friction component acting in the x direction stems from the shearing forces acting on the top (CD) and bottom (C′D′) faces of the cylinder, and this contribution is obtained by evaluating the following integral on these two faces

CDF )

2 Re

( )

∫η

∂Vx dx ∂y

(13)

CD is then simply the sum of these two components. The Nusselt number, Nu, defined as hb/k is similarly evaluated using the temperature field as follows: For the top (CD) and bottom (C′D′) faces of the cylinder

Nux ) -

∂T ∂n

(14a)

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

Nuy ) -

∂T ∂n

(14b)

where n represents the direction normal to the surface of the cylinder. Equation 14 applies for the constant-temperature condition at the solid cylinder. Analogous expressions for the case of the constant heat flux can be obtained by writing the heat balance at the surface of the cylinder. For instance, at each point on the surface of the cylinder, qw ) h(T ′-T∞), which, when made dimensionless using qwb/k as the characteristic temperature difference, yields the following expressions for the local Nusselt number

For the top and the bottom faces of the solid square cylinder Nux )

1 TCD or C′D′(x)

(15a)

For the front and the rear faces of the solid square cylinder Nuy )

1 TCC′ or DD′(y)

(15b)

Such local values were further averaged over each face or over the whole cylinder to obtain the surface averaged values of the Nusselt number and/or the overall mean value of the Nusselt number for the whole cylinder, which is the quantity often required in process engineering design calculations. In addition to the aforementioned parameters, the flow domain was also mapped in terms of the streamlines, isovorticity lines and isotherms, as these facilitate the visualization of the flow and temperature patterns, wake size, etc., to provide further physical insights into the nature of the flow processes. For a fixed geometry, it can easily be shown that there are five dimensionless groups for this problem: Re, Pe, Nu, CD, and n. For a fixed geometry (blockage ratio and upstream and downstream dis-


tances), this study aims to establish the functional dependence of the drag coefficient CD on the Reynolds number and the power-law index, as well as that of the Nusselt number, Nu, on the Reynolds and Peclet numbers and the power-law index. Numerical Solution Procedure The numerical solution procedure used here is an SMAC-type implicit method, as detailed in our previous studies.22,23 Therefore, only the salient features of the procedure are reported here. The SMAC-type implicit scheme was implemented on a staggered grid for the solution of the continuity and momentum equations, i.e., eqs 1, 8a, and 8b. The convective terms in eqs 8a and 8b were discretized using the upwind scheme, whereas the viscous terms were approximated using the standard center-difference approach. The time-stepping solution procedure entailed two steps: The first was a predictor step in which the unknown velocities were calculated (predicted) using an assumed pressure field. Owing to the implicit discretization of the viscous terms, this step was iterative in nature. The second step entailed the repeated correction of the pressure and velocity fields until the equation of continuity was satisfied. This was done by solving the Poisson equation for the pressure correction iteratively within the prescribed limits together with appropriate boundary conditions. The use of the implicit scheme here helped avoid the commonly encountered numerical instability, especially at low Reynolds numbers as is the case in this study. The time-stepping procedure was pseudotransient; starting from arbitrary initial conditions, the time stepping of the momentum equation was continued until the steady-state solution was achieved. The steady-state velocity field was then used to obtain the temperature field by applying an analogous time-stepping scheme to the energy equation, eq 9, with an implicit method using either the upwind or the center-difference scheme for the convective terms. Once again, the conduction terms were approximated by the center-difference approach. Starting with an arbitrary temperature field, the time stepping was continued until a steady-state temperature field was obtained. The resulting pressure, velocity, and temperature fields, in turn, formed the basis of the calculations of the derived variables affording a visualization of flow in terms of isovorticity, streamline, and isotherm patterns and the calculation of the gross engineering parameters such as the drag coefficient and Nusselt number as functions of the pertinent variables. Choice of Numerical Parameters The accuracy and reliability of the numerical results is contingent upon an appropriate choice of the following parameters defining the flow domain: upstream computational domain LU, downstream computational domain LD, and grid size M × N. Naturally, these parameters exert varying levels of influence on the flow and temperature fields, which, in turn, determine the values of the gross engineering parameters. Because no prior results are available for power-law liquids, appropriate values of the aforementioned parameters were selected on the basis of the most severe case with Newtonian fluid behavior, i.e., Re ) 40 and Pr ) 10. Because the main idea here is to elucidate the nonNewtonian effects, and therefore the values of CDP, CDF

and Nu were calculated for a fixed value of b/2h ) 1/8, i.e., for a blockage ratio of 0.125. To establish an appropriate value of the upstream computational domain, the value of LU was varied from 2 to 6, while the downstream distance LD ) 12 was used in all our calculations. This experimentation clearly revealed that the resulting gain in the accuracy of CD and 〈Nu〉 was only marginal (less than approximately 1% in CD and 0.36% in 〈Nu〉) at the expense of an exhortbitant increase in CPU time when LU was increased beyond 6. Similarly, limited calculations with n * 1 suggested the changes in the values of CD and 〈Nu〉 to be even smaller than those quoted above. Thus, on the basis of these considerations, LU ) 6, LD ) 12, and b/2h ) 1/8 were the values used in this study. Similar values have typically also been used by others in the literature for this flow problem. Once these parameters had been fixed, attention was turned to the choice of an appropriate grid. In fact, this exploration was done for three values of n ) 0.5, 1, and 1.4 for the maximum values of the Reynolds number, Re ) 40, and Prandtl number, Pr ) 10, i.e., Pe ) 400. Four grids 76 × 32, 114 × 48, 190 × 80, and 228 × 96 were tested. Once again, weighing the marginal degree of improvement in the values of drag coefficient and Nusselt number obtained with the finest grid, i.e., 228 × 96, against a disproportionately large increase in CPU time, the 190 × 80 grid was regarded to be adequate for the present results to be essentially grid-independent, especially for the values of n different from unity. Thus, to reiterate, all results reported in this study relate to the conditions of LU ) 6, LD ) 12, and b/2h ) 1/ and were obtained using a 190 × 80 grid. It is 8 worthwhile mentioning here that, owing to the twodimensional and steady nature of the flow, the computations were performed only for the upper half of the domain, i.e., 0 e y e h. Results and Discussion Prior to undertaking the detailed presentation and discussion of the new results obtained in this work, it is necessary and desirable to validate the numerical solution procedure used in this study, as this will also help ascertain the level of uncertainty inherent in the new results reported herein for power-law liquids. (i) Validation of Numerical Solution Procedure. Validation of this procedure is generally accomplished by benchmarking the numerical results against the available reliable numerical and/or analytical predictions for the analogous problem. In this work, the prior numerical results on drag due to Breuer et al.10 and Saha et al.5 for Newtonian liquids have been used to benchmark the present results on drag. Saha et al.5 numerically studied the two-dimensional steady flow of an incompressible Newtonian fluid past a square cylinder, for a blockage ratio of 1/10 and upstream and downstream distances of 5.5 and 33.5, respectively. For Re ) 100, they reported the drag coefficient value of CD ) 2.92. This flow (with LU ) 5.5, LD ) 33.5, and b/2h ) 0.1) was simulated in the present study using a uniform grid (402 × 102) and using the first-order upwind discretization approximation for the convection terms in the momentum equation. The present value of the drag coefficient, CD ) 2.941, is indeed in excellent agreement with that of Saha et al.;5 the difference between the two values is only 0.73%, which is not at all uncommon in this kind of work.


Similarly, Breuer et al.10 employed a nonuniform mesh and contrasted the performance of two numerical solution procedures, namely, the finite-volume and the lattice-Boltzmann methods. For a blockage ratio of 1/8 and upstream and downstream distances of 12 and 37, respectively, they reported CD ) 2.00 for Re ) 30. This value also compares favorably with that obtained in the present study, i.e., CD ) 2.06, resulting in a difference of 3%. In fact, the two values differ by even smaller amounts at lower Reynolds number. Clearly, this small difference can safely be ascribed to the slightly different values of LU and LD used, coupled with the different numerics employed in the two studies. It needs to be emphasized here that, because Breuer et al.10 varied the Reynolds number from 0.5 to 300, the required downstream computational domain increased with increasing Reynolds number and, in fact, a value of LD ) 12 is believed to be quite adequate up to about Re ≈ 50. Finally, Mukhopadhyay et al.8 studied the wake structure around a square cylinder in a channel. They reported the flow field upstream of the cylinder to be nearly unaltered by wake formation in the rear of the cylinder. They evaluated the shear stress distribution at the wall of the channel. For fully developed laminar flow, the local skin friction coefficient (Cf‚Re) on the channel walls had a value of 12 in the absence of the cylinder, and clearly, this value increased as a result of the hydrodynamic resistance of the cylinder. For Re ) 50, the difference in the resulting peak values of the skin friction coefficient was on the order of 5% between the present results and those of Mukhopadhyay et al.8 Judging from these comparisons, it is perhaps fair to say that the uniform grid and the first-order upwind scheme employed herein can yield reliable results at the relatively low Reynolds numbers used in this study. Furthermore, the values of drag coefficient presented herein are believed to be accurate to within 3-4%. Likewise, the accuracy of the solution procedure applied to the thermal energy equation, eq 9, to obtain the temperature field was validated using two benchmark cases. In the first instance, the developing thermal boundary layer flow was considered in a two-dimensional channel without the square cylinder. For a Reynolds number (based on the hydraulic diameter) of 150, the computed temperature field was compared with the corresponding analytical solution24 obtained using a uniform grid of 400 × 200 for Pr ) 0.71 at various axial locations. The two values of the temperature were found to be within (0.35% of each other, thereby lending support to the correctness and accuracy of the numerical solver for the unknown temperature. The second benchmarking was done using unpublished results25 for free stream flow across a square cylinder, which were obtained using a hybrid grid, consisting of a fine grid adjacent to the cylinder and a uniform coarse grid away from the cylinder, to study heat transfer between a Newtonian fluid (air) and a square cylinder with a grid size of 323 × 264; a blockage ratio of 1/20 to simulate free stream flow; and upstream and downstream distances of 8.5 and 16.5, respectively. The present values of the Nusselt number averaged over the front, top, and rear faces of the cylinder and the overall mean values are compared with those of Sharma25 in Table 1 for the constant-temperature condition and in Table 2 for the constant-heat-flux condition imposed at the cylinder surface. The present results were obtained using slightly different (but still adequate) values of the

Table 1. Comparison between Present Results and Literature Values25 for the Constant-Temperature Condition for Pr ) 0.70 (Air) Re

ref

〈Nu〉f

〈Nu〉t

〈Nu〉r

〈Nu〉

5

Sharma25 present work Sharma25 present work Sharma25 present work Sharma25 present work

1.684 1.704 2.32 2.33 3.25 3.23 4.597 4.533

1.22 1.23 1.586 1.602 2.06 2.08 2.611 2.653

0.715 0.715 0.798 0.796 0.884 0.878 1.008 0.998

1.21 1.22 1.572 1.583 2.062 2.067 2.707 2.708

10 20 40

Table 2. Comparison of Present Results with Those of Sharma25 for the Constant-Heat-Flux Condition (Pr ) 0.7) Re

ref

〈Nu〉f

〈Nu〉t

〈Nu〉r

〈Nu〉

5

Sharma25 present work Sharma25 present work Sharma25 present work Sharma25 present work

1.580 1.616 2.20 2.22 3.046 3.090 4.249 4.325

1.273 1.297 1.713 1.734 2.316 2.359 3.076 3.150

0.954 0.964 1.140 1.142 1.338 1.333 1.556 1.541

1.272 1.294 1.691 1.708 2.254 2.285 2.989 3.041

10 20 40

upstream and downstream distances of 6 and 12, respectively; a blockage ratio of 1/15 with free-slip boundary condition at the transverse boundary; and a uniform mesh of 190 × 120. An examination of these results shows that the two values differ by at most 1% from each other. Such minor differences are also not uncommon in such studies12 and can easily by attributed to the slightly different values of the blockage ratio and upstream and downstream lengths of the computational domain used in the present work. Conversely, one can argue that the heat-transfer characteristics are relatively insensitive to the blockage ratio and inlet and exit lengths under this range of conditions. This fact also provides a justification for the choice of numerical parameters, including the blockage ratio, mesh, etc., used in this work as described in the previous section. Judging from the aforementioned benchmark comparisons and our previous experience,22,23 it is perhaps reasonable to state that the new values of the total drag coefficients and of the Nusselt number are reliable to within 1-2%. (ii) Drag Phenomena and Flow Field. The hydrodynamic drag force exerted by the fluid on the square cylinder is determined by the individual contributions due to the pressure and the shearing forces acting on the object. As mentioned earlier, for a given geometry, this relationship can readily be expressed in terms of three dimensionless groups, namely, the drag coefficient CD, the Reynolds number Re, and the power-law index n. Furthermore, the total drag coefficient comprises two components, stemming from the pressure forces (CDP) and the shearing forces (CDF). To elucidate the role of the power-law index in an unambiguous manner, the drag coefficient for power-law fluids was normalized with respect to the corresponding value for Newtonian fluids (n ) 1) at the same value of the Reynolds number. Figure 2 depicts the effects of the flow behavior index n and the Reynolds number Re on the individual and total drag coefficients over the ranges 0.5 e n e 1.4 and 5 e Re e 40. Note that the corresponding results for n ) 1 are also shown as a horizontal line with the ordinate value of unity. Broadly speaking, shear-thinning behavior (n < 1) is seen to increase the values of the pressure and total drag coefficients with reference to


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

the corresponding values in a Newtonian fluid, and qualitatively, the opposite kind of behavior is observed for shear-thickening liquids (n > 1), although the effect of n is seen to diminish with increasing value of the Reynolds number in the range used in this work. This behavior is qualitatively consistent with the analogous

results for spherical and spheroidal particles sedimenting in power-law fluids.15,25 The effect of the flow behavior index on the frictional component of the total drag is seen to be the opposite, i.e., shear thinning causes a lowering of the frictional component, whereas shear-thickening behavior augments it. This effect becomes accentuated at increasing value of the Reynolds number. This trend is consistent with the notion that, as the Reynolds number of the flow is gradually increased, the general level of shearing increases, thereby resulting in lower values of the effective viscosity for a shear-thinning fluid and higher values of the effective viscosity for dilatant fluids that then directly influence the frictional component of the drag in the same fashion, as seen in Figure 2. Intuitively, as the value of the Reynolds number is progressively increased, viscous forces make way for inertial forces, so one would expect the effect of the flow behavior index to diminish with increasing value of the Reynolds number. Indeed, this expectation is borne out by Figure 2c, wherein it is clearly seen that the ratio CDNN/CDN ranges from 1.15 to ∼0.88 as the value of n is gradually varied from 0.5 to 1.4. Some further insights can be gained by examining the variation of the relative contributions of the pressure and frictional components of the drag with the Reynolds number and the flow behavior index. A detailed examination of these results reveals that, even at Re ) 5, the frictional contribution to the overall drag hovers around 30-45% as the value of n is gradually increased from 0.5 to 1.4. The slight increase in the value of CDF/CD is consistent with the fact that, in shear-thickening fluids (n > 1), the viscous forces would be more significant than in a shearthinning fluid (n < 1). Furthermore, as the value of the Reynolds number increases, the total drag coefficient includes a progressively smaller contribution from friction. For instance, at Re ) 40, the ratio CDF/CD ranges from 0.5 to 0.21 as the value of n is varied from 0.5 to 1.4. Thus, as expected, overall, in this geometry, the pressure drag increasingly dominates as the value of the Reynolds number is gradually increased, even for shear-thinning and shear-thickening fluids. However, just as in the case of a sphere, non-Newtonian effects would be expected to be more significant at low Reynolds numbers, and this expectation is borne out by the results shown in Figure 2. The intricate interactions between the various physical and kinematic variables can also be seen through the detailed streamline and isovorticity contour plots. Because the main focus of the present study is to highlight the role of the flow behavior index n, Figure 3 shows the representative streamline and isovorticity contour plots for a range of combinations of the values of n (0.5, 1, 1.4) and of the Reynolds number (5, 20, 40). An examination of this figure shows that, for a fixed value of Re, the flow is seen to be faster close to the cylinder in shear-thinning fluids (n < 1) than in a Newtonian medium (n ) 1) and, as expected, it is seen to be impeded in shear-thickening fluids. This is a direct consequence of the dependence of the fluid viscosity on the shear rate. However, moving away from the cylinder in the axial and lateral directions, this effect progressively becomes less prominent. On the other hand, as the value of the Reynolds number is gradually increased, the wake region grows as the fluid behavior undergoes a transition from shear-thinning to Newtonian and, finally, to shear-thickening behavior. In other


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

words, the wake size is seen to be larger in a shearthickening fluid than in a shear-thinning fluid. It needs to be emphasized that there is no inconsistency here and that this counterintuitive result is a direct consequence of the scaling variables used in this study. With a slight rearrangement of the definition of the Reynolds number, it can readily be seen that the representative shear rate is of order Vo/b. Clearly, although this approximation is appropriate close to the cylinder, elsewhere in the flow domain, the shear rate is likely to be of order Vo/2h, i.e., 8 times smaller than Vo/b. Alternatively, one can argue that there is a layer of very viscous fluid surrounding the square cylinder that causes the oncoming fluid stream to veer from its path, which, in turn, results in a larger wake region in shear-thickening fluids than in Newtonian or shearthinning fluids. In view of this result, within the framework of the present scaling scheme, the fluid viscosity is underestimated for shear-thinning fluids, and consequently, the Reynolds number is overestimated. This finding is also consistent with the corresponding results for a sphere falling in power-law liquids, wherein it is seen that the velocity decays much faster in shear-thinning liquids than in Newtonian liquids.16 Such behavior is tantamount to a rapid increase in the fluid viscosity slightly away from the object, and presumably, it delays the formation of a wake in pseudoplastic systems. One can construct a similar argument to explain the presence of a large wake in dilatant systems (n > 1). However, for a constant value of n, as the value of the Reynolds number is increased, inertial forces outweigh the viscous forces.

Although it is difficult to pinpoint the precise value of the Reynolds number at which a wake will form, it is clear that the lower the value of the power-law index, the higher the value of the Reynolds number at which a visible wake appears. For instance, at n ) 0.5, there is no visible wake formed at Re ) 5, but a small wake is present at Re ) 10. As the Reynolds number is gradually increased, the wake region grows in size. This phenomenon is often described in terms of the so-called recirculation length (Lr), which is a measure of the distance from the rear surface of the cylinder to the point of reattachment along the centerline of the wake. This quantity is estimated approximately from the streamline plot at the intersection of the Ψ ) 0 line and the channel centerline i.e., the x axis. Figure 4 shows the variation of the dimensionless recirculation length with the Reynolds number for a range of values of the power-law index. The present results are seen to be consistent with the linear relationship observed for Newtonian fluid behavior reported in the literature.10 This linearity also seems to apply reasonably well for shear-thickening fluids, but the dependence is seen to be slightly weaker in shearthinning fluids, which is, in a sense, consistent with the delayed wake formation occurring in these systems, as mentioned above. Similarly, the increase in the value of the recirculation length seen in Figure 4 is consistent with the larger wake region shown in Figure 3 for shearthickening media, and this is also borne out by the isovorticity contour plots shown in Figure 3. (iii) Heat-Transfer Characteristics. Two additional factors influence the heat-transfer aspects, namely,


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

the Peclet number and the type of thermal boundary condition, i.e., a constant temperature on the surface of the cylinder (case I) or the constant-heat-flux condition (case II). Owing to the underlying inherent differences, the heat-transfer results corresponding to these two conditions are presented and discussed separately in the following sections. (a) Case I. As with the drag results presented in the preceding section, the individual surface-averaged and overall averaged values of the Nusselt number were normalized using the corresponding Newtonian value at the same values of the Reynolds and Peclet numbers. A distinct advantage of this form of representation lies in the fact that it allows for the most direct assessment of the influence of the power-law index on the heattransfer results. Figures 5 (Re ) 5) and 6 (Re ) 40) show representative variations of the normalized Nusselt numbers with the flow behavior index (n) and the Peclet number (Pe). Qualitatively similar results are obtained for the other cases, and thus, these results are not shown here. An examination of Figures 5 and 6 clearly shows the influence of the flow behavior index on heat transfer to be smaller than that on drag as seen in the previous section. The maximum value of the ratio of 〈Nu〉NN/〈Nu〉N is seen to be only about 1.21, thereby showing an enhancement of 21% in heat transfer from the top surface of the cylinder in a shear-thinning fluid. In fact, the maximum enhancement in the overall heat-transfer coefficient is on the order of only 15% in shear-thinning fluids, and the corresponding reductions in shear-thickening media are even smaller, being on the order of only ∼7-8%. Alternatively, one can argue that the effect of the power-law index is adequately embodied in the modified definition of the Reynolds number, and therefore, the normalized Nusselt number, 〈Nu*〉, shows only a weak additional dependence on the flow behavior index. Turning our attention now to heat transfer from the individual surfaces of the square cylinder, the effect of n is seen to vary both qualitatively and quantitatively with increasing values of the Reynolds and Peclet numbers. For instance at low Re () 5) and Pe () 5), the value of 〈Nu*〉t for the top surface (CD in Figure 1) of the cylinder is most strongly influenced by the value of

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

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

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


Figure 7. Effect of the power-law index (n) on the normalized Nusselt numbers corresponding to various faces of the square cylinder for Re ) 5 (case II) for two values of the Peclet number.

corresponding ratios of the Nusselt numbers for the overall cylinder, for the front face and for the rear face, respectively. Under these conditions, because no wake is present, the heat-transfer coefficient hardly varies along this surface. At Re ) 5, as the value of the Peclet number is gradually increased, the values of 〈Nu*〉r are seen to be slightly above unity in shear-thinning media and slightly below unity in shear-thickening fluids. This dependence on n is seen to switch at Re ) 40, so that the effect of n is now reversed (Figure 6). This component is seen to show a reduction in heat transfer in shear-thinning liquids (because of the small wake region) and an enhancement in heat transfer in shearthickening fluids, which is also consistent with the large wake region observed in dilatant media. Irrespective of the value of the Reynolds number and/or Peclet number, the ratio 〈Nu〉NN/〈Nu〉N for the top face always exceeds that for the front face of the cylinder. Further examination of the variation of the local Nusselt number on the surface of the cylinder shows a monotonic increase in the value of the Nusselt number as one traverses from point B to C, followed by a decrease along CD and another slight drop along DE. However, all of these variations appear to be selfcanceling, thereby yielding values of the surface-averaged and overall averaged values of 〈Nu〉 that are only marginally lower or higher than the corresponding Newtonian values at the same values of the Reynolds and Peclet numbers. (b) Case II. Representative analogous results on the normalized Nusselt number for this case are shown in Figures 7 and 8 for a range of values of the Reynolds and Peclet numbers and the power-law index. An inspection of these results shows qualitatively similar trends as seen in Figures 5 and 6 for case I, although

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

there are slight differences in the detailed variation of the Nusselt number along the surface of the cylinder. Once again, however, these differences appear to be selfcompensating in determining the overall mean value of the Nusselt number. Representative isotherm plots for case I (upper half) and II (lower half) are shown in Figure 9 for a range of combinations of Re, Pe, and n. For case I, the temperature field is seen to decay somewhat faster in shearthinning fluids than in Newtonian media, thereby suggesting a thinner boundary layer in these fluids. However, the trend seems to be strongly dependent on the values of the Reynolds and Peclet numbers. Qualitatively, the opposite type of phenomenon is seen in shear-thickening fluids for case I. For case II also, the isotherms show a strong interplay between the values of the Reynolds and Peclet numbers and the value of n. For n ) 0.5, the temperature drop seems to be relatively rapid, especially as the value of the Reynolds number/ Peclet number is progressively increased. Although this effect is accentuated in shear-thinning media, it gets suppressed in shear-thickening fluids. This section is concluded by reiterating the conclusion that the drag coefficient and the Nusselt number normalized with respect to the corresponding Newtonian values show only very weak additional dependences on the power-law index. In fact, if demonstrated for other values of the blockage ratio, this fact offers a convenient scheme for estimating the values of the Nusselt number for shear-thinning and shear-thickening fluids simply from a knowledge of the corresponding Newtonian values for this configuration. Shear-thinning behavior results in a small degree of enhancement in heat transfer, whereas shear thickening has a deleterious effect on it.


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


Concluding Remarks

Subscripts

In this work, the governing equations describing the steady two-dimensional flow over and heat transfer from a square cylinder immersed in a streaming power-law liquid have been solved numerically to obtain detailed velocity and temperature fields for a blockage ratio of 1/ and the following ranges of kinematic and physical 8 parameters: 0.5 e n e 1.4, 5 e Re e 40, 5 < Pe < 400. The role of the type of thermal boundary condition, i.e., constant heat flux and isothermal cylinder, on the overall heat-transfer characteristics has also been studied. Within the range of conditions investigated, the values of the normalized drag coefficient and Nusselt number (both individual and overall) hover around unity, with values slightly above unity in shear-thinning liquids and slightly below unity for dilatant fluids. The weak effect of the flow behavior index n on drag is qualitatively similar to that well known for a sphere. The detailed flow patterns suggest relatively shorter wake regions in shear-thinning liquids and slightly larger recirculation lengths in shear-thickening media. This counterintuitive trend can be ascribed to the scaling procedure used herein. The effect of the Reynolds number on the flow patterns is qualitatively similar to that seen for Newtonian fluids. Similarly, isotherm plots reveal faster decay in the temperature field under certain conditions (high Peclet numbers) in shearthinning liquids, with the reverse behavior being observed in shear-thickening fluids. Unfortunately, no suitable experimental results are available to refute/ substantiate the theoretical predictions presented herein.

f ) front face N ) Newtonian NN ) non-Newtonian r ) rear face t ) top face

Nomenclature b ) side of the square (m) CD ) drag coefficient (-) CDF ) friction drag coefficient (-) CDP ) pressure drag coefficient (-) Cp ) heat capacity of liquid (J/kg K) FD ) drag force per unit length of cylinder (N/m) h ) half-height of the computational domain (m) k ) thermal conductivity of liquid (W/m‚K) LD ) downstream distance (-) Lr ) recirculation length made dimensionless using b (-) LU ) upstream distance (-) m ) power-law consistency coefficient (Pa‚sn) n ) power-law index (-) Nu ) Nusselt number (-) 〈Nu〉 ) average Nusselt number (-) p ) pressure (-) Pe ) Peclet number (-) Pr ) Prandtl number (-) t ) time (s) T ) dimensionless temperature (-) T′ ) fluid temperature (K) T∞ ) free stream fluid temperature (K) Vo ) centerline velocity (m/s) Vx ) x component of velocity (-) Vy ) y component of velocity (-) x y ) rectangular coordinates (m) Greek Letters xx, yy, xy ) components of the rate of deformation tensor (s-1) η ) power-law viscosity (-) F ) fluid density (kg/m3)

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Received for review April 28, 2003 Revised manuscript received August 21, 2003 Accepted September 3, 2003 IE030368F

Two-Dimensional Steady Flow of a Power-Law Fluid Past a Square

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