Experimental investigation of flow induced motion and energy conversion for triangular prism

Journal Pre-proof Experimental investigation of flow induced motion and energy conversion for triangular prism Nan Shao, Jijian Lian, Fang Liu, Xiang Yan, Peiyao Li PII:

S0360-5442(19)32560-5

DOI:

https://doi.org/10.1016/j.energy.2019.116865

Reference:

EGY 116865

To appear in:

Energy

Received Date: 18 March 2019 Revised Date:

31 July 2019

Accepted Date: 26 December 2019

Please cite this article as: Shao N, Lian J, Liu F, Yan X, Li P, Experimental investigation of flow induced motion and energy conversion for triangular prism, Energy (2020), doi: https://doi.org/10.1016/ j.energy.2019.116865. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Experimental Investigation of Flow Induced Motion and Energy Conversion for Triangular Prism Nan Shao1, Jijian Lian1, Fang Liu1, Xiang Yan1*, Peiyao Li1,2 (1. State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, 135 Yaguan Road, Haihe Education Park, Tianjin, 30035, China. 2. Power China Beijing Engineering Corporation Limited, 1 Dingfuzhuangwest Road, Chao Yang District, Beijing, 100024, China.)

Abstract Previous studies proved that the triangular prism would go into galloping branch with high amplitude and low frequency. In order to evaluate the energy conversion capacity, a series experiments of flow induced motion (FIM) for triangular prism with physical springs are conducted in the Reynolds number range of 29,559≤Re≤119,376 by varying load resistance, stiffness, mass ratio and aspect ratio. Selective aspect ratios are applied to enhance the hydrokinetic energy captured by the generator. A physical model is used to control the change of load resistance and spring stiffness for a fast and precise oscillator modeling. The analysis of oscillation responses and energy conversion are carried out based on the statistical evaluation of displacement time-history and voltage signals. The effects of system stiffness, mass ratio, aspect ratio and load resistance on the active power (Pharn) of the triangular prism are presented and discussed. The main conclusions can be summarized as follows: (1) The best branch of the triangular prism energy conversion is galloping branch. (2) In the tests, the maximum active power Pharn=23.37 W and the corresponding efficiency ηharn=5.21 %. The maximum energy conversion efficiency ηharn=6.17 % with the corresponding active power Pharn=2.94 W. (3) With the increase of the stiffness (K) and the reduce of the mass ratio (m*), the Pharn rises up. (4) The higher aspect ratio (α) can be easier self-excited to galloping from the vortex induced vibration (VIV) but has a negative influence on the Pharn of the galloping branch. Keywords Flow induced motion; Triangular prism; Vortex induced vibration; Galloping;

1. Introduction The flow of fluids around structures generates alternating fluid forces applied on the structures’ surface, which cause the structures to vibrate. The reciprocating motion of the structures will influence the flow state in turn, thereby change the fluid force acting on the structures. This phenomenon is called flow induced motion (FIM) [1], which is generally known as destructive phenomenon for many engineering structures such as bridges, offshore risers, vehicle system, offshore platform and overhead transmission lines and so on. Therefore, many previous studies concentrated on the suppression [2-3] of FIM. With the further researches on FIM [4-6], scholars gradually pay more attention on harnessing the energy of FIM. The renewable energy harvesting based on FIM phenomena can be classified into several groups which include flutter, fluttering-autorotation, buffeting, VIV and Galloping [7]. Among them, the energy conversion capacity of VIV upper branch and galloping branch perform better. Galloping is one of the well-known FIMs and hardly observed for isolated smooth circular cylinder but easily observed for non-circular prisms such as rectangular prism [8-9], triangular prism [10-13] and passive turbulence control (PTC) circular cylinder [14] etc. In the study of galloping, the phenomenon of hard galloping (HG1, HG2) and soft galloping (SG) were first discovered for PTC circular cylinder by Park [15] of the University of Michigan. Soft galloping was defined as self-initiated galloping, while hard galloping required an externally imposed initial threshold amplitude. As the continuous improvement of FIM, two types galloping of soft galloping (SG) and hard galloping (HG) were also observed on triangular prism [16-17] and T-section prism [18] by Tianjin University. Lian et al. [16] further proposed the parameters variation on the evolution of hard galloping and soft galloping by a series of experiments of triangular prisms. The linking of VIV to HG and SG responses for triangular prism are shown in Fig. 1. In the Fig. 1(a), the soft galloping can be self-excited by the VIV. In the Fig. 1(b), the triangular prism was applied a threshold initial displacement and dramatically transitioned from VIV to hard galloping, accompanied by a sudden jump in A*.

(a) Evolution processes of Soft Galloping (b) Evolution processes of Hard Galloping Fig. 1. Evolution processes of Hard and Soft Galloping[16] The development of FIM on different section prisms and the evolution of galloping are positive to energy conversion of FIM. So a lot of creative convertors [19-20] extracting FIM energy by using oscillators with different cross sections [21-25] were proposed. The VIVACE [26] proposed by university of Michigan in 2006 is one of the more successful devices. In addition, the virtual damping and spring (Vck) was first proposed in 2011 and resulted in extensive data generation [27], the second generation of Vck was developed and validated in 2015 that made parameters variation more efficient and accurate [28]. With the PTC-cylinder, Vck and VIVACE,

the University of Michigan has achieved fruitful results on energy conversion of FIM. Lee and Bernitsas [29] reported the VIVACE equipped Vck generated hydrokinetic power maximum of 15.85W at 1.11 m/s with single cylinder. As the PTC be found and applied on the VIVACE, the harnessed power climbed to 49.35 W with a power-to-volume density of 341 W/m3 at the velocity of 1.45 m/s [30]. Sun et al [31-32]. analyzed the parameters variation on energy conversion of PTC-cylinder in FIM and conducted a series of experiments to test the converted power of two PTC-cylinder in tandem. The highest efficiency reached 63% of the Betz limit with L/D=2.57 and K=1200N/m in the tests. Furthermore, for four PTC-cylinder, the peak efficiency can achieve 88.6% at flow speed slower than 1.0 m/s, depended on the experiments, the power-to-volume density of 875 W/m3 at 1.45 m/s can be estimated [33]. To the author’s knowledge, the energy conversion of FIM with cylinder and PTC in flow now achieve a lot valuable and systemic results, but the research on FIM and energy conversion of non-circular-section prisms such as triangular prism was not abundant. Zhang et al. [17] of Tianjin University analyzed the oscillation characteristic and energy conversion with different system damping and verified the triangular prisms with high power generation potential by experiment methods. Alonso et al. [10-12] carried out a large number of tests in wind tunnel to investigate the instability of the triangular prism and determined relationship diagram of galloping instability of triangular prism. Iungo and Bruesti [13] further explained the effects of different flow angles and different aspect ratios of triangular prisms in wind tunnel on the FIM responses. Those are some early and basic researches to understand the responses of triangular prism. In recent years, Xu et al. [34] proposed a two-dimensional FIM responses of triangular prism with low Reynolds number (Re=200) by Computational Fluid Dynamics (CFD) to determine the critical transition point between VIV and galloping. Ding et al [35], numerically investigated the FIM responses and the energy conversion of triangular prisms. Until now, there were some researches focused on the oscillation responses and energy conversion of triangular prisms, but the research of influential factors on energy harvesting such as load resistance, stiffness, mass and aspect ratio by experimental methods was sporadic. In order to better understand the energy conversion characteristics of triangular prism, a series of experiments about the following three main points were conducted in this paper. (1) FIM oscillation and energy conversion experiments for triangular prism with different stiffness and load resistance were conducted to study the complete FIM responses and energy conversion rules. (2) Experimental researches on FIM responses of triangular prism were conducted to find out the best branch on the energy utilization. (3) For understanding of the energy conversion characteristics in FIM responses, it is necessary to clarify the variation rules of load resistance, stiffness, mass ratio and section aspect ratio on the energy conversion in each FIM branch of the triangular prism.

2. Experimental methods 2.1 Physical Model All experiments were conducted in one-meter narrow flow channel of recirculating water channel at the State Key Laboratory of Hydraulic Engineering Simulation and Safety (SKL-HESS) of Tianjin University (Fig. 2) with flow velocity of 0.0~1.8 m/s and depth was 1.34 m.

(a) Recirculating water channel system (b) Experimental setup Fig. 2. Physical model system The flow velocity and turbulence were tested by a pitot tube placed 1 m in front of the triangular prisms. The Reynolds numbers of the experiments cover the range of 29,559≤Re≤119,376 (0.376 m/s≤U≤1.517 m/s) in the TrSL3 (20,000
20

0 U=0.62 m/s U=0.84 m/s U=1.22 m/s

40

Oscillation Range

60

Deepth (m)

Deepth (cm)

40

80

80

100

120

120

0.4

0.6

0.8

1.0

Flow Velocity (m/s)

1.2

1.4

Oscillation Range

60

100

0.2

U=0.62 m/s U=0.84 m/s U=1.22 m/s

20

0

10

20

30

40

50

Turbulence (%)

(a) Flow velocity profile (b) Turbulence profile Fig. 3. Incoming flow characteristics In Fig. 4, the test apparatus consists two main parts of the oscillation system and energy conversion system. The apparatus is particularly described in the reference [18].

(a) Test apparatus

(b) Energy conversion system Fig. 4. Physical model

2.2 Test method

(a) Test flow chart

(b) Circuit diagram Fig. 5. Data acquisition system The data of displacement was collected by magnetic induction displacement transducer with the testing range of 0~800 mm, the sensitivity of 0.1 % and the error range of ±0.05 %. For energy conversion tests, output voltage (u) of the generator was collected by data acquisition system of continuous 60 s at 40 Hz sampling frequency, as shown in Fig. 5. While Pharn and ηharn the two main parameters to evaluate the energy conversion capacity were calculated by the following equations. The instantaneous power expression:

P (t ) =

u2 (t ) RL

(1)

where, P(t) is the instantaneous power; u(t) is the instantaneous voltage, and RL is the load resistance. The active power is written as

Pharn

2 1 T 1 T u (t ) = ∫ P ( t )dt = ∫ dt T 0 T 0 RL

where, Pharn is the active power, T is a period of oscillation. The energy conversion efficiency [33] is derived as:

(2)

ηharn % =

Pharn × 100 Pw × BetzLimit

(3)

Where, ηharn is energy conversion efficiency, Betz Limit is the theoretical maximum power that can be extracted from an open flow and is equal to 59.26% (16/27), Pharn is the active power and Pw is the total power in the fluid which is written as

1 Pw = ρU 3 2Amax + D) L 2

(4)

Where, ρ is the water density, U is the incoming flow velocity, Amax is the maximum amplitude in the oscillation period, D is the projection width of the triangular prism in the direction of incoming flow, and L is the prism length. For the time domain, all peak active powers Pharn,max can be extracted and difference coefficient Cv can be calculated as:

Cv = Where,

σP

5

P harn,max

σ P is the square deviation of all peak active powers, Pharn,max is the average of all

peak active powers. 2.3 Variation of System Parameters This section presents methods to adjust system parameters separately including load resistance (RL), stiffness (K), mass ratio (m*) and section aspect ratio (α). 2.3.1 Load resistances

(a) Load resistance diagram (b) Connection resistance diagram Fig. 6. Load resistances The load resistances and connection method are shown in Fig. 6. There are nine load resistance values: 2 Ω, 6 Ω, 11 Ω, 14 Ω, 17 Ω, 20 Ω, 32 Ω, 52 Ω and ∞. All the free decay tests were conducted in air at K=1200 N/m and mo,cal=16.62 kg. The results are listed in Table 1. Table 1 The complete results of free decay test by varying the load resistance values test data

Calculated data

RL (Ω)

ζtotal

fn (Hz)

mosc (kg)

Ctotal (N.s/m)

2

0.358

0.957

33.24

141.53

6

0.272

0.957

33.17

107.53

11

0.228

0.965

32.66

90.14

14

0.211

0.956

33.23

83.41

17

0.184

0.955

33.35

72.74

20

0.164

0.959

33.03

64.83

32

0.139

0.959

33.03

54.95

52

0.127

0.967

32.48

50.21

∞ (O/C)

0.096

0.966

32.56

37.95

Average

-

0.961

32.90

-

2.3.2 Stiffness The components of the variation stiffness system are shown in Fig. 7. The different stiffness cases can be realized by selecting different groups of springs. Additionally, the natural frequencies fn were extracted from the time history of the displacements by fast Fourier transform (FFT) method.

(a) Force diagram; (b) Actual device Fig. 7. Illustration of system stiffness The free decay tests were conducted at the open circuit (O/C) configuration in air. As the system stiffness K varies from 800 N/m to 1600 N/m. The results are listed in Table 2. Table 2 The complete results of free decay test by varying the system stiffness test data

Calculated data

K (N/m)

ζm

fn (Hz)

Ctotal (N.s/m)

mosc (kg)

mo,cal (kg)

800

0.117

0.791

37.69

32.43

16.42

1000

0.106

0.881

38.30

32.63

16.51

1200

0.096

0.961

37.95

32.56

16.62

1400

0.091

1.044

38.85

32.55

16.65

1600

0.087

1.124

39.43

32.10

16.72

Average

ζm

-

38.44

32.45

16.58

2.3.3 Mass ratio In the tests, the variation in mass is achieved by varying additional weight. The results of decay tests are conducted at the open circuit configuration for K=1200 N/m and listed in Table 3. The original mass and mass ratio are calculated by the following expressions.

mo,cal = mosc − madd

(8)

m * = mosc / md

(9)

Where, mo,cal is the original mass; m*is the mass ratio; mosc is the mass of the oscillation system; madd is the mass of the additional weight; md is the displaced mass of the triangular prism. Table 3 The complete results of free decay test by varying the mass of oscillation system test data

Calculated data

madd (kg)

ζm

fn (Hz)

Ctotal (N.s/m)

mosc (kg)

mo,cal (kg)

md (kg)

m*

0.00

0.096

0.961

37.95

32.56

16.62

3.897

8.36

3.91

0.092

0.907

38.76

36.92

16.39

3.897

9.47

7.80

0.089

0.860

39.55

41.08

16.66

3.897

10.54

11.73

0.087

0.819

40.60

45.32

16.96

3.897

11.63

Average

-

-

39.22

-

16.66

2.3.4 Aspect ratio

(a) The physical model and the flow direction to the triangular (b) Different section aspect prism ratios Fig. 8. Triangular prism size diagram In the Fig. 8 (a), the length (l) and the thickness (d) of the prism is 0.9 m and 0.01 m respectively. The triangular prism projection width (D) is 0.1 m and the prism height (H) is 0.1 m. The thickness of the endplate installed at both ends of the prism is 0.01 m. The flow direction with respect to the orientation of the triangular prism is shown in the Fig. 8 (a). In order to research the influence of the aspect ratio (α) of the triangular prism on energy conversion, four triangular prisms with different H (0.15 m, 0.1 m, 0.085 m, 0.05 m) were selected and corresponding aspect ratios (α=H/D) were 1.5, 1.0, 0.85 and 0.5 respectively in Fig. 8 (b). 2.3.5 The system damping The system damping for different stiffness and different load resistance were tested by free decay tests that were performed four times for the respective cases in air. Then ζtotal is calculated by equation (10) and using a simple averaging method. Ctotal is calculated by equation (11). The tests results are listed in Table 4. A ln η 1 (10) ζ total = = ln( i ) 2π 2π Ai +1

Ctotal =2ζ total ⋅ K ⋅ mosc Table 4 The system damping of free decay test by varying stiffness and load resistance

(11)

RL (Ω)

K=800 N/m

K=1000 N/m

K=1200 N/m

K=1400 N/m

K=1600 N/m

ζtotal

Ctatol

ζtotal

Ctatol

ζtotal

Ctatol

ζtotal

Ctatol

ζtotal

Ctatol

2

0.398

128.21

0.380

137.29

0.358

141.53

0.346

147.72

0.335

151.84

6

0.303

97.61

0.285

102.96

0.272

107.53

0.267

113.99

0.258

116.94

11

0.264

85.04

0.241

87.07

0.228

90.14

0.216

92.22

0.209

94.73

14

0.235

75.69

0.219

79.12

0.211

83.41

0.202

86.24

0.194

87.93

17

0.214

68.93

0.192

69.37

0.183

72.34

0.175

74.71

0.168

76.15

20

0.195

62.82

0.176

63.59

0.164

64.83

0.152

64.89

0.146

66.17

32

0.168

54.12

0.152

54.91

0.139

54.95

0.129

55.08

0.122

55.30

52

0.158

50.90

0.139

50.22

0.127

50.21

0.119

50.81

0.112

50.76

∞(O/C)

0.117

37.69

0.106

38.30

0.096

37.95

0.091

38.85

0.087

39.43

3. Results and Discussion The complete FIM responses of triangular prism with different load resistance and different stiffness are firstly analyzed. Subsequently, the active power (Pharn), energy conversion efficiency (ηharn) and quality of electric energy (Cv) for triangular prism in different oscillation branches are compared. The advantageous branch conducive to harvest energy has been identified. Finally, the effects of load resistance, stiffness, mass ratio and aspect ratio on the active power of triangular prism are discussed. 3.1 Oscillation responses Experimental results for amplitude ratio A* (A*=A/D) at different stiffness and different load resistance for triangular prism versus reduced velocity Ur are plotted in Fig. 11. The amplitude A are calculated by averaging the absolute values of the highest values of positive and negative peaks in 60 s test time. The frequency ratio f*=fosc/fn,air of the triangular prism are plotted versus reduced velocity Ur, as shown in Fig. 12. The oscillation frequency fosc is calculated by FFT method from the displacements of time history over the recorded period. In this paper, we conducted the same experimental research method of Lian et al. [16]. As the increases of velocity, the triangular prism experiences VIV initial branch and followed by the VIV-galloping transition branch, and finally enters the galloping branch with the amplitude gradually increases. It can be concluded that the soft galloping is self-excited from VIV. However, if there is an external suppression applied on the prism at Ur=12.25, the prism enters into the VIV lower branch, accompanied by a sudden drop in amplitude, as shown in Fig. 9. Displacement (mm)

300 External Suppression

Displacement

200 100 0 -100 -200 Ur=6

-300

Ur=7.25

Ur=8.5

Ur=9.75

Ur=11

Ur=12.25

Ur=12.25

Ur=10.375

Ur=9.75

Reduced Velocity Ur

Fig. 9. The displacement of different reduced velocities for SG As the flow velocity increases, the prism first undergoes the VIV initial branch then the VIV upper branch and VIV lower branch, however, if a threshold initial displacement is given to push the prism at Ur=11.625~12.875, oscillation is suddenly enhanced and transited from VIV to hard galloping, accompanied by a sudden jump in A*, as shown in Fig. 10.

Displacement (mm)

300 An initial displacement applied

Displacement

200 100 0 -100 -200 -300

Ur=6

Ur=7.25

Ur=8.5

Ur=9.75

Ur=11

Ur=11.625

Ur=11.625

Ur=12.25

Reduced Velocity Ur

Fig. 10. The displacement of different reduced velocities for HG

Ur=12.875

3.1.1 Amplitude response The initial branch of VIV for HG and SG (4.75≤Ur≤7.25) In the branch, the A* grows tardily with flow velocity increases and the higher load resistance values the stronger oscillation, which indicates that as RL decreases, the oscillation becomes more strong regardless the stiffness. Which are consistent with previous experimental results. The VIV upper branch for HG (7.25≤Ur≤11~11.625) The branch can be observed at 1000 N/m≤K≤1600 N/m for 2 Ω≤RL≤20 Ω and K=800 N/m for 2 Ω≤RL≤52 Ω. Compared with the initial branch of VIV, the oscillation of the triangular prism is more stable and the effects of stiffness and load resistance are more obvious in the upper branch. The maximum A* reaches 0.88 (Ur=10.375, K=800 N/m, RL=52 Ω). VIV-galloping transition branch for SG (7.25≤Ur≤11~11.625) The VIV-galloping transition branch which happens for 1000 N/m≤K≤1600 N/m and 32 Ω≤RL≤52 Ω is between VIV initial branch and galloping branch, A* keep growing with a strong uptrend from 0.7 to 2 and the oscillation is self-excited from VIV to galloping. The VIV lower branch for HG and SG (Ur≥11~11.625) For the lower branch, A* drop dramatically and random oscillatory patterns appear as expected. Galloping branch for SG and HG (Ur≥11~11.625) Galloping caused by lift instability is known as a high-amplitude and low-frequency FIM phenomenon. Triangular prism can be subjected to high amplitude galloping oscillation due to the geometric asymmetries with respect to the flow. The maximum A* of the single triangular prism varies from 1.808 (K=800 N/m and RL=11 Ω) at Ur=13.5 (U=1.08 m/s) to 2.515 (K=1400 N/m and RL=52 Ω) at Ur=13.5 (U=1.409 m/s), as shown in Fig. 11. 3.2 K=1000N/m, R L=32Ω, ζtotal =0.152

3.2

K=1000N/m, R L=52Ω, ζtotal =0.139

2.8

K=1200N/m, R L=32Ω, ζtotal =0.139 K=1200N/m, R L=52Ω, ζtotal =0.127

2.8

K=1400N/m, R L=17Ω, ζtotal =0.175

2.4

K=1400N/m, R L=20Ω, ζtotal =0.152

2.4

K=1400N/m, R L=52Ω, ζtotal =0.119

2.0

K=1600N/m, R L=14Ω, ζtotal =0.194

Amplitude Ratio A*

Amplitude Ratio A*

K=1400N/m, R L=32Ω, ζtotal =0.129

K=1600N/m, R L=17Ω, ζtotal =0.168 K=1600N/m, R L=20Ω, ζtotal =0.146

1.6

Soft Galloping Branch

K=1600N/m, R L=32Ω, ζtotal =0.122 K=1600N/m, R L=52Ω, ζtotal =0.112

1.2 VIV Initial Branch

0.8

VIV Lower Branch

K=1400N/m,RL=2Ω, ζtotal =0.346

K=800N/m,RL=6Ω, ζtotal =0.303

K=1400N/m,RL=6Ω, ζtotal =0.267

K=800N/m,RL=11Ω, ζtotal =0.264

K=1400N/m,RL=11Ω, ζtotal =0.21

K=800N/m,RL=20Ω, ζtotal =0.1.95

K=1400N/m,RL=14Ω, ζtotal =0.202

K=800N/m,RL=32Ω, ζtotal =0.1.68

K=1600N/m,RL=2Ω, ζtotal =0.335

K=800N/m,RL=52Ω, ζtotal =0.158

K=1600N/m,RL=6Ω, ζtotal =0.258

K=1000N/m,RL=2Ω, ζtotal =0.380

K=1600N/m,RL=11Ω, ζtotal =0.209

K=1000N/m,RL=6Ω, ζtotal =0.285

2.0

K=1000N/m,RL=11Ω, ζtotal =0.241 K=1000N/m,RL=20Ω, ζtotal =0.176 K=1200N/m,RL=2Ω, ζtotal =0.358

1.6

K=1200N/m,RL=6Ω, ζtotal =0.272 K=1200N/m,RL=11Ω, ζtotal =0.228

1.2

Hard Galloping Branch

K=1200N/m,RL=14Ω, ζtotal =0.211

VIV Upper Branch

VIV Initial Branch

0.8

VIV - Galloping Transition Branch

0.4

K=800N/m,RL=2Ω, ζtotal =0.398

VIV Lower Branch

0.4

0.0 4

5

6

7

8

9

10

Ruduced Velocity Ur

11

12

13

14

0.0 4

5

6

7

8

9

10

11

12

13

14

Ruduced Velocity Ur

(a) Amplitude responses of SG (b) Amplitude responses of HG Fig. 11. Amplitude responses for different stiffness and different load resistance 3.1.2 Frequency response The VIV initial branch for HG and SG (4.75≤Ur≤7.25) The initial branch can also be observed for all stiffness values and load resistance, f* grow with the increase of Ur from 0.4 to 0.8. The VIV upper branch for HG (7.25≤Ur≤11~11.625) For the upper branch of VIV, although the oscillation is gradually stable the system oscillation frequency increase from 0.79 to 1.124 and f* rises from 0.8 to 1.3 with the increase of K.

VIV-galloping transition branch for SG (7.25≤Ur≤11~11.625) For VIV-galloping transition branch, there is a little downtrend of f* and maintain at 0.8 accompanied by a rapid increase in the amplitude of the prism. The VIV lower branch for HG and SG (Ur≥11~11.625) In the lower branch of VIV, it can also be observed that f* increase considerably as the load resistance and the stiffness decrease. Galloping branch for SG and HG (Ur≥11~11.625) In the galloping branch, f* is stable and low, which is an important feature of galloping branch. f* remain close to 0.8 regardless of the load resistance and stiffness, as shown in Fig. 12. 2.4

2.8 2.6

K=1000N/m, RL=52Ω, ζtotal =0.139

2.4

K=1200N/m, RL=52Ω, ζtotal =0.127

2.2

K=1400N/m, RL=32Ω, ζtotal =0.129

Frequency Ratio f *

VIV Lower Branch

K=1400N/m, RL=52Ω, ζtotal =0.119 K=1600N/m, RL=14Ω, ζtotal =0.194

1.4

K=1600N/m, RL=17Ω, ζtotal =0.168 K=1600N/m, RL=20Ω, ζtotal =0.146

1.2

K=1600N/m, RL=32Ω, ζtotal =0.122 K=1600N/m, RL=52Ω, ζtotal =0.112

1.0 0.8

0.2

5

6

7

8

9

10

11

12

13

K=1600N/m,RL=2Ω, ζtotal =0.335 K=1600N/m,RL=6Ω, ζtotal =0.258 K=1600N/m,RL=11Ω, ζtotal =0.209

K=1000N/m,RL=11Ω, ζtotal =0.241

1.8

K=1000N/m,RL=20Ω, ζtotal =0.176

VIV Lower Branch

K=1200N/m,RL=2Ω, ζtotal =0.358

1.6

K=1200N/m,RL=6Ω, ζtotal =0.272 K=1200N/m,RL=11Ω, ζtotal =0.228

1.4

VIV Upper Branch

K=1200N/m,RL=14Ω, ζtotal =0.211

1.2

VIV Initial Branch

0.6 0.4

4

K=1400N/m,RL=14Ω, ζtotal =0.202

K=1000N/m,RL=6Ω, ζtotal =0.285

0.8

Soft Galloping Branch

VIV - Galloping Transition Branch

VIV Initial Branch

0.4

K=1400N/m,RL=11Ω, ζtotal =0.21

K=1000N/m,RL=2Ω, ζtotal =0.380

2.0

1.0

0.6

K=1400N/m,RL=6Ω, ζtotal =0.267

K=800N/m,R L=52Ω, ζtotal =0.158

K=1400N/m, RL=20Ω, ζtotal =0.152

1.6

K=1400N/m,RL=2Ω, ζtotal =0.346

K=800N/m,R L=20Ω, ζtotal =0.1.95 K=800N/m,R L=32Ω, ζtotal =0.1.68

K=1400N/m, RL=17Ω, ζtotal =0.175

1.8

K=800N/m,R L=6Ω, ζtotal =0.303 K=800N/m,R L=11Ω, ζtotal =0.264

K=1200N/m, RL=32Ω, ζtotal =0.139

2.0

Frequency Ratio f *

K=800N/m,R L=2Ω, ζtotal =0.398

K=1000N/m, RL=32Ω, ζtotal =0.152

2.2

14

Hard Galloping Branch 4

5

6

7

8

9

10

11

12

13

14

Ruduced Velocity Ur

Ruduced Velocity Ur

(a) Frequency responses of SG (b) Frequency responses of HG Fig. 12. Frequency responses for different stiffness and different load resistance 3.1.3 Summary With the increase of load resistance and spring stiffness, the triangular prism oscillation gradually changes from VIV to HG and finally SG. The occurrence conditions of oscillation responses for triangular prism are listed in Table 5. Table 5. SG and HG occurrence conditions RL

K=800 N/m

K=1000 N/m

K=1200 N/m

K=1400 N/m

K=1600 N/m

2~6 Ω

VIV

VIV

VIV

VIV

VIV

11 Ω

HG

HG

HG

HG

HG

14 Ω

HG

HG

HG

HG

SG

17 Ω

HG

HG

HG

SG

SG

20 Ω

HG

HG

HG

SG

SG

32~52 Ω

HG

SG

SG

SG

SG

3.2 Advantageous branch 3.2.1 SG (soft galloping) The active power (Pharn), energy conversion efficiency (ηharn), quality of electric energy (Cv) and frequency spectrum of SG are shown in Figs. 13~15. The parameters are listed in Table 6. Table 6 Working condition parameters of SG K 1400 N/m

mosc 32.55 kg

RL 20 Ω

Ctotal -1

64.89 N.s.m

ζtotal

α

fn

0.152

0.85

1.04 Hz

VIV initial branch (4.75≤Ur≤7.25) At the beginning of initial branch of VIV, both Pharn and ηharn are small (Fig. 13), while the Cv of peak active power is high due to the poor oscillation stability (Fig. 14). The basic frequencies in

frequency spectrum of active power are wide (Fig. 15) and the energy is not concentrated. With the increase of Ur, the resonance characteristics are improved and the Pharn and ηharn rise sharply. The Cv is reduced, the frequency band is narrowed, the basic frequency gradually becomes apparent and the electric energy is gradually stabilized. It can be concluded that the energy conversion capacity and quality are not good in the initial branch. VIV-galloping transition branch (7.25≤Ur≤11) The Pharn continue to increase but the Cv of the peak active power decrease. The ηharn grow up slightly at first and then decrease from Ur=8.5, the maximum ηharn=5.27%, as shown in Fig. 13, which illustrates the stability of electrical energy significantly improved in this branch. The basic frequencies of active power are prominent with the frequency band narrow.

8

14 12

6 Galloping

10

VIV Initial VIV-Galloping

Branch

8

Branch

4

Transition

6

Branch VIV Lower

4

Branch

2

Cv

1.6 1.4 1.2 1.0

VIV

Cv

Active Power Pharn (W)

16

Initial

0.8

Branch

VIV-Galloping

VIV

Transition

Lower

Branch

Branch

0.6 0.4 0.2

2 0

1.8

10

Pharn ηharn

18

Energy Conversion Efficiency ηharn (%)

20

4

5

6

7

8

9

10

11

12

13

14

0 15

0.0

Galloping Branch

4

5

6

Reduced velocity Ur

9

10

11

12

13

14

15

Fig. 14. Peak power coefficient (SG) 15

fn.air

fosc

14

13

13

Reducing Reduced

12

Velocity

fosc.third

Reduced Velocity Ur

14

12

Reduced Velocity Ur

8

Reduced velocity Ur

Fig. 13. Active power and Efficiency (SG) 15

7

11

11

fosc.second

10 9 8

fosc

7 6

fn.air

10

Increasing

9 8

Reduced

7

Velocity

6 5

5

4

4 0

1

2 3 4 Frequency Hz

5

0

1

2 3 4 Frequency Hz

5

(a) VIV-Galloping transition branch (b) VIV lower branch Fig. 15. Power spectrum (SG) Galloping branch (Ur≥11) The Pharn increase sharply with the flow velocity increases but the ηharn continue to decrease with a little downtrend. Furthermore, the Cv maintains at 0.07, while the basic frequencies are very

prominent, the frequency band is narrowed and the energy is concentrated. It can be observed that the electrical energy in galloping branch is the best. VIV lower branch (Ur≥11) Both the Pharn and ηharn show a downward trend with an external suppression and the Cv increase significantly from 0.1 to 1. As the basic frequencies are not evident and the frequency band is widened which indicate that the energy is very dispersed. It can be seen that the energy conversion capacity and quality are very poor, which is obviously worse than the other branches. 3.2.2 HG (hard galloping) The active power (Pharn), efficiency (ηharn), quality of electric energy (Cv) and frequency spectrum of hard galloping (HG) branch are shown in Figs. 16~18 for the triangular prism. The working condition parameters are listed in Table 7. Table 7 Working condition parameters of HG K

mosc

1400 N/m

RL

32.81 kg

Ctotal -1

11 Ω

92.22 N.s.m

ζtotal

α

fn

0.216

0.85

1.04 Hz

Differed from SG, there is no VIV-galloping transition branch in HG and only the VIV upper branch exists. For HG, the energy characteristics of the initial branch of VIV, the lower branch of VIV and the galloping branch are similar as those in the SG (Fig. 16). The specific characteristics can be summarized as follows: 24

9

ηharn

8

Active Power Pharn (W)

18 7 15

6

VIV Initial Branch

12

5

Galloping

9

Branch

4

VIV Lower

3

Branch

2

VIV-Upper Branch

6 3

Cv 1.0

0.8 Galloping

Cv

Pharn

Energy Conversion Efficiency ηharn (%)

21

0

1.2

10

0.6

VIV Initial

VIV-Upper

Branch

Branch

Branch

0.4

VIV Lower Branch

0.2

1 4

5

6

7

8

9

10

11

12

13

14

0 15

Reduced velocity Ur

Fig. 16. Active power and Efficiency (HG)

0.0

4

5

6

7

8

9

10

11

12

13

14

Reduced velocity Ur

Fig. 17. Peak power coefficient (HG)

15

15

15

14

14

13

13

fosc

fosc,third

12

Reduced Velocity Ur

12

Reduced Velocity Ur

fosc

fosc,second

11

11 10

fn,air

10

Inceasing 9

Reduced

8

Velocity

7

9 8 7

fn,air

6

6

5

5

4 0

1

2 3 4 Fequency Hz

4

5

0

1

2 3 4 Frequency Hz

5

(a) Complete VIV branch (b) Galloping branch Fig. 18. Power spectrum (HG) VIV upper branch (7.25≤Ur≤11) The Pharn increase slightly with the velocity, but the ηharn reaches the peak value (ηharn 6.07%) at Ur=7.875, as shown in Fig. 16. The Cv maintain at 0.13 and there is no fluctuation with the variation of the flow velocity due to the good resonance characteristics. In addition, the basic frequencies are very distinct, the frequency band is narrowed and the energy is concentrated. 3.2.3 Summary Based on the analysis of section 3.2.1 and section 3.2.2, the characteristics of each branch for the triangular prism are listed in Table 8: Table 8 Characteristics of energy conversion of each branch Branch

VIV Initial

VIV Upper

VIV-Galloping

VIV Lower

Galloping

Ability

Bad

Better

Better

Worse

Best

Quality

Bad

Better

Better

Worse

Best

3.3 Energy conversion 3.3.1 Active power Experimental results for Pharn at different stiffness (K) and load resistance (RL) for triangular prism are plotted versus reduced velocity (Ur) in Fig. 19. The VIV initial branch for HG and SG (4.75≤Ur≤7.25) The Pharn of different K and different RL gradually increase with the growth of Ur. For Ur≥4.75 (U≥0.376 m/s), the energy conversion system starts to output active power and the optimal Pharn of the VIV initial branch is 2.05 W for K=1600 N/m. The VIV upper branch for HG (7.25≤Ur≤11~11.625) The Pharn continue to increase from the initial branch and a steady growth can be observed. The peak Pharn=4.47 W appears at Ur=9.75 for K=1600 N/m and RL=11 Ω.

24

24 K=1000N/m, RL=32Ω, ζtotal =0.152

22

22

K=1000N/m, RL=52Ω, ζtotal =0.139 K=1200N/m, RL=32Ω, ζtotal =0.139

20

20

K=1200N/m, RL=52Ω, ζtotal =0.127 K=1400N/m, RL=17Ω, ζtotal =0.175

Soft Galloping Branch

K=1400N/m, RL=20Ω, ζtotal =0.152 K=1400N/m, RL=32Ω, ζtotal =0.129

16

18

Active Power Pharn (W)

18

Active Power Pharn (W)

20.7W

K=1400N/m, RL=52Ω, ζtotal =0.119 K=1600N/m, RL=14Ω, ζtotal =0.194

14

K=1600N/m, RL=17Ω, ζtotal =0.168 K=1600N/m, RL=20Ω, ζtotal =0.146

12

K=1600N/m, RL=32Ω, ζtotal =0.122

9.99W

K=1600N/m, RL=52Ω, ζtotal =0.112

10 8

VIV - Galloping Transition Branch

VIV Initial Branch

6 4

16

K=800N/m,RL=2Ω, ζtotal =0.398

K=1400N/m,RL=2Ω, ζtotal =0.346

K=800N/m,RL=6Ω, ζtotal =0.303

K=1400N/m,RL=6Ω, ζtotal =0.267

K=800N/m,RL=11Ω, ζtotal =0.264

K=1400N/m,RL=11Ω, ζtotal =0.21

K=800N/m,RL=20Ω, ζtotal =0.1.95

K=1400N/m,RL=14Ω, ζtotal =0.202

K=800N/m,RL=32Ω, ζtotal =0.1.68

K=1600N/m,RL=2Ω, ζtotal =0.335

K=800N/m,RL=52Ω, ζtotal =0.158

K=1600N/m,RL=6Ω, ζtotal =0.258

K=1000N/m,RL=2Ω, ζtotal =0.380

K=1600N/m,RL=11Ω, ζtotal =0.209

K=1000N/m,RL=6Ω, ζtotal =0.285 K=1000N/m,RL=11Ω, ζtotal =0.241

14

K=1000N/m,RL=20Ω, ζtotal =0.176

12

K=1200N/m,RL=2Ω, ζtotal =0.358 K=1200N/m,RL=6Ω, ζtotal =0.272

10

K=1200N/m,RL=11Ω, ζtotal =0.228 K=1200N/m,RL=14Ω, ζtotal =0.211

8 6

2

VIV Lower Branch

0

VIV Upper Branch 4.47W

VIV Initial Branch

4

2.05W

23.37W

Hard Galloping Branch

2

VIV Lower Branch

0 4

5

6

7

8

9

10

Ruduced Velocity Ur

11

12

13

14

4

5

6

7

8

9

10

11

12

13

14

Ruduced Velocity Ur

(a) Active power of SG (b) Active power of HG Fig. 19. Pharn for different stiffness and different load resistance VIV-galloping transition branch for SG (7.25≤Ur≤11~11.625) The oscillation is self-excited from VIV to galloping with high RL and K. The Pharn varies from 2.58W (K=1000 N/m and RL=52 Ω.) to 3.87 W (K=1600 N/m and RL=52 Ω) at Ur=11. And the Pharn varies from 3.87 W (K=1600 N/m and RL=52 Ω.) to 9.99 W (K=1600 N/m and RL=14 Ω) at Ur=11. The lower branch of VIV for HG (Ur≥11~11.625) The Pharn drop sharply to lower than 1W as Ur rises up to 13.5 and the change of Pharn is less obvious. As the growth of Ur, although the oscillation can’t be self-excited from VIV to galloping, the oscillation can directly jump into galloping branch with an external excitation applied on the prism. Galloping branch for SG and HG (Ur≥11~11.625) As the Pharn monotonically increase with Ur and the optimal Pharn=20.7 W appears at RL=14 Ω, K=1600N/m and Ur=13.5 with ηharn=3.88 %. It is noted that Pharn increase sharply with the increase of K and decrease of RL when 11 Ω≤RL≤52 Ω. The maximum Pharn of triangular prism is 23.37W (RL=11 Ω, K=1600N/m, Ur=13.5, U=1.38m/s) which is close to that of PTC circular cylinder Pharn,max=23.54 W (Re=101536, U=1.25m/s) reported by Ding [35]. 3.3.2 Energy conversion efficiency The variation of ηharn versus stiffness (K), load resistance (RL) and reduced velocity (Ur) are plotted in Fig. 20. The ηharn is calculated by equation (3), as shown in Section 2.2. The initial branch of VIV for HG and SG (4.75≤Ur≤7.25) The efficiencies with different K and different RL increase sharply with the increase of Ur. For Ur≥4.75 (U≥0.376 m/s), there is a steady growth of ηharn and the peak ηharn of the VIV initial branch reaches 5.84 % at K=1600 N/m and RL=11 Ω.

10

14

K=1000N/m, RL=32Ω, ζtotal =0.152

K=1600N/m, RL=14Ω, ζtotal =0.194

K=800N/m,RL=2Ω, ζtotal =0.398

K=1200N/m,RL=2Ω, ζtotal =0.358

K=1000N/m, RL=52Ω, ζtotal =0.139

K=1600N/m, RL=17Ω, ζtotal =0.168

K=800N/m,RL=6Ω, ζtotal =0.303

K=1200N/m,RL=6Ω, ζtotal =0.272

K=1200N/m, RL=32Ω, ζtotal =0.139

K=1600N/m, RL=20Ω, ζtotal =0.146

K=800N/m,RL=11Ω, ζtotal =0.264

K=1200N/m,RL=11Ω, ζtotal =0.228

K=1200N/m, RL=52Ω, ζtotal =0.127

K=1600N/m, RL=32Ω, ζtotal =0.122

K=800N/m,RL=20Ω, ζtotal =0.1.95

K=1200N/m,RL=14Ω, ζtotal =0.211

K=1400N/m, RL=17Ω, ζtotal =0.175

K=1600N/m, RL=52Ω, ζtotal =0.112

K=800N/m,RL=32Ω, ζtotal =0.1.68

K=1400N/m,RL=2Ω, ζtotal =0.346

K=800N/m,RL=52Ω, ζtotal =0.158

K=1400N/m,RL=6Ω, ζtotal =0.267

K=1000N/m,RL=2Ω, ζtotal =0.380

K=1400N/m,RL=11Ω, ζtotal =0.21

K=1000N/m,RL=6Ω, ζtotal =0.285

K=1400N/m,RL=14Ω, ζtotal =0.202

K=1000N/m,RL=11Ω, ζtotal =0.241

K=1600N/m,RL=2Ω, ζtotal =0.335

K=1000N/m,RL=20Ω, ζtotal =0.176

K=1600N/m,RL=6Ω, ζtotal =0.258

12

Energy Conversion Efficiency ηharn (%)

Energy Conversion Efficiency ηharn (%)

12

K=1400N/m, RL=20Ω, ζtotal =0.152 K=1400N/m, RL=32Ω, ζtotal =0.129

8

K=1400N/m, RL=52Ω, ζtotal =0.119

Soft Galloping Branch

5.83%

6

4.47%

VIV Initial Branch

4.58%

4

2 VIV - Galloping Transition Branch

0 4

5

6

7

8

9

10

10

8

5.84%

6

4 Hard Galloping Branch 2

VIV Lower Branch VIV Upper Branch

0 12

13

14

5.78%

VIV Initial Branch

VIV Lower Branch 11

K=1600N/m,RL=11Ω, ζtotal =0.209

6.17%

4

5

6

7

Ruduced Velocity Ur

8

9

10

11

12

13

14

Ruduced Velocity Ur

(a) Energy conversion efficiency of SG (b) Energy conversion efficiency of HG Fig. 20. ηharn for different stiffness and different load resistance The VIV upper branch for HG (7.25≤Ur≤11~11.625) In the Fig. 20 (b), ηharn of upper branch increase slowly from the initial branch and then drop to about 2 % at Ur=11. The peak ηharn=6.17 % appears at Ur=8.5 at K=1600 N/m and RL=6 Ω. The VIV-galloping transition branch for SG (7.25≤Ur≤11~11.625) Both higher load resistance of 32 Ω≤RL≤52 Ω with stiffness of 1000 N/m≤K≤1600 N/m and 17 Ω≤RL≤20 Ω with stiffness of 1400 N/m≤K≤1600 N/m are contribute to galloping which is self-excited from VIV. In addition, the SG is also observed for RL=14 Ω with K=1600 N/m. As the Pharn is increasing with the velocity, on the contrary, ηharn shows an overall downward trend, as shown in Fig. 20 (a). The ηharn varies from 1.86 % (K=1600 N/m, RL=52 Ω.) to 4.47% (K=1600 N/m, RL=14 Ω) at Ur=11. The VIV lower branch for HG and SG (Ur≥11~11.625) The ηharn drop sharply to lower than 1% as Ur rises up to 13.5 and the variation of ηharn is less obvious. If an external excitation applied on the triangular prism which can directly jump into galloping branch for HG. Galloping branch for HG and SG (Ur≥11~11.625) The maximum ηharn reaches 5.78 % (15.24 W) with the corresponding load resistance RL=11 Ω, K=1400 N/m and Ur=11.625 (U=1.306 m/s). It is noted that ηharn drops down slowly with the increase of K and decrease of RL for 11 Ω≤RL≤52 Ω. 3.3.3 Summary It can be concluded that galloping branch performs best of FIM on active power but the upper branch of VIV performs best of FIM on energy conversion efficiency. The results are summarized in Table 9. Table 9 Energy conversion of different branches Branch

VIV Initial

VIV Upper

VIV-Galloping

VIV Lower

Galloping

Active Power (W)

0-2.05

0-4.47

3.37-12.95

0-1.92

3.95-23.37

Efficiency (%)

0-5.84

0.74-6.17

1.84-5.81

0-1.83

1.44-5.78

3.4 Parameter analysis The Pharn varies with four parameters including the load resistance (RL), stiffness (K), mass ratio (m*) and aspect ratio (α) of the triangular prism are discussed separately in this section. Variation in parameters are listed in Table 10.

Table 10 Parameters for active power in different conditions Load resistance

Stiffness

Mass ratio

Aspect ratio

Variation in load resistance

2 Ω≤RL≤∞

K=1400 N/m

m*=8.36

α=0.85

Variation in stiffness

RL=20 Ω

800 N/m≤K≤1600 N/m

m*=8.36

α=0.85

Variation in mass ratio

RL=20 Ω

K=1200 N/m

8.36≤m*≤11.63

α=0.85

Variation in aspect ratio

RL=20 Ω

K=1200 N/m

m*=8.36

0.5≤α≤1.5

3.4.1 Load resistance effect The dependence of active power (Pharn) on load resistance (RL) with different reduced velocities (Ur) is shown in Fig. 21. VIV: In the range of 4.75≤Ur≤7.25 for 2 Ω≤RL≤14 Ω, Pharn is small but increases rapidly with the growth of RL. As 7.25≤Ur≤11, Pharn firstly keeps growing with a strong uptrend to 3.2 W (Ur=9.75, RL=14 Ω) and then decreases to about 0.8 W with a slight downward trend. When Ur≥11, Pharn continue to decrease to about 0 W. The maximum Pharn appears at RL=14 Ω, Pharn,max=3.2 W at Ur=9.75. Galloping: The Pharn tends to decrease gradually as the load resistance (11 Ω≤RL≤52 Ω) increases. In addition, galloping does not occur in the responses of the tests with RL=6 Ω and RL=2 Ω. The maximum Pharn appears at RL=11 Ω, Pharn,max=20.87 W at Ur=13.5. It can be observed that, for the larger RL, it is easier to be self-excited from VIV to galloping branch. A lower RL is beneficial to the Pharn of galloping branch. 24 RL=2Ω, ζtotal =0.346 RL=6Ω, ζtotal = 0.267 RL=11Ω, ζtotal = 0.216

20

Active Power Pharn (W)

RL=14Ω, ζtotal = 0.202 RL=17Ω, ζtotal = 0.175 RL=20Ω, ζtotal = 0.152

16

RL=32Ω, ζtotal = 0.129

VIV-Galloping

RL=52Ω, ζtotal = 0.119 RL= ∞ , ζtotal = 0.091

12

VIV Initial Branch

8

Transition Branch

Galloping

and

Branch

VIV Upper Branch

4

VIV Lower Branch

0 4

5

6

7

8

9

10

11

12

13

14

Reduced Velocity Ur

Fig. 21. Active Power for α=0.85, m*=8.36 and K=1400 N/m at various RL. 3.4.2 Stiffness effect The active power curves with different stiffness are plotted in Fig. 22.

VIV: For the complete VIV response and VIV-Galloping transition branch, Pharn increases with the increase of the stiffness (800 N/m≤K≤1200 N/m). The reduced velocities of peak Pharn at different stiffness in VIV region are 9.125, 9.75, 10.375 and the corresponding Pharn are 0.7W (K=800 N/m), 1.43W (K=1000 N/m), 2.33W (K=1200 N/m), respectively. 20 K=800 N/m

18

Galloping Branch

K=1000 N/m K=1200 N/m

16

Active Power Pharn (W)

K=1400 N/m K=1600 N/m

14

VIV-Galloping Transition

12

Branch

10

and VIV Upper Branch

VIV Initial

8

Branch

6 4

VIV Lower Branch

2 0 4

5

6

7

8

9

10

11

12

13

14

Reduced Velocity Ur

Fig. 22. Active power for α=0.85, m*=8.36 and RL=20 Ω at various K Galloping: The Pharn of different stiffness values grow up the flow velocity and increase with the increase of the stiffness (800 N/m≤K≤1600 N/m), as shown in Fig. 22. The optimal active power can be clearly seen to occur for K=1600 N/m, Ur=13.5, Pharn=18W. It can be concluded that the effects of K on the Pharn of each branch are similar. With the same load resistance, the higher stiffness is beneficial to the energy utilization of galloping branch and VIV branch. 3.4.3 Mass ratio effect The Pharn patterns are shown as a function of mass ratio in Fig. 23. VIV: For the VIV responses, the Pharn decrease with the increase of the mass ratio m*. The peak active power of the VIV region grows significantly from 0.65 W at Ur=8.5 of m*=11.63 to 2.27W at Ur=9.75 of m*=8.36. Galloping: The Pharn at different mass ratio m* increase sharply with the growth of the reduced velocity. As Ur≥11, variation in Pharn is obvious with the increase of m*. The optimal Pharn reaches 13.55 W at Ur=13.5 of m*=8.36. It can be concluded that the effects of m* on the Pharn of each branch are the same, the lower m* is beneficial to the energy utilization in the tests.

15 m*=8.36 m*=9.47 m*=10.54 m*=11.63

Active Power Pharn (W)

12

9

VIV Initial

VIV Upper

Branch

Branch

6

Galloping Branch 3

VIV Lower Branch 0 4

5

6

7

8

9

10

11

12

13

14

Reduced Velocity Ur

Fig. 23. Energy conversion efficiency for α=0.85 K=1200 N/m and RL=20 Ω at various m* 3.4.4 Aspect ratio effect The variations of active power (Pharn) versus reduced velocity (Ur) and the aspect ratio (α) are plotted in Fig. 24. VIV As the flow velocity increases, the curves of α=0.5 and α=0.85 occur complete VIV phenomenon. And increasing the α has a positive influence on the Pharn. For α=0.85, the curve crosses with the α=1 and α=1.5 curves at some test points in the VIV regions. The optimal active power is about 2.32 W at Ur=9.75 of α=0.85 which is close to the Pharn of α=1 and α=1.5 at the same velocity. Galloping: The prism with higher aspect ratio (α=1.0, α=1.5) are more likely self-excited from VIV to galloping but the lower α of triangular prism enter galloping by applying a threshold Initial displacement. The peak active power in the galloping region increases sharply, from 9.47 W (Ur=11.625) to 15.81 W (Ur=12.25) of α=0.5. It illustrates that triangular prism with lower α is suitable for extracting the energy of galloping branch, while the higher α is more suitable to upper branch of VIV.

18

α=0.5 α=0.85 α=1 α=1.5

16

Active Power Pharn (W)

14 12

VIV-Galloping Transition Branch and VIV Upper Branch

10 8 VIV Initial Branch

6

Galloping Branch

4 2

VIV Lower Branch

0 4

5

6

7

8

9

10

11

12

13

14

Reduced Velocity Ur Fig. 24. Active power for m*=8.36, K=1200 N/m and RL=20 Ω at various α 3.4.5 summary According to the parameter analysis of active power for triangular prism with the variations of load resistance, stiffness, mass ratio and aspect ratio can be summarized in Fig. 25.

Fig. 25. Evolution of Active power for triangular prism

4. Conclusions The FIM responses and the energy conversion capacity of a single triangular prism for variation in load resistance, stiffness, mass-ratio and aspect ratio are investigated by experimental method. The oscillation responses and energy conversion of a single triangular prism are discussed and presented. The main conclusions as follow: (1) With the increase of load resistance and spring stiffness, oscillation gradually transforms from VIV to HG and finally SG for triangular prism. (2) By increasing the values of RL, A* of VIV responses and galloping branch substantially increase. On the other hand, A* increase considerably as K decrease. (3) For different FIM branches, the galloping branch performs the best but VIV lower branch is the worst for energy conversion. And the electric energy quality of galloping branch is the best. The optimal branch of energy conversion is galloping branch.

(4) The Pharn increase as the RL decrease to a value of approximately as low as 11 Ω, then the Pharn start to decrease. The increase of K and the decrease of m* have a strong positive effect on the Pharn of each branch of VIV and galloping branch. A higher α is beneficial to the improvement of the Pharn of VIV branch and lower α is advantageous to the increase of the Pharn of galloping branch.

Acknowledgement This work is supported by the National Key R&D Program of China (Grant No. 2016YFC0401905). 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] Blevins R.D. Flow-induced Vibration. 2th ed. New York: Van Nostrand Reinhold; 1990. [2] Assi G.R.S, Bearman P.W, Kitney N. Low drag solutions for suppressing vortex induced vibration of circular cylinders. J Fluids Struct 2009;25:666-675. https://doi.org/10.1016/j.jfluidstructs.2008.11.002. [3] Park H, Bernitsas M.M, Ajith Kumar R. Selective roughness in the boundary layer to suppress flow-induced motions of circular cylinder at 30,000≤Re≤120,000. J Offsh Mech Arctic Eng 2012 134:041801. https://doi.org/10.1115/1.4006235. [4] Williamson C.H.K, Roshko A. Vortex formation in the wake of an oscillating cylinder. J Fluids Struct 1988;2:355-381. https://doi.org/10.1016/S0889-9746(88)90058-8. [5] Zdravkovich M.M. in: E. Achenbach (Ed.) Flow Around Circular Cylinders. vol.1. Oxford University Press; Oxford; UK: 1997. [6] Zdravkovich M.M. in: E. Achenbach (Ed.), Flow Around Circular Cylinders, vol.2, Oxford University Press, Oxford, UK, 2002. [7] Rostami A.B, & Armandei M. Renewable energy harvesting by vortex-induced motions: Review and benchmarking of technologies. Renew Sust Energ Rev 2017;70:193-214. doi:10.1016/j.rser.2016.11.202 [8] Mannini C, Marra AM, Massai T, Bartoli G. Interference of vortex-induced vibration and transverse galloping for a rectangular cylinder. J Fluid Struct 2016;66:403-423. https://doi.org/10.1016/j.jfluidstructs.2016.08.002. [9] Zhang B, Wang K.H, Song B, Mao Z, Tian W. Numerical investigation on the effect of the cross-sectional aspect ratio of a rectangular cylinder in FIM on hydrokinetic energy conversion. Energy 2018;165:949-964. https://doi.org/10.1016/j.energy.2018.09.138. [10] Alonso G, Meseguer J, Pérez-Grande I. Galloping instabilities of two-dimensional triangular cross-section bodies. Exp Fluids 2005;38(6):789-95. https://doi.org/10.1007/s00348-005-0974-8. [11] Alonso G, Meseguer J. A parametric study of the galloping stability of two-dimensional triangular cross-section bodies. J Wind Eng Ind Aerod 2006;94(4):241-53. https://doi.org/10.1016/j.jweia.2006.01.009. [12] Alonso G, Meseguer J, & Pérez-Grande I. Galloping stability of triangular cross-sectional bodies: A systematic approach. J Wind Eng Ind Aerod 2007;95(9):928-40. https://doi.org/10.1016/j.jweia.2007.01.012. [13] Iungo G.V, Buresti G. Experimental investigation on the aerodynamic loads and wake flow features of low aspect-ratio triangular prisms at different wind directions. J Fluids Struct 2009;25 (7):1119-1135. https://doi.org/10.1016/j.jfluidstructs.2009.06.004. [14] Chang C.C, Kumar R.A, Bernitsas M.M. VIV and galloping of single circular cylinder with surface roughness at 3.0×104≤Re≤1.2×105. Ocean Eng 2011;38:1713-1732. https://doi.org/10.1016/j.oceaneng.2011.07.013. [15] Park H, Kumar R.A, Bernitsas M.M. Enhancement of fluid induced vibration of rigid circular cylinder on springs by localized surface roughness at 3×104≤Re≤1.2×105. Ocean Eng 2013;72:403-415. https://doi.org/10.1016/j.oceaneng.2013.06.026. [16] Lian J, Yan X, Liu F, Zhang J, Ren Q, Yang X. Experimental Investigation on Soft Galloping and Hard Galloping of Triangular Prisms. Appl Sci 2017;7:198.

https://doi.org/10.3390/app7020198. [17] Zhang J, Liu F, Lian J, Yan X, Ren Q. Flow Induced Vibration and Energy Extraction of an Equilateral Triangle Prism at Different System Damping Ratios. Energies 2016;9:938. https://doi.org/10.3390/en9110938. [18] Shao N, Lian J, Xu G, Liu F, Deng H, Ren Q, Yan X. Experimental Investigation of Flow-Induced Motion and Energy Conversion of a T-Section Prism. Energies 2018;11(8):2035. https://doi.org/10.3390/en11082035. [19] Zhou S, Wang J. Dual serial vortex-induced energy harvesting system for enhanced energy harvesting. AIP Adv 2018;8(7):075221. https://doi.org/10.1063/1.5038884. [20] Zhang Y, Deng S, Wang X. RANS and DDES simulations of a horizontal-axis wind turbine under stalled flow condition using Open FOAM. Energy 2019;167:1155-1163. https://doi.org/10.1016/j.energy.2018.11.014. [21] Lian J, Yan X, Liu F, Zhang J. Analysis on Flow Induced Motion of Cylinders with Different Cross Sections and the Potential Capacity of Energy Transference from the Flow. Shock and Vib 2017;(2017):1-19. https://doi.org/10.1155/2017/4356367. [22] Zhang LB, Dai HL, Abdelkefi A, Wang L. Experimental investigation of aerodynamic energy harvester with different interference cylinder cross-sections. Energy 2019;167:970-981. https://doi.org/10.1016/j.energy.2018.11.059. [23] Zhang B, Song B, Mao Z, Tian W, Li B. Numerical investigation on VIV energy harvesting of bluff bodies with different cross sections in tandem arrangement. Energy 2017;133:723-736. https://doi.org/10.1016/j.energy.2017.05.051. [24] Zhang M, Zhao G, Wang J. Study on Fluid-Induced Vibration Power Harvesting of Square Columns under Different Attack Angles. Geofluids. 2017;2017(2):1-18. https://doi.org/10.1155/2017/6439401. [25] Wang J, Zhou S, Zhang Z, Yurchenko D. High-performance piezoelectric wind energy harvester with Y-shaped attachments. Energy Convers Manage 2019;181:645-652. https://doi.org/10.1016/j.enconman.2018.12.034. [26] Bernitsas M.M, Raghavan K, Ben-Simon Y, & Garcia E.M.H. VIVACE (Vortex Induced Vibration Aquatic Clean Energy): A New Concept in Generation of Clean and Renewable Energy from Fluid Flow. Volume 2: Ocean Engineering and Polar and Arctic Sciences and Technology. 2006. doi:10.1115/omae2006-92645. [27] Lee J.H, Xiros N, & Bernitsas M.M. Virtual damper-spring system for VIV experiments and hydrokinetic energy conversion. Ocean Eng 2011;38(5-6):732-747. doi:10.1016/j.oceaneng.2010.12.014. [28] Sun H, Soo Kim E, Bernitsas MP, Bernitsas M.M. Virtual Spring-Damping System for Flow-Induced Motion Experiments. J Offshore Mech Arct Eng 2015;137:061801. https://doi.org/10.1115/1.4031327. [29] Lee J.H. & Bernitsas M. M. High-damping, high-Reynolds VIV tests for energy harnessing using the VIVACE converter. Ocean Eng 2011;38(16):1697-1712. doi:10.1016/j.oceaneng.2011.06.007. [30] Chang CC. Passive turbulence control for VIV enhancement for hydrokinetic energy harnessing using vortex Induced vibrations, Ph.D. thesis. Ann Arbor, MI: The University of Michigan; 2010. [31] Sun H, Kim E.S, Nowakowski G., Mauer E, & Bernitsas M.M. Effect of mass-ratio, damping,

and stiffness on optimal hydrokinetic energy conversion of a single, rough cylinder in flow induced motions. Renew Energ 2016;99, 936-959. doi:10.1016/j.renene.2016.07.024. [32] Sun H, Ma C, Kim E.S, Nowakowski G, Mauer, E, & Bernitsas M.M. Hydrokinetic energy conversion by two rough tandem-cylinders in flow induced motions: Effect of spacing and stiffness. Renew Energ 2017;107,61-80. doi:10.1016/j.renene.2017.01.043. [33] Kim E.S, & Bernitsas M.M. Performance prediction of horizontal hydrokinetic energy converter using multiple-cylinder synergy in flow induced motion. Appl Energ 2016;170, 92-100. doi:10.1016/j.apenergy.2016.02.116. [34] Xu F, Ou JP, Xiao YQ. CFD numerical simulation of flow-induced transverse vibration of a square cylinder. Journal of Harbin Institute of Technology 2008;40(12):1849-1854. (in Chinese). [35] Ding L. Research on Flow Induced Motion of Multiple Circular Cylinders with Passive Turbulence Control. Chong Qing: Chong Qing University:2013. (in Chinese)

Highlights： 1. The objective is to enhance energy conversion of FIM for triangular prism. 2. Selective aspect ratios are applied to enhance the energy conversion. 3. Energy conversion characteristics of HG and SG are analyzed. 4. The active power for parameters variation are presented.