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Effects of tip speed ratio on wind turbine wake meandering

Published online by Cambridge University Press:  25 June 2025

Yan Wang Open the ORCID record for Yan Wang [Opens in a new window] ,
Guihua Zhao ,
Guoliang Liu ,
Yongze Zhou ,
Mingwei Ge Open the ORCID record for Mingwei Ge [Opens in a new window] ,
Zhen Ouyang ,
Zijing Ding Open the ORCID record for Zijing Ding [Opens in a new window]  and
Ruifeng Hu Open the ORCID record for Ruifeng Hu [Opens in a new window]
Yan Wang
Affiliation:
School of Energy and Power Engineering, Lanzhou University of Technology, Lanzhou 730050, PR China
Guihua Zhao
Affiliation:
School of Energy and Power Engineering, Lanzhou University of Technology, Lanzhou 730050, PR China
Guoliang Liu
Affiliation:
School of Energy and Power Engineering, Lanzhou University of Technology, Lanzhou 730050, PR China
Yongze Zhou
Affiliation:
Center for Particle-Laden Turbulence, Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of Education and College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, PR China
Mingwei Ge
Affiliation:
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, PR China
Zhen Ouyang
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
Zijing Ding
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
Ruifeng Hu*
Affiliation:
Center for Particle-Laden Turbulence, Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of Education and College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, PR China
*
Corresponding author: Ruifeng Hu, hurf@lzu.edu.cn
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Abstract

This study utilises large-eddy simulation with the actuator line model to examine the effects of the tip speed ratio (TSR) on the wake-meandering characteristics of a wind turbine in uniform and turbulent inflows. It is shown that as the TSR grows, the onset position of the wake meandering moves closer to the rotor, and the magnitude of wake oscillation is stronger. This aligns with previous work showing that a higher TSR can accelerate the instability and breakdown of tip vortices. Without a nacelle, the Strouhal number of the wake meandering is found to be independent of the TSR under both the uniform and turbulent inflows. However, with a relatively large nacelle, the Strouhal number first increases and then decreases with TSR. Therefore, the current discovery elucidates the crucial role of the nacelle and clarifies the origin of the TSR dependence of the Strouhal number in wake meandering. In addition, the characteristic frequency of the wake meandering under the turbulent inflow is much smaller than that under the uniform inflow, because of the significant influence of the freestream turbulence. Furthermore, the proper orthogonal decomposition (POD) and spectral POD (SPOD) methods are employed to study the spatiotemporal characteristics of the meandering wake and its TSR dependence. It is found that the tip and root vortices are the prominent wake structures under the uniform inflow, whereas more complex multiscale structures from the interaction between the freestream turbulence and tip/root vortices exist under the turbulent inflow. Moreover, an amplitude modulation phenomenon of the POD time coefficients at the optimal TSR is observed in the uniform inflow case. Finally, a reduced-order model is constructed for predicting the wake dynamics by combining the SPOD and the ‘sparse identification of nonlinear dynamics’ algorithm with high accuracy and interpretability.

JFM classification

Wakes/Jets: Wakes Turbulent Flows: Turbulence simulation Nonlinear Dynamical Systems: Low-dimensional models

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JFM Papers
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Journal of Fluid Mechanics , Volume 1014 , 10 July 2025 , A3
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© The Author(s), 2025. Published by Cambridge University Press

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References

Abraham, A., Dasari, T. & Hong, J. 2019 Effect of turbine nacelle and tower on the near wake of a utility-scale wind turbine. J. Wind Engng Ind. Aerodyn. 193, 103981.CrossRefGoogle Scholar
Aitken, M.L., Čosić, B., Mirocha, J.D. & Lundquist, J.K. 2014 Large eddy simulation of wind turbine wake dynamics in the stable boundary layer using the weather research and forecasting model. J. Renew. Sustain. Energy 6 (3), 033137.CrossRefGoogle Scholar
Bastankhah, M. & Porté-Agel, F. 2015 A wind-tunnel investigation of wind-turbine wakes in yawed conditions. J. Phys.: Conf. Ser. 625, 012014.Google Scholar
Bayron, P., Kelso, R. & Chin, R. 2024 Experimental investigation of tip-speed-ratio influence on horizontal-axis wind turbine wake dynamics. Renew. Energy 225, 120201.10.1016/j.renene.2024.120201CrossRefGoogle Scholar
Bhagwat, M.J. & Leishman, J.G. 2000 Stability analysis of helicopter rotor wakes in axial flight. J. Am. Helicopter Soc. 45 (3), 165–178.10.4050/JAHS.45.165CrossRefGoogle Scholar
Biswas, N. & Buxton, O.R.H. 2024 Effect of tip speed ratio on coherent dynamics in the near wake of a model wind turbine. J. Fluid Mech. 979, A34.10.1017/jfm.2023.1095CrossRefGoogle Scholar
Bongard, J. & Lipson, H. 2007 Automated reverse engineering of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 104 (24), 9943–9948.10.1073/pnas.0609476104CrossRefGoogle ScholarPubMed
Bou-Zeid, E., Meneveau, C. & Parlange, M. 2005 A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows. Phys. Fluids 17 (2), 025105.10.1063/1.1839152CrossRefGoogle Scholar
Braunbehrens, R. & Segalini, A. 2019 A statistical model for wake meandering behind wind turbines. J. Wind Engng Ind. Aerodyn. 193, 103954.10.1016/j.jweia.2019.103954CrossRefGoogle Scholar
Brunton, S.L., Proctor, J.L. & Kutz, J.N. 2016 Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 113 (15), 3932–3937.CrossRefGoogle ScholarPubMed
Calaf, M., Meneveau, C. & Meyers, J. 2010 Large eddy simulation study of fully developed wind-turbine array boundary layers. Phys. Fluids 22 (1), 015110.10.1063/1.3291077CrossRefGoogle Scholar
Chamorro, L.P., Hill, C., Morton, S., Ellis, C., Arndt, R.E.A. & Sotiropoulos, F. 2013 On the interaction between a turbulent open channel flow and an axial-flow turbine. J. Fluid Mech. 716, 658–670.10.1017/jfm.2012.571CrossRefGoogle Scholar
Chamorro, L.P. & Porté-Agel, F. 2010 Effects of thermal stability and incoming boundary-layer flow characteristics on wind-turbine wakes: a wind-tunnel study. Boundary-Layer Meteorol. 136 (3), 515–533.CrossRefGoogle Scholar
De Cillis, G., Cherubini, S., Semeraro, O., Leonardi, S. & De Palma, P. 2021 POD-based analysis of a wind turbine wake under the influence of tower and nacelle. Wind Energy 24 (6), 609–633.10.1002/we.2592CrossRefGoogle Scholar
De Cillis, G., Cherubini, S., Semeraro, O., Leonardi, S. & De Palma, P. 2022 a The influence of incoming turbulence on the dynamic modes of an NREL-5MW wind turbine wake. Renew. Energy 183, 601–616.10.1016/j.renene.2021.11.037CrossRefGoogle Scholar
De Cillis, G., Semeraro, O., Leonardi, S., De Palma, P. & Cherubini, S. 2022 b Dynamic-mode-decomposition of the wake of the NREL-5MW wind turbine impinged by a laminar inflow. Renew. Energy 199, 1–10.10.1016/j.renene.2022.08.113CrossRefGoogle Scholar
Debnath, M., Santoni, C., Leonardi, S. & Iungo, G.V. 2017 Towards reduced order modelling for predicting the dynamics of coherent vorticity structures within wind turbine wakes. Phil. Trans. R. Soc. A: Math. Phys. Engng Sci. 375 (2091), 20160108.CrossRefGoogle ScholarPubMed
Espana, G., Aubrun, S., Loyer, S. & Devinant, P. 2012 Wind tunnel study of the wake meandering downstream of a modelled wind turbine as an effect of large scale turbulent eddies. J. Wind Engng Ind. Aerodyn. 101, 24–33.10.1016/j.jweia.2011.10.011CrossRefGoogle Scholar
Felli, M., Camussi, R. & Felice, F.Di 2011 Mechanisms of evolution of the propeller wake in the transition and far fields. J. Fluid Mech. 682, 5–53.10.1017/jfm.2011.150CrossRefGoogle Scholar
Feng, D., Gupta, V., Li, L.K.B. & Wan, M. 2024 An improved dynamic model for wind-turbine wake flow. Energy 290, 130167.CrossRefGoogle Scholar
Foti, D., Yang, X., Campagnolo, F., Maniaci, D. & Sotiropoulos, F. 2018 a Wake meandering of a model wind turbine operating in two different regimes. Phys. Rev. Fluids 3 (5), 054607.10.1103/PhysRevFluids.3.054607CrossRefGoogle Scholar
Foti, D., Yang, X. & Sotiropoulos, F. 2018 b Similarity of wake meandering for different wind turbine designs for different scales. J. Fluid Mech. 842, 5–25.10.1017/jfm.2018.9CrossRefGoogle Scholar
Foti, D., Yang, X., Guala, M. & Sotiropoulos, F. 2016 Wake meandering statistics of a model wind turbine: insights gained by large eddy simulations. Phys. Rev. Fluids 1 (4), 044407.10.1103/PhysRevFluids.1.044407CrossRefGoogle Scholar
Gupta, B.P. & Loewy, R.G. 1974 Theoretical analysis of the aerodynamic stability of multiple, interdigitated helical vortices. AIAA J. 12 (10), 1381–1387.10.2514/3.49493CrossRefGoogle Scholar
Gupta, V. & Wan, M. 2019 Low-order modelling of wake meandering behind turbines. J. Fluid Mech. 877, 534–560.10.1017/jfm.2019.619CrossRefGoogle Scholar
Hamilton, N., Tutkun, M. & Cal, R.B. 2017 Low-order dynamical system model of a fully developed turbulent channel flow. Phys. Fluids 29 (6), 065107.CrossRefGoogle Scholar
Hamilton, N., Viggiano, B., Calaf, M., Tutkun, M. & Cal, R.B. 2018 A generalized framework for reduced-order modeling of a wind turbine wake. Wind Energy 21 (6), 373–390.10.1002/we.2167CrossRefGoogle Scholar
Heisel, M., Hong, J. & Guala, M. 2018 The spectral signature of wind turbine wake meandering: a wind tunnel and field‐scale study. Wind Energy 21 (9), 715–731.10.1002/we.2189CrossRefGoogle Scholar
Howard, K.B., Singh, A., Sotiropoulos, F. & Guala, M. 2015 On the statistics of wind turbine wake meandering: an experimental investigation. Phys. Fluids 27 (7), 075103.10.1063/1.4923334CrossRefGoogle Scholar
Howland, M.F. & Dabiri, J.O. 2019 Wind farm modeling with interpretable physics-informed machine learning. Energies 12 (14), 2716.CrossRefGoogle Scholar
Hu, H., Yang, Z. & Sarkar, P. 2012 Dynamic wind loads and wake characteristics of a wind turbine model in an atmospheric boundary layer wind. Exp. Fluids 52 (5), 1277–1294.CrossRefGoogle Scholar
Hunt, J.C.R., Wray, A.A. & Moin, P. 1988 Eddies, streams, and convergence zones in turbulent flows. In Proceedings of the 1988 Summer Program. Center for Turbulence Research, Stanford University.Google Scholar
Iungo, G.V., Santoni-Ortiz, C., Abkar, M., Porté-Agel, F., Rotea, M.A. & Leonardi, S. 2015 Data-driven reduced order model for prediction of wind turbine wakes. J. Phys.: Conf. Ser. 625, 012009.Google Scholar
Iungo, G.V., Viola, F., Camarri, S., Porte-Agel, F. & Gallaire, F. 2013 Linear stability analysis of wind turbine wakes performed on wind tunnel measurements. J. Fluid Mech. 737, 499–526.10.1017/jfm.2013.569CrossRefGoogle Scholar
Ivanell, S., Mikkelsen, R., Sørensen, J. & Henningson, D. 2008 Stability analysis of the tip vortices of a wind turbine. Wind Energy 13 (8), 705–715.10.1002/we.391CrossRefGoogle Scholar
Jonkman, J. & Shaler, K. 2021 FAST.Farm User’s Guide and Theory Manual. National Renewable Energy Laboratory, USA.Google Scholar
Kang, S., Yang, X. & Sotiropoulos, F. 2014 On the onset of wake meandering for an axial flow turbine in a turbulent open channel flow. J. Fluid Mech. 744, 376–403.10.1017/jfm.2014.82CrossRefGoogle Scholar
Kleusberg, E., Schlatter, P. & Henningson, D.S. 2020 Parametric dependencies of the yawed wind-turbine wake development. Wind Energy 23 (6), 1367–1380.10.1002/we.2395CrossRefGoogle Scholar
Krogstad, P. & Adaramola, M.S. 2012 Performance and near wake measurements of a model horizontal axis wind turbine. Wind Energy 15 (5), 743–756.CrossRefGoogle Scholar
Kumar, P. & Mahesh, K. 2017 Large eddy simulation of propeller wake instabilities. J. Fluid Mech. 814, 361–396.CrossRefGoogle Scholar
Larsen, G.C., et al. 2007 Dynamic wake meandering modeling. Tech. Rep. Risø-R-1607. Risø National Laboratory, Technical University of Denmark.Google Scholar
Larsen, G.C., Madsen, H.A., Thomsen, K. & Larsen, T.J. 2008 Wake meandering: a pragmatic approach. Wind Energy 11 (4), 377–395.10.1002/we.267CrossRefGoogle Scholar
Li, B., Ge, M., Li, X. & Liu, Y. 2024 A physics-guided machine learning framework for real-time dynamic wake prediction of wind turbines. Phys. Fluids 36 (3), 035143.10.1063/5.0194764CrossRefGoogle Scholar
Li, L., Zhi, H., Ge, M. & Zhang, Q. 2021 A novel three-dimensional analytical model of the added streamwise turbulence intensity for wind-turbine wakes. Energy 238, 121806.10.1016/j.energy.2021.121806CrossRefGoogle Scholar
Li, R., Zhang, J. & Zhao, X. 2022 a Dynamic wind farm wake modeling based on a Bilateral Convolutional Neural Network and high-fidelity LES data. Energy 258, 124845.CrossRefGoogle Scholar
Li, Z., Dong, G. & Yang, X. 2022 b Onset of wake meandering for a floating offshore wind turbine under side-to-side motion. J. Fluid Mech. 934, A29.10.1017/jfm.2021.1147CrossRefGoogle Scholar
Li, Z., Liu, X. & Yang, X. 2022 c Review of turbine parameterization models for large-eddy simulation of wind turbine wakes. Energies 15 (18), 6533.10.3390/en15186533CrossRefGoogle Scholar
Lignarolo, L.E.M., Ragni, D., Krishnaswami, C., Chen, Q., Simão Ferreira, C.J. & van Bussel, G.J.W. 2014 Experimental analysis of the wake of a horizontal-axis wind-turbine model. Renew. Energy 70, 31–46.10.1016/j.renene.2014.01.020CrossRefGoogle Scholar
Lignarolo, L.E.M., Ragni, D., Scarano, F., Simão Ferreira, C.J. & van Bussel, G.J.W. 2015 Tip-vortex instability and turbulent mixing in wind-turbine wakes. J. Fluid Mech. 781, 467–493.10.1017/jfm.2015.470CrossRefGoogle Scholar
Liu, L. & Stevens, R.J.A.M. 2021 Effects of atmospheric stability on the performance of a wind turbine located behind a three-dimensional hill. Renew. Energy 175, 926–935.10.1016/j.renene.2021.05.035CrossRefGoogle Scholar
Loiseau, J.C. & Brunton, S.L. 2016 Constrained sparse galerkin regression. J. Fluid Mech. 838, 42–67.10.1017/jfm.2017.823CrossRefGoogle Scholar
Lumley, J.L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation: Proceedings of the International Colloquium (ed. A.M. Yaglom & V.I. Tatarsky), 15–22 June 1965. Moscow: Nauka, pp.166–178.Google Scholar
Mao, X. & Sørensen, J.N. 2018 Far-wake meandering induced by atmospheric eddies in flow past a wind turbine. J. Fluid Mech. 846, 190–209.CrossRefGoogle Scholar
Martínez-Tossas, L.A. & Meneveau, C. 2019 Filtered lifting line theory and application to the actuator line model. J. Fluid Mech. 863, 269–292.10.1017/jfm.2018.994CrossRefGoogle Scholar
Martínez-Tossas, L., Churchfield, M. & Leonardi, S. 2014 Large eddy simulations of the flow past wind turbines: actuator line and disk modeling. Wind Energy 18 (6), 1047–1060.10.1002/we.1747CrossRefGoogle Scholar
Marusic, I., Mathis, R. & Hutchins, N. 2010 Predictive model for wall-bounded turbulent flow. Science 329 (5988), 193–196.10.1126/science.1188765CrossRefGoogle ScholarPubMed
McTavish, S., Feszty, D. & Nitzsche, F. 2013 Evaluating reynolds number effects in small-scale wind turbine experiments. J. Wind Engng Ind. Aerodyn. 120, 81–90.10.1016/j.jweia.2013.07.006CrossRefGoogle Scholar
Medici, D. & Alfredsson, P.H. 2006 Measurements on a wind turbine wake: 3D effects and bluff body vortex shedding. Wind Energy 9 (3), 219–236.CrossRefGoogle Scholar
Medici, D. & Alfredsson, P.H. 2008 Measurements behind model wind turbines: further evidence of wake meandering. Wind Energy 11 (2), 211–217.10.1002/we.247CrossRefGoogle Scholar
Narasimhan, G., Gayme, D.F. & Meneveau, C. 2022 Effects of wind veer on a yawed wind turbine wake in atmospheric boundary layer flow. Phys. Rev. Fluids 7 (11), 114609.10.1103/PhysRevFluids.7.114609CrossRefGoogle Scholar
Nekkanti, A. & Schmidt, O.T. 2021 Frequency-time analysis, low-rank reconstruction and denoising of turbulent flows using SPOD. J. Fluid Mech. 926, A26.10.1017/jfm.2021.681CrossRefGoogle Scholar
Neunaber, I., Hölling, M. & Obligado, M. 2022 Wind tunnel study on the tip speed ratio’s impact on a wind turbine wake development. Energies 15 (22), 8607.10.3390/en15228607CrossRefGoogle Scholar
Okulov, V.L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319–342.10.1017/S0022112004001934CrossRefGoogle Scholar
Okulov, V.L., Naumov, I.V., Mikkelsen, R.F., Kabardin, I.K. & Sorensen, J.N. 2014 A regular Strouhal number for large-scale instability in the far wake of a rotor. J. Fluid Mech. 747, 369–380.10.1017/jfm.2014.174CrossRefGoogle Scholar
Okulov, V.L. & Sørensen, J.N. 2007 Stability of helical tip vortices in a rotor far wake. J. Fluid Mech. 576, 1–25.10.1017/S0022112006004228CrossRefGoogle Scholar
Paskyabi, M.B., Krutova, M., Nielsen, F.G., Reuder, J. & Guernaoui, O.E. 2020 On stochastic reduced-order and LES-based models of offshore wind turbine wakes. J. Phys. Conf.: Ser. 1669 (1), 012018.10.1088/1742-6596/1669/1/012018CrossRefGoogle Scholar
Picard, C. & Delville, J. 2000 Pressure velocity coupling in a subsonic round jet. Intl J. Heat Fluid Flow 21 (3), 359–364.10.1016/S0142-727X(00)00021-7CrossRefGoogle Scholar
Podvin, B. 2009 A proper-orthogonal-decomposition–based model for the wall layer of a turbulent channel flow. Phys. Fluids 21 (1), 015111.10.1063/1.3068759CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Porté-Agel, F., Bastankhah, M. & Shamsoddin, S. 2020 Wind-turbine and wind-farm flows: a review. Boundary-Layer Meteorol. 174 (1), 1–59.10.1007/s10546-019-00473-0CrossRefGoogle ScholarPubMed
Porté-Agel, F., Meneveau, C. & Parlange, M.B. 2000 A scale-dependent dynamic model for large-eddy simulation: application to a neutral atmospheric boundary layer. J. Fluid Mech. 415, 261–284.CrossRefGoogle Scholar
Posa, A., Broglia, R. & Balaras, E. 2021 Instability of the tip vortices shed by an axial-flow turbine in uniform flow. J. Fluid Mech. 920, A19.10.1017/jfm.2021.433CrossRefGoogle Scholar
Quaranta, H.U., Brynjell-Rahkola, M., Leweke, T. & Henningson, D.S. 2019 Local and global pairing instabilities of two interlaced helical vortices. J. Fluid Mech. 863, 927–955.10.1017/jfm.2018.904CrossRefGoogle Scholar
Rowley, C.W. & Dawson, S.T.M. 2017 Model reduction for flow analysis and control. Annu. Rev. Fluid Mech. 49 (1), 387–417.10.1146/annurev-fluid-010816-060042CrossRefGoogle Scholar
Santoni, C., Carrasquillo, K., Arenas-Navarro, I. & Leonardi, S. 2017 Effect of tower and nacelle on the flow past a wind turbine. Wind Energy 20 (12), 1927–1939.10.1002/we.2130CrossRefGoogle Scholar
Santoni, C., Zhang, D., Zhang, Z., Samaras, D., Sotiropoulos, F. & Khosronejad, A. 2024 Toward ultra-efficient high-fidelity predictions of wind turbine wakes: augmenting the accuracy of engineering models with machine learning. Phys. Fluids 36 (6), 065159.10.1063/5.0213321CrossRefGoogle Scholar
Sarmast, S., Dadfar, R., Mikkelsen, R.F., Schlatter, P., Ivanell, S., Sørensen, J.N. & Henningson, D.S. 2014 Mutual inductance instability of the tip vortices behind a wind turbine. J. Fluid Mech. 755, 705–731.CrossRefGoogle Scholar
Schmidt, M.D. & Lipson, H. 2009 Distilling free-form natural laws from experimental data. Science 324 (5923), 81–85.10.1126/science.1165893CrossRefGoogle ScholarPubMed
Schmidt, O.T., Towne, A., Rigas, G., Colonius, T. & Bres, G.A. 2018 Spectral analysis of jet turbulence. J. Fluid Mech. 855, 953–982.10.1017/jfm.2018.675CrossRefGoogle Scholar
Schümann, H., Pierella, F. & Sætran, L. 2013 Experimental investigation of wind turbine wakes in the wind tunnel. Energy. Proc. 35, 285–296.10.1016/j.egypro.2013.07.181CrossRefGoogle Scholar
Shapiro, C.R., Bauweraerts, P., Meyers, J., Meneveau, C. & Gayme, D.F. 2017 Model-based receding horizon control of wind farms for secondary frequency regulation. Wind Energy 20 (7), 1261–1275.10.1002/we.2093CrossRefGoogle Scholar
Shapiro, C.R., Starke, G.M., Meneveau, C. & Gayme, D.F. 2019 A wake modeling paradigm for wind farm design and control. Energies 12 (15), 2956.10.3390/en12152956CrossRefGoogle Scholar
Sherry, M., Nemes, A., Lo Jacono, D., Blackburn, H.M. & Sheridan, J. 2013 The interaction of helical tip and root vortices in a wind turbine wake. Phys. Fluids 25 (11), 117102.10.1063/1.4824734CrossRefGoogle Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. I. Coherent structures. Q. Appl. Maths 45 (3), 561–571.10.1090/qam/910462CrossRefGoogle Scholar
Stein, V.P. & Kaltenbach, H.-J. 2016 Wind-tunnel modelling of the tip-speed ratio influence on the wake evolution. J. Phys.: Conf. Ser. 753, 032061.Google Scholar
Stevens, R.J.A.M., Martinez-Tossas, L.A. & Meneveau, C. 2018 Comparison of wind farm large eddy simulations using actuator disk and actuator line models with wind tunnel experiments. Renew. Energy 116, 470–478.10.1016/j.renene.2017.08.072CrossRefGoogle Scholar
Stevens, R.J.A.M. & Meneveau, C. 2017 Flow structure and turbulence in wind farms. Ann. Rev. Fluid Mech. 49 (1), 311–339.10.1146/annurev-fluid-010816-060206CrossRefGoogle Scholar
Stevens, R.J.A.M.S., Graham, J. & Meneveau, C. 2014 A concurrent precursor inflow method for large eddy simulations and applications to finite length wind farms. Renew. Energy 68, 46–50.10.1016/j.renene.2014.01.024CrossRefGoogle Scholar
Sørensen, J. & Shen, W.Z. 2002 Numerical modeling of wind turbine wakes. J. Fluids Engng 124 (2), 393–399.10.1115/1.1471361CrossRefGoogle Scholar
Sørensen, J.N. 2011 Instability of helical tip vortices in rotor wakes. J. Fluid Mech. 682, 1–4.10.1017/jfm.2011.277CrossRefGoogle Scholar
Taira, K., Brunton, S.L., Dawson, S.T.M., Rowley, C.W., Colonius, T., McKeon, B.J., Schmidt, O.T., Gordeyev, S., Theofilis, V. & Ukeiley, L.S. 2017 Modal analysis of fluid flows: an overview. AIAA J. 55 (12), 4013–4041.10.2514/1.J056060CrossRefGoogle Scholar
Taylor, G.J., Milborrow, D.J., Mcintosh, D.N. & Swift-Hook, D.T. 1985 Wake measurements on the nibe windmills. In Proceedings of Seventh BWEA Wind Energy Conference, Oxford, UK, British Wind Energy Association, pp.67–73.Google Scholar
Thøgersen, E., Tranberg, B., Herp, J. & Greiner, M. 2017 Statistical meandering wake model and its application to yaw-angle optimisation of wind farms. J. Phys.: Conf. Ser. 854 (1), 012017.Google Scholar
Ti, Z., Deng, X.W. & Yang, H. 2020 Wake modeling of wind turbines using machine learning. Appl. Energy 257, 114025.10.1016/j.apenergy.2019.114025CrossRefGoogle Scholar
Towne, A., Schmidt, O.T. & Colonius, T. 2018 Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J. Fluid Mech. 847, 821–867.10.1017/jfm.2018.283CrossRefGoogle Scholar
Troldborg, N., Sorensen, J.N. & Mikkelsen, R. 2007 Actuator line simulation of wake of wind turbine operating in turbulent inflow. J. Phys.: Conf. Ser. 75, 012063.Google Scholar
Troldborg, N., Sorensen, J.N. & Mikkelsen, R. 2010 Numerical simulations of wake characteristics of a wind turbine in uniform inflow. Wind Energy 13 (1), 86–99.10.1002/we.345CrossRefGoogle Scholar
Wang, Y., Wang, L., Hu, R., Kong, L. & Cheng, J. 2021 Effects of sampling frequency on the proper orthogonal decomposition based reconstruction of a wind turbine wake. IET Renew. Power Gen. 15 (13), 2956–2970.10.1049/rpg2.12228CrossRefGoogle Scholar
Widnall, S.E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54 (4), 641–663.10.1017/S0022112072000928CrossRefGoogle Scholar
Wu, Y.-T., Lin, C.-Y. & Hsu, C.-M. 2020 An experimental investigation of wake characteristics and power generation efficiency of a small wind turbine under different tip speed ratios. Energies 13 (8), 2113.10.3390/en13082113CrossRefGoogle Scholar
Wu, Y.-T. & Porté-Agel, F. 2011 Large-eddy simulation of wind turbine wakes: evaluation of turbine parameterizations. Boundary-Layer Meteorol. 138 (3), 345–366.10.1007/s10546-010-9569-xCrossRefGoogle Scholar
Yang, X. 2020 Towards the development of a wake meandering model based on neural networks. J. Phys.: Conf. Ser. 1618 (6), 062026.Google Scholar
Yang, X., Hong, J., Barone, M. & Sotiropoulos, F. 2015 Coherent dynamics in the rotor tip shear layer of utility-scale wind turbines. J. Fluid Mech. 804, 90–115.10.1017/jfm.2016.503CrossRefGoogle Scholar
Yang, X. & Sotiropoulos, F. 2019 a A review on the meandering of wind turbine wakes. Energies 12 (24), 4725.10.3390/en12244725CrossRefGoogle Scholar
Yang, X. & Sotiropoulos, F. 2019 b Wake characteristics of a utility-scale wind turbine under coherent inflow structures and different operating conditions. Phys. Rev. Fluids 4 (2), 024604.10.1103/PhysRevFluids.4.024604CrossRefGoogle Scholar
Yang, X.I.A., Chen, P.E.S., Hu, R. & Abkar, M. 2022 Logarithmic-linear law of the streamwise velocity variance in stably stratified boundary layers. Boundary-Layer Meteorol. 183 (2), 199–213.CrossRefGoogle Scholar
Zhang, J. & Zhao, X. 2020 A novel dynamic wind farm wake model based on deep learning. Appl. Energy 277, 115552.10.1016/j.apenergy.2020.115552CrossRefGoogle Scholar
Zhang, W., Markfort, C.D. & Porté-Agel, F. 2012 Near-wake flow structure downwind of a wind turbine in a turbulent boundary layer. Exp. Fluids 52 (5), 1219–1235.10.1007/s00348-011-1250-8CrossRefGoogle Scholar