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On the vortex-induced vibration of a low mass ratio circular cylinder near a planar boundary Sina Daneshvar , Chris Morton * Department of Mechanical and Manufacturing Engineering, University of Calgary, Alberta, Canada

A R T I C L E I N F O

A B S T R A C T

Keywords: Vortex-induced vibrations Boundary layer vortex Amplitude response Low-order model Particle image velocimetry

Vortex-induced vibration of a circular cylinder near a plane boundary has been investigated experimentally for 2; 500 � ReD � 10; 000, 3:0 � U� � 12:0, 0:3 � S� � 8, and m� ¼ 5:3, where ReD is the Reynolds number based on cylinder diameter, U� is the reduced velocity, S� is the gap distance between the planar wall and the cylinder, and m� is the system’s mass ratio. The experimental data includes simultaneous amplitude response and wake measurements via time-resolved planar particle image velocimetry (PIV). From experimental raw data, frequency content of the oscillations as well as wake velocity and vorticity fields are analyzed. A new vortex shedding flow regime map is proposed, illustrating dominant vortex shedding patterns in the wake across the investigated nondimensional parameter space. The quantitative relationship between the cylinder’s motion and instantaneous vortex dynamics is shown.

1. Introduction Vortex-induced vibrations (VIV) of cylindrical structures is of inter est to many engineering applications. For example, VIV influences the dynamic loading on offshore risers transporting oil from the seabed to the surface; it is important to the design of such engineering structures as off-shore platforms, wind turbine masts, cable-supported bridge decks, pipelines at water crossings, and tethered structures in the ocean. Numerous practical applications of VIV in engineering problems has led to large number of studies in this field, many of which are presented in thorough reviews of Williamson & Govardhan (Williamson and Govardhan, 2004), Sarpkaya (2004), and Bearman (1984). Common non-dimensional parameters used in the characterization of the response for a single-degree-of-freedom (1DOF) VIV system include the response amplitude (A� ¼ A=D where D is the cylinder diameter and A is the oscillation amplitude), frequency (f � ¼ f= fN , where f is the dominant peak from the spectra of the cylinder displacement and fN is the natural frequency of the system), reduced velocity (U� ¼ U∞ =fN D, where U∞ is the free stream velocity), mass ratio (m� ¼ 4m=πρD2 L, where m is the mass of the system), and damping ratio pffiffiffiffiffiffi (ζ ¼ c=2 km, where k is the system stiffness, and c is the damping co efficient). Feng (1968) investigated the amplitude response of a high mass ratio system (m� ¼ 248) by measuring the oscillations of a circular cylinder in a wind tunnel. The corresponding results can be found in

Fig. 1a. For low mass ratio systems, e.g. the results from Williamson & Govardhan (Williamson and Govardhan, 2004) at m� ¼ 2:4, significant changes to the response are observed. In particular, the amplitude response can be divided into four main response regimes including initial branch (U� � 4:5), upper branch (4:5 � U� � 7:5), lower branch (7:5 � U� � 10), and de-synchronization region (U� � 10) as illustrated in Fig. 1. It is noted that the specific U� range corresponding to each branch has been shown to depend upon both m� and ζ (e.g. see (Riches and Morton, 2018)). Distinct changes in the frequency response of the cylinder occur within each response regime. For the initial branch, it has been shown that two dominant frequencies define the cylinder’s oscillation: the Strouhal vortex shedding and system’s natural frequency (Fig. 1b). The transition between the initial and the upper branch occurs as the Strouhal frequency reaches the natural frequency of the system in water (fo ¼ fN;water ) (Williamson and Govardhan, 2004). The oscillation fre quency in the upper branch continues to increase with an increase in U� up to U� � 7 (Fig. 1b). At U� � 7, as the oscillation frequency reaches the natural frequency of the system in vacuum (fo ¼ fN;vacuum ) (Fig. 1b), and this corresponds to the transition to the lower branch (Fig. 1a). In the lower branch, the oscillation frequency is locked at a frequency higher than the natural frequency of the system. This is attributed to decreasing added mass effect in the lock-in range (Williamson and Govardhan, 2004). For about U� � 10, the amplitude response begins to

* Corresponding author. E-mail address: [email protected] (C. Morton). https://doi.org/10.1016/j.oceaneng.2020.107109 Received 26 June 2019; Received in revised form 6 November 2019; Accepted 10 February 2020 Available online 5 March 2020 0029-8018/© 2020 Elsevier Ltd. All rights reserved.

S. Daneshvar and C. Morton

Ocean Engineering 201 (2020) 107109

is accompanied by a 2T mode of vortex shedding (Jauvtis and Wil liamson, 2004). The 2T mode involves three vortices being shed from the cylinder each half-oscillation cycle, in contrast with the 2P or 2Po shedding patterns observed in 1DOF system response and illustrated in Fig. 2. It is recognized that the present work focuses on 1DOF con strained systems, such as that employed in the VIVACE energy converter (Lee and Bernitsas, 2011). In many engineering applications, VIV of a structure takes place in the vicinity of a wall boundary, and thus, different dynamics, in terms of both the vibration response and the wake, are expected. For conceptual energy harvesting systems which make use of VIV (Lee and Bernitsas, 2011), the cylinder is placed in the vicinity to a river bed or sea floor. Within the oil and gas industry, pipelines at river crossings may undergo VIV. The pipelines are buried in the riverbed, however, with the passage of time, they may become fully or partially exposed to water flow due to soil erosion (Yang et al., 2008). The exposure may result in flow-induced fluctuating loading along the pipeline, and lead to structure’s fatigue failure. Such hazardous incidents are detrimental from both financial and environmental perspectives. While numerous studies have investigated different aspects of VIV in free-stream, there is a lack of detailed information on VIV of cylindrical bodies placed near a planar boundary, such as a riverbed. With the introduction of a planar boundary, a new non-dimensional parameter is introduced as the gap ratio, S� ¼ DS , (in which S is the distance between the wall and cylinder’s tangent). S� plays an important role in the cyl inder’s response as well as wake dynamics. A summary of experimental and numerical studies pertaining to flow-induced vibrations of a circular cylinder near a wall boundary is presented in Table 1. A review of the pertinent literature has revealed the following key dynamical features of the wake and cylindrical body. For S� � 2 3, wall influence on the amplitude and frequency response is negligible, as no significant changes in vortex shedding frequency and amplitude are observed (Grass et al., 1984; Yang et al., 2006, 2009). For smaller gap ratios (S� � 2), the vortex shedding frequency increases progressively as the gap ratio decreases (Grass et al., 1984). For S� < 1, significant asymmetry in the wake is induced, and therefore, different flow behavior is observed. Fig. 3 qualitatively il lustrates how the instantaneous flow behavior is influenced by the ex istence of a planar boundary. The sketch in Fig. 3 depicts 3 vortices (V1 , �rma �n V2 , and V3 ). Vortices V1 and V2 comprise the traditional von Ka wake vortices, while V3 is a vortex formed from wall boundary layer separation. The separation of the wall boundary layer is induced by a

Fig. 1. (a) Amplitude response of low and high mass ratio VIV systems (b) dominant frequency of the oscillations for low mass ratio VIV systems (Wil liamson and Govardhan, 2004).

decay into the de-synchronization region. The cylinder motion is no longer synchronized with the vortex shedding, giving rise to a two fre quencies defining the cylinder oscillation: the Strouhal frequency and lock-in frequency. The amplitude and the frequency response of the system are linked with the vortex dynamics in the wake (Williamson and Govardhan, 2004). In the initial branch, the dominant vortex shedding pattern is 2S (Fig. 2a), while the wake topology for the lower branch is 2P (Fig. 2b), as observed by (Govardhan and Williamson, 2000; Cagney and Balabani, 2013). The wake in the upper branch is referred to as 2P0 mode (Govardhan and Williamson, 2000), wherein the secondary vortex in each pair (P2 and P4 in Fig. 2b) is of a significantly lower strength than the leading vortex (P1 and P3 in Fig. 2b). In the de-synchronization region, no clear vortex pattern is synchronized to the cylinder motion (Khalak and Williamson, 2006). For the case of vortex-induced vibration for a two-degree of freedom (2DOF) cylinder with low mass ratio and damping, experiments carried out by Jauvtis and Williamson (2004) have shown the typical three-branch response of a 1DOF system (Williamson and Govardhan, 2004) is modified such that the characteristic 2S wake mode within the initial branch transitions to a super-upper branch (this is in contrast with the upper branch classification for a 1DOF system). In the super-upper branch, the vibration response reaches its highest amplitude, and this

Table 1 Summary of studies on VIV near a planar wall boundary.

Fig. 2. (a) 2S vortex shedding pattern. (b) 2P and 2P0 vortex shedding pattern. 2

Study

m�

U�

S�

Re

δ=D

(Yang et al., 2006, 2009) (Chung, 2016) (Zang and Zhou, 2017) (Zhao and Cheng, 2011) (Wang et al., 2013) (Fu et al., 2014) (Sumer et al., 1994) (Li et al., 2016) (Manuel et al., 2017)

1.3–3.9

2.0–11.0

0.1–4.7

3,000–16,000

NA

2.0

2.0–11.0

0.06–31.5

100

NA

1.47

2.0–10.0

0.05–1.5

12,000–60,000

3.4

2.6

1–15

0.02–0.3

1,000–15,000

NA

1.0

1.5–6.6

0.05–2.5

3,000–13,000

0.4

NA

14.5

0.1–0.9

200,000

NA

1.0

5.0–5.6

0.35–1

50,000–60,000

0.02–0.19

10

3–9

0.9 - ∞

200

0.6

5.5

3–12

0–5

7250–15500

NA

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Ocean Engineering 201 (2020) 107109

2. Methodology 2.1. Data acquisition Experiments were carried out in the water tunnel facility at the University of Calgary. An experimental VIV apparatus was designed, fabricated, and placed in the water tunnel. This experimental rig is also employed by Riches et al. (refer to (Riches and Morton, 2018) and (Riches et al., 2018)) where additional information can be found. As can be seen in Fig. 4a, the apparatus is composed of an acrylic cylinder (D ¼ N ) as well as two 25.4 mm (1”)) attached to two linear springs (k ¼ 36 m air bushings to facilitate near frictionless 1 DOF oscillation in the transverse direction. The mass ratio of the system is m� ¼ 5:3. The natural frequency of the system is fN ¼ 1:2Hz (in water). The damping ratio was estimated to be ζ ¼ 0:0013 (in air) via free-decay testing. The aspect ratio of (the immersed portion of) the cylinder is 17. A rigid smooth aluminum plate with the size of 20D � 0:125D � 17D was mounted vertically to investigate the wall effects on vortex dynamics as well as amplitude response. The wall laminar boundary layer had a thickness of δ=D � 0:05 at the cylinders stream-wise location (x=D ¼ 0). A 3-D rendering of the experimental setup including VIV apparatus, the planar wall, high speed camera, range finder, and laser generator is shown in Fig. 4b. A summary of the parameters pertaining to the experimental setup is shown Table 2. The system was validated in a previous experimental study con ducted by Riches et al. (2018) through amplitude and frequency response measurements in the water tunnel facility. Results are compared with those of other low mass ratio systems, as shown in Fig. 5. Riches et al. (2018) explains that the amplitude response of the system within the initial and upper branches are in good agreement with Wong et al. (2017) due to the closeness of the mass ratio in the two studies. Within the lower branch and desynchronization regions, the results do not match with that of Wong et al. (2017). This is due to the substantially lower structural damping in the present study. Previous works of Klamo et al. (2006), Riches et al. (Riches and Morton, 2018) and others have shown that a decrease in structural damping extends the range of syn chronization significantly for a given mass ratio. The cylinder’s sinusoidal oscillations were recorded directly using a laser range finder sensor, model CP24MHT80, with a resolution of 20 μm. Planar Particle Image Velocimetry (PIV) was conducted at cylinder’s mid-span. A Phantom Lab M340 high speed camera was used to capture particle images in the cylinder’s wake. A laser sheet was generated by a high energy Photonics DM30 Nd:YLF pulsed laser to illuminate the particles in the wake. 10 μm hollow spherical glass particles were used as flow seeding for PIV. A LaVision high speed controller is used to trigger the laser and the camera simultaneously. The laser was operated in double-pulse mode such that two laser pulses are fired within a small

Fig. 3. Wall effects on vortex shedding.

coupling (synchronization) between the shear layer shed from the inner side of the cylinder and the wall boundary layer (Price et al., 2002). In this case, decreasing the gap ratio further yields the following: (a) larger reduced velocity at which the maximum vibration amplitude takes place, (b) larger size of lock-in region, and (c) smaller maximum vi bration amplitude (Chung, 2016). When the gap ratio is smaller than a critical gap ratio (S� � 0:2 0:75), the cylinder impacts the wall. Chung (2016) found no significant difference in terms of wake topology between oscillation-free scenarios and physical impact cases, while Zhao & Cheng (Zhao and Cheng, 2011) observed a new vortex shedding pattern (i.e., vortex-shedding-after-bounce-back) as a result of distortion of regular vortex shedding patterns. Although the amplitude response tends to decline as the gap ratio decreases, once a physical impact between the wall and the cylinder takes place, both amplitude and frequency response of the system increases (Manuel et al., 2017). In some cases, vortex suppression is observed for smaller gap ratios, resulting in a drastic decrease in the amplitude of oscillation. Lei et al. (1999) analyzed the lift force coefficient, and identified that vortex suppression is linked to the wall’s boundary layer thickness, resulting in vortex suppression at S� � 0:2 0:3. The vortex shedding suppression is attributed to viscous wall effects and the separation bubble formed on the wall which impedes the shear layer roll up into a vortex. In addition, having an opposite sign vorticity, the separation bubble annihilates the vorticity in the inner shear layer of the cylinder, suppressing vortex formation for small gap ratios (Grass et al., 1984; Li et al., 1063; Mar tinuzzi et al., 2003). Li et al. (Li et al., 2016; Li et al., 1063) looked at the flow physics in the wake of a 2-DOF system and mentioned that similar to stationary cylinder, a separation bubble forming on the wall sup presses the vortex shedding from the inner side of the cylinder, and consequently results in oscillation damping in the transverse direction and “stream-wise frequency lock-in and eventually enhanced stream-wise oscillations.” The results of previous experimental and numerical studies has revealed that no consensus has been reached on the critical gap ratio for a 1DOF system, in which a physical impact between the cylinder and the wall takes place, and the gap ratio at which vortex suppression is observed. The lack of consensus may be attributed to different driving parameters in these studies such as mass-damping ratio and wall’s boundary layer thickness. To date, no study has investigated the fluidstructure coupling of an oscillating cylinder near a planar wall bound ary for a comprehensive range of U� and S� , so that all oscillation branches are investigated thoroughly for different wall positions with respect to the cylinder. In addition, in the previous literature, vortex shedding patterns are investigated qualitatively and no detailed quan titative study in this regard has been performed. In view of these gaps in the literature, the present study investigates 1DOF VIV of an elastically mounted circular cylinder in the vicinity of a wall boundary for a wide range of dimensionless parameters: S� , U� , m� , and ζ. The goal is to provide a quantitative characterization of the wake (including dominant vortex shedding patterns) and identify key aspects of the wake development near the critical gap ratio.

Fig. 4. (a) schematic and (b) 3-D rendering of the VIV setup. c is the damping pffiffiffiffiffiffi coefficient (c ¼ 2ζ km, where ζ is the damping ratio) and k is the stiffness. 3

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Ocean Engineering 201 (2020) 107109

measurements was quantified according to correlation statistics method (Wieneke, 2015) and is less than 1% in the free-stream and less than 7% in the wake, based on instantaneous velocity fields. A decomposition of the flow field data was completed using the raw vector fields. Using triple decomposition (Hussain, 1986), the velocity field in the cylinder’s wake at a given location and time can be written as the sum of the average velocity and the velocity of coherent and inco herent structures in the fluctuating field:

Table 2 Summary of experimental parameters pertaining to the present study. Parameter

Range

Units

m�

5.3

–

U�

3.0–12.0

–

S�

0.3–8.0

–

ζ ReD

0.0013 2,500–10,000

– –

δ= D

0.05

–

k fN

36.0 1.2

N/m Hz

L= D

17

–

Uðx; tÞ ¼ UðxÞ þ u’c ðx; tÞ þ u’ic ðx; tÞ

(1)

Proper Orthogonal Decomposition (POD) is a popular tool used to distinguish between coherent and incoherent structures in the wake. Using snapshot method (Sirovich, 1987), an orthonormal basis of the velocity field is obtained such that each space-time realization of the fluctuating velocity field u’i ðx; tÞ ¼ u’c ðx; tÞ þ u’ic ðx; tÞ, where the subscript ‘i’ denotes the Cartesian tensor component can be expanded as the sum of the products of spatial functions (φi ) and corresponding temporal coefficients (ai ) (Tropea et al., 2007):

time frame (dt). Following the recommendations by Raffel et al. (1989), this time difference is selected so that the particle displacement between two consecutive images is approximately 7 pixels in the free-stream. The camera recorded the particle images pairs at the frame rate of 12 Hz in a rectangular field of view, with a 7D � 6D dimension. For a given the camera frame rate and oscillation frequency of the system, 10–12 images were acquired per oscillation cycle. Two data sets, each containing 1, 462 particle image pairs, were collected for each of the test cases.

N X

u’i ðx; tÞ ¼

ai ðtÞφi ðxÞ

(2)

i¼1

Sorting the modes according to their contribution to Turbulent Ki netic Energy (TKE), one is able to discern those modes associated with vortex shedding and other coherent phenomena, and those corre sponding to stochastic fluctuations. After reorganizing the modes based on this criterion, using only the first few modes extracted from POD analysis, one is able to reconstruct a Low-Order Model (LOM) (Liberge and Hamdouni, 2010) of the flow field which provides a more concise and potentially more informative description of the wake behavior as only the dominant modes contributing to the coherent wake fluctuations are retained, while turbulent fluctuations are removed (Schmid, 2010). The final relevant component to the flow analysis is the fluctuating

2.2. Data processing Using Fourier analysis, the frequency content of the oscillation was computed as follows. Each amplitude response VIV time series data set is divided into smaller segments, each 30 s long with 50% overlap (a total of 14 segments per data set). A Fourier transform was performed on each time series segment, and the ensemble average of the spectra is pre sented as the oscillation frequency response. The frequency resolution is �0:01Hz. LaVision DaVis 8.3 was utilized to process images using an iterative multi-grid cross-correlation algorithm (Scarano and Riethmuller, 1999). In order to obtain the velocity fields from the particle images, two passes with interrogation window size of 48 � 48 pixels followed by three passes with the window size of 24 � 24 pixels are performed. The overlap between the adjacent windows was selected to be 75 % yielding a final vector spacing of 0.03D. Universal outlier detection (Westerweel and Scarano, 2005) was used such that a vector with small peak ratio (Charonko and Vlachos, 2013) (Q � 1:8) is removed and replaced via interpolation and a 3 � 3 smoothing. The uncertainty of PIV

turbulent kinetic energy (k ¼ 12 ðu0 2 þ v0 2 Þ). It is the mean kinetic energy associated with eddies in turbulent flows. Equation (3) represents the total Turbulent Kinetic Energy (TKE) encompassed in the PIV domain (Ω) (Mohebi et al., 2017). Z Z Z Z � (3) TKE ¼ kdA � u’2 þ v’2 dA Ω

Ω

It is important to identify a quantitative metric to determine how

Fig. 5. System validation by comparing VIV amplitude response with (Khalak and Williamson, 1997) and (Wong et al., 2017). 4

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many modes to include in a POD based low order model. In the present study, two criteria are used: (i) The contribution of modes to TKE should be more than 2%, and (ii) the fluctuating velocity components (u’ and v’), associated with the modes, are correlated. If the contribution of a mode to shear Reynolds stresses in any region of the wake is less than 2%, then the mode is assumed to be mainly linked with incoherent structures and hence, is not used in the model. The thresholds selected above for the POD series truncation are based on previous experimental studies linking POD modes to coherent and important dynamical fea tures of cylinder wakes (e.g. (Riches et al., 2018),) The stream-wise and transverse components of the fluctuating ve locity field (u’ and v’) associated with incoherent structures are expected to be uncorrelated (Delville et al., 1999). Therefore, the shear Reynolds stress (u0 v0 ) associated with the incoherent modes is very small throughout the wake. The residual of the shear Reynolds stress between LOM’s containing the first n and n þ 1 POD modes is expressed in Equation (4). This re sidual is used in determining an appropriate number of modes to include in a POD based low order representation, and was carried out on all data sets. u~’~v’ ¼ u’v ’jnþ1

3. Results 3.1. Cylinder’s response Fig. 7 illustrates the maximum amplitude response of the system (A� ) as a function of the flow reduced velocity (U� ) for different gap ratios (S� ). In the present study, S� ¼ 8 is considered approximately equivalent to an unbounded free-stream. The results for S� ¼ 8 are in agreement with previous studies conducted in free-stream as shown previously in Fig. 5. For the case of an open free stream it is relatively straightforward to classify the response of a low mass-damping system by its typical three branches and desynchronization region (Williamson and Govard han, 2004). The transition from the initial to the upper branch is marked by an abrupt increase in the amplitude response of the system and accompanied by a change in the frequency response (the cylinder oscillation frequency approaches the natural frequency of the system). The transition from the upper to the lower branch is identified based on a decrease in the amplitude response of the system, and this is linked to a change in the wake vortex shedding mode (from 2Po to 2P shedding pattern). The frequency response in the lower branch exhibits a single dominant peak due to the complete synchronization between the cyl inder oscillation and the flow. Finally, the transition from the lower branch to the desynchronization region is associated with a decay in the amplitude response and a change in frequency response. The flow development is no longer synchronized to the cylinders motion, giving rise to a Strouhal vortex shedding signature in the frequency domain of the system response. Therefore, for S� ¼ 8, the U� ranges for the iden tified response branches are: (i) initial branch, 3 < U� � 5; (ii) upper branch, 5 < U� � 7; (iii) lower branch, 7 < U� � 10:5, and (iv) de-synchronization, U� > 9:5. These results are approximately matching S� ¼ 3, indicating that wall boundary effects do not appreciably affect

(4)

u’v’jn

~’~v ’) Fig. 6a–c illustrate the residuals of shear Reynolds stresses (u between the reconstructed wake using n and nþ1 number of modes, defined in Equation (4), at S� ¼ 8 for U� ¼ 4:0 (initial branch), U� ¼ 5:3 (upper branch), and U� ¼ 6:6 (lower branch). For the initial branch the contribution to shear Reynolds stress drops below 2% at n ¼ 6, while for the upper as well as the lower branches n ¼ 5. Therefore, using only limited number of modes for these specific cases, one can resolve the majority of the contribution to the coherent Reynolds stresses, while minimizing stochastic contributions.

Fig. 6. Normalized residual of shear Reynolds stresses (u~U2~v ) between the reconstructed wake using first n and nþ1 first POD modes at S� ¼ 8 for (a) U� ¼ 4:0 (initial ∞ branch) and (b) U� ¼ 5:3 (upper branch). For the initial branch n ¼ 6, while for the upper as well as the lower branches n ¼ 5. The highest difference occurs in the near wake and is less than 2%. 0 0

5

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Fig. 7. Amplitude response of the system for different gap ratios, where A� is the average amplitude of oscillation. The branches are color coded as follows: blue initial branch, green - upper branch, magenta - lower branch, black - de-synchronization branch, and red - impact branch. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

the system response for approximately S� � 3. For S� < 3, the vibrations are damped significantly across all branches. In particular, for S� ¼ 1, the same criteria as the free stream case can be used to identify the branches of motion. For S� ¼ 1, the upper branch begins at U� � 4:8, the lower branch begins at U� � 5:5, and the de-synchronization branch beings at U� � 10 (as indicated via color coding in Fig. 7). For S� ¼ 0:5 and S� ¼ 0:3, the cylinder starts impacting the wall at U� � 4:2. The impact persists through a wide range of U� (4:2 � U� � 7:0). Hence, a new branch, corresponding to 4:2 � U� � 7:0, is identified and named as the ‘impact’ branch, also known as the ‘bounce back’ branch in previous works (e.g (Zang and Zhou, 2017), (Chung, 2016)). In this branch, the amplitude response does not change and is equal to the gap ratio (A� ¼ S� ). For 7 � U� � 9, the amplitude response decreases slightly such that no impact with the wall takes place. For U� � 9, the system enters the de-synchronization region in which the oscillation amplitude decreases significantly. Fig. 8 presents the dominant frequency response normalized by the natural frequency of the system in water. Representative spectra (power spectral density of the amplitude response) are shown in Fig. 9. For all cases of S� � 1 in the initial branch (3 � U� � 5), there are two domi nant frequencies: one corresponding to the Strouhal vortex shedding frequency (indicated by a dashed line in Fig. 8) and the other corre sponding to the dominant oscillation frequency. It is observed that in the free-stream case (S� ¼ 8), f � increases monotonically with U� . At U� � 5, f � approaches 1 and the system enters the upper branch (5 � U� � 7). In

this branch, only one dominant frequency (the oscillation frequency) is detectable in the amplitude response data. In addition, the oscillation frequency increases almost linearly with U� for S� ¼ 8; 3; 1. For about U� � 7, the oscillation frequency locks-in to a constant frequency (f �lock in � 1:15) across all S� . In agreement with previous literature on VIV, two dominant frequencies are observed as the system enters desynchronization: One frequency associated to the lock-in (f �lock in ), and the other at the Strouhal vortex shedding frequency (not shown in Fig. 8 for its magnitude was near the noise level). For S� ¼ 1, the frequency response follows the same general trend as that observed for S� ¼ 3 and 8 (cf. 9a and 9b). Two dominant frequencies are observed in the initial branch frequency response. In all other branches, only a single dominant frequency is observed. For S� ¼ 0:5, in the initial branch, the oscillation frequency approximately matches with the Strouhal frequency of sta tionary cylinder in free-stream. Unlike S� ¼ 8 and S� ¼ 1, only one dominant frequency (f � � 0:95) is identified at U� ¼ 4:0. When impact between the cylinder and the wall boundary takes place, significant changes are observed in the power spectra of the amplitude response, where an impact frequency (f �impact ) is the dominant frequency and the secondary peaks appear at its harmonics (Fig. 9c). In general, the impact frequency is observed to increase linearly with increasing U� up to about U� ¼ 7:2 (Fig. 8). As can be seen in Fig. 7, an impact between the cylinder and the wall boundary occurs for U� ¼ 5:3 as well as U� ¼ 6:5. In the impact branch, several peaks in the spectra are detectable, with second as well as third

Fig. 8. Dominant frequency of the oscillations for different gap ratios. (blue-initial branch, green-upper branch, magenta-lower branch, black-de-synchronization, and red-impact branch.) The dashed line represent the Strouhal number variation with respect to reduced velocity for a rigid cylinder, following the empirical equations presented by Norberg (Norberg, 2003). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) 6

Ocean Engineering 201 (2020) 107109

S. Daneshvar and C. Morton

Fig. 9. Power spectra of the amplitude response for different U� at (a) S� ¼ 8, (b) S� ¼ 1, and (c) S� ¼ 0:5.

harmonics of the impact frequency identifiable in Fig. 9b. At U� ¼ 9:0, corresponding to the de-synchronization region, the dominant fre quency decreases to f �lock in � 1:16. 3.2. Low-order reconstruction of vortex shedding Planar PIV measurements were carried out in order to gain a better understanding of the wake dynamics associated to the cylinder’s response. The instantaneous velocity fields obtained from PIV mea surements and their corresponding vorticity fields were analyzed using Proper Orthogonal Decomposition (POD) as outlined in Section 2.2. The goal is to extract the coherent motions from the turbulent field and identify the dominant vortex shedding modes associated to each response regime illustrated in Fig. 7. The distribution of Turbulent Ki netic Energy (TKE) for the first 10 most energetic modes obtained from POD analysis as well as their cumulative contribution are presented in Fig. 10 at U� ¼ 4:0 and 5.3, corresponding to the initial as well as upper branch, respectively. These modes were analyzed following the pro cedure outlined in Section 2.2 to establish a metric by which dynami cally important modes could be selected for low-order reconstruction. The results of this analysis revealed the following: There are 5 dynam ically relevant modes in the initial and desynchronization branches as well as 6 dynamically important modes in the upper, lower, and impact branches. The energy level of the first two POD modes is expected to be significantly influenced by the dominant vortex shedding pattern in the wake as well as the coherence of the vortex shedding (Jeon and Gharib, 2001). As can be seen in Fig. 10a, at U� ¼ 4:0 and S� ¼ 0:5, 44% of TKE is resolved within the first two modes, while for higher S� , these modes contain approximately 60% of TKE. This difference is even more apparent for S� ¼ 0:3 such that the cumulative TKE of the first two modes is less than 20% in the initial branch. Given the low contribution of these modes to TKE in the wake and the fact these modes have been confirmed to represent the fundamental mode pair associated with vortex shedding from the cylinder (spatial eigenfunctions have been omitted for brevity, see Riches et al. (Riches et al., 2018)) for further details at S� ¼ 8, one can deduce that the vortex shedding is likely significantly altered at S� ¼ 0:3 and warrants further investigations through Low-Order Model (LOM) reconstruction. The mode energy distribution in the upper branch or impact branch (Fig. 10b) shows a completely different trend, with cumulative energy in the first two modes at S� ¼ 0:3 being higher than all other cases at 57%. The results suggest that upon vortex-wake and cylinder synchronization, the ‘impact branch’ contains coherent motions of similar relative contribu tion as the upper and lower branches, consistent with the results shown by Riches et al. (2018). POD analysis in the upper branch or lower branch yields mode pairs with disparate energy content (Fig. 10b). Fig. 11 shows the LOM of the wake, depicting ‘2S’ vortex shedding

Fig. 10. Contribution of the first 10 POD modes to TKE at (a) U� ¼ 4:0 and (b) U� ¼ 5:3.

for S� ¼ 8 and U� ¼ 4:0. As shown in Fig. 11a, at t � ¼ 0, the lower shear layer starts to roll-up to form the vortex V1 . As the cylinder is moving in þy direction, V2 gains strength from the shear layer on the opposite side of the cylinder. At t� ¼ 0:5, when the cylinder reaches the maximum amplitude in its oscillation in the þy direction, the growing vortex V2 cuts off V1 from its source of vorticity (lower shear layer); and the separated vortex is advected downstream (Fig. 11b). At t� ¼ 1, V2 sep arates from the shear layer, as the growing vortex V3 cuts off the supply of vorticity to V2 (Fig. 11c). The process is periodic with the vortex shedding synchronized to cylinder’s transverse location. This behavior is in agreement with the description provide by Brika (Brika and Laneville, 1993) as well as Williamson and Roshko (1988) and others. Given their trajectories observed in Fig. 11, shed vortices tend to advect inward toward the wake-centerline. Fig. 12 shows ‘2P0’ vortex shedding for S� ¼ 8 and U� ¼ 5:3, which is observed in the upper and lower branches. At t � ¼ 0, the cylinder has 7

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Fig. 11. Vortex shedding for S� ¼ 8 and U� ¼ 4:0 at (a) t � ¼ 0, (b) t � ¼ 0:5, (c) t � ¼ 1, and (d) t � ¼ 1:5. (e) Schematic of the cylinder position at each step.

Fig. 12. ‘2P0’ Vortex shedding for S� ¼ 8 and U� ¼ 5:3 at (a) t � ¼ 0, (b) t � ¼ 0:25, (c) t � ¼ 0:5, (d) t � ¼ 0:75, and (e) t � ¼ 1. (f) Schematic of cylinder position at each time step. 8

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reached the maximum amplitude of oscillation in þy direction and starts to move in -y direction. At this instant, vortex V1 is separated from the shear layer on the upper side of the cylinder (Fig. 12a). The cylinder moves in -y direction and at t � ¼ 0:25, vortex V2 is cut off from its source of vorticity (Fig. 12b). Counter-rotating vortices V1 and V2 form a vortex pair, and are advected together downstream. The process is periodic with counter-rotating pairs of vortices being shed each half cycle of cylinder oscillation with synchronization between the wake and cylin der motion. Similar results are observed within the lower branch ‘2P’ type vortex shedding and are omitted for brevity. This behavior is in agreement with the description provide by Brika (Brika and Laneville, 1993) as well as Williamson and Roshko (1988). The vortex shedding at S� ¼ 1 and U� ¼ 4:0 appears to be similar to that for S� ¼ 8 and U� ¼ 4:0. Fig. 13a shows the roll-up of lower vortex, V1 at t� ¼ 0. When the cylinder reaches the maximum amplitude response in þy direction, V1 is cut off from the shear layer and V2 with opposite orientation starts forming on the other side of the cylinder. Fig. 13b reveals the formation of a vortex on the wall (V3 ), synchronized with the separation of V1 from the shear layer. At t � ¼ 1, when the cylinder reaches the maximum amplitude of oscillation in -y direction, V2 is separated from the shear layer and is shed from the cylinder. Hence, in each shedding cycle three vortices are introduced into the wake and the dominant shedding pattern can be considered to be ‘2S þ �rma �n vortex shedding S’, wherein, the ‘2S’ pertains to traditional von Ka and the ‘S’ is associated with the roll-up of the boundary layer into a single vortex in the wake. Fig. 14 depicts the vortex shedding for S� ¼ 1 and U� ¼ 5:3. At t� ¼ 0 (Fig. 14a), while the cylinder reaches its peak amplitude in -y direction, V1 is shed from the lower side of the cylinder and induces the separation 1:5. of the wall’s boundary layer vortex, V2 , at x=D � 2 and y= D � Separated vortices V1 and V2 are convected downstream. At t� ¼ 0:5 (Fig. 14b), vortex V4 is shed from the opposite side of the cylinder as the cylinder reaches the maximum amplitude response in þy direction starts

to move in the opposite direction. Note that at t� ¼ 0:25 and t � ¼ 0:75, respectively, V3 and V5 are shed from the cylinder and form a pair with leading vortices V1 and V4 . However, the corresponding figures are not illustrated for the sake of brevity. At t� ¼ 1 (Fig. 14c), similar to t� ¼ 0, V6 is shed from the lower side of the cylinder, leading to the separation of the planar wall’s boundary layer. In addition, with respect to convective streamlines, the wall’s boundary layer roll-ups to a single vortex, V3 , and is separated from the wall. Therefore, the dominant vortex shedding pattern is found to be ‘2P0 þ S’. Vortex shedding in a typical shedding cycle for S� ¼ 0:5 and U� ¼ 4:0 (corresponding to the initial branch) is presented in Fig. 15 to investi gate the effect of boundary layer on vortex shedding in detail. At t� ¼ 0, as the cylinder is moving in -y direction, vortex V1 is forming on the outer side of the cylinder (Fig. 15a). When the cylinder reaches its maximum amplitude in -y direction at t � ¼ 0:5, V1 is completely formed and about to be separated from the upper shear layer due to roll-up of vortex V2 on the opposite side of cylinder. The small gap between the cylinder and the wall results in the roll-up of the planar wall’s boundary layer into the vortex V3 at x=D � 2 and y=D � 1. The roll-up is syn chronized with vortex shedding such that V3 and V2 separate simulta neously. It appears that V3 inhibits the full formation of V2 from the shear layer (Fig. 15c), perhaps through annihilation. In addition to disturbance to V2 formation, as can be seen in Fig. 15c, V3 connects to V1 and supplies it with vorticity. At t � ¼ 1:5, strength of V2 is decayed such that it is not detectable in the corresponding vorticity field (Fig. 15d). Therefore, in each shedding cycle, one of the shed vortices is annihilated via mixing with the counter-rotating boundary layer vortex, resulting in the ‘S’ vortex shedding pattern. Vortex shedding for S� ¼ 0:5 and U� ¼ 5:3, corresponding to the impact branch, is presented in Fig. 16. At t � ¼ 0, as the cylinder passes the wake centerline, vortex V1 is forming from the upper shear layer. Concurrently, forming from the lower shear layer, V2 induces the sep aration of V3 from the planar wall at x=D � 1. At t � ¼ 0:25, the cylinder

Fig. 13. ‘2S þ S’ Vortex shedding for S� ¼ 1 and U� ¼ 4:0 at (a) t � ¼ 0, (b) t � ¼ 0:5, and (c) t � ¼ 1. (d) Schematic cylinder position at each time step. 9

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Fig. 14. ‘2P0þS’ Vortex shedding for S� ¼ 1 and U� ¼ 5:3 at (a) t � ¼ 0, (b) t � ¼ 0:5, (c) t � ¼ 1, and (d) t � ¼ 1:5. (e) Schematic of cylinder position at each time step.

Fig. 15. ‘S’ Vortex shedding for S� ¼ 0:5 and U� ¼ 4:0 at (a) t � ¼ 0, (b) t � ¼ 0:5, (c) t � ¼ 1, (d) t � ¼ 1:5, and (e) t � ¼ 2. (f) Schematic cylinder position for S� ¼ 0:5 and U� ¼ 4:0. 10

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impacts the wall and vortices V2 and V3 are separated from their shear layers At t� ¼ 1, counter-rotating vortices V2 and V3 are not clearly detectable in the near wake, suggesting an interaction between them, resulting in mixing and annihilation. At t� ¼ 1:5, only V1 can be detected in the wake. As the vortices in the ‘2P0’ pattern are advected outward away from the wake centerline, it is unlikely that V3 interacts with V2 . Therefore, the suggested a shedding pattern classification is ‘P0’, as only the outer pair of shed vortices are detectable in the far wake. Note that since the oscillation amplitude is limited in this case, the formation of the trailing vortex is interrupted by a collision with a wall, and hence, the trailing vortex is much weaker when compared to higher S� . Based on the above discussion, vortex shedding patterns can be categorized with respect to U� and S� , as can be seen in Fig. 17. For VIV of a circular cylinder in free-stream (high values of S� ), three patterns are observed: 2S, 2P0 , and 2P. Bringing the wall closer to the cylinder, the interference between the shed vortices from the cylinder and the wall’s boundary layer results in the roll up of a boundary layer vortex. The onset of this single vortex in the wake results in different patterns which are referred to as 2S þ S, 2P0 þ S, and 2P þ S in this study. For even lower values of S� , the interaction of shed vortices and the boundary layer vortex leads to vortex annihilation, introducing new vortex shedding patterns: S, P0 , and P. Schematics of the aforementioned shedding patterns are presented in Fig. 18, and supplementary video files provide direct access to low order representations of the vorticity shedding process. The critical dependency of the nature of the boundary layer vortex and the corresponding shedding pattern observed in the wall boundary layer indicates that vorticity flux contained in the wall boundary layer as well as distribution of the vorticity are critical pa rameters in determining the system’s response and wake patterns. Further investigations are needed to investigate the effects of the boundary layer as well as energy exchange between the wall boundary

Fig. 17. Vortex shedding map as a function of U� and S� . For consistency, the symbols match those in Figs. 7 and 8. The branches are color coded as follows: blue - initial branch, green - upper branch, magenta - lower branch, black - desynchronization branch, and red - impact branch. Refer to Fig. 18 to find the schematics of the vortex shedding patterns.

and the cylinder. 4. Conclusion Synchronized amplitude and PIV measurements were conducted to characterize 1DOF VIV of a elastically mounted circular cylinder in the vicinity of a planar wall boundary. The results indicate that the existence of the wall results in changes to the response of the system and its

Fig. 16. ‘P0’ Vortex shedding for S� ¼ 0:5 and U� ¼ 5:3 at (a) t � ¼ 0, (b) t � ¼ 0:25, (c) t � ¼ 0:75, (d) t � ¼ 1, (e) t � ¼ 1:5. (f) Schematic cylinder position at each time step. 11

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CRediT authorship contribution statement Sina Daneshvar: Methodology, Software, Formal analysis, Writing original draft. Chris Morton: Conceptualization, Investigation, Data curation, Writing - review & editing, Supervision, Funding acquisition, Project administration. Acknowledgements The authors gratefully acknolwedge the Natural Sciences and Engi neering Research Council of Canada (NSERC) and Canada Foundation for Innovation (CFI) for the funding of this work (NSERC Application ID RGPIN-2016-04079 and CFI Project 34707). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.oceaneng.2020.107109. References Bearman, P.W., 1984. Vortex shedding from oscillating bluff bodies. Annu. Rev. Fluid Mech. 16 (1), 195–222. Brika, D., Laneville, A., 1993. Vortex-induced vibrations of a long flexible circular cylinder. J. Fluid Mech. 250, 481–508. Cagney, N., Balabani, S., 2013. Wake modes of a cylinder undergoing free streamwise vortex-induced vibrations. J. Fluid Struct. 38, 127–145. Charonko, J.J., Vlachos, P.P., 2013. Estimation of uncertainty bounds for individual particle image velocimetry measurements from cross-correlation peak ratio. Meas. Sci. Technol. 24 (6), 65301. https://doi.org/10.1088/0957-0233/24/6/065301. Chung, M.H., 2016. Transverse vortex-induced vibration of spring-supported circular cylinder translating near a plane wall. Eur. J. Mech. B Fluid 55, 88–103. Delville, J., Ukeiley, L., Cordier, L., Bonnet, J.P., Glauser, M., 1999. Examination of largescale structures in a turbulent plane mixing layer. Part 1. Proper orthogonal decomposition. J. Fluid Mech. 391, 91–122. Feng, C.C., 1968. The Measurement of Vortex Idunced Effects in Flow Past Stationary and Oscillating Circular and D-Section Cylinders, Master’s Thesis. University of British Columbia. Fu, S., Xu, Y., Chen, Y., 2014. Seabed effects on the hydrodynamics of a circular cylinder undergoing vortex-induced vibration at high Reynolds number. J. Waterw. Port, Coast. Ocean Eng. 140 (3), 1–12. https://doi.org/10.1061/(ASCE)WW.19435460.0000241. Govardhan, R., Williamson, C., 2000. Modes of vortex formation and frequency response of a freely vibrating cylinder. J. Fluid Mech. 420, 85–130. Grass, A.J., Raven, P.W.J., Stuart, R.J., Bray, J.A., 1984. The influence of boundary layer velocity gradients and bed proximity on vortex shedding from free spanning pipelines. J. Energy Resour. Technol. 106 (1), 70. Hussain, A.K.M.F., 1986. Coherent structures and turbulence. J. Fluid Mech. 173, 303–356. https://doi.org/10.1017/S0022112086001192. Jauvtis, N., Williamson, C., 2004. The effect of two degrees of freedom on vortex-induced vibration at low mass and dampoing. J. Fluid Mech. 509, 23–62. Jeon, D., Gharib, M., 2001. On circular cylinders undergoing two-degree-of-freedom forced motion. J. Fluid Struct. 16 (15), 533–541. https://doi.org/10.1006/ js.2000.0365. Khalak, A., Williamson, C.H.K., 1997. Investigation of the relative effects of mass and damping in vortex-induced vibration of a circular cylinder. J. Wind Eng. Ind. Aerod. 69–71, 341–350. Khalak, A., Williamson, C., 2006. Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. J. Fluid Struct. 13, 813–851. Klamo, J., Leonard, A., Roshko, A., 2006. The effects of damping on the amplitude and frequency response of a freely vibrating cylinder in cross-flow. J. Fluid Struct. 22, 845–856. Lee, J., Bernitsas, M.M., 2011. High-damping, high-Reynolds VIV tests for energy harnessing using the VIVACE converter. Ocean Eng. 38, 1697–1712. Lei, C., Cheng, L., Kavanagh, K., 1999. Re-examination of the effect of a plane boundary on force and vortex shedding of a circular cylinder. J. Wind Eng. Ind. Aerod. 80 (3), 263–286. https://doi.org/10.1016/S0167-6105(98)00204-9. Z. Li, R. K. Jaiman, B. C. Khoo, Z. Li, R. K. Jaiman, B. C. Khoo, Coupled dynamics of vortex-induced vibration and stationary wall at low Reynolds number Coupled dynamics of vortex-induced vibration and stationary wall at low Reynolds number, Phys. Fluids 093601. doi:10.1063/1.4986410. Li, Z., Yao, W., Yang, K., Jaiman, R.K., Khoo, B.C., 2016. On the vortex-induced oscillations of a freely vibrating cylinder in the vicinity of a stationary plane wall. J. Fluid Struct. 65, 495–526. https://doi.org/10.1016/j.jfluidstructs.2016.07.001. Liberge, E., Hamdouni, A., 2010. Reduced order modelling method via proper orthogonal decomposition (POD) for flow around an oscillating cylinder. J. Fluid Struct. 26 (2), 292–311. https://doi.org/10.1016/J.JFLUIDSTRUCTS.2009.10.006. Manuel, J., Barbosa, D.O., Qu, Y., Metrikine, A.V., Lourens, E.-m., 2017. Vortex-induced vibrations of a freely vibrating cylinder near a plane boundary : experimental

Fig. 18. Vortex shedding patterns schematics. Note that the 2P0 vortex shed ding pattern is similar to 2P pattern. Dashed lines indicate vortices which are annihilated at formation.

synchronization with the wake. Decreasing S� leads to a decrease in the oscillation amplitude. At a critical S� , a physical impact between the cylinder and the wall takes place, resulting in a distinct frequency response. POD analysis has revealed that a small number of modes (between 5 and 6 modes) are dynamically important to describe the wake. A low-order representation of the coherent motions in the wake are captured in order to describe the vortex shedding process together with synchronized motion of the cylinder. In total, six distinct vortex shedding patterns have been observed for the range of 3 � U� � 12 and 0:3 � S� � 8. The vortex shedding pattern for S� ¼ 3 resembles that in free-stream: 2S and 2P0 ð2PÞ patterns in the initial and upper (lower) branches, respectively. For S� ¼ 1, shed vortices from the cylinder influence the planar wall’s boundary layer such that it rolls up into a single vortex on each cylinder oscillation cycle, resulting in different vortex shedding patterns, referred to as 2Sþ S and 2P þ S. For S� ¼ 0:5, the interaction between the boundary layer vortex and the counter-rotating shed vortex from the cylinder results in vorticity annihilation via mixing. As a result of annihilation, for each oscillation cycle, a single vortex or pair of vortices are observed in the wake, which are referred to as S and P vortex shedding patterns. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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