The effects of submergence depth on Vortex-Induced Vibration (VIV) and energy harvesting of a circular cylinder

Journal Pre-proof The effects of submergence depth on Vortex-Induced Vibration (VIV) and energy harvesting of a circular cylinder

Mengfan Gu, Baowei Song, Baoshou Zhang, Zhaoyong Mao, Wenlong Tian PII:

S0960-1481(19)31779-3

DOI:

https://doi.org/10.1016/j.renene.2019.11.086

Reference:

RENE 12634

To appear in:

Renewable Energy

Received Date:

30 October 2018

Accepted Date:

16 November 2019

Please cite this article as: Mengfan Gu, Baowei Song, Baoshou Zhang, Zhaoyong Mao, Wenlong Tian, The effects of submergence depth on Vortex-Induced Vibration (VIV) and energy harvesting of a circular cylinder, Renewable Energy (2019), https://doi.org/10.1016/j.renene.2019.11.086

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Journal Pre-proof

The effects of submergence depth on Vortex-Induced Vibration (VIV) and energy harvesting of a circular cylinder Mengfan Gu a, Baowei Song a, Baoshou Zhang a, b,*, Zhaoyong Mao a, Wenlong Tian a

a

School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China

b

Department of Civil and Environmental Engineering, University of Houston, Houston, TX 77204, USA

* Corresponding author: Baoshou Zhang. E-mail address: [email protected] / [email protected]

0

Journal Pre-proof

ABSTRACT Vortex-Induced Vibration (VIV) is a kind of high-energy phenomena and can be used to harvest energy from ocean/river currents. In this paper, a spring-mounted circular cylinder in VIV was numerically investigated with 2-dimensional simulations to examine the effects of submergence depth on the energy conversion. The flow speed changes from 0.2m/s to 1.3m/s (1.61×1049.7×104), the oscillation amplitude of the cylinder decreases to zero. In this case, the oscillation response doesn’t belong to VIV.

KEYWORDS VIV (Vortex Induced Vibration); Energy harvesting; Submergence depth; Numerical simulations; Circular cylinder.

Nomenclature U

Flow speed

TOSC

Period of oscillation

L

Length of cylinder

A

Amplitude of oscillation

K

Spring stiffness

A *=A/D

Amplitude ratio

M

Total mass of VIV system

S

Submergence depth

D

Diameter of the cylinder

𝑆 ∗ = 𝑆/𝐴

Submergence depth ratio

Ctotal

System damping coefficient

m *= M (Vcylinder   water )

Mass-ratio

fosc

Oscillating cylinder frequency

madd

Added mass



Water density

𝐹fluid,y(𝑡)

Total fluid force

v

Kinematic viscosity

Re =UD⁄v

f n , water  1 2   K ( M  m add )

𝑈 = 𝑈 (𝑓𝑛,𝑤𝑎𝑡𝑒𝑟 ∙ 𝐷) ∗

Reynolds number

1

Cylinder’s natural frequency Reduced velocity

Journal Pre-proof 1. INTRODUCTION As a result of the recent development of renewable energy conversion technologies, many different energy converters have been built to harvest renewable energy. Vortex Induced Vibration (VIV) phenomenon has also been utilized to harness energy from slow-speed ocean and river currents [1-4]. VIV of a circular cylinder is regarded as an induced oscillatory motion by a passing flow that provides alternating lift on the cylinder [5-8]. Particularly, Bernitsas et al. [4, 9, 10] have invented VIVACE (Vortex Induced Vibration Aquatic Clean Energy) converter. The VIV phenomenon has been used to harvest energy from low flow speed currents [2, 11, 12]. Based on Ref. [4], in which the cost comparisons between the VIV energy converter and other energy conversion systems were reported, as shown in Fig. 1. Obviously, the VIV energy conversion technology can generate good economic returns. The research about the VIV energy converter was carried out by Lee et al. [13], Sun et al. [3, 14, 15] and Park et al. [16] using experiments; and by Ding et al. [8, 12] using numerical methods.

Fig. 1 Cost comparison between the VIV energy converter and other energy conversion systems (e.g. VIV energy converter: 0.055$/kWh; Solar: 0.489$/kWh; Wind energy converter: 0.069$/kWh). The data comes from Ref. [4]

In earlier works, Williamson et al. [17-19] and Zhao et al [20] have extensively studied the VIV responses of circular cylinders and square cylinders. They found three different branches (initial branch, upper branch and lower branch) of the VIV responses. Bernitsas et al. [2] and Kim et al. [21-22] also came to similar conclusions about the branches of VIV. After a series of optimization investigations, Lee et al. [23] reported that power density of the VIV energy converter is 98.2W/m3 at about 1.0m/s. Besides, numerical studies based on different CFD methods have been carried out to study the VIV responses. Zhang et al [6] studied the VIV responses of a rectangular cylinder using Fluent. Ding et al. [8, 12] analyzed the VIV responses of non-circular cross-sectional cylinders via CFD method based on OpenFOAM. Zhao et al. [24, 25] studied FIV (Flow Induced Vibration) of cylinders in an inline arranged square array by 2-dimensional simulations. Similarly, Mao et al. [26] analyzed the VIV of the circular underwater moored platform based on Fluent. The earlier investigations were mainly focused on the VIV responses in deep waters or in single-phase flow. The effects of submergence depth on the VIV responses were always ignored. However, the submergence depth is essential since the wake structure and vortex patterns may be changed [27]. In the experiments completed in our lab with a circular cylinder in VIV near a free surface, we found a significant wave-making phenomenon 2

Journal Pre-proof (free surface deformation). When the VIV energy converter is in close proximity to the free surface, even though the VIV energy converter is fully submerged, it is supposed that the submergence depth will affect the VIV energy conversion. Until now, there are few studies on the effects of the submergence depth. Reichl et al. [28] completed a series of numerical investigations of flow past a fixed cylinder when the cylinder is close to the free surface. Carberry et al. and Zhu et al. [29, 30] studied the effects of free surface on the VIV responses only for forced vibration. The closest relevant researches about the effects of free surface or submergence depth are limited to studies by Raghavan et al and Bemitsas et al [27, 31]. It was reported that the synchronization range of VIV reduces drastically as the submergence depth reduces. However, the above mentioned studies are focused on the VIV amplitude and frequency responses, not in energy conversion. In addition, the vortex patterns associated to the submergence depth is also not known. Consequently, it is worthy of studying the effects of submergence depth and the disturbance caused by the free surface deformation. To study the effects of submergence depth on VIV energy conversion and obtain an appropriate depth, a series of 2-dimensional numerical simulations were carried out. The VIV responses of a circular cylinder were studied under various submergence depths (0.1m-0.5m). The flow speed is 0.2m/s
2. GEOMETRY CONFIGURATION 2.1. Description of the VIV Energy Conversion System The VIV energy conversion system is modeled as an elastically supported cylinder [2, 32], as shown in Fig. 2(a). To better understand this concept, a simplified sketch is shown in Fig. 2(b). The cylinder is fully submerged and constrained to oscillate freely using linear bearings. The circular cylinder is parallel to the seabed and free surface. Based on Ref. [2, 8], it was reported that the transverse VIV amplitude is higher than the streamwise VIV amplitude. Thus, only the transverse response is considered. 𝑆 represents the submergence depth; 𝑆 ∗ = 𝑆/𝐷 is the submergence depth ratio. 𝐶𝑡𝑜𝑡𝑎𝑙 is the total system damping from the generator and transmission mechanisms. 𝐾 is the spring stiffness constant. The vortices alternately separate from the upside and underside of the cylinder.

3

Journal Pre-proof Springs Springs

Air

Ocean current

Free Surface (Sea Level)

Circular cylinder

Submergence depth S

C total

K

Vortices Transmission Mechanism

Vortices

Ocean current

Vortices

y x

Generator Seabed (a)

(b)

Fig. 2 (a) VIV hydrokinetic energy conversion system and (b) Simplified sketch near the free surface

The background of this study comes from the towing tank tests for VIV in our laboratory, as shown in Fig. 3. When the VIV hydrokinetic energy converter is towed in close proximity to the free surface, especially at low submergence depths (𝑆 < 0.3m), free surface deformation is observed, which means the submergence depth is an important factor and has a great impact on VIV. However, in these experiments, it is difficult to record the free surface deformation. Based on earlier works, numerical methods can reasonably study the VIV responses and measure the free surface deformation. Therefore, the numerical method is applied for analyzing the VIV responses near a free surface. Towing Cradle Transmission Mechanism

o Moti

n

ectio

n Dir

Wave

Springs (b)

Slide System Towing Tank

(c)

Wave-making

Cylinder

Motion Direction

Wave

Generator

Motion Direction

Water

ect io

n

Oscillation

Air

Mo

tion

Dir

Flow Direction

(d)

(a)

Fig. 3 (a) Towing tank test for VIV; (b) & (c) Wave-making phenomenon and (d) Schematic representation of Wave-making phenomenon in VIV (Videos for the wave-making phenomenon [33, 34])

Bernitsas and Sun et al. [2, 3, 16] have studied the VIV responses of circular cylinders and provided a lot of reference data. Therefore, similar design parameters of the VIV energy conversion system were determined based on these studies. The main parameters are listed in Table 1. The independent variable 𝑆 increases from 0.1m to 0.5m. The test flow speed changes from 0.2m/s to 1.3m/s.

4

Journal Pre-proof Table 1 Physical parameters of the VIV hydrokinetic energy conversion system Description

Symbol

Spring stiffness per unit length

K

( N m )

1063

Submergence depth

S

(mm)

100-500

Length of the cylinder

L

(mm)

1000

Water density

 (kg/m³)

1000

Kinematic viscosity of water



1.14  10-6

Mass-ratio

m

Total oscillating mass of VIV system

M

Damping

𝐶𝑡𝑜𝑡𝑎𝑙

(m2/s)

Value

1.725 (kg)

13.63 30

2.2. Equation of motion In this section, the mathematical model for the VIV energy conversion system is presented. To define the responses of the circular cylinder [6, 8], a typical combined mass-spring-damper oscillator model is introduced here. In y-direction, the motion of the cylinder is defined by: 𝑀𝑦 + 𝐶𝑡𝑜𝑡𝑎𝑙𝑦 +𝐾𝑦 = 𝐹fluid,y(𝑡)

(1)

where 𝑀 represents the total oscillating mass; 𝐹fluid,y(𝑡) represents the total fluid force. According to Fluent solver, 𝐹fluid,y(𝑡) can be solved directly. The VIV converted power 𝑃VIV of the cylinder is defined as: 1

𝑇

𝑃VIV = 𝑇𝑜𝑠𝑐∫0𝑜𝑠𝑐𝐹fluid,y 𝑦𝑑𝑡

(2)

where 𝑇𝑜𝑠𝑐 represents one cycle of oscillation. From Eq. (1), we have 1

𝑇

𝑃VIV = 𝑇𝑜𝑠𝑐∫0𝑜𝑠𝑐(𝑀𝑦 + 𝐶𝑡𝑜𝑡𝑎𝑙𝑦 + 𝐾𝑦) 𝑦𝑑𝑡

(3)

According to the previous studies in Ref. [2, 4, 8], the 𝑀𝑦 and 𝐾𝑦 terms can be simplified. More details are not discussed here. Finally, we have 1

𝑇

𝑃VIV = 𝑇𝑜𝑠𝑐∫0𝑜𝑠𝑐𝐶𝑡𝑜𝑡𝑎𝑙 𝑦2𝑑𝑡

(4)

Since the velocity 𝑦 will be obtained based on numerical results in Fluent, 𝑃VIV can be solved directly. Based on the Bernoulli’s equation, in the area swept by the circular cylinder, the power of the fluid 𝑃fluid can be defined as 1

𝑃fluid = 2𝜌𝑈3𝐿(𝐻 + 2𝐴𝑚𝑎𝑥)

(5)

where 𝑈 is the flow speed. The energy conversion efficiency 𝜂VIV is the ratio between the output 𝑃VIV of the VIV energy conversion system and the input energy 𝑃fluid. The Betz Limit represents the maximum energy conversion efficiency that can be harvested from the flow. Betz Limit also should be considered. According to Ref. [3, 35], the factor 16/27 is the Betz Limit coefficient. Finally, 𝜂VIV is defined as 𝑃VIV

𝜂VIV = 𝑃fluid × Betz Limit × 100%

3. NUMERICAL METHOD 5

(6)

Journal Pre-proof 3.1. Mathematical Model Two-dimensional RANS equations accompanied with the k-  SST turbulence model are used to simulate the two-phase flow around the oscillating cylinder, based on Fluent. In Fluent solver, User-Defined Function (UDF) module was used to solve the motion of the cylinder. Newmark-𝛽 method is always used to solve differential equations, such as the mass-spring-damper oscillator model Eq. (1) [6, 36, 37]. Therefore, Newmark-𝛽 method was embedded in the UDF module. The displacement, 𝑦, velocity, 𝑦 , and acceleration, 𝑦, of the cylinder are defined as 𝑦𝑡 + ∆𝑡 = (𝑦𝑡 + 𝑦 ∆𝑡) +[

(12 ― 𝛽)𝑦 +𝛽 𝑦

𝑡 + ∆𝑡]

𝑡

𝛾

(

1

1

𝛾

)

∆𝑡2

(

1

(7)

)

𝑦𝑡 + ∆𝑡 = 𝛽 ∆𝑡2(𝑦𝑡 + ∆𝑡 ― 𝑦𝑡) ― 1 ― 𝛽 𝑦𝑡 ― 1 ― 2𝛽 𝑦𝑡∆𝑡 1

𝑦𝑡 + ∆𝑡 = 𝛽∆𝑡2(𝑦𝑡 + ∆𝑡 ― 𝑦𝑡) ― 𝛽∆𝑡𝑦𝑡 ―(2𝛽 ―1)𝑦𝑡

(8) (9)

where the 𝑡 and ∆𝑡 represent the time and the time step. 𝛾 and 𝛽, are two weighting constants. Based on Ref. [36, 37], 𝛾 and 𝛽 are 0.5 and 0.25. This setting gave second-order accurate and unconditionally stable for the solution. Considering Eq. (1) at time 𝑡 + ∆𝑡, we have 𝑀 𝑦𝑡 + ∆𝑡 + 𝐶𝑡𝑜𝑡𝑎𝑙 𝑦𝑡 + ∆𝑡 +𝐾 𝑦𝑡 + ∆𝑡 = 𝐹fluid,y(𝑡)𝑡 + ∆𝑡

(10)

By combining the last three equations (Eqs. (7)-(9), 3 unknown items (𝑦𝑡 + ∆𝑡, 𝑦𝑡 + ∆𝑡and𝑦𝑡 + ∆𝑡) can be obtained at time 𝑡 + ∆𝑡. Finally, Eq. (10) can be further simplified as

[𝐾] 𝑦𝑡 + ∆𝑡 = [𝐹]

(11)

where

[𝐾 ] = 𝐾 + [𝐹] = 𝐹fluid,y(𝑡)𝑡 + ∆𝑡 +

[

1 𝑦 𝛽 ∆𝑡2 𝑡

1

+ 𝛽 ∆𝑡𝑦𝑡 +

1

𝛾

𝑀 + 𝛽 ∆𝑡𝐶𝑡𝑜𝑡𝑎𝑙 𝛽 ∆𝑡 2

(2𝛽1 ― 1)𝑦𝑡]𝑀 + [𝛽 ∆𝑡𝛾 𝑦𝑡 ― (1 ― 𝛽𝛾)𝑦𝑡 ― (1 ― 2𝛽1 )𝑦𝑡]𝐶𝑡𝑜𝑡𝑎𝑙. 2

(12a) (12b)

Finally, computer algorithms, written with C++, are applied to solve the Eqs. (8), (9) and (11). More details about this method can be found in previous papers [32, 38-39]. The process of the numerical method is shown in a flowchart, as shown in Fig. 4.

6

Journal Pre-proof Start

Initialize flow field

Based on the k-w SST turbulence model, solve RANS equations

Based on Fluent solver and embedded UDF code

Firstly. Obtain the fluid force Secondly. Based on the current state, solve the next state accordinng to Eq. (1) using Newmark-β method Finally. Output new speed and position etc. of the cylinder

Update the mesh of computational domain based on the new position of the cylinder

Whether or not the simulation time is completed

No

Yes End

Fig. 4 Numerical method in a flowchart

3.2. Computation Domains and Boundary Conditions Numerical modeling and simulation is a suitable and emerging approach for the two-phase flow. The parameters varied in the tests are the submergence depth 𝑆 and flow speed. The computational domains and grids are shown in Fig. 5. The length and width of the computational domain are set to be 50𝐷 and 30𝐷, where 𝐷 is the cylinder diameter. Thus, the blockage ratio is only 3.3%. The computational domain consists of Sub-Domain, Slipway and External Flow Field. The length and width of the Sub-Domain is 4.5𝐷. Four kinds of boundaries are included in the domain, such as inlet, outlet, side-wall, and cylinder. The two-phase flow is set at the inlet: Phase-1 Air and Phase-2 Water. The flow speed changes from 0.2m/s to 1.3m/s to simulate actual ocean/river currents. The pressure outlet boundary was used at the outlet of the domain. More details about the computational domains have also been described in previous studies [6, 32]. According to the requirements of the k-𝜔 SST turbulence model, the near-wall grid resolution around the cylinder should be high enough. Consequently, the value of y+ is about 1 on the surface of the cylinder. Thus, the minimum height of the first layer grids is about 1.5×10-5m-3×10-5m under the circumstances with different flow speeds.

7

Journal Pre-proof Side-Wall Slipway

Phase-1 Air

Moving Layer

Submergence depth S Moving Layer

30D

Boundary Layer

C

Outlet

Velocity-inlet

Free Surface (Sea Level)

Phase-2 Water Sub-Domain External Flow Field

y x

Slipway

Side-Wall 50D

(b)

(a)

Fig. 5 (a) Computational domain and (b) Grid configuration for the cylinder

3.3. Numerical Method Verification To verify the convergence and accuracy of the numerical model, the amplitude and frequency of the CFD results are compared with the experimental results. According to previous experiments [18, 40] for an oscillating circular cylinder, the design parameters of the new VIV system are determined and listed in Table 2. The range of reduced velocity is 2< 𝑈 ∗ <14. The numerical and experimental results are compared in Fig. 6. When the reduced velocity increases to 4, the amplitude increases. This is the initial branch of VIV. Then, VIV is in the upper branch. After that, the VIV responses keep decreasing. When the reduced velocity 𝑈 ∗ >12, the VIV responses disappear. According to the frequency response, the natural frequency and vortex shedding frequency combine to affect the frequency of VIV. In particular, two curves are observed. Curve_A is close to the natural frequency and mainly affected by natural frequency. Curve_B is close to the vortex shedding frequency. In summary, Table 2 Main parameters of the VIV system for validation of the numerical method Description

Symbol

Value

Diameter of the cylinder

D (m)

0.0508

Spring stiffness

K ( N m )

43.5

Mass-ratio

m

2.4

𝑈

Velocity ratio

∗

2-14

Natural frequency

f n,water

0.4

Kinematic viscosity

 (m2/s)

1.14  10-6

Water density

 (kg/m³)

1000

In general, three branches of VIV all appear when compared with the experimental data. The comparison shows an acceptable match for the numerical method. It was also found that the upper branch is relatively narrow. There are two main reasons leading to the difference. The first reason is in the reference experiments,

8

Journal Pre-proof the parameters of the VIV system are given in non-dimensional or normalized format, such as the mass ratio m*, the damping ratio 𝜉*, and the reduced velocity U*, etc. Thus, we have to deduce the actual parameters based on the non-dimensional parameters, which may lead to the difference. The second reason is the current study was conducted using the two-dimensional assumption. This simulates ideal three-dimensional fluid without lateral flow. The assumption limits the robustness of the resonance point between the vortex shedding frequency and the natural frequency. In particular, the VIV upper branch (the strongest resonance region) means the vortex shedding frequency coincides with the natural frequency of the VIV system. However, for the ideal three-dimensional fluid, the coincidence point of the vortex shedding frequency and the natural frequency is ideal and very sensitive. Therefore, the synchronization range around the coincidence point is relatively narrow. Accordingly, the upper branch is relatively narrow and low around the coincidence point. Similar conclusions were also drawn in other literature [2, 41]. In addition to this difference in the upper branch, all other VIV branches are in good agreement with the experimental data. The initial branch, upper branch, and lower branch appear at the same flow speeds when compared with the experimental results. In this case, the numerical method is convergent and acceptable. Amplitude ratio 4

m*=2.4, CFD m*=2.4, EXP (Ref.) m*=2.4, CFD (Ref.)

Lower fosc/fn,water

A/D

0.6 0.5 0.4

1.4

3 2.5

r Cu

2

Curve_A

1

Initial

0

2

4

6

8

10

12

14

16

18

0

B e_

Cu 0

2

4

1 0.8 0.6

rv

0.5

0.1

Curve_A

1.2

1.5

0.3 0.2

_B ve

rve _B

0.7

3.5

6

8

U*

10

12

14

16

18

0.4

Cu

Upper

0.8

1.6 m*=2.4,CFD m*=2.4,EXP (Ref.) m*=2.4,CFD (Ref.)

fosc/fn,water

0.9

0

Frequency ratio

Frequency ratio

1

2

4

6

U*

8 U*

10

12

14

Fig. 6 Comparison of VIV responses for the numerical results and the experimental results

4. RESULTS AND DISCUSSION 4.1 Amplitude and Frequency The VIV amplitudes and frequencies are compared for the 5 submergence depths (𝑆=0.1m; 𝑆=0.2m; 𝑆 =0.3m; 𝑆=0.5m; 𝑆=++m) in Fig. 7. Meanwhile, two tables are also given to summary the amplitude and frequency versus the flow speed and submergence depth, as shown in Table 3 and Table 4. ‘𝑆=++m’ means the submergence depth is infinite (in single-phase flow), which is considered as a reference. When analyzing the VIV amplitude, the inputs are the flow speed and submergence depth, and the output is the value of amplitude. Sensitivity analysis is the study of how the uncertainty in the output of a mathematical system can be divided into its inputs. The global sensitivity coefficients for the amplitude are calculated based on the FAST (Fourier amplitude sensitivity test) method which is a variance-based global sensitivity analysis method. The global sensitivity coefficients of the flow speed and submergence depth are 0.8865 and 0.0535, which indicates the amplitude is sensitive to the flow speed in the global region. The following observations can be made: (1) The VIV amplitude shows an increasing trend as the submergence depth increases, and the VIV range of the cylinder becomes broad. Therefore, it is concluded proximity to the free surface suppresses the VIV responses of the circular cylinder. However, when the submergence depth 𝑆 increases to 0.5m, the increasing 9

Journal Pre-proof trend of amplitude is not observed, which means VIV is no longer affected by the free surface. (2) Re<36 200: VIV initial branch. The VIV responses gradually become strong. For all the submergence depths, the increasing trend is similar; in addition to the submergence depth 𝑆=0.1m, all other frequency curves show an increasing trend. (3) 36 20097 000: the VIV responses disappear. In this region, the amplitude decreases to zero. The decrease in amplitude is accompanied by weak growth in the VIV frequency. The vibration no longer belongs to VIV, which will not be discussed in the following sections. It is easy to conclude that the VIV responses will be weakened as the submergence depth decreases. It is hypothesized to be associated with the effects of the vortices induced by the free surface deformation. Therefore, further explanations will be given to discuss detailed VIV responses near the free surface in the following sections. 1

0.09

Initial

0.05 0.7

0.6 0.04 0.5 0.03

0.8

0.5 0.4

1

0.2

Disappear

0.1 0.4

2

3

0.6

4

0.8 U (m/s)

5

6

1

7

Reynolds number

8

0

1.2

9

2

1.5

0.3

Cease prematurely

0.3 0.01 0.2 0 0.10.2

0.9 1.4

2.5

0.6

Lower

0.4 0.02

3

0.7 Efficiency (%) Frequency (Hz)

0.06 0.8

H=0.1m H=0.2m H=0.3m H=0.5m H=+∞m

0

1.2 0.7 0.6 1 0.5 0.8 0.4

0

0.2 0.4 3 0.2 4 0.1 0

10 4

Initial

0.8

H=0.1m H=0.2m H=0.3m H=0.5m H=+∞m

0.3 0.6

0.5

2

H=0.1m H=0.1m H=0.2m H=0.2m H=0.3m H=0.5m H=0.3m H=0.1m H=0.5m H=++m H=0.2m H=++mDisappear H=0.3m Lower H=0.5m H=+∞m

Upper

1.6 1

A/D

Amplitude (m) Efficiency (%)

0.07 0.9

1.8 3.5

0.9

Amplitude (m)

0.08 1

0

H=0.1m H=0.2m H=0.3m H=0.5m H=++m

Upper

2

0.5

5 0.4

6 0.6 7 80.8 9 1 1.5 Reynolds number U (m/s) U (m/s) 3 4 5 6 7

110 2 8

1.2 4

x 10

9

Reynolds number

x 10

(a)

(b)

Fig. 7 (a) Amplitude response and (b) frequency response of the cylinder for different submergence depths

10

2.5

10 4

x 10

Journal Pre-proof

Table 3 The VIV amplitude vs the flow velocity and submergence depth S

Submergence depth

(m)

0.1

0.2

0.3

0.5

++

Flow

0.2

0.004

0.0041

0.0023

0.0040

0.0041

velocity

0.25

0.007

0.0065

0.0070

0.0068

0.0068

(m/s)

0.3

0.0115

0.012

0.0105

0.0105

0.0105

0.35

0.0188

0.0153

0.0154

0.0155

0.018

0.4

0.025

0.028

0.0225

0.021

0.0258

0.45

0.0415

0.0435

0.035

0.0375

0.0373

0.5

0.0574

0.0686

0.0679

0.065

0.0614

0.54

0.0585

0.0763

0.0745

0.0729

0.0729

0.55

0.0577

0.0719

0.0768

0.0753

0.0752

0.56

0.0577

0.0671

0.0781

0.0771

0.077

0.57

0.0571

0.063

0.0701

0.0724

0.075

0.6

0.0537

0.0537

0.0607

0.0625

0.0626

0.65

0.0509

0.0532

0.0543

0.0543

0.0554

0.7

0.0475

0.0534

0.0542

0.0547

0.0548

0.75

0.0422

0.0528

0.0541

0.0546

0.0549

0.8

0.0338

0.0505

0.0530

0.0545

0.0548

0.85

0.0205

0.0467

0.0529

0.0536

0.0543

0.9

0.0110

0.0352

0.0508

0.0526

0.053

0.95

0.003

0.0243

0.047

0.0501

0.0515

1

0.0015

0.0065

0.0402

0.0466

0.0482

1.05

0.001

0.0005

0.0316

0.0425

0.044

1.1

0.0006

0.001

0.0114

0.0325

0.0348

1.15

0.0004

0.0015

0.0035

0.017

0.0135

1.2

0.0007

0.0012

0.002

0.005

0.002

1.25

0.0005

0.0007

0.0012

0.0025

0.0013

1.3

0.0005

0.0041

0.0007

0.0015

0.0011

11

Journal Pre-proof Table 4 The VIV frequency vs the flow velocity and submergence depth S

Submergence depth

(m)

0.1

0.2

0.3

0.5

++

Flow

0.2

0.543

0.5371

0.5371

0.5371

0.5371

velocity

0.25

0.6836

0.6836

0.6836

0.6348

0.6348

(m/s)

0.3

0.7324

0.7324

0.7324

0.7324

0.7324

0.35

0.8301

0.8789

0.8789

0.8301

0.8301

0.4

0.9277

0.9766

0.8789

0.9277

0.9033

0.45

1.172

0.9766

0.9766

0.9766

0.9766

0.5

1.318

1.074

1.074

1.074

1.0742

0.54

1.367

1.172

1.172

1.074

1.0986

0.55

1.416

1.221

1.123

1.074

1.0986

0.56

1.465

1.367

1.172

1.172

1.123

0.57

1.465

1.367

1.367

1.27

1.2451

0.6

1.465

1.416

1.416

1.3674

1.3916

0.65

1.465

1.465

1.465

1.465

1.4648

0.7

1.465

1.416

1.465

1.465

1.4404

0.75

1.416

1.465

1.416

1.465

1.416

0.8

1.367

1.465

1.416

1.465

1.416

0.85

1.367

1.465

1.416

1.465

1.416

0.9

1.465

1.465

1.416

1.465

1.416

0.95

1.465

1.465

1.416

1.465

1.4404

1

1.465

1.465

1.4656

1.465

1.4404

1.05

1.391

1.465

1.465

1.465

1.4648

1.1

1.3675

1.465

1.465

1.465

1.4648

1.15

1.4846

1.465

1.465

1.465

1.4893

1.2

1.476

1.514

1.514

1.563

1.4893

1.25

1.5105

1.563

1.563

1.563

1.5137

1.3

1.51

1.514

1.563

1.563

1.5137

4.2 VIV Responses 4.2.1 VIV at low submergence depth For low submergence depths (S=0.1m and S=0.2m), in the VIV initial branch and upper branch, the power and efficiency of S=0.2m is close to that of S=0.5. However, in the VIV lower branch, the responses of S=0.2m are different from that of S=0.5. In particular, when in the VIV lower branch, the flow speed is relatively high. In this case, large-scale free surface deformation will appear, and the flow structure of S=0.2 is much closer to that of S=0.1 rather than S=0.5. Since the VIV responses with S=0.2m is not typical, the case with S=0.1m is mainly analyzed and regarded as an example at low submergence depths. 3 typical flow speeds (𝑈=0.3m/s, 0.55m/s 0.85m/s) are selected to analyze the VIV responses. When the flow speed 𝑈=0.3m/s (Re = 24200), the VIV response is in the initial branch. The local sensitivity coefficients of the flow speed and submergence depth are 0.1092 and 0.8844, which indicates the amplitude is sensitive to the submergence depth. The VIV frequency is 0.7324Hz. The amplitude is only

12

Journal Pre-proof 0.01m. The lift force and amplitude etc. are compared in Fig. 8. The oscillation frequency is calculated by Fast Fourier Transform (FFT) of the displacement of the cylinder. The development and variation rules of the amplitude are similar to the lift force. 20

0.02 Amplitude (m)

Lift (N)

10 0 -10

Lift (N)

8

5

10

15 t (s) (a)

20

25

30

0.01

(0.73242Hz)

4 2 0

2

4 Frequency (Hz) (c)

Unstable Vibration

0

6

0

0 -0.01 -0.02

0

Amplitude (m)

-20

0.01

6

5

10

15 t (s) (b)

20

25

30

6

7

(0.73242Hz)

0.005

0

8

0

1

2

3 4 5 Frequency (Hz) (d)

Fig. 8 (a) Lift force versus time; (b) Amplitude response versus time; (c) Frequency spectrum for lift force and (d) Frequency spectrum for amplitude of the circular cylinder with submergence depth 𝑆= 0.1m at U=0.3m/s (𝑅𝑒 = 2.42 × 104)

The vorticity field and streamlines around the cylinder can be observed after CFD post-processing. Fig. 9 shows the vorticity field and streamlines. 2S vortex pattern can be observed. 2S means two single of vortices shed in one cycle of oscillation. By the way, in earlier works, the vortex shedding pattern has been designated by the number of vortices shed in one cycle of vibration. The different vortex patterns associated with specific VIV or FIM (Flow-Induced Motion) regimes have been named as 2S, 2P, and P + S etc. where 2S and 2P stand for two single and two pairs of vortices shed in one cycle. In our previous paper [6], the vortex shedding pattern was concluded as nP+2S. n can be calculated by

( ) ―1

n = ROUND

𝑓2 𝑓1

(13)

where 𝑓1 represents the frequency of VIV or FIM and in general is close to the natural frequency. 𝑓2 is the vortex shedding frequency. ROUND() provides the normal rounding functionality, rounding a number up or down based on any existing decimals. 2 single vortices (2S) are observed at top and bottom positions; n pairs of vortices (nP) are observed in the upward and downward journey. This formula illustrates the connection between the vortex shedding patterns and the frequencies. For VIV, the vortex shedding pattern is always 2S since the amplitude is relatively low. For galloping, due to the high-amplitude response, nP vortices will appear in the upward and downward journey. Thus, the vortex shedding pattern is always nP+2S for galloping or high-amplitude response. In a word, the vortex shedding pattern and the frequency are sensitive to the flow speed and amplitude. At the submergence depth 𝑆=0.1m, the cylinder is close to the free surface. The cylinder has a long wake including a series of vortices. The vortices are close to a free surface. Since the flow speed and the VIV amplitude is very low, the free surface deformation is not significant. Accordingly, the effects of submergence 13

Journal Pre-proof depth are not obvious. Phase-1 Air

Phase-1 Air Free surface

Free surface Original Location

S

Original Location

S

VIV

VIV

Phase-2 Water

Phase-2 Water (a)

(b)

Fig. 9 (a) Vorticity field and (b) streamlines around the cylinder with submergence depth 𝑆= 0.1m at U=0.3m/s (𝑅𝑒 = 2.42 × 104)

When the flow speed is 0.55m/s, the VIV response is in the upper branch. The local sensitivity coefficients of the flow speed and submergence depth are 0.0005 and 0.9924, which indicates the amplitude is sensitive to the submergence depth. The lift force, amplitude and frequency spectrum are compared in Fig. 10. The VIV amplitude is about 0.057m, which is significantly higher than the VIV amplitude in the initial branch. The frequency is 1.416Hz. Amplitude (m)

Lift (N)

20 0 -20 0

5

10

0.1 0.05 0 -0.05 -0.1

15

0

5

t (s) (a)

Amplitude (m)

(1.416Hz) Lift (N)

15

0.06

10

5

0

10 t (s) (b)

0

2

4 6 Frequency (Hz) (c)

8

(1.416Hz) 0.04 0.02 0

10

0

2

4 6 Frequency (Hz) (d)

8

10

Fig. 10 (a) Lift force versus time; (b) Amplitude response versus time; (c) Frequency spectrum for lift force and (d) Frequency spectrum for amplitude of the circular cylinder with submergence depth 𝑆= 0.1m at U=0.55m/s (𝑅𝑒 = 4.43 × 104)

In our experiments, the free surface deformation (wave-making phenomenon) becomes stronger as the flow speed increases. As shown in Fig. 11, the free surface deformation can be clearly observed as the flow speed increases to 0.55m/s. The blockage effect and Coanda effect cause the wave-making phenomenon. The vortices caused by the free surface deformation reduce the vortices generated from the upper the cylinder, since these two kinds of vortices are in opposite directions (two opposite directions are denoted by two different colors: blue and red.). This results in suppressing the VIV responses. This phenomenon is more obvious at high flow speeds. In summary, the submergence depth and flow speed all affect the free surface deformation and vorticity distribution. 14

Journal Pre-proof Phase-1 Air

Phase-1 Air

Vortices from free surface deformation

Original Location Wave

Free surface

Wave Original Location

Free surface

S

S

VIV

VIV

Phase-2 Water

Phase-2 Water (a)

(b)

Fig. 11 (a) Vorticity field and (b) streamlines around the cylinder with submergence depth 𝑆= 0.1m at U=0.55m/s (𝑅𝑒 = 4.43 × 104)

As the flow speed increases to 0.85m/s, the VIV response is located in the lower branch. The local sensitivity coefficients of the flow speed and submergence depth are 0.0165 and 0.9764. The lift force and amplitude are presented in Fig. 12. The VIV amplitude is only 0.02m. The frequency is 1.425Hz. As the flow speed increases, the shedding frequency keeps increasing. Meanwhile, the frequency has a strong tendency to synchronize with the shedding frequency and will move further away from the natural frequency, thus the

20

0.04

10

0.02

Amplitude (m)

Lift (N)

amplitude is suppressed and jumps to a low value in the VIV lower branch.

0 -10 -20

0

5

10

0 -0.02 -0.04

15

0

5

t (s) (a)

15

0.03

4

Amplitude (m)

Lift (N)

6

(1.425Hz)

2 0

10 t (s) (b)

0

2

4 Frequency (Hz) (c)

6

(1.425Hz) 0.02 0.01 0

8

0

2

4 Frequency (Hz) (d)

6

8

Fig. 12 (a) Lift force versus time; (b) Amplitude response versus time; (c) Frequency spectrum for lift force and (d) Frequency spectrum for amplitude of the circular cylinder with submergence depth 𝑆= 0.1m at U=0.85m/s (𝑅𝑒 = 6.86 × 104)

The vorticity field and streamlines are presented in Fig. 13. In this case, large-scale surface deformation is observed. The shape of free surface deformation is very similar to the experimental results in our lab and the published studies [27, 31]. It was clearly observed that two kinds of vortices are induced by the free surface deformation and the cylinder. More specifically, the direction of rotation of these two kinds of vortices is opposite. Accordingly, the vortices shed from the top of the cylinder are reduced by the vortices caused by the free surface deformation. Therefore, the lift force caused by the vortex shedding is reduced. That is why the

15

Journal Pre-proof VIV amplitude is compressed when the submergence depth decreases. Phase-1 Air

Phase-1 Air

Vortices from free surface deformation

Opposite

Wave

Free surface

Wave S

S

Free surface

Original Location

VIV Original Location

VIV Vortices from cylinder

Phase-2 Water

Phase-2 Water (a)

(b)

Fig. 13 (a) Vorticity field and (b) streamlines around the cylinder with submergence depth 𝑆= 0.1m at U=0.85m/s (𝑅𝑒 = 6.86 × 104)

According to the results above mentioned, the VIV amplitude is relatively low when at low submergence depths. The reasons are as follows. In general, two kinds of vortices (i.e. the vortices shed from the shear layer of the cylinder and the vortices generated from the deformed free surface) coexist in the flow. The first kind of vortices provides the power that drives the VIV responses of the cylinder. The deformed free surface is caused by pressure fluctuations between the free surface and cylinder, and produces a string of vortices which belong to the second kind of vortices. The second kind of vortices is the disturbance factor. There are two reasons leading to the decrease in the VIV responses and energy collection when the large-scale free surface deformation appears. Firstly, the shedding frequency of the vortices generated from the free surface doesn't synch with the vortices shed from the cylinder. Secondly, the vortices generated from the free surface and the upper side vortices generated from the cylinder are in opposite directions. As shown in Fig. 11 and Fig. 13, these opposite vortices are marked by different colors. Therefore, these two kinds of vortices gradually mix together resulting in the decrease in the intensity of vorticity around the cylinder. In other words, the deformed free surface and mixed vortices change the pressure distribution around the cylinder and reduce the lift force acting on the cylinder resulting in the decline in the VIV responses and energy conversion. 4.2.2 VIV at high submergence depth When the submergence depth exceeds 0.5m, the VIV amplitude and the energy conversion no longer increase and approach to the case with infinite submergence depth, which means the effects of submergence depth or the influence of the free surface can be ignored when S>0.5m. Therefore, the case with S=0.5m is selected as an example at high submergence depths. 3 typical flow speeds (𝑈=0.3m/s, 0.55m/s 1.05m/s) are selected. Due to the increase of submergence depth, the cylinder is far away from the free surface. Therefore, the motion of the cylinder will be not significantly affected by the free surface. At 𝑈=0.3m/s (Re = 24200), the VIV response is in the initial branch. The local sensitivity coefficients of the flow speed and submergence depth are 0.9400 and 0.0533. This indicates the amplitude is sensitive to the flow speed at high submergence depths, which is different from the cases at low submergence depth. In Fig. 14, the lift force, amplitude and frequency spectrum are compared. Regular VIV motion is observed. The VIV amplitude is about 0.01m. The VIV frequency is 0.7324Hz. 16

Journal Pre-proof 0.02 Amplitude (m)

20 Lift (N)

10 0 -10 -20 0

10 t (s) (a)

15

(0.73242Hz)

6 4 2 0

0

2

4 6 Frequency (Hz) (c)

8

0 -0.01 -0.02

20

Amplitude (m)

Lift (N)

8

5

0.01

0.01

5

10 t (s) (b)

15

20

(0.73242Hz)

0.005

0

10

0

0

2

4 6 Frequency (Hz) (d)

8

10

Fig. 14 (a) Lift force versus time; (b) Amplitude response versus time; (c) Frequency spectrum for lift force and (d) Frequency spectrum for amplitude of the circular cylinder with submergence depth 𝑆= 0.5m at U=0.3m/s (𝑅𝑒 = 2.42 × 104)

As shown in Fig. 15, the vorticity field and streamlines around the cylinder are presented. Two vortices are shed in one cycle of oscillation. Free surface deformation cannot be observed, which also means the effects of free surface are very weak. Phase-1 Air

Phase-1 Air Free surface

Free surface Phase-2 Water

Phase-2 Water

S

S

Original Location

Original Location

VIV

VIV

(a)

(b)

Fig. 15 (a) Vorticity field and (b) streamlines around the cylinder with submergence depth 𝑆= 0.5m at U=0.3m/s (𝑅𝑒 = 2.42 × 104)

When the flow speed is 0.55m/s, the VIV response is located in the upper branch. The local sensitivity coefficients of the flow speed and submergence depth are 0.8072 and 0.1899. In Fig. 16, the VIV amplitude is about 0.08m. The VIV frequency is 1.123Hz. The amplitude is significantly higher than that of low submergence depths (see Fig. 10).

17

40

0.1

20

0.05

Amplitude (m)

Lift (N)

Journal Pre-proof

0 -20 -40

0

5

10

0 -0.05 -0.1

15

0

5

t (s) (a)

10

15

t (s) (b) 0.08

Lift (N)

Amplitude (m)

(1.123Hz)

30 20 10 0

0

2

4 Frequency (Hz) (c)

6

0.04 0.02 0

8

(1.123Hz)

0.06

0

2

4 Frequency (Hz) (d)

6

8

Fig. 16 (a) Lift force versus time; (b) Amplitude response versus time; (c) Frequency spectrum for lift force and (d) Frequency spectrum for amplitude of the circular cylinder with submergence depth 𝑆= 0.5m at U=0.55m/s (𝑅𝑒 = 4.43 × 104)

The vorticity field and streamlines are presented in Fig. 17. A typical VIV wake structure is observed. Compared to Fig. 11, the wave-making phenomenon (free surface deformation) can not be observed, which means the effects of free surface are not obvious in this case. Phase-1 Air

Phase-1 Air

Free surface

Free surface

Phase-2 Water

Phase-2 Water S

S Original Location

Original Location

VIV

(a)

VIV

(b)

Fig. 18 (a) Vorticity field and (b) streamlines around the cylinder with submergence depth 𝑆= 0.5m at U=0.55m/s (𝑅𝑒 = 4.43 × 104)

As the flow speed increases to 1.05m/s, the VIV response is in the lower branch. The local sensitivity coefficients of the flow speed and submergence depth are 0.0008 and 0.9921. This indicates the amplitude is sensitive to the submergence depth in this case. The lift force and amplitude etc. are plotted in Fig. 18. The VIV frequency is 1.4648Hz. In the lower branch, the shedding frequency keeps increasing and becomes desynchronized from the natural frequency, resulting in the decreasing of the VIV amplitude. Consequently, the VIV amplitude is only 0.04m.

18

Journal Pre-proof 40 Amplitude (m)

Lift (N)

20 0 -20 -40

0

2

4

6 t (s) (a)

8

10

0.05 0 -0.05

12

0

2

4

6 t (s) (b)

8

10

12

10 Amplitude (m)

Lift (N)

(1.4648Hz) 5

0

0

2

4 6 Frequency (Hz) (c)

8

0.02

0

10

(1.4648Hz)

0.04

0

2

4 Frequency (Hz) (d)

6

8

Fig. 18 (a) Lift force versus time; (b) Amplitude response versus time; (c) Frequency spectrum for lift force and (d) Frequency spectrum for amplitude of the circular cylinder with submergence depth 𝑆= 0.5m at U=0.85m/s (𝑅𝑒 = 6.86 × 104)

As the flow speed increases, the free surface deformation is observed again, as shown in Fig. 19, since the blockage effect of the cylinder became prominent. However, the effects of free surface on the VIV cylinder are still weak. The VIV responses have been very close to the VIV responses at infinite submergence depth (see Fig. 7). In conclusion, when the submergence depth reaches up to 0.5m, its effects are very weak and can be ignored. Therefore, S=0.5m is considered as the critical value of the submergence depth.

Phase-1 Air

Phase-1 Air

Free surface

Phase-2 Water

Free surface

Phase-2 Water

Weak wave-making phenomenon

S

S

Original Location

Original Location

VIV

VIV

(a)

(b)

Fig. 19 (a) Vorticity field and (b) streamlines around the cylinder with submergence depth 𝑆= 0.5m at U=0.85m/s (𝑅𝑒 = 6.86 × 104)

4.3 Energy Conversion For the VIV energy conversion system, it is important to understand the effects of submergence depth on energy conversion. In Fig. 20, the converted power is plotted with the submergence depths of 0.1m, 0.2m, 0.3m, 0.5m, ++m. The observations are as follows: a) In the VIV initial branch, the converted power gradually increases for all submergence depths. The power curves are similar with each other, which means the effects of submergence depth on the converted power are not significant.

19

Journal Pre-proof b) In the VIV upper branch, the power graphs all exhibit a maximum, being the maximum power P=3.65W, 4.26W, 4.47W, 4.51W and 4.47W at the submergence depth S=0.1, 0.2m, 0.3m, 0.5m and ++m, respectively. The VIV upper branch is the most suitable region for harvesting energy. c) In the lower branch, as mentioned earlier, as the submergence depth decreases, the VIV amplitude decreases and results in a similar decreasing in energy conversion. In addition, as the submergence depth decreases, the synchronization range of VIV is also reduced and becomes narrow. d) In general, the converted power at small submergence depths is less than high submergence depths. This result can be attributed to the free surface deformation reducing the vorticity and lift force acting on the cylinder. Meanwhile, the free surface deformation also absorbs the kinetic energy of fluid. As a result, the available energy is lost. Therefore, the converted power decreases as the submergence depth decreases.

AmplitudePower (m) (W)

5

Upper

4.5 1 4 0.9 3.5 0.8 3 0.7 2.5 0.6 2 0.5 1.5 0.4 1 0.3 0.5 0.2 0 0.10.2 0

S=0.1m S=0.2m S=0.3m H=0.1m S=0.5m H=0.2m S=+∞m H=0.3m

Initial

H=0.5m H=++m

Lower Disappear

0.4 2

3

0.6 4

5

0.8 U (m/s) 6

1 7

Reynolds number

8

1.2 9

10 4

x 10

Fig. 20 Converted power of the circular cylinder for different submergence depths

The energy conversion efficiencies are compared in Fig. 21. At first, the efficiency keeps increasing in the initial branch. The maximum efficiency is achieved in the upper branch. The maximum efficiency is=28.4%, 31.5%, 33.5%, 34.7% and 34.3% at submergence depth S=0.1, 0.2m, 0.3m, 0.5m and ++m, respectively. Then, the efficiency begins to decrease. In the VIV initial branch, the efficiency curves are close to each other, which means the effects of submergence depth are not significant. In the VIV upper and lower branches, as the submergence depth increases, the energy conversion efficiency keeps increasing, since the disturbance from free surface becomes weak. When the submergence depth increases to 0.5m, the efficiency no longer increases and approaches to the case with the infinite submergence depth. The efficiency curve of S=0.5m has coincided many times with the efficiency curve of S=++m, which means the effects of submergence depth can be ignored. The results suggest that S=0.5m is considered as the critical submergence depth.

20

Journal Pre-proof 40

Upper

H=0.1m H=0.2m H=0.3m H=0.1m H=0.5m H=0.2m H=+∞m H=0.3m H=0.5m H=++m

35 1

Amplitude (m) Efficiency (%)

30 0.9

Initial

0.8 25

Lower

0.7 20 0.6

15 0.5 10 0.4

Disappear

0.3 5 0.2 0 0.10.2 0

0.4

2

3

0.6

4

5

0.8 U (m/s) 6

1

7

8

1.2

9

Reynolds number

10 4

x 10

Fig. 21 Energy conversion efficiency of the circular cylinder for different submergence depths

5. CONCLUSIONS In this paper, the VIV responses are extensively studied with different submergence depths. A series of simulations were carried out to explore more thoroughly the VIV responses within a wide range of flow speed (0.2m/s-1.3m/s). The simulations are based on Fluent and embedded UDF codes. Experimental results are used to verify the numerical solutions. Main conclusions include: (1) In general, as the submergence depth decreases, the VIV amplitude decreases and results in a similar decrease in the hydrokinetic energy conversion. Compared to high submergence depths, when the submergence depth is 0.1m, the VIV amplitude is significantly suppressed. The maximum amplitude is only 0.057m. Meanwhile, the synchronization range of VIV is shortened as the submergence depth decreases. (2) The VIV initial branch is first observed as the flow speed increases. The VIV responses gradually become strong. The vortex shedding is 2S pattern. In the VIV upper branch, the maximum amplitude reaches up to 0.08m (A*=0.9). In the VIV lower branch, the VIV amplitude decreases to zero. Accordingly, the converted power and energy conversion efficiency approaches zero. In this case, the oscillating cylinder produces a small amount of energy, which poses no application value in practical application. It is noted that when the flow speed exceed 1.2m/s (Re>9.7 × 104), the oscillation responses no longer belong to VIV. (3) For all the cases in the initial branch of VIV, the converted power increases as the flow speed increases. The maximum power appears in the upper branch and reaches up to about 4.51W. The converted power starts decreasing in the lower branch. As the submergence depth increases with the same velocity, the converted power keeps increasing. Therefore, a cylinder at high submergence depths can be positively exploited for the VIV energy conversion system. (4) The energy conversion efficiency increases with the increasing flow speed. After the VIV upper branch, the efficiency starts to decrease. As the submergence depth increases from 0.1 to 0.5, the maximum efficiency keeps increasing. Finally, the maximum energy conversion efficiency reaches up to 34.7%. The efficiency curve at the submergence depth S=0.5m coincides many times with the efficiency curve at the submergence depth S=++m, which means the effects of submergence depth has become weak when the 21

Journal Pre-proof submergence depth reaches 0.5m. In a word, these conclusions are believed to be of interest from both the scientific and the practicing engineering point of view. In the future, the other design parameters of the VIV hydrokinetic energy conversion system should be further improved and optimized.

ACKNOWLEDGEMENT This research is supported by the scholarship from China Scholarship Council (CSC) (Grant No. 201706290076) and the National Science Foundation of China (Grant No. 61572404).

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Journal Pre-proof [31] Bemitsas, M. M„ Raghavan, K., and Maroulis, D. Effect of Free Surface on VIV for Energy Harnessing at 8x10^3
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Highlights Proximity to free surface has negative influence on vortex shedding. The VIV amplitude is gradually suppressed as the submergence depth decreases. The maximum energy conversion efficiency is achieved in the VIV upper branch. The disturbance from free surface can be ignored when the submergence depth>0.5m. The oscillation response no longer belongs to VIV when the flow speed>1.2m/s.