# The Influence of Intra-Array Wake Dynamics on Depth-Averaged Kinetic Tidal Turbine Energy Extraction Simulations

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}) of 0.58–0.69 (RMSE = 7.16–8.28%) using a k-Ɛ turbulence closure scheme. Various array configurations at device scale are simulated and compared with an equivalent array-averaging approach by analyzing channel flux differential. Parametrization of array-averaging energy extraction techniques can misrepresent simulated energy transfer and removal. The potential peak error in channel flux exceeds 0.5% when the number of turbines n

_{TECs}≈ 25 devices. This error exceeds 2% when simulating commercial-scale turbine array farms (i.e., >100 devices).

## 1. Introduction

- Actual energy captured by TECs requiring consideration of device ‘water to wire’ efficiency factor that accounts for hydrodynamic, mechanical and electrical losses.
- Intra-array spacing of devices relative to the plan area of model cells.
- Support structure form drag associated with each TEC.
- Any associated energy losses in downstream wake mixing and free stream velocity interaction between devices.

- To incorporate tidal energy extraction at device scale.
- To test the performance of the modelled hydrodynamics.
- To compare array-averaging with device-scale results.
- To quantify associated uncertainty.

## 2. Materials and Methods

#### 2.1. Seabed Stress Term

^{−1}s

^{−2}or N m

^{−2}), is a decelerating force exerted per unit area of seabed due to the flow. The implementation of turbines enhances or retards the total seabed shear stress within the local region. Numerical 2D models represent this process via an additional stress term in the momentum equations. A simple representation of seabed shear stress, assuming a gently sloping seabed is represented by the quadratic friction law [32]:

^{−1}), by the quadratic friction law based on the drag coefficient; C

_{D}. ρ is water density (kg m

^{−3}); h is water depth (m); g is acceleration due to gravity (9.81 m s

^{−2}); and n (s m

^{−1/3}) is a Manning coefficient value that can be related to bed roughness and grain size. Bed shear stress can be used to determine sediment transport rates, based on predetermined flow thresholds for sediment resuspension and settlement. For some model applications, it may be important to accurately parameterize both spatial and temporal variations in seabed roughness and, hence, bed shear stress. TEC array developments, for example, both affect and are influenced by differences in substrate type and water column turbidity levels [33].

#### 2.2. Blockage Ratio

#### 2.3. Mesh Resolution Dependency

#### 2.4. Turbulence Closure Schemes

- Zero equation models (constant viscosity, mixing-length, free-shear layer etc.)
- One equation models (Eddy viscosity concept, Bradshaw et al. etc.)
- Two equation models (k-Ɛ, k-ω, ASM etc.)
- Seven equation model (Reynolds turbulent stress/flux)

- Constant viscosity (CV, the default in Telemac)
- Elder model
- k-epsilon (k-Ɛ) model
- Smagorinski model

#### 2.5. Constant Viscosity Scheme

^{−6}, corresponding to the molecular viscosity of water. When this approach is applied to simulations, the optimal velocity diffusivity coefficient value must be determined. Values for horizontal velocity diffusivity (m

^{2}s

^{−1}) applicable to coastal applications are difficult to source, although values of between 0.1 and 50 are suggested for fluvial environments [39]. Haverson et al. [40,41] use the CV TCS and apply coefficient values of 10 and 10

^{−6}, respectively, whereas Fallon et al. [42] use a value of 1.

^{−6}to 1. This sensitivity test is calibrated against the empirical data (Appendix B).

#### 2.6. Depth-Averaged K-Epsilon Scheme

^{−6}) should always be applied when using this scheme.

#### 2.7. Energy Extraction Term

_{D}) of 0.6 [13,20]. Published coastal modelling studies have used an increased (i.e., rougher) value of C

_{D}, for example Martin-Short et al. [19] use 0.7, whereas other studies [13,20,41] use 0.9. Previous research has also assumed that biofouling on structures might increase drag by up to 50% [45]. However, further research may be required in order to justify such an assumption.

_{T}) values that vary with change in velocity (Figure 3). Below device cut-in and above cut-out velocity, thrust (or flow backpressure when the turbine rotates and generates electrical power) is zero (Equation (5)). Between cut-in and rated velocity, C

_{T}is continuous at 0.85 (Equation (6)), when transitioning from rated to cut-out velocity, C

_{T}follows a polynomial curve fit based on the values presented by Baston et al. [46] (Equation (7)). This simulates feathering of the turbine blades to maintain optimal generation and rated power as flow velocity increases.

_{T}and F

_{S}(kg m s

^{−2}or N) are:

_{r}(m s

^{−1}) is a reference upstream velocity and θ (°) is the angle of orientation of the central axis of the TEC to the flow, having a positive clockwise value when rotated from the x-axis as defined by Joly et al. [47]. A

_{T}is the turbine rotor swept area and A

_{S}is the exposed area of the TEC support structure to flow (Equation (10)). When implementing the decelerating force terms (Equations (11) and (12)) into the model, we calculate the defined area of seabed, A, containing each turbine (i) based on the number of mesh nodes (j) and their associated element size. For this reason, regular grid elements are utilized that facilitate equidistant spacing in order to minimize errors associated with mesh density [48]. The calculated total stress is thus distributed evenly across the total number of mesh nodes within the defined device area (i.e., 20 m

^{2}, see Table 1). We thus define the stress term (N m

^{−2}) to be:

^{−1}):

- number of TECs
- center position coordinates of each TEC
- the length and width of the grid area occupied by each TEC
- orientation of center axis of TEC to x axis
- TEC rotor radius
- TEC upstream reference velocity distance
- TEC structural drag coefficient

_{P}) value occurs as this is calculated within the subroutine in a similar manner to C

_{T}, i.e., to enable more realistic estimation of time varying outputs with change in velocity. The standard turbine power, P (W) equation is:

_{P}value for each device as:

_{RAT}is the maximum power generated at rated velocity. C

_{P}is assumed to be the water to wire coefficient value that includes all losses due to energy transfer (i.e., mechanical, electrical, hydrokinetic etc.). It should be noted that for this study, we utilize the same generic turbine coefficient values for all TECs simulated. In addition, whereas array-averaged simulations exhibit an inherent inability to account for device power alteration in relation to rotor orientation to flow direction, using a device-scale approach does not. Published studies [6,16] have considered the effect that device rotor face offset, or yaw misalignment to flow has in combination with other effects, such as flow asymmetry and ambient turbulent intensity. The influence of device orientation offset will affect available power generated at each rotor due to changes in channel flow power density caused by intra-array turbulence. A convenient measure of change in available array power is capacity factor (CF), which is the average power generated over a given period, divided by the rated peak power. It is, therefore, not possible to precisely determine CF using an array-averaged approach.

#### 2.8. Method Validation

^{−3}. A time step of 0.1 s was chosen based on the Courant stability criterion of the finest mesh (although clearly a larger time step could be used for the coarser mesh simulations).

#### 2.9. Axial Wake Centreline Velocity Deficit Profile

_{def}):

_{0}is a measured undisturbed velocity taken at a distance equal to five rotor diameters upstream of the rotor position. U

_{x}(x = 1−20) are a series of measurements taken equidistantly one rotor diameter apart in the downstream axial centerline wake (Figure 4). A polynomial fit and nearest neighbour extrapolation is applied to approximate the mean profile from published data (Figure 5). Error estimates (±1 s.d) about the mean provide a realistic range of values across downstream positions. This type of validation accounts for the effects that axial turbine thrust [49], turbine proximity to boundaries and turbulent intensity levels [49,53,56] impart on measurements for various rotor designs (e.g., two or three blades).

#### 2.10. Transverse Wake Profile

#### 2.11. Array-Scale Evaluation

^{2}s

^{−1}) were calculated using a 5 km wide discretized channel domain that incorporates applied energy extraction for various array layouts (Figure 6). The differential channel width integrated flux between an upstream and downstream section spaced a kilometer either side of the array central position is analyzed. This highlights changes caused by intra-array effects that alter the flow through the channel, where:

_{TECs}. It is hypothesized that existential difference in channel flow will be exhibited due to the nuances of the applied methodologies (Figure 7).

## 3. Results

#### 3.1. Single Device Validation

^{2}) and root mean squared error (RMSE) values (Table 3). Determinant accuracy of fit ranges by 11% (R

^{2}) and RMSE by 1.12%, when altering mesh size. The finest grid discretization provides the most accurate energy extraction determined from axial centerline deficit profiles (R

^{2}= 0.69, RMSE = 7.16%). Coarser grids provide a less accurate fit (R

^{2}= 0.58 and 0.61, RMSE = 8.28% and 8.02%, respectively). Maximum velocity deficit deviation from ambient flow conditions occurs immediately behind a turbine. The mean axial centerline velocity deficit (Figure 5) is approximately 52% one rotor diameter downstream, and 17% at 10 rotor diameters. In comparison, maximum simulated velocity deficit (30–38%) occurs 2–4 rotor diameters downstream (Figure 8).

#### 3.2. Array-Scale Analysis

_{TECs}= 24 for inline turbine rows and n

_{TECs}≈ 26 devices for staggered turbine rows.

## 4. Discussion

^{2}). The subsequent RMSE is 7.16%, 8.28% and 8.02% dependent upon mesh resolution (1, 5 and 10 m respectively). Simulations are based on a 20 m rotor diameter, ensuring that a minimum of two computational nodes fall within the defined device seabed area for the coarse mesh and twenty at finest resolution.

_{T}), (i.e., the backpressure exerted on flow) and the associated intra-array device wake interactions that result, greatly influence turbulent mixing and energy dissipation and transfer within the hydrokinetic system. Comparison of the difference between undisturbed inflow conditions to the downstream, disturbed flux for device-scale and array-averaged simulations, illustrates that wake interactions enhance impacts as the number of devices in the array increases.

_{TECs}≈ 25 devices for inline and staggered rows of turbines.

_{P}. Flow velocity is affected by system feedbacks from energy extraction terms based on drag (C

_{D}) and thrust (C

_{T}) parameterization. Therefore, individual device power capture is independent of the overall associated energy loss.

## 5. Conclusions

^{2}) and 7.16–8.28% (RMSE) was achieved when compared with empirical, scaled tank tests. If using a constant viscosity TCS, a velocity diffusivity coefficient of 0.5 must be assigned to achieve similar accuracy.

^{°}to the flow. This increases from 1.00% to 1.71% and 1.81% when the offset is 20°. However, power captured by an array is independent of energy extraction governed by drag and thrust terms, but dependent upon intra-array dynamics and associated velocity manipulation from device-wake feedbacks. Difference in CF is negated or improved (0.02%, 0.01% and −0.11%, −0.45% for five and fifteen devices, respectively) when devices are configured in staggered rows.

_{TECs}≈ 24–26 devices. This suggests that quantification of the uncertainty associated with utilizing an array-averaging approach, particularly when upscaling to commercial tidal array configurations, may be pertinent. Our results indicate that the uncertainty associated with upscaling must necessarily be established for environmental assessments that negate intra-array effects, as this will exceed 2% error when n

_{TECs}≥100 devices.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Modified format of external data input file providing TEC characteristics for Telemac DRAGFO subroutine.

Formatted Data File 2 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

N | ||||||||||||||

X1 | Y1 | L1 | W1 | THETA1 | R1 | DD1 | CD1 | EFFIN1 | EFFRAT1 | VIN1 | VRAT1 | VOUT1 | MH1 | MD1 |

X2 | Y2 | L2 | W2 | THETA2 | R2 | DD2 | CD2 | EFFIN2 | EFFRAT2 | VIN2 | VRAT2 | VOUT2 | MH2 | MD2 |

: | : | : | : | : | : | : | : | : | : | : | : | : | : | : |

XN | YN | LN | WN | THETAN | RN | DDN | CDN | EFFINN | EFFRATN | VINN | VRATN | VOUTN | MHN | MDN |

## Appendix B

**Figure A1.**Simulated Telemac-2D along channel axial centerline velocity deficit profiles using the constant viscosity scheme at 1 m grid resolution with varying coefficient values. Solid line represents mean empirical test results, grey shading highlights ±1 s.d (see Figure 5).

**Table A2.**Evaluation of CV turbulence closure scheme fit to mean observed physical modelling data for various velocity diffusivity coefficient values.

Turbulence Scheme | Mesh Resolution (m) | Coefficient | R^{2} | RMSE (%) |
---|---|---|---|---|

Constant Viscosity | 1 | 10^{−6} | <0.01 | 17.03 |

1 | 0.1 | <0.01 | 12.84 | |

1 | 0.5 | 0.67 | 7.34 | |

1 | 1.0 | 0.13 | 11.92 |

## References

- Roche, R.C.; Walker-Springett, K.; Robins, P.E.; Jones, J.; Veneruso, G.; Whitton, T.A.; Piano, M.; Ward, S.L.; Duce, C.E.; Waggitt, J.J.; et al. Research priorities for assessing potential impacts of emerging marine renewable energy technologies: Insights from developments in Wales (UK). Renew. Energy
**2016**, 99, 1327–1341. [Google Scholar] [CrossRef] - Magnaga, D.; Uihlein, A. Ocean energy development in Europe: Current status and future perspectives. Int. J. Mar. Energy
**2015**, 11, 84–104. [Google Scholar] [CrossRef] - Neill, S.P.; Litt, E.J.; Couch, S.J.; Davies, A.G. The impact of tidal stream turbines on large-scale sediment dynamics. Renew. Energy
**2009**, 34, 2803–2812. [Google Scholar] [CrossRef] - Robins, P.E.; Neill, S.P.; Lewis, M.J. Impacts of tidal-stream arrays in relation to the natural variability of sedimentary processes. Renew. Energy
**2014**, 72, 311–321. [Google Scholar] [CrossRef] - The European Marine Energy Centre Ltd. (EMEC). Assessment of Tidal Energy Resource—Marine Renewable Energy Guides; BSI: London, UK, 2009; Available online: http://www.emec.org.uk/assessment-of-tidal-energy-resource/ (accessed on 29 May 2017).
- Frost, C.H.; Evans, P.S.; Harrold, M.J.; Mason-Jones, A.; O’Doherty, T.; O’Doherty, D.M. The impact of axial flow misalignment on a tidal turbine. Renew. Energy
**2017**, 113, 1333–1344. [Google Scholar] [CrossRef] - Chen, Y.; Lin, B.; Lin, J.; Wang, S. Experimental study of wake structure behind a horizontal axis tidal stream turbine. Appl. Energy
**2017**, 196, 82–96. [Google Scholar] [CrossRef] - Uihlein, A.; Magagna, D. Wave and tidal current energy—A review of the current state of research beyond technology. Renew. Sustain. Energy Rev.
**2016**, 58, 1070–1081. [Google Scholar] [CrossRef] - Work, P.A.; Haas, K.A.; Defne, Z.; Gay, T. Tidal sream energy site assessment via three-dimensional model and measurements. Appl. Energy
**2013**, 102, 510–519. [Google Scholar] [CrossRef] - Fang, H.; Xianwen, B.; Benxia, L.; Qianqian, L. The assessment of extractable tidal energy and the effect of tidal energy turbine deployment on the hydrodynamics in Zhoushan. Acta Oceanol. Sin.
**2015**, 34, 86–91. [Google Scholar] - Yang, Z.; Wang, T.; Copping, A.E. Modeling tidal stream energy extraction and its effects on transport processes in a tidal channel and bay system using a three-dimensional coastal ocean model. Renew. Energy
**2013**, 50, 605–613. [Google Scholar] [CrossRef] - Neill, S.P.; Jordan, J.R.; Couch, S.J. Impact of tidal energy converter (TEC) arrays on the dynamics of headland sand. Renew. Energy
**2012**, 37, 387–397. [Google Scholar] [CrossRef] - Plew, D.R.; Stevens, C.L. Numerical modelling of the effect of turbines on currents in a tidal channel—Tory Channel, New Zealand. Renew. Energy
**2013**, 57, 269–282. [Google Scholar] [CrossRef] - Lewis, M.; Neill, S.P.; Robins, P.E.; Hashemi, M.R. Resource Assessment for future generations of tidal-sream energy arrays. Energy
**2015**, 83, 403–415. [Google Scholar] [CrossRef] [Green Version] - Robins, P.E.; Neill, S.P.; Lewis, M.J.; Ward, S.L. Characterising the spatial and temporal variability of the tidal-stream energy resource over the northwest European shelf seas. Appl. Energy
**2015**, 147, 510–522. [Google Scholar] [CrossRef] [Green Version] - Piano, M.; Neill, S.P.; Lewis, M.J.; Robins, P.E.; Hashemi, M.R.; Davies, A.G.; Ward, S.L.; Roberts, M.J. Tidal stream resource assessment uncertainty due to flow asymmetry and turbine yaw misalignment. Renew. Energy
**2017**, 114, 1363–1375. [Google Scholar] [CrossRef] - Piano, M.; Ward, S.; Robins, P.; Neill, S.; Lewis, M.; Davies, A.G.; Powell, B.; Wyn-Owen, A.; Hashemi, M.R. Characterizing the tidal energy resource of the West Anglesey Demonstration Zone (UK), using Telemac-2D and field observations. In Proceedings of the Telemac & Mascaret Users Conference, Daresbury, UK, 15–16 October 2015. [Google Scholar]
- Neill, S.P.; Hashemi, M.R. Fundamentals of Ocean Renewable Energy: Generating Electricity from the Sea, 1st ed.; Academic Press: Cambridge, MA, USA, 2018; ISBN 9780128104484. [Google Scholar]
- Martin-Short, R.; Hill, J.; Kramer, S.C.; Avdis, A.; Allison, P.A.; Piggott, M.D. Tidal resource extraction in the Pentland Firth, UK: Potential impacts on flow regime and sediment transport in the Inner Sound of Stroma. Renew. Energy
**2015**, 76, 596–607. [Google Scholar] [CrossRef] - Perez-Ortiz, A.; Pescatore, J.; Bryden, I. A systematic approach to undertake tidal energy resource assessment with Telemac-2D. In Proceedings of the European Wave and Tidal Energy Conference (EWTEC), Aalborg, Denmark, 2–5 September 2013. [Google Scholar]
- International Electrotechnical Commission. IEC 62600-201 TS: Marine Energy-Wave, Tidal and Other Water Current Converters-Part 201: Tidal Energy Resource Assessment and Characterization; IEC & BSI: Vancouver, BC, Canada, 2014; Available online: https://webstore.iec.ch/publication/22099 (accessed on 3 June 2016).
- Divett, T.; Vennell, R.; Stevens, C. Optimisation of multiple turbine arrays in a channel with tidally reversing flow by numerical modelling with adaptive mesh. Philos. Trans. R. Soc. London, Ser. A: Math. Phys. Eng. Sci.
**2013**, 371, 1–12. [Google Scholar] [CrossRef] [PubMed] - Masters, I.; Williams, A.; Croft, T.N.; Togneri, M.; Edmunds, M.; Zangiabadi, E.; Fairley, I.; Karunarathna, H. A Comparison of Numerical Modelling Techniques for Tidal Stream Turbine Analysis. Energies
**2015**, 8, 7833–7853. [Google Scholar] [CrossRef] [Green Version] - Ramos, V.; Carballo, R.; Alvarez, M.; Sanchez, M.; Iglesias, G. Assessment of the impacts of tidal stream energy through high-resolution numerical modelling. Energy
**2013**, 61, 541–554. [Google Scholar] [CrossRef] - Shives, M.; Crawford, C.; Hiles, C.; Walters, R. Combining numerical methods for basin and turbine scales for improved modelling of in-situ turbine arrays. In Proceedings of the European Wave and Tidal Energy Conference (EWTEC), Aalborg, Denmark, 2–5 September 2013. [Google Scholar]
- Shives, M.; Crawford, C.; Grovue, S. A tuned actuator cylinder approach for predicting cross-flow turbine performance with wake interaction and channel blockage effects. Int. J. Mar. Energy
**2017**, 18, 30–56. [Google Scholar] [CrossRef] - Ahmed, A.; Afgan, I.; Apsley, D.D.; Stallard, T.; Stansby, P.K. CFD simulations of a full-scale tidal turbine: comparison of LES and RANS with field data. In Proceedings of the European Wave and Tidal Energy Conference (EWTEC), Nantes, France, 6–11 September 2015. [Google Scholar]
- Stallard, T.; Collings, R.; Feng, T.; Whelan, J. Interactions between tidal turbine wakes: experimental study of a group of three-bladed rotors. Philos. Trans. Ser. A Math. Phys. Eng. Sci.
**2013**, 371. [Google Scholar] [CrossRef] [PubMed] - Kramer, S.C.; Piggott, M.D. A correction to the enhanced bottom drag parameterisation of tidal turbines. Renew. Energy
**2016**, 92, 385–396. [Google Scholar] [CrossRef] - Villaret, C.; Hervouet, J.-M.; Kopmann, R.; Merkel, U.; Davies, A.G. Morphodynamic modeling using the Telemac finite-element system. Comput. Geosci.
**2013**, 53, 105–113. [Google Scholar] [CrossRef] - Supercomputing Wales. Available online: https://www.supercomputing.wales/#hero (accessed on 30 Augest 2018).
- Soulsby, R. Dynamics of Marine Sands, 1st ed.; Thomas Telford Publications: London, UK, 1997; ISBN 072772584X. [Google Scholar]
- Guillou, N.; Thiebot, J. The impact of seabed rock roughness on tidal stream power extraction. Energy
**2016**, 112, 762–773. [Google Scholar] [CrossRef] - Gaurier, B.; Germain, G.; Facq, J.V.; Johnstone, C.M.; Grant, A.D.; Day, A.H.; Nixon, E.; de Felice, F.; Constanzo, M. Tidal energy “Round Robin” tests—comparisons between towing tank and circulating tank results. Int. J. Mar. Energy
**2015**, 12, 87–109. [Google Scholar] [CrossRef] [Green Version] - Hervouet, J.M. Hydrodynamics of Free Surface Flows, 1st ed.; John Wiley and Sons: New York, NY, USA, 2007; ISBN 9780470035580. [Google Scholar]
- Melvin, J.; Kaufman, R.; Lim, H.; Kaman, T.; Rao, P.; Glimm, J. Macro and micro issues in turbulent mixing. Sci. Ch. Technol. Sci.
**2013**, 56, 2355–2360. [Google Scholar] [CrossRef] - Hanjalic, K. Closure Models for Incompressible Turbulent Flows. Delft University of Technology, Delft, 2004. Available online: https://www.researchgate.net/file.PostFileLoader.html?id=53898c5fd039b1bd3a8b45c0&assetKey=AS%3A273543839846420%401442229341759 (accessed on 4 November 2017).
- Rodi, W. Turbulence Models and Their Application in Hydraulics, 3rd ed.; A.A. Balkema: Rotterdam, The Netherland, 2000. [Google Scholar]
- Fischer, H.B.; List, E.J.; Koh, R.C.; Imberger, J.; Brooks, N.H. Mixing in Inland and Coastal Waters, 1st ed.; Academic Press: New York, NY, USA, 1979. [Google Scholar]
- Haverson, D.; Bacon, J.; Smith, H. Modelling the far field impacts of a tidal energy development at Ramsey Sound. In Proceedings of the XXI Telemac and Mascaret User Club, Grenoble, France, 15–17 October 2014. [Google Scholar]
- Haverson, D.; Bacon, J.; Smith, H.C.; Venugopal, V.; Xiao, Q. Cumulative Impact Assessment of Tidal Stream Energy Extraction in the Irish Sea. Ocean Eng.
**2017**, 137, 417–428. [Google Scholar] [CrossRef] - Fallon, D.; Hartnett, M.; Olbert, A.; Nash, S. The effects of array configuration on the hydro-environmental impacts of tidal turbines. Renew. Energy
**2014**, 64, 10–25. [Google Scholar] [CrossRef] [Green Version] - Rastogi, A.K.; Rodi, W. Predictions of heat and mass transfer in open channels. J. Hydraul. Div. ASCE
**1978**, HY3, 397–420. [Google Scholar] - Shives, M.; Crawford, C. Tuned actuator disk approach for predicting tidal turbine performance with wake interaction. Int. J. Mar. Energy
**2017**, 17, 1–20. [Google Scholar] [CrossRef] - Batten, W.M.; Bahaj, A.S.; Molland, A.F.; Chaplin, J.R. The prediction of the hydrodynamic performance of marine current turbines. Renew. Energy
**2008**, 33, 1085–1096. [Google Scholar] [CrossRef] - Baston, S.; Waldman, S.; Side, J. Modelling Energy Extraction in Tidal Flows: TerraWatt Position Paper Revision 3.1. TerraWatt Consortium, July 2015. Available online: https://www.researchgate.net/profile/Susana_Baston/publication/283008185_Modelling_energy_extraction_in_tidal_flows_Terawatt_position_paper_revision_31/links/5652efa108aefe619b18e368/Modelling-energy-extraction-in-tidal-flows-Terawatt-position-paper-revision-31.pdf (accessed on 5 January 2016).
- Joly, A.; Pham, C.-T.; Andreewsky, M.; Saviot, S.; Fillot, L. Using the DRAGFO subroutine to model Tidal Energy Converters in Telemac-2D. In Proceedings of the XXII Telemac & Mascaret Users Conference, Daresbury, UK, 15–16 October 2015. [Google Scholar]
- Waldman, S.M.; Genet, G.; Baston, S.; Side, J. Correcting for mesh size dependency in a regional model’s representation of tidal turbines. In Proceedings of the 11th European Wave and Tidal Energy Conference (EWTEC), Nantes, France, 6–11 September 2015. [Google Scholar]
- Bahaj, A.S.; Myers, L.E.; Thomson, M.D.; Jorge, N. Characterising the wake of horizontal axis marine current turbines. In Proceedings of the 7th European Wave and Tidal Energy Conference (EWTEC), Porto, Portugal, 11–14 September 2007. [Google Scholar]
- Myers, L.; Bahaj, A.S. Near wake properties of horizontal axis marine current turbines. In Proceedings of the 8th European Wave and Tidal Energy Conference, Uppsala, Sweden, 7–10 September 2009. [Google Scholar]
- Maganga, F.; Pinon, G.; Germain, G.; Rivoalen, E. Wake properties characterisation of marine current turbines. In Proceedings of the 3rd International Conference on Ocean Energy, Bilbao, Spain, 6–8 October 2010. [Google Scholar]
- Gomez, M.G.; Grant, A. Marine Current Turbines: Array Effects. Energy Systems Research Unit. Master’s Thesis, University of Strathclyde, Glasgow, UK, September 2008. [Google Scholar]
- Myers, L.E.; Bahaj, A.S. Experimental analysis of the flow field around horizontal axis tidal turbines by use of scale mesh disk rotor simulators. Ocean Eng.
**2010**, 37, 218–227. [Google Scholar] [CrossRef] - Abolghasemi, M.A.; Piggott, M.D.; Spinneken, J.; Vire, A.; Cotter, C.J.; Crammond, S. Simulating tidal turbines with multi-scale mesh optimisation techniques. J. Fluids Struct.
**2016**, 66, 69–90. [Google Scholar] [CrossRef] - Stallard, T.; Feng, T.; Stansby, P.K. Experimental study of the mean wake of a tidal stream rotor in a shallow turbulent flow. J. Fluids Struct.
**2015**, 54, 235–246. [Google Scholar] [CrossRef] - Mycek, P.; Gaurier, B.; Germain, G.; Pinon, G.; Rivolean, E. Experimental study of the turbulence intensity effects on marine current turbines behaviour. Part I: One single turbine. Renew. Energy
**2014**, 66, 729–746. [Google Scholar] [CrossRef] - Roc, T.; Greaves, D.; Thyng, K.M.; Conley, D.C. Tidal turbine representation in an ocean circulation model: Towards realistic applications. Ocean Eng.
**2014**, 78, 95–111. [Google Scholar] [CrossRef] - Guillou, N.; Chapalain, G.; Neill, S. The influence of waves on the tidal kinetic energy resource at a tidal stream energy site. Appl. Energy
**2016**, 180, 402–415. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Plan view of the discretized tidal channel. Zoomed section illustrates various mesh density resolutions applied in order to implement a single TEC (solid black polygon) in the center of the domain. The associated system calculated, seabed area (shaded dashed polygon) over which the enhanced stress term is applied is thus element mesh size dependent (Table 1).

**Figure 2.**Depth-averaged velocity (dashed vertical line) values typically represent those occurring at 2/5th of the vertical water column, regardless of the applied power law. Even distribution of the profile occurs approximately 1/5th to 3/5th of the vertical water column.

**Figure 3.**(

**a**) TEC thrust coefficient curve; (

**b**) Resultant thrust and structural drag force; (

**c**) Formulated normalized stress for the parameterized generic device.

**Figure 4.**Plan view of simulated steady state flow and subsequent wake velocity profiles for (

**a**) 1 m (

**b**) 5 m and (

**c**) 10 m mesh density discretization. Direction of flow is shown and the undisturbed velocity, U

_{0}is taken at five rotor diameters upstream. Downstream along channel measurement points, U

_{x}(red dots) are equidistantly spaced at one rotor diameter apart and transverse wake measurements (black dashed lines) are analyzed at 1, 2, 4, 6, 8 and 10 rotor diameters downstream.

**Figure 5.**Mean downstream axial centerline velocity deficit profile (solid line) ascertained from published empirical scaled physical tank test modelling studies. Associated standard deviation error bars provide a realistic range of values.

**Figure 6.**Configuration plan providing an indication of seabed area occupied and used for calculations when individual turbines (solid polygons) and array-averaged (dashed polygons) approaches are implemented.

**Figure 7.**An example of simulated channel flow for array-averaged and individual device resolving energy extraction methods using configuration layout E.

**Figure 8.**Simulated Telemac-2D along channel axial centerline velocity deficit profiles using the k-Ɛ turbulence closure scheme for various mesh densities. Solid line represents mean empirical test results, grey shading highlights ±1 s.d. (see Figure 5).

**Figure 9.**Simulated transverse wake velocity deficit profile: (

**a**) Analyzed at U

_{x}

_{(n)}for n = 1, 2, 4, 6, 8, 10; (

**b**) The mid- to far-field normalized transverse wake velocity deficit profiles for measurements at >4 rotor diameters (solid line indicates the ideal normalized Gaussian profile; Equation (22)).

**Figure 10.**(

**a**) Comparison of channel-width integrated flux differential when energy extraction between array-averaged and device-scale configurations (Figure 6) are analyzed; (

**b**) Number of TECs simulated, n

_{TECs}, versus peak percentage error in simulations for both single and multiple turbine rows.

**Figure 11.**(

**a**) Power curve of a single generic turbine simulated with zero, 10° and 20° rotor misalignment to flow; (

**b**) CF for inline (circles) and staggered (crosses) array configurations both with and without TEC offset to flow.

**Table 1.**Mesh resolution dependent Telemac geometrical approximation of seabed area compared with user defined area for a single TEC.

Grid Resolution (m) | Defined TEC Area (m^{2}) | Model Approximated Area (m^{2}) | Error (%) |
---|---|---|---|

10 | 20 | 133.3 | 566.7 |

5 | 20 | 83.3 | 316.7 |

1 | 20 | 20.7 | 3.4 |

Parameter | Value |
---|---|

Monopile diameter, ${\varnothing}_{M}$ (m) | 5.00 |

Hub height, H (m) | 22.50 |

Rotor diameter,${\varnothing}_{R}$ (m) | 20.00 |

Cut-in speed, V_{IN} (m s^{−1}) | 1.00 |

Cut-out speed, V_{OUT} (m s^{−1}) | 4.00 |

Rated speed, V_{RAT} (m s^{−1}) | 2.50 |

Cut-in efficiency, Eff_{IN} | 0.35 |

Rated efficiency, Eff_{RAT} | 0.40 |

Structural drag coefficient, C_{D} | 0.60 |

Upstream reference velocity distance, DD (m) | ${\varnothing}_{R}$ |

Turbulence Scheme | Mesh Resolution (m) | Coefficient | R^{2} | RMSE (%) |
---|---|---|---|---|

k-epsilon | 1 | 10^{−6} | 0.69 | 7.16 |

5 | 10^{−6} | 0.58 | 8.28 | |

10 | 10^{−6} | 0.61 | 8.02 |

Array Layout | CF Change (%) | ||
---|---|---|---|

n_{TECs} | Offset 10° | Offset 20° | |

1 | +0.04 | +1.00 | |

Inline–single row | 5 | +0.18 | +1.71 |

15 | +0.42 | +1.81 | |

Staggered–dual row | 5 | +0.02 | +0.01 |

15 | −0.11 | −0.45 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Piano, M.; Robins, P.E.; Davies, A.G.; Neill, S.P.
The Influence of Intra-Array Wake Dynamics on Depth-Averaged Kinetic Tidal Turbine Energy Extraction Simulations. *Energies* **2018**, *11*, 2852.
https://doi.org/10.3390/en11102852

**AMA Style**

Piano M, Robins PE, Davies AG, Neill SP.
The Influence of Intra-Array Wake Dynamics on Depth-Averaged Kinetic Tidal Turbine Energy Extraction Simulations. *Energies*. 2018; 11(10):2852.
https://doi.org/10.3390/en11102852

**Chicago/Turabian Style**

Piano, Marco, Peter E. Robins, Alan G. Davies, and Simon P. Neill.
2018. "The Influence of Intra-Array Wake Dynamics on Depth-Averaged Kinetic Tidal Turbine Energy Extraction Simulations" *Energies* 11, no. 10: 2852.
https://doi.org/10.3390/en11102852