# Analytical Model for Phase Synchronization of a Pair of Vertical-Axis Wind Turbines

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## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Numerical Results

#### 3.1. Characteristics of Rotors

#### 3.2. Phase Synchronization

#### 3.3. Dependence of the Synchronized Angular Velocity on the Gap

#### 3.4. Oscillation Period of the Difference in Angular Velocities

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${D}_{i}$ | Diameter of rotor i |

${R}_{i}$ | Radius of rotor i |

${g}_{ij}$ | Gap between rotors i and j |

${n}_{i}$ | Number of blades on rotor i |

x | Coordinate in streamwise direction |

y | Coordinate in spanwise direction |

${x}_{ik}$ | x coordinate of blade k of rotor i |

${y}_{ik}$ | y coordinate of blade k of rotor i |

${\psi}_{ik}$ | Azimuthal angle of blade k of rotor i |

${c}_{ik}$ | Chord length of blade k of rotor i |

${S}_{ik}$ | Projected area of blade k of rotor i along rotation direction |

${W}_{ik,j\ell}$ | Distance between blade k of rotor i and blade ℓ of rotor j |

${\varphi}_{ij}$ | Angle of rotor j observed from rotor i |

V | Upstream flow speed |

${V}_{i}$ | Effective flow velocity at rotor i |

${I}_{i}$ | Moment of inertia of rotor i |

${\omega}_{i}$ | Angular velocity of rotor i |

${Q}_{i}$ | Rotor torque on rotor i |

${L}_{i}$ | Load torque on rotor i |

${Q}_{\mathrm{p}i}$ | Torque due to pressure fluctuation on rotor i |

${\psi}_{i}$ | Azimuthal angle of representative blade of rotor i |

${Q}_{\mathrm{max}}\left({V}_{i}\right)$ | Maximum rotor torque for given ${V}_{i}$ |

${\omega}_{0}\left({V}_{i}\right)$ | No-load angular velocity for given ${V}_{i}$ |

${F}_{n}\left(\psi \right)$ | Torque-modulation function for n-blades rotor |

${c}_{1}$ | Parameter for ${Q}_{\mathrm{max}}\left({V}_{i}\right)$ |

${c}_{2}$ | Parameter for ${\omega}_{0}\left({V}_{i}\right)$ |

${\lambda}_{i}$ | Tip-speed ratio of rotor i |

$\psi $ | Azimuthal angle fixed in space |

${\sigma}_{i}$ | Solidity of rotor i |

${F}_{1}\left(\psi \right)$ | Torque-modulation function for single-blade rotor |

${a}_{i}$ | Parameter for self-induced velocity |

${\Gamma}_{j}$ | Circulation of rotor j |

${c}_{3}$ | Parameter for ${\Gamma}_{j}$ |

${c}_{4}$ | Parameter for ${L}_{i}$ |

${V}_{\mathrm{av}}$ | Average flow velocity $({V}_{1}+{V}_{2})/2$ |

$\delta V$ | Change in flow velocity |

$\delta p$ | Pressure fluctuation |

p | Air pressure |

$\alpha $ | Parameter controlling strength of interaction between rotors |

${\omega}_{\mathrm{SI}}$ | Steady-state angular velocity of single rotor by CFD using DFBI model |

${Q}_{\mathrm{SI}}$ | Steady-state rotor torque of single rotor by CFD using DFBI model |

${\omega}_{\mathrm{av}}\left(0\right)$ | Average of initial angular velocities $\left(\right|{\omega}_{1}\left(0\right)|+|{\omega}_{2}\left(0\right)\left|\right)/2$ |

$\delta {\psi}_{k}$ | Phase differences between representative blade of rotor 1 and blade k of rotor 2 |

$\Delta \omega $ | Difference of angular velocities $|{\omega}_{1}|-|{\omega}_{2}|$ |

${C}_{qi}$ | Torque coefficient of rotor i |

${\widehat{Q}}_{\mathrm{max}}$ | Normalized maximum rotor torque |

${\lambda}_{0}$ | No-load tip-speed ratio |

$\rho $ | Mass density of air |

${A}_{i}$ | Swept area of rotor i |

${\widehat{\Gamma}}_{j}$ | Normalized circulation of rotor j |

${C}_{Li}$ | Normalized load torque for rotor i |

${c}_{L}$ | Parameter for ${C}_{Li}$ |

## Appendix A. Verification of Parameter Dependence

**Figure A3.**Torque dependence on the azimuthal angle obtained by CFD. Equation (6) tries to simulate this dependence.

## Appendix B. Normalized Expression

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**Figure 3.**Modulation function ${F}_{n}\left(\psi \right)$ of the rotor torque on the azimuthal angle of the blade. The curves $n=1$ and $n=3$ plot the modulation for a single blade and a summation over three blades, respectively. The average of ${F}_{n}\left(\psi \right)$ over $\psi $ is taken to be unity.

**Figure 4.**The rotor torques for $V=6,8,10$, and $12\phantom{\rule{0.166667em}{0ex}}\mathrm{m}/\mathrm{s}$ and load torques are plotted against the angular velocity of the rotor.

**Figure 5.**Time evolution of ${\omega}_{i}$ for $\alpha =0,0.05,0.1$, and $0.2$ in the CD layout with a $10\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ gap.

**Figure 6.**Time evolution of phase differences $\delta {\psi}_{k}=({\psi}_{11}+{\psi}_{2k})\phantom{\rule{0.166667em}{0ex}}\mathrm{mod}\phantom{\rule{0.166667em}{0ex}}2\pi $ for $\alpha =0.05$ and $0.2$ in the CD layout with $10\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ gap.

**Figure 7.**Time evolution of ${\omega}_{i}$ for $\alpha =0.05$ and $0.2$ in the CO layout with $10\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ gap. Induced velocities are not taken into account.

**Figure 8.**Time evolution of ${\omega}_{i}$ for the (

**a**) CO and (

**b**) CD layouts with $\alpha =0.1$ and ${g}_{12}=10\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ when the rotor torque of rotor 2 is 95% that of rotor 1. The induced velocities are not taken into account.

**Figure 9.**The angular velocity in the phase-synchronized steady state is plotted against the gap ${g}_{12}/D$. The dashed line around $\omega \simeq 372\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}/\mathrm{s}$ shows the angular velocity at the steady state without the mutually induced velocity.

**Figure 10.**Time evolution of difference of angular velocities $\Delta \omega :=|{\omega}_{1}|-|{\omega}_{2}|$ is shown. The oscillation period is longer for larger gap distances, as well as for the CU layout compared with the CD layout.

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**MDPI and ACS Style**

Furukawa, M.; Hara, Y.; Jodai, Y. Analytical Model for Phase Synchronization of a Pair of Vertical-Axis Wind Turbines. *Energies* **2022**, *15*, 4130.
https://doi.org/10.3390/en15114130

**AMA Style**

Furukawa M, Hara Y, Jodai Y. Analytical Model for Phase Synchronization of a Pair of Vertical-Axis Wind Turbines. *Energies*. 2022; 15(11):4130.
https://doi.org/10.3390/en15114130

**Chicago/Turabian Style**

Furukawa, Masaru, Yutaka Hara, and Yoshifumi Jodai. 2022. "Analytical Model for Phase Synchronization of a Pair of Vertical-Axis Wind Turbines" *Energies* 15, no. 11: 4130.
https://doi.org/10.3390/en15114130