Journal of Fluids and Structures 39 (2013) 1–14
Contents lists available at SciVerse ScienceDirect
Journal of Fluids and Structures
journal homepage: www.elsevier.com/locate/jfs
Vortex-induced vibration of a collinear array of bottom fixed
flexible cylinders
F. Oviedo-Tolentino a, R. Romero-Méndez a, A. Hernández-Guerrero b,n,
F.G. Pérez-Gutiérrez a
a
Facultad de Ingenierı́a, Universidad Autónoma de San Luı́s Potosı́, Av. Dr. Manuel Nava 8, Zona Universitaria Poniente, San Luı́s Potosı́,
S.L.P. 78290, México
b
Departamento de Ingenierı́a Mecánica, Universidad de Guanajuato, Carretera Salamanca-Valle de Santiago Km. 3.5 þ 1.8, Comunidad de
Palo Blanco, Salamanca, Gto. 36730, México
a r t i c l e i n f o
abstract
Article history:
Received 8 November 2010
Accepted 18 February 2013
Available online 21 March 2013
An experimental study of vortex-induced vibrations of a collinear array of 10 identical
flexible cylinders was conducted between 140 o Re o 450. The experiments were
performed in a water tunnel using 0.13 mass-damping and 1% blockage ratio. Each
cylinder response, for tested Reynolds number range, was determined tracking the
cylinder free-end by means of a particle tracking velocimetry technique. Free-end
displacements, free-end orbits, phase angle between cylinders as well as each cylinder
frequency were obtained with data post-processing. The natural frequencies of one of the
cylinders, in still water and still air, were measured as a reference throughout the lock-in
region. The results show that the maximum displacement in the cross-flow direction for
the first cylinder in the array is 28% higher compared to when it is isolated.
Synchronization is reached from cylinder 2 throughout cylinder 8 for 340 o Re o 450.
Under these conditions the cylinders have a CCW rotation and the cylinder frequency
matches very accurately the cylinder natural frequency in still water. Moreover, the
phase angle with respect to cylinder 2 shows a gradual delay as the Reynolds number is
increased. The phase angle and the cylinder response are correlated with the vortex wake
behavior and with the number of coherent vortices in the gaps between cylinders,
respectively.
& 2013 Elsevier Ltd. All rights reserved.
Keywords:
Vortex-induced vibration
Flexible cylinder array
Phase angle
1. Introduction
The vortex-induced vibrations of circular cylinders has been a subject of interest for a long time. Engineering
applications such as marine risers, tall buildings, suspension bridges, high-voltage transmission lines, heat exchangers, etc.
are some examples where vortex-induced vibrations may be present. Many of the phenomena in vortex-induced vibration
are simultaneously discussed in the reviews by Bearman (1984) and Gabbai and Benaroya (2005). There are countless
studies made on vortex-induced vibration of isolated circular cylinders. These studies focus on the following parameters:
mass damping ratio, added mass, hysteretic behavior, blockage ratio, vortex shedding frequency, cylinder frequency,
n
Corresponding author. Tel.: þ52 464 6479940x2382.
E-mail addresses: [email protected], [email protected] (A. Hernández-Guerrero).
0889-9746/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jfluidstructs.2013.02.016
2
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
Nomenclature
Ax
Ay
An
B
CW
CCW
D
fo
fc
Fa
Fw
H
Cylinder in-line amplitude
Cylinder cross-flow amplitude
Dimensionless amplitude An ¼ A=D
D
Blockage ratio B ¼ W
Clockwise rotation
Counterclockwise rotation
Cylinder diameter
Vortex shedding frequency
Cylinder frequency
Natural cylinder frequency in still air
Natural cylinder frequency in still water
Relative position along the cylinder
L
mc
md
mn
P
Re
St
U
Ur
W
f
n
za
zw
Total longitude of the cylinder
Cylinder mass
Displaced fluid mass
Mass ratio mn ¼ mc =md
Distance between cylinders
Reynolds number Re ¼ UD=n
Strouhal number St ¼ f o D=U
Free stream velocity
Reduced velocity U r ¼ U=F w D
Spanwise tunnel test section
Phase angle between cylinders
Kinematic viscosity
Damping ratio in still air
Damping ratio in still water
vortex modes shapes and cylinder free-end paths. The relationship between the aforementioned parameters and the
cylinder response (lock-in) has been considered in many investigations, some of which are described below.
Khalak and Williamson (1996, 1997a, 1997b, 1999) performed experimental studies for a very low mass-damping
parameter ðmn z ¼ 0:013Þ, high Reynolds numbers ð5000 oRe o16000Þ and cross-free vibration. Their experimental model
was built using a spring damper system attached to the base of the cylinder. They showed that the cylinder response is not
correctly characterized by the mass ratio parameter for a given value of mass damping ratio. For low mass ratio values, the
cylinder oscillates to higher amplitudes. Their cylinder response was formed by three regions: initial, upper and lower
branches. Using flow visualization, they associated the initial branch with a 2S vortex shedding mode (two single vortices
shed per cycle), whereas the lower branch was associated with a 2P vortex shedding mode (two pairs of vortices shed per
cycle).
Vikestad et al. (2000) carried out experiments on an elastically mounted rigid cylinder. They found that the added mass
decreased with the Reynolds number and for this reason the cylinder frequency can take a different value from the
cylinder natural frequency in still fluid. This finding gives good reasons to think that the cylinder response will be related
to the natural frequency in still air and in another still fluid medium, and also by the vortex shedding frequency due to the
flow around a stationary cylinder. In their experimental study, Kheirkhah et al. (2012) found that the cylinder orbiting
response is related to the natural frequency in the lock-in region. They showed that both the streamwise and in-line
vibrations lock onto the natural frequency of the structure, describing an elliptical orbit response. On the other hand,
Govardhan and Williamson (2000) discussed that the upper branch in the lock-in zone, is clearly delimited by the natural
frequency in the still fluid media and the natural frequency in vacuum. This finding dictates that as the difference between
the natural frequency in still fluid media and the natural frequency in vacuum is greater, the lock-in range increases.
Étienne and Pelletier (2012) conducted a numerical study for a cylinder with zero mass ratio value. This condition
increases the cylinder natural frequency range in the lock-in region and consequently the reduced velocity range increases.
The lock-in region has shown a hysteretic behavior in the initial and lower branches, however Prasanth et al. (2006) and
Prasanth and Mittal (2008) numerically calculated that, for blockages of 1% or less, the hysteretic behavior in the initial
branch is completely eliminated and for that case the cylinder response is formed only by two branches. Up to now, the
relationship between the cylinder natural frequency in still fluid, vacuum, lock-in zone and the vortex shedding frequency
is not completely clear. Sarpkaya (1995) defined synchronization region as the matching of the frequency of the periodic
wake vortex mode with the body oscillation frequency.
The study of vortex-induced vibration on cylinder arrays is important in industrial applications where operational cost
needs to be reduced. Paidoussis (1980) cited various industrial problems due to vortex-induced vibration in heat
exchangers and nuclear reactors. A frequent configuration consists of two cylinders in tandem disposition. The effect of the
cylinder spacing in tandem arrangement was studied by Papaioannou et al. (2008); they showed that the downstream
cylinder response is affected by the distance between cylinders. In their numerical study, they showed that the flow
behavior in their tandem array was like an isolated cylinder for cylinder separation P=D o3:5. There are various
mechanisms that cause vibration in a cylinder array, most of them are discussed by Goyder (2002). The fluid-elastic
instability is considered the most severe form of induced vibration. Moreover, in a cylinder array, greater amplitudes of
vibration can be reached; according to Tanaka and Takahara (1981), who suggest that the fluid dynamic forces on a
cylinder are induced by the neighbor cylinders and the cylinder itself.
The flow excitation mechanisms in typical tube array patterns in tube and shell heat exchangers were analyzed by
Weaver and Fitzpatrick (1988). They give design guidelines and research requirements for each excitation mechanisms.
They point out the destructive nature of the fluid-elastic instability in a heat exchanger and suggest that more
fundamental investigations are needed. A systematic investigation on six different arrays was conducted by Tsun-kuo
and Ming-huei (2005), varying the frequency of the surrounding cylinders. They found that there is almost no effect of the
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
3
difference in the natural frequency on the critical velocity, nevertheless it has a strong effect on the vibration amplitude
above the critical velocity. Recently, Zhao and Cheng (2012) conducted a numerical study on a square configuration. Their
study was focused on how the inlet flow angle influences the lock-in range of the array. They showed that as the flow inlet
angle increases up to 301, the lock-in range is also increased; after this flow inlet angle the lock-in range is reduced.
This paper aims at finding the vortex-induced vibrational response of a collinear array of bottom-fixed flexible cylinders
subjected to cross-flow. All cylinders have the same natural frequency in still air and still water. Those natural frequencies
are used as a reference to find what is the cylinder preferential frequency in the lock-in zone. According to this purpose,
the free-end path, cylinder frequencies and phase angle between cylinders were obtained by means of a particle tracking
velocimetry system of the free-end of the cylinders.
2. Description of experiment
2.1. Experimental model
A sketch of the experimental model is shown in Fig. 1. The model consists of a horizontal acrylic flat plate (9 mm thick,
95 cm long and 37.5 cm wide) in which 10 long cylindrical bars were placed normal to the plate inserting them into a
drilled hole in a collinear arrangement. The cylinders were 2.40 mm in diameter and 40 cm in height; they were attached
to the flat plate equally spaced by interference and chloroform injection. The first cylinder was located 70 cm from the
leading edge. A distance of 6D between cylinder centerlines was chosen to ensure vortex shedding in the gap between
cylinders, in agreement with the results of Papaioannou et al. (2008). The other end side of the cylinders was free to move
as the fluid flows. The material selected for the cylinder was a copper and zinc alloy with an elastic modulus of
10.5 1010 Pa. A second model with only one cylinder was built in order to compare the behavior of an isolated cylinder
Fig. 1. Experimental setup.
Fig. 2. Water tunnel.
4
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
1.5
1.5
1
1
Normalized Amplitude, A*
Normalized Amplitude, A*
and the behavior of multiple interacting cylinders. The model was placed in a water tunnel whose test section (38.1 cm
wide, 50.8 cm high and 1.5 m long) allows a free stream flow aligned with the cylinder array. According to the water depth
and the test section width, the gap between the cylinder free end and the water free surface was 5.7 cm and the blockage
ratio was B¼ 1%. In order to reduce flow perturbations along the plate, the leading edge of the plate was sharpened. The
experiments were performed for a mass damping parameter, mn zw ¼ 0:13.
The experiments were performed in the water tunnel schematically shown in Fig. 2. The test section of the tunnel is
38.1 cm wide, 50.8 cm high and 1.5 m long. With that experimental test section size a 45.7 cm deep water free stream was
used for all the experiments. The side and bottom walls of the test section are made of tempered glass for visual access to
the model. The water tunnel temperature in the tunnel was 22 1C and may be operated at velocities ranging from 0.01 to
0.3 m/s. The downstream end of the delivery plenum of the water tunnel has a section with flow-conditioning elements.
The first is a perforated stainless steel plate, which reduces the turbulence to a small scale, followed by a fiberglass screen
that further reduces the turbulence level. The last element is a honeycomb flow straightener. The contraction section of the
water tunnel has an area ratio of 6:1. This geometry provides good velocity distribution, turbulence reduction to less than
1% RMS at the inlet of the test section, and avoids local separation and vorticity development. The water tunnel is equipped
with a velocimetry system at the inlet of the test section. The cylinder free-end position was recorded with a high speed
video camera equipped with a special zoom lens which is focused manually for short focal distances. The camera has an
internal memory (2 GB) in which images can be stored digitally at a rate of 506 frames per second at full resolution,
0.5
0
-0.5
-1
-1.5
0.5
0
-0.5
-1
0
2
4
6
8
-1.5
10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time[s]
Time[s]
Fig. 3. Natural response in (a) still air and (b) still water.
4
5
6
7
8
9
0.09
0.9
Normalized In-Line Direction, A*x
Normalized Cross-Flow Direction, A*y
1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
4
5
6
7
8
9
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.1
0
140 160 180 200 220 240 260 280 300 320 340 360
0
140 160 180 200 220 240 260 280 300 320 340 360
Re
Re
Fig. 4. Effect of the surrounding cylinders: amplitudes of oscillation of an isolated cylinder and of the first cylinder in the row arrangement.
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
5
1280 1024 px. For lower resolutions the frame rate speed can be increased proportionally up to 112 000 frames
per second. This camera has an excellent light sensitivity of 2500 ISO. The shutter time can be adjusted from 2 ms to 1 s.
The camera has an Ethernet interface (Gigabit Ethernet) to download the image frame or video in a BMP, RAW or AVI file
format. With these features, the free-end cylinder displacements could be measured up to an accuracy of the order of
11 mm.
2.2. Experimental methodology
The displacement of each cylinder free-end was captured during 20 s at 240 fps and 100 ms exposure time using the
video camera described aided by external illumination. Plots of the cylinder free-end position as a function of time were
obtained after a particle tracking velocimetry post-processing technique was applied to the captured videos. In order to
find the shift angle between cylinders it was necessary to capture the signal of cylinder pairs. To find the relationship
between cylinders 1 and 3, the signals between cylinders 1 and 2 and then 2 and 3 were processed simultaneously. The
time signals were processed in a spectral analysis to determine the oscillating frequency of the cylinders. The Reynolds
number was gradually increased between 140 oRe o450. Each experiment was repeated at least twice to make sure that
Ur
3
Normalized Cross-Flow Amplitude, A*y
1.6
1.4
1.2
4
5
6
7
8
9
10
11
12
Cylinder 1
Cylinder 2
Cylinder 3
Cylinder 4
Cylinder 5
Cylinder 6
Cylinder 7
Cylinder 8
Cylinder 9
Cylinder 10
1
0.8
0.6
0.4
0.2
0
100
150
200
250
300
350
400
450
500
Re
Fig. 5. Cross-flow amplitude of oscillation of each one of the cylinders of the array.
Ur
3
Normalized In-line Amplitude,A*x
0.7
0.6
0.5
4
5
6
7
8
9
10
11
12
Cylinder 1
Cylinder 2
Cylinder 3
Cylinder 4
Cylinder 5
Cylinder 6
Cylinder 7
Cylinder 8
Cylinder 9
Cylinder 10
0.4
0.3
0.2
0.1
0
100
150
200
250
300
350
400
450
500
Re
Fig. 6. In-line amplitude of oscillation of each one of the cylinders of the array.
6
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
each cylinder response showed a consistent behavior. The comparison between cylinder response for both, the isolated
cylinder and the first cylinder in the array, was determined using the maximum amplitude of the free tip displacement, in
cross-flow and in-line direction, of each cylinder. The flow behavior of the vortex wake was determined using flow
visualization experiments which were achieved by seeding the water with 14 mm spherical hollow particles. The particles
reflect the light emitted from a 30 mW He–Ne laser. The flow pattern is recorded digitally with the same video camera
using long shutter times.
Fig. 7. Vortex wake patterns, cylinders 1–4 from left to right: (a) Re ¼160, (b) Re ¼180, (c) 190, (d) Re ¼ 200, and (e) Re ¼ 236.
0.24
0.22
0.2
St
0.18
0.16
0.14
0.12
0.1
0.08
50
F
F
Cylinder 1
Cylinder 2
Cylinder 3
Cylinder 4
Cylinder 5
Stationary
100
150
200
250
300
350
Re
Fig. 8. Strouhal number, cylinders 1–5.
400
450
500
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
7
3. Results
3.1. Amplitude response
Fig. 3 shows the natural response of the isolated cylinder, in still air and water after a perturbation. Such perturbation
was produced inducing an unitary impulse to the cylinder free end in the cross flow direction. With this experiment, the
cylinder natural frequency and the damping ratio in still fluid were measured. The first was obtained by means of a
spectral analysis of the time trace while the second was measured using the logarithmic decrement. For still water
F w ¼ 6:75 Hz and zw ¼ 1:55%, while for still air F a ¼ 7:15 Hz and za ¼ 0:29%, respectively.
Fig. 4 shows the maximum vibrational amplitude as a function of the Reynolds number and the reduced velocity;
in cross-flow (4a) and in-line direction (4b), for both an isolated cylinder and the first cylinder in the proposed array.
By comparing these two graphs, it should be noted that the maximum vibrational amplitude in the cross-flow direction
was greater than that in the in-line direction by about an order of magnitude. Fig. 4a shows that although cylinder 1 in the
array starts vibrating at the lowest Reynolds number, it is the isolated cylinder that reaches the greatest oscillation
amplitude in the range 160 o Re o 188. However, as Reynolds number was increased, cylinder 1 vibrational maximum
amplitude (0.93D) was 29% greater than that of the isolated cylinder (0.72D). In contrast, Fig. 4b shows that the in-line
vibrational amplitude is greater for the isolated cylinder. Moreover from both responses, it should be pointed out that the
lock-in region for cylinder 1 in the array is slightly wider than that of the isolated cylinder and the isolated cylinder
registers its maximum amplitude at Re ¼197 ðU r ¼ 5Þ. The added mass effect modifies the cylinder natural frequency in the
lock-in region and consequently the principal branches in the cylinder lock-in may represent the cylinder frequency
changes. Specifically, Govardhan and Williamson (2000) discussed that the upper branch is clearly delimited by the
natural frequency in still working fluid and the natural frequency in vacuum. This phenomena is shown in our results by
the soft transition from the maximum cross flow amplitude to the lower branch, extending the range of the upper branch.
These observations suggest that for experiments with no effects of the added mass the upper branch tends to disappear.
The neighbor cylinders amplify the maximum vibrational response in cross-flow direction for the cylinder in the array.
This result agrees with the hypothesis made by Tanaka and Takahara (1981), who suggest that the fluid dynamic forces on
a cylinder are induced by both the neighbor cylinders and the cylinder itself. In contrast, the in-line response is reduced for
the cylinder 1 in the array when compared with the isolated cylinder.
Fig. 5 shows the cross-flow amplitude as a function of the Reynolds number of the 10 cylinders in the interval
140 oRe o450. Cylinders 2–7 reach greater amplitude than external cylinders (cylinder 1, 8, 9 and 10) and cylinder 2
shows vibrational onset at a lower Reynolds number. Interestingly, at Re 200 cylinder 2 reduces its amplitude response
to a local minimum and cylinder 1 reaches its maximum amplitude. Cylinders 4–10 show the same lock-in onset at
Re 200. Fig. 6 shows the in-line response as a function of the Reynolds number; the cylinders show in-line direction
lock-in onset at the same Reynolds numbers than cross-flow direction. Cylinders 4–6, register largest in-line amplitude.
The in-line response is associated with the vibration modes of the free-end path.
From the cross-flow response (Fig. 5) it should be noted that, in the Reynolds number interval tested, cylinders 1–7
show a higher dependence on the Reynolds number compared to cylinders 8, 9 and 10. This behavior is due to the presence
of a vortex accumulation within the gaps between cylinders. Fig. 7 shows flow visualization in the gaps between cylinders
1–4 for different Reynolds numbers; flow direction is from left to right. At Re ¼160 (Fig. 7a) there is only one coherent
vortex in the gap between cylinders 1 and 2. In contrast, in the gap between cylinders 2 and 3, there are two coherent
0.24
0.22
0.2
St
0.18
0.16
0.14
F
F
Cylinder 2
Cylinder 3
Cylinder 4
Cylinder 5
Cylinder 6
Cylinder 7
Cylinder 8
Stationary
0.12
0.1
0.08
50
100
150
200
250
300
350
Re
Fig. 9. Strouhal number, cylinders 2–8.
400
450
500
8
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
vortices. This flow behavior causes cylinder 2 to reach higher vibration amplitude than cylinder 1. As the Reynolds number
is increased, the vortex formation in the gap between cylinders 1 and 2 increases (Fig. 7b), and consequently cylinder 1
increases its cross flow vibration amplitude. The increased number of coherent vortices in the gap between cylinders 1 and
2 (Fig. 7c and d) seems to amplify the vibrational response of cylinder 1; however, cylinder 2 reduces its vibrational
amplitude to a local minimum, coinciding with a reduction in the number of coherent vortices in the gap between
cylinders 2 and 3 at Re ¼200. At Re¼ 236, the gap between cylinders 2 and 3 increases again the number of coherent
vortices, with the corresponding growth of vibrational amplitude of cylinder 2. It worth to note that at this Reynolds
number, there is only one coherent vortex in the gap between cylinders 1 and 2, when a reduction of the vibrational
amplitude of cylinder 1 occurs. From these observations it can be seen that the first local maximum and local minimum in
the vibrational amplitude of cylinders 1 and 2 is related to the number of coherent vortices in the gaps downstream of each
cylinder. We hypothesized that the number of coherent vortices in the gaps modifies the vortex shedding frequency,
leading the cylinders to increase their vibration amplitude.
1.5
1.5
Cylinder 1
1
0.5
0
-0.5
0.5
0
-0.5
-1
-1
Re = 206
-1.5
-1.5
-0.5
0
0.5
In-Line Direction, A*x
1.5
-0.5
-1
-1.5
Re = 295
CCW and CW rotation
1.5
Cylinder 4
1
0.5
0
-0.5
-1.5
-0.5
0
0.5
In-Line Direction, A*x
Re = 276
CCW and CW rotation
-0.5
0
0.5
In-Line Direction, A*x
Cylinder 3
-1
Re = 377
CCW rotation
-0.5
-1.5
Cross-Flow Direction, A*y
0
0
-1
1
Cross-Flow Direction, A*y
Cross-Flow Direction, A*y
1
0.5
0.5
-0.5
0
0.5
In-Line Direction, A*x
Cylinder 4
Cylinder 3
1
Cross-Flow Direction, A*y
Cross-Flow Direction, A*y
Cross-Flow Direction, A*y
1
1.5
1.5
Cylinder 2
0.5
0
-0.5
-1
Re = 385
CCW rotation
-0.5
0
0.5
In-Line Direction, A*x
-1.5
Re = 442
CCW rotation
-0.5
0
0.5
In-Line Direction, A*x
Fig. 10. Free-end paths of some cylinders at different Reynolds numbers.
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
9
3.2. Frequency and free-end paths
Figs. 8 and 9 show Strouhal numbers computed using the measured tube frequency ðf c D=UÞ as a function of Reynolds
number for cylinders 1–5 and 2–8, respectively. Inserted as a reference are, the non-dimensional natural frequency in still
air ðF a D=UÞ, and in still water ðF w D=UÞ for a single isolated cylinder. Also plotted in these graphs is the Strouhal number for
a single stationary cylinder calculated from the empirical equation St ¼ 0:2125:35=Re. Cylinder 1 shows a tendency to
vibrate at the natural frequency in still air in the range 220o Re o 340, and its vibration amplitude highly reduced at
Re 4 340, in agreement with Fig. 5. The motion of cylinder 1 influences the frequency of cylinders 2–5 stimulating them to
vibrate at a similar frequency for Re o 240. Fig. 9 shows that synchronization of all cylinders is not attained within the
entire range of Reynolds numbers tested. However, cylinders 2–8 synchronize (vibrate at the same non-dimensional
frequency) in the range 340 oRe o450, and the oscillation frequency is very close to the natural frequency in still water.
It is worth mentioning that the synchronization from cylinder 2 to 8 began when cylinder 1 was not longer in the lock-in
region at Re ¼340.
Fig. 10 shows the characteristic free-end paths described by some cylinders of the array at specific values of Reynolds
number. Due to the small in-line oscillation amplitude of cylinder 1, a line path in the cross-flow direction is observed.
However the remaining cylinders show stable elliptical or variable paths. The stable elliptical paths have counterclockwise
(CCW) preferential rotation and almost a constant maximum amplitude. The elliptical behavior dictates that both signals,
cross-flow and in-line amplitudes, have the same frequency but different shift angle between signals for the same cylinder.
A vertical elliptical path has a phase angle of 901 between cross-flow and in-line response. This phase angle and maximum
amplitude are in general constant for the synchronization in the range 340 oRe o450, for cylinders 2–8. Variable paths
show both CCW rotation and CW rotation with variable maximum amplitude and variable phase angle between cross-flow
and in-line signals.
3.3. Phase angle between cylinders
Fig. 11 shows the phase angle, f, between cylinders 1 and 2 and the dimensionless frequencies at which these cylinders
are in synchronization. The vibration frequency of the synchronized pair 1 and 2 changes from the natural frequency in
still water to the natural frequency in still air for Re 200. The vibration mode of cylinder 2 changes from CW rotation to
CCW rotation, and the phase angle between cylinders also changes significantly due to the effect of the change in
added mass.
The phase angle between cylinder pairs 2–8 is shown in Fig. 12. The phase angle was determined between cylinder
pairs moving exactly at the same frequency. There are Reynolds number ranges at which no dimensionless frequencies are
presented because the cylinder pairs were moving at different frequencies. The change in added mass at Re 200 also
affects the phase angle of cylinder pairs 2–3, 3–4 and 4–5. The synchronization in the Reynolds number range
340 oRe o450 occurs at the natural frequency in still water. In this synchronization range, the added mass shows no
dependence on the Reynolds number. On the other hand, the phase angle between cylinder pairs 2–5 shows a gradual
delay as the Reynolds number increases and the phase angle shows a Reynolds number dependence in all the
synchronization range.
The phase angle change is correlated with the vortex wake path, at Re ¼180 cylinders 2 and 3 have 01 phase angle and
both are located in the same side of the vortex wake (Fig. 7b). In contrast, for Re ¼200, cylinders 1 and 2 have 1801 phase
0.22
Cylinders 1 and 2
0.2
1.5
0.16
1
Cross-Flow Direction, A*y
0.14
0.12
0.1
360
φ
270
180
1.5
Cylinder 2
Re = 190
CW rotation
0.5
0
-0.5
Cylinder 2
Re = 236
CCW rotation
1
Cross-Flow Direction, A*y
St
0.18
Fw
Fa
fc
-1
0.5
0
-0.5
-1
90
0
150
-1.5
200
250
300
Re
350
400
450
500
-1.5
-0.5
0
0.5
In-Line Direction, A*x
Fig. 11. Phase angle and cylinder response between cylinders 1 and 2.
-0.5
0
0.5
In-Line Direction, A*x
10
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
0.22
Cylinders 2 and 3
0.2
0.18
St
0.16
0.14
0.12
0.12
0.1
0.1
360
360
270
270
180
180
90
90
200
250
300
350
400
450
0
150
500
200
250
Re
0.22
Cylinders 4 and 5
0.22
Fw
Fa
fc
400
450
500
450
500
450
500
St
Fw
Fa
fc
0.16
0.14
0.14
0.12
0.12
0.1
0.1
360
360
270
270
180
180
90
200
0.22
250
300 350
Re
400
Cylinders 6 and 7
Fw
Fa
fc
0.2
0.18
450
0
150
500
200
0.22
250
300 350
Re
400
Cylinders 7 and 8
Fw
Fa
fc
0.2
0.18
St
0.16
0.16
0.14
0.14
0.12
0.12
0.1
0.1
360
360
270
270
180
180
φ
St
Cylinders 5 and 6
0.18
0.16
90
φ
350
0.2
φ
φ
St
0.18
90
0
150
300
Re
0.2
0
150
Fw
Fa
fc
0.16
0.14
0
150
Cylinders 3 and 4
0.2
φ
φ
St
0.18
0.22
Fw
Fa
fc
90
200
250
300 350
Re
400
450
500
0
150
200
250
Fig. 12. Phase angle between subsequent cylinders.
300 350
Re
400
A*y
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
2
1
0
-1
-2
2
3
4
5
6
7
8
t=0s
0
2
4
6
8
10
12
14
A*y
A*x
2
1
0
-1
-2
t = 5/240 s
0
2
4
6
8
10
12
14
A*y
A*x
2
1
0
-1
-2
t = 10/240 s
0
2
4
6
8
10
12
14
A*x
2
t = 15/240 s
A*y
1
0
-1
-2
0
2
4
6
8
10
12
14
A*x
2
t = 20/240 s
A*y
1
0
-1
-2
0
2
4
6
8
10
12
14
A*x
2
t = 25/240 s
A*y
1
0
-1
-2
0
2
4
6
8
10
12
14
A*x
2
t = 30/240 s
A*y
1
0
-1
-2
0
2
4
6
8
10
12
14
A*x
2
t = 35/240 s
A*y
1
0
-1
-2
0
2
4
6
8
10
12
14
A*x
Fig. 13. Cylinder position at different times, from cylinder 2 to 8 at Re¼ 441.
11
12
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
angle and both are located in opposite sides of the vortex wake (Fig. 7c). In addition, for Re ¼236, both cylinders have 2701
phase angle, but the vortex wake is bifurcated past cylinder 2 (Fig. 7e).
Fig. 13 shows the position of cylinders 2–8 (numbered from left to right) at different times for Re ¼441, a
Reynolds number in which these cylinders were in synchronization with a period of 35/240 s. Each frame shown in
1.5
Cylinder 2
0.5
0
-0.5
-1
0.5
0
-0.5
-1
CCW rotation
-0.5
0
CCW rotation
-1.5
-0.5
In-Line Direction, A*x
0
0
-0.5
1.5
Cylinder 6
1
CCW rotation
-1.5
-0.5
0
Cross-Flow Direction, A*y
0.5
0
-0.5
-1
CCW rotation
0
-0.5
1.5
Cylinder 7
0
-0.5
CCW rotation
-0.5
0
Cylinder 8
0.5
0
-0.5
CCW rotation
-1.5
0.5
-0.5
In-Line Direction, A*x
0
360
φ
270
180
90
Cylinder 3
Cylinder 4
Cylinder 5
Cylinder 6
Cylinder 7
Cylinder 8
410
415
420
425
430
435
0.5
In-Line Direction, A*x
Fig. 14. Free-end paths for cylinder 2–8 at Re ¼ 441.
0
405
440
445
Re
Fig. 15. Phase angle with respect to cylinder 2.
450
0
0.5
In-Line Direction, A*x
-1
-1.5
In-Line Direction, A*x
-0.5
1
0.5
0.5
CCW rotation
-1.5
In-Line Direction, A*x
-1
-0.5
0
0.5
1
-1.5
0.5
-1
0.5
In-Line Direction, A*x
1.5
0.5
-1
0.5
Cylinder 5
1
Cross-Flow Direction, A*y
-1.5
1.5
Cylinder 4
1
Cross-Flow Direction, A*y
Cross-Flow Direction, A*y
1
Cross-Flow Direction, A*y
Cross-Flow Direction, A*y
1
1.5
Cylinder 3
Cross-Flow Direction, A*y
1.5
455
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
13
the figure corresponds to a time delay of 5/240 s with respect to the previous frame. As a complement, Fig. 14 shows
the simultaneous free-end path of the cylinders 2–8 at Re ¼441. All these cylinders have a preferential frequency,
very close to the natural frequency in still water and a CCW rotation. Fig. 15 shows the phase angle as a function of
the Reynolds number; for cylinders 3–8 with respect to cylinder 2 within the range 405 oRe o450. It can be seen
from this figure that the phase angle of cylinders 3–8 is not constant even if they have the same oscillation
frequency. Moreover it can be noted that the phase angle has a gradual delay, which means that the initial phase
angle decreases as the Reynolds number increases. This delay is greater for downstream cylinders. From these
observations it can be concluded that the synchronization between cylinders for a specific Reynolds number occurs
at only one phase angle between them and this angle is not necessarily the same phase angle for another Reynolds
number.
4. Conclusions
Results have been presented for the fluid-structure interaction between a flexible collinear cylinder array with
low mass damping cylinder factor ðmn zw ¼ 0:13Þ and blockage ratio ðW=D o 1%Þ. The cylinders are fixed at one end and
are free to move at the other end. The flow studied is in the range 140o Re o 450. It was found that the maximum
amplitude of oscillation of the first cylinder in the proposed array was 28% greater than the maximum amplitude of
the isolated cylinder. This is due to the effect of the neighboring cylinders. As suggested by Tanaka and Takahara
(1981), the cylinder response is amplified by the vibrational response of the neighbor cylinders and this effect is
greater when neighbor cylinders and the cylinder itself vibrate at the same frequency. The first cylinder in the
arrangement has a tendency to vibrate at the natural frequency in still air in the range 220 o Re o 340, making the
following four cylinders, to vibrate at this frequency when they synchronize with cylinder 1 in this Reynolds number
range. The sudden changes in the vibrational response occur when the cylinder changes its frequency response,
rotational sense or phase angle with respect to subsequent cylinders. There is synchronization from cylinder 2 to
cylinder 8 in the range 340 oReo 450. While in synchronization, all cylinders involved have a CCW rotation and
vibrate at the natural frequency in still water. This synchronization began when cylinder 1 was not longer in the lockin region at Re ¼340. The cylinder frequency changes are associated with changes in the added mass and according
with this the added mass has no dependence of the Reynolds number in the synchronization region. The phase angle
between cylinders shows a gradual delay with respect to cylinder 2 as the Reynolds number increases and is greater
for downstream cylinders. The local maximum vibrational amplitudes are related with the number of coherent
vortices in the downstream cylinder gap.
Acknowledgements
The authors thank CONACyT (México) for support from projectCB-2007/84618, and PROMEP (México) for a grant from
the extraordinary funds of PROMEP-UASLP-12-CA04 and PROMEP/103.5/12/7964.
References
Bearman, P.W., 1984. Vortex shedding from oscillating bluff bodies. Annual Reviews Fluid Mechanics 16, 195–222.
Étienne, S., Pelletier, D., 2012. The low Reynolds number limit of vortex-induced vibrations. Journal of Fluids and Structures 31, 18–29.
Gabbai, R.D., Benaroya, H., 2005. An overview of modeling and experiments of vortex induced vibration of circular cylinders. Journal of Sound and
Vibration 282, 575–616.
Govardhan, R., Williamson, C.H.K., 2000. Modes of vortex formation and frequency response of a freely vibrating cylinder. Journal of Fluid Mechanics 420,
85–130.
Goyder, H.G.D., 2002. Flow-induced vibration in heat exchangers. IChemE—Institution of Chemical Engineers Trans 80 (Part A), 226–232.
Khalak, A., Williamson, C.H.K., 1996. Dynamics of a hydroelastic cylinder with very low mass and damping. Journal of Fluids and Structures 10, 455–472.
Khalak, A., Williamson, C.H.K., 1997a. Fluid forces and dynamics of a hydroelastic structure with very low mass and damping. Journal of Fluids and
Structures 11, 973–982.
Khalak, A., Williamson, C.H.K., 1997b. Investigation of relative effects of mass and damping in vortex-induced vibration of a circular cylinder. Journal of
Wind Engineering and Industrial Aerodynamics 69–71, 341–350.
Khalak, A., Williamson, C.H.K., 1999. Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. Journal of Fluids and
Structures 13, 813–851.
Kheirkhah, S., Yarusevych, S., Narasimhan, S., 2012. Orbiting response in vortex-induced vibrations of a two-degree-of-freedom pivoted circular cylinder.
Journal of Fluids and Structures 28, 343–358.
Paidoussis, M.P., 1980. Flow induced vibration in nuclear reactors and heat exchangers: practical experiences and state of the art knowledge. In: IUTAM/
IAHR Symposium on Practical Experiences with Flow Induced Vibrations, Karlsruhe, West Germany, pp. 1–80.
Papaioannou, G.V., Yue, D.K.P., Triantafyllou, M.S., Karniadakis, G.E., 2008. On the effect of spacing on the vortex-induced vibrations of two tandem
cylinders. Journal of Fluids and Structures 24, 833–854.
Prasanth, T.K., Behara, S., Singh, S.P., Kumar, R., Mittal, S., 2006. Effect of blockage on vortex-induced vibrations at low Reynolds numbers. Journal of Fluids
and Structures 22, 865–876.
Prasanth, T.K., Mittal, S., 2008. Vortex-induced vibrations of a circular cylinder at low Reynolds numbers. Journal of Fluid Mechanics 594, 463–491.
Sarpkaya, T., 1995. Hydrodynamic damping, flow-induced oscillations, and biharmonic response. ASME—Journal of Offshore Mechanics and Arctic
Engineering 117, 232–238.
Tanaka, H., Takahara, S., 1981. Fluid elastic vibration of tube array in cross flow. Journal of Sound and Vibration 77 (1), 19–37.
14
F. Oviedo-Tolentino et al. / Journal of Fluids and Structures 39 (2013) 1–14
Tsun-kuo, L., Ming-huei, Y., 2005. An experimental study on the cross-flow vibration of a flexible cylinder in cylinder arrays. Experimental Thermal and
Fluid Science 29, 523–536.
Vikestad, K., Vandiver, J.K., Larsen, C.M., 2000. Added mass and oscillation frequency for a circular cylinder subjected to vortex-induced vibrations and
external disturbance. Journal of Fluids and Structures 14, 1071–1088.
Weaver, D.S., Fitzpatrick, J.A., 1988. A review of cross-flow induced vibrations in heat exchanger tube arrays. Journal of Fluids and Structures 2, 73–93.
Zhao, M., Cheng, L., 2012. Numerical simulation of vortex-induced vibration of four circular cylinders in a square configuration. Journal of Fluids and
Structures 31, 125–140.