Experimental investigation on the flow induced vibration of an equilateral triangle prism in water

Applied Ocean Research 61 (2016) 92–100

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Experimental investigation on the flow induced vibration of an equilateral triangle prism in water Jun Zhang, Guobin Xu, Fang Liu, Jijian Lian ∗ , Xiang Yan State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China

a r t i c l e

i n f o

Article history: Received 17 July 2015 Received in revised form 10 August 2016 Accepted 12 August 2016 Keywords: Equilateral triangle prism Flow induced vibration Vortex induced vibration Transition branch from VIV to galloping Galloping

a b s t r a c t A series of flow induced vibration (FIV) experiments for an equilateral triangle prism elastically mounted in a water channel are performed with different system stiffness at constant damping and mass. An amplitude variation coefficient is proposed to describe FIV stationarity in the present study. The FIV of the prism can be divided into three primary regions based on the amplitude and frequency responses, which are the vortex induced vibration (VIV) branch, the transition branch from VIV to galloping, and the galloping branch. The transition branch occurs at the reduced velocity in the range of 7.8 < Ur = U/(fn,air ·D) < 10.4, accompanied with a relatively rapid increase in amplitude and a precipitous drop in frequency and vibration stationarity. In addition, the reduced velocity where the transition region is initiated is independent of the system stiffness. The maximum amplitude reaches 3.17 D in the galloping branch. The ratio of the response frequency to the natural frequency of the prism in air remains locked to approximately 0.65 throughout the fully developed galloping branch. Large amplitude responses in an infinite range of flow velocities, excellent vibration stationarity and steady vibration frequencies, which are characteristics of the galloping of the prism, have a positive impact on improving energy conversion. © 2016 Published by Elsevier Ltd.

1. Introduction Flow induced vibration (FIV), which is commonly encountered in engineering fields, such as long-spanned or high-rise buildings, has been a subject of interest for the past several decades. Vortex induced vibration (VIV) and galloping are the most common FIV phenomena. VIV occurs due to the alternating shedding of vortices from either side of the bluff cylinder [1]. The alternating shedding of vortices results in periodic changes in the pressure distribution on the cylinder surface. Hence, the VIV of bodies are self-excited motions. However, galloping, characterized as the large amplitude and low frequency oscillation normal to the flow, is caused by forces acting on a cylinder as it is subjected to periodic variations in the angle of attack of the flow [2]. Usually the transverse motion of a body gives rise to the periodically varying angle of attack. The VIV of an elastically mounted circular cylinder has been studied extensively by Sarpkaya [3], Bearman [4], Williamson and Govardhan [5], Khalak and Williamson [6] and others. Sarpkaya [3] and Williamson [5] reported that the oscillation amplitude of a cylinder in VIV is self-limiting, as excessively large vibrations break

∗ Corresponding author at: Address: School of Civil Engineering, Tianjin University, No.92, Weijin Road, Nankai District, Tianjin, 300072, China. Tel./ E-mail address: tju [email protected] (J. Lian). http://dx.doi.org/10.1016/j.apor.2016.08.002 0141-1187/© 2016 Published by Elsevier Ltd.

the wake mode and vortex pattern. They also indicated that oscillations with large amplitude of a cylinder in VIV occur in a certain range of reduced velocities Ur = U/(fn ·D), where fn is the natural frequency of the vibration system, and D is the diameter of the cylinder. This range of reduced velocities is defined as the synchronization range or lock-in range. The parameters, including mass ratio m* (the ratio of oscillating mass to displaced fluid mass), system damping ratio  and combined-parameter m* , have a great impact on VIV [6]. The VIV regimes for an elastically mounted cylinder with low m*  are classified into four branches on the basis of amplitude response: the initial branch, upper branch, lower branch and desynchronization branch [7]. Williamson and Roshoko [8] summarized the experimental results and classified three types of vortex shedding patterns, 2S, 2P, and P + S, by the number of vortices shed per vibration cycle of cylinder. The different wake patterns are associated with specific VIV regimes. 2S and 2P patterns represent two single and two pairs of vortices shedding from the body surface per cycle, respectively. It is well known that flow around an elastically mounted circular cylinder can induce periodic vibration of the cylinder. However, noncircular section bodies with symmetric sharp edges, such as rectangular section prisms as well as isosceles or equilateral triangular cylinders, could be more prone to experiencing FIV because the symmetric apex angles in their geometries appear to be helpful

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for the flow to separate from the body surface. In the last decades, many researchers have devoted great effort to study the FIV features of square or rectangular section bodies [9–17]. Both VIV and galloping of rectangular section prisms with a low mass ratio were observed by Nemes et al. [16] and Barrero and Fernandez [17]. Compared with the studies concerning the FIV of rectangular cylinders, there are only limited investigations related to triangular cylinders immersed in a uniform flow. Iungo and Buresti [18] examined the influences of wind direction and cross-sectional geometry on the flow around a triangular cylinder. A series of triangular prisms with different triangular cross-section apex angles and aspect ratios were tested in a wind tunnel by varying the wind direction. They demonstrated that the fluctuating cross-flow forces connected with vortex shedding vary significantly with incident flow orientation and aspect ratio. Similar studies were also conducted by Camarri et al. [19] and Johansson et al. [20]. Furthermore, Alonso et al. [2,21,22] based on the Glauert-Den Hartog criterion and dynamical tests, comprehensively studied the transverse galloping stability of triangular prisms and concluded that the stability to transverse galloping of triangular cross-section cylinders depends on the cross-sectional geometry and the angle of attack. In addition, FIV perpendicular to the flow of an elastically mounted equilateral triangular prism was numerically researched by Lin Ding et al. [23]. The results showed that the FIV of an elastically mounted equilateral triangular cylinder can be divided into the initial and upper branches of VIV, the transition branch from the VIV branch to the galloping branch, and the galloping branch, regarding the amplitude and frequency responses. Feng Xu and Jinping Ou [24] also performed a numerical investigation of the FIV of elastically mounted cylinders with different sections and confirmed that the vibrations of the square and triangle cylinders change from vortex induced vibration to galloping as the frequency ratio fn /fst (the ratio of the natural frequency to the Strouhal frequency) decreases. Contrary to suppressing the FIV usually encountered in engineering fields, Bernitsas et al. [25,26] developed the Vortex Induced Vibration for Aquatic Clean Energy converter (VIVACE). The VIVACE, which utilizes FIV of a cylinder or cylinder array to capture hydrokinetic energy, is an innovative energy harnessing device with a high energy density, low operable flow velocities and low maintenance cost. This device satisfies all of the requirements set by the California Energy Commission and the U.S. DOE [26]. For the VIVACE, enhancement of FIV is required to convert more hydrokinetic energy to mechanical energy and subsequently to electrical energy over a broad velocity range. Therefore, Chang et al. [27], Kim et al. [28] and Park et al. [29] effectively enhanced the FIV of the circular cylinder by altering the cylinder surface roughness with the help of Passive Turbulence Control (PTC). Besides, in order to make the change of spring and damping efficient and accurate, Lee et al. [30] built a device VCK to replace the physical damping and spring of the VIVACE Converter. Sun et al. [31] introduced an Arduino embedded board to control a servomotor instead of Labview, which significantly reduces the controller-induced lag compared to the first generation of VCK . Apart from the cylinder with PTC, the elastically mounted triangular prism can experience strong FIV in an extremely broad range of flow velocities, as has been reported by some researchers. The prism enters into galloping mode when the velocity of the incident flow exceeds a certain critical value, and the response amplitude in galloping increases with increasing flow velocities. Accordingly, the use of the galloping as an alternative to harvest energy from a fluid flow has obvious advantages. However, as far as we know, almost all of the studies on flow induced vibration of triangular prisms were performed in wind tunnels [2,18,21,22] or by means of numerical simulations [19,20,23,24]. The characteristics of FIV of elastically mounted triangle cylinders immersed in the water flow, which is

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the objective of the present study, have not been experimentally investigated to date. In this paper, the motion equation followed by mathematical models for the power extracted from a fluid flow by the elastically mounted equilateral triangle cylinder is derived in Section 2. In Section 3, the experimental apparatus along with the physical model is described. The results of free decay experiments are presented in Section 4. In Section 5, the results of FIV tests are discussed. The conclusions are summarized at the end. 2. Mathematical model For the case of the one-degree-of-freedom cylinder, a classical mass-spring-damper oscillator model has been presented in the literature [26]. It can be assumed that the FIV of the onedegree-of-freedom cylinder approximates the maximum response at resonance. The x-axis is chosen in the free-stream direction and the y-axis in the vertical direction, z-axis is defined as the axis of the cylinder perpendicular to the direction of the free-stream and y-axis. The one-degree-of-freedom equation of vibration is given as mosc y¨ + csystem y˙ + Ky = Fy

(1)

where mosc is the oscillating system mass, y is the displacement perpendicular to the incident flow and the cylinder axis, K is the system spring stiffness, csystem is the damping coefficient of the oscillating system, and Fy is the force exerted by the fluid on the body surface in the y-direction. In this model, the fluid force can be divided into viscous and inviscid components [5,26] as follows: Fy,

= −ma y¨ √ 3 = cy (t)U 2 DL 4

(2)

inviscous

Fy, viscous

(3)

where ma is the added mass (which is equal to added mass coefficient Ca multiplied by displaced fluid mass md ), cy (t) is the lift coefficient,  is the fluid density, U is the flow velocity, D is the characteristic dimension of the prism in the direction transverse to the incident flow, and L is the length of the cylinder. By substituting Eq. (2) and Eq. (3) into Eq. (1), the motion equation can be rewritten as √ 3 (4) cy (t)U 2 DL (mosc + ma ) y¨ + csystem y˙ + Ky = 4 At this stage, the mass of the displaced fluid of an equilateral triangle prism is introduced, which is defined as √ 3 D2 L md = (5) 4 Thus, Eq. (4) can be reduced to (mosc + ma )¨y + csystem y˙ + Ky =

U2 cy (t)md D

(6)

In the present study, the reduced velocity Ur is introduced, which is defined as Ur =

U fn,air D

(7)

where fn,air is the natural frequency of the vibration system in air. The reduced velocity Ur is often used as the x-axis on plots in investigations on the FIV of a circle cylinder because it properly collapses many test cases onto the same plot. The mass ratio of an equilateral triangle prism, m* , is defined as the total oscillating mass of the prism in FIV mosc divided by the displaced fluid mass md . m∗ =

mosc md

(8)

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Table 1 Main components of the vibration system. Part No.

Description

1 2 3

Equilateral triangle prism Endplates Side strut used for connecting the prism and linear bearing Linear bearing playing a role in limiting the system to vibration in the vertical direction Tension spring Displacement transducer Steel frame used for fixing the linear bearing, tension spring and displacement transducer

4 5 6 7

The damping ratio of the vibration system is defined as system =

c

system

2

(9)

mosc K

Eq. (6) can be nondimensionalized by introducing the dimensionless variables, including Ur , m* , Ca , and  system . The natural frequency of vibration system in vacuum or air is used. After substituting the dimensionless values into Eq (6), the motion equation becomes





(m∗ + Ca ) · y¨ ∗ + 4system y˙ ∗ + 42 y∗ = cy (t)Ur 2

(10)

∗

2 ∗ ¨ where y¨ = y/(Df n,air ), and y =ymax /D. As shown in Eq. (10), the FIV of the prism is affected by the dimensionless parameters including the mass ratio m* and the system damping ratio  system . The dimensionless parameter  system reflects the effects of the mass, damping and stiffness on the FIV of the prism. In the present investigation, a series of FIV tests for an equilateral triangle prism elastically mounted in a water channel are performed with different system stiffness at constant damping and mass to get a preliminary understanding of the FIV characteristics of the prism immersed in the water flow.

3. Experimental setup and physical model

Fig. 1. Equilateral triangle cylinder.

3.1. Experimental apparatus The experiments are conducted in the recirculating water channel at the State Key Laboratory of Hydraulic Engineering Simulation and Safety of Tianjin University. The channel has a main test section of 1 m in width and 1.5 m in depth, and the flow velocity ranges from 0 to approximately 1.6 m/s, corresponding to the Reynolds Number (Re = U·D/ where  is the kinematic viscosity of water and D = 0.06 m) in the range from 0 to 8.2 × 104 . The equilateral triangle prism in the present study is made of polymethyl methacrylate. Endplates with dimensions of 0.13 × 0.08 × 0.01 m are attached on the both ends of the prism to induce parallel shedding at high Reynolds number tests [33–35] as depicted in Fig. 1. Fig. 2 shows the vibration system with Parts 1–7 identified and listed in Table 1. The test prism is constrained to move in the vertical direction (perpendicular to the flow direction). The side struts are made of duralumin plate, and the mass of each side strut is 0.708 kg. Besides, the upper and the lower ends of the spring are fixed on the vertical points to keep the spring in vertical condition. The spring is always under tension in the test process. The vertical displacement y(t) is measured using a non-contact magnetic induction displacement transducer with the measuring range from 0 to 800 mm. The accuracy of the displacement transducer is within ±0.05%, giving a displacement measurement resolution of 0.001%FS. The flow velocity measurement is performed using the Pitot tube with a differential pressure transmitter, which is positioned 1 m in front of the prism. The accuracy of the

differential pressure transmitter is within ±0.1% of the 6 kPa linear range available, and its resolution is within 0.01% FS. All the data are recorded for a time interval of 30 s at a 50 Hz sampling rate. 3.2. Model parameters The cross section of the triangular prism in this investigation is a regular triangle with side length D = 0.06 m, and one of the side edges is placed in the direction transverse to the incident flow, as shown in Fig. 3, because Alonso and Meseguer [2] noted that triangular cross-section cylinders are prone to galloping instability when the base of the triangle is facing the flow. To comprehensively study the characteristics of the flow induced vibration (FIV) of an equilateral triangle cylinder elastically mounted in a water channel, a series of FIV experiments with different system spring stiffness are performed at a constant damping coefficient csystem (not taking account into the damping coefficient of the springs). The system spring stiffness of the oscillating system varies from 300 to 800 N/m with an interval of 100 N/m. The springs are customized according to the stiffness, and the spring stiffness is verified by mounting iron weight before the free decay experiment. Fig. 4 presents the validation result for K = 100 N/m. All the validation results indicate that the springs are linear enough, and the measured spring stiffness of is very close to calibration value. The maximum error is as low as ± 0.5%. Other model parameters are given in Table 2.

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Fig. 2. Vibration system with an equilateral triangle cylinder.

Table 2 Physical model parameters. Side length of the cross section D

Length of the prism L

Aspect ratio of the prism ␣=L/D

Displaced fluid mass md

Oscillating mass mosc

Mass ratio m*

0.06m

0.5m

8.33

0.779 kg

3.283 kg

4.21

of the vibration system. The spring stiffness K of the vibration system varies from 300 to 800 N/m. For each K case, free decay experiments are performed four times for the respective cases of in air. The damping ratio and the natural frequency are determined using a simple averaging method. Depending on the test results, the damping  air can be determined using the logarithmic decrement method, which is expressed as system =

ln  1 A = ln( i ) 2 2 Ai+1

(11)

where Ai denotes the amplitude of the ith peak. The expression for the damping coefficient of the vibration system can be obtained from Eq. (9), as mentioned in Section 2. Fig. 3. Simple schematic of the equilateral triangle prism elastically mounted in the channel.

csystem = 2system

4. Free decay experiment Free decay experiments with different spring stiffness values in air are conducted to obtain the damping ratio and natural frequency

mosc K

(12)

The natural frequency of the vibration system is extracted using FFT for the time histories of the displacement of free decay test. In addition to using the free decay test, the natural frequency of the vibration system in air can also be calculated by Eq (13). fn,air

Fig. 4. The validation of the spring for K = 100 N/m.



1 = 2



K mosc

(13)

In the present study, we calculate the natural frequencies of the vibration system in air using Eq. (13). To verify the reliability of the experiment apparatus, the natural frequency measured by the free decay experiment is compared with the calculated value, which is presented in Fig. 5. As shown in Fig. 5, the measured natural frequencies of the vibration systems in air are in good agreement with the calculated values, and the maximum error is as low as 4.7%, which indicates that the text setup and method are reliable. The damping coefficient csystem for all K along with a trend curve are plotted in Fig. 6. As shown in Fig. 6, csystem for all K remain approximately constant with the maximum relative error of ±1.7%.

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Fig. 5. The calculated and measured natural frequencies of the vibration systems in air.

Fig. 7. Amplitude ratio and frequency ratio versus reduced velocity Ur for different  system .

Fig. 6. Damping coefficient csystem for all K.

Table 3 Free decay test results. K (N/m)

fn,air (Hz)

csystem (N s/m)

300 400 500 600 700 800

1.468 1.673 1.883 2.090 2.273 2.440

8.147 8.380 8.387 8.354 8.324 8.159

The experiment results are listed in Table 3. Note that the measured natural frequency is adopted in the following discussions in this paper. 5. Results and discussion of the FIV experiments To investigate the characteristics of the FIV of an equilateral triangle cylinder elastically mounted in a water channel, a series of FIV experiments for the prism with different system spring stiffness are performed at velocities ranging from 0 to1.6 m/s in the present study while the system damping coefficient and the oscillating mass remain constant, and the variation of the system spring stiffness varies from 300 to 800 N/m. The FIV response of the prism, which is influenced by extensive vibration system parameters, including mass ratio, damping and stiffness, has a large impact on the energy output. In the present investigation, amplitude, frequency and amplitude variation coefficient are measured or calculated to characterize FIV. Besides, the potential advantages of the equilateral triangle prism as an alternative oscillator is discussed based on the FIV characteristics.

Fig. 8. Amplitude and frequency ratios versus Ur of the circular cylinder reported by Raghavan [38].

5.1. Amplitude and frequency responses The amplitude ratio A∗ = A/D and the frequency ratio f* = fosc /fn,air versus reduced velocity Ur = U/(fn,air ·D) for all K cases are plotted in Fig. 7(a) and (b), respectively. Note that the maximum amplitude ratio A* = A/D is the average of all of the peaks of oscillation over 30 s, and the response frequency of the prism is extracted using FFT for the time histories of displacement. Besides, due to the overlap of symbols of f* for all K cases, a trend curve for f* is given in Fig. 7(b). In order to discuss the difference of the FIV characteristics between an equilateral triangle prism and a smooth circular cylinder, the typical response of a smooth circular cylinder with the maximum A∗ = 1.90 reported by Raghavan [38] are presented in Fig. 8. Note that the value of 1.90 is the maximum response amplitude ratio of a smooth circular cylinder obtained from FIV experiments to date. Fig. 7 shows that, for each system spring stiffness K, the amplitude ratio A* of the prism monotonically increases with increasing reduced velocity Ur , and the frequency ratio f* is less than 1 throughout the test Ur range revealing that the oscillation frequency is lower than the natural frequency of the vibration sys-

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tem in air. As Ur increases up to 13.86, the amplitude ratio A* still exhibits an increasing trend. The amplitude and frequency responses indicate that the FIV of the prism can be divided into three primary regions: the vortex induced vibration (VIV) branch, the transition branch from VIV to galloping, and the galloping branch. Compared with the typical response of the circular cylinder consisting of an initial branch followed by an upper branch and ending in lower branch as shown in Fig. 8, for FIV of the equilateral triangle cylinder, the transition branch from VIV to galloping and the galloping branch follow the VIV branch thus expanding dramatically the FIV range. This FIV response pattern is consistent with results of the equilateral triangle cylinder reported by Lin Ding et al. [23] and is similar to the observations in previous studies related to the square triangle [16,36,37]. The characteristics of the amplitude and frequency responses for the equilateral triangle prism in each branch are summarized as follows: (1) VIV. At reduced velocity in the range of 4.32 ≤ Ur <6, the amplitude ratio A* and the frequency ratio f* are small and increase rapidly with Ur indicating the oscillation response of the prism is in the VIV initial branch. For Ur in the range from 6 to 7.8, A* for each K case continues growing at a slower pace, and f* remains approximately constant, which indicate that the prism goes into the VIV upper branch. A* in the upper branch is not constant; on the contrary, it increases with increasing Ur , as shown in Fig. 7(a). The Ur range of the VIV branch for the prism is approximately from 4.32 to 7.8. VIV of the prism is driven by the imposed lift on the body from vortex shedding [1]. (2) Transition from VIV to galloping. For the reduced velocity Ur in the range from 7.8 to 10.4, the amplitude ratio A* for each K case displays a relatively rapid increase with Ur accompanied by a sharp decline in the frequency ratio f* indicating the end of the VIV branch. The prism goes into the transition branch from VIV to galloping. Note that the reduced velocity where the transition is initiated is independent of K, as shown in Fig. 7. The driving mechanisms of oscillatory lift due to vortex shedding and instability inducing oscillatory lift due to shear layer motion coexist in this transition region [16,17,23,27,29]. The reduced velocity Ur along with the corresponding fluid velocity U and Reynolds number Re for the commencement and the end of the transition branch are listed in Table 4. (3) Galloping. For reduced velocity Ur > 10.4, the amplitude ratio A* continues to increase with Ur while the frequency ratio f* levels off at low values. And A* curve increases more gradually compared with the transition branch from VIV to galloping. The prism goes into fully developed galloping mode. The maximum A* value of 3.17 is achieved at Ur = 13.40 for K = 300 N/m. Note that A* = 3.17 is the maximum value limited by the conditions in present experiments. In addition, the maximum A* = 3.17 for the equilateral triangular prism is much higher than the maximum A* = 1.90 for a single circular cylinder as shown in Fig. 8. f* appears approximately equal to 0.65 throughout the fully developed galloping branch. Accordingly, the frequency response of the equilateral triangle prism elastically mounted in water can be predicted based on the system natural frequency in air. Galloping oscillation of the prism is mainly driven by lift instability, while the effect of vortex shedding on the motion in this region is negligible [23]. In this section, three typical sets of frequency spectra for the cases of K = 300, 400 and 500 N/m are presented in Figs. 9–11, respectively. Note that each set of Fourier transform magnitudes is divided by a fixed value, and then the spectra for each K case are superposed onto a vertical Ur axis. The evolution of the dom-

Fig. 9. Frequency spectra for K = 300 N/m (fn,air = 1.468).

inant frequency and the second dominant frequency are lined out in each set spectrum plot, along with the natural frequency. Other cases have similar frequency spectral characteristics. Figs. 9–11 show that the frequency spectra present different characteristics in three branches. (1) VIV. As seen in Figs. 9–11, when the FIV of the prism starts, the frequency peaks are wide band, indicating the oscillation in the VIV initial branch is quasi-periodic. For Ur in the range of 6 < Ur < 7.8, the frequency band becomes narrow, and the dominant frequency is evident, which indicates that the prism goes into the VIV upper branch with harmonic oscillations. (2) Transition from VIV to galloping. For each K case at Ur in the range of 7.8 < Ur < 10.4, there is at least one frequency spectrum displaying two dominate frequencies. This behavior occurs because both of the forcing mechanisms of vortex shedding and lift instability exist to drive the prism oscillation in the transition region from VIV to galloping [23]. Correspondingly, one of the peaks is for VIV, and the other is for galloping. (3) Galloping. Figs. 9–11 show that oscillation response is narrow band and the dominant frequency is obvious for Ur >10.4. In other words, the oscillation of the prism in the galloping branch is periodic. In addition, the spectra in the galloping region present small peaks (probably due to vortex shedding) at the second and third harmonic components of the oscillation frequency, which implies that the frequency of the vortex shedding is much higher than the oscillation dominant frequency of the prism. This supposition is supported by Lin Ding [23] who reported that as many as 10 vortices (2T + 2P where 2T pattern represent three vortices is shed during each half cycle) are shed from an equilateral triangular prism per cycle of galloping oscil-

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Table 4 Ur U and Re for the commencement and the end of the transition branch. K (N/m)

300 400 500 600 700 800

the commencement of the transition branch

the end of the transition branch

Ur

U

Re

Ur

U

Re

7.88 8.90 8.98 8.51 8.37 8.30

0.694 0.894 1.015 1.067 1.142 1.215

36544 47083 53451 56208 60136 63990

9.31 10.31 9.61 9.07 8.83 8.84

0.816 1.035 1.086 1.138 1.205 1.294

42964 54540 57186 59926 63451 68143

Fig. 10. Frequency spectra for K = 400 N/m (fn,air = 1.673).

lation. It could be inferred that the forcing mechanism of lift instability plays a dominant role in driving the prism galloping oscillation. 5.2. Displacement time history and vibration stationarity Typical examples of the time histories of displacement for the case of the system spring stiffness K = 300 N/m are shown in Fig. 12. Other cases have similar response characteristics. Fig. 12(a) shows that the prism experiences quasi-periodic oscillation at the onset of VIV (Ur = 4.32). At a reduced velocity of Ur = 6.27 (see Fig. 12(b)), the oscillation in the upper branch is steadier than that in the initial branch (see Fig. 12(a)). Fig. 12(c) depicts the oscillation in the transition branch from VIV to galloping oscillation (Ur = 9.31). The oscillation response in this transition region is less steady due to the coexistence of the driving mechanisms of vortex shedding and the lift instability [16,17,23,27,29]. As shown in Fig. 12(d), the prism goes into fully developed galloping at Ur =10.21 with very steady oscillation.

Fig. 11. Frequency spectra for K = 500 N/m (fn,air = 1.883).

Stationarity of the FIV of the prism, which is an important property that describes the prism dynamics, is discussed in the present study. Furthermore, excellent vibration stationarity of FIV is important for energy extraction from a fluid flow. Vibration stationarity can be represented as the variation of peaks of the time history of displacement in a certain period in which the flow velocity U is constant. Fig. 12 shows that peaks in the time history of displacement always vary at the same U. Hence, to quantitatively describe vibration stationarity, the amplitude variation coefficient CVA is proposed in this section, which is defined as

  N

2  1 Ai − A N

CVA =

A A

=

i=1

(14) A

¯ where  A denotes the standard deviation of peak amplitudes, Ais the average of peak amplitudes, N is the number of peak amplitudes, and Ai is the amplitude of the ith peak.

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Fig. 12. Typical time histories of the displacement for K = 300 N/m.

tion in the transition region is severe, and the modulation is a manifestation of the transition from VIV to galloping. (3) For Ur > 10.4, CVA levels off at low values indicating the prism maintains steady oscillation in fully developed galloping branch.

Fig. 13. Amplitude variation coefficient and its trend curve versus Ur .

The physical interpretation of the amplitude variation coefficient CVA is that higher CVA signifies poorer vibration stationarity. The amplitude variation coefficient versus reduced velocity is plotted in Fig. 13. Similar to the frequency ratio f* shown in Fig. 7(b), the symbols of CVA for all K cases nearly collapses. Thus, a curve for describing the variation trend of CVA with Ur is also plotted in Fig. 13. As shown in Fig. 13, the prism in different branches experiences FIV with different stationarity:

Note that all of the observations from the trend curve for CVA are supported by the time histories of displacement shown in Fig. 12, i.e., using an amplitude variation coefficient to describe the vibration stationarity of the prism in FIV is feasible and effective. The FIV test results show that the vibration system with an equilateral triangle prism on springs in water can maintain a significant FIV response over an extremely broad range of velocities. Contrary to the vortex induced vibration phenomenon of a smooth circle cylinder, where significant oscillation occurs for a narrow range of flow velocities and with limited oscillation amplitudes, the galloping of an equilateral triangle prism on springs develops in an infinite range of flow velocities, without a self-limited response beyond the critical flow velocity. For extracting energy from a fluid flow, this behavior is clearly advantageous because, if the galloping oscillation appears at moderate flow velocities, the device could be oscillating significantly and thus harvest energy from low to high flow velocities. In addition, an approximately constant response frequency and excellent vibration stationarity of the prism in the galloping region can significantly improve the electricity quality. 6. Conclusions

(1) For the reduced velocity Ur < 6, the CVA is relatively high, indicating the vibration stationarity in the VIV initial branch is poor. However, CVA for Ur in the range of 6 < Ur < 7.8 is lower and exhibits an approximately constant trend, which means the oscillation in the VIV upper branch is steady. (2) The CVA in the transition branch from VIV to galloping increases significantly at Ur in the range of 7.8 < Ur < 9.4 and then decreases rapidly with increasing reduced velocity until Ur ≈ 10.4. This behavior indicates that the amplitude modula-

The characteristics of the FIV of an equilateral triangle prism elastically mounted in a water channel were studied experimentally at a mass ratio m* = 4.2 for the system spring stiffness varying from 300 to 800 N/m. The motion equation of an elastically mounted equilateral triangle prism is deduced. In addition, an amplitude variation coefficient was proposed to describe FIV stationarity of the prism in the present study. The responses of amplitude and frequency of the prism were discussed in the paper,

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along with the advantages of the energy conversion for the oscillating prism. The following conclusions can be drawn from the results presented in this paper. (1) The amplitude ratio A* of the prism monotonically increases with increasing reduced velocity Ur . The FIV response of the prism can be divided into three major regions based on amplitude and frequency responses: the VIV branch (including the initial and upper branches), the transition branch from VIV to galloping, and the galloping branch. The transition branch occurs at the reduced velocity in the range of 7.8 < Ur < 10.4 accompanied with a relatively rapid increase in amplitude and a precipitous drop in frequency and vibration stationarity. The reduced velocity Ur at which the transition branch is initiated is independent of the system spring stiffness. The maximum amplitude ratio A* reaches 3.17 at Ur = 13.40 in the galloping branch. In addition, the frequency spectra in the transition branch exhibit two dominant peaks due to the coexistence of the driving mechanisms of vortex shedding and lift instability, while the FIV in the galloping region can be regarded as harmonic oscillations. The frequency ratio f * remains locked to approximately 0.65 in the fully developed galloping branch. (2) The time histories of displacement and the amplitude variation coefficient indicate that the prism undergoes quasi-periodic oscillations in the VIV initial branch and the transition branch from VIV to galloping, while it can maintain steady oscillations in the VIV upper branch and the galloping branch. (3) The characteristics of the galloping of the equilateral triangle cylinder, including significant amplitude responses in an infinite range of flow velocities, excellent vibration stationarity and steady vibration frequencies are beneficial for improving energy conversion. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 51209158), the Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51021004), and the National Natural Science Foundation of China (Grant No. 51179119). All workers from the State Key Laboratory of Hydraulic Engineering Simulation and Safety of Tianjin University are acknowledged. The authors are also grateful for the assistance of the anonymous reviewers. References [1] R.D. Blevins, Flow-Induced Vibration, 3th ed., Van Nostrand Reinhold, New York, 1990. [2] G. Alonso, J. Meseguer, A parametric study of the galloping stability of two dimensional triangular cross-section bodies, J. Wind Eng. Ind. Aerodyn. 94 (2006) 241–253. [3] T. Sarpkaya, A critical review of the intrinsic nature of vortex-induced vibrations, J. Fluid Struct. 19 (2004) 389–447. [4] P.W. Bearman, Circular cylinder wakes and vortex-induced vibrations, J. Fluid Struct. 27 (2011) 648–658. [5] C.H.K. Williamson, R. Govardhan, Vortex-induced vibrations, Annu. Rev. Fluid Mech. 36 (2004) 413–455. [6] A. Khalak, C.H.K. Williamson, Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping, J. Fluids Struct. 13 (1999) 813–851. [7] C.H.K. Williamsona, R. Govardhanb, A brief review of recent results in vortex-induced vibrations, J. Wind Eng. Ind. Aerodyn. 96 (2008) 713–735. [8] C.H.K. Williamson, A. Roshko, Vortex formation in the wake of an oscillating cylinder, J. Fluid Struct. 2 (1988) 355–381.

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