# Numerical and Experimental Identification of the Aerodynamic Power Losses of the ISWEC

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## Abstract

**:**

## 1. Introduction

## 2. Geometry

## 3. Semi-Empiric Model

#### 3.1. Theoretical Background

- Axial contribution: closed-form analytic solution of the NSE for the flow field within the radial gap between two cylinders (Taylor–Couette flow);
- Transversal contribution: closed-form analytic solution of the NSE for the motion field of a fluid on a rotating disk.

#### 3.2. Axial Contribution—In Housing

#### 3.3. Transversal Contribution—In Housing

#### 3.4. Axial Contribution—Free

#### 3.5. Transversal Contribution—Free

## 4. Case Study: Semi-Empiric Model

#### 4.1. In Housing

#### 4.2. Free

## 5. CFD Model

#### 5.1. RANS Turbulence Model

#### 5.2. Numerical Grid

#### 5.3. CFD Tests Map and Results

## 6. Experimental Test Rig

#### 6.1. Test Rig Operation Mode

#### 6.2. Data Acquisition

#### 6.3. Seals Friction

#### 6.4. Bearings Friction

- (a)
- If there is no gyroscopic effect ($\dot{\epsilon}=0$), E and ${M}_{bear}$ are insensitive to $\dot{\phi}$. Consequently, the power needed to overcome the bearings friction is proportional to $\dot{\phi}$;
- (b)
- If the gyroscope is activated by the pitch motion of the device, $\dot{\epsilon}\ne 0$ and the power losses become proportional to ${\dot{\phi}}^{2}$.

## 7. Experimental and Numerical Models Results Comparison

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

- Convective terms (C):$$\begin{array}{c}{C}_{2,1}=\frac{\partial {u}_{1}^{\ast}{u}_{2}^{\ast}}{\partial {x}_{1}^{\ast}}\approx \frac{{u}_{1}^{\ast}{u}_{2}^{\ast}}{\Delta {x}_{1}^{\ast}},\hfill \\ {C}_{2,2}=\frac{\partial {u}_{2}^{\ast}{u}_{2}^{\ast}}{\partial {x}_{2}^{\ast}}\approx \frac{{u}_{2}^{\ast}{u}_{2}^{\ast}}{\Delta {x}_{2}^{\ast}},\hfill \\ {C}_{2,3}=\frac{\partial {u}_{3}^{\ast}{u}_{2}^{\ast}}{\partial {x}_{3}^{\ast}}\approx \frac{{u}_{3}^{\ast}{u}_{2}^{\ast}}{\Delta {x}_{3}^{\ast}},\hfill \\ {C}_{3,1}=\frac{\partial {u}_{1}^{\ast}{u}_{3}^{\ast}}{\partial {x}_{1}^{\ast}}\approx \frac{{u}_{1}^{\ast}{u}_{3}^{\ast}}{\Delta {x}_{1}^{\ast}},\hfill \\ {C}_{3,2}=\frac{\partial {u}_{2}^{\ast}{u}_{3}^{\ast}}{\partial {x}_{2}^{\ast}}\approx \frac{{u}_{2}^{\ast}{u}_{3}^{\ast}}{\Delta {x}_{2}^{\ast}},\hfill \\ {C}_{3,3}=\frac{\partial {u}_{3}^{\ast}{u}_{3}^{\ast}}{\partial {x}_{3}^{\ast}}\approx \frac{{u}_{3}^{\ast}{u}_{3}^{\ast}}{\Delta {x}_{3}^{\ast}},\hfill \end{array}$$
- Pressure terms (P):$$\begin{array}{c}{P}_{2}=\frac{\partial P}{\partial {x}_{2}^{\ast}}\approx \frac{\Delta P}{\Delta {x}_{2}^{\ast}},\hfill \\ {P}_{3}=\frac{\partial P}{\partial {x}_{3}^{\ast}}\approx \frac{\Delta P}{\Delta {x}_{3}^{\ast}},\hfill \end{array}$$
- Viscous terms (V):$$\begin{array}{c}{V}_{2,1}=\frac{1}{Re}\left(\frac{{\partial}^{2}{u}_{2}^{\ast}}{\partial {x}_{1}^{\ast 2}}\right)\approx \frac{1}{Re}\frac{{u}_{2}^{\ast}}{\Delta {x}_{1}^{\ast 2}},\hfill \\ {V}_{2,2}=\frac{1}{Re}\left(\frac{{\partial}^{2}{u}_{2}^{\ast}}{\partial {x}_{2}^{\ast 2}}\right)\approx \frac{1}{Re}\frac{{u}_{2}^{\ast}}{\Delta {x}_{2}^{\ast 2}},\hfill \\ {V}_{2,3}=\frac{1}{Re}\left(\frac{{\partial}^{2}{u}_{2}^{\ast}}{\partial {x}_{3}^{\ast 2}}\right)\approx \frac{1}{Re}\frac{{u}_{2}^{\ast}}{\Delta {x}_{3}^{\ast 2}},\hfill \\ {V}_{3,1}=\frac{1}{Re}\left(\frac{{\partial}^{2}{u}_{3}^{\ast}}{\partial {x}_{1}^{\ast 2}}\right)\approx \frac{1}{Re}\frac{{u}_{3}^{\ast}}{\Delta {x}_{1}^{\ast 2}},\hfill \\ {V}_{3,2}=\frac{1}{Re}\left(\frac{{\partial}^{2}{u}_{3}^{\ast}}{\partial {x}_{2}^{\ast 2}}\right)\approx \frac{1}{Re}\frac{{u}_{3}^{\ast}}{\Delta {x}_{2}^{\ast 2}},\hfill \\ {V}_{3,3}=\frac{1}{Re}\left(\frac{{\partial}^{2}{u}_{3}^{\ast}}{\partial {x}_{3}^{\ast 2}}\right)\approx \frac{1}{Re}\frac{{u}_{3}^{\ast}}{\Delta {x}_{3}^{\ast 2}},\hfill \end{array}$$
- Fictitious force terms (B):$$\begin{array}{c}{B}_{1}=0,\hfill \\ {B}_{2,1}=-{x}_{3}^{\ast}{\theta}_{0}{\omega}^{2}sin\omega t\approx -R{\theta}_{0}\frac{4{\pi}^{2}}{{T}^{2}},\hfill \\ {B}_{2,2}=+{x}_{2}^{\ast}{\theta}_{0}{\omega}^{2}{cos}^{2}\omega t\approx \frac{H}{2}{\theta}_{0}^{2}\frac{4{\pi}^{2}}{{T}^{2}},\hfill \\ {B}_{2,3}=+2{u}_{3}^{\ast}{\theta}_{0}\omega cos\omega t\approx 2{u}_{3}^{\ast}{\theta}_{0}\frac{2\pi}{T},\hfill \\ {B}_{3,1}=+{x}_{2}^{\ast}{\theta}_{0}{\omega}^{2}sin\omega t\approx \frac{H}{2}{\theta}_{0}\frac{4{\pi}^{2}}{{T}^{2}},\hfill \\ {B}_{3,2}=+{x}_{3}^{\ast}{\theta}_{0}^{2}{\omega}^{2}{cos}^{2}\omega t\approx R{\theta}_{0}\frac{4{\pi}^{2}}{{T}^{2}},\hfill \\ {B}_{3,3}=-2{u}_{2}^{\ast}{\theta}_{0}\omega cos\omega t\approx -2{u}_{2}^{\ast}{\theta}_{0}\frac{2\pi}{T}.\hfill \end{array}$$

C[%] | V[%] | P[%] | B[%] | |
---|---|---|---|---|

Momentum eq. - ${x}_{2}$ | 0 | 0 | 94 | 6 |

Momentum eq. - ${x}_{3}$ | 80 | 0 | 20 | 0 |

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**Figure 3.**Axial component of the fluid friction torque as a function of the pressure at different rotational velocities.

**Figure 4.**Transversal component on each disk of the fluid friction torque as a function of the pressure at different rotational velocities.

**Figure 5.**Total fluid friction torque as a function of the pressure at different rotational velocities.

**Figure 6.**Total fluid friction torque as a function of the rotational velocity at different pressure values.

**Figure 8.**Fluid velocity field for the in housing configuration, with ${p}_{i}$ of 1 kPa and $\dot{\phi}$ of 200 rpm and 800 rpm, on the left and right of the figure, respectively.

**Figure 13.**Loading scheme of the bearings for different precession angles, equal to 0, 45 and 90 degrees, from

**left**to

**right**.

**Figure 15.**Aerodynamic power: Free configuration results comparison against available experimental data.

**Figure 16.**Aerodynamic power: In housing configuration results comparison against available experimental data.

Low Turbulent | $\mathit{R}{\mathit{e}}_{\mathit{\varphi}\mathit{m}}\le {10}^{4}$ | $\mathit{C}}_{\mathit{M}}^{\mathit{a}}=1.03{\left(\frac{\mathit{d}}{{\mathit{R}}_{1}}\right)}^{0.3}\mathit{R}{\mathit{e}}_{\mathit{\varphi}\mathit{m}}^{-0.5$ |

High Turbulent | $R{e}_{\varphi m}>{10}^{4}$ | $C}_{M}^{a}=0.065{\left(\frac{d}{{R}_{1}}\right)}^{0.3}R{e}_{\varphi m}^{-0.2$ |

Laminar | $\mathit{R}{\mathit{e}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{k}}<{10}^{4}$ | $\mathit{C}}_{\mathit{M}}^{\mathit{t}\mathit{i}}=\frac{\mathit{\pi}{\mathit{R}}_{1}}{\mathit{s}}\frac{1}{\mathit{R}{\mathit{e}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{k}}$ |

Low Turbulent | $10}^{4}<R{e}_{disk}<2\ast {10}^{5$ | $C}_{M}^{ti}=\frac{1.334}{R{e}_{disk}^{1/2}$ |

High Turbulent | $R{e}_{disl}>2\ast {10}^{5}$ | $C}_{M}^{ti}=\frac{0.0311}{R{e}_{disk}^{1/5}$ |

Laminar | $\mathit{R}{\mathit{e}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{k}}<3\ast {10}^{5}$ | $\mathit{C}}_{\mathit{M}}^{\mathit{t}\mathit{f}}=\frac{1.935}{\sqrt{\mathit{R}{\mathit{e}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{k}}}$ |

Turbulent | $R{e}_{disk}>3\ast {10}^{5}$ | $C}_{M}^{tf}=\frac{0.073}{R{e}_{disk}^{1/5}$ |

In Housing | Free | |||
---|---|---|---|---|

Min | Max | Min | Max | |

$R{e}_{disk}$ | $1.5661\xb7{10}^{4}$ | $6.2643\xb7{10}^{6}$ | $1.5661\xb7{10}^{4}$ | $6.2643\xb7{10}^{6}$ |

$R{e}_{\phi m}$ | $1.0198\xb7{10}^{3}$ | $4.0791\xb7{10}^{5}$ | - | - |

Number of prism layers | 15 |

Prismatic layer stretching | 1.3 |

Prismatic layer total thickness | 0.01 m |

Number of cells (In housing) | 4.93$\xb7{10}^{5}$ |

Number of cells (Free) | 1.51$\xb7{10}^{6}$ |

**Table 6.**Ratio of the aerodynamic torque in the free configuration and the in housing configuration, for different rotational speeds and depressions in the sealed container. Absolute values are shown in Figure 9.

$\dot{\mathit{\phi}}$ [rpm] | |||
---|---|---|---|

Torque (Nm) | 200 | 800 | |

${p}_{i}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$ (Pa) | 10 | 1.79 | 1.89 |

5.05 | 3.04 | 3.24 | |

0.1 | 32.47 | 58.82 |

Gyroscope Units Main Characteristics | |
---|---|

Flywheel | |

Mass | 10,000 kg |

Axial moment of inertia | 8164 kg m${}^{2}$ |

Electric Motor | |

Rated angular velocity | 750 rpm |

Rated Torque | 300 Nm |

Rated Power | 23.5 kW |

Bearings | |

Axial spherical roller thrust bearings | SKF 29416E |

Radial spherical roller bearings | SKF 223366CCK/W33 |

Seals | |

Radial shaft seals | DOMSEL B 380x420x20 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sirigu, A.S.; Gallizio, F.; Giorgi, G.; Bonfanti, M.; Bracco, G.; Mattiazzo, G.
Numerical and Experimental Identification of the Aerodynamic Power Losses of the ISWEC. *J. Mar. Sci. Eng.* **2020**, *8*, 49.
https://doi.org/10.3390/jmse8010049

**AMA Style**

Sirigu AS, Gallizio F, Giorgi G, Bonfanti M, Bracco G, Mattiazzo G.
Numerical and Experimental Identification of the Aerodynamic Power Losses of the ISWEC. *Journal of Marine Science and Engineering*. 2020; 8(1):49.
https://doi.org/10.3390/jmse8010049

**Chicago/Turabian Style**

Sirigu, Antonello Sergej, Federico Gallizio, Giuseppe Giorgi, Mauro Bonfanti, Giovanni Bracco, and Giuliana Mattiazzo.
2020. "Numerical and Experimental Identification of the Aerodynamic Power Losses of the ISWEC" *Journal of Marine Science and Engineering* 8, no. 1: 49.
https://doi.org/10.3390/jmse8010049