The motion of a single and multiple neutrally buoyant elliptical cylinders in
plane Poiseuille flow
Shih-Di Chen, Tsorng-Whay Pan, and Chien-Cheng Chang
Citation: Phys. Fluids 24, 103302 (2012); doi: 10.1063/1.4757387
View online: http://dx.doi.org/10.1063/1.4757387
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PHYSICS OF FLUIDS 24, 103302 (2012)
The motion of a single and multiple neutrally buoyant
elliptical cylinders in plane Poiseuille flow
Shih-Di Chen,1 Tsorng-Whay Pan,2,a) and Chien-Cheng Chang1,3,a)
1
Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan
Department of Mathematics, University of Houston, Houston, Texas 77204, USA
3
Center for Advanced Studies in Theoretical Sciences, National Taiwan University,
Taipei 106, Taiwan
2
(Received 16 January 2012; accepted 10 September 2012; published online 12 October 2012)
In this article, we investigate the motion of neutrally buoyant elliptical cylinders
in plane Poiseuille flow of a Newtonian fluid. The method of distributed Lagrange
multiplier/fictitious domain was used to solve the Navier-Stokes equations as well
as for the motion of elliptical cylinders. The motion of a single elliptical cylinder is
shown to be dependent on the channel Reynolds number Re, the particle size ratio K
= a* /H* , and the aspect ratio A = a* /b* of the cylinder, where H* is the half height
of the channel, a* and b* are the lengths of the semi-major axis and semi-minor axis
of the cylinder, respectively. It is found that there is a critical Reynolds number, Rec
∼ 3, which distinguishes the lateral migration of a single elliptical cylinder below and
above it. As Re is increased, the equilibrium position of the elliptical cylinder shifts
towards the wall when Re ≤ Rec or shifts closer to the central axis when Re ≥ Rec .
Moreover, there are interesting correlations between the center-of-mass trajectories
and the orientation dynamics, which depend on the ranges of K and Re. The motion of
multiple elliptical cylinders is also affected by the total solid area fraction φ T , which
is defined to be the proportion of the area occupied by the cylinders in the domain of
computation. For a few elliptical cylinders (the number of cylinders ND = 16 and the
corresponding φ T = 3.77%), the cylinders may scatter into several groups at lower
Re (≤ 100), and each group fluctuates about an averaged position. At the higher Re
(= 1000), the cylinders may converge to an equilibrium position on each side of the
channel center. For a larger number of cylinders (ND = 36, 54, 72, 108, and the
corresponding φ T = 8.48%–25.45%), we observed a significant rheological behavior
in the velocity profiles. In addition, there exists a particle-free layer next to each wall,
and the thickness of the particle-free layers is increased as A (or K) or Re is increased.
C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4757387]
I. INTRODUCTION
A flow with rigid neutrally buoyant particles in a simple shear or Poiseuille flow between
parallel plates can be simulated to investigate issues such as the fibres in a pulp (ellipse),1 the
glutaraldehyde-fixed (rigid) red blood cells in the blood flow (biconcave),2, 3 and other particles of
moderate aspect ratios in a similar flow. Understanding the motion of rigid particles in neutrally
buoyant flow is important for fundamental developments in suspension rheology.
The orientation dynamics of a single particle have been studied widely, from the Stokes limit
to finite Reynolds number ranges for a simple shear flow. Jeffery4 considered a single ellipsoid in
an unbounded shear flow, neglecting the inertia effect (the particle Reynolds number Rep = Re · K
1), and obtained a set of differential equations for the rotation of the ellipsoid depending on
a) Authors to whom correspondence should be addressed. Electronic addresses: [email protected] and
[email protected].
1070-6631/2012/24(10)/103302/25/$30.00
24, 103302-1
C 2012 American Institute of Physics
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103302-2
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
the initial conditions. The rotation of an ellipsoid is indeterminate, so he hypothesized that “the
particle will tend to adopt that motion which, of all the possible motions under the approximated
equations, corresponds to ‘the least dissipation of energy.’” However, several later studies by other
authors reported cases of maximum dissipation. Karnis, Goldsmith, and Mason5 demonstrated in
experiments that the orbits of disks and rods assume maximum dissipation in the Couette flow at
small particle Reynolds numbers (3 × 10−4 ≤ Rep ≤ 7 × 10−3 ). Harper and Chang6 demonstrated
that even with the restricted class of flows considered by Jeffery, the minimum dissipation hypothesis
does not hold. These authors considered the motion of a dumb-bell shape particle in an unbounded
shear flow by including the inertia effects at low particle Reynolds numbers (Rep 1). The particle
was shown to have a preferred periodic orbit that corresponds to maximum dissipation. Ding and
Aidun7 studied numerically the dynamics of an elliptical cylinder suspended in a Couette flow at
various particle Reynolds numbers (0.08 ≤ Rep ≤ 40), and showed a transient behavior from being
rotary to stationary as the particle Reynolds number is increased. Their numerical findings were also
demonstrated experimentally by Zettner and Yoda,8 who studied the motion of an elliptical cylinder
in a plane Couette flow apparatus. Subramanian et al.9, 10 investigated the effect of fluid inertia
on the orientation of a fibre in a simple shear flow, and also found that above a critical Reynolds
number, the fibre ceases to rotate and inclines at a stable orientation. The orientation dynamics of
an elliptical cylinder or ellipsoid in the Stokes limit have been studied for not only the Couette
flow but also the Poiseuille flow. Chwang and Wu11, 12 solved the Stokes equations by assuming
small particle Reynolds numbers (Rep 1) to determine the motion of an ellipsoid in unbounded
quadratic flows. They showed that the ellipsoid moves in a straight line parallel to the flow direction,
without any side drift, at a variable speed that is governed by a trajectory equation. Sugihara-Seki13
studied numerically the motion of a neutrally buoyant elliptical cylinder in a two-dimensional (2D)
Poiseuille flow at small Reynolds numbers (Re 1, Rep 1), and found that an elliptical cylinder
is either rotary or oscillatory depending on the particle-to-channel size ratio K, the aspect ratio A of
the elliptical cylinder, and the initial condition.
On the other hand, the center-of-mass trajectories of a single particle at finite Reynolds number
ranges have been investigated more extensively for the Poiseuille flow. Segre and Silberberg14, 15
experimentally studied the lateral migration of dilute suspensions of neutrally buoyant spheres
in Poiseuille flow through a tube. The spheres migrate away from the wall and the centerline to
accumulate at about 0.6 of the tube radius from the centerline at small but finite Reynolds numbers
(Re < 30). Ho and Leal16 examined the motion of a rigid circular cylinder in a 2D Poiseuille flow
with inclusion of the inertia effects at small Reynolds numbers (Rep 1) by a regular perturbation
method. The circular cylinder reaches a stable lateral equilibrium position independent of the initial
location of release, and the position is the Segre–Silberberg position − 0.6 of the channel half-width
from the centerline for the Poiseuille flow. Schonberg and Hinch17 studied the inertial migration of
a sphere in a Poiseuille flow at the Reynolds numbers of order unity (Re = 1–75, Rep 1), and
found that the Segre–Silberberg position moves towards the wall as Reynolds number is increased.
In general, it is believed that the migration away from the centerline of the channel is due to an
effect of the curvature of velocity profile.18 If we make the approximation that a particle set in the
Poiseuille flow has a zero mean velocity relative to the fluid, then owing to the curvature of the
velocity field, the fluid velocity will be (absolutely) higher on the wall side than on the centerline.
This dissymmetry will cause an excess lower pressure on the side where the velocity of the fluid
is higher, leading the particle to migrate away from the axis. For a particle moving parallel to a
wall, the pressure field around the particle will be greater on the wall side, so the wall tends to
repel the particle. The particle will therefore reach the equilibrium position when the forces due to
the curvature effect and the wall repulsion strike a balance in the Poiseuille flow.19 Feng et al.18
investigated the motion of neutrally buoyant and non-neutrally buoyant 2D circular cylinders in plane
Poiseuille flows using a finite-element method and obtained qualitative agreements with the results
of perturbation theories and experiments. The center-of-mass trajectories of an elliptical cylinder or
ellipsoid have been investigated less than that of a circular one or sphere in previous studies. Zhao
and Sharp20 investigated the rotational stability of an elliptical cylinder in 2D Poiseuille flow for
different aspect ratios, transverse locations, and Reynolds numbers (Re = 1.5–30) by calculating the
lift and torque. They found that stationary orientations could be reached for a higher aspect ratio, and
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103302-3
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
also suggested that increased Re may result in more stationary orientations. Pan et al.21 simulated the
motion of an ellipsoid in neutrally buoyant 3D Poiseuille flows and found that its rotation exhibits
distinctive states depending on the Reynolds number ranges and the shape of the ellipsoid. In spite
of these previous studies, the relationship between the orientation dynamics and the center-of-mass
trajectory is not clear.
When the number of the particles is increased, the particle-particle interactions can cause a rheological effect. Leighton and Acrivos22 reported on the particle-particle interaction in a concentrated
suspension of neutrally buoyant spheres, and provided the first evidence for the existence of a nonuniform equilibrium concentration distribution in a Couette flow at zero particle Reynolds number
(Rep < 10−4 ). Nott and Brady23 simulated the pressure-driven flow in a channel for the suspensions
of neutrally buoyant spheres at zero particle Reynolds number (Rep 1). They observed that even
if one starts from a random distribution of sphere locations, the spheres gradually migrate towards
the center of the channel, resulting in an inhomogeneous distribution and a blunting of the particle
velocity profile. Han and Kim24 investigated the motion of spheres in the tube flow of suspensions for
a wide range of solid volume fractions and particle Reynolds numbers (Rep = 0.05–0.376) experimentally. They found that as the total solid volume fraction is increased, the flow velocity profile and
the particle velocity profile are more blunted than the initial parabolic profiles at higher solid volume
fraction conditions, and the spheres concentrate round the center of the tube. Cohen et al.25 studied
particle interactions of the hard-sphere colloidal particles in a highly confined geometry by confocal
microscopy, and showed that geometric confinement leads to the ordered but highly non-equilibrium
structure. Humphry et al.26 also studied the effects of particle concentration and channel geometry
on inertial focusing, and showed that both the location and the number of focusing positions depend
on the number of particles per unit length along the channel. Matas et al.27 studied the migration of
dilute suspensions of spheres (φ T < 1%) in a tube flow at Reynolds numbers Re = 76–1700. They
observed other inner equilibrium positions closer to the center, and found that most of the spheres
cluster round either the Segre–Silberberg equilibrium positions or the inner equilibrium position.
However, they were unable to determine whether this inner equilibrium position is a real equilibrium
position in a steady state or simply a transient feature. Recent advances in computational sciences
allow direct simulations of complicated solid-fluid interactions. Direct numerical simulations have
been used to study the motion of particles in shear flows. Pan and Glowinski28 simulated the motion
of circular cylinders in neutrally buoyant 2D flows, and confirmed that collisions between cylinders
play a key role in driving cylinders to the central region of plane Poiseuille flow. For a multiple array
of particles, the interactions between the particles in the rear and front are more complicated, but
mainly because a rear particle in the wake of a front one may experience less drag29 and touch the
front particle at later times.28
This study is aimed to investigate the motion of a single and multiple neutrally buoyant elliptical
cylinders via mutual interactions. In this article, we consider the effects of the aspect ratio (A) and
the cylinder-to-channel size ratio (K) on the equilibrium position, translational velocity and angular
velocity of a single cylinder under various Reynolds numbers (Re). The present interest also extends
to multi-elliptical cylinders under various conditions: the number of cylinders ND varying from 4 to
108 (the total solid area fraction φ T = 0.94%–25.45%) with A from 1 to 3.333, K from 0.11 to 0.2,
and Re from 10 to 1000. The rest of the article is structured as follows. In Sec. II, we introduce the
governing equations of the model problem concerning a single or multiple rigid neutrally buoyant
elliptical cylinders moving freely in a two-dimensional Poiseuille flow. Then in Sec. III, we briefly
describe the method used in this study – the distributed Lagrange multiplier/fictitious domain method
(DLM/FD) developed by Glowinski et al.28, 30–32 In Sec. IV, we present and discuss the numerical
results of the cases of a single and multiple elliptical cylinders in five parts. Section IV A provides
the validation of a single elliptical cylinder in plane Couette flow. In Sec. IV B, we focus on the
motion of a single elliptical cylinder in plane Poiseuille flow and its dependence on the effects
of the aspect ratio A, the size ratio K of the cylinder, and the Reynolds number Re. The motion
of a few elliptical cylinders (the number of cylinders ND = 16, and the total solid area fraction
φ T = 3.77%) in plane Poiseuille flow and its dependence on the effects of the Reynolds number Re
are investigated in Sec. IV C. We examine, in Sec. IV D, the motion of many elliptical cylinders
(ND = 72, φ T = 16.96%) in plane Poiseuille flow and its dependence on the effects of the aspect
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103302-4
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
FIG. 1. (a) Model for validation: Schematic of single rigid elliptical cylinder in plane Couette flow. (b) Focus problem
of this study: Schematic of multiple rigid elliptical cylinders in plane Poiseuille flow with the domain of computation
D* = (L* × 2H* ).
ratio A and the Reynolds number Re with particular emphasis on the rheological behaviors, and in
Sec. IV E, the thickness of the particle-free layer yf on the effects of the total solid area fraction φ T
(= 0.94%–25.45% and ND = 4–108), the aspect ratio A, and the Reynolds number Re. The main
results are summarized in the Concluding Remarks.
II. DESCRIPTION OF THE PROBLEM
Figures 1(a) and 1(b) show the validation model of a single rigid neutrally buoyant elliptical
cylinder in a Couette flow (as shown in Fig. 1(a)) and the focus problem of multiple rigid neutrally
buoyant elliptical cylinders in plane Poiseuille flow (as shown in Fig. 1(b)). In these plots, a* and b*
are the lengths of the semi-major axis and semi-minor axis of the cylinder; A = a* /b* is the aspect
∗
is
ratio of the cylinder; H* is the half height of the channel; L* is the length of the channel; Umax
the maximum velocity of Couette flow or plane Poiseuille flow (without the existence of cylinders);
(X ∗p , y ∗p ) is the center location of each cylinder; θ denotes the orientation of the major axis, or the
angle of inclination (i.e., the angle the semi-major axis makes with the x-axis); and ∗p is the angular
velocity of the elliptical cylinder.
The fluid-particle system is governed by the Navier-Stokes equations for the fluid and (the
∗
∗
, 2H* /Umax
, and
Euler-)Newton’s equations for rigid body motions. In this study, we take 2H* , Umax
∗ ∗
*
μ f Umax /2H to be the characteristic length, translational speed, time, and pressure, respectively,
where μ∗f is the dynamic viscosity of the fluid. A normalized physical quantity or variable is denoted
by the same letter for the dimensional quantity or variable but without an asterisk. The dimensionless
governing equations are given by30
∂u
+ (u · ∇) u = −∇ p + ∇ 2 u,
Re
(1)
∂t
dU p
6
= Geg +
α Re
(2)
[− pl + τ ] · nd,
dt
πK3 60
dω
=
x − X p × ([− pl + τ ] · n) d,
(3)
α Re
5
dt
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103302-5
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
where denotes the boundary of the solid cylinder. In these equations, u is the fluid velocity; Up is
the particle velocity; ω(= p ) is the speed of rotation of the cylinder; τ is the shear stress; p is the
pressure; α = ρ ∗p /ρ ∗f is the mass density ratio, where ρ ∗p and ρ ∗f are the mass densities of the cylinder
and fluid, respectively; eg is the unit vector in the direction of gravitation; and K = a* /H* is the size
ratio. In addition, we defined two dimensionless parameters as follows:
Re =
and
G=
∗
(2H ∗ )
ρ ∗f Umax
μ∗f
, the Reynolds number,
ρ ∗p − ρ ∗f g ∗ (2H ∗ )2
∗
μ∗f Umax
, the gravity number,
(4)
(5)
where g* is the gravitation acceleration. Since the cylinders are neutrally buoyant in plane Couette
and Poiseuille flow in this study, G is zero. The flow is periodic in the x-direction, and the no-slip
condition is applied on the top and bottom walls. Equations (2) and (3) are described for a single
elliptical cylinder; their generalization to multiple cylinders is straightforward.
III. NUMERICAL METHOD
In this section, we briefly describe the numerical method used in this study for simulating
particulate flows, and refer the readers to the cited references for details. The method of solution is
actually a combination of a distributed Lagrange-multiplier-based fictitious domain method (DLM)
and the operator splitting methods. The basic idea is to imagine that the fluid fills the entire space
inside as well as outside the particle boundaries. The fluid-flow problem is then posed on a larger
domain (the “fictitious domain”). This larger domain is simpler, allowing a simple regular mesh to
be used, which in turn renders use of some specialized fast solution techniques. The larger domain
is also time-independent, so the same mesh can be used for the entire simulation, eliminating the
need for repeated re-meshing and projection.
The fluid inside the particle boundary must exhibit a rigid-body motion. This constraint is
enforced using the distributed Lagrange multiplier, which represents the additional body force per
unit volume needed to maintain the rigid-body motion inside the particle boundary, much like the
pressure in incompressible fluid flow, whose gradient is the force required to maintain the constraint
of incompressibility. For space discretization, we use P1 -iso-P2 and P1 finite elements for the velocity
field and pressure, respectively (like in Bristeau et al.33 ). In time advancing, many operator splitting
schemes can be applied to the problem based on the Lagrange multiplier/fictitious domain method.
We apply the Lie scheme to obtain a sequence of sub-problems for each time step (see Ref. 28 for
details). The computational method has been validated in our previous study28 for the motion of
neutrally buoyant disks in Poiseuille flow. In this paper, we extend the method to the case of elliptical
cylinders moving in Poiseuille flow.
IV. RESULTS AND DISCUSSION
A. Validation of an elliptical cylinder in plane Couette flow
We consider the motion of a neutrally buoyant elliptical cylinder in plane Couette flow, which
was considered by Ding and Aidun.7 Their results were obtained by the lattice Boltzmann equation.
The domain of computation is D = (5 × 1), the aspect ratio A = 2, and the size ratio K = 0.2 for
this study (note that K = 0.1 for Ding’s work). The flow is periodic in the x-direction, and the top
and bottom boundaries are the no-slip u(x,0) = 1 and u(x,1) = 0. The mesh size is h = 1/320, and
the time step is t = 0.001. The definition of the particle Reynolds number in the Couette flow is
the same as Ding and Aidun,7 Rep = Re · K.
In Fig. 2, we can see that the computational results are in good agreement with Jeffery’s
solution4 for Rep = 0 and the ones obtained by Ding and Aidun for Rep = 0.08 and 1.0, respectively.
Figure 3 shows the normalized minimum angular velocity ωmin /π and the period of rotation T versus
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103302-6
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
FIG. 2. Validation of the computational results: The figure shows comparisons in both angle of inclination and angular
velocity of an elliptical cylinder. Orientation θ /π versus time t: (a) our result (solid line) at Rep = 0.08; (b) Ding’s result
(dashed line) at Rep = 0.08; (c) our result at Rep = 1; (d) Ding’s result at Rep = 1. Angular velocity ω/π versus time t: (e)
our result at Rep = 0.08; (f) Ding’s result at Rep = 0.08; (g) our result at Rep = 1; (h) Ding’s result at Rep = 1. Jeffery’s
solutions for θ p versus t: Jθ = θ /π , and for ω versus t: Jω = ω/π at Rep = 0 are also plotted for comparisons. The domain
of computation is D = (5 × 1); the size ratio K = 0.2 for this study and K = 0.1 for Ding’s work; the aspect ratio A = 2.
FIG. 3. Normalized minimum angular velocity ωmin /π (straight line) and period of rotation T (curve) versus Rep of an
elliptical cylinder. It is noted that T increases to infinity as Rep approaches the critical value Rep,c ∼ 29. The domain of
computation is D = (5 × 1); the size ratio K = 0.2 for this study and K = 0.1 for Ding’s work; the aspect ratio A = 2.
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103302-7
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
Rep . The normalized minimum angular velocity ωmin /π decreases as Rep is increased with a nearly
straight line relationship, where Rep,c = 29 is the critical particle Reynolds number, above which the
rotational motion is stopped. It is also noted that the period of rotation T of the elliptical cylinder
with the larger size ratio (K = 0.2) is shorter than that of the smaller one (K = 0.1) at Rep = 28; the
cross of the two curves of K = 0.1 and 0.2 shows that a larger size ratio promotes rotation beyond a
certain Re ∼ 25.
B. The motion of a single elliptical cylinder in plane Poiseuille flow
We consider the motion of a single neutrally buoyant elliptical cylinder in a pressure-driven
plane Poiseuille flow. The domain of computation is D = (L × 1); A = 1.875, and K = 0.15 of
the cylinder. The flow is periodic in the x-direction, and the pressure gradient is 8, which gives a
driving force to the flow field. The fluid and cylinder are initially at rest. The initial y-coordinate
of the center of the elliptical cylinder is 0.4. The mesh size is h = 1/256, and the time step is t
= 0.01. We calculate the lateral migration of the elliptical cylinder by changing the length of the
channel (L = 1, 2, 4, 6) at the Reynolds numbers Re = 10, 100, 1000, respectively. The results show
that in the shorter channels (L = 1, 2), the equilibrium position of the elliptical cylinder is much
lower than in other cases (L = 4, 6). Since we assume the periodic boundary in the x-direction,
the mutual interactions between cylinders in neighboring periods are not negligible for the short
channels (L = 1, 2), which is insufficient to recover the flow condition for a single cylinder.34 The
cases of L = 4 and L = 6 have about the same equilibrium position at each Re, which means that
the hydrodynamic interactions across the periodic boundary do not affect the cylinder motion much.
In the following simulations, we choose the length of channel L = 4 to achieve good accuracy and
reduce the computational cost.
1. The effect of the shape of the cylinder (A, K)
In this section, we vary the shape of a neutrally buoyant elliptical cylinder in a pressure-driven
Poiseuille flow. We keep the cross section constant and vary the aspect ratio A, and therefore the size
ratio K of the elliptical cylinder. We chose A = 3.333, 2.5, 1.875, 1.2, 1.0, and the corresponding
K = 0.2, 0.17, 0.15, 0.12, 0.11 at Re = 10.
Figure 4 shows the temporal development of the dimensionless height Yp of the elliptical
cylinder for different aspect ratios. Considering the long-time behavior, we observe that the cylinder
is eventually fluctuating about an averaged height, denoted by Yeq = Yp . Correspondingly, we
define the radial equilibrium position,
req = 1/2 − Yeq ,
in the lower half channel, which measures the average distances of the mass center to the central
line of the channel. The relation of the averaged equilibrium position and the aspect ratio of the
elliptical cylinder fitted by the least-squares gives req = 0.4801 − 0.002(A − 1) − 0.0025(A − 1)2 ,
which shows that the equilibrium position of the cylinder with a higher A is closer to the central
axis. The mechanism of this lateral migration is related to the balance among the wall repulsion
due to lubrication, inertial lift related to shear slip, and lift due to the curvature of the velocity
profile in the Poiseuille flow.16, 18, 19, 35 Figures 5(a) and 5(b) show the translational velocities for
cylinders of different shapes. The translational velocity of an elliptical cylinder Up in one rotation
period reaches the maximum when its major axis is aligned with the direction of flow, and reaches
the minimum when its major axis is perpendicular to the direction of flow. We also found that the
time-averaged translational velocity Up,avg between 2400 ≤ t ≤ 2500 and the amplitude of oscillation
in the translational velocity Up,osc increase with increases in the aspect ratio A as shown in Fig. 6.
Figures 7(a) and 7(b) show the normalized angular velocity ω/π of an elliptical cylinder between
2490 ≤ t ≤ 2500, and its averaged value ωs /π over this period for different aspect ratios. The
elliptical cylinder is subject to a large torque when it is at a large angle of inclination to the flow
direction.36 The higher A corresponds to a lower ωs /π , because the higher A increases the moment
arm associated with the pressure force, which produces a relatively larger torque; therefore, ωs /π
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103302-8
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
0.4
A=3.333, K=0.20
A=2.500, K=0.17
A=1.875, K=0.15
A=1.200, K=0.12
A=1.000, K=0.11
0.38
0.36
Y
p
0.34
0.32
0.3
0.28
A
0.26
0.24
0
500
1000
1500
2000
2500
t
FIG. 4. Temporal development of lateral migration of an elliptical cylinder with different A and K at Re = 10.
is decreased.29 By the least-squares fitting, we obtain the relation of the averaged angular velocity
ωs /π and the aspect ratio of cylinder A as ωs /π = 0.3051 − 0.0355(A − 1) − 0.033(A − 1)2 , which
shows that the elliptical cylinder with a higher aspect ratio rotates more slowly on average; this
tendency is similar to the results of Chwang12 obtained for an ellipsoid in an unbounded quadratic
flow.
From Jeffery’s orbit, we can obtain the maximum angular velocity ωmax = γ a2 /(a2 +b2 ) at the
angles of inclination θ = (n+1/2)π (when the major axis is perpendicular to the flow direction),
and the minimum angular velocity ωmin = γ b2 /(a2 +b2 ) at the angles of inclination θ = nπ (when
the major axis is aligned with the flow direction), where γ is the shear rate, and n is an integer. We
also examine an elliptical cylinder of aspect ratio A = 3 at Re = 10 for different size ratios K =
0.2, 0.4, 0.5, 0.6. In our results, the maximum and minimum angular velocities do occur at angles of
inclination close to θ = (n+1/2)π and θ = nπ , respectively; however the values of ωmax and ωmin
are somewhat different from Jeffery’s orbits when the size ratio K is greater than 0.4, as shown in
Fig. 8.
2. The effect of Reynolds number (Re)
In this section, we consider the motion of an elliptical cylinder with A = 1.875 and K = 0.15
at different Reynolds numbers from Re = 0.5 to 1000. Figures 9(a) and 9(b) show the equilibrium
position versus Reynolds number with different aspect ratios, and the present results are compared
to other studies of 2D and 3D particles. Figure 9(a) shows the equilibrium positions of an elliptical
cylinder with the aspect ratio A = 3.333, 1.875, 1 and the size ratio K = 0.2, 0.15, 0.11 at low Re;
they all have the same trend. For these three cases, the transition is observed to occur at about Rec
= 3, below which the Segre–Silberberg position shifts away from the centerline with increasing Re,
and above which the trend is the opposite. Figure 9(b) shows the equilibrium position versus Re with
the present result and the other studies of 2D and 3D particle for comparison. The tendency of the
equilibrium position versus Reynolds number is different for a circular cylinder (2D) and a sphere
(3D). The main features are summarized as follows: (1) The sphere moves towards the wall as Re is
increased (i.e., the results () obtained by Schonberg and Hinch17 ). (2) Figures 9(a) and 9(b) show
the present results of a circular cylinder with K = 0.11 (●) at Re = 1–1000. The equilibrium position
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103302-9
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
1
A=3.333, K =0.20
A=2.500, K =0.17
A=1.875, K =0.15
A=1.200, K =0.12
A=1.000, K =0.11
0.95
U
p
0.9
0.85
0.8
0.75
0.7
0
500
1000
1500
2000
2500
2440
2450
t
(a)
0.8
A=3.333
0.795
A=2.500
0.79
A=1.875
0.785
U
p
0.78
0.775
0.77
0.765
0.76
A=1.200
A=1.000
0.755
0.75
2400
2410
2420
2430
t
(b)
FIG. 5. (a) Translational velocity Up versus time t of an elliptical cylinder for different A and K at Re = 10. (b) The local
plot at 2400 ≤ t ≤ 2450.
moves towards the wall with increasing Re when Re ≤ Rec (∼3), and then the equilibrium position
moves closer to the central axis with increasing Re when Re ≥ Rec . (3) Yang et al.37 simulated the
motion of a circular cylinder with K = 0.1 (◦) at Re = 12.5–1000. The equilibrium position moves
closer to the channel center with increasing Re, as shown in Fig. 9(b). (4) Feng et al.36 simulated the
motion of a circular cylinder with K = 0.25 () at Re = 40–200. The equilibrium position moves
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103302-10
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
0.8
0.795
6.7e−3
0.79
0.785
Up,avg
0.78
4.4e−3
0.775
2.9e−3
0.77
9e−4
0.765
0.76
U
=1e−4
p,osc
0.755
0.75
0.5
1
1.5
2
2.5
3
3.5
4
A
FIG. 6. Average translational velocity Up,avg and its amplitude of oscillation Up,osc versus the aspect ratio A at Re = 10 when
it fluctuates about the averaged equilibrium position. Amplitude of oscillation is marked as the error bar.
towards the wall with increasing Re, as shown in Fig. 9(b). These results indicate that the relation
of the equilibrium position and the Reynolds number may significantly depend on the size ratio K.
We also consider the motion of an elliptical cylinder at a higher Re; Figure 10 shows the temporal
development of lateral migration of the cylinder. The equilibrium position becomes closer to the
central axis of the channel as Re is increased, which is similar to the case for the circular cylinder.38
We further examine the effects of the presence of walls. In our study, we do observe that the
center of the elliptical cylinder undergoing the Poiseuille flow oscillates about its mean equilibrium
position. The amplitude of oscillation Yosc versus Re and versus K are presented in Fig. 11. It is
shown that Yosc becomes smaller with decreasing Re or K; this long-term dynamic is consistent with
the results of Chwang12 for an ellipsoid. Then, we examine the effect of the size ratio K and the
0.32
0.6
A=3.333
A=2.5
A=1.875
0.5
0.3
0.28
A=1.2
0.4
0.26
ωs /π
ω /π
A=1
0.3
0.24
0.22
0.2
0.2
0.1
0.18
0
2490
0.16
2492
2494
2496
t
(a)
2498
2500
1
1.5
2
2.5
3
3.5
A
(b)
FIG. 7. (a) Angular velocity ω/π versus time t between 2490 < t < 2500; (b) averaged angular velocity ωs /π versus time t
of an elliptical cylinders for different aspect ratios A at Re = 10.
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103302-11
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
Our ωmax
0.9
Jeffery’s ω
max
0.8
Our ω
0.7
Jeffery’s ωmin
ω
max
/π , ω
min
/π
min
[θ/π]=0.5167
0.6
0.5131 0.5092
0.4918
0.5
0.4
0.3
0.2
[θ/π]=0.9982
0.9961 0.9973
0.1
0.9979
0
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
K
FIG. 8. Maximum angular velocity ωmax /π and minimum angular velocity ωmin /π for an elliptical cylinder of A = 3 at Re
= 10 for different K = 0.2, 0.4, 0.5, 0.6. Jeffery’s solutions are also plotted for comparisons. The notation [θ /π ] denotes the
decimal part of θ /π calculated in the present method.
particle Reynolds number Rep on the rotation of an elliptical cylinder for (a) K = 0.4, A = 3 and
(b) K = 0.76, A = 2. It is found that the oscillatory orientation does occur at a sufficiently high
Rep , in particular when K is large. Figure 12(a) shows that the elliptical cylinder with K = 0.4 is
always rotary at Rep = 4, 16, 40, and becomes oscillatory in orientation at Rep = 80. Figure 12(b)
shows that the elliptical cylinder with the larger K = 0.76 exhibits an oscillatory orientation at
Rep = 30, 152, 380, but becomes stationary at the largest Rep = 418. Figure 13 shows the maximum
and minimum normalized angles of inclination θ max /π , θ min /π versus Re for the elliptical cylinders
with K = 0.4, A = 3 and K = 0.76, A = 2, respectively. We have the following observations. (1)
0.28
A=3.333, K=0.2
A=1.875, K=0.15
A=1, K=0.11
0.275
A=3.333, K=0.2
A=1.875, K=0.15
A=1, K=0.11
0.4
Yang
0.35
0.27
(2D), K=0.1
Feng
(2D), K=0.25
Yang
(3D), K=0.1
Schonberg
(3D)
Yeq
Yeq
0.3
0.265
0.25
0.26
0.2
0.255
0.25
0.15
0
5
10
Re
(a)
15
20
0.1
0
50
100
150
200
Re
(b)
FIG. 9. (a) Equilibrium position Yeq versus Reynolds number Re for the elliptical cylinder with A = 3.333, K = 0.2 (),
and A = 1.875, K = 0.15 (); and the circular cylinder with K = 0.11 (●). (b) The present results compared to a sphere by
Schonberg and Hinch17 (), a circular cylinder with K = 0.1 (◦) and a sphere with K = 0.1 () by Yang et al.,32, 37 and a
lager circular cylinder with K = 0.25 () by Feng et al.36
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103302-12
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
0.42
Re=1000
0.4
0.38
0.36
Re=200
Y
p
0.34
0.32
0.3
Re=100
0.28
Re=40
0.26
0.24
Re=20
Re=10
0
500
1000
1500
2000
2500
t
FIG. 10. Temporal development of lateral migration Yp of an elliptical cylinder with A = 1.875, K = 0.15 at different Re.
The curves at t = 2500 from the bottom to the top correspond to Re = 10, 20, 40, 100, 200, 1000, respectively.
In a range of lower Reynolds numbers (Re < 200 for K = 0.4), the elliptical cylinder is rotary
all the time. (2) In a range of moderate Reynolds numbers (Re ≥ 200 for K = 0.4, Re < 550 for
K = 0.76), the elliptical cylinder exhibits an oscillatory orientation dynamics; the maximum and
minimum normalized angles of inclination θ max /π and θ min /π approach 0.9 and 0.1, respectively, as
Re is increased. This behavior is independent of the size ratio K. (3) In a range of higher Reynolds
0.012
K=0.15, A=1.875
K=0.2, A=3.333
K=0.4, A=3
0.01
Y
osc
0.008
0.006
0.004
0.002
0
0
20
40
60
80
100
Re
FIG. 11. The amplitude of oscillation Yosc versus Re of an elliptical cylinder when it fluctuates about the averaged equilibrium
position for (1) K = 0.15 and A = 1.875 (◦), (2) K = 0.2 and A = 3.333 (), and (3) K = 0.4 and A = 3 ().
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103302-13
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
0.5
Re=10, Re =4
p
0.4
Re=40, Re =16
p
Re=100, Re =40
0.3
p
Re=200, Re =80
p
0.2
ω /π
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
150
200
250
300
350
t
(a)
0.3
Re=40, Re =30
p
0.25
Re=200, Rep=152
0.2
Re=500, Re =380
0.15
Re=550, Rep=418
p
ω /π
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
0
50
100
150
200
250
300
t
(b)
FIG. 12. Angular velocity ω/π versus time t of an elliptical cylinder with (a) small size ratio K = 0.4 and aspect ratio A
= 3, (b) large size ratio K = 0.76 and aspect ratio A = 2 at different Rep . Note that for the smaller K = 0.4, the elliptical
cylinder keeps rotating at Rep = 4–40, but oscillates in orientation at Rep = 80. For the larger K = 0.76, the elliptical cylinder
oscillates in orientation at Rep = 30–380, but becomes stationary in orientation with θ = 0 at Rep = 418.
numbers (Re ≥ 550 for K = 0.76), the elliptical cylinder becomes stationary, and the angle θ is
approaching zero.
In order to examine the correlations between the center-of-mass trajectory and the orientation
dynamics of an elliptical cylinder, we present in Figs. 14(a) and 14(b) the height of the elliptical
cylinder corresponding to Figs. 12(a) and 12(b). Figure 14(a) shows that the center-of-mass for
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103302-14
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
1
0.9
θ
max
/π , θ
min
/π
K =0.4, A=3
K =0.76, A=2
0.1
0
0
200
400
600
800
1000
Re
FIG. 13. The maximum and minimum normalized angles of inclination θ max /π , θ min /π versus Re for the elliptical cylinders
with K = 0.4, A = 3 and K = 0.76, A = 2, respectively.
K = 0.4 oscillates mildly about an averaged equilibrium position for Rep = 4–40, but it oscillates
about the channel center at the higher Rep = 80. Figure 14(b) shows that the elliptical cylinder with
the larger K = 0.76 oscillates about the channel center for Rep = 30–380, but it moves to the channel
center at the higher Rep = 418. It is of great interest to observe the following correlations between
the center-of-mass trajectory and the orientation dynamics. (1) In the cases where the center-of-mass
trajectory moves to the channel center, the elliptical cylinder is stationary in orientation with the
angle of inclination θ = 0. (2) In the cases where the center-of-mass trajectory oscillates about
the channel center, the elliptical cylinder also exhibits oscillatory orientation dynamics. (3) In the
0.8
Re=10, Re =4
Re=40, Re =30
0.7
p
p
Re=40, Re =16
Re=200, Re =152
p
0.7
p
Re=100, Re =40
Re=500, Re =380
0.65
p
p
Re=200, Re =80
Re=550, Re =418
p
p
0.6
Yp
Y
p
0.6
0.5
0.55
0.5
0.4
0.45
0.3
0.4
0.2
0
100
200
300
t
(a)
400
500
0
50
100
150
200
250
300
t
(b)
FIG. 14. Lateral migration of an elliptical cylinder with (a) small size ratio K = 0.4 and aspect ratio A = 3, (b) large size
ratio K = 0.76 and aspect ratio A = 2 at different Rep . Note that for the smaller K = 0.4, the center-of-mass is oscillating
mildly about an equilibrium position for Rep = 4–40, but will eventually oscillate about the channel center at the higher Rep
= 80. For the larger K = 0.76, the elliptical cylinder oscillates about the channel center for Rep = 30–380, and eventually
moves to the channel center at the higher Rep = 418.
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103302-15
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
1
0.9
0.8
0.7
1
p
Y
Y
0.6
0.5
0.5
0.4
0
0
0.5
1
1.5
2
2.5
3
3.5
4
0.3
X
0.2
(a)
0.1
0
0
100
200
300
400
500
600
700
800
900
1000
120
140
160
180
200
t
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
p
1
0.5
Y
Y
p
(b)
1
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
100
200
300
400
500
600
700
800
900
1000
0
0
20
40
60
80
100
t
t
(c)
(d)
FIG. 15. (a) The initial positions of ND = 16 (φ T = 3.77%) cylinders with A = 1.875 and K = 0.15, the initial setup is
sx = 0.25, sy = 0.25, and random θ 0 . Temporal development of lateral migration Yp of ND = 16 cylinders at (b) Re = 10,
(c) Re = 100, and (d) Re = 1000.
cases where the center-of-mass trajectory oscillates mildly about the Segre–Silberberg equilibrium
position, the elliptical cylinder is rotary all the time.
C. The cases of a few elliptical cylinders (ND = 16, φ T = 3.77%)
In this section, we consider the motion of 16 (the corresponding φ T = 3.77%) neutrally buoyant
elliptical cylinders with A = 1.875 and K = 0.15 in a pressure-driven plane Poiseuille flow at
Re = 10, 100, 1000. The domain of computation is D = (4 × 1). The initial setups are similar to
those in Fig. 15(a): the initial distance between two cylinder centers is sx = 0.25 in the x-direction,
and sy = 0.25 in y-direction, and the initial angles of inclination of the particles θ 0 are chosen
randomly. The other parameters are the same as in Sec. IV B.
1. The effect of the Reynolds number (Re)
Figures 15(b)–15(d) show the temporal developments of lateral migrations at Re = 10, 100,
and 1000. At the lower Re = 10, the collisions between cylinders repel them off each other
(Fig. 15(b)). The cylinders exhibit a bifurcation behavior by scattering into eight groups, with
each group fluctuating about a time-averaged position Yavg . At the higher Re = 100, the cylinders
tend to cluster near the equilibrium positions (Fig. 15(c)). At the even higher Re = 1000, the cylinders
move to the equilibrium position on each side of the central axis (Fig. 15(d)).
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103302-16
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
1
0.9
0.8
0.7
1
p
Y
y
0.6
0.5
0.5
0.4
0
0
0.5
1
1.5
2
2.5
3
3.5
4
x
0.3
(a)
0.2
0.1
0
0
500
1000
1500
2000
2500
t
(b)
FIG. 16. (a) The initial positions of ND = 16 (φ T = 3.77%) cylinders with A = 1.875 and K = 0.15, the initial setup is
sx = 0.5, sy = 0.2, and θ 0 = π /4. (b) Temporal development of lateral migration Yp of ND = 16 cylinders at Re = 10.
2. The effect of the initial distributions of cylinders
We also consider the effect of the initial positions for 16 elliptical cylinders at Re = 10.
Figure 16(a) shows the initial setup: the initial distance between two cylinder centers is sx = 0.5
in the x-direction, and sy = 0.2 in the y-direction, and initial angles of inclination θ 0 = π /4.
Figure 16(b) shows the temporal development of lateral migration of 16 cylinders at Re = 10. The
particle-particle collisions occur much later if the initial positions of the cylinders are farther away
from each other (compared to Fig. 15(b), where sx = 0.25, sy = 0.25). Eventually, the cylinders are
divided into eight groups, and fluctuate about a time-averaged position. The results show that the bifurcation behavior is independent of the initial positions of the cylinders. Then, we consider the effects
of the initial orientations for 16 elliptical cylinders at Re = 10 and 1000. The initial orientations are
either chosen randomly, or fixed at θ 0 = 0, π /4, or π /2. For the cases at Re = 10, the bifurcation in the
averaged equilibrium positions remains almost the same for all the initial orientations. For the cases at
Re = 1000, all the cases have the same equilibrium position at about req = 0.13. These results show
that the bifurcation behaviors in the time-averaged equilibrium positions are independent of the
initial positions and orientations of the elliptical cylinders.
D. The cases for many cylinders (ND = 72, φ T = 16.9%)
In this section, we consider the motion of 72 neutrally buoyant cylinders in a pressure-driven
Poiseuille flow at Re = 1000; here the total solid area fraction φ T is increased to 16.9%. The initial
distance between two cylinder centers is sx = 0.167 in the x-direction, sy = 0.167 in the y-direction
with θ 0 = π /4 for the larger K = 0.2 (A = 3.333), and θ 0 randomly chosen for the smaller K = 0.15
(A = 1.875) and 0.11 (A = 1). The other parameters are the same as in Sec. IV B.
1. The effect of the aspect ratio (A)
In the first three cases, there are ND = 72 (φ T = 16.9%) cylinders with aspect ratios of A
= 3.333, 1.875, 1 and the corresponding K = 0.2, 0.15, 0.11, respectively. Figures 17(a)–17(c)
and 18(a)–18(c), and 19(a)–19(c) show the cylinder positions at t = 1000, the distributions of the
translational velocity of the fluid (u) and of the cylinders (Up ), and the time-averaged solid area
distribution φ during t = 900 and 1000 for 72 cylinders. The elliptical cylinders with the higher
A = 3.333 have faster Up ’s among the three cases, being more closely concentrated towards the
central axis. The cylinders with the moderate A = 1.875 have slower Up ’s, and a relatively lower
distribution φ can be observed in the middle channel. The cylinders with the smallest A = 1 have
the slowest Up ’s, and the cylinders are more widely spread across the channel. The cylinders and
flow velocity profiles are more blunted as A is increased, yet there is no velocity blunting near the
wall due to the insignificant rheological effect there. We also observe a lower distribution φ in the
middle channel; this situation is similar to the experimental result obtained for spheres with a small
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103302-17
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
y
y
(a)
1
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0
0.1
0.2
0.3
0.4
0.5
Up, u
φ
(b)
(c)
0.6
0.7
0.8
0.9
1
FIG. 17. The behaviors of 72 cylinders (φ T = 16.9%) with A = 3.333, K = 0.2 at Re = 1000: (a) The cylinder positions at
t = 1000; (b) the distribution of the translational velocities of cylinders Up (●), the averaged flow velocity profile u (solid
line) and that of the Poiseuille flow (without cylinders) (dashed line); (c) the time-averaged solid area distribution φ during t
= 900 and 1000.
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
y
y
(a)
1
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
Up, u
(b)
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
φ
0.6
0.7
0.8
0.9
1
(c)
FIG. 18. The behaviors of 72 cylinders (φ T = 16.9%) with A = 1.875, K = 0.15 at Re = 1000: (a) The cylinder positions at
t = 1000; (b) the distribution of the translational velocities of cylinders Up (●), the averaged flow velocity profile u (solid
line) and that of the Poiseuille flow (without cylinders) (dashed line); (c) the time-averaged solid area distribution φ during t
= 900 and 1000.
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103302-18
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
y
y
(a)
1
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Up, u
(b)
0
0
0.1
0.2
0.3
0.4
0.5
φ
0.6
0.7
0.8
0.9
1
(c)
FIG. 19. The behaviors of 72 cylinders (φ T = 16.9%) with A = 1, K = 0.11 at Re = 1000: (a) The cylinder positions at
t = 1000; (b) the distribution of the translational velocities of cylinders Up (●), the averaged flow velocity profile u (solid
line) and that of the Poiseuille flow (without cylinders) (dashed line); (c) the time-averaged solid area distribution φ during
t = 900 and 1000.
solid volume fraction by Han and Kim.24 The cylinders with a lower A (1.875 and 1) tend to cluster
about two peak positions. In contrast to the elliptical cylinders (A = 1.875), the circular cylinders
interact against this converging tendency to scatter the cylinders more widely across the channel.
For each cylinder, we consider its time-averaged angular velocity ωs /π during t = 900 and 1000.
Figure 20 shows the distribution of ωs /π for the 72 cylinders. At these aspect ratios (A = 3.333,
1.875, 1), the cylinders near the walls have higher ωs /π , while the ones near the channel center have
almost zero ωs /π . In other words, the cylinders are not rotating except those at the outer regions
(near the walls), since the cylinders closer to the central axis of the channel are constrained by the
neighboring cylinders. This constraint of rotation is more significant for the cylinders with a higher
aspect ratio, since the cylinders are more concentrated near the channel center.
2. The effect of the Reynolds number (Re)
We consider the motion of ND = 72 (φ T = 16.9%) cylinders with A = 1.875 and K = 0.15
in a pressure-driven Poiseuille flow at Re = 500, 1000, and 2000, respectively. The domain of
computation is D = (4 × 1), and the pressure gradient is 8. The fluid and particles are at rest
initially. The initial setup is sx = 0.167, and sy = 0.167, with θ 0 chosen randomly. The mesh size is
h = 1/256, and the time step is t = 0.01.
In order to see the scaling relationships with respect to Re, we normalize the translational
velocity by Reα . Figure 21 shows good overlaps of u/Reα and Up /Reα of 72 cylinders during t
= 900 and 1000 for the different Reynolds numbers, with α = 1.068. The cylinders are more
densely concentrated around the central axis of the channel as Re is increased. Figures 22(a)–22(c)
show the time-averaged solid area distribution φ during t = 900 and 1000 of the 72 cylinders for
the different Reynolds numbers. It is observed that there are no longer clearly defined averaged
equilibrium positions. Instead, a particle-free layer exists near each wall. We define the thickness
yf of the layer to be the distance between the wall and the clustering of cylinders (or the thickness
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103302-19
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
1
0.9
0.8
0.7
y
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
| ωs/π |
1.5
2
2.5
FIG. 20. The distribution of the time-averaged angular velocities ωs /π during t = 900 and 1000 for the 72 cylinders
(φ T = 16.9%) with A = 3.333 and K = 0.2 (●), A = 1.875 and K = 0.15 (◦), A = 1 and K = 0.11 (×).
of the time-averaged solid area distribution φ = 0 region next to the wall). As Re is increased, yf
also increases, while the central axis of the channel has a relatively larger local minimum in φ. The
cylinders at a higher Re are distributed closer to the central axis, while at low Re, the more frequent
particle-particle interactions cause the cylinders to scatter more widely across the channel. It is noted
1
0.9
0.8
0.7
y
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
Up/Reα , u/Reα or u/Re
4
5
−4
x 10
FIG. 21. The normalized flow velocity profile u/Reα (solid line) and the normalized distributions of the translational velocities
Up /Reα with α = 1.068 of 72 cylinders (φ T = 16.9%, A = 1.875, K = 0.15) at Re = 500 (●), 1000 (), 2000 () during t = 900
and 1000. The normalized Poiseuille flow velocity (without cylinders) u/Re is denoted by the dashed line for comparison.
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103302-20
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
TABLE I. The number of equilibrium positions (if it is 2) or the averaged equilibrium positions (if it is greater than 2) and
the thickness of the particle-free layers yf .
A
NDa (or φ T b )
36
54
(8.48)
(12.72)
Re
4
(0.94)
8
(1.88)
16
(3.77)
10
4
4
8
8
100
2
2
4
500
2
2
2
1000
2
2
2–3
10
4
4
8
100
2
2
4
500
2
2
2
1000
2
2
2
10
100
500
4
2
2
4
2
2
8
4
2
1000
2
2
2
3.333
1.875
1
72
(16.96)
90
(21.21)
108
(25.45)
No
[0.0935]
No
[0.2336]
No
[0.2991]
8
Noc
[0.0093]d
No
[0.0748]
No
[0.1963]
No
[0.2336]
8
No
[0.0093]
No
[0.0561]
No
[0.1495]
No
[0.2150]
8
No
[0.0654]
No
[0.1589]
No
[0.2430]
8
6–8
No
[0.1121]
No
[0.1589]
No
[0.0374]
No
[0.1121]
No
[0.1776]
9
6–8
No
[0.0564]
No
[0.1121]
No
[0.0187]
No
[0.0748]
No
[0.1215]
9
8–9
No
[0.0374]
No
[0.0841]
No
[0.0093]
No
[0.0374]
No
[0.1028]
No
[0.1963]
No
[0]
No
[0.0093]
No
[0.0561]
No
[0.0841]
9
9
No
[0.0187]
No
[0.0374]
No
[0.0093]
No
[0.0280]
No
[0.0748]
No
[0.1682]
No
[0]
No
[0]
No
[0.0280]
No
[0.0654]
9
9
No
[0.0093]
No
[0.0187]
a ND
is the number of cylinders.
is the total solid area fraction (%).
c “No” means no averaged position identified.
d [y ] represents the thickness of the particle-free layer.
f
T
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
y
1
y
y
bφ
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0.1
0.2
φ
(a)
0.3
0.4
0.5
0
0
0.1
0.2
φ
0.3
(b)
0.4
0.5
0
0
0.1
0.2
φ
0.3
0.4
0.5
(c)
FIG. 22. The time-averaged solid area distribution φ for 72 cylinders (φ T = 16.9%) with A = 1.875, K = 0.15 at Re = (a)
500, (b) 1000, (c) 2000 during t = 900 and 1000.
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103302-21
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
TABLE II. The relations between the thickness of the particle-free layer and the total solid area fraction of the cylinders,
where 8.48% < φ T < 25.45%.
100
Re
500
1000
0.0002φ T 2 − 0.0100φ T + 0.1364
0.0001φ T 2 − 0.0065φ T + 0.1439
0.0003φ T − 0.0173φ T + 0.2335
0.0002φ T 2 − 0.0150φ T + 0.2691
0.0001φ T 2 − 0.0122φ T + 0.3346
0.0003φ T − 0.0204φ T + 0.3084
0.0004φ T 2 − 0.0223φ T + 0.3888
0.0003φ T 2 − 0.0147φ T + 0.4984
A
1
1.875
3.333
2
2
that the transition flow occurs when Re > 2300 for the channel flow in the absence of cylinders,39, 40
but the transition to turbulence could occur at lower Re in the presence of cylinders.41 However, our
results are mainly for the cases at Re = 500, 1000, and only one case at Re = 2000.
E. The effect of the total solid area fraction (φ T = 0.94%–25.45%, ND = 4–108)
We consider a wide range in the number of elliptical cylinders with (1) A = 3.333 and K = 0.2,
(2) A = 1.875 and K = 0.15, and (3) A = 1 and K = 0.11 in a pressure-driven Poiseuille flow. The
numbers of cylinders are ND = 4, 8, 16, 36, 54, 72, 90, and 108, which correspond to the total solid
area fractions φ T = 0.94, 1.88, 3.77, 8.48, 12.72, 16.96, 21.21, and 25.45 (%), respectively. The
Reynolds numbers include Re = 10, 100, 500, and 1000. The initial distance between two cylinder
centers in the x-direction and y-direction are sx = sy = 0.25 for ND = 4–36, and sx = sy = 0.167
for ND = 54–108. The initial angles of inclination are θ 0 = π /4 for the larger K = 0.2, and θ 0 is
random for the smaller K = 0.15 and 0.11. The other parameters are the same as in Sec. IV B.
Table I lists the number of time-averaged equilibrium positions of the cylinders for different
aspect ratios A and the dimensionless thickness of the particle-free layers yf at various Reynolds
numbers Re. The equilibrium positions can be clearly identified when ND is low. As ND is increased,
0.35
0.3
0.25
y
f
0.2
0.15
0.1
0.05
0
8
10
12
14
16
18
φT (%)
20
22
24
26
FIG. 23. The thickness of the particle-free layer yf versus the total solid area fraction φ T of the cylinders with A = 3.333 and
K = 0.2 (●), A = 1.875 and K = 0.15 (◦), A = 1 and K = 0.11 () at Re = 100 (dotted line), 500 (dashed line), 1000 (solid
line).
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103302-22
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
1
1
0.9
0.9
0.8
0.8
0.7
0.7
φT=0.12
φ =0.16
T
φ =0.25
T
0.6
ND
0.5
y
y
0.6
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
p
φ
(a)
(b)
U or u
0.6
0.8
1
FIG. 24. (a) The time-averaged distribution of the translational velocities of cylinder Up for ND = 54 (●), 72 (), 108 ()
circular cylinders with K = 0.11 (φ T = 0.12, 0.16, 0.25, respectively), and averaged flow velocity profile u (solid line) during
t = 100 and 200 at Re = 10 (Rep = 1.1). The flow velocity profiles without particles are denoted by dashed line. (b) The
time-averaged solid area distribution φ.
the rheological effect is also more significant, and the equilibrium positions can no longer be defined;
instead, a particle-free layer is found near each wall of the channel. Figure 23 shows the dimensionless
thickness of the particle-free layers yf versus the aspect ratio A (= 3.333, 1.875, 1) at three different
Re (= 100, 500, 1000). As φ T is increased, the particle-particle interactions are intensified, causing
deviation of the cylinders from the equilibrium positions and thus a decrease in yf . It is also found that
yf increases as Re or A is increased, because the particle-free layers are easier to produce at higher
Re or for cylinders with a larger A. At a lower Re (= 10, 100), the cylinders fluctuate about several
averaged equilibrium positions, and the particle-free layer cannot be formed. The relations between
yf and φ T of the cylinders for A = 3.333, 1.875, 1 at Re = 100, 500, 1000 in the range of 8.48 < φ T
< 25.45 (%) are shown in Table II. Basically, these relations are linear, but they are modified by a
small quadratic term, which derives the linear relationships only at higher total solid area fraction
φ T ’s. The decreasing rate of the linear term increases as Re is increased. Figures 24(a) and 24(b)
show the translational velocity profiles (K = 0.11) at Re = 10 (Rep = 1.1) and the corresponding
averaged flow velocity profiles for ND = 54, 72, 108 (φ T = 12.72%, 16.96%, 25.45%), respectively.
The blunting of the velocity profile uaxial /u = 1.50 (uaxial and u are the space-averaged fluid
velocities in the channel center and in the domain of computation, respectively) for φ T = 16.96%,
which is close to uaxial /u = 1.46 for φ T = 15%, obtained by Nott and Brady.23 As φ T is increased,
the Up and u profiles deviate more from the Poiseuille flow profile (without cylinders), especially at
places near the channel center. Basically, the distribution of φ has a major maximum near each wall
and a minor maximum in the channel center. This behavior is more pronounced as φ T is increased.
V. CONCLUDING REMARKS
This study provides a detailed investigation of the motion of a single and multiple neutrally
buoyant elliptical cylinders in plane Poiseuille flow. As a validation, we consider a single elliptical
cylinder in Couette flow, and obtained a critical particle Reynolds number Rep,c = 29, which is in
good agreement with Ding and Aidun’s solution.7
We summarize the results for the motion of a single elliptical cylinder in two parts: (a) the
smaller size ratio range K ≤ 0.2, and (b) the larger size ratio range K ≥ 0.4.
(a)
For the smaller size ratio range K ≤ 0.2, we consider the cases with K = 0.2, 0.17, 0.15,
0.12, 0.11 and corresponding aspect ratios A = 3.333, 2.5, 1.875, 1.2, 1 in the Poiseuille
flow at Re = 10. It was found that the cylinder with a higher A moves closer to the central
line, and has a higher translational velocity Up but a lower average angular velocity ω. By
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103302-23
(b)
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
least-squares fitting, we obtain the equilibrium position req and the time-averaged angular
velocity ωs versus A as req = 0.4801 − 0.002(A − 1) − 0.0025(A − 1)2 and ωs /π = 0.3051
− 0.0355(A − 1) − 0.033(A − 1)2 , respectively. We also investigated the effect of Re (1–1000)
on a single elliptical cylinder with A = 1.875 and K = 0.15, and found that there is a critical
Reynolds number Rec ∼ 3, below which req shifts towards the wall with increasing Re, and
above which the trend is the opposite. This trend and Rec are identical with a circular cylinder
with the same size ratio. The center-of-mass of an elliptical cylinder (A = 1.875, 3.333) can
oscillate about its averaged equilibrium position, and the amplitude of oscillation becomes
larger with increasing Re or K.
For the larger size ratio range K ≥ 0.4, we consider the cases with K = 0.4, 0.76 and
corresponding A = 3, 2 in the range of Re = 10–550 (Rep = 4–418). For K = 0.4, the cylinder
always is rotary at Re = 10, 40, 100 (Rep = 4, 16, 40), and becomes oscillatory in the angle
of inclination in θ at Re = 200 (Rep = 80). For K = 0.76, the cylinder exhibits an oscillatory
behavior in inclination at Re = 40, 200, 500 (Rep = 30, 152, 380), and becomes stationary in
θ = 0 at the largest Re = 550 (Rep = 418). It is found that the center-of-mass trajectories and
the orientation dynamics have the following correlations. (1) In the cases where the centerof-mass trajectory eventually moves to the channel center, the elliptical cylinder becomes
stationary in orientation with the angle of inclination θ = 0. (2) In the cases where the
center-of-mass trajectory oscillates about the channel center, the elliptical cylinder exhibits
an oscillatory orientation dynamics. (3) In the cases where the center-of-mass trajectory is
eventually oscillating mildly about the Segre–Silberberg equilibrium position, the elliptical
cylinder is rotary all the time.
We also summarize the results for the motion of multiple elliptical cylinders in two parts: (a) a few
cylinders, ND = 16 (the corresponding total solid area fraction φ T = 3.77%) and (b) a larger number,
ND = 36–108 (φ T = 8.48%–25.45%).
(a)
(b)
For a smaller ND = 16 (φ T = 3.77%), we consider the cylinders with A = 1.875 and K = 0.15
at Re = 10, 100, 1000. It is shown that at the small Re = 10, the cylinders scatter into four
groups on each side, with each group fluctuating about an averaged equilibrium position, and
this bifurcation behavior is shown to be independent of the initial positions and orientations
of the cylinders. The particle-particle interactions are less dominant as Re is increased, and
there are fewer averaged equilibrium positions (four positions on each side of the channel
center at Re = 10; two at Re = 100; one at Re = 1000).
For the larger ND = 36, 54, 72, 90, and 108 (φ T = 8.48, 12.72, 16.96, 21.21, and 25.45
(%), respectively) in the range of Re = 10–1000, we observed the rheological effect due to
the motion of cylinders in fluid, which affects the migration of the cylinders dramatically.
There are no longer clear equilibrium positions, and the particle-free layers near the walls are
formed instead. The cylinders and flow velocity profiles are more blunted as φ T is increased,
yet there is no velocity blunting near the walls due to the insignificant rheological effect there.
The number of cylinders ND = 36 (φ T = 8.48%) is a critical case in which the Reynolds
number has a great impact. At lower Re (<100), the cylinders are widely spread over the
entire channel, while at higher Re (>100), there are particle-free layers next to the walls.
Interesting remarks are also available for more circular cylinders (ND = 54, 72, 108 with φ T
= 12.72%, 16.96%, 25.45%. At the low Re = 10, the distribution of φ has a major maximum
near each wall and a minor maximum in the channel center. At the high Re = 1000, the
maximum-φ positions shift away from the walls. If we consider a smaller φ T (12.72% or
16.96%), the channel center becomes the location of a local minimum of φ. However, at the
higher φ T = 25.45%, the φ-distribution in the middle channel is a high plateau with thinner
particle-free layers next to the walls. The results for A = 3.333, 1.875, 1 on 72 cylinders (φ T
= 16.96%) at Re = 1000 showed that the cylinders with higher A are more densely concentrated
around the channel center. The general trend is that the thickness of the particle-free layer yf
increases as A (or K) or Re is increased. In addition, an elliptical cylinder has a higher angular
velocity when it is close to the particle-free layer than to the channel center.
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103302-24
Chen, Pan, and Chang
Phys. Fluids 24, 103302 (2012)
ACKNOWLEDGMENTS
The work was supported in part by the National Science Council (Taiwan) under Contract Nos.
NSC97-2221-E-002-223-MY3, NSC99-2628-M-002-003, and NSC100-2221-E-002-152-MY3.
T.-W. Pan acknowledges the support of the US NSF (Grant No. DMS-0914788).
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