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).

References

[1] Bearman, P.W. & Wadcock, AJ., The interaction between a

pair of circular cylinder normal to a stream, Journal of Fluid

Mechanics, 61, 1973, 499–511.

[2] Jeong, J. & Hussain, F., On the identification of a vortex,

Journal of Fluid Mechanics, 285, 1995, 69–74.

[3] Kim H.J. & Durbin P.A., Investigation of the flow between a

pair of cylinders in the flopping regime, Journal of Fluid

Mechanics, 196, 1988, 431–48.

[4] Lee, S.J., Lee, S.I. & Park, C.W., Reducing the drag on a

circular cylinder by upstream installation of a small control

rod, Fluid Dyn. Res., 34, 2004, 233–250.

[5] Strykowski, P.J. & Sreenivasan, K.R., On the formation and

suppression of vortex shedding at low Reynolds numbers,

Journal of Fluid Mechanics, 218, 1990, 71–107.

[6] Tsutsui, T., Igarashi, T. & Kamemoto, K., Interactive flow

around two circular cylinders of different diameters at close

proximity: experiment and numerical analysis by vortex

method, J. Wind Eng. Ind. Aerodynamic 69, 1997, 279–291.

[7] Williamson CHK., Evolution of a single wake behind a pair

of bluff bodies, Journal of Fluid Mechanics, 159, 1985, 1–

18.

[8] Zdravkovich MM., The effects of interference between

circular cylinders in cross flow, Journal of Fluid and

Structures, 1, 1987, 239–61.

[9] Zhao M., Cheng L., Teng B. and Liang D., Numerical

simulation of viscous flow past two circular cylinders of

different diameters, Appl. Ocean Res, 27, 2005, 39–55.

[10] Zhao M., Cheng L., Teng B. and Dong G., Hydrodynamic

forces on dual cylinders of different diameters in steady

currents, Journal of Fluid and Structures, 23, 2007, 59-83.

[11] Zhao M., Cheng L. and Zhou T., Direct numerical

simulation of three-dimensional flow past a yawed circular

cylinder of infinite length, Journal of Fluid and Structures,

25, 2009, 831–847.

[12] Zhao M., Thapa J., Cheng L. and Zhou T., Three-

dimensional transition of vortex shedding floe around a

circular cylinder at right and oblique attacks, Physics of

Fluids, 25, 2013, 014105