# Research on the Hydrodynamic Performance of a Vertical Axis Current Turbine with Forced Oscillation

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## Abstract

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## 1. Introduction

_{t}(Turbine Thrust Coefficient) and C

_{p}(Turbine Power Output Efficiency) when the turbine was mounted closer to the free surface. The time-averaged C

_{t}and C

_{p}decreased as the immersion depth reduced.

## 2. Numerical Method for Vertical Axis Current Turbine under Forced Oscillation

#### 2.1. Turbine Load

_{A}, and hydrodynamic forces acting on the blades provide torque for axis O. The turbine rotates counter clockwise with a constant angular speed ω (the rotation dire). O-XYZ is the global coordinate system (absolute coordinate system). The origin of the global coordinate O is located at the center of the main shaft of the turbine. The X-axis is parallel to the incoming flow direction. The Y-axis is perpendicular to the X axis. The local coordinate system o-xyz (moving coordinate system) moves together with the blades. Origin o of the local coordinate system is located at the center of the blade autorotation axis. The x-axis is parallel to the chord line. The positive direction of the x-axis is from the airfoil leading edge to the trailing edge. The y-axis is perpendicular to the chord line. The azimuth angle of the blade is $\theta $. The pitch angle of the blade is $\phi $. The attack angle of the air foil is $\alpha $. $\alpha $ is positive in the counterclockwise direction.

_{e}―Reynolds number; σ―the compactness of the turbine; C―chord length; H―blade span length; C

_{P}―power capture efficiency; μ―kinematic viscosity of water; R―turbine radius; and D―turbine diameter;

#### 2.2. Hydrodynamic Analysis of the Vertical Axis Current Turbine under Forced Oscillation

- The VACT is connected to the floater through its shaft and the shaft is fixed onto the floater. The motion of the whole system is harmonic with a small oscillation amplitude.
- The spokes and main shaft of the turbine are assumed to have no effects on the turbine’s hydrodynamic characteristics.
- The diameter of the rotor is small compared to the wave length. The velocity of water particles caused by the incident wave is constant at water depth direction.
- VACT can be seen as a slender body and strip theory could be used.

_{A}. The boundary conditions of the whole flow domain are illustrated in Figure 5, while Figure 6 illustrates the details of meshes in sub-domains.

^{5}.

^{−4}. In the transient simulation setup adopted in this manuscript, the transient solver is used. The minimum number of coefficient loops is four at each time step. Adaptive time steps are chosen in the solution process. The time step is set to 1.0 × 10

^{−3}s. The maximum number of coefficient loops within each time step is 15. Detailed discussions of the validation of the time step are shown in Section 3.2. The high resolution second-order backward Euler scheme is used in the transient scheme option.

## 3. Validation of CFX Results

#### 3.1. Validation of Grid Independence

^{+}continues decreasing (from mesh 2 to mesh 3). Therefore, the density of mesh 2 can meet the mesh independence requirements in the numerical simulation. The detailed parameters of mesh 2 in Table 2 are adopted in the simulation process.

#### 3.2. Validation of Time Step

#### 3.3. Comparision between Experiments and Numerical Simulation

_{P}is obtained by Equation (13) in Section 2.1. As is shown in Figure 10, exp represents the value of experimental results. Sim represents the value of CFX simulation results. Previous papers [36,37,38] gave the scale coefficient at different arm installation locations and also gave the correlation between turbine span length and maximum power efficiency. The ratio between the span and radius of the turbine used in this experiment is 1.5. Li and Calisal pointed out that the power output could decrease as much as 19.5% in a numerical simulation when the ratio between the span and radius is 1.5. The arm used in this turbine can lead to a decrease of the power output as high as 20%. The cor curve represents the correction curve of sim according to the referenced research results.

## 4. Results from Hydrodynamic Analysis of the Rotor Under Forced Oscillation

#### 4.1. Flow Field Analysis Under Forced Oscillation

#### 4.2. Analysis of Loads on the Rotor of the Turbine

#### 4.3. Analysis of Loads on the Blade of the Rotor

_{A}= 3.5 m/s and λ = 2.0. Meanwhile, the rotor oscillates with the amplitude ξ

_{0}= 0.6 m and frequency ω

_{e}= 0.2 rad/s. As Figure 14c shows, the normal force coefficient is larger than the tangential force coefficient, which means that the normal force is the dominant part of the loads acting on the blade. Under an oscillation condition, the cyclic variation trend of the normal force curve and the tangential force coefficient curve is similar to the trend without oscillation, but the peak values change due to oscillation.

#### 4.4. Effects of Oscillating Frequency

#### 4.5. Effects of Oscillating Amplitude

#### 4.6. Analysis of Torque

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\overrightarrow{V}}_{R}$ | Incoming flow velocity |

$L$ | Lift force of a single blade |

${D}_{r}$ | Drag force of a single blade |

${M}_{o}$ | Torque acting on the o axis of the airfoil |

${f}_{x}$ | Force along the chord line |

${f}_{y}$ | Force perpendicular to the chord line |

${f}_{t}$ | Blade tangential force |

${f}_{n}$ | Radial blade force |

${F}_{X}$ | Thrust of the rotor |

${F}_{Y}$ | Lateral force of the rotor |

$q$ | Torque of a single blade |

SST | Shear Stress Turbulence |

N-S | Navier-Stokes |

LES | Large eddy simulation |

RANS | Reynolds-averaged Navier–Stokes |

Y plus | Dimensionless wall distance |

CEL | Computer expression language |

Cp | Power output efficiency |

$\overline{\mu}$ | Mean thrust of the rotor |

$\widehat{\mu}$ | Fluctuation of the amplitude |

$\sigma $ | Solidarity of the turbine |

TSR | Tip speed ratio |

MCT | marine current turbine |

VAW | Ts vertical axis wind turbines |

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**Figure 5.**Sub-domains and boundary conditions for simulation of forced oscillation in surge. (

**a**) Sub-domains for simulation of forced oscillation in surge. (

**b**) Boundary conditions.

**Figure 6.**Meshes in sub-domains. (

**a**) Meshes around blades. (

**b**) Meshes in stationary domain. (

**c**) Meshes in oscillatory domain and stationary domain. (

**d**) Meshes in rotational domain.

**Figure 9.**Experimental equipment. (

**a**) Circulation Water Tunnel. (

**b**) Supporting Platform. (

**c**) Two-bladed vertical axis turbine.

**Figure 11.**Comparison between experiement and numerical simulation results. (

**a**) Turbine normal force coefficient. (

**b**) Turbine tangential force coefficient.

**Figure 12.**Cloud distribution of vortex in the flow field of the turbine. (

**a**) Cloud distribution of vortex when the turbine only rotates. (

**b**) Cloud distribution of vortex when the turbine rotates under forced oscillation in sway. (

**c**) Cloud distribution of vortex when the turbine moves along the incident flow. (

**d**) Cloud distribution of vortex when the turbine moves against the incident flow.

**Figure 13.**Load curves of the rotor. (

**a**) Thrust coefficient of the rotor when it does not oscillate. (

**b**) Coefficient of tangential force when the rotor does not oscillate. (

**c**) Thrust coefficient of the rotor when it oscillates in surge. (

**d**) Coefficient of tangential force when the rotor oscillates in surge. (

**e**) Thrust coefficient of the rotor when it oscillates in sway. (

**f**) Coefficient of tangential force when the rotor oscillates in sway.

**Figure 14.**Normal and tangential force coefficient of one blade. (

**a**) Normal and tangential force coefficient of one blade when the rotor is static. (

**b**) Normal and tangential force coefficient of one blade when the rotor is in surge motion. (

**c**) Normal and tangential force coefficient of one blade when the rotor is in sway motion.

**Figure 15.**Time-averaged value and amplitude of load coefficients of the rotor vs. oscillating frequency. (

**a**) Time-averaged value and amplitude of load coefficients of the rotor under surge oscillation. (

**b**) Time-averaged value and amplitude of load coefficients of the rotor under sway oscillation.

**Figure 16.**Time-averaged value and amplitude of load coefficients of the rotor vs. oscillating amplitude. (

**a**) Time-averaged value and amplitude of load coefficients of the rotor under surge oscillation. (

**b**) Time-averaged value and amplitude of load coefficients of the rotor under sway oscillation.

**Figure 17.**Time-averaged value and amplitude of the torque coefficient of the rotor. (

**a**) Effects of oscillating frequency. (

**b**) Effects of oscillating amplitude.

Number of Blades | Turbine Diameter (m) | Chord Length (m) | Aspect Ratio | Compactness |
---|---|---|---|---|

2 | 6 | 0.9 | 6.5 | 0.0955 |

Mesh | Total Mesh Quantity (×10^{3}) | Y^{+} | Solution Time | Total Boundary Mesh Layer | Thickness of the First Layer |
---|---|---|---|---|---|

1 | 45 | 21.5–38.2 | 4 h | 30 | 0.0004 m |

2 | 119 | 2.65–4.76 | 12 h | 30 | 0.0001 m |

3 | 274 | 0.83–1.96 | 26 h | 30 | 0.00005 m |

Diameter (m) | Numer of Blades | Cord Length (m) | Air Foil | Aspect Ratio |
---|---|---|---|---|

0.8 m | 2 | 0.12 | NACA0018 | 1.5 |

Diameter D/m | Blades Number Z | Chord Length C/m | Airfoil | Rotation Speed $\mathit{\omega}$/rad ^{−1} |
---|---|---|---|---|

1.22 | 2 | 0.0914 | NACA0012 | 0.749 |

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**MDPI and ACS Style**

Ma, Y.; Hu, C.; Li, Y.; Deng, R.
Research on the Hydrodynamic Performance of a Vertical Axis Current Turbine with Forced Oscillation. *Energies* **2018**, *11*, 3349.
https://doi.org/10.3390/en11123349

**AMA Style**

Ma Y, Hu C, Li Y, Deng R.
Research on the Hydrodynamic Performance of a Vertical Axis Current Turbine with Forced Oscillation. *Energies*. 2018; 11(12):3349.
https://doi.org/10.3390/en11123349

**Chicago/Turabian Style**

Ma, Yong, Chao Hu, Yulong Li, and Rui Deng.
2018. "Research on the Hydrodynamic Performance of a Vertical Axis Current Turbine with Forced Oscillation" *Energies* 11, no. 12: 3349.
https://doi.org/10.3390/en11123349