# VPP Coupling High-Fidelity Analyses and Analytical Formulations for Multihulls Sails and Appendages Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Performance Prediction Model

#### 2.1. Boat Global Forces and Moment Equilibrium

**X**

**equilibrium**

**equilibrium**(assuming to neglect the lift generated by the boat)

**equilibrium**

**equilibrium**

#### 2.1.1. Hull Forces Modelling

#### 2.1.2. Appendages Forces Modelling

#### 2.2. Closure of the Performance Solution Problem

- In the first iteration the sail lift curve slope ${\left(\frac{\partial {C}_{L}}{\partial \alpha}\right)}_{S}$ and the induced drag factor ${e}_{S}$ are estimated from literature as function of sail aspect and taper ratio. The value of zero-lift drag coefficient ${C}_{D0}{}_{S}$ is roughly guessed. The sail lift and drag coefficients ${C}_{L}{}_{S}$ and ${C}_{D}{}_{S}$, obtained from the CFD analysis, are used to estimate ${C}_{L0}{}_{S}$ from Equation (25) and ${k}_{S}$ from Equation (26).
- In the second iteration the additional CFD solution is used to complete the analytical lift curve formulation adjusting the values of the lift curve slope ${\left(\frac{\partial {C}_{L}}{\partial \alpha}\right)}_{S}$ and zero-incidence lift coefficient ${C}_{L0}{}_{S}$. The parameters updated in the polar curve are ${C}_{D0}{}_{S}$ and ${k}_{S}$ while the value of ${e}_{S}$ is still guessed.
- In the third iteration the analytical drag polar formulation is completed with the computation of the induced drag factor ${e}_{S}$ which is last unknown parameter. The lift curve is updated connecting a quadratic formulation to the previous computed linear part.
- In all the following iterations the sailing condition estimation are performed modelling the polars regions under investigation updating both curves by a generic quadratic formulation using the closest three solutions.

## 3. Optimization Environment

#### 3.1. Sail Parametric Geometric Module

#### 3.2. Sail CFD Analysis Module Implemented Adopting Commercial Software

#### 3.3. CFD Analysis Module Based on Open-Source Tools

- CAD import and pre-processing;
- Geometry meshing;
- Flow field solving;
- Data visualisation and post-processing.

- Conversion from CAD to STL;
- Mesh generation and CFD configuration update;
- CFD run and solutions export;
- Post-processing and results extraction.

#### 3.4. Implementation of the Optimization Environment

## 4. Test of the Analysis Modules

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\alpha $ | Angle of incidence |

$\beta $ | Leeway angle |

$\gamma $ | Rudder angle |

$\delta $ | Appendage dihedral angle |

$\lambda $ | Aspect ratio |

$\phi $ | Heeling angle |

${\rho}_{w}$ | Sea water density |

$AWA$ | Apparent wind angle |

$AWS$ | Apparent wind speed |

$b$ | Draft of appendage |

$d$ | Distance between hulls centrelines |

${C}_{D}$ | Drag coefficient |

${C}_{D0}$ | Drag coefficient at zero incidence |

${C}_{f}$ | Friction drag coefficient |

${C}_{L}$ | Lift coefficient |

${C}_{L0}$ | Lift coefficient at zero incidence |

${C}_{w}$ | Wave drag coefficient |

$D$ | Drag |

${D}_{{B}_{x}}$ | $X$ component of the boat aerodynamic drag |

${D}_{{B}_{y}}$ | $Y$ component of the boat aerodynamic drag |

${D}_{H}$ | Hull drag |

${D}_{{M}_{x}}$ | $X$ component of the crew aerodynamic drag |

${D}_{{M}_{y}}$ | $Y$ component of the crew aerodynamic drag |

$e$ | Oswald efficiency factor |

${F}_{h}$ | Sail heeling force |

${F}_{t}$ | Sail thrust force |

$h$ | Appendage aerodynamic centre |

${h}_{B}$ | Height of boat centre of gravity |

${h}_{h}$ | Height of sail centre of effort |

${h}_{g}$ | Height of the boat centre of gravity |

$L$ | Lift |

${L}_{H}$ | Hull side force (parallel to the sea plane) |

${l}_{M}$ | Arm of crew righting moment |

$p$ | perimeter of the appendage (excluded root) |

$r$ | Daggerboard stagger angle |

${R}_{N}$ | Reynolds number |

$S$ | Reference surface |

$TWA$ | True wind angle |

$TWS$ | True wind speed |

$V$ | Boat speed |

${W}_{BE}$ | Boat empty weight |

${W}_{BO}$ | Boat operative weight |

${W}_{M}$ | Crew weight |

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**Figure 9.**Example of a CFD solution of the hull with appendage used to fine tune the foils analytical models.

Reference Surfaces | Side Force | Wave Drag | Leeway Drag |
---|---|---|---|

${k}_{{S}_{H}{}_{1}}$ = 0.00437 | ${k}_{H}{}_{1}$ = 6 × 10^{−7} | ${w}_{w}$ = 80 kg | ${k}_{\beta}$ = 2 × 10^{−6} |

${k}_{{S}_{H}{}_{2}}$ = 0.07 | ${k}_{H}{}_{2}$ = 1.3 × 10^{−4} | ${k}_{{w}_{1}}$ = 2.16 × 10^{−6} | ${\tau}_{\beta}$ = 1.5 |

${\tau}_{{S}_{H}}$ = 0.83 | ${\tau}_{{H}_{1}}$ = 1.3 | ${k}_{{w}_{2}}$ = −8.3 × 10^{−6} | ${w}_{\beta}$ = 400 kg |

${k}_{{S}_{w}{}_{1}}$ = 0.00876 | ${\tau}_{{H}_{2}}$ = 0.2 | ${k}_{{w}_{3}}$ = 9 × 10^{−6} | |

${k}_{{S}_{w}{}_{2}}$ = 0.95 | |||

${\tau}_{{S}_{w}}$ = 0.5 | shape factor $k$ = 0.01 | ||

${W}_{B{O}_{0}}$ = 94 kg |

ANSYS Fluent | OpenFOAM | |
---|---|---|

Spanner | 35 deg | 41.7 deg |

Sail setting | 5.9 deg | 6 deg |

Thrust force | 238.2 N | 232.5 N |

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**MDPI and ACS Style**

Cella, U.; Salvadore, F.; Ponzini, R.; Biancolini, M.E.
VPP Coupling High-Fidelity Analyses and Analytical Formulations for Multihulls Sails and Appendages Optimization. *J. Mar. Sci. Eng.* **2021**, *9*, 607.
https://doi.org/10.3390/jmse9060607

**AMA Style**

Cella U, Salvadore F, Ponzini R, Biancolini ME.
VPP Coupling High-Fidelity Analyses and Analytical Formulations for Multihulls Sails and Appendages Optimization. *Journal of Marine Science and Engineering*. 2021; 9(6):607.
https://doi.org/10.3390/jmse9060607

**Chicago/Turabian Style**

Cella, Ubaldo, Francesco Salvadore, Raffaele Ponzini, and Marco Evangelos Biancolini.
2021. "VPP Coupling High-Fidelity Analyses and Analytical Formulations for Multihulls Sails and Appendages Optimization" *Journal of Marine Science and Engineering* 9, no. 6: 607.
https://doi.org/10.3390/jmse9060607