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Ind. Eng. Chem. Res. 2007, 46, 3820-3840
Two-Dimensional Steady Poiseuille Flow of Power-Law Fluids Across a Circular Cylinder in a Plane Confined Channel: Wall Effects and Drag Coefficients Ram Prakash Bharti and R. P. Chhabra* Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India
V. Eswaran Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India
The Poiseuille flow of incompressible power-law fluids past a circular cylinder placed midway between two parallel plates has been investigated numerically by solving the continuity and momentum equations using FLUENT (Version 6.2). Extensive results highlighting the roles of the Reynolds number (Re), the power-law index (n), and the blockage ratio (β) on the global and detailed flow characteristics have been presented over wide ranges of conditions (1 e Re e 40, 0.2 e n e 1.9, and 1.1 e β e 4). For a fixed value of the blockage ratio, the drag coefficient increases as the shear-thickening (n > 1) tendency of the fluid increases, whereas shear-thinning (n < 1) fluid behavior shows the opposite dependence. At small Re, this effect is observed to be very strong and it gradually diminishes as Re increases. The effect of Re diminishes for n > 1 with a decrease in β, whereas the dependence becomes stronger for n < 1. Individual drag coefficients also show qualitatively similar dependence on Re, n, and β. In addition, the streamline and pressure profiles have also been presented to provide further physical insights into the detailed kinematics of the flow. The wake size is observed to increase as the flow behavior index (n) decreases. Because of wall effects, the flow separation seems to be delayed in shear-thickening fluids, whereas the opposite trend was observed in shear-thinning fluids. While the pressure profiles are observed to be similar to that for an unconfined flow, the flattening of the pressure curve in the rear portion of the cylinder suggests sluggish pressure recovery due to wall effects. In contrast to an unconfined flow, the front stagnation pressure coefficient values can be negative in magnitude, under certain conditions. The dependence of the pressure coefficient on the flow behavior index intensifies in shear-thickening fluids with a decrease in the blockage ratio. 1. Introduction Because of its fundamental and pragmatic significance, considerable research efforts have been devoted to the study of crossflow of fluids past cylinders of circular and noncircular cross sections. Typical examples include the flow in tubular and pin heat exchangers, in the resin-transfer molding (RTM) process of manufacturing fiber-reinforced composites, in filtration screens and aerosol filters, etc. Consequently, a voluminous body of knowledge is now available on the transverse flow of Newtonian fluids over a circular cylinder in an unconfined flow configuration (e.g., see, Zdravkovich1,2), although much less is known about the effect of confining walls on the flow phenomenon, even for Newtonian fluids. The severity of confinement is characterized by defining a blockage ratio (β or λ). For a channel of height H and circular cylinder of diameter D, the blockage ratio is defined as follows:
H D
(1a)
D 1 ) H β
(1b)
β) or
λ)
Zdravkovich2 classified the blockage effects for the Newtonian fluid flow over a circular cylinder as follows: * To whom correspondence should be addressed. Tel.: +91-512259 7393. Fax: +91-512-259 0104. E-mail:
[email protected].
(1) For λ < 0.1, the wall effects are small and may be ignored. (2) For 0.1 e λ e 0.6, the blockage modifies the flow and the correction is necessary. (3) For λ > 0.6, the blockage radically alters the flow around the cylinder and the correction of data is unjustified. This rather crude classification is applicable for all flow regimes except in the two-dimensional laminar flow. At very low Reynolds number (Re) values in the two-dimensional laminar flow regime, the wall effects are known to be important, even for λ < 0.001, because of the slow decay of the flow (velocity) field. All in all, a reasonable body of information is now available on the crossflow of Newtonian fluids over a circular cylinder. On the other hand, it is readily acknowledged that many substances encountered in industrial practice (pulp and paper, food, polymer and process engineering applications) display shear-thinning and/or shear-thickening behavior.3 Because of their high viscosity levels, non-Newtonian substances are generally processed under laminar flow conditions. Admittedly, many non-Newtonian fluids (notably, polymeric systems) display viscoelastic behavior; the available scant literature both for the creeping flow past a single cylinder and over a periodic array of cylinders seems to suggest the viscoelastic effects to be minor in this flow configuration.4 Furthermore, the fluid relaxation time often decreases with shear rate. Thus, the relaxation time will also decrease as Re increases and, hence, a suitably defined Deborah number would be small. Therefore, the viscoelastic effects are not expected to be significant in this case.
10.1021/ie070166+ CCC: $37.00 © 2007 American Chemical Society Published on Web 05/02/2007
Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 3821
B2 ) 1.26653975 B4 ) -0.91804 B6 ) 1.87710 B8 ) -4.66549
Figure 1. Schematic representation of a channel confined (Poiseuille) flow over a circular cylinder.
Similarly, the currently available numerical simulations examine the role of viscoelasticity in the absence of shear-dependent viscosity and predict very little change in the value of drag coefficient and, thus, it seems that the steady flow resistance is determined primarily by the viscous properties of the fluid.4 Therefore, it seems reasonable to begin with the flow of purely viscous power-law type fluids as long as the power-law constants are evaluated in the shear-rate range that is appropriate for the flow over a cylinder and the level of complexity can be built up gradually to accommodate the other non-Newtonian characteristics. To our knowledge, there has been no prior study on the crossflow of power-law fluids past a circular cylinder confined in a channel. This work intends to fill this gap in the literature. At the outset, it is instructive; however, to briefly recount the available limited work on the flow of power-law fluids past an unconfined circular cylinder to facilitate the subsequent presentation of the new results for the flow of power-law fluids past a confined circular cylinder. 2. Previous Work The problem of Poiseuille flow (see Figure 1 for the schematics) of Newtonian fluids past a cylinder placed symmetrically between two fixed parallel walls was studied analytically by Faxen5 for λ e 0.5. Using the method of images to obtain the series expansion, he presented the following expression for drag:
CD )
FD(λ) 6π ) µUavg f(λ) + g(λ)
where
f(λ) ) A0 - (1 + 0.5λ2 + A4λ4 + A6λ6 + A8λ8) ln(λ) g(λ) ) B2λ2 + B4λ4 + B6λ6 + B8λ8 and
A0 ) -0.9156892732 A4 ) 0.05464866 A6 ) -0.26462967 A8 ) 0.792986
(2)
For small values of λ, Faxen’s solution is quite accurate. Some other attempts have also been made to solve the Oseen’s linearized equations.6,7 For λ < 0.2, the numerical results show good agreement with the limited experimental results of White.8 On the other hand, the extent of wall effects on drag coefficient in an uniform flow past a cylinder placed symmetrically between two plane walls (moving with the uniform stream) has been investigated numerically by Huang and Feng9 for 0.1 e Re e 10 and 1.25 e β e 500, by Ben Richou et al.10 for 0.01 e λ e 0.6 at very low Reynolds numbers (10-4 e Re e 1) and by Chakraborty et al.11 in the range 1.54 e β e 20 and 0.1 e Re e 200, respectively. They also presented an equation for the drag coefficient as a function of the blockage ratio β and the Reynolds number Re. Using a stream functionvorticity formulation, in conjunction with the finite element method, Anagnostopoulos et al.12 presented results on the influence of blockage ratio (λ ) 0.05, 0.15, and 0.25) on the steady and unsteady wake characteristics at a fixed value of the Reynolds number (Re ) 106). Recently, the influence of wall blockage (0.01 e λ e 0.6) on the drag force at very low Reynolds numbers (10-4 e Re e 1) has been investigated in Poiseuille flow by Ben Richou et al.13 They reported a good match with the predictions of Faxen and with their experimental results. Aside from these experimental and numerical studies, the effects of blockage on the flow and heat transfer have also been investigated analytically using the standard boundary layer approximation.14,15 Vaitiekunas et al.14 investigated the effect of channel blockage on the dimensionless shear stress, the location of Umax (which is the point of boundary layer detachment), and the local heat-transfer coefficient. Using the experimental results available in the literature,16 they approximated the velocity distribution outside the boundary layer by the modified Hiemenz polynomial, as a function of the wall blockage. Recently, Khan et al.15 have presented a functional dependence of the drag and the Nusselt number (Nu) on the Reynolds number and blockage ratio using the modified von Karman-Pohlhausen method, which uses the fourth-order velocity profile inside the boundary layer. Outside the boundary layer, they obtained the potential flow velocity using the method of images. They assumed a third-order temperature profile to solve the corresponding thermal energy equation for the two commonly used thermal boundary conditions. Thus, in summary, only very limited information is available on the role of confining walls, even for the flow of Newtonian fluids past a circular cylinder. On the other hand, reliable results are now available on the power-law fluid flow and the associated convection past an unconfined circular cylinder (e.g., see refs 17-29). Overall, the currently available numerical results go up to Re ) 40 and power-law indexes in the range of 0.2 e n e 2. The results obtained using different numerics and domains, etc., show excellent correspondence, thereby confirming their reliability and accuracy. However, to the best of our knowledge, there has been no prior study elucidating the role of confining (planar) walls on
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Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 Table 3. Comparison of the Present Total Drag Values with the Available Literature Values for a Channel Confined Flow Across a Circular Cylinder Present Total Drag Value β)2
source
Re ) 0.01 132.26 138.48 133.23
present work Faxen5,a Ben Richou et al.13,b
β)4
β ) 10
31.933 31.943
13.358 13.359 13.745
Re ) 1 present work Ben Richou et al.13,b a
Figure 2. Schematics representation of a two-dimensional nonuniform computational grid (a) in the full domain and (b) in the region close to the cylinder. Table 1. Effect of Downstream Length (Ld) on the Drag Coefficientsa n ) 0.2 Ld/D
CDP
CDF
n)1 CD
CDP
CDF
n ) 1.9 CD
CDP
CDF
CD
40 60 80
Re ) 1, β ) 100 0.19 -1.48 -0.31 -0.05 -0.05 -0.05 -0.08 -0.09 -0.09 -1.35 1.71 -0.44 -0.04 -0.03 -0.03 -0.07 -0.08 -0.07 -0.21 0.32 -0.06 -0.03 -0.03 -0.03 -0.07 -0.08 -0.08
40 60 80
Re ) 40, β ) 100 0.03 0.01 0.03 -0.10 -0.10 -0.10 0.04 0.01 0.03 -0.10 -0.09 -0.10 -0.21 -0.04 -0.19 0.04 0.01 0.03 -0.10 -0.09 -0.09
40 60 80
Re ) 1, β ) 1.1 -0.01 -0.01 -0.01
