12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

A FLUID-STRUCTURE INTERACTION STUDY ON
WAKE-INDUCED PENDULUM MOTION OF TANDEM CYLINDERS
Hyunjun Kim
Department of Naval Architecture and Ocean Eng.
Pusan National University
2, Busandaehakro 63beon-gil, Geumjeong-gu,
Busan 46241, Republic of Korea
junkim@pusan.ac.kr

Youngjin Jang
Hyundai Maritime Research Institute
Hyundai Heavy Industry
1000,Bangeojinsunhwan-doro,Dong-gu,
Ulsan 44032, Republic of Korea
andrewjang3774@hanmail.net

Inwon Lee
Department of Naval Architecture and Ocean Eng. Pusan National University
2, Busandaehakro 63beon-gil, Geumjeong-gu, Busan 46241, Republic of Korea
inwon@pusan.ac.kr

ABSTRACTThis paper presents an experimental investigation of the
response characteristics of the pendulum motion of tandem
cylinders undergoing wake-induced vibrations (WIV), which
was first devised by Lee et al. (2021). In this pendulum systemof tandem cylinders (PSTC), the downstream cylindersuspended from a pivot on the spanwise axis of the upstream
cylinder oscillates in the circumferential direction in response to
the incoming wake flow from the upstream cylinder. The
governing parameters of the PSTC are the torsional spring
stiffness, the cylinder spacing, and the moment of inertia of thesystem. The response characteristics for the WIV of the PSTC
were measured experimentally by varying these main
parameters. In addition, the relationship between two parameters,
moment of inertia and torsional spring stiffness, was observedby changing individual values.

In a recent study by the authors (Lee et al. 2021), a novel
swinging WIV was devised and analyzed numerically throughlarge eddy simulation (LES). While the existing WIV system
consists of a tandem arrangement of cylinders in translationalmotion, the present swinging WIV system, called a “pendulum
system of tandem cylinders (PSTC),” exhibits an oscillatory,
revolving motion of the downstream cylinder pivoted to the
upstream cylinder. In a numerical simulation by Lee et al. (2021),
the present PSTC was observed to have higher energy converting
capability compared to that of the conventional “translational
system of tandem cylinders (TSTC).” The higher efficiency of
the PSTC was attributed to an increase in the oscillation
frequency due to the additional restoring force contribution fromdrag force.
In this study, the response characteristics of the PSTC are
investigated experimentally in detail by varying parameters suchas torsional spring stiffness, the weight of the downstream
cylinder, and the distance between the cylinders. To assess themechanical energy conversion performance of the present PSTC,
the mechanical power was calculated based on the measured
torque and the angular velocity. This power can be utilized as acomprehensive measure to evaluate the spectral characteristics
of torque and angular displacement of the present pendulum
oscillation system.

INTRODUCTION
When a cylinder is immersed in an incoming flow, vortices
shed periodically from the cylinder exert an alternating lift force
on the cylinder. If the cylinder is mounted on an elastic support,
it undergoes periodic vibration, which is called “vortex-induced
vibration (VIV).” If a cylinder is placed in front of a moving
cylinder undergoing VIV at a distance close enough for the
wakes from both cylinders to interact with each other, the motionof the downstream cylinder is affected by the wake generated by
the upstream cylinder. This motion is called “wake-induced
vibration (WIV).” This phenomenon can usually be realized byplacing a stationary cylinder in front of the existing VIV system
in a tandem arrangement. Many experimental studies have been
attempted to observe and analyze this WIV phenomenon. Assiet al. (2010, 2013) conducted experiments with a staggered
arrangement of two cylinders and found that WIV is influenced
by the mean lift force by the wake within a specific linear range
along the cross-flow direction, showing no resonancecharacteristics of VIV. This linearity of mean lift force along thecross-flow direction is called “wake stiffness,” and was provenin the WIV experimental setup without a spring. For tandem
cylinders with different diameters, Assi (2014) showed that
displacement in the cross-flow direction of the downstream
cylinder became smaller as the downstream cylinder becomeslarger.

EXPERIMENTAL SETUP
The experimental apparatus used in this study is shown in
Fig. 1. It was mounted on the measurement rail of the carriage
of a towing tank in Pusan National University. The towing tankdimensions are 100 m (length) × 8 m (width) × 3.5 m (depth).
Uniform inflow to the PSTC was generated by moving thecarriage. In this study, the carriage speed was limited to 4 m/s
due to the excessive VIV of the fixed upstream cylinder beyond
this speed.
The rotating pipe pivots on the fixed inner cylinder, which
extends downward to form the upstream cylinder of the PSTC.
The downstream cylinder is supported at both ends byconnecting plates that connect the downstream cylinder to therotating pipe. Thus, the pipe rotates in accordance with theswinging motion of the downstream cylinder, whose angulardisplacement is measured by a potentiometer. In order to amplify
the potentiometer response, multiplying gear with a gear ratio of
5 is installed between the rotating pipe and the potentiometer.

112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
The external hydrodynamic torque acting on the rotating pipe ismeasured by the dynamometer. The connecting plates have fourholes, by which the cylinder spacing can be varied as ⁄ =2, 3, 4, 5.

Table 2. Variation of the cylinder spacing
No. Case name [N⋅m/rad]
K29-S28 M290-I045F40
K29-S39 M130-I045F40
0.293
K29-S410 M073-I045F40
K29-S511 M047-I045F40

CHARACTERISTICS OF WIV RESPONSE
The response characteristics of the present pendulum system(PSTC) depend on three main parameters: the cylinder spacing, the moment of inertia and the natural frequency of therotating system . In this study, a total of 14 cases were selected
to investigate the effect of the three parameters on the pendulum
motion response, as shown in Table 1. Each case name consistsof five strings (case identifiers), which signify the approximate
values of spring stiffness (KXX), nondimensional cylinder
spacing ⁄ (SX), mass of the downstream cylinder (MXX),
moment of inertia (IXX) and natural frequency (FXX).

No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14

⁄

[kg] [kg⋅m2]

[Hz]

2

2.90

0.02603

1.30

0.0321
0.045

0.404

4

0.73

0.03305

0.47

0.0385

Table 3. Variation of spring stiffness

Table 1. List of test cases
Case name
K00-S4-M047-I035-F00
K00-S4-M230-I108-F00
K10-S4-M047-I035-F27
K10-S4-M230-I108-F15
K21-S4-M047-I035-F39
K26-S4-M047-I035-F43
K26-S4-M230-I108-F24
K29-S2-M290-I045-F40
K29-S3-M130-I045-F40
K29-S4-M073-I045-F40
K29-S5-M047-I045-F40
K40-S4-M047-I035-F54
K40-S4-M117-I063-F40
K40-S4-M230-I108-F31

No. Case name

[N⋅m/
rad]

K00-S42 M230-I108- 0.000
F00
K10-S44 M230-I108- 0.0976
F15
K26-S47 M230-I108- 0.2553
F24
K40-S414 M230-I108- 0.4022
F31⁄[kg] [kg⋅m2][Hz]0.000 0.0070

0.151 0.0077
4

2.30

0.108
0.245 0.0062

0.307 0.0077

Figure 2 presents 2D contour plots of four major WIV
response variables, i.e., the root mean square of angulardisplacement , dominant frequency , torque coefficientand power as a function of Reynolds number
and thenatural frequency . Here, the dominant frequency is defined as
the peak frequency where the power spectrum of angulardisplacement becomes maximum. The torque coefficient andthe power can be calculated as follows:

The 14 cases can be subdivided into four groups. In the firstgroup listed in Table 2, the nondimensional cylinder spacing⁄ is varied. Here, the mass of the downstream cylinder foreach case was chosen such that the moment of inertia remains
unchanged in spite of the varying cylinder spacings. In addition,
the spring stiffness was kept constant, thereby making thenatural frequency constant. In this manner, the effect of ⁄ on
the WIV response was able to be isolated. Test cases listed inTable 3 correspond to the variation of the natural frequency
by means of changing spring stiffness .
Assi et al. (2013) showed that the wake behind the upstream
cylinder can operate as a stiffness element and called this “wakestiffness” in the WIV response. Here, the word “stiffness” isused because the mean lift changes linearly with the position of
the downstream cylinder in the cross-flow direction in the wakeof the upstream cylinder. With the wake stiffness being
dominant, the amplitude of the lift force and consequently the
response of the downstream cylinder becomes linearly
proportional to the reduced velocity. Assi et al. (2013) noted that
the WIV response characteristics can be divided into three
regions, the VIV response region, the wake stiffness region, andthe pure wake region. It is noteworthy that the swinging WIV
response for the present PSTC shows similar behaviors as thoseobserved for the translational WIV of Assi et al. (2013).

=
=

(1)

.

!
$ %&
! "!# !#

' %()

(2)

Here, &̂ represents the root mean square of torque fluctuations.
In Fig. 2 (a), it is observed that angular displacement
becomes larger in the lower right corner of the domain, i.e., ahigh Reynolds number and low natural frequency. On the other
hand, the dominant frequency is maximized for highandhigh , as in Fig. 2 (b). Although both and increase withincreasing
in general, the tendency of increase appears
differently for each natural frequency case. For instance, forthe low natural frequency region + ≤ 0.1/01, the increase inwith increasing
occurs rather abruptly for≥ 1.2 × 104 ,while the increase in appears more gradually for higher natural
frequencies, as shown in Fig. 2 (a). In Fig. 2 (b), a low frequency
domain with ≤ 0.45/0 for low natural frequencies is marked.

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022with the dark grey cells representing dominant frequency
component. It is worthwhile to mention that the frequency axis
for the 2D contours is defined differently depending on theexistence of a torsional spring. In every plot, a constant Strouhal⁄< = 0.2 is drawn to visualize the lock-in
number line ) =region in the 2D velocity-frequency space.References to original
sources for cited material should be listed together at the end of
the paper; footnotes should not be used for this purpose.
Figure 5 demonstrates the frequency signature plots for the
first group of test cases involved in the variation of cylinder
spacing . For the smallest spacing ⁄ = 2 in Fig. 5 (a), themajor frequency component for is found along the horizontal
axis, i.e., ⁄ = 0. In other words, the angular displacement forthis case hardly presents periodic oscillation, which is inaccordance with the noticeably small values of the RMS angulardisplacement in Fig. 5 (a). In addition, the –contour on theleft and the &–contour on the right are poorly correlated, which
accounts for the near-zero power for this case shown in Fig. 5(d). For the case of ⁄ = 3 in Fig. 5 (b), it is found thatfrequency components are spread in a rather wide frequency
range. This is in contrast to the narrow, sharp branches observed
for the cases ⁄ = 4 in Fig. 5 (c) and ⁄ = 5 in Fig. 5 (d). In
all cases in this group, the major branches do not follow theconstant Strouhal number line ) = 0.2.

This implies that the frequency of WIV does not necessarily
follow monotonically increasing trend.
In summary, it is observed that the swinging WIV response
depends not only on the natural frequency but also on otherparameters such as the moment of inertia and Reynolds number
in rather complicated manners.
Effect of nondimensional cylinder spacing S/D withconstant natural frequency
Figure 3 plots the RMS angular displacement , dominantfrequency , torque coefficient and power as a function of
Reynolds number. It is worthwhile to mention that this power is
not the actual power harnessed from the generator, but themechanically converted energy from the hydrokinetic energy.
For ⁄ = 2, the cylinder spacing is too close for the gap flow
to develop. Thus, case S2 shows a much smaller motion responsethan other cases do. The RMS angular displacement becomelargest for cases S3 and S4. While S3 and S4 have almost thesame magnitude of RMS angular displacement, S4 is found tohave a slightly higher frequency than S3. As a result, case S4
shows the largest power, as in Fig. 3 (d). For the sake of brevity,
the nondimensional spacing is fixed as ⁄ = 4 in thefollowing parametric investigations.
Effect of spring stiffness with high moment of inertia
In this group of cases, the moment of inertia was increased
about three times ( = 0.108 kg ∙ m; ) and other parameters were
kept the same as the previous group. The results are plotted inFig. 4. As seen, a noticeable change was found in with respect
to . As seen in Fig. 4 (a), the motion response for the casewithout a spring (K00 case) becomes largest for≥ 1.5 × 104,
which is in contrast to the smallest response for≤ 1.1 × 104.
On the other hand, the case with highest spring stiffness (K40
case) shows a significantly smaller response for≥ 1.2 ×104 . In Fig. 14 (b), the dominant frequency
follows the⁄< = 0.2 in theconstant Strouhal number line ) =Reynolds number up to= 0.6 × 104 . Beyond this, thedominant frequency becomes larger with increasing stiffness,
which follows the same trends as the natural frequency =? / . Similar to before, the power becomes smallest for the;>highest stiffness case.

CONCLUSIONSIn this study, the swinging WIV response characteristics ofthe pendulum system of tandem cylinders (PSTC) devised by
Lee et al. (2021) were investigated with a series of towing tankmeasurements. The aim of the present study was to analyze theeffect of the system parameters on the WIV response
characteristics. It is observed that the RMS angular displacement
becomes smaller and the dominant frequency
becomeslarger with increasing spring stiffness, and this characteristic
becomes clearer as the moment of inertia becomes large. This is
attributable to the smaller damping ratio associated with thelarge moment of inertia cases. It can be concluded that the effectof the moment of inertia is larger than that of the spring stiffness.

ACKNOWLDEGEMENT
This work was supported by the National Research
Foundation of Korea (NRF) grant funded by the Ministry of
Science and ICT of Korea (No. 2022R1A2C2010821), to which
deep gratitude is expressed.

FREQUENCY SIGNATURES OF DOWNSTREAMCYLINDER
For the first group of test conditions explained above, thenormalized power spectral density (PSD) for the angulardisplacement and the torque & acting on the downstream
cylinder are plotted in Fig. 5. Here, the PSD is defined as theFourier transform of autocorrelation functions of each quantity(angular displacement and torque) measured at each velocity as
follows:AA +∞1 = $"∞AA +)1"B;>C! () .

REFERENCES
Assi, G. R. S., Bearman, P. W., Carmo, B. S., Meneghini, J.
R., Sherwin, S. J., Willden, R. H. J., 2013, "The role of wake
stiffness on the wake-induced vibration of the downstream
cylinder of a tandem pair," Journal of Fluid Mechanics, Vol. 718,
pp. 210-245.
Assi, G. R. S., 2014. "Wake-induced vibration of tandem
cylinders of different diameters," Journal of Fluids andStructures, Vol. 50, pp. 329-339.
Lee, C. M., Paik, K-J., Kim, E. S., Lee, I., 2021, "A fluid–
structure interaction simulation on the wake-induced vibrationof tandem cylinders with pivoted rotational motion, " Physics of
Fluids, Vol. 33, 045107(3)Here, the subscript D denotes the quantity underconsideration, which corresponds to either the angulardisplacement or the torque &. The normalized PSD is thencalculated by dividing AA + 1 by maximum value AA,EFA ;therefore, its value ranges from 0 to 1. Such normalized PSDs ateach velocity are gathered together to represent the variation of
dominant frequency components in 2D velocity-frequency space3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

0.55

0.30
0.35
0.40

0.50
.50
.55

0.11

0.12
0.13

0.07

0.09

3
0.0

0.02
0.01

0.0
6

0.10
0.12

0.1
4
0.
15

0.10

0.1 2
0.13

16

0.06

0.16

0.08

0.05
0.05

0.07
0.090.08

0.01

0.60

0.90

0.
60

0.50

0.40
.35

0.32

0.15

0

0.20

0.2

0.26
0.2
8
.3 0.30. 0 0
30

0.22
0.24

0.6

0

0.
60

0.40
0.45

0

0.16

0.16

0.06
06

0.04
2.1.400
12.8
0

0.90

0.11

1.20

f0 [Hz]

0

0 .2
0.02

f0 [Hz]

0.20

8
0.1

0.16

08

0.12 0.10

0.

0.1

0.02

0.0 4

f0 [Hz]

0.90

0.13

1.80
1.50

0.11

3.00
2.70
2.40
2.10

0.14
0.14
0.12

0.09

0.0

1
1.5
Re x 104

Figure 2. Contour plots of swinging WIV response with respect toand natural frequency
displacement , (b) dominant frequency , (c) torque coefficient , (d) power

4

0.1

0.07

0.02

0.5

2

0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15

P [W]

8

0.04

0.1
0

0.0

3
0.0

0.2

0.10

0.09

0.06

f0 [Hz]

2

0.6

0.4

0.3
4
0.0

2

0.4
0.03

1
1.5
Re x 104

0.04

0.5

0.
05

0.01

1.50
1.80

0

0.45

02
0.

0
3.0

0.1

5

0.60

1.20

0.2

0

2.40

4.20
3.90
3.60
3.30
3.00
2.70
2.40
2.10
1.80
1.50
1.20
0.90
0.60
0.30
0.00

5

1
1.5
Re x 104

0.5

Cτ
0
0.6

1.5

0.3

0.5

(d)
0.60

0.60

0.90
1.20
1.50
1.80
2.10

0.41.20

0

0.5

0.80

0

0.4

2

0.1

0

0.7

0.50

1
1.5
Re x 104

0 .5

0.30

(c)
0.5

0.20
0.18

0.5

0.0
0.0

0

0.2

0.35

0.1

.25

0.24

0.12
0
0.104.1

08
0.

4
0.2

0.3 0.15

5
0.7

2

0.18

0.06

0.2

22
0.

0.65

6

0.80

0.45

1
0.

0 .2

4
0.1

0.12

0.4

0.20 08fθ [Hz]
0.
90 0
.90

0.60

8
0.1

0.0

0.5

0.85

6

0.3

0.32
0.30
0.28
0.26
0.24
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02

(b)

0.60

0.0

0.4

θ [rad]

0.55
0.50

14
0.

^

0.15

0.5

0.20

(a)

0.70

Figure 1. The PSTC apparatus

0.1

5

0.16
0.15
0.14
0.13
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01

: (a) RMS angular12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

(a)

(b) 2

K29-S2-M290-I045-F40
K29-S3-M130-I045-F40
K29-S4-M073-I045-F40
K29-S5-M047-I045-F40

0.4

K29-S2-M290-I045-F40
K29-S3-M130-I045-F40
K29-S4-M073-I045-F40
K29-S5-M047-I045-F40

1.5<fθ[Hz]

θ[rad]

0.3

0.2

St=0.2

0.5

0.1

0
0

1

0.5

1

1.5

0
0

2

0.5

Rex104(c) 6(d)K29-S2-M290-I045-F40
K29-S3-M130-I045-F40
K29-S4-M073-I045-F40
K29-S5-M047-I045-F40

5

1.5

2

K29-S2-M290-I045-F40
K29-S3-M130-I045-F40
K29-S4-M073-I045-F40
K29-S5-M047-I045-F40

0.2

4

0.15

P[W]

Cτ

1

Rex104

3

0.1

2
0.05

1
0
0

0.5

1

4

1.5

0
0

2

0.5

1

1.5

2

4
Rex10
Rex10
Figure 3. WIV response for varying cylinder spacings with constant natural frequency: (a) RMS angular displacement ,(b) dominant frequency , (c) torque coefficient , (d) power

(a)

(b) 2

K00-S4-M230-I108-F00
K10-S4-M230-I108-F15
K26-S4-M230-I108-F24
K40-S4-M230-I108-F31

0.4

K00-S4-M230-I108-F00
K10-S4-M230-I108-F15
K26-S4-M230-I108-F24
K40-S4-M230-I108-F31

1.5<fθ[Hz]

θ[rad]

0.3

0.2

St=0.2

0.5

0.1

0
0

1

0.5

1

1.5

0
0

2

0.5

Rex104(c) 6(d)K00-S4-M230-I108-F00
K10-S4-M230-I108-F15
K26-S4-M230-I108-F24
K40-S4-M230-I108-F31

5

1.5

2

K00-S4-M230-I108-F00
K10-S4-M230-I108-F15
K26-S4-M230-I108-F24
K40-S4-M230-I108-F31

0.2

4

0.15

P[W]

Cτ

1

Rex104

3

0.1

2
0.05

1
0
0

0.5

1

4

1.5

0
0

2

0.5

1

1.5

2

4
Rex10
Rex10
Figure 4. WIV response for varying spring stiffness with large moment of inertia ( = 0.108 kg · m; ): (a) RMS angulardisplacement , (b) dominant frequency , (c) torque coefficient , (d) power5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
5

K29-S2-M290-I045-F40

Sθθ/Sθθ, max

0.9

3
2
1

K29-S2-M290-I045-F40

Sττ/Sττ, max

4

f/f0

f/f0

4

5
0.0

0.0

0.9

3
2
1

St=0.2

00

5

10

U*

15

20

St=0.2

00

25

5

10

U*

15

20

25

(a) K29-S2-M290-I045-F40
5K29-S3-M130-I045-F40

Sθθ/Sθθ, max

0.9

3
2
1

K29-S3-M130-I045-F40

Sττ/Sττ, max

4

f/f0

f/f0

4

5
0.0

0.0

0.9

3
2
1

St=0.2

00

5

10

U*

15

20

St=0.2

00

25

5

10

U*

15

20

25

(b) K29-S3- M130-I045-F40
5K29-S4-M073-I045-F40

Sθθ/Sθθ, max

0.9

3
2
1

K29-S4-M073-I045-F40

Sττ/Sττ, max

4

f/f0

f/f0

4

5
0.0

0.0

0.9

3
2
1

St=0.2

00

5

10

U*

15

20

St=0.2

00

25

5

10

U*

15

20

25

(c) K29-S4-M073-I045-F40
5K29-S5-M047-I045-F40

Sθθ/Sθθ, max

0.0

0.9

3
2
1
00

K29-S5-M047-I045-F40

Sττ/Sττ, max

4

f/f0

f/f0

4

5

0.0

0.9

3
2
1

St=0.2
5

10

U*

15

20

25

00

St=0.25(d) K29-S5-M047-I045-F40
Figure 5. Normalized power spectral density (PSD) of the angular displacement
spacings S

6

10

U*

15

20

and torque & for various cylinder25