# Study of Hydrodynamic Interference of Vertical-Axis Tidal Turbine Array

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

_{1}, y

_{1}) and as (x

_{2}, y

_{2}) for turbine B. The relevant parameters in Figure 1 are expressed as

_{1}, y

_{1}) and (x

_{2}, y

_{2}).

_{1}of the turbine in time t, θ = ωt, in which ω is the angular velocity about the rotation center O

_{1}.

**n**

_{b}is the normal of the blade surface, S

_{b};

**r**is the vector from the origin, o, to any surface point in the coordinate system, $\xi o\eta $. At infinity, $\Phi \left(p,t\right)$ satisfies the velocity potential for uniform incoming flow according to

_{0}is the initial time.

_{p}

_{1}, the power coefficient of turbine B as C

_{p}

_{2}and the power coefficient of a single turbine under the same conditions as C

_{p}. The relative power coefficient of double turbines array is

## 3. Results

#### 3.1. Issues of Turbine Array

#### 3.2. Validation of the Method

#### 3.3. Analysis of Results

#### 3.3.1. Cases of Ψ = 0°

#### 3.3.2. Cases of Ψ = 90°

#### 3.3.3. Case of Arbitrary Ψ

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

BEMT | Blade Element Momentum Theory |

CFD | Computational Fluid Dynamics |

LES | Large Eddy Simulation |

RANS | Reynolds Average Navier-Stokes |

BEMT | Blade Element Momentum Theory |

Xd, Yd | coordinate spacing of the rotation centers of turbines in twin-turbine system |

$\psi $ | angle between the connection of the rotation centers of turbines and the X axis |

L | distance of the rotation centers of turbines |

t | time |

t_{0} | initial time |

Ω | rotating angular velocity about the origin o |

U | translational velocity of the origin o |

V | velocity of uniform incoming flow at infinity |

θ | azimuthal angle |

$\omega $ | rotating angular velocity about the rotation center of turbine |

$\Phi $ | velocity potential |

${\tau}_{e}$ | fluid domain |

$\varphi $ | perturbation velocity potential |

P | pressure on the blade surface |

f | force on the blade |

q | moment of the blade |

Cq | moment coefficient of a single blade on turbine |

F | hydrodynamic force acting on the turbine |

Q | hydrodynamic torque acting on the turbine rotor shaft |

$\overline{F}$ | resultant force coefficient of the turbine |

CQ | average torque coefficient of the turbine rotor shaft |

CP | power coefficient of the turbine |

D | diameter of turbine |

R | radius of turbine |

b | wingspan length of blade |

C | chord length of blade |

Z | blade number of a single turbine |

λ | tip speed ratios |

${{C}_{p}}^{\prime}$ | relative power coefficient of double turbines array |

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**Figure 3.**Central path of trailing vortices of turbine array; tip speed ratio λ = 2. The mutual interference of wake of the turbines is shown. (

**a**) Three turbines in “-” type arrangement; (

**b**) Three turbines in “Z” type arrangement.

**Figure 4.**The variation of relative power coefficient with turbine spacing of double turbines in parallel connection. (

**a**) λ = 1.7; (

**b**) λ = 2.75.

**Figure 6.**Influence of spacing on turbine moment coefficient where Z = 1. (

**a**) The impact on downstream turbine; (

**b**) The impact on upstream turbine.

**Figure 8.**The force of the rotor varies with the azimuthal angle in the case of double turbines in series where Z = 3 and λ = 2.

**Figure 9.**The relative power coefficient varies with the spacing in parallel connection where λ = 2.5.

**Figure 10.**The variation law of single blade moment coefficient with azimuthal angle in parallel turbine where λ = 2.5.

**Figure 11.**The comparison of the turbine torque coefficient in the parallel and single situation where Z = 3.

**Figure 12.**The comparison of the turbine torque coefficient in the parallel and single situation where Z = 4.

**Figure 13.**The comparison of the turbine resultant force coefficient in the parallel and single situation where Z = 3 and λ = 2.5.

**Figure 14.**Influence of deflection angle on hydrodynamic characteristics of hydraulic turbine where Z = 3 and L = 4R.

**Figure 15.**Influence of deflection angle on hydrodynamic characteristics of hydraulic turbine where Z = 2.

**Figure 16.**Influence of deflection angle on hydrodynamic characteristics of hydraulic turbine where Z = 2.

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**MDPI and ACS Style**

Li, G.; Chen, Q.; Gu, H.
Study of Hydrodynamic Interference of Vertical-Axis Tidal Turbine Array. *Water* **2018**, *10*, 1228.
https://doi.org/10.3390/w10091228

**AMA Style**

Li G, Chen Q, Gu H.
Study of Hydrodynamic Interference of Vertical-Axis Tidal Turbine Array. *Water*. 2018; 10(9):1228.
https://doi.org/10.3390/w10091228

**Chicago/Turabian Style**

Li, Guangnian, Qingren Chen, and Hanbin Gu.
2018. "Study of Hydrodynamic Interference of Vertical-Axis Tidal Turbine Array" *Water* 10, no. 9: 1228.
https://doi.org/10.3390/w10091228