# On the Accuracy of Three-Dimensional Actuator Disc Approach in Modelling a Large-Scale Tidal Turbine in a Simple Channel

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## Abstract

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## 1. Introduction

## 2. Background

#### 2.1. Description of Telemac3D

#### 2.2. Theory of the RANS Actuator Disc

_{i}(i = u, v, w) is the fluid’s velocity component averaged over time t, P is the mean pressure, $\rho $ is the fluid density, $\mu $ is the dynamic viscosity, ${u}^{\prime}$ is an instantaneous velocity fluctuation in time during the time step $\delta t$, ${x}_{i}$ (i = x, y, z) is the spatial geometrical scale, $-\rho \overline{{u}_{i}^{\prime}}\overline{{u}_{j}^{\prime}}$ is the Reynolds stresses that must be solved using turbulence model, ${g}_{i}$ is the component of the gravitational acceleration, and ${S}_{i}$ is an added source term for the ith (where i = x, y or z) momentum equations. In the present paper, the k-$\epsilon $ turbulence model is utilized to close the RANS equations and solve for the Reynolds stresses.

_{T}, open area ratio, $\theta $, and induction factor, a have been discussed in [16,17,18] where:

#### 2.3. Limitation of the Actuator Disc Approach

- The overall turbine structure is not being represented and thus affecting the turbulence in the near wake region, known to be 2–5 rotor diameters downstream of the turbines.
- Kinematics of turbulence, such as vortices trailing from the edges of a blade cannot be replicated, and thus, they must be properly parameterized.
- Energy extraction due to mechanical motion of the turbine rotor cannot be reproduced, instead the energy removed from the disc will be converted into small scale turbulence eddies behind the disc. However, the influence of swirl on the far region is assumed to be minimal.
- This concept cannot be used to investigate the performance of a turbine (e.g., maximum power produced) since it does not include the blades.

#### 2.4. Benchmarking and Data Validation

- Flume dimensions are: 21 m long, 1.35 m wide, and tank depth of 0.3 m.
- Perforated disc with diameter of 0.1 m was used, where the porosity ranged from 0.48 m to 0.35 m (corresponding to coefficient of thrust, ${C}_{T}=0.61\text{}\mathrm{to}\text{}0.97)$.
- The flow speed was approximately set to 0.3 m/s, with mean U velocity component of 0.25 m/s.
- The vertical velocity profile was developed to closely match the 1/7th power law, with uniform velocity near the open surface.
- An acoustic Doppler velocimeter was employed to measure downstream fluid velocities, starting from 3 to 20-disc diameters in a longitudinal direction, as well as up to a 4-disc diameter in the lateral axis.

## 3. Methodology

#### 3.1. Actuator Disc Representation in Telemac3D

- (a)
- The turbine arrangement and the overall dimensions of the domain used in this study is presented in Figure 1, where the disc was located 250 m from the channel inlet. Additionally, its z and y axis centreline were fixed at a 30-m mid-depth and 70-m from the side wall.
- (b)
- The use of a structured grid at the turbine position was chosen as it would allow for a better representation of the turbine shape, as well as maintaining the distance between nodes for refinement purposes. Figure 2 provides the graphical information on the dimensions and pertinent parameters concerning the implementation of the structured grid in the domain. The size of the structured grid (i.e., lx = 26 m, ly = 40 m and lz = 60 m) were deliberately set to be larger than the turbine diameter (D = 20 m) and its width ($\mathsf{\Delta}{x}_{t}=$ 2 m) to allow for numerical tolerance upon the execution of the momentum sink in the TRISOU subroutine. The grid element spacing within this structured grid are denoted by $\mathsf{\Delta}x$, $\mathsf{\Delta}y$ and Δz in the x, y, and z directions, respectively. For the simulations, lx, ly, and lz were kept constant, but the dimensions of $\mathsf{\Delta}x$, $\mathsf{\Delta}y$, and $\mathsf{\Delta}z$ were varied and their impact on the wake characteristics was investigated and the results are presented in Section 4.
- (c)
- The location of the disc (i.e., the turbine) in the domain was specified by four nodes in the horizontal plane (see Figure 2a), denoted as a, b, c, and d, which will act as the enclosure for the turbine. The coordinate of each node was represented by a pair of x and y. The distance between $y\left(4\right)$ and $y\left(1\right)$ refers to the turbine diameter, D, while the distance between $x\left(1\right)$ and $x\left(2\right)$ corresponds to the disc thickness, $\mathsf{\Delta}{x}_{t}$.
- (d)
- Although several mesh transformation options are available in the Telemac3D module, the sigma coordinate system was chosen to represent the depth due to its simplicity, as shown in Figure 3a. In fact, the interval between the vertical planes, $\mathsf{\Delta}z$ as well as the mesh density in the y direction, $\mathsf{\Delta}y$ must be carefully selected since the intersections between the z and y axis nodes will determine the accuracy of disc frame. Coarser mesh density in both the y and z axis will result in a limited number of nodes available within the disc surface area, as shown in Figure 3b. Whereas, Figure 3c portrays unbalanced concentration of the nodes when one of the axis uses a very fine grid resolution compare to the other. Section 4.3.2 will elaborate further on this subject matter.
- (e)
- Once the optimal resolution for both $\mathsf{\Delta}z$ and $\mathsf{\Delta}y$ was established, the momentum source term (see Equation (6)) was applied into the model through the existing nodes within the 10-m radius from the disc centre, Figure 3b. For this purpose, Equation (9) was employed to locate all the relevant nodes that formulate the disc’s 20-m frame.

#### 3.2. Model Set-Up

_{T}and K has been shown previously using Equation (7). In this paper, K is set to a constant value of 2, which correspond to C

_{T}= 0.89.

#### 3.3. Boundary Condition

^{3}/s was imposed at the channel inlet, where Q is equal to the surface area of the inlet (60 m × 140 m) multiply by the mean flow velocity (2.6 m/s). Next, the downstream boundary was set to equal the channel water depth of 60 m to enable flow continuity. Initial condition was set to “PARTICULAR” (an option available in Telemac3D for defining the initial condition), where the initial depth of 60 m was specified in the CONDIM subroutine. A commonly used method of defining inflow velocities is to use a power-law (1/nth) profile. Although it is possible to use any value for n to approximate the flow conditions, a comparison of several values for n was considered to be beyond the scope of this study. Thus, the vertical velocity profile in the domain was imposed using one of the most commonly used power laws, 1/7th, so as to be similar to a full-scale tidal site [24]. In addition, the Chezy formulation with a friction coefficient of 44 was applied to the bottom to reduce the flow velocity as well as to increase the shear near the bed. This value was chosen as it has been used previously in [25] to represent the bottom roughness in the Pentland Firth region. Although not shown in the present paper, a wide range of friction coefficient values have been examined, and their influence on the model’s output were shown to be negligible. In this study, both hydrostatic and non-hydrostatic code were tested, and are discussed in Section 4.5.

#### 3.4. Turbulence Input

## 4. Models’ Sensitivity and Validation

#### 4.1. Validation Metric

#### 4.2. Mesh Dependence Test

#### 4.3. Sensitivity of the Structured Grid at Turbine Location

#### 4.3.1. Influence of the Disc Thickness, $\mathsf{\Delta}{x}_{t}$

#### 4.3.2. Resolution of the Structured Grid ($\mathsf{\Delta}x\text{}\text{}\mathsf{\Delta}y$)

#### 4.3.3. Grid Resolution for $\mathsf{\Delta}y$

#### 4.4. Sensitivity of the Vertical Resolutions, $\Delta z$

#### 4.5. Hydrostatic vs. Non-Hydrostatic Pressure Models

_{z}are the three-dimensional component of the velocity and sink term in the vertical direction, respectively. To examine the influence of both assumptions, models with $\mathsf{\Delta}x=2\text{}\mathrm{m},\text{}\mathsf{\Delta}y$ = 2 m, $\mathsf{\Delta}{x}_{t}=$ 2 m, and $\mathsf{\Delta}z$ = $24\sigma $ layers were simulated in both hydrostatic and non-hydrostatic solvers. Interestingly, the impact of the non-hydrostatic code on the velocity component was less pronounced as illustrated in Figure 10, where both models produced a nearly identical velocity reduction at 1.5 < y/D < 3. However, for the bottom half of the channel, the hydrostatic model somewhat showed a closer agreement with the measurement points than the model using the non-hydrostatic solver, especially in the near bed region. This was apparent in the 4D and 7D regions, where the wake velocity from the non-hydrostatic model showed a slight divergent from the measurement data approximately before y/D = 0.7. This trend continued to persist until the far wake regions, where the non-hydrostatic model slightly overestimated the wake velocity up to y/D = 1.5.

## 5. Conclusions

- The results demonstrated that the numerical model was highly sensitive to the mesh refinement upstream and downstream of the turbine, where coarser models tended to underestimate both the velocity retardation and TI.
- Altering the thickness of the turbine ($\mathsf{\Delta}{x}_{t}$) had negligible impact on the downstream wakes and turbulence mixings.
- Numerical accuracy of the model was found to be highly susceptible to changes in the grid density of the turbine enclosure. The optimal structured grid density of Δx = 2 m, and Δy = 2 m satisfactorily modelled both the velocity deficit and the TI.
- The importance of the grid spacing in y direction ($\mathsf{\Delta}y$) in characterising the thrust and also the shape of the disc was also highlighted.
- The influence of vertical resolutions (Δz) in representing the depth of the channel on the model was investigated to find a balance between computational efficiency and numerical accuracy. The findings indicated that appropriate adjustment on both the horizontal and vertical planes must be attained to accomplish the optimal ratio between the nodes resolution in both z and y orientation. Note that the optimal structured grid density found in this study was limited to the computational resources available to the authors. The methodology presented in this article, however, are still valid for a more refined grid implementation.
- The impact of the hydrostatic and non-hydrostatic pressure assumptions on the predicted output were examined, where both models exhibit nearly indistinguishable flow retardation characteristics behind the simulated disc. However, for the TI, only the non-hydrostatic model was able to match the experimental result, while the hydrostatic solver failed to properly resolve the turbulence mixing in the wake regions between 4D and 15D.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram showing views from the inlet (

**a**) and channel side (

**b**). The front view displays the swept area of the turbine of diameter, D = 20 m.

**Figure 2.**Graphical information of the implementation of the actuator disc approach on the structured grid. Δx, Δy, and Δz are the grids spacing used at the turbine location in x, y and z, directions respectively: (

**a**) dimensions of the embedded structured grid in the domain; (

**b**) x–y plane displaying the “enclosure” of the disc, where the momentum sink is applied within the specified quadrants (i.e., a, b, c, and d). Note that these figures are not drawn to scale.

**Figure 3.**Graphical information exhibiting the influence of the vertical resolutions (24 layers) on the model. Sigma layers and their corresponding depth are presented in drawing (

**a**), where the respective planes that cross the disc’s surface area are clearly shown. Detailed illustration on the comparison of the structured grid density nodes at the turbine swept area (y–z axis) are respectively given in the (

**b**) structured grid with minimal resolution and in the (

**c**) structured grid with higher density. $Y{\left(I\right)}_{i}$ refers to the nodes in the y orientation of the structured grid, while $Z{\left(I\right)}_{i}$ corresponds to the ith vertical planes imposed on the model. Note that these figures are not drawn to scale.

**Figure 4.**x–y horizontal plane illustrating the geometry mesh. The structured grid is used to define the enclosure of the actuator disc, while the rest of the domain is enforced using the unstructured mesh. The turbine is located at 250-m from the channel inlet. Selected downstream nodes that were used in the data extraction and validation purposes are represented by the red points along the turbine centreline in terms of the turbine diameter, D. The dotted rectangle outlines the zone (

**a**) where the mesh refinement (min = 1 m, max = 10 m) was administered.

**Figure 5.**Mesh dependency study for the administered refinement zone at increasing distances downstream of the turbine. The mesh edge length within the refinement zone is varied according to the cases being explored.

**Figure 6.**The influence of the disc thickness, $\mathsf{\Delta}{x}_{t}$ on the flow at increasing distances downstream of the turbine. The value of $\mathsf{\Delta}{x}_{t}$ was varied, while the vertical resolution $\mathsf{\Delta}z$ and the size of the structured grid ($\mathsf{\Delta}x\text{}\text{}\mathsf{\Delta}y$) were maintained at 24 sigma layers and 2 m, respectively for all cases.

**Figure 7.**The influence of the structured grid density, $(\mathsf{\Delta}x\text{}\text{}\mathsf{\Delta}y$) on the flow at increasing distances downstream of the turbine. The vertical resolution,$\mathsf{\Delta}z$, and the disc thickness, $\mathsf{\Delta}{x}_{t}$, were maintained at 24 sigma layers and 2 m, respectively for all cases.

**Figure 8.**The influence of $\mathsf{\Delta}y$ resolution on the flow at increasing distances downstream of the turbine. The value of $\mathsf{\Delta}\mathrm{y}$ was varied, while $\mathsf{\Delta}x$ and $\mathsf{\Delta}z$ were maintained at 2 m and 24 sigma layers, respectively for all cases.

**Figure 9.**The influence of the vertical resolution $\mathsf{\Delta}z$ on the flow at increasing distances downstream of the turbine. The value of Δz was varied, while the size of the structured grid (Δx & Δy) was set to 2 m.

**Figure 10.**Comparison between hydrodynamic and non-hydrodynamic assumptions on the models. The vertical resolution $\mathsf{\Delta}z$ and the size of the structured grid ($\mathsf{\Delta}x\text{}\text{}\mathsf{\Delta}y)$ were maintained at 24 sigma layers and 2 m, respectively for all cases.

**Figure 11.**The influence of the structured grid density $(\mathsf{\Delta}x\text{}\text{}\mathsf{\Delta}y)$ on the flow using the non-hydrostatic approximations. The vertical resolution $\mathsf{\Delta}z$ and the disc thickness $\mathsf{\Delta}{x}_{t}$ were maintained at 24 sigma layers and 2 m, respectively for all cases.

Telemac3D Subroutines | Function |
---|---|

CONDIM | To set the initial condition for the model’s depth and velocity profile |

VEL_PROF_Z | To specify the vertical velocity profile at the channel inlet |

KEPCL3 and KEPINI | To be used with k-epsilon turbulence model |

TRISOU | To implement the source terms for the momentum equations |

**Table 2.**Default values of the numerical parameters employed in the simulation of the actuator disc.

Numerical Parameters | Input/Values |
---|---|

Law of the bottom friction and the corresponding friction coefficient | Chezy (44) |

Turbulence model | k-$\epsilon $ turbulence models |

Hydrostatic assumption | True |

Initial condition | “PARTICULAR” where the initial elevation is set to 60 m |

Vertical resolutions | 24 sigma layers |

Boundary condition on the bottom | Slip condition |

Boundary forcing | Inlet: prescribed flowrate, Q = 21,840 m^{3}/s Outlet: prescribed elevation, H = 60 m |

Resistance coefficient, K | 2 (corresponding to C_{T} = 0.89) |

Refinement Details | Centreline Velocity at Various Longitudinal Positions from Actuator Disc (m/s) | |||||
---|---|---|---|---|---|---|

Mesh Size | Number of Elements | 5D Upstream | 4D Downstream | 7D Downstream | 11D Downstream | 20D Downstream |

Case 1 (1 m) | 925,536 | 2.683 | 1.515 | 1.903 | 2.153 | 2.418 |

Case 2 (2 m) | 316,584 | 2.680 | 1.735 | 2.092 | 2.315 | 2.554 |

Case 3 (5 m) | 101,496 | 2.690 | 1.853 | 2.186 | 2.391 | 2.600 |

Case 4 (10 m) | 72,936 | 2.694 | 1.846 | 2.247 | 2.464 | 2.659 |

Sigma Layers | Distance between Planes (in Meter) | Number of Mesh Elements |
---|---|---|

24 | 2.61 | 317,928 |

18 | 3.53 | 238,446 |

15 | 4.29 | 198,705 |

10 | 6.66 | 132,470 |

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## Share and Cite

**MDPI and ACS Style**

Rahman, A.; Venugopal, V.; Thiebot, J.
On the Accuracy of Three-Dimensional Actuator Disc Approach in Modelling a Large-Scale Tidal Turbine in a Simple Channel. *Energies* **2018**, *11*, 2151.
https://doi.org/10.3390/en11082151

**AMA Style**

Rahman A, Venugopal V, Thiebot J.
On the Accuracy of Three-Dimensional Actuator Disc Approach in Modelling a Large-Scale Tidal Turbine in a Simple Channel. *Energies*. 2018; 11(8):2151.
https://doi.org/10.3390/en11082151

**Chicago/Turabian Style**

Rahman, Anas, Vengatesan Venugopal, and Jerome Thiebot.
2018. "On the Accuracy of Three-Dimensional Actuator Disc Approach in Modelling a Large-Scale Tidal Turbine in a Simple Channel" *Energies* 11, no. 8: 2151.
https://doi.org/10.3390/en11082151