Vortex-induced vibration of a bottom fixed flexible circular beam

Ocean Engineering 88 (2014) 463–471

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Vortex-induced vibration of a bottom fixed flexible circular beam F. Oviedo-Tolentino a,n, F.G. Pérez-Gutiérrez a, R. Romero-Méndez a, A. Hernández-Guerrero b a b

Facultad de Ingeniería, Universidad Autónoma de San Luis Potosí, Av. Dr. Manuel Nava 8, Zona Universitaria Poniente, San Luis Potosí, S.L.P. 78290, Mexico Department of Mechanical Engineering, University of Guanajuato, Mexico

art ic l e i nf o

a b s t r a c t

Article history: Received 23 October 2013 Accepted 7 July 2014 Available online 25 July 2014

In this paper, we study the vortex-induced vibration of a bottom fixed circular cylinder with a 0.126 mass damping ratio and blockage ratio lower than 1%. The experiments were performed in a water tunnel at low Reynolds numbers (90 o Re o 350). The free end path of the cylinder was analyzed using a particle tracking velocimetry technique, while the shedding vortex modes were analyzed using a dye injection flow visualization technique. The results show a lock-in zone characterized by three recognizable regions which correspond to the: (1) initial, (2) upper and (3) lower branches. In the first region, the vortexshedding can occur at different cylinder frequencies for a fixed Reynolds number. It is also observed that the combined fluid-cylinder response is dominated mainly by the natural frequency of the cylinder in still water and by the vortex-shedding frequency of a stationary cylinder. In the second region, the combined responses are dominated by the natural frequencies in both still water and still air. Finally, in the last region the vortex shedding and the cylinder frequency match very accurately with the natural frequency in still air. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Vortex-induced vibration Fluid–structure interaction Lock-in region

1. Introduction Vortex-induced vibrations (VIV) are present in many applications such as risers, oil ducts in deep water, tall buildings, bridges, traffic lights, and high voltage lines that extend for long distances. This kind of fluctuations may cause fatigue in the structures leading sometimes to failure. Particularly, VIV of cylindrical structures are the most common and therefore the most interesting. VIV are due to forces generated by an alternating vortex shedding. The structure vibrations interact with the flow, changing the fluid force acting on the structure and a non-linear problem with multi-frequency structure response takes place (Blevins, 1994). The fluid–structure interaction of vortex-induced vibrations is characterized by a lock-in region. The lock-in or synchronization was properly defined by Sarpkaya (1995) as the reduced velocity range in which the frequency of the driving force locks to the frequency of the vibrating body. This phenomenon depends on the structure geometry and material and fluid properties; its main characterizing parameters are the mass damping ratio, the added mass, the aspect ratio, Reynolds number and reduced velocity. Studies of VIV of circular cylinders have focused on elastic, pivoted, and flexible cylinders, with one or two degrees of freedom. The most frequently studied model is the elastic cylinder

n

Corresponding author. Tel.: þ 52 444 8262330x2115. E-mail address: [email protected] (F. Oviedo-Tolentino).

http://dx.doi.org/10.1016/j.oceaneng.2014.07.012 0029-8018/& 2014 Elsevier Ltd. All rights reserved.

with one degree of freedom. The effect of the mass–damping ratio has been well documented for elastic cylinders undergoing one degree of freedom. Cylinders with high mass–damping ratio show a two branches response (initial and lower), often referred as the Feng-type response (Feng, 1968). Khalak and Williamson (1996, 1997a, 1997b, 1999) performed experimental studies for a very low n mass–damping parameter (m ζ w ¼ 0:013); they found that the cylinder response was formed by a three branches response: initial, upper and lower branches. The branches are separated by discontinuous mode transitions. 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). Govardhan and Williamson (2000) confirm the 2S and 2P vortex shedding modes for the initial and the lower branch respectively and furthermore report a 2S vortex shedding mode for the upper branch. Recently, numerical studies have shown dependency of the vortex shedding mode with the cylinder boundary conditions and the Reynolds number range. Navrose and Mittal (2013) conducted a numerical study of free vibrations at Re¼1000, they differ in the vortex shedding mode in the upper branch. They report a 2S mode in the upper branch and attribute that this difference is an effect of the Reynolds number. Zhao and Cheng (2014) conducted a numerical study of vortex-induced vibrations of a circular cylinder of finite length for five aspect ratios (L=D ¼ 1, 2, 5, 10 and 20). They report 2S vortex shedding mode in all the range of the reduced velocity except for

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Nomenclature A An B D fc fo fs Fa Fw H L

cylinder amplitude dimensionless amplitude An ¼ A=D blockage ratio B ¼D/W cylinder diameter cylinder frequency vortex shedding frequency vortex shedding frequency, fixed cylinder natural cylinder frequency in still air natural cylinder frequency in still water relative position along the cylinder total length of the cylinder

U r ¼ 4, besides they report a flow variation in the wave along the cylinder span. They found a phase angle (phase angle: difference between the response displacement and the lift coefficient) evolution of 1801 in vortex shedding mode between the top end and the free-end of the cylinder. Zhao and Cheng (2011) performed a numerical study of a circular cylinder close to a plane boundary from Re¼1000 to Re¼15,000. In their results, they show a clearly vortex shedding mode dependence with the reduced velocity. The cylinder response is strongly influenced by the mass– damping parameter. The results obtained by Khalak and Williamson (1999) showed that the mass–damping ratio controls the peak amplitude whereas the lock-in range is mainly controlled by the mass ratio. Govardhan and Williamson (2000) derived an expression to determine the cylinder frequency as a function of the mass ratio and showed the existence of a critical mass ratio, below which the lock-in range extends to infinity, in accordance to Sarpkaya (1978) who relates the changes of cylinder frequency within the lock-in region with the changes in the added mass. Studies on the added mass in the lock-in have been conducted by Vikestad et al. (2000) and Willden and Graham (2001); both investigations show that the added mass decreases monotonically as the reduced velocity is increased. Willden and Graham (2001) results show that once the cylinder vibrates at a frequency higher than its natural frequency, the added mass becomes negative and tends to reach an almost constant value. These observations support the idea of Govardhan and Williamson (2000) to define the cylinder frequency in the lower branch, where the added mass is not assumed to change and therefore the cylinder frequency shows a constant value. As it has been discussed, the changes in cylinder frequency within the lock-in region are related to changes in added mass. The effect of the added mass is more evident at low mass– damping ratios, where cylinders display three-branch response. At this condition there should be a difference between the cylinder frequency in still fluid medium and the cylinder frequency in vacuum. These observations provide good reasons to relate the cylinder frequency in the lock-in region with the cylinder frequency in still fluid medium, still air (for simplicity) and with the vortex shedding frequency of a stationary cylinder. In this investigation we study the vibrational response of a bottom fixed, slender, flexible cylinder with a low blockage ratio. This geometrical disposition is inspired on numerous engineering applications. The cylinder response is measured via a non-intrusive visualization technique. The vortex shedding frequency is measured and compared with the cylinder frequency in still water and in still air, in this way it is possible to find a relationship between the cylinder response, the vortex shedding frequency and the cylinder frequency throughout the lock-in region.

mc md mn Re St U Ur W x y

ν ζa ζw

structural mass displaced fluid mass mass ratio mn ¼ mc =md Reynolds number Re ¼ UD=ν Strouhal number St ¼ f o D=U free stream velocity reduced velocity U r ¼ U=F w D spanwise tunnel test section in-line cylinder oscillation cross-flow cylinder oscillation kinematic viscosity damping ratio in still air damping ratio in still water

2. Materials and methods 2.1. Experimental model A sketch of the experimental model is shown in Fig. 1. The model consisted of a horizontal Plexiglas acrylic flat plate (9 mm thick, 90 cm long and 30 cm wide), in which a 2.40 mm in diameter and 40 cm length cylindrical bar was inserted by interference in the normal direction into a drilled hole at a location 70 cm from the leading edge; the other end side of the bar was left free to move as the fluid flows. The material selected for the bar was a copper and zinc alloy with an elastic modulus of 10.5  1010 Pa. The model was placed inside the test section of a water tunnel where the cylinder-flow interaction was analyzed. In order to reduce perturbations that could affect the development of the boundary layer along the plate, the leading edge of the plate was streamlined. The experiments were performed for a low mass damping parameter, mn ζ w ¼ 0:126 (mn ¼ 8:13 and ζ w ¼ 1:55%). 2.2. Experimental equipment The experiments were performed in the water tunnel shown schematically in Fig. 2. The test section of the tunnel is 0.381 m wide, 0.508 m high and 1.5 m long; its walls are made of tempered glass for visual access to the model being tested. The water tunnel operates 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 three flow-conditioning elements. (1) A perforated stainless steel plate followed by (2) a fiberglass screen and (3) a honeycomb flow straightener; these three elements reduce the turbulence level to a small scale. 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

Fig. 1. Experimental model.

F. Oviedo-Tolentino et al. / Ocean Engineering 88 (2014) 463–471

the inlet of the test section, and avoids local separation and vorticity development. The water tunnel is equipped with a dye injection system and a flow velocity control system. 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 a 2 GB internal memory in which images can be stored digitally at 506 frames per second (fps) at a full resolution of 1280  1024 pixels; for lower resolutions the frame rate speed can be increased proportionally up to 112,000 fps. This camera has an excellent 2500 ISO light sensitivity, in addition, external illumination was used. The shutter time can be adjusted from 2 μs to 1 s and BMP or AVI video files can be downloaded through a Gigabit Ethernet interface. With these features the free-end cylinder displacements could be measured with an accuracy in the order of 11 μm. 2.3. Experimental methodology The cylinder free tip displacements were captured during 20 s at 240 fps with a 100 μs exposure time. The frames captured were processed using the principle of particle tracking velocimetry. The time signal was processed in a spectral analysis to determine the

oscillating frequency of the cylinder. The amplitude at the cylinder free end tip was measured using both the maximum and the RMS cylinder amplitude in the lock-in region. The reduced velocity was gradually increased , from U r ¼ 2:31 to U r ¼ 9:00 at a constant kinematic viscosity (ν ¼ 1  10  6 m2 =s). The velocity at the inlet of the test section was measured by means of a flowmeter and was verified using the particle tracking technique described in OviedoTolentino et al. (2008). For the Reynolds number interval, the boundary layer thickness varies from 2.16 to 1.10 cm at the base of the cylinder location. Determination of vortex frequencies was performed via a dye injection flow visualization technique at four positions along the cylinder span, that is, for H=L ¼ 0:20, 0.40, 0.60 and 0.80. The vortex shedding frequency was recorded at 90 fps during 12 s. In order to determine when the pattern was repeated, a cross-correlation function was applied followed by a spectral analysis. The flow structures formed in the lock-in region were identified using the vortex motion classification made by Williamson and Roshko (1988).

3. Results 3.1. Cylinder response Fig. 3 shows the cylinder natural response in still air and in still water. With this experiment the cylinder natural frequency and the damping ratio in different media can be measured: for water medium F w ¼ 6:75 Hz; and ζ w ¼ 1:55%, while for air medium F a ¼ 7:15 Hz and ζ a ¼ 0:29%. The damping ratio is obtained by using the logarithmic decrement. The difference between the cylinder frequency in still water and that in still air accounts for the effect of the added mass. This frequency variation is in part responsible for the branches formed in the lock-in region. In addition to these two measured frequencies, we believe that the vortex shedding frequency of the flow around a stationary cylinder is related to the cylinder response in the lock-in region. Therefore, experimental results of the vortex shedding frequency, measured by means of vorticity levels with a LDV technique, of a stationary cylinder from Hammache and Gharib (1991) are included for the analysis. The lock-in zone is usually classified as (see for example Khalak and Williamson, 1999): (i) initial, (ii) upper and (iii) lower branches. These branches are separated by discontinuous mode

Fig. 2. Water tunnel.

1.5

1.5

1

1

0.5

0.5

y/D

y/D

465

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5 0

2

4

6

8

10

0

0.2

0.4

time[s]

Fig. 3. Cylinder vibrational response: (a) still air and (b) still water.

0.6

0.8 time[s]

1

1.2

1.4

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transitions in the cylinder response. Such mode transitions may be easily identified on a plot of amplitude response as a function of the reduced velocity. In the present work, the response amplitude versus reduced velocity plot shows a smooth behavior, which made it difficult to identify the branches. In contrast, relating the cylinder frequency in the lock-in region with the cylinder frequency in still water and still air, three regions were identified. Such regions correspond to the initial, upper and lower branches. Fig. 4 shows the dimensionless vibration amplitude as a function of the Reynolds number and reduced velocity using the classification for the lock-in zone (cross-flow direction in Fig. 4(a) and in-line flow direction in Fig. 4(b)). Region I (initial branch) comprises frequency values between the vortex shedding frequency of a stationary cylinder and the cylinder natural frequency in still water. In this region the cylinder response grows rapidly until it reaches the natural frequency in still water; at this point the cylinder registers the maximum amplitude in the lock-in zone. This region is also characterized by a multi-frequency response, in agreement with the results of Prasanth and Mittal (2008) and Blevins (1994); however, only the dominant frequency response is reported here. In Region II (upper branch) the cylinder frequency response lies between the values of the natural frequency in still water and the natural frequency in still air. In this region, the cylinder amplitude response decreases until it reaches the value of the natural frequency in still air. Region III (lower branch) is established from this point; here the cylinder amplitude of oscillation has a gradual variation with the Reynolds number until it decays abruptly to zero. In this region the cylinder frequency is equal to the natural frequency value in still air. The cylinder response identification presented here agrees remarkably well with the deductions made by Govardhan and Williamson (2000), who suggest that the upper branch is delimited by the cylinder frequency in the still fluid medium and by the cylinder frequency in vacuum. The maximum cross-flow amplitude registered in this investigation was 0.72D. This value agrees with the maximum amplitude predicted by Skop and Balasubramanian (1997) for flexible cylinders on the Griffin plot where Anmax  0:77D for the case of

4

5

6

Ur

7

ðmn þ C A Þζ w ¼ 0:14, where CA is the potential added mass coefficient (C A ¼ 1:0 for circular cylinders). The dominant vibration of the cylinder is in the direction transversal to the incoming flow. Fig. 4b shows the maximum dimensionless amplitude measured in the inlet flow direction. For this case the maximum amplitude value is 0.08D, representing only 11% of the maximum amplitude reached for the transversal direction. Time traces displacement in the cross-flow direction is shown in Fig. 5. From this figure it can be observed that the Region I starts with nonregular cylinder oscillations and ends with regular cylinder oscillations. However, Regions II and III are completely characterized by regular cylinder oscillations for each Reynolds number case. It should be noted that the regular oscillations start when the cylinder frequency reaches the cylinder frequency in still water (Region II onset). Willden and Graham (2001) reported a rapid decrease of the added mass with increases of the reduced velocity up to the point where the cylinder vibrates at its natural frequency. These observations suggest that the nonregular cylinder oscillations in Region I may be associated with the high dependence of the added mass on the reduced velocity. The normalized cylinder free-end path at different values of reduced velocity within Region I is shown in Fig. 6. It can be observed that for Region I the cylinder presents variations in the cross-flow and in the in-line vibration; this is noticeable because the free-end does not have a preferential path. When the cylinder reaches the natural frequency in still water the free-end has a very well defined elliptical orbit, with the maximum cross-flow and in-line oscillations (Fig. 4). Such elliptical response dictates that the inline and cross-flow cylinder oscillations have the same frequency with a phase angle between signals. Kheirkhah et al. (2012) associated the elliptical path with a high structural coupling between vibrations in the cross-flow and in-line direction. When the cylinder vibrates at a frequency different from natural frequency in still water the cylinder oscillations are almost only in the cross-flow direction. Thus, it is concluded that the cylinder response is mainly in the cross-flow direction, and the in-line response effect is only present when the cylinder frequency vibrates at the natural frequency in the still working fluid medium.

8

4

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6

Ur

7

A*max A*rms

A*max A*rms 0.08

0.7

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fc = fo = Fw

0.6

fc = fo = Fa

0.06

0.5

fc = fo = Fw Fw < fo, fc < Fa

Region I 0.4

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Region III

fc = fo = Fa

A*

*

Fw < fo, fc < Fa

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A

8

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Re

Fig. 4. Lock-in region in (a) cross-flow and (b) in-line direction. Region I (initial branch) comprises frequency values between the vortex shedding frequency of a stationary cylinder and the cylinder natural frequency in still water. In Region II (upper branch) the cylinder frequency response lies between the values of the natural frequency in still water and the natural frequency in still air. In Region III (lower branch) the cylinder frequency is equal to the natural frequency value in still air.

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0.25

467

0.4 Ur = 4.90

0.2

Ur = 4.45

0.3

0.15 0.2 0.1 0.1 y/D

y/D

0.05 0 -0.05

0 -0.1

-0.1 -0.2 -0.15 -0.3

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-0.4 0

2

4

6

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0

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8

time[s]

0.4

0.6

Ur = 4.55

0.3

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14

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time[s]

Ur = 4.65

0.4 0.2 0.2 y/D

y/D

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Ur = 5.06

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1 Ur = 5.86

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y/D

10 time[s]

time[s]

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Fig. 5. Cylinder time traces displacement in the cross-flow direction: figures from (a) to (h) correspond to Region I where the cylinder response starts with nonregular oscillations and ends with regular oscillations; figures (g) and (h) correspond to Regions II and III respectively; in both regions the cylinder response always shows regular oscillations.

3.2. Vortex shedding patterns The vortex shedding classification follows the patterns described by Williamson and Roshko (1988). They used the following classification: 2S mode (two single vortices shed per cycle), 2P mode (two pairs of vortices shed per cycle) and PþS mode (one pair and one single vortex shed per cycle). The vortex shedding patterns are shown in Fig. 7. At low reduced velocities (U r o 3:86) the flow behavior is similar to the one found around a stationary cylinder and the vortex shedding pattern is the classical von Kármán vortex street (2S mode), see Fig. 7a. However, when

the cylinder vibrates, the vortex pattern is modified. Now there are two lines of vortices, one at each side of the cylinder, as shown in Fig. 7b. This behavior is similar to the 2P mode found by Williamson and Roshko (1988) where they reported a formation of vortex pairs that convect outwards from the centerline wake. At H=L ¼ 0:20, the cylinder oscillation is so small that the vortex shedding pattern is the 2S mode throughout the lock-in region. As the position moves up to the value H=L ¼ 0:40 the vortex shedding begins to show some changes. The vortex shedding in this case switches between a 2S mode to an expanding and stretching consecutive single vortex mode, as shown in Fig. 8. This behavior is

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1

1 Ur=4.40

Ur=4.47

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y/D

y/D

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-0.5

Fig. 6. Orbits of the free-end of the cylinder in the initial branch (4:40 o U r o 5:06), Region I.

found in the range 4:96 o U r o 5:84, corresponding mostly to Region II. At H=L ¼ 0:60 the vortex shedding is modified earlier. The vortex mode is 2S, except in the range 4:68 o U r o 5:40 where

the (2P) mode is present. The progress of the vortex mode is shown in Fig. 9. This figure shows how the 2S mode undergoes changes and then recovers the 2S mode again. The case H=L ¼ 0:80

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469

Fig. 7. Vortex shedding modes: (a) H=L ¼ 0:40 and (b) H=L ¼ 0:80.

Fig. 8. Vortex shedding modes at different times, H=L ¼ 0:40 and Ur ¼ 5.48.

Fig. 9. Vortex shedding modes in the lock-in region at H=L ¼ 0:60.

shows a 2P mode in the range, 4:55 o U r o 6:07. This is due to the fact that at this position the cylinder displacements are greater than for the later cases. The vortex shedding modes at four cylinder positions are shown in Fig. 10 for U r ¼ 5:2 (in the upper branch region), where according to Govardhan and Williamson (2000) a 2P vortex shedding mode is expected. However, in this investigation a 2S mode is reported for the lower cylinder part (H=L ¼ 0:20 and H=L ¼ 0:40) and a 2P mode for the upper cylinder

part (H=L ¼ 0:60 and H=L ¼ 0:80). This difference may be attributed to the bottom fixed end of the cylinder, which differentiates the elastic cylinder in Govardhan and Williamson (2000) from the flexible cylinder in this investigation. The fact that the oscillation amplitude be related with the vortex shedding modes has been documented by Williamson and Roshko (1988) where a vortex wake modes map was developed as a function of the reduced velocity and the oscillation amplitude.

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Fig. 10. Vortex shedding modes along the cylinder span at U r ¼ 5:2: (a) H=L ¼ 0:20, (b) H=L ¼ 0:40, (c) H=L ¼ 0:60 and (d) H=L ¼ 0:80. 2S vortex shedding mode for the lower part of the cylinder (“a” and “b”) and 2P mode for the upper part of the cylinder (“c” and “d”).

2

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5 Ur

Fa Fw fc fo at H/L = 0.20 Stationary Hammache and Gharib (1991)

0.4

0.35

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8

Fa Fw fc fo at H/L = 0.40 Stationary Hammache and Gharib (1991)

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0.45 Fa Fw fc fo at H/L = 0.60 Stationary Hammache and Gharib (1991)

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Re

Fig. 11. Non-dimensional frequencies in the lock-in region at different positions along the cylinder span. At H=L 4 0.40 the vortex shedding frequency locks on to the cylinder frequency. The cylinder frequency in the lock-in region is delimited by the cylinder frequency in still water, still air and the vortex shedding frequency of a stationary cylinder.

F. Oviedo-Tolentino et al. / Ocean Engineering 88 (2014) 463–471

3.3. Cylinder and vortex shedding frequencies Fig. 11 shows the non-dimensional frequencies encountered in this work at different positions along the cylinder span. This figure shows that the cylinder frequency is not constant throughout the lock-in region. At H=L ¼ 0:20 the vortex shedding frequency does not match the cylinder frequency throughout Region I and the first part of Region II. This case shows a behavior like that observed for stationary cylinders. The vortex shedding matches the cylinder frequency when the stationary frequency is equal to the natural frequency in still water. After that point the cylinder and vortex shedding frequencies progress to the cylinder frequency in still air value. Moving along the cylinder span direction away from the base, an earlier matching between the vortex shedding and the cylinder frequency is observed. For H=L ¼ 0:40 and above, the vortex shedding frequency tends to match the cylinder frequency throughout the lock-in region. This behavior supports the lock-in region definition made by Sarpkaya (1995) who aptly defined the lock-in region as the matching between the frequency of the driving force and the frequency of the vibrating body. In addition, the vortex shedding and cylinder frequency are restricted by: (i) the vortex shedding of a stationary cylinder and the natural frequency in still water for Region I, (ii) by the natural frequency in still water and the natural frequency in still air for Region II, and (iii) by the natural frequency (only in still air) for Region III. The results in this section agree remarkably well with the results of Govardhan and Williamson (2000) who delimited the upper branch with the cylinder frequency in still fluid and the cylinder frequency in vacuum. This experiment suggests that for fluid media which cause lower natural frequencies, a wider Region II is expected. At the same time these results explain why at high mass–damping ratios the lock-in region displays only a two branch response, as the classical Feng-type response. 4. Conclusions Results have been presented for the fluid–structure interaction between a bottom fixed flexible cylinder with low mass damping factor (mn ζ w ¼ 0:126) and blockage ratio (B o1%), and a fluid moving at reduced velocity range ð3:86 o U r o 8:74Þ. For these parameters, the cylinder exhibits three response regions which correspond to the initial, upper and lower branches. The cylinder frequency in the lock-in region is governed by the cylinder natural frequency in still water, in still air, and the vortex shedding frequency of the stationary cylinder. The cylinder response may be classified based on these frequencies. The maximum cylinder amplitude occurs when the cylinder frequency matches the natural frequency of the cylinder in still water (U r ¼ 5:06). For H=L ¼ 0:20 the vortex shedding frequency is composed of the stationary cylinder frequency in the range 2:44 rU r r 5:33 and of the cylinder frequency for U r 45:33, slightly after the onset of Region II. For cases H=L ¼ 0:40, 0.60, and 0.80 the vortex shedding frequency shows a gradual tendency to match the cylinder frequency; this behavior supports the synchronization definition

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made by Sarpkaya (1995). The cylinder amplitude response can be clearly classified as follows: Region I f s oðf o ¼ f c Þ o F w , Region II F w o ðf o ¼ f c Þ o F a and Region III f o ¼ f c ¼ F a . For a fluid medium which gives lower natural frequencies (lower mass–damping ratios) a wider Region II is expected. For higher mass–damping ratios Region II disappears and only two response branches are expected, as the classical Feng-type response. The dominant vortex modes are 2S and 2P. The vortex mode 2P appears for high amplitudes and lower dimensionless cylinder frequency.

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