# A Numerically Efficient Parametrization of Turbulent Wind-Turbine Flows for Different Thermal Stratifications

## Abstract

The wake characteristics of a wind turbine in a turbulent atmospheric boundary layer under different thermal stratifications are investigated by means of large-eddy simulation with the geophysical flow solver EULAG. The turbulent inflow is based on a method that imposes the spectral energy distribution of a neutral boundary-layer precursor simulation, the turbulence-preserving method. This method is extended herein to make it applicable for different thermal stratification regimes (convective, stable, neutral) by including suitable turbulence assumptions, which are deduced from velocity fields of a diurnal-cycle precursor simulation. The wind-turbine-wake characteristics derived from simulations that include the parametrization result in good agreement with diurnal-cycle-driven wind-turbine simulations. Furthermore, different levels of accuracy are tested in the parametrization assumptions, representing the thermal stratification. These range from three-dimensional matrices of the precursor-simulation wind field to individual values. The resulting wake characteristics are similar, even for the simplest parametrization set-up, making the diurnal-cycle precursor simulation non-essential for the wind-turbine simulations. Therefore, the proposed parametrization results in a computationally fast, simple, and efficient tool for analyzing the effects of different thermal stratifications on wind-turbine wakes by means of large-eddy simulation.

## Keywords

Atmospheric boundary layer Diurnal cycle Large-eddy simulation Turbulence Wind-turbine wake## Notes

### Acknowledgements

The authors thank Fernando Porté-Agel and Gert-Jan Steeneveld for the constructive discussion on the turbulence preserving method. Further, we thank Mark Zagar for providing the airfoil data of the 10-MW reference wind turbine from DTU. This work was performed within the project LIPS, funded by the Federal Ministry of Economy and Energy on the basis of a resolution of the German Bundestag under the contract number 0325518.

### Funding

The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de).

## References

- Abkar M, Porté-Agel F (2014) The effect of atmospheric stability on wind-turbine wakes: a large-eddy simulation study. J Phys Conf Ser 524(1):012,138. https://doi.org/10.1088/1742-6596/524/1/012138 CrossRefGoogle Scholar
- Abkar M, Sharifi A, Porté-Agel F (2016) Wake flow in a wind farm during a diurnal cycle. J Turbul 17(4):420–441. https://doi.org/10.1080/14685248.2015.1127379 CrossRefGoogle Scholar
- Aitken ML, Kosović B, Mirocha JD, Lundquist JK (2014) Large eddy simulation of wind turbine wake dynamics in the stable boundary layer using the Weather Research and Forecasting model. J Renew Sust Energy 6:1529–1539CrossRefGoogle Scholar
- Bhaganagar K, Debnath M (2014) Implications of stably stratified atmospheric boundary layer turbulence on the near-wake structure of wind turbines. Energies 7(9):5740–5763. https://doi.org/10.3390/en7095740 CrossRefGoogle Scholar
- Bhaganagar K, Debnath M (2015) The effects of mean atmospheric forcings of the stable atmospheric boundary layer on wind turbine wake. J Renew Sust Energy 7(1):013,124. https://doi.org/10.1063/1.4907687 CrossRefGoogle Scholar
- Chamorro LP, Porté-Agel F (2009) A wind-tunnel investigation of wind-turbine wakes: boundary-layer turbulence effects. Boundary-Layer Meteorol 132:129–149CrossRefGoogle Scholar
- Dörenkämper M, Witha B, Steinfeld G, Heinemann D, Kühn M (2015) The impact of stable atmospheric boundary layers on wind-turbine wakes within offshore wind farms. J Wind Eng Ind Aerodyn 144:146–153CrossRefGoogle Scholar
- Doyle JD, Gaberšek S, Jiang Q, Bernardet L, Brown JM, Dörnbrack A, Filaus E, Grubišic V, Kirshbaum DJ, Knoth O, Koch S, Schmidli J, Stiperski I, Vosper S, Zhong S (2011) An intercomparison of t-rex mountain-wave simulations and implications for mesoscale predictability. Mon Weather Rev 139:2811–2831CrossRefGoogle Scholar
- Englberger A, Dörnbrack A (2017a) Impact of neutral boundary-layer turbulence on wind-turbine wakes: a numerical modelling study. Boundary-Layer Meteorol 162(3):427–449CrossRefGoogle Scholar
- Englberger A, Dörnbrack A (2017b) Impact of the diurnal cycle of the atmospheric boundary layer on wind-turbine wakes: a numerical modelling study. Boundary-Layer Meteorol 166(3):423–448CrossRefGoogle Scholar
- Fröhlich J (2006) Large Eddy simulation turbulenter Strömungen. Teubner Verlag/GWV Fachverlage GmbH, Wiesbaden, p 414 (in German)Google Scholar
- Grimsdell AW, Angevine WM (2002) Observations of the afternoon transition of the convective boundary layer. J Appl Meteorol 41(1):3–11. https://doi.org/10.1175/1520-0450(2002)041%3c3C0003:OOTATO%3e3E2.0.CO;2 CrossRefGoogle Scholar
- Kelley CL, Ennis BL (2016) Swift site atmospheric characterization. Technical report, Sandia National Laboratories (SNL-NM), Albuquerque, NM, USAGoogle Scholar
- Kühnlein C, Smolarkiewicz PK, Dörnbrack A (2012) Modelling atmospheric flows with adaptive moving meshes. J Comput Phys 231:2741–2763CrossRefGoogle Scholar
- Lee JC, Lundquist JK (2017) Observing and simulating wind-turbine wakes during the evening transition. Boundary-Layer Meteorol 164(3):449–474CrossRefGoogle Scholar
- Mann J (1994) The spatial structure of neutral atmospheric surface-layer turbulence. J Fluid Mech 273:141–168CrossRefGoogle Scholar
- Margolin LG, Smolarkiewicz PK, Sorbjan Z (1999) Large-eddy simulations of convective boundary layers using nonoscillatory differencing. Phys D Nonlinear Phenom 133(1):390–397CrossRefGoogle Scholar
- Medici D, Alfredsson PH (2006) Measurements on a wind turbine wake: 3D effects and bluff body vortex shedding. Wind Energy 9:219–236CrossRefGoogle Scholar
- Mirocha J, Kirkil G, Bou-Zeid E, Chow FK, Kosović B (2013) Transition and equilibration of neutral atmospheric boundary layer flow in one-way nested large-eddy simulations using the weather research and forecasting model. Mon Weather Rev 141:918–940CrossRefGoogle Scholar
- Mirocha J, Kosovic B, Aitken M, Lundquist J (2014) Implementation of a generalized actuator disk wind turbine model into the weather research and forecasting model for large-eddy simulation applications. J Renew Sust Energy 6:013104CrossRefGoogle Scholar
- Muñoz-Esparza D, Kosović B, Mirocha J, van Beeck J (2014) Bridging the transition from mesoscale to microscale turbulence in numerical weather prediction models. Boundary-Layer Meteorol 153:409–440CrossRefGoogle Scholar
- Naughton JW, Heinz S, Balas M, Kelly R, Gopalan H, Lindberg W, Gundling C, Rai R, Sitaraman J, Singh M (2011) Turbulence and the isolated wind turbine. In: 6th AIAA theoretical fluid mechanics conference, 27–30 June 2011, Honolulu, Hawaii, pp 1–19Google Scholar
- Prusa JM, Smolarkiewicz PK, Wyszogrodzki AA (2008) EULAG, a computational model for multiscale flows. Comput Fluids 37:1193–1207CrossRefGoogle Scholar
- Sathe A, Mann J, Barlas T, Bierbooms W, Bussel G (2013) Influence of atmospheric stability on wind turbine loads. Wind Energy 16(7):1013–1032. https://doi.org/10.1002/we.1528 CrossRefGoogle Scholar
- Schmidt H, Schumann U (1989) Coherent structure of the convective boundary layer derived from large-eddy simulations. J Fluid Mech 200:511–562. https://doi.org/10.1017/S0022112089000753 CrossRefGoogle Scholar
- Shapiro A, Fedorovich E (2010) Analytical description of a nocturnal low-level jet. Q J R Meteorol Soc 136(650):1255–1262Google Scholar
- Smolarkiewicz PK, Charbonneau P (2013) EULAG, a computational model for multiscale flows: an MHD extension. J Comput Phys 236:608–623CrossRefGoogle Scholar
- Smolarkiewicz PK, Dörnbrack A (2008) Conservative integrals of adiabatic durran’s equations. Int J Numer Method Fluid 56:1513–1519CrossRefGoogle Scholar
- Smolarkiewicz PK, Margolin LG (1993) On forward-in-time differencing for fluids: extension to a curviliniear framework. Mon Weather Rev 121:1847–1859CrossRefGoogle Scholar
- Smolarkiewicz PK, Margolin LG (1998) MPDATA: a finite-difference solver for geophysical flows. J Comput Phys 140:459–480CrossRefGoogle Scholar
- Smolarkiewicz PK, Prusa JM (2002) Forward-in-time differencing for fluids: simulation of geophysical turbulence. In: Turbulent flow computation, Kluwer, Boston, pp 279–312Google Scholar
- Smolarkiewicz PK, Prusa JM (2005) Towards mesh adaptivity for geophysical turbulence: continuous mapping approach. Int J Numer Method Fluid 47:789–801CrossRefGoogle Scholar
- Smolarkiewicz PK, Pudykiewicz JA (1992) A class of semi-Lagrangian approximations for fluids. J Atmos Sci 49:2082–2096CrossRefGoogle Scholar
- Smolarkiewicz PK, Sharman R, Weil J, Perry SG, Heist D, Bowker G (2007) Building resolving large-eddy simulations and comparison with wind tunnel experiments. J Comput Phys 227:633–653CrossRefGoogle Scholar
- Stull RB (1988) An introduction to boundary layer meteorology. Kluwer, DordrechtCrossRefGoogle Scholar
- Troldborg N, Sørensen JN, Mikkelsen R (2007) Actuator line simulation of wake of wind turbine operating in turbulent inflow. J Phys Conf Ser 75:012,063CrossRefGoogle Scholar
- Vanderwende B, Lundquist JK (2012) The modification of wind turbine performance by statistically distinct atmospheric regimes. Environ Res Lett 7(3):034,035CrossRefGoogle Scholar
- Wedi NP, Smolarkiewicz PK (2004) Extending Gal–Chen and Somerville terrain-following coordinate transformation on time-dependent curvilinear boundaries. J Comput Phys 193:1–20CrossRefGoogle Scholar
- Wedi NP, Smolarkiewicz PK (2006) Direct numerical simulation of the plumb-McEwan laboratory analog of the QBO. J Atmos Sci 63:3226–3252CrossRefGoogle Scholar
- Wharton S, Lundquist JK (2012) Atmospheric stability affects wind turbine power collection. Environ Res Lett 7(1):014,005CrossRefGoogle Scholar
- Witha B, Steinfeld G, Heinemann D (2014) Wind energy—impact of turbulence. In: Hölling M, Peinke J, Ivanell S (eds) High-resolution offshore wake simulations with the LES model PALM. Springer, Oldenburg, pp 175–181Google Scholar
- Wu YT, Porté-Agel F (2012) Atmospheric turbulence effects on wind-turbine wakes: an LES study. Energies 5:5340–5362CrossRefGoogle Scholar
- Zhang W, Markfort CD, Porté-Agel F (2012) Near-wake flow structure downwind of a wind turbine in a turbulent boundary layer. Exp Fluids 52:1219–1235CrossRefGoogle Scholar