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A momentum-conserving wake superposition method for wind-farm flows under pressure gradient

Published online by Cambridge University Press:  12 November 2024

Bowen Du Open the ORCID record for Bowen Du [Opens in a new window] ,
Mingwei Ge Open the ORCID record for Mingwei Ge [Opens in a new window] ,
Xintao Li  and
Yongqian Liu
Bowen Du
Affiliation:
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, 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
Xintao Li
Affiliation:
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, PR China
Yongqian Liu
Affiliation:
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, PR China
*
†Email address for correspondence: gemingwei@ncepu.edu.cn
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Abstract

Pressure gradient over topography will significantly affect wind-farm flow. However, knowledge gaps still exist on how to superpose wind-turbine wakes in the wind-farm flow analytical model to account for this effect, leading to systematic errors in evaluating wind-farm wake effects. To this end, we derive an implicit momentum-conserving wake superposition method under pressure gradient (PG-IMCM) based on the total momentum deficit equation, which is linearised by the convection velocity introduced by Zong & Porté-Agel (J. Fluid Mech., vol. 889, 2020, A8). The PG-IMCM method consists of the linear-weighted sum of individual velocity deficits, the sum of the individual pressure correction terms and the total pressure correction term. Based on a sensitivity analysis, we demonstrate that the last two terms nearly cancel out and, thus, can be neglected, resulting in a simplified form, which has the same form as its counterpart under zero pressure gradient but with the single-wake quantities redefined based on the wake model under pressure gradient. This motivates us to further examine the performance of the combination of five empirical superposition methods and the stand-alone wake model under pressure gradient. Validation results based on large-eddy simulation show that PG-IMCM has an overall satisfactory performance in both the magnitude and shape of the velocity-deficit profiles, provided that the stand-alone turbine wake can be modelled accurately, which is virtually identical with its simplified form. Further comparison with empirical superposition methods shows that local linear and wind product superposition methods based on the updated base flow also have comparable performance, with only discernable differences with the PG-IMCM method in the near-wake region of downstream turbines. Therefore, they are two attractive methods for engineering applications.

JFM classification

Wakes/Jets: Wakes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 999 , 25 November 2024 , A27
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Allaerts, D. & Meyers, J. 2019 Sensitivity and feedback of wind-farm-induced gravity waves. J. Fluid Mech. 862, 990–1028.CrossRefGoogle ScholarPubMed
Archer, C.L., Vasel-Be-Hagh, A., Yan, C., Wu, S., Pan, Y., Brodie, J.F. & Maguire, A.E. 2018 Review and evaluation of wake loss models for wind energy applications. Appl. Energy 226, 1187–1207.CrossRefGoogle Scholar
Bastankhah, M. & Porté-Agel, F. 2014 A new analytical model for wind-turbine wakes. Renew. Energy 70, 116–123.CrossRefGoogle Scholar
Bastankhah, M. & Porté-Agel, F. 2016 Experimental and theoretical study of wind turbine wakes in yawed conditions. J. Fluid Mech. 806, 506–541.CrossRefGoogle Scholar
Bastankhah, M., Welch, B.L., Martínez-Tossas, L.A., King, J. & Fleming, P. 2021 Analytical solution for the cumulative wake of wind turbines in wind farms. J. Fluid Mech. 911, A53.CrossRefGoogle Scholar
Bay, C.J., Fleming, P., Doekemeijer, B., King, J., Churchfield, M. & Mudafort, R. 2023 Addressing deep array effects and impacts to wake steering with the cumulative-curl wake model. Wind Energy Sci. 8 (3), 401–419.CrossRefGoogle Scholar
Blondel, F. 2023 Brief communication: a momentum-conserving superposition method applied to the super-Gaussian wind turbine wake model. Wind Energy Sci. 8 (2), 141–147.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Brogna, R., Feng, J., Sørensen, J.N., Shen, W.Z. & Porté-Agel, F. 2020 A new wake model and comparison of eight algorithms for layout optimization of wind farms in complex terrain. Appl. Energy 259, 114189.CrossRefGoogle Scholar
Carbajo Fuertes, F., Markfort, C.D. & Porté-Agel, F. 2018 Wind turbine wake characterization with nacelle-mounted wind lidars for analytical wake model validation. Remote Sens. 10 (5), 668.CrossRefGoogle Scholar
Chester, S., Meneveau, C. & Parlange, M.B. 2007 Modeling turbulent flow over fractal trees with renormalized numerical simulation. J. Comput. Phys. 225 (1), 427–448.CrossRefGoogle Scholar
Crespo, A., Hernandez, J. & Frandsen, S. 1999 Survey of modelling methods for wind turbine wakes and wind farms. Wind Energy 2 (1), 1–24.3.0.CO;2-7>CrossRefGoogle Scholar
Dar, A.S., Gertler, A.S. & Porté-Agel, F. 2023 An experimental and analytical study of wind turbine wakes under pressure gradient. Phys. Fluids 35 (4), 045140.CrossRefGoogle Scholar
Dar, A.S. & Porté-Agel, F. 2022 An analytical model for wind turbine wakes under pressure gradient. Energies 15 (15), 5345.CrossRefGoogle Scholar
Dar, A.S. & Porté-Agel, F. 2024 Wind turbine wake superposition under pressure gradient. Phys. Fluids 36 (1), 015145.CrossRefGoogle Scholar
Du, B., Ge, M. & Liu, Y. 2022 A physical wind-turbine wake growth model under different stratified atmospheric conditions. Wind Energy 25 (10), 1812–1836.CrossRefGoogle Scholar
Du, B., Ge, M., Zeng, C., Cui, G. & Liu, Y. 2021 Influence of atmospheric stability on wind-turbine wakes with a certain hub-height turbulence intensity. Phys. Fluids 33 (5), 055111.CrossRefGoogle Scholar
Fan, X., Ge, M., Tan, W. & Li, Q. 2021 Impacts of coexisting buildings and trees on the performance of rooftop wind turbines: an idealized numerical study. Renew. Energy 177, 164–180.CrossRefGoogle Scholar
Farrell, A., King, J., Draxl, C., Mudafort, R., Hamilton, N., Bay, C.J., Fleming, P. & Simley, E. 2021 Design and analysis of a wake model for spatially heterogeneous flow. Wind Energy Sci. 6 (3), 737–758.CrossRefGoogle Scholar
Feng, J., Shen, W.Z. & Li, Y. 2018 An optimization framework for wind farm design in complex terrain. Appl. Sci. 8 (11), 2053.CrossRefGoogle Scholar
Finnigan, J., Ayotte, K., Harman, I., Katul, G., Oldroyd, H., Patton, E., Poggi, D., Ross, A. & Taylor, P. 2020 Boundary-layer flow over complex topography. Boundary-Layer Meteorol. 177, 247–313.CrossRefGoogle Scholar
Gambuzza, S. & Ganapathisubramani, B. 2023 The influence of free stream turbulence on the development of a wind turbine wake. J. Fluid Mech. 963, A19.CrossRefGoogle Scholar
Ge, M., Yang, H., Zhang, H. & Zuo, Y. 2021 A prediction model for vertical turbulence momentum flux above infinite wind farms. Phys. Fluids 33 (5), 055108.CrossRefGoogle Scholar
Ge, M., Zhang, S., Meng, H. & Ma, H. 2020 Study on interaction between the wind-turbine wake and the urban district model by large eddy simulation. Renew. Energy 157, 941–950.CrossRefGoogle Scholar
Gupta, V. & Wan, M. 2019 Low-order modelling of wake meandering behind turbines. J. Fluid Mech. 877, 534–560.CrossRefGoogle Scholar
Hunt, J.C.R., Leibovich, S. & Richards, K.J. 1988 Turbulent shear flows over low hills. Q. J. R. Meteorol. Soc. 114 (484), 1435–1470.CrossRefGoogle Scholar
Ishihara, T. & Qian, G.-W. 2018 A new Gaussian-based analytical wake model for wind turbines considering ambient turbulence intensities and thrust coefficient effects. J. Wind Engng Ind. Aerodyn. 177, 275–292.CrossRefGoogle Scholar
Katic, I., Højstrup, J. & Jensen, N.O. 1986 A simple model for cluster efficiency. In European Wind Energy Association Conference and Exhibition, vol. 1, pp. 407–410. A. Raguzzi Rome.Google Scholar
Lanzilao, L. & Meyers, J. 2022 A new wake-merging method for wind-farm power prediction in the presence of heterogeneous background velocity fields. Wind Energy 25 (2), 237–259.CrossRefGoogle Scholar
Li, L., Wang, B., Ge, M., Huang, Z., Li, X. & Liu, Y. 2023 A novel superposition method for streamwise turbulence intensity of wind-turbine wakes. Energy 276, 127491.CrossRefGoogle Scholar
Li, Z., Dong, G. & Yang, X. 2022 Onset of wake meandering for a floating offshore wind turbine under side-to-side motion. J. Fluid Mech. 934, A29.CrossRefGoogle Scholar
Lin, M. & Porté-Agel, F. 2022 Large-eddy simulation of a wind-turbine array subjected to active yaw control. Wind Energy Sci. 7 (6), 2215–2230.CrossRefGoogle Scholar
Lissaman, P.B.S. 1979 Energy effectiveness of arbitrary arrays of wind turbines. J. Energy 3 (6), 323–328.CrossRefGoogle Scholar
Liu, L. & Stevens, R.J.A.M. 2020 a Effects of two-dimensional steep hills on the performance of wind turbines and wind farms. Boundary-Layer Meteorol. 176, 251–269.CrossRefGoogle Scholar
Liu, L. & Stevens, R.J.A.M. 2020 b Wall modeled immersed boundary method for high Reynolds number flow over complex terrain. Comput. Fluids 208, 104604.CrossRefGoogle Scholar
Liu, Z., Lu, S. & Ishihara, T. 2021 Large eddy simulations of wind turbine wakes in typical complex topographies. Wind Energy 24 (8), 857–886.CrossRefGoogle Scholar
Ma, H., Ge, M., Wu, G., Du, B. & Liu, Y. 2021 Formulas of the optimized yaw angles for cooperative control of wind farms with aligned turbines to maximize the power production. Appl. Energy 303, 117691.CrossRefGoogle 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
Messmer, T., Hölling, M. & Peinke, J. 2024 Enhanced recovery caused by nonlinear dynamics in the wake of a floating offshore wind turbine. J. Fluid Mech. 984, A66.CrossRefGoogle Scholar
Niayifar, A. & Porté-Agel, F. 2016 Analytical modeling of wind farms: a new approach for power prediction. Energies 9 (9), 741.CrossRefGoogle Scholar
Porté-Agel, F., Bastankhah, M. & Shamsoddin, S. 2020 Wind-turbine and wind-farm flows: a review. Boundary-Layer Meteorol. 174 (1), 1–59.CrossRefGoogle ScholarPubMed
Shamsoddin, S. & Porté-Agel, F. 2017 Turbulent planar wakes under pressure gradient conditions. J. Fluid Mech. 830, R4.CrossRefGoogle Scholar
Shamsoddin, S. & Porté-Agel, F. 2018 a A model for the effect of pressure gradient on turbulent axisymmetric wakes. J. Fluid Mech. 837, R3.CrossRefGoogle Scholar
Shamsoddin, S. & Porté-Agel, F. 2018 b Wind turbine wakes over hills. J. Fluid Mech. 855, 671–702.CrossRefGoogle Scholar
Shapiro, C.R., Gayme, D.F. & Meneveau, C. 2018 Modelling yawed wind turbine wakes: a lifting line approach. J. Fluid Mech. 841, R1.CrossRefGoogle Scholar
Shapiro, C.R., Gayme, D.F. & Meneveau, C. 2019 Filtered actuator disks: theory and application to wind turbine models in large eddy simulation. Wind Energy 22 (10), 1414–1420.CrossRefGoogle Scholar
Stevens, R.J.A.M., 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.CrossRefGoogle Scholar
Stevens, R.J.A.M., Martínez-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.CrossRefGoogle Scholar
Stevens, R.J.A.M. & Meneveau, C. 2017 Flow structure and turbulence in wind farms. Annu. Rev. Fluid Mech. 49, 311–339.CrossRefGoogle Scholar
Sun, H., Yang, H. & Gao, X. 2023 Investigation into wind turbine wake effect on complex terrain. Energy 269, 126767.CrossRefGoogle Scholar
Teng, J. & Markfort, C.D. 2020 A calibration procedure for an analytical wake model using wind farm operational data. Energies 13 (14), 3537.CrossRefGoogle Scholar
Thomas, F.O. & Liu, X. 2004 An experimental investigation of symmetric and asymmetric turbulent wake development in pressure gradient. Phys. Fluids 16 (5), 1725–1745.CrossRefGoogle Scholar
Vahidi, D. & Porté-Agel, F. 2022 A physics-based model for wind turbine wake expansion in the atmospheric boundary layer. J. Fluid Mech. 943, A49.CrossRefGoogle Scholar
Van Kuik, G.A.M., et al. 2016 Long-term research challenges in wind energy – a research agenda by the European academy of wind energy. Wind Energy Sci. 1 (1), 1–39.CrossRefGoogle Scholar
Voutsinas, S., Rados, K. & Zervos, A. 1990 On the analysis of wake effects in wind parks. Wind Engng 14 (4), 204–219.Google Scholar
Wang, D., Feng, D., Peng, H., Mao, F., Doranehgard, M.H., Gupta, V., Li, L.K.B. & Wan, M. 2023 Implications of steep hilly terrain for modeling wind-turbine wakes. J. Clean Prod. 398, 136614.CrossRefGoogle Scholar
Wu, Y.-T. & Porté-Agel, F. 2011 Large-eddy simulation of wind-turbine wakes: evaluation of turbine parametrisations. Boundary-Layer Meteorol. 138, 345–366.CrossRefGoogle Scholar
Yang, H., Ge, M., Abkar, M. & Yang, X.I.A. 2022 Large-eddy simulation study of wind turbine array above swell sea. Energy 256, 124674.CrossRefGoogle Scholar
Zhang, H., Ge, M., Liu, Y. & Yang, X.I.A. 2021 a A new coupled model for the equivalent roughness heights of wind farms. Renew. Energy 171, 34–46.CrossRefGoogle Scholar
Zhang, S., Du, B., Ge, M. & Zuo, Y. 2022 a Study on the operation of small rooftop wind turbines and its effect on the wind environment in blocks. Renew. Energy 183, 708–718.CrossRefGoogle Scholar
Zhang, S., Yang, H., Du, B. & Ge, M. 2021 b Effects of a rooftop wind turbine on the dispersion of air pollutant behind a cube-shaped building. Theor. Appl. Mech. Lett. 11 (5), 100296.CrossRefGoogle Scholar
Zhang, Z., Huang, P., Bitsuamlak, G. & Cao, S. 2022 b Large-eddy simulation of wind-turbine wakes over two-dimensional hills. Phys. Fluids 34 (6), 065123.Google Scholar
Zong, H. & Porté-Agel, F. 2020 A momentum-conserving wake superposition method for wind farm power prediction. J. Fluid Mech. 889, A8.CrossRefGoogle Scholar