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19th Australasian Fluid Mechanics Conference
Melbourne, Australia
8-11 December 2014
Three-Dimensional Simulation of Flow past Two Circular Cylinders of Different Diameters Jitendra Thapa, Ming Zhao and Shailesh Vaidya
School of Computing, Engineering and Mathematics University of Western Sydney, Penrith, NSW 2751, Australia
Abstract
The vortex shedding flow past two circular cylinders of different
diameters is investigated numerically by solving the three-
dimensional Naiver-stokes equations using the Petrov-Galerkin
finite element method (PG-FEM). The Reynolds number based
on the free stream velocity (U) and the diameter of the large
cylinder (D) is Re=1000. Simulations are carried out for a
constant gap of 0.0625D and a constant diameter ratio of 0.45.
The study is focused on the effect of position angle of small
cylinder relative to the larger one on the three-dimensional flow,
the force coefficients, the vortex shedding frequencies from the
two cylinders and flow characteristics. As observed in the
previous experimental studies, biased flow in the wake of the gap
is observed when the two cylinders are in nearly side-by-side
arrangement and it leads to significant reduction of the oscillation
of the forces on the cylinders.
Introduction
Offshore pipelines of different diameters are of
engineering interests because they are widely used in the offshore
oil and gas engineering. One small pipeline (secondary pipeline)
and a large (main) pipeline are often bundled together to reduce
the cost of the installation and stabilisation. The two pipelines are
strapped together at certain intervals along their axial direction
during the installation. The pipeline bundle with two pipelines of
different diameters, the most popular arrangement, as shown in
Figure1 is often referred as piggyback pipeline in the offshore oil
and gas industry. A piggyback pipeline is modelled by two
circular cylinders separated by a small gap in this study.
Inflow
α
D
d
Small cylinder
Large cylinder
x
y
G
α
Figure 1. Definition Sketch of flow past two circular cylinders of
different diameters
Flow past two circular cylinders in a side-by-side or
tandem arrangement signifies a remarkably complex flow
configuration. Due to the proximity between the two circular
cylinders, variety of flow patterns characterised by the wake
behaviour may be discerned. The resulting forces and vortex
shedding patterns have been found to be completely different
from those on a single body when more than one body was
placed in a fluid flow (Zdravkovich 1987). In case of two side-
by-side cylinders of identical diameters, a single wake was found
behind the cylinders if the distance between the centres of the
cylinders (L) is less than 2.2 times of the cylinder diameter (D)
(Bearman and Wadkock 1973; Williamson 1985; Kim and
Durbin 1988). When the distance is less than 2D in a side-by-side
arrangement, repulsive force between the cylinders was observed.
For the tandem arrangements, negative drag and the vortex
shedding on the downstream cylinder were observed if the
distance between the centres of the two cylinders is less than 3D
(Meneghini et al., 2001). Lee et al. (2004) demonstrated that a
small control rod can control hydrodynamic forces on a circular
cylinder. The vortex shedding behind a cylinder could be reduced
or suppressed over a limited range of Reynolds numbers by
placing a small secondary control rod (Strykowski and
Sreenivasan 1990). Using experimental and numerical method,
Tsutsui et al. (1997) investigated the behaviour of an interactive
flow around two circular cylinders of different diameters at close
proximity. The shear layer separated from the main cylinder was
found to re-attach and adhere to the rear surface of the main
cylinder. Moreover, numerical simulations of the intermittent re-
attachment and time-averaged fluid forces agreed well with those
of previous experiments, and the qualitative characteristics of
calculated Strouhal numbers coincided with those of experiments
(Tsutsui et al., 1997).
X
Y
-1.5 -1 -0.5 0 0.5 1 1.5 2
-1.5
-1
-0.5
0
0.5
1
1.5
(a)
(b)
Figure 2. Computational mesh near two circular cylinders for α= 135°
In this study, flow past two circular cylinders of
different diameters is investigated numerically. As shown in
Figure 1, the large cylinder represents the main pipeline and the
small cylinder represents the piggyback pipeline in a piggyback
pipeline system. Figure 1 (b) is zoomed in view of the mesh in
the gap between the cylinders to visualise the mesh structure
clearly. The position of small cylinder is determined by the gap
between the cylinders G, its diameter d and a position angle α.
The study is focused on the effect of the position angles of small
cylinder relative to large cylinder on the flow and hydrodynamic
forces. The diameter ratio (d/D) is 0.45. The Reynolds number is
1000 for the main cylinder and 450 for the secondary cylinder
and the constant gap between two cylinders is 0.0625D. The
position angles of small cylinder relative to the flow direction are
90°, 135° and 180°.
Numerical Method
The three-dimensional Navier-Stokes (NS) equations are
solved in order to simulate the flow. The non-dimensional NS
equations are written as
,0Re
12
2
x
u
x
p
x
uu
t
u i
ij
ij
i (1)
,0
i
i
x
u (2)
where (x1, x2, x3) = (x, y, z) are the Cartesian co-ordinates, ui is the
fluid velocity component in the xi-direction, t is the time and p is
the pressure. The Reynolds number is defined as Re=UD/ν with ν
being the kinematic viscosity of the fluid. The governing
equations are solved by the Petrov-Galerkin finite element model
developed by Zhao et al. (2009). A 60×40×9.6 computational
domain is used to perform the simulation. All simulations were
performed on a cluster computers located in the advanced
computing facility in Western Australia (iVEC). Each simulation
was conducted at least up to the non-dimensional time of Ut/D =
550 to ensure the fully development of vortex shedding is
achieved and 64 central processing unit (CPU) were used for
each of the simulations. The whole computational domain was
divided into 192 layers of 8-noded hexahedron tri-linear finite
elements along the axial direction of the cylinders. 130 and 96
elements were distributed on the circumferences of large and
small cylinders, respectively. The non-dimensional
computational time step Δt was set to 0.003.
Results and Discussion
The study is focused on the effect of the position angle of
small circular cylinder relative to the large circular cylinder with
a small gap between the cylinders is 0.0625D. Numerical
simulations were carried out at constant Reynolds number of Re
= 1000 based on large cylinder and Re = 450 based on small
cylinder with the position of small cylinder was arranged as 90°,
135° and 180° where the ratio of diameters (d/D) was 0.45.
Figure 2 shows the computational mesh around the two cylinders
at α = 135°. The quality of computational meshes for other cases
of α are same as shown in Figure 2.
Figure 3 shows the time histories of the computed mean drag
and lift coefficients for both cylinders at three position angles of
the small cylinder which are defined as
22
2
1,
2
1UD
FC
UD
FC L
LD
D
where ρ is the fluid density, FD and FL is total drag and lift
coefficient. The oscillation of the force coefficients due to the
vortex shedding can be seen in the time histories of the force
coefficients. Three-dimensionality of the flow appeared after
non-dimensional time Ut/D ≥ 100 for all three cases and the
analysis was carried out after this non-dimensional time Ut/D ≥
100. The non-dimensional time Ut/D=0 in Figure 3 is actually the
time when full three-dimensional flow has developed, The mean
drag coefficient is found to be decreased as the position angle of
small cylinder increases.
The vortex flow structure can be identified by the iso-surface
of the second negative eigenvalue of the tensor 22ΩΨ , where
Ψ and Ω are the symmetric and the anti-symmetric parts of the
velocity-gradient tensor, respectively. This second eigenvalue,
say e2, represents the location of the vortex core (Jeong and
Hussain, 1995). Figure 4 shows the iso-surfaces of e2=-0.5 for
three position angles of the small cylinder. The vortex flows at
the left and right columns of Figure 4 are those at the instants
when the lift coefficient is the maximum and minimum,
respectively. Fully developed three-dimensional vortex shedding
can be seen clearly for all three cases based on the iso-surface of
e2 =-0.5. The development of spanwise vortices immediately
downstream the cylinders in the wake can be seen in Figure 5,
whereas the streamwise vortices dominate the flow further
downstream for all cases. Due to the transformation of energy
from spanwise vortices to streamwise vortices, the spanwise
vortices dissipate quickly in all cases. It has been found that flow
past a single cylinder at Re=1000 is in Mode B, characterized by
a wake dominated by the streamwise vortices. It can be seen in
Figure 4 that the wake flow for the two cylinder system is
dominated by the streamwise vortices, which is in mode B.
-1.2
-0.6
0.0
0.6
1.2
1.8
0 100 200 300 400 500
Fo
rce
coef
fici
ents
Ut/D
CD
CLCylinder 1
0.0
0.2
0.4
0.6
0.8
0 100 200 300 400 500
Fo
rce
coef
fici
ents
Ut/D
CD
CL
Cylinder 2
-1.2
-0.6
0.0
0.6
1.2
1.8
0 100 200 300 400 500F
orc
e co
effi
cien
tsUt/D
CD
CLCylinder 1
-0.2
0.0
0.2
0.4
0.6
0 100 200 300 400 500
Fo
rce
coef
fici
ents
Ut/D
CD
CLCylinder 2
-0.2
0.2
0.6
1.0
1.4
0 100 200 300 400 500
Fo
rce
coef
fici
ents
Ut/D
CD
CLCylinder 1
-0.1
0.0
0.1
0.2
0 100 200 300 400 500
Fo
rce
coef
fici
ents
Ut/D
CD
CL
Cylinder 2
Figure 3. Time histories of force coefficients for G/D= 0.0625. (a) α =
90°; (b) α = 135° and (c) α = 180°
Figure 5 shows the contours of axial vorticity at the mid-
section of both cylinders at two instants when the lift coefficient
is the maximum and the minimum. It can be seen that the vortex
shedding is dominated by the shear layers from the outer sides of
two cylinders. The vortex shedding between the large and small
cylinders was not observed and obviously the vortices shed from
the outer side of the two cylinders. However, the jet flow from
the gap has significant effect on the vortex shedding. The flow
pattern in Figure 5 (a) and (b) agree well with the experiments by
Tsutsui et al. (1997) where the separated shear layer from the
large cylinder reattaches to the rear face of large cylinder by the
formation of wall jet. The gap flow is biased to the rear face of
large cylinder which is referred as reattachment by Tsutsui et al.
(1997). The jet flow in Figure 5 (a) and (b) prevent the shear
layer from the bottom side of the large cylinder re-attaching to
the back surface of the cylinder, leading to significant decrease in
the lift coefficient oscillation. Single wake is formed from the
(a)
(b)
(c)
both cylinders for all the simulated position angles of the small
cylinder. The formation of single wake from two cylinders was
also observed in Zhao et al. (2005) for α = 90° and the gap 0.05.
The vortices are found to dissipate very quickly when they are
convected downstream.
Figure 4. Vortex structures at maximum (left) and minimum (right) lift
presented by the iso-surface of e2= -0.5.
The variation of drag and lift coefficients of the large and
small cylinders with angle of α and its comparison with
experimental data and the two-dimensional numerical result from
Zhao et al.(2007) are compared with the experimental data in
Figure 6. For the large cylinder, CD1 decreases with increasing α.
The maximum value of CD1 is observed to be 1.49 at α=90°. The
small cylinder hides in the wake of main cylinder that’s why the
drag coefficient becomes lower than that of single cylinder. The
drag coefficient of small cylinder is also found to be decreased
with increasing α. At α=180°, the value of CD2 becomes zero
because of the symmetry of the cylinder system. The variations
of the mean drag and lift coefficients with the positional angle are
agreed well with the observation by Takayuki et al. (1997) and
Zhao et al. (2007).
x
y
-1 0 1 2 3 4 5 6 7 8 9 10 11-3
-2
-1
0
1
2
3
2
1.6
1.2
0.8
0.4
-0.4
-0.8
-1.2
-1.6
-2
z=4.8, 281.1
x
y
-1 0 1 2 3 4 5 6 7 8 9 10 11-3
-2
-1
0
1
2
3
2
1.6
1.2
0.8
0.4
-0.4
-0.8
-1.2
-1.6
-2
z=4.8, t=333.9
x
y
-1 0 1 2 3 4 5 6 7 8 9 1 0 1 1-3
-2
-1
0
1
2
3
2
1 .6
1 .2
0 .8
0 .4
-0 .4
-0 .8
-1 .2
-1 .6
-2
z=4.8, t=569.7
x
y
-1 0 1 2 3 4 5 6 7 8 9 10 11-3
-2
-1
0
1
2
3
2
1.6
1.2
0.8
0.4
-0.4
-0.8
-1.2
-1.6
-2
z=4.8, t=483.3
x
y-1 0 1 2 3 4 5 6 7 8 9 10 11
-3
-2
-1
0
1
2
3
2
1.6
1.2
0.8
0.4
-0.4
-0.8
-1.2
-1.6
-2
z=4.8, t=481.2
x
y
-1 0 1 2 3 4 5 6 7 8 9 10 11-3
-2
-1
0
1
2
3
2
1.6
1.2
0.8
0.4
-0.4
-0.8
-1.2
-1.6
-2
z=4.8, t=510.9
Figure 5. Contour of axial vorticity at mid-section of both cylinders at
maximum and minimum lift: (a) α = 90°; (b) α = 135° and (c) α = 180°
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
90 112.5 135 157.5 180
Mea
n d
rag c
oef
fici
ent
α
CD1, This study CD1, Exp.
Zhao et al.2007 CD2, This study
CD2, Exp. Zhao et al.2007
(a)
-1.0
-0.6
-0.2
0.2
0.6
1.0
90 112.5 135 157.5 180
Mea
n l
ift
coef
fici
ent
α
CL1, This study CL1, Exp.
Zhao et al.2007 CL2, This study
CL2, Exp. Zhao et al.2007
(b)
Figure 6. Comparison of force coefficients with experimental and
numerical result for both cylinders
(a)
(b)
(c)
Figure 7 shows the variation of RMS lift coefficient with α
and the comparison of Strouhal number with the results by
Takayuki et al. (1997) and Zhao et al. (2007). The RMS lift of
the large cylinder is found to be decreased with increasing α. The
RMS lift coefficient for an isolated single cylinder is 0.177 at
Re=1000 (Zhao et al. 2013). Significant reduction of RMS lift
coefficient is observed when a small cylinder is placed near a
large cylinder for all the studied position angles (α). The RMS lift
of small cylinder is also reduced significantly compared to that of
a single cylinder as in Figure 7 (a). The Strouhal number, defined
by St = fD/U, where f is the frequency of the lift coefficient is
shown in Figure 7 (b). The maximum Strouhal number occurs as
α=180°. The Strouhal number as the two pipelines are in a side-
by-side arrangement is smaller than that when the small cylinder
is in the wake of the large cylinder.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
90 112.5 135 157.5 180
RM
S L
ift
coef
fici
ent
α
RMS_CL1
RMS_CL2
(a)
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
90 112.5 135 157.5 180
St
α
St1, This study
St1, Exp.
Zhao et al.2007
(b)
Figure 7. Variation of RMS lift coefficients for both cylinder with α and
comparison of Strouhal number with experimental and numerical data
Conclusions
Three-dimensional numerical simulations are carried out to
investigate the effect of angular position of small cylinder
relative to main cylinder in the two cylinder system at constant
gap of 0.0625D. Only one wake behind two cylinders was
formed in the wake of the cylinders. The mean drag coefficients
on the large and small cylinder follow the similar trend as in
Zhao et al. (2007) and Tsutsui et al. (1997). The RMS lift
coefficients of both cylinders are found to be reduced
significantly compared with that of a single cylinder.
Acknowledgments
The author would like to acknowledge the support from
Australian Research Council through ARC Discovery Project
Program Grant No. DP110105171. The calculations were carried
out on Australia’s Supercomputer facility (iVEC in Western
Australia).
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