# Optimization Study of Marine Energy Harvesting from Vortex-Induced Vibration Using a Response-Surface Method

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## Abstract

**:**

## 1. Introduction

## 2. Optimization Methodology

#### 2.1. Concept of Response-Surface Method

#### 2.2. Feasibility Verification of Response-Surface Method

## 3. Numerical Methods of Modeling Fluid–Structure Interaction

#### 3.1. Computational Fluid Dynamics

#### 3.2. Rigid Body Motions

## 4. Test Case Description and Solution Verification

## 5. Simulation Design Using a Response-Surface Method

#### 5.1. Simulation Design and Statistical Analysis

#### 5.2. Verification of Optimum Parameters Using CFD Simulations

## 6. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ANOVA | Analysis of variance |

BBD | Box–Behnken design |

CFD | Computational-fluid dynamics |

CFL | Courant–Friedrichs–Lewy number |

DOE | Design-of-experiment |

DOF | Degree of freedom |

RANS | Reynolds-averaged Navier–Stokes |

RSM | Response-surface method |

RSM-BBD | Box–Behnken design response surface method |

VIV | Vortex-induced vibration |

VIVACE | Vortex-induced vibration aquatic clean energy |

Nomenclature | |

$\Delta t$ | Time step |

$\Delta x$ | Grid size |

${\delta}_{D}$ | Numerical error |

$\u03f5$ | Error variable |

$\zeta $ | Damping ratio |

$\eta $ | Energy capture efficiency |

$\nu $ | Kinematic viscosity |

$\rho $ | Density |

${\varphi}_{i}$ | Solution obtained on grid i |

${\varphi}_{\infty}$ | Numerical benchmark result |

${A}^{*}$ | Amplitude ratio |

c | Structural damping |

${c}_{d}$ | Drag coefficient |

${c}_{l}$ | Lift coefficient |

d | Diameter |

${f}_{n}$ | Natural frequency |

k | Structural stiffness |

$\mathbf{F}$ | Force vector |

$\mathbf{g}$ | Gravity vector |

l | Length |

m | Structural mass |

${m}^{*}$ | Mass ratio |

$\mathbf{n}$ | Surface normal vector |

p | Pressure |

P | Ratio of convergence |

${p}_{RE}$ | Observed order of accuracy |

${p}_{th}$ | Theoretical order of convergence |

R | Convergence ratio |

$Re$ | Reynolds number |

T | Oscillatory period |

${U}_{D}$ | Uncertainty |

v | Free stream velocity |

$\mathbf{v}$ | Fluid velocity field vector |

${v}_{r}$ | Reduced velocity |

${x}_{i}$ | Operating variable |

${y}^{+}$ | Dimensionless grid height for the first-layer grid |

## References

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**Figure 1.**The Box–Behnken design as derived from a cube, consisting of the central point and the middle points of the edges.

**Figure 2.**The response-surface images computed from results collected from [38] where the interaction effects of operating variables on the energy conversion efficiency are demonstrated (the data was from [38]). (

**a**) Interaction of spacing ratio and diameter ratio. (

**b**) Interaction of spacing ratio and reduced velocity. (

**c**) Interaction of diameter ratio and reduced velocity.

**Figure 3.**Perspective views of the computational domain, mesh topology, and grid details near the considered cylinder. (

**a**) Sketch of computational domain. (

**b**) Overview of mesh topology. (

**c**) Grid details near cylinder.

**Figure 4.**Convergencestudy, in terms of time-step and grid-spacing sizes, using the predicted amplitude ratios.

**Figure 5.**Response amplitude ratio and lift coefficient in the time and frequency domains, at its optimal operating condition ($Re=4.97\times {10}^{4}$).

**Figure 6.**Example distributions of velocity and vorticity for the considered cylinder, at its optimal operating condition ($Re=4.97\times {10}^{4}$). (

**a**) Distribution of velocity magnitude. (

**b**) Distribution of vorticity magnitude.

**Figure 7.**The predicted energy capture ratio by the BBD algorithm versus the actual energy capture ratio, simulated using CFD.

Run | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | OP |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\eta $ [%] | 29.87 | 18.33 | 35.41 | 21.66 | 29.87 | 27.08 | 14.16 | 29.87 | 25.25 | 16.25 | 22.54 | 9.58 | 28.33 | 29.87 | 16.25 | 29.87 | 26.66 | 36.10 |

**Table 2.**Estimated errors and uncertainties of response ratio (${A}^{*}$) based on various time-step and grid-spacing sizes.

Property | ${\mathit{A}}_{1}^{*}$ [-] | ${\mathit{A}}_{2}^{*}$ [-] | ${\mathit{A}}_{3}^{*}$ [-] | R [-] | ${\mathit{A}}_{\mathit{\infty}}^{*}$ [-] | ${\mathit{\u03f5}}_{1}$ [%] | ${\mathit{\u03f5}}_{2}$ [%] | ${\mathit{\u03f5}}_{3}$ [%] | U [%] |
---|---|---|---|---|---|---|---|---|---|

$\Delta t$ | 0.5625 | 0.6043 | 0.6075 | 0.077 | 0.6080 | −7.48 | −0.61 | −0.08 | 0.41 |

$\Delta x$ | 0.5950 | 0.6075 | 0.6087 | 0.096 | 0.6089 | −2.29 | −0.76 | −0.04 | 0.16 |

Variables | Symbols | Level −1 | Level 0 | Level 1 |
---|---|---|---|---|

Velocity [m/s] | A | 0.55 | 0.65 | 0.75 |

Stiffness [N/m] | B | 300 | 400 | 500 |

Mass [kg] | C | 2.60 | 3.00 | 3.40 |

Run | Factor A Velocity [m/s] | Factor B Stiffness [N/m] | Factor C Mass [kg] | Response $\mathit{\eta}$ Efficiency [-] |
---|---|---|---|---|

1 | 0.55 | 300 | 3.0 | 0.140 |

2 | 0.75 | 300 | 3.0 | 0.127 |

3 | 0.55 | 500 | 3.0 | 0.166 |

4 | 0.75 | 500 | 3.0 | 0.113 |

5 | 0.55 | 400 | 2.6 | 0.139 |

6 | 0.75 | 400 | 2.6 | 0.095 |

7 | 0.55 | 400 | 3.4 | 0.128 |

8 | 0.75 | 400 | 3.4 | 0.112 |

9 | 0.65 | 300 | 2.6 | 0.120 |

10 | 0.65 | 500 | 2.6 | 0.129 |

11 | 0.65 | 300 | 3.4 | 0.124 |

12 | 0.65 | 500 | 3.4 | 0.116 |

13 | 0.65 | 400 | 3.0 | 0.124 |

Source | Coefficient | Sum of Squares | DOF | Mean Square | f-Value | p-Value |
---|---|---|---|---|---|---|

Model | 0.1240 | 0.0034 | 9 | 0.0004 | 55.50 | <0.0001 |

A | −0.0158 | 0.0020 | 1 | 0.0020 | 290.9 | <0.0001 |

B | 0.0016 | 0.0000 | 1 | 0.0000 | 3.100 | 0.1218 |

C | −0.0004 | 1.1 × 10${}^{-6}$ | 1 | 1.1 × 10${}^{-6}$ | 0.165 | 0.6968 |

AB | −0.0100 | 0.0004 | 1 | 0.0004 | 58.64 | 0.0001 |

AC | 0.0070 | 0.0002 | 1 | 0.0002 | 28.73 | 0.0011 |

BC | −0.0042 | 0.0001 | 1 | 0.0001 | 10.59 | 0.0140 |

A${}^{2}$ | 0.0044 | 0.0001 | 1 | 0.0001 | 11.81 | 0.0109 |

B${}^{2}$ | 0.0081 | 0.0003 | 1 | 0.0003 | 40.75 | 0.0004 |

C${}^{2}$ | −0.0099 | 0.0004 | 1 | 0.0004 | 60.19 | 0.0001 |

Residual | - | 0.0000 | 7 | 6.8 × 10${}^{-6}$ | - | - |

Adj. R${}^{2}$ | 0.9684 | - | - | - | - | - |

Pre. R${}^{2}$ | 0.7789 | - | - | - | - | - |

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## Share and Cite

**MDPI and ACS Style**

Xu, P.; Jia, S.; Li, D.; el Moctar, O.; Jiang, C.
Optimization Study of Marine Energy Harvesting from Vortex-Induced Vibration Using a Response-Surface Method. *J. Mar. Sci. Eng.* **2023**, *11*, 668.
https://doi.org/10.3390/jmse11030668

**AMA Style**

Xu P, Jia S, Li D, el Moctar O, Jiang C.
Optimization Study of Marine Energy Harvesting from Vortex-Induced Vibration Using a Response-Surface Method. *Journal of Marine Science and Engineering*. 2023; 11(3):668.
https://doi.org/10.3390/jmse11030668

**Chicago/Turabian Style**

Xu, Peng, Shanshan Jia, Dongao Li, Ould el Moctar, and Changqing Jiang.
2023. "Optimization Study of Marine Energy Harvesting from Vortex-Induced Vibration Using a Response-Surface Method" *Journal of Marine Science and Engineering* 11, no. 3: 668.
https://doi.org/10.3390/jmse11030668