# 3D Simulation with Flow-Induced Rotation for Non-Deformable Tidal Turbines

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Experimental Setup

#### 2.1.1. Horizontal Axis Tidal Turbine Experimental Setup

#### 2.1.2. Vertical Axis Tidal Turbine Experimental Setup

#### 2.2. Governing Equations

`PimpleFoam`was used as a fluid solver, and it is described in the OpenFoam documentation [26] as a “Transient solver for incompressible, turbulent flow of Newtonian fluids on a moving mesh”. It is based on an algorithm coupling velocity to the pressure equations. Equation (8) was used to construct a first estimate of the velocity ${U}_{i}\ast $. The pressure, P, was deduced from Equation (9) using ${U}_{i}\ast $.The velocity matrix could then be corrected. This operation was repeated until the solution reached the convergence criteria, which were set to ${10}^{-6}$ for the error, before moving to the next time step (Figure 3).

`sixDoFRigidBodyMotion`, which was added to

`PimpleFoam`to model the fluid–structure interaction. This module computes the forces acting on the blades and converts them into point displacement for the close-rotor mesh. Without this module, these point displacements are fixed with a constant value that constrains the rotation of the rotor at a certain speed. In our “forced rotation” cases, the

`sixDoFRigidBodyMotion`module was disabled and the rotation speed was set to obtain a certain tip speed ratio. The forces on the blade could be retrieved (Figure 4). When the rotation was induced by the flow, the forces and momentum were calculated from the flow characteristics using Equations (10) and (11). They were then converted into acceleration or moments of inertia depending on the system constraints using Equation (12). When the system was stabilized at the final free speed, all the energy brought by the fluid was converted into rotation speed. The forces acting on the blades were, therefore, equal to zero.

`PimpleFoam`in order to represent physical motions with accuracy. After the initialization phase of the flow, the forces and displacements were computed for each

`PimpleFoam`iteration. When all the variables reached their convergence criteria, the model started a new time step. A more accurate description of the solver is available in [9]. The difference between forced and flow-induced approaches is shown in Figure 5.

#### Boundary and Initial Conditions

`PimpleFoam`is a velocity–pressure solver, the pressure was solved with the velocity propagation.

#### 2.3. 3D Geometries and Mesh

`SnappyHexMesh`of OpenFoam. They were unstructured except on walls where structured layers were added to control the boundary layer. A numerical flume was built to avoid side effects and to capture the turbulent wake. The AMI (arbitrary mesh interface) method used in [12] was activated to avoid mesh deformation during the rotor rotation.

#### 2.3.1. Horizontal Axis Turbine

#### 2.3.2. Vertical Axis Turbine

#### 2.4. Numerical Setup

## 3. Results and Discussion

#### 3.1. Mesh Convergence

#### 3.2. Validation with Experimental Data

#### 3.2.1. Horizontal Axis Turbine

#### 3.2.2. Vertical Axis Turbine

#### 3.3. Flow-Induced Model Validation

#### 3.3.1. Free Horizontal Tidal Turbine

#### 3.3.2. Free Vertical Tidal Turbine

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Parameters | Definitions | Units |

D | Rotor diameter | $\mathrm{m}$ |

R | Rotor radius | $\mathrm{m}$ |

$\delta {\mathsf{\Omega}}_{1}$ | Computational domain inlet surface | - |

$\delta {\mathsf{\Omega}}_{2}$ | Computational domain outlet surface | - |

$\delta {\mathsf{\Omega}}_{3,4,5,6}$ | Computational domain side surfaces | - |

$\mathbf{n}$ | Surface normal vector | - |

${\mathbf{n}}_{blade}$ | Surface normal vector in the blade referential | - |

$\mathbf{t}$ | Surface tangential vector in the blade referential | - |

$\nu $ | Kinematic viscosity | ${\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ |

$\omega $ | Rotor angular velocity | $\mathrm{rad}\xb7{\mathrm{s}}^{-1}$ |

$\mathsf{\Omega}$ | Computational domain | - |

p | Fluid pressure | $\mathrm{Pa}$ |

$\rho $ | Fluid density | $\mathrm{kg}\xb7{\mathrm{m}}^{-3}$ |

${U}_{\infty}$ | Inlet velocity magnitude | $\mathrm{m}\xb7{\mathrm{s}}^{-1}$ |

$\lambda $ | Tip speed ratio | $\mathrm{m}\xb7{\mathrm{s}}^{-1}$ |

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**Figure 1.**General characteristics (

**a**) and design (

**b**) of the IFREMER-LOMC turbine [21], University of Le Havre, 2013.

**Figure 2.**Geometry of the Darrieus tidal turbine [22], University of Grenoble, 2011.

**Figure 3.**Scheme of one iteration of

`PimpleFoam`performed and repeated until the reaching of the convergence criteria. ${\mathrm{U}}_{\mathrm{n}}\ast $ and ${\mathrm{P}}_{\mathrm{n}}\ast $ are, respectively, the estimate of the velocity matrix and pressure matrix. ${\mathrm{P}}_{\mathrm{n}+1}$ and ${\mathrm{U}}_{\mathrm{n}+1}$ are the corrected pressure and velocity fields at iteration n.

**Figure 5.**Energy recovery chain for real life (red box), the forced rotation (blue box) and the flow induced rotation (green box). Black arrows represent the energy transfer.

**Figure 6.**Three-dimensional views of the horizontal tidal turbine with a hub: front (left panel) and sideways (right panel) views.

**Figure 8.**View of the X–Y plane of the computational domain with boundary conditions. D is the rotor diameter. U is the flow velocity in $\mathrm{m}\xb7{\mathrm{s}}^{-1}$, P is the pressure in $\mathrm{Pa}$, and ${\mathrm{P}}_{\mathrm{local}}$ is the pressure at the calculated point closest to the wall. The red square delimits the arbitary mesh interface (AMI) zone, and the blue circle shows the rotor position. The two black lines (1.2D and 2D) and the black cross are the extraction lines and the probe, respectively, used in the model validation in Section 3.2.

**Figure 9.**Blade section for a level 5 refinement—the coarsest mesh (#2.1) with a close-up on leading edge (

**a**), trailing edge (

**c**) and upper surface (

**b**). Distorted elements are due to the visualization section.

**Figure 10.**Blade section for a level 9 refinement—the finest mesh (#2.4) with a close-up on leading edge (

**a**), trailing edge (

**c**) and upper surface (

**b**). Distorted elements are due to the visualization section.

**Figure 11.**Three-dimensional geometry of the blades of the vertical axis tidal turbine without hub for a sideways view.

**Figure 12.**View of X–Y plane for the computational domain with boundary conditions (in green). D is the rotor’s diameter in $\mathrm{m}$. U is the flow velocity in $\mathrm{m}\xb7{\mathrm{s}}^{-1}$, and P is the pressure in $\mathrm{Pa}$. The two black lines (2D and 4D) and the black cross are the extraction lines and the probe, respectively, used in the model validation in Section 3.2.

**Figure 13.**${u}^{\ast}$ magnitude along ${y}^{\ast}$ for #1, #2.0, #2.4 and #3 at ${x}^{\ast}$ = 1.2.

**Figure 15.**Time evolution of the vorticity magnitude close to the rotor, inside the AMI zone (${x}^{\ast}$ = 2), for coarse (#1 with red line) and refined (#2 with blue line) meshes.

**Figure 16.**Time evolution of the power coefficient (Cp) with coarse (red line) and fine (blue line) mesh refinement in the wake.

**Figure 17.**Time evolution of Cp for the cases with (#3 red line) and without a hub (#2.0 blue line).

**Figure 18.**Cp time evolution for cases #2.0 (blue), #2.1 (red), #2.2 (green), #2.3 (pink) and #2.4 (cyan).

**Figure 19.**Relative difference between Cp after 1 s of simulation for cases #2.0 (blue lozenge), #2.1 (red dot), #2.2 (green square), #2.3 (pink plus), #2.4 (cyan cross) and #2.4.

**Figure 21.**${C}_{d}$ time evolution according to the level of refinement for cases #2.0 (blue), #2.1 (red), #2.2 (green), #2.3 (pink) and #2.4 (cyan).

**Figure 22.**Relative difference in ${C}_{d}$ according to the level of refinement for cases #2.0 (blue lozenge), #2.1 (red dot), #2.2 (green square), #2.3 (pink plus), #2.4 (cyan cross) and #2.4.

**Figure 23.**Non-dimensional velocity profiles according to the non-dimensional position for experimental data (red lozenges) and model (#2.4) as ${x}^{\ast}=1.2$. The mismatch at ${y}^{\ast}=0$ was due to the missing hub in the numerical case.

**Figure 24.**Same legend as for previous figure, but profiles were taken at ${x}^{\ast}=2$. The mismatch at ${y}^{\ast}=0$ was due to the missing hub in the numerical case.

**Figure 25.**${u}^{\ast}$ depending on the ${y}^{\ast}$ position comparison between case #5 (blue line) and experimental data (red lozenges) at ${x}^{\ast}=2$.

**Figure 26.**${u}^{\ast}$ depending on the ${y}^{\ast}$ position comparison between case #5 (blue line) and experimental data (red lozenges) at ${x}^{\ast}=4$.

**Figure 27.**Time evolution of the power coefficient Cp for case #5 (blue line with crosses) and experimental data (red line with plus). The black line is the Cp mean value starting at 0.5 s.

**Figure 28.**Time evolution of the drag coefficient ${C}_{d}$ for case #5 (blue line with crosses) and experimental data (red line with plus). The black line is the Cp mean value starting at 0.5 s.

**Figure 29.**Time evolution of the angular velocity for cases #4.0 (red line), #4.1 (green line) and #4.2 (blue line).

**Figure 30.**Time evolution of the vorticity magnitude at the probe position (${x}^{\ast}=1$, ${y}^{\ast}=0.5$) after phase shift correction for #4.0 (red line), #4.1 (green line) and #4.2 (blue line).

**Figure 31.**Frequency-based spectra of vorticity for the cases #4.0 (red line), #4.1 (green line) and #4.2 (blue line).

**Figure 32.**Time evolution of the angular velocity for forced (blue line, #6.1) and induced (red line, #6.2) rotation.

**Figure 33.**Frequency-based spectra of the vorticity for forced (blue line, #6.1) and induced (red line, #6.2) rotation cases.

**Figure 34.**Time evolution of the fluid vorticity at the probe location for forced (blue line, #6.1) and induced (red line, #6.2) rotation.

Levels | Characteristic Length |
---|---|

Level 1 | $5.6\times {10}^{-2}$ |

Level 2 | $2.8\times {10}^{-2}$ |

Level 3 | $1.4\times {10}^{-2}$ |

Level 4 | $7\times {10}^{-3}$ |

Level 5 | $3.5\times {10}^{-3}$ |

Level 6 | $1.175\times {10}^{-3}$ |

Level 7 | $8.75\times {10}^{-4}$ |

Level 8 | $4.375\times {10}^{-4}$ |

Level 9 | $2.1875\times {10}^{-4}$ |

**Table 2.**Summary of test cases for the horizontal axis turbine according to the level of refinement on the blades and in the wake, the presence or absence of a hub, and the total number of points in the mesh (Nb).

Mesh | Level on Blades | Level in Wake | Hub | Nb ($\times {10}^{6}$) |
---|---|---|---|---|

Reference case (#1) | Level 6 | Level 1 | No | 1 |

Wake case (#2.0) | Level 6 | Level 3 | No | 1.9 |

Blade 5 (#2.1) | Level 5 | Level 3 | No | 1.6 |

Blade 7 (#2.2) | Level 7 | Level 3 | No | 3.5 |

Blade 8 (#2.3) | Level 8 | Level 3 | No | 9 |

Blade 9 (#2.4) | Level 9 | Level 3 | No | 12 |

Hub case (#3) | Level 6 | Level 3 | Yes | 2.2 |

Flow-induced L6 (#4.0) | Level 6 | Level 3 | No | 1.9 |

Forced free rotation speed L6 (#4.1) | Level 6 | Level 3 | No | 1.9 |

Flow-induced L7 (#4.2) | Level 7 | Level 3 | No | 3.5 |

**Table 3.**Summary of test cases for vertical axis turbine according to the level of refinement on the blades and in the wake, the presence or absence of a hub, and the total number of points in the mesh (Nb).

Mesh | Level on Blades | Level in Wake | Hub | Nb ($\times {10}^{6}$) |
---|---|---|---|---|

Forced for validation (#5) | Level 6 | Level 3 | No | 1.2 |

Forced free speed (#6.1) | Level 6 | Level 3 | No | 1.2 |

Flow induced (#6.2) | Level 6 | Level 3 | No | 1.2 |

**Table 4.**Table of the physical parameters used in simulations for horizontal and vertical axis cases.

Parameter | Horizontal Axis Turbine | Vertical Axis Turbine | Unit |
---|---|---|---|

$\rho $ | 1025 | 998 | $\mathrm{kg}\xb7{\mathrm{m}}^{-3}$ |

$\nu $ | $1.3\times {10}^{-6}$ | $1.05\times {10}^{-6}$ | ${\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ |

${U}_{\infty}$ | $0.8$ | $2.3$ | $\mathrm{m}\xb7{\mathrm{s}}^{-1}$ |

$\mathsf{\Omega}$ | $9.143$ | $52.2$ | $\mathrm{rad}\xb7{\mathrm{s}}^{-1}$ |

$\lambda $ | 4 | 2 | - |

**Table 5.**Summary of the Cp values after 1 s of simulation depending on the level of refinement on the blades. The relative difference compared to the level 9 simulation is given in the last column.

Level | Case | Cp | Relative Difference |
---|---|---|---|

5 | #2.1 | $0.4728$ | 0.2688 |

6 | #2.0 | $0.4227$ | 0.1341 |

7 | #2.2 | $0.4020$ | 0.0787 |

8 | #2.3 | $0.3691$ | 0.0095 |

9 | #2.4 | $0.3724$ | 0 |

**Table 6.**Summary of the Cd values after 1 s of simulation depending on the level of refinement on the blades. The relative difference compared to the level 9 simulation is given in the last column.

Level | Case | Cd | Relative Difference |
---|---|---|---|

5 | #2.1 | 0.5698 | 0.0501 |

6 | #2.0 | 0.6120 | 0.0.0203 |

7 | #2.2 | $0.5984$ | 0.0025 |

8 | #2.3 | $0.5878$ | 0.0020 |

9 | #2.4 | $0.5998$ | 0 |

Coefficient | Value | Relative Difference |
---|---|---|

${C}_{p}$ | 0.3729 | 0.070 |

${C}_{p}^{\ast}$ | 0.3887 | 0.031 |

${C}_{d}$ | 0.5998 | 0.1552 |

${C}_{d}^{\ast}$ | 0.6418 | 0.09 |

Coefficient | Experimental | Numerical | Relative Difference |
---|---|---|---|

${C}_{p}$ | 0.24 | 0.2779 | 0.1576 |

${C}_{d}$ | 1.13 | 1.3806 | 0.1800 |

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## Share and Cite

**MDPI and ACS Style**

Robin, I.; Bennis, A.-C.; Dauvin, J.-C.
3D Simulation with Flow-Induced Rotation for Non-Deformable Tidal Turbines. *J. Mar. Sci. Eng.* **2021**, *9*, 250.
https://doi.org/10.3390/jmse9030250

**AMA Style**

Robin I, Bennis A-C, Dauvin J-C.
3D Simulation with Flow-Induced Rotation for Non-Deformable Tidal Turbines. *Journal of Marine Science and Engineering*. 2021; 9(3):250.
https://doi.org/10.3390/jmse9030250

**Chicago/Turabian Style**

Robin, Ilan, Anne-Claire Bennis, and Jean-Claude Dauvin.
2021. "3D Simulation with Flow-Induced Rotation for Non-Deformable Tidal Turbines" *Journal of Marine Science and Engineering* 9, no. 3: 250.
https://doi.org/10.3390/jmse9030250