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Linear stability analysis of wind turbine wakes performed on wind tunnel measurements

Published online by Cambridge University Press:  27 November 2013

G. V. Iungo*
Wind Engineering and Renewable Energy Laboratory, École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland
F. Viola
Wind Engineering and Renewable Energy Laboratory, École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland
S. Camarri
Department of Civil and Industrial Engineering, University of Pisa, Pisa, 56122, Italy
F. Porté-Agel
Wind Engineering and Renewable Energy Laboratory, École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland
F. Gallaire
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland
Email address for correspondence:


Wind tunnel measurements were performed for the wake produced by a three-bladed wind turbine immersed in uniform flow. These tests show the presence of a vorticity structure in the near-wake region mainly oriented along the streamwise direction, which is denoted as the hub vortex. The hub vortex is characterized by oscillations with frequencies lower than that connected to the rotational velocity of the rotor, which previous works have ascribed to wake meandering. This phenomenon consists of transversal oscillations of the wind turbine wake, which might be excited by the vortex shedding from the rotor disc acting as a bluff body. In this work, temporal and spatial linear stability analyses of a wind turbine wake are performed on a base flow obtained with time-averaged wind tunnel velocity measurements. This study shows that the low-frequency spectral component detected experimentally matches the most amplified frequency of the counter-winding single-helix mode downstream of the wind turbine. Then, simultaneous hot-wire measurements confirm the presence of a helicoidal unstable mode of the hub vortex with a streamwise wavenumber roughly equal to that predicted from the linear stability analysis.

©2013 Cambridge University Press 

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Alfredsson, P. H. & Dahlberg, J. A. 1979 A preliminary wind tunnel study of windmill wake dispersion in various flow conditions. Technical Note AU-1499, Part 7, FFA, Stockholm, Sweden.Google Scholar
Antkowiak, A. 2005 Short-term dynamics of an isolated vortex. PhD thesis, IMFT–Université Paul Sabatier.Google Scholar
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 37 (5), 750756.CrossRefGoogle Scholar
Burton, T., Sharpe, D., Jenkins, N. & Bossanyi, N. 2001 Wind Energy Handbook. Wiley.CrossRefGoogle Scholar
Camarri, S., Fallenius, B. E. G. & Fransson, J. H. M. 2013 Stability analysis of experimental flow fields behind a porous cylinder for the investigation of the large-scale wake vortices. J. Fluid Mech. 715, 499536.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1993 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Chamorro, L. P. & Porté-Agel, F. 2009 A wind-tunnel investigation of wind-turbine wakes: boundary-layer turbulence effects. Boundary-Layer Meteorol. 132 (1), 129149.Google 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), 515533.Google Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.Google Scholar
Couairon, A. & Chomaz, J.-M. 1999 Fully nonlinear global modes in slowly varying flows. Phys. Fluids 11 (12), 36883703.Google Scholar
Felli, M., Camussi, R. & DiFelice, F. 2011 Mechanisms of evolution of the propeller wake in the transition and far fields. J. Fluid Mech. 682, 553.Google Scholar
Gallaire, F. & Chomaz, J. M. 2003 Mode selection in swirling jet experiments: a linear stability analysis. J. Fluid Mech. 494, 222224.CrossRefGoogle Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic instability. J. Fluid Mech. 14, 222224.Google Scholar
Gupta, B. P. & Loewy, R. G. 1974 Theoretical analysis of the aerodynamic stability of multiple, interdigitated helical vortices. AIAA J. 12, 13811387.Google Scholar
Huerre, P. 1998 Hydrodynamic Instabilities in Open Flows. Cambridge University Press.Google Scholar
Iungo, G. V. & Lombardi, E. 2011 A procedure based on proper orthogonal decomposition for time-frequency analysis of time series. Exp. Fluids 51, 969985.Google Scholar
Ivanell, S., Mikkelsen, R., Sørensen, J. N. & Henningson, D. 2010 Stability analysis of the tip vortices of a wind turbine. Wind Energy 13, 705715.Google Scholar
Ivanell, S., Sørensen, J. N., Mikkelsen, R. & Henningson, D. 2009 Analysis of numerically generated wake structures. Wind Energy 12, 6380.Google Scholar
Joukowski, N. E. 1912 Vortex theory of a rowing screw. Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lubitelei Estestvoznaniya 16 (1)in Russian.Google Scholar
Juniper, M. P., Tammisola, O. & Lundell, F. 2011 The local and global stability of confined planar wakes at intermediate Reynolds number. J. Fluid Mech. 686, 218238.Google Scholar
Kerswell, R. R. & Davey, A. 1996 On the linear instability of elliptic pipe flow. J. Fluid Mech. 316, 307324.Google Scholar
Klein, R., Majda, A. J. & Damodaran, K. 1995 Simplified equations for the interaction of nearly parallel vortex filaments. J. Fluid Mech. 288, 201248.Google Scholar
Leibovich, S. & Stewartson, K. 1983 A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335356.Google Scholar
Leontini, J. S., Thompson, M. C. & Hourigan, K. 2010 A numerical study of global frequency selection in the time-mean wake of a circular cylinder. J. Fluid Mech. 645, 435446.CrossRefGoogle Scholar
Levy, H. & Forsdyke, A. G. 1928 The steady motion and stability of a helical vortex. Proc. R. Soc. Lond. A 120, 670690.Google Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Lu, H. & Porté-Agel, F. 2011 Large-eddy simulation of a very large wind farm in a stable atmospheric boundary layer. Phys. Fluids 23 (6), 065101.Google Scholar
Massouh, F. & Dobrev, I. 2007 Exploration of the vortex wake behind of wind turbine rotor. J. Phys.: Conf. Ser. 75 012036.Google Scholar
Medici, D. & Alfredsson, P. H. 2006 Mesurements on a wind turbine wake: 3D effects and bluff body vortex shedding. Wind Energy 9 (3), 219236.Google Scholar
Medici, D. & Alfredsson, P. H. 2008 Measurements behind model wind turbines: further evidence of wake meandering. Wind Energy 11, 211217.Google Scholar
Meliga, P., Sipp, D. & Chomaz, J.-M. 2009 Elephant modes and low frequency unsteadiness in a high Reynolds number, transonic afterbody wake. Phys. Fluids 21 (5), 054105.CrossRefGoogle Scholar
Monkewitz, P. A. 1988 A note on vortex shedding from axisymmetric bluff bodies. J. Fluid Mech. 192, 561575.Google Scholar
Oberleithner, K., Sieber, M., Nayeri, C. N., Paschereit, C. O., Petz, C., Hege, H.-C., Noack, B. R. & Wygnanski, I. 2011 Three-dimensional coherent structures in a swirling jet undergoing vortex breakdown: stability analysis and empirical mode construction. J. Fluid Mech. 679, 383414.CrossRefGoogle Scholar
Okulov, V. L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319342.Google Scholar
Okulov, V. L. & Sørensen, J. N. 2007 Stability of helical tip vortices in a rotor far wake. J. Fluid Mech. 576, 125.Google Scholar
Olendraru, C. & Sellier, A. 2002 Viscous effects in the absolute/convective instability of the Batchelor vortex. J. Fluid Mech. 459, 371396.Google Scholar
Olendraru, C., Sellier, A., Rossi, M. & Huerre, P. 1999 Inviscid instability of the Batchelor vortex: absolute-convective transition and spatial branches. Phys. Fluids 11, 18051820.Google Scholar
Ortega, J. M., Bristol, R. L. & Savas, Ö. 2003 Experimental study of the instability of unequal-strength counter-rotating vortex pairs. J. Fluid Mech. 474, 3584.Google Scholar
Pier, B. & Huerre, P. 2001 Nonlinear synchronization in open flows. J. Fluids Struct. 15, 471480.Google Scholar
Porté-Agel, F., Wu, Y-T., Lu, H. & Conzemius, R. J. 2011 Large-eddy simulation of atmospheric boundary layer flow through wind turbines and wind farms. J. Wind Engng Ind. Aerodyn. 99 (4), 154168.Google Scholar
Sarpkaya, T. 1971 On stationary and travelling vortex breakdowns. J. Fluid Mech. 45, 545559.Google Scholar
Schito, P. 2012 Large eddy simulations of wind turbines: interaction with turbulent flow. PhD thesis, Department of Mechanical Engineering, Politecnico di Milano.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Sherry, M., Sheridan, J. & LoJacono, D. 2010 Horizontal axis wind turbine tip and root vortex measurements. In 15th International Symposium on Applications of Laser Techniques to Fluid Mechanics.Google Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333358.Google Scholar
Sørensen, J. N. 2011 Instability of helical tip vortices in rotor wakes. J. Fluid Mech. 682, 14.CrossRefGoogle Scholar
Vermeer, L. J., Sørensen, J. N. & Crespo, A. 2003 Wind turbine wake aerodynamics. Prog. Aerosp. Sci. 39 (6/7), 467510.Google Scholar
Whale, J., Papadopoulos, K. H., Anderson, C. G., Helmis, C. G. & Skyner, D. J. 1996 A study of the near wake structure of a wind turbine comparing measurements from laboratory and full-scale experiments. Solar Energy 56 (6), 621633.Google Scholar
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54 (4), 641663.Google 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), 12191235.Google Scholar