# Numerical Study of the Dynamic Stall Effect on a Pair of Cross-Flow Hydrokinetic Turbines and Associated Torque Enhancement Due to Flow Blockage

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Blockage Effect on a Single Turbine

#### 2.1. Materials and Methodology

#### 2.1.1. Turbine Geometry

#### 2.1.2. Pre-Processing

#### 2.1.3. Turbulence Model, Boundary Conditions, and Numerical Schemes

#### 2.2. Sensitivity Analysis

#### 2.2.1. Convergence Criterion

#### 2.2.2. Iterative Spatial and Temporal Sensitivity Analysis

#### 2.3. Results and Discussion

#### 2.3.1. Comparison with Experimental Results

#### 2.3.2. The Laboratory Frame of Reference

#### 2.3.3. The Blade Frame of Reference

- (1)
- Before the maximum point, the torque generation was due the attached fast flow on the inner of the blade, which created an associated low pressure region and suction force. As the azimuthal angle increased, the relative angle between the incoming flow and the blade chord also increased and resulted in higher lift and torque. Higher blockage ratio created faster flow on the suction side and smaller area of the blade wake.
- (2)
- After reaching the maximum torque, the blade experienced the dynamic stall effect and the lift force was maintained by a vortex shed from the leading edge and attached on the suction side. Higher blockage ratio created a bigger and stronger vortex, which maintained higher lift and torque.

**Figure 13.**The velocity fields in the blade reference frame before it reached the maximum torque of 4 blockage ratios: $19.7\%$ (

**a**), $25.3\%$ (

**b**), $33.8\%$ (

**c**), and $50.6\%$ (

**d**). On each figure, the lower surface is the turbine inner side and the upper surface is the turbine outer side.

**Figure 14.**The velocity fields in the blade reference frame after it reached the maximum torque of 4 blockage ratios: $19.7\%$ (

**a**), $25.3\%$ (

**b**), $33.8\%$ (

**c**), and $50.6\%$ (

**d**). On each figure, the lower surface is the turbine inner side and the upper surface is the turbine outer side.

## 3. Twin Turbine Configurations

#### 3.1. Materials and Methodology

#### 3.1.1. System Geometry

#### 3.1.2. Simulation Setup

#### 3.2. Results and Discussion

#### 3.2.1. The Laboratory Frame of Reference

#### 3.2.2. The Blade Frame of Reference

## 4. Summary and Conclusions

- (1)
- At the same blockage ratio, moving the turbine closer a fixed wall created positive impact on the positive torque phase.
- (2)
- Placing a moving object near the turbine could have both positive and negative impact depending on the movement direction of the object. Specifically, another turbine blade moving against the incoming freestream flow would have reduced the torque while movement in the same direction of the freestream created a stronger and bigger LEV to maintain higher hydrodynamic torque.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MHK | Marine hydrokinetic turbine; |

VAWT | Vertical axis wind turbine; |

PIV | Particle image velocimetry; |

CFD | Computational fluid dynamics. |

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**Figure 2.**An example of the convergence behavior of ${C}_{P}$. This example shows the results on a 1.3 million cell mesh at a time step of $1.25\times {10}^{-4}$ s.

**Figure 3.**An iterative sensitivity analysis. The first and third steps are shown in (

**a**) and the second step of the process is displayed in (

**b**). The points enclosed in same style circles are equivalent on the 2 figures. The final spatial and temporal resolution used for the rest of this article is enclosed in the red rectangle.

**Figure 4.**The numerically calculated and corrected power coefficient ${C}_{P}$ as a function of tip-speed-ratio $\lambda $. The purple dotted lines enclosed the interested points for later torque analyses.

**Figure 5.**The experimental and numerically corrected power coefficient ${C}_{P}$ as a function of tip-speed-ratio $\lambda $ at 4 blockage ratios.

**Figure 6.**The numerical (blue turbine) and experimental (red turbine) near-wake results. The CFD flow pictures are instantaneous snapshots at 2 phase angles after the convergence criterion was satistfied (

**a1**,

**a2**,

**b1**,

**b2**). The experimental pictures are the phase-averaged results discussed in [16] at the relevant phase angle (

**a3**,

**b3**).

**Figure 7.**The torque as a function of the turbine azimuthal angle at 4 blockage ratios and 2 tip-speed-ratios $\lambda =0.85$ (

**a**) and $\lambda =1.42$ (

**b**).

**Figure 8.**An example of the velocity field around the blade during its positive torque phase (

**a**–

**e**) and at an azimuthal angle after it switched into the negative torque zone (

**f**). The analysis focused on the blade enclosed in the red dashed lines in (

**a**).

**Figure 9.**The velocity, vorticity, and pressure fields around one of the turbine blades at its maximum torque angle of 4 blockage ratios: $19.7\%$ (

**a**), $25.3\%$ (

**b**), $33.8\%$ (

**c**), and $50.6\%$ (

**d**).

**Figure 10.**The pressure loading on one of the turbine blades at its maximum torque angle of 4 blockage ratios: $19.7\%$ (

**a**), $25.3\%$ (

**b**), $33.8\%$ (

**c**), and $50.6\%$ (

**d**). The relevant blade is colored in white on the right side.

**Figure 11.**The velocity, vorticity, and pressure fields around one of the turbine blades after reaching its maximum torque angle of 4 blockage ratios: $19.7\%$ (

**a**), $25.3\%$ (

**b**), $33.8\%$ (

**c**), and $50.6\%$ (

**d**).

**Figure 12.**The pressure loading on one of the turbine blades after reaching its maximum torque angle of 4 blockage ratios: $19.7\%$ (

**a**), $25.3\%$ (

**b**), $33.8\%$ (

**c**), and $50.6\%$ (

**d**). The relevant blade is colored in white on the right side.

**Figure 15.**The turbulent viscosity field at $\beta =25.3\%$ (

**a**) and $\beta =50.6\%$ (

**b**) during the generation of the leading edge vortex.

**Figure 16.**The interested turbine configurations for this computational fluid dynamic study: Single turbine at $\beta =25.3\%$ (

**a**), single turbine at $\beta =50.6\%$ (

**b**), forward counter-rotating turbines at $\beta =50.6\%$ (

**c**), and backward counter-rotating turbine at $\beta =50.6\%$ (

**d**). The green arrows denote the rotating direction and the solid black lines illustrate the water tunnel walls.

**Figure 18.**Examples of the convergence behavior of the 2 turbines for 2 cases: Forward at ${\omega}_{t}=12.5$ rad/s (

**a**) and forward at ${\omega}_{t}=14.5$ rad/s (

**b**).

**Figure 19.**The power curves of the configurations displayed in Figure 16 and their associated blockage ratios.

**Figure 20.**The torque curves of the single and twin turbine configurations at $\lambda =1.42$ (

**a**) and $\lambda =1.56$ (

**b**).

**Figure 21.**The velocity, vorticity, and pressure fields around one of the turbine blades around its maximum torque angle of the 3 configurations: Single turbine (

**a**), forward counter-rotating turbines (

**b**), and backward counter-rotating turbines (

**c**).

**Figure 22.**The pressure loading on one of the turbine blades around its maximum torque angle of the 3 configurations: Single turbine (

**a**), forward counter-rotating turbines (

**b**), and backward counter-rotating turbines (

**c**). The relevant blade is colored in white on the right side.

**Figure 23.**The velocity, vorticity, and pressure fields around one of the turbine blades after reaching its maximum torque angle of the 3 configurations: Single turbine (

**a**), forward counter-rotating turbines (

**b**), and backward counter-rotating turbines (

**c**).

**Figure 24.**The pressure loading on one of the turbine blades after reaching its maximum torque angle of the 3 configurations: Single turbine (

**a**), forward counter-rotating turbines (

**b**), and backward counter-rotating turbines (

**c**). The relevant blade is colored in white on the right side.

**Figure 25.**The velocity fields in the blade reference frame before it reached the maximum torque of the 3 configurations: single turbine (

**a**), forward counter-rotating turbines (

**b**), and backward counter-rotating turbines (

**c**). On each figure, the lower surface is the turbine inner side and the upper surface is the turbine outer side. The figures on the left side show the position of the blade of interest colored in white.

**Figure 26.**The velocity fields in the blade reference frame after it reached the maximum torque of the 3 configurations: single turbine (

**a**), forward counter-rotating turbines (

**b**), and backward counter-rotating turbines (

**c**). On each figure, the lower surface is the turbine inner side and the upper surface is the turbine outer side. The figures on the left side show the position of the blade of interest colored in white.

Parameter | Symbol | Value (Unit) |
---|---|---|

Blade chord | c | 2.54 (cm) |

Solidity | $\sigma $ | 0.39 (-) |

Reynolds number (diameter based) | $R{e}_{D}$ | 20,000 (-) |

Water density | $\rho $ | 998 (kg/m${}^{3}$) |

Freestream velocity | ${U}_{\infty}$ | 0.3 (m/s) |

Blockage ratio | $\beta $ | 0.197/0.253/0.338/0.506 (-) |

**Table 2.**Summary of the meshes used in the sensitivity analysis. ${N}_{x}$ and ${N}_{y}$ are the number of coarse cells in the streamwise and transverse direction.

Case | ${\mathit{N}}_{\mathit{x}}$ | ${\mathit{N}}_{\mathit{y}}$ | Total Number of Cells |
---|---|---|---|

1 | 222 | 30 | 588,649 |

2 | 296 | 40 | 1,026,883 |

3 | 333 | 45 | 1,304,364 |

4 | 370 | 50 | 1,605,068 |

5 | 444 | 60 | 2,305,123 |

Case | $\mathit{\beta}$ | Note |
---|---|---|

T1 Experiment | 19.7% | $\lambda =0.85$—Optimal power output |

Numerical Model 1 | 19.7% | Same blockage ratio as the experiment |

Numerical Model 2 | 25.3% | Same physical dimensions as the experiment |

Case | $\mathit{\lambda}$ | ${\mathit{C}}_{\mathit{P}\mathbf{,}\mathit{max}}$ | Difference from the Baseline |
---|---|---|---|

Single turbine, $\beta =25.3\%$ (baseline) | 1.25 | 0.19 | N/A |

Single turbine, $\beta =50.6\%$ | 1.53 | 0.41 | 115% |

Forward counter-rotating | 1.65 | 0.47 | 144% |

Backward counter-rotating | 1.53 | 0.32 | 65.7% |

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**MDPI and ACS Style**

Doan, M.N.; Obi, S.
Numerical Study of the Dynamic Stall Effect on a Pair of Cross-Flow Hydrokinetic Turbines and Associated Torque Enhancement Due to Flow Blockage. *J. Mar. Sci. Eng.* **2021**, *9*, 829.
https://doi.org/10.3390/jmse9080829

**AMA Style**

Doan MN, Obi S.
Numerical Study of the Dynamic Stall Effect on a Pair of Cross-Flow Hydrokinetic Turbines and Associated Torque Enhancement Due to Flow Blockage. *Journal of Marine Science and Engineering*. 2021; 9(8):829.
https://doi.org/10.3390/jmse9080829

**Chicago/Turabian Style**

Doan, Minh N., and Shinnosuke Obi.
2021. "Numerical Study of the Dynamic Stall Effect on a Pair of Cross-Flow Hydrokinetic Turbines and Associated Torque Enhancement Due to Flow Blockage" *Journal of Marine Science and Engineering* 9, no. 8: 829.
https://doi.org/10.3390/jmse9080829