Two-dimensional unsteady laminar flow of a power law fluid across a square cylinder

J. Non-Newtonian Fluid Mech. 160 (2009) 157–167

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Two-dimensional unsteady laminar flow of a power law fluid across a square cylinder Akhilesh K. Sahu a , R.P. Chhabra a,∗ , V. Eswaran b a b

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

a r t i c l e

i n f o

Article history: Received 5 August 2008 Received in revised form 13 February 2009 Accepted 24 March 2009 Keywords: Power law fluid Vortex shedding Square cylinder Drag

a b s t r a c t The two-dimensional and unsteady free stream flow of power law fluids past a long square cylinder has been investigated numerically in the range of conditions 60 ≤ Re ≤ 160 and 0.5 ≤ n ≤ 2.0. Over this range of Reynolds numbers, the flow is periodic in time. A semi-explicit finite volume method has been used on a non-uniform collocated grid arrangement to solve the governing equations. The global quantities such as drag coefficients, Strouhal number and the detailed kinematic variables like stream function, vorticity and so on, have been obtained for the above range of conditions. While, over this range of Reynolds number, the flow is known to be periodic in time for Newtonian fluids, a pseudo-periodic flow regime displaying more than one dominant frequency in the lift is observed for shear-thinning fluids. This seems to occur at Reynolds numbers of 120 and 140 for n = 0.5 and 0.6, respectively. Broadly speaking, the smaller the value of the power law index, lower is the Reynolds number of the onset of the pseudoperiodic regime. This work is concerned only with the fully periodic regime and, therefore, the range of Reynolds numbers studied varies with the value of the power law index. Not withstanding this aspect, in particular here, the effects of Reynolds number and of the power law index have been elucidated in the unsteady laminar flow regime. The leading edge separation in shear-thinning fluids produces an increase in drag values with the increasing Reynolds number, while shear-thickening fluid behaviour delays this separation and shows the lowering of the drag coefficient with the Reynolds number. Also, the preliminary results suggest the transition from the steady to unsteady flow conditions to occur at lower Reynolds numbers in shear-thinning fluids than that in Newtonian fluids. © 2009 Elsevier B.V. All rights reserved.

1. Introduction The phenomenon of vortex shedding from bluff bodies has long been of interest to the fluid dynamics community and has been extensively studied. Many flows of theoretical and practical interest, e.g., in tubular and pin type heat exchangers, flow dividers in polymer processing applications, aerodynamics, cooling of electronic components, etc. display the phenomena of vortex shedding under appropriate conditions. A bulk of the work reported in the literature deals with the flow of Newtonian liquids past circular cylinders, rectangular cylinders and flat plates. Excellent reviews summarizing the current state of the art and areas meriting further exploration are available in the literature for the extensively studied case of a circular cylinder [1–4]. In contrast, the analogous flow over a cylinder of square crosssection has been investigated less extensively, even for Newtonian fluids. For instance, Davis and Moore [5] reported a 2D numerical

∗ Corresponding author. Tel.: +91 512 2597393; fax: +91 512 2590104. E-mail address: [email protected] (R.P. Chhabra). 0377-0257/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2009.03.010

simulation of the unconfined flow past a cylinder of rectangular cross-section for Reynolds number in the range of 100–2800. They reported that the lift and drag coefficient and Strouhal number show a strong dependence on the Reynolds number. Okajima [6] conducted an experimental study for this flow in the Reynolds number range of 70– 2 × 104 for a range of rectangular cylinders. Subsequently, he also carried out numerical simulations by a finite difference method and a discrete vortex method and showed the existence of a critical value of the Reynolds number where the Strouhal number changes are accompanied by a drastic change in the flow patterns [7]. The problem of laminar vortex shedding from a square cylinder for Re ≤ 300 was further studied numerically by Franke et al. [8], using the finite volume method. They presented the time dependence of a number of flow parameters such as drag, lift, Strouhal number, etc. Tamura and Kuwahara [9] computed 2D and 3D flows past a square cylinder for various length-to-diameter ratios at high-Reynolds numbers to ascertain the extent of end effects. Kelkar and Patankar [10] performed a linear stability analysis to find the critical Reynolds number at which the flow over a square cylinder becomes unsteady but is still two-dimensional. Zaki et. al. [11] carried out numerical calculations for the flow

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over a square cylinder at Reynolds number ≤ 250 and experimental study for Reynolds number in the range of 100–10,000. They showed the variation of the vortex-shedding frequency with the orientation of the cylinder to the oncoming flow stream. Sohankar et al. [12] studied the onset of vortex shedding and the influence of outlet boundary conditions for the flow over an inclined square cylinder at angles of incidence varying in the range 0– 45◦ and Re = 45–200. Subsequently, Sohankar et al. [13] also carried out 2D and 3D unsteady simulations for the same configuration at moderate Reynolds numbers (Re = 150–500). They found a stable 2D vortex-shedding (laminar) flow regime at Re = 150, and 3D flow at Re = 200, without, however, establishing the exact values of the Reynolds number at which the transition from 2D to 3D occurs. Saha et al. [14] have also numerically analyzed the force coefficients and the frequency of vortex shedding in the wake of a square cylinder exposed to a uniform shear flow for the Reynolds number in the range of 250–1500. Subsequently they [15] also presented the spatial evolution of vortices and transition to three-dimensionality in the wake of a square cylinder for the range of Reynolds number 150–500. Sharma and Eswaran [16] numerically investigated the flow structure and heat transfer characteristics of an isolated square cylinder in the cross-flow of air for both steady and unsteady periodic laminar flow in the 2D regime, for Reynolds numbers of 1–160 and a Prandtl number of 0.7. They reported that the flow for Re ≤ 40 is steady, while that for Re ≥ 50 is unsteady and periodic in nature, with the transition to unsteadiness occurring between Re = 40 and 50. In terms of separation, three different ranges of Reynolds number have been predicted: the first is the onset of separation between Re = 1 and 2; the second, the onset of vortex shedding (with trailing-edge separation) between Re = 40 and 50; and the third, the onset of leading-edge separation between Re = 100 and 110. Suffice it to say here that adequate information is available on global momentum characteristics for Newtonian fluids, albeit the available body of knowledge is less extensive and coherent than that for a circular cylinder. It is well known that many materials encountered in industrial applications exhibit more complex rheological behaviour than Newtonian fluids [17]. Typical examples include high molecular weight polymers and their solutions, suspensions and slurries, foams and froths, etc. These materials are encountered frequently in polymer, food and scores of other process engineering applications. In spite of their wide occurrence, only few studies have been reported on the flow of non-Newtonian fluids past bluff bodies. Admittedly most industrial fluids display a range of non-Newtonian flow characteristics including shear-dependent viscosity, yield stress, viscoelasticity, etc., it is readily acknowledged that the simplest and perhaps also the commonest characteristics is the shear-thinning viscosity. Until recently, shear-thickening behaviour was considered to be less relevant, but with the growing importance of thick pastes and suspensions in chemical, food and other processing applications, it is no longer so [18]. Therefore, it seems reasonable to begin with this type of fluid behaviour, and then the level of complexity can gradually be built up by incorporating other characteristics like yield-stress, viscoelasticity, etc. Therefore, this work is concerned with the flow of power law fluids past a square cylinder in the time-dependent laminar flow regime. The corresponding body of knowledge for the simplest class of non-Newtonian fluids, namely power law fluids, is much less extensive than that for Newtonian fluids, even for a circular cylinder. The flow of power law fluids across a circular cylinder in the steady laminar flow regime was initially studied by Soares et al. [19]. Recently, Bharti et al. [4,20] have reported extensive flow and heat transfer results for this problem. For the case of a square cylinder, only a few studies are available. Paliwal et al. [21] investigated the 2D steady and unconfined flow of power law liquids (0.5 ≤ n ≤ 1.4) past a square cylinder for Re = 5, 10, 20, 30 and 40 for the range of

Peclet numbers 5–400 for the constant wall temperature and the constant heat flux thermal boundary conditions by using the finite difference method. However, they employed an uniform and relatively coarse grid which compromised the accuracy of their results. Subsequently, Dhiman et al. [22] re-visited this problem using a much finer grid and over a wide range of Reynolds numbers as 1 ≤ Re ≤ 45 and power law index (0.5 ≤ n ≤ 2) by using the finite volume method.They used a higher order discretization scheme for the convective terms and a very fine computational grid structure, specially near the cylinder and therefore, their results are believed to be more reliable than that of Paliwal et al. [21]. Broadly speaking, the role of power law index gradually diminishes as the Reynolds number is progressively increased in the steady flow regime. Preliminary results on the role of confining boundaries on the flow and heat transfer from a square cylinder are also available in the literature [23,24]. To the best of our knowledge, no prior numerical study exists in the literature pertinent to the vortex-shedding characteristics of a square cylinder in power law fluids and this work aims to fill this gap. As stated earlier here and noted by others [25], while no experimental results are available thus far in the literature on the cross-flow of non-Newtonian fluids over a cylinder of square cross-section, it is perhaps useful to review briefly the pertinent experimental literature for the flow over a circular cylinder. The available body of knowledge can be categorized into three classes. First, some of the early studies such as that of James and co-workers [26,27], Sarpkaya et al. [28], etc., dealt with the flow behaviour of drag reducing polymer solutions over a circular cylinder. Admittedly, it is likely that these dilute solutions do exhibit non-Newtonian characteristics, but their shear viscosities are nearly constant, and depending upon the concentration and molecular weight of the polymer, these values are only slightly higher than that of water. It is therefore not at all uncommon to use the viscosity of water in calculating the Reynolds number of flow. The major thrust of such studies [26,27] was on drag and heat transfer measurements over the Reynolds number range (50 ≤ Re ≤ 200). They showed that beyond a critical value of the Reynolds number, both drag coefficient and Nusselt number were influenced solely by the value of a suitably defined Weissenberg number. The increase in drag at low-Reynolds numbers documented by James and Acosta [26] was also observed in surfactant solutions recently [29]. However, Ogata et al. [29] also observed drag reduction beyond a critical Reynolds number. The value of the critical Reynolds number was found to increase with the diameter of cylinder. Ogata et al. [29] corroborated the different types of drag coefficient–Reynolds number relationships in terms of the formation of wide stagnation zones. On the other hand, the major focus of the study of Sarpkaya et al. [28] was to explore the effect of polymer degradation on the surface pressure profile, drag, vortex-shedding characteristics, angle of separation, etc., in the range 5 × 104 ≤ Re ≤ 3 × 105 . The second category of studies aims to elucidate the role of fluid elasticity on the phenomena of vortex shedding. Some of the early studies [30,31] are based on the so-called drag reducing polymer solutions of polyethylene oxide. For instance, Gadd [30] reported a reduction in the vortex-shedding frequency with the increasing polymer solution in PEO solutions, but it was nearly independent of the concentration for the other drag reducing polymers like guar gum and polyacrylamide. This work relates to a constant value of Reynolds number, Re = 240 and of cylinder diameter, d = 150 ␮m. Subsequently, this work has been extended to higher values of the Reynolds number (up to 400) [31]. Similarly, Usui et al. [32] also studied the behaviour of PEO solutions and they were able to correlate the reduction in Strouhal number with a composite parameter 0.5 (Re/We) for cylinders of large diameters employed in their study. Most of the aforementioned studies have alluded to the role of cylinder diameter way beyond the occurrence of diameter in the

A.K. Sahu et al. / J. Non-Newtonian Fluid Mech. 160 (2009) 157–167

global parameters like Reynolds number, drag coefficient and Nusselt number, etc. This is in stark contrast to the behaviour observed in Newtonian fluids. Secondly, the only rheological measurements made were that of the shear viscosity (found to be nearly constant, i.e., no shear thinning) and the severity of viscoelasticity effect was established by evaluating the fluid relaxation time using the Rouse formula which is also applicable to dilute solutions. It is therefore reasonable to conclude that most of the aforementioned studies elucidate the role of viscoelasticity in the absence of shear-thinning viscosity. Finally, in the third category are the extensive studies of Coelho and co-workers [25,33,34]. In the three part series, extensive results have been reported on the identification of various flow regimes and the corresponding values of the transition Reynolds number, and on the pressure measurements in well-characterized Newtonian, shear-thinning fluids with weak and strong viscoelasticity. This study clearly shows the complex interplay between shear-thinning and viscoelasticity. This interaction gets further accentuated depending upon the diameter, length-to-diameter ratio of the cylinder and the type of ends of the cylinder. For instance, Coelho et al. [33] explored in a systematic manner to arrive at an appropriate choice of characteristic viscosity (or a characteristic shear rate) to define the Reynolds number so that the transition will occur almost at the same value as that in Newtonian fluids. Their study embraced the Reynolds number (based on a representative shear rate of U∞ /2d) in the range 50 ≤ Re ≤ 9000. The fluid elasticity was found not only to lower the critical Reynolds number, but also to shorten the transition range. While the shear-thinning behaviour was seen to increase the vortex shedding frequency thereby increasing the vortex filaments. They have proposed three Reynolds number values denoting the onset of vortex shedding, onset and end of the transition regimes. Similarly based on the detailed surface pressure measurements [34], they reported increasing drag reduction with the increasing polymer concentration which was attributed to the increasing levels of elastic forces acting on the shear layers. In summary, it is thus fair to conclude that the flow characteristics of polymer solutions over a circular cylinder are influenced in an intricate manner by their shear-dependent viscosity, viscoelasticity as well as by the geometric details of the cylinders such as their length-to-diameter ratio and the shape of the end faces. As seen here, while some results have been obtained on the onset of vortex shedding from circular cylinders, no analogous information is available on the other transitions such as 2D steady to 2D unsteady on one hand, or from the 2D unsteady to 3D regime on the other hand to enclose the vortexshedding regime. Similarly, as far as known to us, there has been no such experimental study on the flow of non-Newtonian fluids past a cylinder of square cross-section and therefore, it remains to be seen whether the various flow regime transitions for a square cylinder would be even qualitatively similar or not to that for circular cylinder as seen above. Finally, based on the above discussion of the literature, it can be thus concluded that adequate information is available for the flow and heat transfer in Newtonian fluids from a square cylinder over a wide range of Reynolds numbers in the cross-flow regime. Limited results – experimental and numerical – are also available for the flow of power law fluids over circular cylinders. In contrast, the corresponding literature for power law fluids is limited to the steady flow regime (1 ≤ Re ≤ 45) for both circular and square cylinders, except for a recent study on laminar vortex shedding from a circular cylinder in power law fluids [35]. No prior experimental or numerical results are available on the vortex-shedding characteristics (laminar) and the other related fluid mechanical aspects for the flow past a square cylinder. In view of the complex interplay between fluid shear-thinning and elasticity seen above for the circular cylinder case, it seems reasonable to first study the flow

159

Fig. 1. Schematics of the flow around a square cylinder.

characteristics of power law fluids to elucidate the role of shearthinning in an unambiguous manner and the level of complexity can gradually be built up in a systematic manner to incorporate viscoelastic effects. This study aims to fill this gap in the literature. In particular, extensive numerical results are presented herein for the time dependent flow of power law fluids past a square cylinder encompassing the following ranges of conditions 0.5 ≤ n ≤ 2 and 60 ≤ Re ≤ 160. However, the maximum value of the Reynolds number is chosen, based on the value of the power law index (n < 1), so that the simulations are limited to the fully-periodic regime. 2. Problem statement and mathematical formulation The physical problem considered here is the two-dimensional flow of an incompressible power law fluid around a square cylinder of size B, placed in a uniform stream having velocity u∞ (Fig. 1). Though the aim of this work is to study the unconfined flow, it is really not possible to simulate truly unconfined flow condition in numerical studies. Therefore, it is customary to postulate the presence of fictitious boundaries to make the problem computationally feasible. Naturally a prudent choice of the height of the computational domain H, upstream length LU and downstream length LD is necessary to obtain the results which are free from the domain effects. Based on the previous extensive studies [16,22], the values of H, LU and LD used in this work are consistent with the literature values. The governing equations, namely, the continuity and momentum, have been made dimensionless using the following scaling variables: B for length variables, U∞ for velocities, B/U∞ for time, 2 for pressure, m(U /B)n−1 for viscosity and U /B for the comU∞ ∞ ∞ ponents of the rate of deformation tensor. For an incompressible, 2D and laminar flow, the dimensionless integral forms of continuity, the x- and y-components of Cauchy’s equations are given below: Continuity:



V · dS = 0

(1)

S

x-Momentum:

∂ ∂t





u d˝ + ˝

2 pˆi · dS + Re S

S

y-Momentum:

∂ ∂t



uV · dS = −





v d˝ +

vV · dS = −

˝

S



2 pˆj · dS + Re S

 (iiˆi + ij ˆj) · dS S

(2)

 (jiˆi + jj ˆj) · dS S

(3)

where the dimensionless Reynolds number is defined as (n−2)

Re =

Bn U∞ m



(4)

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Here m and n are the constants of the power law viscosity model. ˆ s dS (n ˆ s is the unit normal vector to In Eqs. (2) and (3), dS is given by n the surface dS) and ˆi, ˆj are the unit vectors in the x- and y-directions, respectively. For an incompressible fluid, the extra stress tensor is related to the components of the rate of deformation tensor,  as

expected to be functions of the Reynolds number and the power law index. This relationship is investigated in this work.

ij = 2ij

In the present work, the general finite volume method of Eswaran and Prakash [36] and of Sharma and Eswaran [37] initially developed for complex 3D geometries on a non-staggered grid has been used here in its simplified form for 2D flows. In brief, the semi-explicit method is used to solve the Cauchy equations in conjunction with the first-order explicit scheme for performing time integration. The momentum equations are discretized in an explicit manner, with the exception of pressure terms which are treated implicitly. Two steps are implemented at each time level: first, a predicted velocity is obtained from the discretized momentum equation using the previous time-level pressure field; the second corrector step consists of the iterative solution of the pressure-correction equation and in obtaining the corresponding velocity corrections such that the final velocity field satisfies the continuity equation to the prescribed limit. The convective terms are discretized using the QUICK [16] scheme while the diffusive terms are discretized using the central difference scheme. The iterative process at each time step is continued until the divergence-free velocity field is obtained. The flow calculations are advanced by using a short time step (dt) to get the time dependent flow characteristics. In the present study, the time step is taken to be 1 × 10−4 . The use of smaller values of the time-step did not produce any significant change in the root mean square (rms) and average values of the lift and drag coefficients. In the present range of conditions, time period of vortex shedding varies from 5 to 7 units of non-dimensional time. Therefore, the time step 1 × 10−4 used in this study is sufficiently small to capture all time dependent effects. The simulation at a given Reynolds number is initiated either from rest (zero velocity everywhere) or from the converged flow field relating to the nearby values of the Reynolds number and power law index. Both initial conditions lead to the same flow field, only the computational time to achieve the stable periodic flow differs in these two cases. The grid structure used in the present work is identical to that used recently [38] and is shown here in Fig. 2. The grid is divided into five separate zones in both directions, and uniform as well as non-uniform grid distributions are employed. The grid distribution was made fine and uniform (cell size ı) around the cylinder to adequately capture wake-wall interactions in both directions. Zones far away from the cylinder are made up of uniform coarse cells (cell size ), where the flow is supposed to be same as mainstream flow and the gradients are therefore expected to be small. The hyperbolic tangent function has been used for stretching the cell sizes between these limits of and ı. Six different grid structures have been prepared to check the grid independence with ı = 0.02B, 0.0167B, 0.0125B, 0.01B, 0.0083B and 0.005B. In the first five structures, the size of coarse cell ( ) is 0.25B and in the sixth grid it is 0.125B. The details of the non-uniform grids used in grid independence study are presented in Table 1.

(5)

where ij = (1/2)(∂j Vi + ∂i Vj ). The non-Newtonian viscosity behaviour of the fluid is modeled here by the power law model. This model expresses the apparent viscosity  (˙  ) (ratio of shear stress to shear rate) as a function of the shear rate (˙  ) as follows.  = m˙ (n−1)

(6)

where m and n are the power law consistency index and flow behaviour index, respectively. Finally, the non-dimensional power law viscosity is calculated as follows:  = (2ij · ji )(n−1)/2

(7)

The boundary conditions for this flow may be written as follows: • Free slip boundary condition for the unconfined case at y = 0 and y = H (artificial boundaries) so that there is no viscous effect at these boundaries: ∂u = 0; ∂y

v=0

(8)

• No slip boundary condition on the surface of the square cylinder:

v=0

u = 0;

(9)

• The inlet boundary is characterized by a uniform x-velocity with no flow in the lateral direction, i.e.,

v=0

u = 1;

(10)

• At the outlet boundary, the convective boundary condition (Orlanski condition) is specified as follows: ∂ ∂ + Uc =0 ∂t ∂x

(11)

where Uc is assumed to be unity (the area average outflow velocity) and  is a dependent variable, u or v. The numerical solution of Eqs. (1)–(3) along with the abovenoted boundary conditions yields the velocity field which is further used to deduce the global characteristics like drag and lift coefficient and Strouhal number as follows: • Viscous drag coefficient CDf =



2 Re

1

[t (x) + b (x)] dx

(12)

0

• Pressure drag coefficient



1

CDp = 2

[pr (y) − pf (y)] dy

(13)

0

• Lift coefficient 2 CL = Re



1



[f (y) + r (y)] dy + 2 0

1

[pt (x) − pb (x)] dx

(14)

0

3. Numerical details

Table 1 Non-uniform grids used in grid independence study. S. No.

No. of uniform CV on the each face of the cylinder

Cell size (ı)

Grid size

1 2 3 4a 5 6

50 60 80 100 120 200

0.02B 0.0167B 0.0125B 0.01B 0.0083B 0.005B

223 × 179 246 × 200 295 × 241 325 × 271 365 × 303 649 × 541

• Strouhal number St =

fB U∞

(15)

Here, in Eqs. (12)–(14), the subscripts t, b, f and r denote respectively the top, bottom, front and rear surfaces of the cylinder. In Eq. (15), f is the frequency of vortex shedding. All these parameters are

a

Used in this work.

A.K. Sahu et al. / J. Non-Newtonian Fluid Mech. 160 (2009) 157–167

161

Fig. 3. Grid independence results for CD , CDp , CLrms and St at Re = 160 for n = 2.0.

Table 2 Comparison of CD , Lr , CLrms and St values in unsteady flow regime with literature. Re Fig. 2. Non-uniform computational grid with 325 × 271 grid points; (inset) enlarged view of grid near the cylinder.

40

n

Source

CD

Lr

CLrms

St

0.6

Dhiman et al. [22] Present Dhiman et al. [22] Present Dhiman et al. [22] Present Robichaux et al. [40] Present Sharma and Eswaran [16] Present

1.5986 1.5949 1.7616 1.7668 1.9396 1.8822 1.53 1.4895 1.4681 1.4613

2.1999 2.0969 2.8295 2.8261 4.07 4.5371 – – – –

– – – – – – – – 0.3183 0.3066

– – – – – – 0.154 0.1485 0.1596 0.1598

1.0

In order to obtain reliable and accurate results, it is important to choose carefully the length and width of the computational domain and grid size. The influence of the computational domain for the flow of Newtonian and power law fluids past a square cylinder has been extensively explored by Sharma and Eswaran [16] and Dhiman et al. [22]. Based on theses two studies, the upstream face distance (dimensionless) of the square cylinder from inlet, LU , is taken as 8.5 and the downstream face distance of the cylinder from the outlet, LD , is set at 16.5. The distance between the lower and upper slip boundaries, H, is kept at 20. However, a thorough grid independence study for the present problem has been done at the highest values of the Reynolds number of 160 and of the power law index of 2.0. Six non-uniform grids 223 × 179 (G1), 246 × 200 (G2), 295 × 241 (G3), 325 × 271 (G4), 365 × 303 (G5) and 649 × 541 (G6) have been used for this purpose. The effect of grid size on the major parameters characterizing the flow, namely, CD , CDp , CLrms and St is shown in Fig. 3. The percentage change in the values of the pressure and total drag coefficients, rms value of lift coefficient and the Strouhal number for the last two grids G5 (365 × 303) and G6 (649 × 541) are 0.8%, 0.6%, 1.1%, 0.07%, respectively. The corresponding changes between the grid G4 (325 × 271) and finest grid G6 (649 × 541) are 0.9%, 0.7%, 1.2% and 0.07%, respectively. Grid G6 has twice as many cells as G4 in both x- and y-directions. To estimate the uncertainty of Grid G4, the grid convergence index (GCI) proposed by Roache [39] has been evaluated assuming the present numerical method is to be first order, and using the factor of safety, Fs , equal to unity. The resulting values of GCI for grid G4 are 2%, 1.4%, 2.6% and 0.14% for CDp , CD , CLrms and St, respectively. The computation time with grid G6 is nearly five times of that with grid G4 while with grid G5 it is twice of that of G4. So one can conclude that the grid G4 (325 × 271) denotes a good compromise between the accuracy and the computational effort and is used in all further computations. The relatively low value of GCI for grid G4 gives confidence in the results reported in this work. The numerical method used here has been extensively validated and benchmarked by Sharma and Eswaran [16] for the flow of Newtonian fluids in the unsteady flow regime and by Dhiman et al. [22]

1.5 100

1.0

160

1.0

for the steady flow of power law fluids for a square cylinder. The present values of the key parameters including pressure drag coefficient, CDp , total drag coefficient, CD , in the steady flow regime, with rms value of lift, CLrms , and drag coefficient, CDrms , and the Strouhal number, St, in the unsteady flow regime are compared with those of Robichaux et. al. [40] and Sharma and Eswaran [16] and Dhiman et al. [22] in Table 2. As expected, an excellent match is seen to exist between the present and previous results, even though the diffusion terms in the momentum equations are treated slightly differently here than that in [22]. The validity of the present numerical solution methodology is further demonstrated by solving the well-known cavity flow problem for power law fluids. A comparison in terms of the peak values of u and v with the results presented by Neofytou [41] is presented in Table 3, where once again, the correspondence between the two sets of results is seen to be very good. The excellent agreements seen in Tables 2 and 3 not only inspires confidence in the use of this solver to study the effect of the Reynolds number and power law index on momentum transfer characteristics of an unconfined square cylinder in the 2D unsteady flow regime, but also indicates that the present results are probably more accurate than that suggested by the GCI values.

Table 3 Comparison of umin , vmax and vmin with Neofytou (2005) at Re = 100. n

Source

umin

vmax

vmin

0.75

Neofytou [41] Present Neofytou [41] Present

−0.1750 −0.1811 −0.2393 −0.2386

0.1400 0.1408 0.2278 0.2260

−0.2238 −0.2236 −0.2683 −0.2657

1.5

162

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Fig. 4. (A–C) Instantaneous streamlines during one cycle of vortex shedding behind a square cylinder at Re = 100 for n = 0.5, 1.0 and 2.0.

4. Results and discussion In this work, extensive computations have been carried out for Re = 60–160 in the step of 20 and for n = 0.5, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8 and 2.0 to elucidate the influence of shear-thinning (n < 1) and shear-thickening (n > 1) behaviour on the flow in the time-dependent flow regime. However, as mentioned earlier, the maximum value of the Reynolds number used here varies with the power law index n to steer away from the pseudo-periodic flow regime, as detailed in the ensuing sections. In order to keep the number of simulations at a reasonable level, the value of n was varied in steps of 0.2, while it might not provide specific information about the flow characteristics at the intermediate values (for instance at n = 0.9 or 1.1), but this choice is believed to capture the overall behaviour. Thus, for instance, if different flow phenomena is observed at n = 1.2, it is possibly also present at n = 1.05 or 1.1. Therefore the values of power law index covered herein are intended to qualitatively capture the role of fluid behaviour on flow characteristics and the present results have been discussed in terms of shear-thinning (n < 1) and shear-thickening (n > 1) fluid behaviours. It has been shown that flow is 2D and steady at Re ≤ 45 for the values of n considered here [22]. For the flow of Newtonian fluids (n = 1), linear stability analysis of Kelkar and Patankar [10] suggests the flow to become time dependent for Re > 53, whereas Sohankar et al. [12] found vortex shedding to begin at Re = 51.2 ± 1.0. Discrepancies of this order in the critical values of the Reynolds number are not at all uncommon in such situations. In the case of the flow over a circular cylinder, the vortex shedding is seen to be delayed as the fluid behaviour changes from shear-thickening to Newtonian and finally to shear-thinning. In other words, shear-thinning fluids transit to unsteadiness at higher and shear-thickening fluids at lower critical Reynolds numbers than that seen in Newtonian fluids. In order to support the range of unsteady flow regime, we have carried out some simulations in the steady and transition zone and found that for all values of power law index (shear-thinning, Newtonian and shear-thickening), the flow over a square cylinder is time dependent at Re ≥ 60. Fig. 4 presents the representative instantaneous streamlines in the vicinity of the cylinder for n = 0.5, 1.0 and 2.0 and Re = 100. Also shown in these Fig. 4(A)–(C) is the phenomenon of vortex shedding

at eight successive moments of time which span one time period of the cycle of shedding (first moment will be repeated after the eighth moment for the next cycle of vortex shedding). The vortex forming on the top of rear face grows and breaks off in figure (a)–(d) in all three cases. Similar phenomenon also occurs in the next half of the vortex-shedding cycle at the bottom of the rear face (figure (e)–(h)). This alternate breaking-off of vortices from the rear face of the cylinder makes the flow periodic. Fig. 4 shows that the vortex-shedding process for shear-thinning and shear-thickening fluids is qualitatively similar to that in Newtonian fluids. However, the leading edge separation is observed at lower Reynolds numbers for shear-thinning fluids, as seen in instantaneous streamline plots (Fig. 4(A)) for n = 0.5 compared to the only trailing edge separation for n = 1.0 and 2.0 at Re = 100. Sharma and Eswaran [16] reported the onset of leading edge separation to occur at Re > 100 in Newtonian fluids. Therefore, the present results are inline with their findings in this regard. Typical time-averaged streamlines for different values of the power law index and a range of Reynolds numbers are presented in Fig. 5. A detailed examination of the results shows that the shearthickening behaviour delays the onset of leading edge separation while for shear-thinning fluids it occurs at Re < 100 depending upon the value of the power law index. Clearly, all else being equal, as the value of the power law index is gradually increased, the effective viscosity of the fluid close to the cylinder increases, and tends to stabilize the flow. Shear-thinning fluids exhibit three different types of vortex shedding flow: no separation from sides; separation and reattachment on the sides; separation at the leading edge without reattachment on the side, depending upon the values of the power law index and the Reynolds number which is qualitatively similar to the behaviour seen in Newtonian fluids, though the transitional Reynolds numbers seem to be different in the two cases. In contrast, no leading edge separation is observed for n > 1 in the range of conditions used in this study. The length of the recirculation region of the time-averaged flow is shown in Fig. 6. Irrespective of the type of fluid, the mean wake length decreases with the increasing Reynolds number. Further examination of Fig. 6 reveals that though the gradient of Lr –Re plots decreases for all values of n (n > 1 and n < 1), but in the case of shear-thinning fluids n < 1 at high-Reynolds number an increase in gradient is observed. In shear-thinning fluids, in addi-

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Fig. 5. Time averaged streamlines representing the trailing and leading edge separation at Re = 60, 100 and 160 for n = 0.5, 0.8, 1.0 and 2.0.

Fig. 6. Variation of mean wake length with Reynolds number in shear-thickening and shear-thinning fluids.

tion to the wake in the rear of the cylinder, a separate wake forms over the top and bottom faces of the cylinder due to the leading edge separation. As the Reynolds number is increased further, the flow does not reattach itself and therefore the two wakes merge thereby increasing the width of the wake, which is accompanied by a concomitant decrease in the length of the recirculation zone in the x-direction. This manifests itself as a sharp decrease in Lr seen in Fig. 6. In particular, for n = 0.8, the gradient of Lr –Re plot decreases up to Re = 140 and then increases between Re = 140 and 160. The minimum Reynolds number at which the leading edge separation is first observed is evidently dependent upon the value of the power law index. Broadly, in comparison with Newtonian fluids, the leading edge separation is delayed in shear-thickening fluids to larger values of Reynolds number, and occurs at lower values of the Reynolds number in shear-thinning fluids, as seen in Table 4. This can possibly be ascribed to the relatively low effective viscos-

ity of a shear-thinning fluid close to the cylinder and hence the local Reynolds number would be, in effect, larger than the nominal global value. Furthermore, at high values of the Reynolds number and low values of the power law index, shear-thinning fluids exhibit a quasiTable 4 Reynolds number (approximate value) at which leading edge separation occurs. n

Re

0.5 0.6 0.7 0.8 1.0

60 60 80 100 120

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Fig. 7. Lift coefficient spectra at Re = 160 for n = 0.6 and 0.7.

periodic flow. It is useful to recall here that for the case of Newtonian fluids, a similar quasi-periodicity sets in at Re > 210 [42]. It is customary to analyze this phenomenon in terms of a Fourier transform (FT). The FTs of lift coefficient at Re = 160 and n = 0.6 and 0.7 are presented in Fig. 7. In this figure, the single peak of power spectra at n = 0.7 shows that the flow is truly periodic in nature with a single dominant frequency, while the number of peaks seen at n = 0.6 indicates the quasi-periodic nature of the flow under these conditions. In the present study, this type of quasi-periodicity is observed for Re > 120 and n < 0.6. The frequency of vortex shedding is measured from the plots of temporal variation of lift coefficients. The non-dimensional frequency of oscillations in lift coefficient is termed as the Strouhal number. The frequency determined from the Fourier transform of

the lift coefficient is found to be same as that obtained from the frequency seen in the time series. Fig. 8 shows the variation of Strouhal number with the Reynolds number for shear-thinning and shearthickening fluids. For shear-thickening fluids the Strouhal number is seen to increase monotonically with the increasing Reynolds number. On the other hand, this dependence is seen to be quite different in shear-thinning fluids. In particular, depending upon the value of the power law index, the value of St increases to a maximum value, followed by a sharp drop as the Reynolds number is incremented. This is so presumably due to the leading edge separation in shear-thinning fluids under these circumstances which does not happen in shear-thickening fluids over this range of Reynolds numbers. The leading edge separation (without re-attachment) will lead to the widening of the wake in the y-direction and thereby lowering its frequency of shedding. For power law fluids, one can also n whereas the inertial argue that, the viscous forces scale as ∼U∞ 2 . Thus, for a shear-thinning fluid, viscous forces will scale as ∼U∞ forces diminish as the value of the power law index is gradually decreased whereas for a shear-thickening fluid, the viscous force will increase hand in hand with the inertial forces. Thus, it is likely that some of non-monotonic trends seen in Figs. 8 and 9 arise from the interaction between these two non-linear terms which scale differently with velocity. The drag force exerted by the fluid on the square cylinder is made up of two components: viscous drag which acts only on the top and bottom surfaces of the cylinder in the x-direction and the pressure drag which acts on the front and rear surfaces of the cylinder. The corresponding coefficients are given by Eqs. (12) and (13) and the total drag coefficient is simply the sum of these two components. The variation of the time-average values of the total drag coefficient and the pressure drag coefficient with Re for both shear-thinning and shear-thickening fluid behaviours is shown in Figs. 9 and 10, respectively. For shear-thinning fluids, the recirculation at the top and bottom surfaces of the cylinder makes the friction drag coefficient (CDf = CD − CDp )negative, and both CD and CDp increase with the increasing Reynolds number. Shear-thickening fluids show the opposite trend, i.e., both CD and CDp decrease with the increasing Reynolds number. The variation of the rms (root mean square) values of the lift and drag coefficient with the Reynolds number Re shown in Figs. 11 and 12 respectively gives an indication of the amplitude of variations present in the flow. The rms value represents the amplitude of oscillations. Similar to Newtonian fluids, both shear-thinning and shear-thickening fluids show an increase in both these quantities

Fig. 8. Variation of Strouhal number with Reynolds number for shear-thickening and shear-thinning fluids.

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Fig. 9. Variation of time-averaged total drag coefficient with Reynolds number for shear-thickening and shear-thinning fluids.

Fig. 10. Variation of time-averaged pressure coefficient with Reynolds number for shear-thickening and shear-thinning fluids.

with the increasing Reynolds number in the present range of conditions. Finally, it is also appropriate to add here that further examination of the velocity-time behaviour for the quasi-periodic flow conditions revealed that as one moves away from the cylinder, the

spectrum broadens and the dominating frequency becomes weaker than that closer to the cylinder. This clearly suggests the flow to be more periodic near the cylinder in this regime. However, a more detailed discussion of quasi-periodicity as well as the issue of 2D to 3D transition still remains to be explored. This will, presumably,

Fig. 11. Variation of rms value of lift coefficient with Reynolds number for shear-thickening and shear-thinning fluids.

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Fig. 12. Variation of rms value of drag coefficient with Reynolds number for shear-thickening and shear-thinning fluids.

entail 3D simulations and hopefully will be addressed in future studies. Since, as of now, no experimental data is available on this problem, the numerical results presented herein can serve as an useful starting point to stimulate experimental work in this area. 5. Conclusions The unsteady flow of power law fluids past an unconfined square cylinder is numerically investigated. In particular, the effects of the Reynolds number and the power law index on the flow structure, drag and lift coefficients, and Strouhal number have been studied for the range of Reynolds numbers, 60 ≤ Re ≤ 160 and power law index, 0.5 ≤ n ≤ 2.0. Similar to the Newtonian fluid, shear-thickening and shear-thinning fluids exhibit vortex shedding over the range of conditions. But the transition values of Reynolds number denoting the onset of leading edge separation in shearthinning fluids is lower than the value for Newtonian fluids. On the other hand, no leading edge separation is seen to occur in shearthickening fluids, at least in the present range of Reynolds number and power law index. While the drag coefficient decreases with the increasing Reynolds number in shear-thickening fluids. In the case of shear-thinning fluids, recirculation at the top and bottom surfaces of the cylinder at high-Reynolds number and/or in highly shear-thinning fluids leads to an increase in the total drag coefficient. Furthermore, in the present range of conditions, the flow of shear-thickening fluids is truly periodic in nature while in the case of shear-thinning fluids, it becomes pseudo-periodic at highReynolds numbers and/or at small values of power law index, i.e., in highly shear-thinning fluids. Acknowledgements We are grateful to two anonymous reviewers for making numerous constructive suggestions which have led to significant improvements. References [1] C.H.K. Williamson, Vortex dynamics in the cylinder wake, Annu. Rev. Fluid Mech. 28 (1996) 477–539. [2] M.M. Zdravkovich, Flow Around Circular Cylinders Volume 1, Fundamentals, Oxford University Press, New York, 1997. [3] M.M. Zdravkovich, Flow Around Circular Cylinders Volume 2, Applications, Oxford University Press, New York, 2003. [4] R.P. Bharti, R.P. Chhabra, V. Eswaran, A numerical study of the steady forced convection heat transfer from an unconfined circular cylinder, Heat Mass Transfer 43 (2007) 639–648.

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