# Marine Turbine Hydrodynamics by a Boundary Element Method with Viscous Flow Correction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model

#### 2.1. Trailing Wake Model

#### 2.2. Viscous-Flow Correction Model

`SE`+,

`SE`-), while stall occurs when lift drops as $\alpha $ increases in absolute value and drag rises abruptly (point

`ST`).

## 3. Case Studies for Validation of Computational Model

#### 3.1. Fixed Pitch Turbine

#### 3.2. Variable Pitch Turbine

## 4. Fixed Pitch Turbine Study

## 5. Variable Pitch Turbine Study

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Description | Units |

c | Turbine blade chord | [m] |

${C}_{p}$ | Pressure coefficient | [-] |

${C}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{P}}$ | Power coefficient | [-] |

${C}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{F}}$ | Friction coefficient | [-] |

${C}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{Q}}$ | Torque coefficient | [-] |

${C}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{T}}$ | Thrust coefficient | [-] |

D | Turbine diameter, $2R$ | [m] |

D | Drag | [N] |

${\mathcal{K}}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{D}}$ | Drag correction factor | [-] |

${\mathcal{K}}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{L}}$ | Lift correction factor | [-] |

L | Lift | [N] |

n | Turbine rotational speed | [s${}^{-1}$] |

P | Turbine power | [W] |

${p}_{0}$ | Reference pressure | [Pa] |

Q | Turbine torque | [Nm] |

R | Turbine radius | [m] |

$R{e}_{r}$ | Reynolds number, Equation (6) | [-] |

T | Turbine thrust | [N] |

TSR | Tip Speed Ratio | [-] |

V | Freestream velocity | [ms${}^{-1}$] |

$\alpha $ | angle of attack | [deg] |

$\nu $ | Kynematic viscosity | [m${}^{2}$s${}^{-1}$] |

$\phi $ | Velocity scalar potential | [m${}^{2}$s${}^{-1}$] |

$\mathsf{\Omega}$ | Turbine rotational speed | [rads${}^{-1}$] |

$\varphi $ | Wake (linear) pitch | [m] |

$\mathsf{\Phi}$ | Blade pitch | [deg] |

$\rho $ | Water density | [kgm${}^{-3}$] |

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**Figure 1.**Sketch of the frame of reference associated to the solid boundary describing an isolated turbine and the surface of trailing wake shed by one blade.

**Figure 2.**Streamtube radius downstream of rotor plane from Equation (9) and comparative data from experiments.

**Figure 3.**Inflow velocity components and hydrodynamic force components on turbine blade section at radius r.

**Figure 4.**Notional lift and drag curves of a two-dimensional profile: viscous flow and inviscid flow conditions with flat plate drag correction compared.

**Figure 6.**IFREMER-FP turbine. Calculated thrust and torque coefficients as a function of discretization parameter ${M}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{B}}$.

**Left**and

**center**: thrust and torque by non-corrected BIEM;

**right**: torque by BIEM-VFC.

**Figure 8.**IFREMER-FP turbine. Trailing wake geometry of BIEM model at different operating conditions: TSR = 2 (

**top left**), TSR = 3 (

**top right**), TSR = 6 (

**bottom**).

**Figure 9.**IFREMER-FP turbine. Reynolds number $R{e}_{r}$ (

**left**) and effective angle of attack ${\alpha}_{e}$ (

**right**) as a function of radius r and of TSR.

**Figure 10.**NACA 63-421 2D foil: lift (

**left**) and drag (

**right**) coefficients calculated by X-Foil and from experiments [23].

**Figure 11.**IFREMER-FP turbine. Correction factors for radial contributions to lift (

**left**) and drag (

**right**) as a function of radius r and of TSR.

**Figure 12.**IFREMER-FP turbine performance predictions by BIEM and BIEM-VFC compared to experimental data in [11]: thrust (

**top left**); torque (

**top right**) and power (

**bottom**) coefficients.

**Figure 13.**IFREMER-FP turbine. Pressure distribution evaluated by inviscid-flow BIEM, TSR = 3.3 (peak power condition).

**Left**: pressure side;

**right**: suction side.

**Figure 14.**IFREMER-FP turbine. Pressure distribution evaluated by inviscid-flow BIEM at radial section at 70% of blade span. From

**top left**to

**bottom right**: TSR = 3.3, 5, 6, 7.4.

**Figure 15.**UoS-VP turbine. Three-dimensional model and computational grid for BIEM analysis.

**Left**: front view;

**right**: details of hub and blade roots.

**Figure 16.**UoS-VP turbine. Axial induced velocity distribution at axial position corresponding to blade trailing edge and 70% of blade span. Different pitch settings $\mathsf{\Phi}$ compared.

**Figure 17.**UoS-VP turbine. Wake geometry of BIEM model at different operating conditions. From

**left**to

**right**, TSR = 3, 6, 9. Design pitch setting, $\mathsf{\Phi}={20}^{\circ}$.

**Figure 18.**UoS-VP turbine. Reynolds number $R{e}_{r}$ as a function of radius r and of turbine operating condition (TSR).

**Figure 19.**UoS-VP turbine. Reynolds number $R{e}_{r}$ at radius $r/R=0.7$ for pitch settings corresponding to the highest inflow speed ($V=1.73$ m/s, $\mathsf{\Phi}={20}^{\circ}$), and for the lowest inflow speed ($V=1.3$ m/s, $\mathsf{\Phi}={27}^{\circ}$).

**Figure 20.**UoS-VP turbine. Effective angle of attack ${\alpha}_{e}$ as a function of radius r and of turbine operating condition (TSR). Design pitch setting $\mathsf{\Phi}={20}^{\circ}$.

**Figure 21.**UoS-VP turbine. Effective angle of attack ${\alpha}_{e}$ at radius $r/R=0.7$ for different pitch settings and TSR.

**Figure 22.**NACA 63-815 2D foil: lift (

**left**) and drag (

**right**) coefficients used for the viscous flow correction of BIEM.

**Figure 23.**UoS-VP turbine. Correction factors for radial contributions to lift (

**left**) and drag (

**right**) as a function of radius r and of operating condition (TSR). Pitch setting $\mathsf{\Phi}={20}^{\circ}$.

**Figure 24.**UoS-VP turbine. Lift correction factor ${\mathcal{K}}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{L}}$ (

**top**) and drag correction factor ${\mathcal{K}}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{D}}$ (

**bottom**) at radius $r/R=0.7$ for different Pitch settings.

**Figure 25.**UoS-VP turbine performance predictions by BIEM and BIEM-VFC compared to experimental data in [13]. Thrust coefficient at pitch settings $\mathsf{\Phi}={15}^{\circ},{20}^{\circ},{25}^{\circ},{27}^{\circ},{30}^{\circ}$.

**Figure 26.**UoS-VP turbine performance predictions by BIEM and BIEM-VFC compared to experimental data in [13]. Power coefficient at pitch settings $\mathsf{\Phi}={15}^{\circ},{20}^{\circ},{25}^{\circ},{27}^{\circ},{30}^{\circ}$.

**Figure 27.**UoS-VP turbine. Effect of pitch setting $\mathsf{\Phi}$: from

**top**to

**bottom**,

**left**to

**right**, ${C}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{{T}_{max}}},{C}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{{P}_{max}}}$ and corresponding TSR values [email protected]${C}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{{T}_{max}}}$, [email protected]${C}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{{P}_{max}}}$.

**Figure 28.**UoS-VP turbine performance predictions by BIEM-VFC compared to results of BEM models from [25]: thrust coefficient ${C}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{T}}$. Pitch settings $\mathsf{\Phi}={15}^{\circ}$ to 27°.

**Figure 29.**UoS-VP turbine performance predictions by BIEM-VFC compared to results of BEM models from [25]: power coefficient ${C}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{P}}$. Pitch settings $\mathsf{\Phi}={15}^{\circ}$ to 27${}^{\circ}$.

**Figure 30.**UoS-VP turbine performance predictions by BIEM-VFC compared to results from the BIEM model in [7]: thrust coefficient ${C}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{T}}$ (

**left**); and power coefficient ${C}_{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\phantom{\rule{3.33333pt}{0ex}}}_{P}}$ (

**right**). Pitch settings $\mathsf{\Phi}={20}^{\xb0}$ to 27°.

Rotor diameter, D | 700 [mm] |

Blades number, Z | 3 |

Pitch angle at 70% span, $\mathsf{\Phi}$ | 7.3 [deg] |

Thickness ratio, 75% span, $t/c$ | 0.21 |

Hub/rotor diameter ratio | $0.131$ |

Blade section profile | NACA 63-4xx |

Rotor diameter, D | 800 [mm] |

Blades number, Z | 3 |

Pitch angle at 20% span, $\mathsf{\Phi}$ | $15,20,25,27,30$ [deg] |

Thickn. ratio, 75% span, $t/c$ | 0.151 |

Hub/rotor diameter ratio | $0.125$ |

Blade section profile | NACA 63-8xx |

Blade pitch setting, $\mathsf{\Phi}$ [deg] | 15 | 20 | 25 | 27 | 30 |

Inflow speed, V [m/s] | 1.40 | 1.73 | 1.54 | 1.30 | 1.54 |

© 2018 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**

Salvatore, F.; Sarichloo, Z.; Calcagni, D. Marine Turbine Hydrodynamics by a Boundary Element Method with Viscous Flow Correction. *J. Mar. Sci. Eng.* **2018**, *6*, 53.
https://doi.org/10.3390/jmse6020053

**AMA Style**

Salvatore F, Sarichloo Z, Calcagni D. Marine Turbine Hydrodynamics by a Boundary Element Method with Viscous Flow Correction. *Journal of Marine Science and Engineering*. 2018; 6(2):53.
https://doi.org/10.3390/jmse6020053

**Chicago/Turabian Style**

Salvatore, Francesco, Zohreh Sarichloo, and Danilo Calcagni. 2018. "Marine Turbine Hydrodynamics by a Boundary Element Method with Viscous Flow Correction" *Journal of Marine Science and Engineering* 6, no. 2: 53.
https://doi.org/10.3390/jmse6020053