# Hydrodynamic Effects of Tidal-Stream Power Extraction for Varying Turbine Operating Conditions

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## Abstract

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## 1. Introduction

- Influence of the turbine’s wake induction factor in the thrust and power coefficients calculation.
- Flow rate effect due to power extraction for increasing values of the blockage ratio.
- Evaluation of plausible ranges of the turbine’s wake induction factor and blockage ratio values and their effect on the local hydrodynamics through the examination of the elevation and velocity profiles.
- Assessment of the tidal-stream potential energy resource considering optimal conditions of turbine performance.

## 2. Modelling Approach

#### 2.1. Hydrodynamic Models

#### 2.1.1. Gradually Varying Flows

#### 2.1.2. Rapidly Varying Flows

#### 2.1.3. Grid Structures

#### 2.2. Turbine-Array Representation

#### 2.3. Domain and Array Configuration

## 3. Hydrodynamic Effects of Power Extraction

#### 3.1. Wake-Induction Factor

#### 3.2. Power Extraction and Flow Rate Affectation

#### 3.3. Wake-Induction Factor Influence on the Tidal-Stream

#### 3.4. Array Configuration and Blockage Ratio Influence on the Tidal-Stream

## 4. Energy Resource Assessment

#### 4.1. Relative Head Drop and Turbine Efficiency

#### 4.2. Power Metrics

## 5. Discussion and Conclusions

- The turbine’s operating conditions played an important role in determining the available power for electrical energy generation.
- The maximisation of power extraction for electrical generation required the use of the optimum turbine-wake induction factor and an adequate blockage ratio, so that the power loss due to the turbine’s wake mixing was reduced.
- Situations where limiting values of the turbine’s operating conditions are used should be avoided, as they led to negligible power available.
- –
- A low wake induction factor was related to high thrust forces, characterised by low porosity turbines. These conditions produced (i) high downstream turbine-wake mixing and, consequently, a high power loss, (ii) strong velocity reduction, and (iii) significant head drop.
- –
- A high wake induction factor indicated low thrust forces, produced by high porosity turbines. This limit indicated (i) less flow disturbance, (ii) small velocity reduction, and (iii) negligible head drop.
- –
- Large blockage ratios also produced high thrust forces, which reduced the flow rate, intensified the magnitudes of flow bypassing the turbines, and enhanced turbine-wake mixing dissipation, reducing the amount of power extracted by the turbines.

- An accurate evaluation of the turbine’s operating conditions’ effect on local hydrodynamics was provided by an RVF solver. Particularly satisfactory results were obtained for a partial-fence.
- For scenarios where power was extracted uniformly across a channel (fence configuration), the GVF solver was a computationally economical tool to assess the resource; however, prudence should be taken as the solver underestimated the velocity reduction produced by power extraction.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sketches of the horizontal grid structure, alternating direction implicit (ADI) model: staggered grid (

**a**), total variation diminishing (TVD) model: non-staggered grid (

**b**).

**Figure 2.**Sketches of the 2D shallow waters model (

**a**), the tidal-stream power extraction with an actuator disc in open channel flows model (

**b**), and turbine’s thrust force $\overrightarrow{{F}_{T}}$ exerted on the incident tidal-stream (

**c**).

**Figure 3.**Tidal channel with a constant cross-section, which connected two large basins. An array of turbines was deployed in the middle of the channel.

**Figure 4.**Time-averaged thrust and power coefficients obtained from a plausible range of the wake-induction factor, ADI (x marker), and TVD (square marker).

**Figure 5.**Normalised maximum power removed by the turbines against the normalised maximum flow rate for increasing values of B. TVD (unfilled markers), ADI (filled markers), and the analytical solution (continuous line). Reproduced from [55] with permission from Elsevier.

**Figure 6.**Plan view of the channel and stream-wise transect indicator (

**a**). Transverse view of normalised water elevation profiles along the stream-wise transect obtained with ADI (

**b**,

**c**) and TVD (

**d**,

**e**). Nat Sta, natural state.

**Figure 7.**Selected values of normalised water elevation profiles and the Y component of the velocity along the channel centreline obtained with ADI (

**a**,

**b**) and TVD (

**c**,

**d**).

**Figure 8.**Stream-wise profile of the water elevation $\zeta $ and Y component of the velocity, along the channel centreline, for a fence (continuous line) and partial-fence (dashed line) configuration obtained with ADI (

**a**,

**b**) and TVD (

**c**,

**d**).

**Figure 9.**Effect of the blockage ratio on the maximum head drop (

**a**) and the turbine’s efficiencies (

**b**); results obtained for a fence configuration. Reproduced from [55] with permission from Elsevier.

**Figure 10.**Effect of the blockage ratio on ${P}_{T}$ (

**a.***), ${P}_{*}$ (

**b.***), and ${P}_{W}$ (

**c.***). Solutions from the analytical model (filled markers), TVD (dashed line), and ADI (continuous line). Reproduced from [55] with permission from Elsevier.

Model | Configuration | Scenario | B | ${\mathit{\alpha}}_{4}$ | $\mathbf{\Delta}\mathit{t}$ (s) |
---|---|---|---|---|---|

ADI | Fence | 1 | $0\le B\le 0.8$ | ${\alpha}_{4}=1/3$ | 12 |

2 | B = 0.2 | 0 $<{\alpha}_{4}<$1 | 12 | ||

Partial-Fence | 3 | $0\le B\le 0.8$ | ${\alpha}_{4}=1/3$ | 12 | |

TVD | Fence | 4 | $0\le B\le 0.8$ | ${\alpha}_{4}=1/3$ | 1.5 |

5 | B = 0.2 | 0 $<{\alpha}_{4}<$1 | 1.5 | ||

Partial-Fence | 6 | $0\le B\le 0.8$ | ${\alpha}_{4}=1/3$ | 1.5 |

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## Share and Cite

**MDPI and ACS Style**

Flores Mateos, L.; Hartnett, M.
Hydrodynamic Effects of Tidal-Stream Power Extraction for Varying Turbine Operating Conditions. *Energies* **2020**, *13*, 3240.
https://doi.org/10.3390/en13123240

**AMA Style**

Flores Mateos L, Hartnett M.
Hydrodynamic Effects of Tidal-Stream Power Extraction for Varying Turbine Operating Conditions. *Energies*. 2020; 13(12):3240.
https://doi.org/10.3390/en13123240

**Chicago/Turabian Style**

Flores Mateos, Lilia, and Michael Hartnett.
2020. "Hydrodynamic Effects of Tidal-Stream Power Extraction for Varying Turbine Operating Conditions" *Energies* 13, no. 12: 3240.
https://doi.org/10.3390/en13123240