Chemical Engineering Science 62 (2007) 1682 – 1702
www.elsevier.com/locate/ces
Steady flow of power-law fluids across an unconfined elliptical cylinder
P. Sivakumar, Ram Prakash Bharti, R.P. Chhabra ∗
Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India
Received 3 July 2006; received in revised form 25 November 2006; accepted 27 November 2006
Available online 14 December 2006
Abstract
The momentum transfer characteristics of the power-law fluid flow past an unconfined elliptic cylinder is investigated numerically by solving
continuity and momentum equations using FLUENT (version 6.2) in the two-dimensional steady cross-flow regime. The influence of the
power-law index (0.2 n 1.8), Reynolds number (0.01 Re 40) and the aspect ratio of the elliptic cylinder (0.2 E 5) on the local and
global flow characteristics has been studied. In addition, flow patterns showing streamline and vorticity profiles, and the pressure distribution
on the surface of the cylinder have also been presented to provide further physical insights into the detailed flow kinematics. For shear-thinning
(n < 1) behaviour and the aspect ratio E > 1, flow separation is somewhat delayed and the resulting wake is also shorter; on the other hand,
for shear-thickening (n > 1) fluid behaviour and for E < 1, the opposite behaviour is obtained. The pressure coefficient and drag coefficient
show a complex dependence on the Reynolds number and power-law index. The decrease in the degree of shear-thinning behaviour increases
the drag coefficient, especially at low Reynolds numbers. While the aspect ratio of the cylinder exerts significant influence on the detailed
flow characteristics, the total drag coefficient is only weakly dependent on the aspect ratio in shear-thickening fluids. The effect of the flow
behaviour index, however, diminishes gradually with the increasing Reynolds number. The numerical results have also been presented in terms
of closure relations for easy use in a new application.
䉷 2007 Elsevier Ltd. All rights reserved.
Keywords: Power-law fluids; Elliptical cylinders; Drag coefficients; Pressure coefficients; Reynolds number; Shear-thinning; Shear-thickening
1. Introduction
Owing to its fundamental and pragmatic significance, considerable research efforts have been devoted to the study of
cross-flow of fluids past cylinders of circular and non-circular
cross-sections. Typical examples include the flow in tubular
and pin heat exchangers, in the RTM process of manufacturing fibre 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, e.g., see, Zdravkovich (1997, 2003). In contrast, much less is known about the role of the cross-sectional
shape of the cylinder on the flow phenomenon. For instance,
only limited results available for the flow of Newtonian fluids
past a cylinder of square and rectangular cross-sections have
been investigated numerically in the low-to-moderate Reynolds
∗ Corresponding author. Tel.: +91 512 2597393; fax: +91 512 2590104.
E-mail address: [email protected] (R.P. Chhabra).
0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2006.11.055
number range during the last 10–20 years (Dhiman et al., 2005,
2006a; Sharma and Eswaran, 2004, 2005; Chhabra, 2006,
and references therein). Likewise, the tubes of elliptic crosssection are used commonly in tubular heat exchangers as these
offer good space economy without sacrificing heat transfer-flow
resistance benefits.
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 behaviour (Chhabra and Richardson, 1999).
Owing to their high viscosity levels, non-Newtonian substances
are generally processed in laminar flow conditions. Though
many non-Newtonian fluids display viscoelastic behaviours,
the available scant literature both for the flow past a single
cylinder and a periodic array of cylinders seems to suggest the
viscoelastic effects to be minor in this flow configuration
(Chhabra, 2006; Liu et al., 1998; Talwar and Khomami,
1995). Furthermore, the fluid relaxation time often shows a
dependence on the shear rate, which is similar to the sheardependence of viscosity. Thus, the relaxation time will also
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
decrease with the increasing value of the Reynolds number and
hence, a suitably defined Deborah number would also be small.
On this count, the viscoelastic effects are not expected to be
significant in this case. Unfortunately, most numerical simulations examine the role of viscoelasticity in the absence of sheardependent viscosity and predict very little change in the value
of drag coefficient. Thus, it appears that the steady flow resistance is determined primarily by the flow geometry (shape of
the cylinder) and the viscous properties of the fluid. 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 appropriate for the flow over
a elliptical cylinder, and the level of complexity can be built
up gradually to accommodate other non-Newtonian characteristics. Admittedly, the power-law model does not predict the socalled zero shear viscosity in the limit of zero shear rate, but the
available numerical and experimental studies for the flow over
a sphere and a circular cylinder clearly show that despite this
deficiency, the use of this model yields satisfactory predictions
of the flow parameters (Chhabra, 2006; Schowalter, 1978).
As far as known to us, there has been no prior study of the
cross-flow of power-law fluids over an elliptic cylinder. This
work aims 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 circular cylinders
and on Newtonian fluid flow past elliptic cylinders to facilitate
the subsequent presentation of the new results for the flow of
power-law fluids past an elliptic cylinder.
2. Previous work
The flow of Newtonian fluids past an unconfined circular
cylinder has been studied extensively, e.g., see Zdravkovich
(1997, 2003). All in all, a reasonable body of experimental
and/or numerical results is now available on almost all aspects
of the cross-flow of Newtonian fluids past a circular cylinder
over most conditions of interest.
In contrast, only limited information is available for the
Newtonian flow over an elliptical cylinder, even in the steady
cross-flow regime. Imai (1954) and Hasimoto (1958) studied
the steady flow past an elliptic cylinder at very low Reynolds
numbers by using the Oseen’s linearized equations and presented an expression for drag. Lugt and Haussling (1974)
numerically solved the flow equations for the laminar flow past
an abruptly started elliptic cylinder at 45◦ incidence. In particular, time required to reach the steady or quasi-steady state
flow was investigated by following the evolution of streamlines
and lines of constant vorticity and of drag, lift and moment
coefficients with time. Meller (1978) numerically solved the
vorticity-stream function formulation of the Navier–Stokes
equations using the finite difference method and the matrix pivotal condensation method for only two values of the Reynolds
number of 20 and 40 and for an aspect ratio of 10 at an angle
of attack of 30◦ . Subsequently, Patel (1981) presented a semianalytical solution of the Navier–Stokes equations to investigate the viscous, incompressible flow around an impulsively
started elliptic cylinder at various angles of incidence (0, 30, 45
1683
and 90◦ ) at Re = 200. D’Alessio and Dennis (1994) employed
the vorticity-stream function form of the Navier–Stokes equations in terms of transformed coordinates to enforce the correct
decay of vorticity at large distances from the elliptic cylinder.
They presented the values of drag and lift coefficients for Re=5
and 20 for different inclinations of the elliptic cylinder ranging
from 0◦ to 90◦ . Subsequently, this flow was revisited by Dennis
and Young (2003) who studied the steady two-dimensional flow
past an elliptic cylinder at various inclinations to the oncoming
stream by solving the Navier–Stokes equations in the range of
the Reynolds number (1 Re 40) and for a range of inclinations (0–90◦ ) for one value of the aspect ratio E = 5. Using the
spectral-element method, Johnson et al. (2001) reported the
effects of the Reynolds number (30 Re 200) and of aspect
ratio (0.01 E 1) on the Strouhal number, drag coefficients
and on the onset of vortex shedding. They reported the conditions for the steady flow regime to be relatively insensitive to the
value of aspect ratio E, whereas the span of the other regimes
narrows down with a decrease in the value of E. For E < 1,
the steady regime was reported to end at Re ≈ 35–40 which is
comparable to the commonly used value of ∼ 40–45 for circular and square cylinders. Apart from the above-referenced numerical studies, Khan et al. (2005) have recently employed the
boundary layer approximation of the momentum and energy
equations using the Von Karman–Pohlhausen integral method
for flow and heat transfer from an elliptical cylinder (E 1).
Scant experimental results are also available for the flow and
heat transfer from elliptic cylinder, but most of these relate to
high Reynolds number, e.g., see Ilgarbuis and Butkus (1988),
Nishiyama et al. (1988a,b), Ota et al. (1987), Seban and Drake
(1953), etc. and these have focused on time-dependent flow
characteristics. Apart from these single cylinder studies, some
results are also available for Newtonian fluid flow and heat
transfer in bundles of elliptic cylinders (Epstein and Masliyah,
1972; Masliyah, 1973). Therefore, only limited information is
available even for the flow of Newtonian fluids over an elliptic
cylinder.
Even less is known about the power-law fluid flow over a
circular cylinder and as far as known to us, no prior results
are available for elliptic cylinders. Since the detailed discussions of these for a circular cylinder are available elsewhere
(Bharti et al., 2006, 2007; Sivakumar et al., 2006), only the
salient features are recapitulated here. The three creeping flow
studies (Ferreira and Chhabra, 2004; Tanner, 1993; Whitney
and Rodin, 2001) are in excellent agreement with each other
for n < 1, as far as the values of the total drag coefficients are
concerned; the corresponding values of the limiting Reynolds
number denoting the cessation of the creeping flow regime have
been delineated recently (Sivakumar et al., 2006). Likewise,
the two independent numerical studies in the two-dimensional
steady flow regime for finite values of the Reynolds number
(Re 40) (Bharti et al., 2005, 2006, 2007; Chhabra et al., 2004;
Soares et al., 2005) are also in good agreement with each other.
For the sake of completeness, it is also appropriate to mention here that limited results for the steady flow of powerlaw fluids past confined and unconfined square (Dhiman et al.,
2006b; Gupta et al., 2003; Paliwal et al., 2003) and rectangular
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P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
(Nitin and Chhabra, 2005) cylinders are also available. In the
latter cases, the drag coefficient normalized using the corresponding Newtonian values shows only weak additional dependence on the power-law index and over the Reynolds number
range of 5–40. Before leaving this section, it is appropriate to
add here that there has been one study for the creeping flow of
power-law fluids through periodic arrays of elliptic cylinders
(Woods et al., 2003). In principle, these results should reduce
to the limit of a single elliptic cylinder as the solid volume
fraction approaches zero. However, since the drag coefficient
shows a non-linear and rather strong dependence on volume
fraction, this approach yields reliable values only when the
results are available for extremely small values of the solid volume fraction.
Thus, it is safe to conclude that there has been no prior
study on the steady flow of power-law fluids past an unconfined elliptical cylinder. The present work is concerned
with the determination of the detailed kinematics and macroscopic hydrodynamic characteristics in the two-dimensional
steady cross-flow regime for the range of Reynolds number
as 0.01Re 40, aspect ratio as 0.2 E 5, and power-law
index as 0.2 n 1.8, thereby covering both shear-thinning
(n < 1) and shear-thickening (n > 1) fluid behaviours.
Consider the two-dimensional, steady flow of an incompressible power-law liquid with an uniform velocity U∞ across a
long elliptical cylinder of aspect ratio E. The unconfined flow
condition is simulated here by enclosing the elliptic cylinder
in a circular outer boundary of diameter D∞ , as shown in
Fig. 1. The diameter of the outer circular boundary D∞ is
taken to be sufficiently large to minimize the boundary effects
(Fig. 1). The continuity and momentum equations for this flow
configuration are written as
continuity equation : ∇.u = 0,
(1a)
momentum equation : (u.∇u − f) − ∇. = 0,
(1b)
where, , u, f and are the density, velocity (Ux and Uy ), body
force and the stress tensor, respectively. The stress tensor, sum
of isotropic (pressure, p) and the deviatoric stress tensor (), is
given by
(2)
The rheological equation of state for power-law fluids is given
by
= 2(u),
(3)
where (u), the components of the rate of strain tensor, is given
by
(u) = 21 {(∇u) + (∇u)T }.
For a power-law fluid, the viscosity, , is given by
(n−1)/2
I2
=m
,
2
I2 = (2xx + 2yy ) + (2xy + 2yx )
(4)
(5)
(6)
and the components of the rate of strain tensor are related to
the velocity components as follows:
jUy
jUy
jUx
1 jUx
, yy =
, xy = yx =
+
.
xx =
jx
jy
2 jy
jx
(7)
The physically realistic boundary conditions for this flow may
be written as follows:
• At the inlet boundary: The condition of the uniform flow is
imposed, i.e.,
Ux = U∞
and
Uy = 0.
(8a)
• On the surface of the elliptical cylinder: The standard no-slip
condition is used, i.e.,
Ux = 0
3. Problem statement
= −pI + .
where m is the power-law consistency index and n is the powerlaw index of the fluid (n < 1: shear-thinning; n = 1: Newtonian;
and n > 1: shear-thickening fluids) and I2 is the second invariant
of the rate of strain tensor () given by
and
Uy = 0.
(8b)
• At the exit boundary: The default outflow boundary condition
in FLUENT, which assumes a zero diffusion flux for all flow
variables, was used. Physically this condition implies that the
conditions of the outflow plane are extrapolated from within
the domain and have negligible impact on the upstream flow
conditions. The extrapolation procedure used by FLUENT
updates the outflow velocity and pressure in a manner that
is consistent with a fully developed flow assumption, when
there is no area change at the outflow boundary. It is important
to note here that gradients in the cross-stream direction may
exist at the outflow boundary. Only the diffusion fluxes in
the direction of flow at the exit plane are assumed to be zero.
Also, the use of this condition obviates the need to prescribe
any boundary condition for pressure.
This is similar to the homogeneous Neumann condition,
given by
jUx
=0
jx
and
jUy
= 0.
jx
(8c)
• At the plane of symmetry, i.e., centre line (y = 0): The symmetric flow condition has been used. It can be written as
follows:
jUx
=0
jy
and
Uy = 0.
(8d)
Owing to the symmetry of the flow over the range of conditions
studied herein, the solution is obtained only in the upper half of
the domain y 0 (Fig. 1). The numerical solution of Eqs. (1a)
and (1b) along with the above-noted boundary conditions yields
the velocity (Ux and Uy ) and pressure (p) fields and these, in
turn, are used to deduce the local and global characteristics like
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
a
b
c
D∞
D∞
D∞
a
a
Uniform Velocity U∞
1685
b
Outflow
a
b
Uniform Velocity U∞
Uniform Velocity U∞
Outflow
b
Outflow
Cylinder
Cylinder
y
Cylinder
y
y
x
Outer Boundary
Outer Boundary
x
x
Outer Boundary
Fig. 1. Schematic representation of the unconfined flow around a elliptical cylinder: (a) E < 1(a > b), (b) E = 1 (a = b, circular cylinder), (c) E > 1(a < b).
stream-function, vorticity, wake length, pressure coefficients,
drag coefficients, as outlined below.
At this point, it is appropriate to introduce some definitions.
• The Reynolds number (Re) for power-law fluids is defined
as follows:
Re =
2−n
(2a)n U∞
.
m
(9)
• Surface vorticity, , is computed by the following expression:
jUr
=
,
js
(10)
where Ur = (Ux sin + Uy cos ), and is the angular displacement from the front stagnation point and s is the unit
vector normal to the surface of the cylinder.
• Surface pressure coefficient, Cp , the ratio of the static to
dynamic pressure on the surface of the cylinder, is defined
as follows:
p() − p∞
Cp = 2
,
(11)
2
U∞
where p() is the surface pressure at an angle and p∞ is
the free stream pressure.
• Recirculation (or wake) length, Lw , is measured from the
rear stagnation point ( = 180◦ ) to the point of reattachment
for the near closed streamline (Ux = Uy = 0) on the line of
symmetry (y = 0) in the downstream section. It is expressed
by the following relation:
Lw =
l−b
,
2b
(12)
where l is the distance from the centre of the cylinder to the
point of reattachment for the near closed streamline.
• Total drag coefficient, CD , is defined as
CD =
2FD
= CDP + CDF ,
2 a
U∞
(13)
where FD is the drag force per unit length of the cylinder.
The pressure and friction drag coefficients, CDP and CDF ,
are calculated using the following definitions:
2FDP
CDP =
(14)
= Cp nx dS ∗ ,
2 a
U∞
S
where S ∗ is the surface area.
2FDF
2n+1
=
(∗ .ns ) dS ∗
CDF =
2 a
U∞
Re S
=
2n+1
Re
S
(∗xx nx + ∗xy ny ) dS ∗ ,
(15)
where ns is the direction vector normal to the surface of the
cylinder, given as
(x/a 2 )ex + (y/b2 )ey
ns = = n x ex + n y e y ,
(x/a 2 )2 + (y/b2 )2
(16)
where ex and ey are the x- and y-components of the unit
vector, respectively, and ∗ , the dimensionless shear stress,
is expressed as
∗ij = ∗
jUj
jUi
+
jj
ji
=
I2∗
2
(n−1)/2 jUj
jUi
+
,
jj
ji
(17)
where ∗ is the dimensionless viscosity and I2∗ is the dimensionless second invariant of the rate of strain tensor. The
shear stress, viscosity and second invariant of the rate of
strain tensor are scaled with m(U∞ /a)n , m(U∞ /a)(n−1)/2
and (U∞ /a)2 , respectively.
Since the governing equations in the Cartesian coordinates
have been solved using FLUENT, hence all the above-noted
expressions are written in Cartesian coordinates only. Corresponding expressions, especially for the individual drag
coefficients, in other coordinate systems are available in the
literature, e.g., see Dennis and Young (2003), D’Alessio and
Dennis (1994), Khan et al. (2005), Kiya and Arie (1975),
Sugihara-Seki (1993), etc.
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P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
4. Numerical methodology
The present numerical study has been carried out using
FLUENT (version 6.2). The unstructured ‘quadrilateral’ cells
of non-uniform spacing were generated using GAMBIT. The
grid near the surface of the cylinder was sufficiently fine to
resolve the boundary layer over the cylinder. Furthermore,
the two-dimensional, steady, laminar segregated solver was
used with second order upwinding scheme for the convective
terms in the momentum equation. The semi-implicit method
for the pressure linked equations (SIMPLE) scheme was used
for pressure–velocity coupling and non-Newtonian power-law
model was used for viscosity. A convergence criterion of 10−10
was used for x- and y-components of the velocity and the
residuals of continuity.
5. Choice of numerical parameters
Naturally, the accuracy and reliability of the numerical results is contingent upon a prudent choice of optimal domain and
grid. In this work, the domain is characterized by the diameter
(D∞ ) of the faraway cylindrical envelope of fluid. Evidently, it
is not possible to simulate truly unconfined flow in any numerical study. A very large value of D∞ will warrant enormous
computational resources and a small value will unduly influence the results. The effect of this parameter is examined in the
next section, followed by that of the grid size to arrive at their
optimal values.
5.1. Domain independence study
The available scant literature shows that the smaller the value
of the Reynolds number, the larger the value of D∞ required for
the results to be free from domain effects. This is so presumably
due to the thickness of the boundary layer which decreases
with the increasing Reynolds number. Bearing in mind this
feature, several values of D∞ /(2a) ranging from 200 to 1200
have been used in this study to minimize the so-called domain
effects. For three values of n = 0.2, 1 and 1.8, three values of
Re = 0.01, 5 and 40 and for the extreme values of E = 0.2
and 5, the results are shown in Table 1. It can be seen that an
increase in the domain size from 1000 to 1200 at Re = 0.01
alters the drag values by 1.14% and 1.35% at n = 1 for E = 0.2
and 5, respectively. The increases in the domain size from 700
to 1200 and from 900 to 1200 at n = 0.2 and 1.8 show the
maximum change in the drag values of 0.004% and 0.34%
for E = 0.2, the corresponding values are 0% and 0.47% for
E = 5 at Re = 0.01. Furthermore, it was found that for Re = 5
and 40, the change in the computational domain size from 200
to 300 and 250 to 300 yields very small change in the drag
values (maximum value being 0.47% and 0.60% for E = 0.2
and 5, respectively) for n = 0.2, 1 and 1.8, respectively. These
results clearly suggest a faster decay in velocity with distance
in both shear-thinning and shear-thickening fluids than that in
a Newtonian fluid. Therefore, the domain sizes of 1200 and
300 are believed to be adequate in the Reynolds number range
of 0.01Re5 and 5Re 40, respectively, over the power-
law index range (0.2 n 1.8) considered here, to obtain the
results which are essentially free from domain effects.
5.2. Grid independence study
Having fixed the domain size, the grid independence study
was carried out using three non-uniform grids for the three
values of the power-law index of 0.2, 1 and 1.8, at Re = 0.01
and 40 for E = 0.2 and 5, as shown in Table 2. Obviously, in
moving from the grid G2 to G3, the results remain virtually
unchanged (except for the friction drag for E = 0.2, n = 0.2 at
Re=40, which changes by 3.8%) accompanied by an enormous
increase (up to 2–3 fold in extreme cases) in the computational
time. In spite of this seemingly large change, the corresponding
change in the value of the total drag coefficient is less than 1%.
Therefore, the grid G2 is believed to be sufficiently refined to
resolve the flow phenomena with acceptable levels of accuracy
within the range of conditions of interest here.
6. Results and discussion
In this work, steady flow computations have been carried
out by systematically varying the value of the power-law index in the range as 0.2 n 1.8 in the steps of 0.2, for the
Reynolds number of 0.01, 1, 5, 10, 20, 30, 40 and for five
values of E = 0.2, 0.5, 1, 2 and 5. The results presented
herein are based on the domain size, D∞ /(2a), of 1200 and
300, respectively, for 0.01 Re 5 and 5 Re 40 and with
the grid size of G2. The flow of Newtonian fluids is known
to be two-dimensional and steady with or without two symmetric vortices over this range of Reynolds number (Dennis
and Young, 2003; Lugt and Haussling, 1974; Williamson,
1996; Zdravkovich, 1997) and this is assumed to be so for
power-law fluids also. However, prior to presenting the new
results, it is appropriate to validate the solution procedure to
ascertain the accuracy and reliability of the results presented
herein.
6.1. Validation of results
Since extensive validation for the flow of Newtonian and
power-law fluids over a circular cylinder (E =1) has been dealt
with elsewhere (Bharti et al., 2006; Sivakumar et al., 2006),
only the additional comparison for the flow of Newtonian fluids
past an elliptic cylinder is reported here (Table 3). The agreement between the present and prior results is seen to be excellent, except that for E = 0.2 and Re = 40. While the exact
reasons for this discrepancy are not known, it needs to be emphasized here that the present results are based on the use of
a much larger domain, i.e., D∞ /(2a) of 300 (5 Re40) and
1200 (0.01 Re 5), whereas the corresponding values are 25
(Dennis and Young, 2003) and 7.5 (values due to Dennis and
Young (2003) in Dennis and Chang, 1969) and the grid employed in the present work (G2 grid) is also finer than that used
by Dennis and Young (2003) and D’Alessio and Dennis (1994).
This is particularly puzzling since the agreement for Re = 5 and
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
1687
Table 1
Domain independence study for the elliptic cylinders (E = 0.2 and 5)
Domain size
CDP
CDF
D∞ /(2a)
n = 0.2
500
700
900
1000
1200
E = 0.2, Re = 0.01
2367.0033
–
2366.9577
360.7814
–
348.4848
–
344.0414
2366.8682
340.1520
100
200
300
E = 0.2, Re = 5
5.3891
5.3891
5.3891
n=1
n = 1.8
n = 0.2
CD
n=1
n = 1.8
n = 0.2
n=1
n = 1.8
–
37.9013
37.5075
37.3819
–
145.4178
143.8668
–
–
143.8606
–
75.6175
73.0404
72.1091
71.2940
–
12.9681
12.8333
12.7903
–
2512.3756
2510.8274
–
–
2510.7288
–
436.399
421.5253
416.1505
411.446
–
50.8695
50.3408
50.1723
–
3.2602
3.2268
3.2161
2.5220
2.4837
2.4720
0.2757
0.2757
0.2757
0.6196
0.6138
0.6119
0.7699
0.7583
0.7547
5.6648
5.6649
5.6649
3.8799
3.8407
3.8280
3.2919
3.2421
3.2268
250
300
E = 0.2, Re = 40
1.5311
1.4747
1.5308
1.4735
1.4367
1.4351
0.01916
0.01916
0.1594
0.1593
0.3086
0.3082
1.5503
1.5499
1.6341
1.6328
1.7453
1.7433
500
700
900
1000
1200
E = 5, Re = 0.01
1286.0257
–
1286.0439
84.2902
–
80.8679
–
79.5933
1286.0439
78.5353
–
9.7702
9.6437
9.5987
–
2911.2636
2911.2591
–
–
2911.2591
–
427.8410
410.4807
404.0138
398.6456
–
12.9681
12.8333
12.7903
–
4197.2893
4197.303
–
–
4197.303
–
512.1312
491.3487
483.6072
477.1810
–
55.9885
55.2629
55.0051
–
100
200
300
E = 5, Re = 5
2.8317
2.8318
2.8318
–
1.0137
1.0085
0.7418
0.7272
0.7228
5.9005
5.9006
5.9005
–
4.0817
4.063
3.0461
2.9877
2.9700
8.7323
8.7324
8.7324
–
5.0954
5.0715
3.7880
3.7149
3.6929
200
300
E = 5, Re = 40
0.5395
0.5395
0.4412
0.4403
0.4050
0.4038
0.8068
0.8068
1.2025
1.2003
1.3025
1.2987
1.3463
1.3463
1.6437
1.6406
1.7076
1.7026
n=1
n = 1.8
Table 2
Grid independence study for Re = 0.01 and 40
CDP
Grid details
Grid
Ncell
Nc
/(2a)
n = 0.2
CDF
n=1
n = 1.8
n = 0.2
CD
n=1
n = 1.8
n = 0.2
G1
G2
G3
10 000
24 000
30 000
100
200
200
0.0100
0.0018
0.0013
Re = 40, E = 0.2
1.5094
1.4621
1.5327
1.5044
1.5333
1.5071
1.4166
1.4810
1.4849
0.0369
0.0218
0.0210
0.2042
0.1671
0.1650
0.3888
0.3269
0.3238
1.5463
1.5545
1.5543
1.6663
1.6716
1.6721
1.8055
1.8079
1.8087
G1
G2
G3
16 000
22 000
33 600
100
200
240
0.0070
0.0015
0.0010
Re = 40, E = 5
0.5472
0.4411
0.5393
0.4444
0.5390
0.4442
0.4031
0.4094
0.4088
0.8017
0.8067
0.8076
1.2131
1.2097
1.2103
1.3235
1.3157
1.3168
1.3489
1.3460
1.3467
1.6543
1.6541
1.6545
1.7266
1.7251
1.7256
G2
G3
107 800
128 000
200
200
0.0018
0.0013
Re = 0.01, E = 0.2
2374.6020
333.5823
2377.2439
334.5221
36.4118
36.3234
134.6189
132.3215
70.9433
70.0080
12.6726
12.6461
2509.221
2509.5654
404.5256
404.5302
49.0845
48.9696
G2
G3
61 200
65 200
200
240
0.0015
0.0010
Re = 0.01, E = 5
1295.9740
77.6424
1302.2717
78.4666
9.5332
9.5578
2901.1760
2894.9470
394.1143
393.2763
45.0962
45.0709
4197.1510
4197.2187
471.7567
471.7429
54.6294
54.6288
Ncells is the number of cells in the computational domain; Nc is the number of points on the surface of the cylinder and is the grid spacing in the vicinity
of the cylinder.
20 is excellent, but the two values show a significant difference
only for Re = 40. No more explanation can be given for this
discrepancy at this stage. Bearing in mind the aforementioned
factors coupled with the fact that the numerical predictions for
power-law fluids tend to be intrinsically less accurate (owing to
the interaction between the two non-linear terms), it is perhaps
1688
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
Table 3
Comparison of Newtonian (n = 1) drag results for elliptical cylinders (E = 0.2
and 5)
E = 0.2
Re
Present
Dennis and Young (2003)
D’Alessio and Dennis (1994)
5
20
40
3.8289
2.0802
1.6319
3.854
2.119
1.876
3.862
2.140
–
Dennis and Young (2003)
8.096
2.712
1.7650
1.169
0.7890
Dennis and Chang (1969)a )
8.222
–
1.848
1.228
0.794
E=5
Re
Present
1
8.1105
5
2.7361
10
1.7681
20
1.1680
40
0.7860
a Values
as cited by Dennis and Young (2003).
reasonable to expect the present results for power-law fluids to
be probably reliable to within ±2–3%.
6.2. Drag phenomena
For unconfined flow, the scaling of the field equations and
the boundary conditions suggests the individual and total drag
coefficients to be functions of the Reynolds number (Re), flow
behaviour index (n) and the aspect ratio (E) of the elliptical
cylinder. These dependences have been explored in the ensuing
sections.
6.2.1. Pressure drag coefficient (CDP )
Table 4 shows the dependence of the pressure and friction
drag coefficients on the Reynolds number, power-law index
and the aspect ratio of the elliptical cylinder. For a fixed value
of the Reynolds number, the value of the pressure drag coefficient, CDP decreases as the fluid behaviour changes from
shear-thinning (n < 1) to Newtonian (n = 1), and finally to
shear-thickening (n > 1), irrespective of the shape of the cylinder. Furthermore, the influence of the power-law index gradually diminishes as the value of the Reynolds number is progressively increased. Conversely, the effect of the flow behaviour
index is more prominent at low Reynolds numbers than that
at high Reynolds numbers, a trend which is also consistent
with that seen for spheroidal and spherical particles (Chhabra,
2006). This is simply due to the fact that with the increasing
value of the Reynolds number, inertial forces far outweigh the
viscous forces whence it hardly matters whether the fluid is
shear-thinning or shear-thickening or Newtonian. For a fixed
value of the power-law index, the value of CDP is seen to increase with a decrease in the Reynolds number for all values
of aspect ratio E. For instance, the value of CDP at n = 0.2 increases by a factor of 1537, 1960 and 2392 for E = 0.2, 1 and 5
as the Reynolds number is decreased from 40 to 0.01. The corresponding changes in the value of CDP at n = 1 are almost an
order of magnitude smaller than these values and even smaller
changes are seen for an extremely shear-thickening fluid with
n = 1.8. It can clearly be seen that the dependence of the pres-
sure drag coefficient on the Reynolds number, power-law index
and the aspect ratio is highly non-linear. The present numerical
data for the pressure drag coefficient (CDP ) over the range of
conditions (0.01 Re 40, 0.2 n 1.8 and 0.2 E 5) can
be best represented by the following relationship:
log(10CDP Rem ) = f1 (n)f2 (Rem , n)f3 (Rem , Em ).
(18)
The functions f1 (n), f2 (Rem , n) and f3 (Rem , Em ) are given as
f1 (n) = 7.55 − 1.35n2 + 0.40n3 ,
f2 (Rem , n) = 0.998 1.145 − 0.32n + 0.26n2
0.118 0.0655
,
−0.055n +
−
X
X2
3
f3 (Rem , Em ) = − 0.366 + 0.103X + 0.032X 2
0.758 0.032
− 0.002X 3 +
−
,
2
Em
Em
(19)
(20)
(21)
where X = log(10Rem ) and the modified Reynolds number
(Rem ) and modified aspect ratio (Em ) are represented as
follows:
√
100
Rem =
(22)
and Em = 1 + E.
Re
Eq. (18) shows maximum and average deviations (max and
avg ) of 14.72% and 4.37%, respectively, for the 360 individual
data points. There are, however, only 39 data points exhibiting
deviations greater than 10%, generally relating to combination
of high Re, low n and small value of E.
6.2.2. Friction drag coefficient (CDF )
The variation of the friction drag coefficient with the
Reynolds number, power-law index and aspect ratio is also
included in Table 4. For a fixed value of the Reynolds number, the dependence of the friction drag coefficient on the
power-law index shows a non-monotonous behaviour in both
shear-thinning and shear-thickening fluids. The key trends can
be summarized as follows: for a fixed value of the aspect
ratio and for weakly non-Newtonian fluids, the friction drag
coefficient is nearly independent of the power-law index at
low Reynolds numbers. As the Reynolds number is gradually
increased, the friction drag coefficient transits via an almost
power-law index independent behaviour to an upward trend
wherein the value of CDF increases slowly with the increasing
value of the power-law index over the entire range. Broadly,
the larger the Reynolds number, weaker is the dependence on
the power-law index. These trends are clearly due to the nonlinear nature of the viscosity over the surface of the cylinder
(Bharti et al., 2006). The viscosity for a shear-thinning fluids
(n < 1) becomes very large as the shear rate decreases and
hence it tends to infinity faraway from the cylinder. Therefore,
the viscous effects dominate even faraway from the cylinder.
It is, however, appropriate to add here that this stems from
the unrealistic fluid behaviours predicted by the simple powerlaw fluid in the limit of zero shear rate. However, on the
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
1689
Table 4
Dependence of the pressure and friction drag coefficients on the Reynolds number (Re), power-law index (n) and aspect ratio (E)
n
Re = 0.01
20
30
40
Re = 0.01
E = 0.2, pressure drag coefficient (CDP )
0.2 2375.9478 237.5592 23.9572 5.3884 3.1428
0.6 1346.3304 134.8532 14.3732 3.9490 2.5917
1
333.5823 44.7364 8.1027 3.1779 2.3274
1.4
86.8510 19.9672 5.5220 2.7318 2.1454
1.8
36.4118 11.5126 4.2218 2.4355 1.9996
2.0690
1.8738
1.8079
1.7450
1.6763
1.7258
1.6119
1.5917
1.5628
1.5197
1.5453
1.4645
1.4615
1.4475
1.4169
E = 0.2, friction drag coefficient (CDF )
134.7223 13.4686 1.3479 0.2593 0.1067 0.0401 0.0257 0.0191
170.2879 17.0571 1.8124 0.4610 0.2619 0.1436 0.1008 0.0788
70.9433
9.5105 1.7035 0.6121 0.4004 0.2570 0.1952 0.1596
24.6542
5.6621 1.5386 0.6971 0.4984 0.3503 0.2805 0.2373
12.6726
4.0046 1.4298 0.7533 0.5709 0.4257 0.3532 0.3064
E = 0.5, pressure drag coefficient (CDP )
0.2 2186.3272 218.6630 22.0423 4.8996 2.8187
0.6 1145.1463 114.6440 12.2512 3.3947 2.2306
1
266.1707 36.0942 6.5975 2.6169 1.9253
1.4
68.3911 15.8394 4.4109 2.1982 1.7364
1.8
28.5659
9.0728 3.3411 1.9397 1.6009
1.7678
1.5974
1.5037
1.4255
1.3544
1.4390
1.3677
1.3311
1.2872
1.2383
1.2866
1.2411
1.2289
1.2015
1.1639
E = 0.5, friction drag coefficient (CDF )
367.2389 36.7219 3.6769 0.7288 0.3424 0.1375 0.0805 0.0568
398.4208 39.8847 4.2423 1.0725 0.6143 0.3448 0.2430 0.1895
138.0381 18.7110 3.3707 1.2111 0.7945 0.5166 0.3973 0.3278
43.2766 10.0226 2.7454 1.2482 0.8960 0.6372 0.5165 0.4419
20.6677
6.5797 2.3684 1.2551 0.9564 0.7214 0.6054 0.5309
E = 1, pressure drag coefficient
0.2 1951.2650 195.1465 19.6615
0.6 924.8634 92.6154 9.9524
1
206.7774 27.9680 5.1847
1.4
52.8495 12.1910 3.4327
1.8
21.9883
7.0059 2.5970
(CDP )
4.3221
2.8040
2.0897
1.7245
1.5125
2.4388
1.8440
1.5411
1.3657
1.2508
1.4842
1.3040
1.2007
1.1240
1.0616
1.1516
1.1003
1.0607
1.0174
0.9742
0.9954
0.9899
0.9790
0.9524
0.9195
E = 1, friction drag coefficient
723.8972 72.3863 7.2447
681.9876 68.2901 7.2864
210.5813 28.4644 5.1708
61.8726 14.2860 3.9494
28.0521
8.9808 3.2567
(CDF )
1.4534
1.8465
1.8613
1.7967
1.7264
0.7178
1.0650
1.2241
1.2911
1.3171
0.3394
0.6139
0.8047
0.9249
0.9997
0.2069
0.4406
0.6262
0.7567
0.8457
0.1443
0.3469
0.5223
0.6537
0.7479
E = 2, pressure drag coefficient
0.2 1645.7317 164.5780 16.5779
0.6 674.0869 67.5313 7.3550
1
141.3820 19.5365 3.7597
1.4
36.4728
8.6388 2.4962
1.8
15.6546
5.0639 1.9058
(CDP )
3.6262
2.1456
1.5597
1.2730
1.1143
2.0132
1.4153
1.1597
1.0068
0.9195
1.1840
0.9900
0.8893
0.8235
0.7767
0.8965
0.8220
0.7781
0.7414
0.7102
E = 2, friction drag coefficient
0.7461 1369.7311 136.9786 13.7023
0.7271 1075.3721 107.7219 11.5905
0.7126 287.8216 39.7249 7.4070
0.6912
80.0214 18.9939 5.3515
0.6689
35.5244 11.5752 4.2452
(CDF )
2.7638
2.9717
2.6877
2.4364
2.2443
1.3956
1.7262
1.7714
1.7496
1.7087
0.7020
1.0134
1.1734
1.2581
1.2999
0.4657
0.7424
0.9219
1.0359
1.1053
0.3455
0.5943
0.7761
0.9012
0.9834
E = 5, pressure drag coefficient
0.2 1295.9744 129.6308 13.0755
0.6 402.9162 40.4019 4.5800
1
77.6424 11.0763 2.3102
1.4
20.9627
5.1494 1.5731
1.8
9.5332
3.1774 1.2359
(CDP )
2.8535
1.4367
1.0085
0.8210
0.7228
1.5725
0.9571
0.7456
0.6430
0.5871
0.9017
0.6645
0.5668
0.5142
0.4839
0.6653
0.5436
0.4879
0.4549
0.4347
E = 5, friction drag coefficient (CDF )
0.5417 2901.1761 290.1007 29.0330 5.8819 2.9996
0.4737 1736.2927 174.0588 19.1987 5.0654 2.9700
0.4403 394.1143 56.0330 10.9978 4.0630 2.6872
0.4183 103.8442 25.5858 7.4535 3.4049 2.4412
0.4038
45.0962 15.1635 5.6513 2.9700 2.2509
1.5469
1.7731
1.7913
1.7569
1.7091
1.0549
1.3172
1.4166
1.4510
1.4555
0.8053
1.0687
1.2003
1.2673
1.2987
0.1
1
5
10
other hand, even if one uses the other fluid models (like Ellis or
Carreau), the value of the viscosity will approach the so-called
zero shear viscosity faraway from the cylinder. Of course, the
shear-thickening fluid will exhibit maximum viscosity near the
cylinder. Due to this large viscosity (faraway from the cylinder)
in shear-thinning fluids, the value of the friction drag coefficient
was seen to be larger than that in shear-thickening fluids at
low Reynolds numbers. On the other hand, for shear-thickening
(n > 1) fluids, the inertial effects dominate faraway from the
cylinder, which therefore shows an opposite dependence of the
drag coefficient on the power-law index in the low Reynolds
number region. The present numerical data for the friction drag
coefficient (CDF ) over the range of conditions (0.01 Re 40,
0.2 n 1.8 and 0.2 E 5) can be best represented by the
following closure relationship:
log(50CDF Rem ) = f1 (n)f2 (Rem , n)f3 (Rem , Em ).
(23)
The functions f2 (Rem , n) and f3 (Rem , Em ) are given as
f2 (Rem , n) = 0.586[−0.329 + (0.42 − 0.015n)X
+ 0.007X 2 + 0.19n + 0.01n2 ],
(24)
0.1
1
5
10
20
30
2.495 0.0868
−
f3 (Rem , Em ) = − 0.386 +
X X2
3.44
2.295
1
+ 3.59 −
−
,
2
Em
X
Em
40
(25)
where X = log(10Rem ) and the function f1 (n), the modified
Reynolds number (Rem ) and the modified aspect ratio (Em )
are given by Eqs. (19) and (22), respectively. Eq. (23) shows
maximum and average deviations (max and avg ) of 60.26%
and 3.68%, respectively, for the 360 individual data points.
There are, however, 15 data points exhibiting deviations greater
than 15%, only one data point with > 50%, relating to Re =
40, n = 0.2 and E = 0.2, respectively. Further examination
of the residuals revealed these points showing large deviations relate to certain combinations involving high values of Re
(∼ 30–40) and/or small values of n (∼ 0.2) and/or small values of E (∼ 0.2). If these 15 data points are excluded, the
maximum and average deviations reduce drastically to 14.92%
and 3.05%, respectively, for the remaining 345 individual data
points.
6.2.3. Total drag coefficient (CD )
The total drag coefficient (sum of the pressure and friction
drag coefficients, Eq. (13)) is plotted in Fig. 2 for a range of
1690
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
Fig. 2. Effect of Reynolds number (Re), power-law index (n) and aspect ratio
(E) on the total drag coefficient (CD ).
Fig. 3. Effect of Reynolds number (Re), power-law index (n) and aspect ratio
(E) on the drag ratio (CDP /CDF ).
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
1691
values of the Reynolds number, power-law index and aspect
ratio. Once again, the dependence is seen to be similar to that
of CDP and CDF shown in Table 4. The dependence of the
total drag coefficient, CD on the power-law index is seen to
be stronger in shear-thinning (n < 1) fluids than that in shearthickening (n > 1) fluids. Consequently, at low Reynolds numbers (0.01 Re 1), significant variation in the drag values can
be seen in shear-thinning fluids than that in Newtonian and in
shear-thickening fluids. At high values of the Reynolds number
(20 Re 40), the value of CD increases with an increase in
the power-law index for 0.2 E 5; however, this effect gets
accentuated at E = 2 and 5. These trends are qualitatively consistent with the literature values for a circular cylinder (Bharti
et al., 2006; Chhabra et al., 2004; Soares et al., 2005). The total drag coefficient (CD ) over the range of the Reynolds number (0.01 Re 40), power-law index (0.2 n1.8) and aspect ratio (0.2 E 5) can be best represented by the following
relation:
log(CD Rem ) = f1 (n)f2 (Rem , n)f3 (Rem , Em ),
(26)
where the functions f2 (Rem , n) and f3 (Rem , Em ) are given as
f2 (Rem , n) = 0.26X + 0.03n − 0.06,
2
f3 (Rem , Em ) = 1.74 − 1.17Em + 0.41Em
+
Fig. 4. Effect of Reynolds number (Re), power-law index (n) and aspect ratio
N ).
(E) on the normalized total drag coefficient (CD
(27)
0.106
,
X
(28)
where X = log(Rem ). The function f1 (n), the modified
Reynolds number (Rem ) and the modified aspect ratio (Em ) are
given by Eqs. (19) and (22), respectively. Eq. (26) reproduces
360 numerical data points with the maximum and average
deviations (max and avg ) of 33.93% and 4.77%, respectively.
There are, however, only 11 data points exhibiting deviations
greater that 10%. Further examination of the residuals revealed
these points to relate to the same combinations of high values
of Re (∼ 30–40) and/or small values of n (∼ 0.2) and/or small
values of E (∼ 0.2), as noted earlier. If these 11 data points
are excluded, the maximum and average deviations reduce to
8.83% and 3.48%, respectively, for the remaining 349 individual data points. It needs to be stressed here that no prior
predictive equation exist in the literature even for the flow of
Newtonian fluids over an elliptical cylinder over these ranges
of Re and E. Admittedly, some attempts have been made to
consolidate the results for power-law fluid flow in tube bundles
by Bruschke and Advani (1993) and Woods et al. (2003) by
using different length scales with limited success. Not only the
length scale depends upon the solid volume fraction of arrays
and/or on n, but these approaches have been tested for n < 1
and/or E < 1 and in the creeping flow limit (Re = 0). In the
present case, the results relate to n < 1 and n > 1, Reynolds
number up to 40 and 0.2 E 5, it is far from obvious whether
a single choice of length scale will work under all conditions.
On the other hand, the correlations developed here based
on the length and velocity scales which emerge naturally by
the non-dimensionalization of the governing equations and
boundary conditions. Also, these expression embrace much
1692
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
a
b
Y
0.6
n=0.2
c
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.5
0
0.5
1
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
n=0.2
-0.5
0
0.5
n=0.2
-0.2 -0.1
1
0
0.1
0.2
0.3
X
0.6
0.6
n=1.0
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.5
0
0.5
0.6
-0.5
1
n=1.8
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.5
0
0.5
1
0
0.5
0.6
0.4
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
n=1.0
1
n=1.8
-0.5
0
0.5
1
n=1.0
-0.2
0
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
0.2
n=1.8
-0.2
0
0.2
Fig. 5. Streamline (upper half) and vorticity (lower half) profiles for Re = 0.01. (a) E = 0.2, (b) E = 1, (c) E = 5.
wider range of Reynolds number (Re) than that in the study
of Bruschke and Advani (1993), Re = 0, E = 1 and of Woods
et al. (2003), Re = 0, E < 1.
Further examination of the present results in terms of the
variation of the drag ratio (CDP /CDF ) on the Reynolds number
(Re), power-law index (n) and aspect ratio (E) revealed this
ratio (CDP /CDF ) to decrease with an increase in the power-law
index (Fig. 3) for the ranges of conditions studied in this work.
Qualitatively, this effect can be explained as follows: for a fixed
n . For a shearvalue of E and Re, the viscous forces scales as U∞
thinning fluid (n < 1), as the value of n is gradually increased,
the viscous forces will rise and hence the friction will increase.
Similarly, for a shear-thickening fluid (n > 1), viscous forces
will rise even more steeply. In either event, one would expect the
ratio (CDP /CDF ) to decrease as seen in Fig. 3. A rapid decrease
in the ratio with an increase in power-law index was seen at low
Reynolds numbers and/or aspect ratio in shear-thinning fluids
thereby suggesting rather strong dependence on the aspect ratio,
E. Broadly, the value of the ratio (CDP /CDF ) was > 1 for
E 1, which changes to (CDP /CDF ) 1 as the value of the
aspect ratio E exceeds 1. This shows the role of shape. Thus,
the pressure drag dominates for E 1 which acts as a more
of a bluff body than a circular cylinder thereby resulting in
the sudden and significant bending of streamlines and showing
poor pressure recovery. On the other hand, E > 1 behaves more
like a streamlined surface wherein a fluid element can easily
follow the cylinder surface whence reducing the role of pressure
forces.
Finally, in order to elucidate the role of power-law rheology
on the cylinder drag in an unambiguous manner, drag coefficient for an elliptic cylinder in power-law fluids is normalized
N =C
using the corresponding Newtonian value (CD
D,n /CD,n=1 )
at the same values of the Reynolds number and aspect ratio.
Fig. 4 shows the dependence of the normalized drag coefficient
on the Reynolds number and power-law index. It is seen that the
shear-thinning behaviour causes an increase in the value of the
drag coefficient which diminishes with the increasing Reynolds
number. As expected, not only this trend flips over in shearthickening fluids, but the effect is also seen to be somewhat
suppressed. This figure clearly shows the effect of power-law
index to be much more prominent at low Reynolds numbers
than that at high Reynolds numbers. Also, the value of the drag
coefficient is seen to be enhanced much more than that for a
sphere where this ratio reaches a maximum value of ∼ 1.5 at
n ≈ 0.4 as opposed to the value of 9 seen here for the extreme
case of n = 0.2.
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
a
b
Y
0.6
n=0.2
c
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.5
0
0.5
1693
1
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
n=0.2
-0.5
0
0.5
n=0.2
-0.2 -0.1
1
0
0.1
0.2
0.3
X
0.6
0.6
n=1.0
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.5
0
0.5
0.6
-0.5
1
n=1.8
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.5
0
0.5
1
0
0.5
0.6
0.4
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
n=1.0
1
n=1.8
-0.5
0
0.5
1
n=1.0
-0.2
0
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
0.2
n=1.8
-0.2
0
0.2
Fig. 6. Streamline (upper half) and vorticity (lower half) profiles for Re = 1. (a) E = 0.2, (b) E = 1, (c) E = 5.
6.3. Detailed flow kinematics
Evidently, the dependence of the individual and total
drag coefficients seen in the previous section should also
be reflected in similar differences in the detailed kinematics of the flow close to the cylinder. In particular,
the streamline and vorticity profiles, recirculation length,
variation of pressure coefficients over the surface of the
cylinder, and their values at the stagnation points are analysed to gain further physical insights into the nature of the
flow.
6.3.1. Streamline and vorticity profiles
Representative streamline (upper half) and vorticity (lower
half) profiles close to the elliptical cylinder (E = 0.2, 1 and 5)
for four values of the Reynolds number of 0.01, 1, 10 and 40
and for three values of the power-law index of 0.2, 1 and 1.8 are
shown in Figs. 5–8. For a fixed value of power-law index (n),
flow is seen to be somewhat retarded as the value of the aspect
ratio E increases. This is clearly due to fact that with increasing
deviation in the value of E from 1, the shape of the cylinder
corresponds to a vertical (E → 0) or to a horizontal (E → ∞)
flat plate-like obstructions to the flow. For a fixed value of E,
the flow is seen to accelerate with an increase in the power-law
index and/or Reynolds number and/or both. The effect is more
visible in shear-thinning (n < 1) fluids and/or for E < 1 than
that in shear-thickening (n > 1), or in Newtonian (n=1) and/or
for E 1. As expected. there is no evidence of flow separation at Re = 0.01 for the range of power-law index and aspect
ratios considered here. These figures also show the decreasing
values of the Reynolds number, at which the separation occurs, with a decrease in the value of the aspect ratio E for both
shear-thinning and shear-thickening fluids. For instance, for a
cylinder with E = 0.2, a significant wake can clearly be seen at
Re = 1, while no flow separation is seen for E = 1 and 5 at this
value of the Reynolds number, Fig. 6. Irrespective of the type
of fluid behaviour, based on the loss of fore and aft symmetry and/or flow separation, the limiting value of the Reynolds
number denoting the end of the creeping flow regime shows a
drastic reduction as the value of the aspect ratio E is decreased.
As noted earlier, this behaviour is clearly due to the fact
that a cylinder with E < 1 behaves like a bluff body whereas
E > 1 configuration shows some features of streamline body.
Furthermore, shear-thinning behaviour delays the separation
1694
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
a
b
n=0.2
0.2
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
n=0.2
0.4
0.2
0
Y
c
0.6
0
-0.2
-0.2
-0.4
-0.4
-0.5
0
0.5
1
-0.5
0
0.5
n=0.2
-0.2 -0.1
1
0
0.1
0.2
0.3
X
0.6
n=1.0
0.2
0.4
0
0.2
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
n=1.0
0
-0.2
-0.2
-0.4
-0.4
-0.5
0
0.5
n=1.8
0.2
-0.5
1
0
0.5
0.6
1
n=1.8
0.4
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.5
0
0.5
1
-0.5
0
0.5
1
n=1.0
-0.2
0
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
0.2
n=1.8
-0.2
0
0.2
Fig. 7. Streamline (upper half) and vorticity (lower half) profiles for Re = 10. (a) E = 0.2, (b) E = 1, (c) E = 5.
while it occurs earlier in shear-thickening fluids than that in
Newtonian fluids for a fixed value of the aspect ratio. This
feature is qualitatively consistent with that seen for a circular
cylinder. Qualitatively, it can be rationalized as follows: Since
the effective shear rate is maximum close to the cylinder, the
viscosity is minimum in this region for a shear-thinning fluids
(and of course, maximum for a shear-thickening fluid). The
velocity field is known to decay faster in a shear-thinning fluid
which leads to a drastic reduction in shear rate, i.e., high viscosity. In other words, even slightly away from the cylinder, highly
viscous slow moving liquid meets with the low viscosity fluid
in a thin layer encapsulating the cylinder. Thus, the separation is
somewhat delayed in this case. On the other hand, for n > 1, the
viscosity is maximum near the obstacle and lower elsewhere so
this less viscous fast moving fluid stream leads to an early flow
separation.
6.3.2. Recirculation length (Lw )
The phenomena of flow recirculation and separation in the
rear of the cylinder are often quantified in terms of the recirculation (or wake) length and the angle of separation, respectively. Fig. 9 shows the dependence of the recirculation
length on the power-law index (0.2 n1.8), Reynolds num-
ber (0.01 Re 40) and aspect ratio (0.2 E 5) of elliptical
cylinders. For a fixed value of the Reynolds number, the length
of the recirculation zone decreases with an increase in the value
of the aspect ratio E, irrespective of the type of flow behaviour
of the fluid. No flow separation was observed at Re = 0.01 for
any value of power-law index (n) and aspect ratio (E) studied
here (Figs. 5 and 9), at Re = 1 for all values of value of powerlaw index n and the aspect ratio E 1 (except at n = 1.8 and
E = 1, see Figs. 6 and 9) and for E = 5 in shear-thinning fluids
(E = 5, Fig. 9) for all values of the Reynolds number. For a
fixed value of the aspect ratio E, the value of the recirculation
length (Lw ) decreases with the decreasing power-law index for
shear-thinning fluids, whereas it increases with the increasing
degree of shear-thickening. The wake grows faster with an increase in the value of E for E > 1 than that for E < 1 as the
fluid behaviour changes from Newtonian to shear-thickening;
the change in behaviour from Newtonian to shear-thinning
shows the opposite dependence on E. In summary, the wake
tends to be smaller in shear-thinning fluids and/or for E < 1
than that in shear-thickening fluids and/or E > 1 under otherwise similar conditions. These trends are obviously consistent
with the plausible considerations presented in the preceding
section.
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
a
b
1
n=0.2
0.4
0.5
Y
c
0.6
n=0.2
0.2
0
0
-0.2
-0.5
1695
-0.4
n=0.2
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-1
0
0.5
1
1.5
2
2.5
-0.5
0
0.5
-0.2 -0.1
1
0
0.1
0.2
0.3
X
1
0.6
n=1.0
n=1.0
0.4
0.5
0.2
0
0
-0.2
-0.5
-0.4
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
n=1.0
-1
0
0.5
1
1.5
1
2
2.5
-0.5
0
0.5
0.6
n=1.8
n=1.8
0.4
0.5
0.2
0
0
-0.2
-0.5
-0.2 -0.1
1
-0.4
0
0.1
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
0.2
0.3
n=1.8
-1
0
0.5
1
1.5
2
2.5
-0.5
0
0.5
1
-0.2 -0.1
0
0.1
0.2
0.3
Fig. 8. Streamline (upper half) and vorticity (lower half) profiles for Re = 40. (a) E = 0.2, (b) E = 1, (c) E = 5.
6.3.3. Surface pressure coefficients (Cp )
Representative variation of the pressure coefficient over the
surface of an elliptic cylinder is shown in Figs. 10–12 for a
range of values of the power-law index, for four values of the
Reynolds number as Re = 0.01, 1, 10 and 40 and for three
values of the aspect ratio as E = 0.2, 1 and 5, respectively.
These figures suggest the stronger dependence of the pressure
profiles on the shape (E) of the cylinder than that on Reynolds
number (Re) and/or on the power-law index (n). This finding
is also consistent with that for spheroidal particles (Chhabra,
2006). The effect of power-law index (n) on the value of Cp
is seen to be stronger in shear-thinning (n < 1) fluids and/or in
flow without separation than that in shear-thickening (n > 1)
fluids and/or the high Reynolds number flow. The curves show
qualitatively similar patterns irrespective of the aspect ratio of
the elliptical cylinder and the behaviour of the fluid. At low
Reynolds numbers and for E < 1, it can be seen from Fig. 10
that surface pressure is almost constant in the front and the
rear half of cylinder with a step change at = /2. In contrast
for E = 1 (circular cylinder), the pressure profile patterns are
significantly different (Fig. 11) than that for E < 1 (Fig. 10) for
the flow with separation in the rear of the cylinder. In this case,
the pressure is seen to decrease from its maximum value at the
front stagnation point ( = 0)) along the surface of the cylinder
followed by an increase due to the flow recirculation in the
rear side of the cylinder. In the absence of separation, pressure
continues to decrease up to the rear stagnation point (=). For
E > 1, the pressure decreases rapidly near the front stagnation
point to a minimum (Fig. 12), which varies very little thereafter
in the rear side thereby following the surface of the streamlined
shape of the elliptical cylinder. In the front of the cylinder, the
rate of decrease of Cp increases with an increase in the value
of E which is clearly due to the varying extents of the bending
of streamlines. The fore and aft symmetry in the variation of
Cp in the rear and front of the cylinders can be seen under the
conditions without flow separation, Figs. 10–12, respectively,
i.e., the crossover (Cp =0) occurs at =/2 of the cylinder. This
crossover point gradually shifts forward with the increasing
Reynolds number thereby suggesting early separation. For a
fixed value of the aspect ratio (E) and Reynolds number (Re),
the shear-thinning (n < 1) fluids always show a higher value of
the pressure in the front of the cylinder than that for Newtonian
fluids with a flip over in the downstream side of the cylinder,
e.g., see Figs. 10–12 (left side). The shear-thickening (n > 1)
fluids show the opposite dependence of the Cp profiles on the
power-law index. The smaller the value of the power-law index
1696
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
Fig. 9. Dependence of recirculation length (Lw ) on Reynolds number (Re), power-law index (n) and aspect ratio (E).
(n), the higher is the value of Cp (see Figs. 10–12 (right side)).
For a fixed value of the power-law index (n) and the aspect ratio
(E), the value of the pressure coefficient is seen to be higher
at low Reynolds numbers, which decreases with an increase in
the Reynolds number (Re).
Fig. 13(a) shows the dependence of the front stagnation pressure coefficient Cp (0) on the Reynolds number, power-law
index and the aspect ratio. The maximum value of Cp (0) was
seen at low Reynolds numbers and low power-law index. For
a fixed value of the Reynolds number (Re) and the aspect ratio
(E), Cp (0) is seen to decrease with an increase in the powerlaw index at low Reynolds numbers, irrespective of the type of
the fluid behaviour. As the Reynolds number is progressively
increased, the rate of increase in the value of Cp (0) with an
increase in the power-law index was seen to slow down and
only a slight change can be seen at higher Reynolds numbers.
At higher values of Re, Cp (0) is seen to be weakly dependent on the power-law index. The rate of change in the front
stagnation pressure coefficient with power-law index was seen
to be almost constant at low value of the Reynolds number.
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
1697
Fig. 10. Effect of Reynolds number and power-law index on Cp over the surface of the elliptical cylinder (E = 0.2).
For a fixed value of power-law index (n) and the Reynolds
number (Re), Cp (0) shows an increase with an increase in the
value of the aspect ratio (E), irrespective of the fluid behaviour.
This increase was seen to be higher in shear-thickening fluids and/or E > 1 than that in shear-thinning and Newtonian
fluids and/or E < 1. The functional dependence of the front
stagnation pressure coefficient, Cp (0) on the Reynolds number
(0.01 Re 40), power-law index (0.2 n 1.8) and aspect
ratio (0.2 E 5) can be best represented by the following
relation:
log[Cp (0)Rem ] = f1 (n)f2 (Rem , n)f3 (Rem , Em ),
(29)
1698
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
Fig. 11. Effect of Reynolds number and power-law index on the variation of Cp over the surface of the elliptical cylinder (E = 1).
where the functions f2 (Rem , n) and f3 (Rem , Em ) are given as
f2 (Rem , n) = 0.96 − 0.133n + 0.12n2 +
0.02
,
X
f3 (Rem , Em ) = 0.126 + 0.177X + 0.015X 2
0.31 0.174
−
+
,
2
Em
Em
(30)
(31)
where X=log(Rem ). The function f1 (n), the modified Reynolds
number (Rem ) and the modified aspect ratio (Em ) are expressed
by Eqs. (19) and (22), respectively. Eq. (29) shows maximum
and average deviations (max and avg ) of 13.29% and 3.59%,
respectively, for the 360 individual data points. Only 4 data
points show > 10%.
The dependence of the rear stagnation pressure coefficient,
Cp () on the Reynolds number, power-law index and the aspect ratio, shown in Fig. 13(b) is qualitatively similar, except for the fact that it is negative. In addition, the crossover
of the values can be seen with an increasing Reynolds number. The functional dependence of the rear stagnation pressure
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
1699
Fig. 12. Effect of Reynolds number and power-law index on the variation of Cp over the surface of the elliptical cylinder (E = 5).
coefficient, Cp () on the Reynolds number (0.01 Re 40),
power-law index (0.2 n1.8) and aspect ratio (0.2 E 5)
can be best represented by the following relation:
log[−10Cp ()Rem ] = f1 (n)f2 (Rem , n)f3 (Rem , Em ),
(32)
where the functions f2 (Rem , n) and f3 (Rem , Em ) are given as
f2 (Rem , n) = 0.99 1.162 − 0.16n + 0.05n2
0.65 0.47 0.257n
−
+ 2 +
,
(33)
X
X
X
1700
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
is increased, the flow is governed by two non-linear terms,
namely, inertial and viscous, which scale differently with velocity. For instance, one can argue that the viscous forces will
n whereas the inertial forces as ∼
approximately scale as ∼ U∞
2
U∞ . Thus, keeping everything else fixed, the decreasing value
of power-law index (n) will suggest diminishing importance of
the viscous effects for shear-thinning fluids, while the inertial
2 ). On the other hand, viscous efterm will still scale as (∝ U∞
fects are likely to grow with the increasing value of the powerlaw index (n) for a shear-thickening fluid. For the extreme case
1.8 , almost
of n = 1.8, the viscous terms will also scale as ∼ U∞
identical to the inertial term. It is believed that these different
kinds of dependencies on the velocity and power-law flow index are also responsible for the non-monotonous behaviour of
drag coefficients. Finally, the highly non-linear form of the momentum equations also poses enormous challenges in choosing appropriate numerics and numerical parameters even in the
steady flow regime. Hence, the drag and flow patterns show
a much more intricate dependence on the Reynolds number
(Re) and power-law index (n) than that seen for a Newtonian
fluid.
7. Concluding remarks
Fig. 13. Effect of Reynolds number, power-law index and aspect ratio of the
elliptical cylinders on the (a) front and (b) rear stagnation pressure coefficient.
f3 (Rem , Em ) = 0.06 + 0.153X + 0.0095X 2
2
+ 0.03Em X − 0.143Em + 0.004Em
,
(34)
where X = log(10Rem ). The function f1 (n), the modified
Reynolds number (Rem ) and the modified aspect ratio (Em )
are expressed by Eqs. (19) and (22), respectively. Eq. (32)
shows maximum and average deviations (max and avg ) of
7.52% and 1.78%, respectively, for the 360 individual data
points. Only 7 data points show > 5%.
In summary, the detailed kinematics of the flow and resulting values of the individual and total drag coefficients are seen
to be influenced in an complex manner by the values of the
Reynolds number, the power-law index and the aspect ratio of
the elliptical cylinder. This interplay is further accentuated by
the fact that even at low Reynolds numbers, the viscous terms
are non-linear for power-law fluids. As the Reynolds number
The flow characteristics of the two-dimensional steady flow
of incompressible power-law fluids over an elliptical cylinder
have been investigated numerically for the range of Reynolds
number as 0.01 Re 40, power-law index as 0.2 n 1.8 and
for the aspect ratio as 0.2 E 5 including the special case of
a circular cylinder (E = 1). Extensive results on flow patterns
in terms of the streamline and vorticity profiles adjacent to the
cylinder are presented and discussed to gain physical insights
into the nature of flow. The onset of wake formation (thereby
marking the end of the creeping flow regime) was seen to be
delayed with an increase in the aspect ratio (E), i.e., streamlining of the shape and/or with a decrease in the power-law index
(n). The early separation was observed in shear-thickening fluids, as well as for the cylinders of E < 1. The shear-thinning
behaviour (n < 1) of the fluid decreases the size of the wake;
on the other hand, the shear-thickening fluids (n > 1) show the
opposite behaviour. Distinct maximum values of the pressure
coefficient were seen at the front stagnation point for the cylinders of E 1, while it was nearly constant in the front side of
the cylinder for E < 1 for all values of the power-law index
(n) and the Reynolds number (Re). At low Reynolds numbers,
the front stagnation pressure coefficient was always higher in
shear-thinning fluids than that in shear-thickening fluids. The
pressure drag coefficient always decreases with an increase
in the power-law index, irrespective of the value of Reynolds
number, fluid behaviour, or shape of the cylinder. The friction
drag coefficient shows a complex dependence on the Reynolds
number (Re), power-law index (n) and the aspect ratio (E).
Therefore, the total drag coefficient decreases with an increase
in power-law index at low Reynolds numbers. In the creeping flow region, the shear-thinning behaviour results in higher
value of the drag coefficient than that in shear-thickening fluids. The effect of aspect ratio, E on the drag coefficient in
P. Sivakumar et al. / Chemical Engineering Science 62 (2007) 1682 – 1702
shear-thickening fluids was also modest as compared to that in
shear-thinning fluids. Overall, the aspect ratio of the cylinder
plays a significant role in influencing the detailed flow patterns
and the resulting gross behaviour. The functional dependence
of the flow characteristics (drag and stagnation pressure coefficients) on the dimensionless parameters (Re, E and n) has also
been presented.
Notations
a
b
CD
CDF
CDP
N
CD
Cp
Cp (0)
Cp ()
D∞
e x , ey
E
FD
FDF
FDP
I2
I2∗
Lw
m
n
nx , ny
ns
Nc
p
p∞
p()
Re
U∞
Ux , Uy
x, y
semi-axis of the elliptical cylinder normal to the
flow, m
semi-axis of the elliptical cylinder along the
flow, m
drag coefficient, dimensionless
frictional component of drag coefficient, dimensionless
pressure component of drag coefficient, dimensionless
normalized drag coefficient, CD,non-Newtonian /
CD,Newtonian , dimensionless
pressure coefficient, dimensionless
value of Cp at front stagnation ( = 0), dimensionless
value of Cp at rear stagnation ( = ), dimensionless
diameter of the outer boundary, m
x- and y-component of the unit vector, dimensionless
aspect ratio of the elliptical cylinder, =b/a, dimensionless
drag force per unit length of the cylinder, N/m
frictional component of the drag force per unit
length of the cylinder, N/m
pressure component of the drag force per unit
length of the cylinder, N/m
second invariant of the rate of the strain tensor,
s−2
dimensionless second invariant of the rate of the
strain tensor dimensionless, =I2 /(U∞ /a)2
recirculation length, dimensionless
power-law consistency index, Pa sn
power-law flow behaviour index, dimensionless
x- andy-component of the direction vector normal to the surface of the cylinder, dimensionless
direction vector normal to the surface of the
cylinder, dimensionless
number of grid points on the surface of the cylinder, dimensionless
pressure, Pa
free stream pressure, Pa
surface pressure at an angle , Pa
Reynolds number, dimensionless
free stream velocity, m/s
x- and y-components of the velocity, m/s
streamwise and transverse coordinates, m
1701
Greek symbols
∗
∗
∗xx
∗xy
component of the rate of the strain tensor, s−1
viscosity, Pa s
dimensionless viscosity, =/[m(U∞ /a)(n−1)/2 ]
angular displacement from the front stagnation,
degree
density of the fluid (kg/m3 )
shear stress, Pa
dimensionless shear stress, =/[m(U∞ /a)n ]
x- component of the dimensionless shear stress,
=xx /[m(U∞ /a)n ]
y-component of the dimensionless shear stress,
=xy /[m(U∞ /a)n ]
vorticity, s−1
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