On the Accuracy of Three-Dimensional Actuator Disc Approach in Modelling a Large-Scale Tidal Turbine in a Simple Channel
2.1. Description of Telemac3D
2.2. Theory of the RANS Actuator Disc
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, .
- 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.
3.1. Actuator Disc Representation in Telemac3D
- 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.
- 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 ( 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 , and Δz in the x, y, and z directions, respectively. For the simulations, lx, ly, and lz were kept constant, but the dimensions of , , and were varied and their impact on the wake characteristics was investigated and the results are presented in Section 4.
- 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 and refers to the turbine diameter, D, while the distance between and corresponds to the disc thickness, .
- 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, as well as the mesh density in the y direction, 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.
- Once the optimal resolution for both and 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
3.3. Boundary Condition
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,
4.3.2. Resolution of the Structured Grid ()
4.3.3. Grid Resolution for
4.4. Sensitivity of the Vertical Resolutions,
4.5. Hydrostatic vs. Non-Hydrostatic Pressure Models
- 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 () 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 () 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.
Conflicts of Interest
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|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|
|Law of the bottom friction and the corresponding friction coefficient||Chezy (44)|
|Turbulence model||k- turbulence models|
|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 m3/s |
Outlet: prescribed elevation, H = 60 m
|Resistance coefficient, K||2 (corresponding to CT = 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|
|Number of Mesh Elements|
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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
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/en11082151Chicago/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