Skip to main content Accessibility help
×
Home

Mutual inductance instability of the tip vortices behind a wind turbine

  • Sasan Sarmast (a1) (a2), Reza Dadfar (a1), Robert F. Mikkelsen (a2), Philipp Schlatter (a1), Stefan Ivanell (a1) (a3), Jens N. Sørensen (a2) and Dan S. Henningson (a1)...

Abstract

Two modal decomposition techniques are employed to analyse the stability of wind turbine wakes. A numerical study on a single wind turbine wake is carried out focusing on the instability onset of the trailing tip vortices shed from the turbine blades. The numerical model is based on large-eddy simulations (LES) of the Navier–Stokes equations using the actuator line (ACL) method to simulate the wake behind the Tjæreborg wind turbine. The wake is perturbed by low-amplitude excitation sources located in the neighbourhood of the tip spirals. The amplification of the waves travelling along the spiral triggers instabilities, leading to breakdown of the wake. Based on the grid configurations and the type of excitations, two basic flow cases, symmetric and asymmetric, are identified. In the symmetric setup, we impose a 120° symmetry condition in the dynamics of the flow and in the asymmetric setup we calculate the full 360° wake. Different cases are subsequently analysed using dynamic mode decomposition (DMD) and proper orthogonal decomposition (POD). The results reveal that the main instability mechanism is dispersive and that the modal growth in the symmetric setup arises only for some specific frequencies and spatial structures, e.g. two dominant groups of modes with positive growth (spatial structures) are identified, while breaking the symmetry reveals that almost all the modes have positive growth rate. In both setups, the most unstable modes have a non-dimensional spatial growth rate close to $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\pi /2$ and they are characterized by an out-of-phase displacement of successive helix turns leading to local vortex pairing. The present results indicate that the asymmetric case is crucial to study, as the stability characteristics of the flow change significantly compared to the symmetric configurations. Based on the constant non-dimensional growth rate of disturbances, we derive a new analytical relationship between the length of the wake up to the turbulent breakdown and the operating conditions of a wind turbine.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Mutual inductance instability of the tip vortices behind a wind turbine
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Mutual inductance instability of the tip vortices behind a wind turbine
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Mutual inductance instability of the tip vortices behind a wind turbine
      Available formats
      ×

Copyright

The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .

Corresponding author

Email address for correspondence: henning@mech.kth.se

References

Hide All
Alfredsson, P. H. & Dahlberg, J. A.1979 A preliminary wind tunnel study of windmill wake dispersion in various flow conditions. Tech. Note AU-1499, Part 7.
Aubry, N. 1991 On the hidden beauty of the proper orthogonal decomposition. Theor. Comput. Fluid Dyn. 2 (5), 339352.
Bhagwat, M. J. & Leishman, J. G. 2001 Stability, consistency and convergence of numerical algorithms for time-marching free-vortex wake analysis. J. Am. Helicopter Soc. 46 (1), 5970.
Chen, K. K., Tu, J. H. & Rowley, C. W. 2012 Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci. 22 (6), 887915.
Chevalier, M., Schlatter, P., Lundbladh, A. & Henningson, D. S.2007 SIMSON: a pseudo-spectral solver for incompressible boundary layer flows. Tech. Rep. TRITA-MEK 2007:07. Royal Institute of Technology, Stockholm, Sweden.
Felli, M., Camussi, R. & Di Felice, F. 2011 Mechanisms of evolution of the propeller wake in the transition and far fields. J. Fluid Mech. 682, 553.
Frederich, O. & Luchtenburg, D. M.2011 Modal analysis of complex turbulent flow. In 7th Intl Symp. on Turbulence and Shear Flow Phenomena (TSFP-7), Ottawa, Canada.
Gupta, B. P. & Loewy, R. G. 1974 Theoretical analysis of the aerodynamic stability of multiple, interdigitated helical vortices. AIAA J. 12 (10), 13811387.
Ivanell, S., Mikkelsen, R., Sørensen, J. N. & Henningson, D. S. 2010 Stability analysis of the tip vortices of a wind turbine. Wind Energy 13 (8), 705715.
Jain, R., Conlisk, A. T., Mahalingam, R. & Komerath, N. M.1998 Interaction of tip-vortices in the wake of a two-bladed rotor. In 54th Annual Forum of the Am. Helicopter Soc., Washington, DC.
Joukowski, N. E. 1912 Vortex theory of screw propeller. Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lubitelei Estestvoznaniya 16 (1), 131.
Lamb, H. 1932 Hydrodynamics, 6th edn Cambridge University Press.
Leishman, G., Bhagwat, M. J. & Ananthan, S. 2004 The vortex ring state as a spatially and temporally developing wake instability. J. Am. Helicopter Soc. 49 (2), 160175.
Levy, H. & Forsdyke, A. G. 1927 The stability of an infinite system of circular vortices. Proceedings R. Soc. Lond. A 114 (768), 594604.
Levy, H. & Forsdyke, A. G. 1928 The steady motion and stability of a helical vortex. Proc. R. Soc. Lond. A 120 (786), 670690.
Leweke, T., Bolnot, H., Quaranta, U. & Dizès, S. Le.2013 Local and global pairing in helical vortex systems. Intl Conf. on Aerodynamics of Offshore Wind Energy Systems and Wakes, Lyngby, Denmark.
Lumley, J. L. 1970 Stochastic Tools in Turbulence. Academic.
Manhart, M. & Wengle, H. 1993 A spatiotemporal decomposition of a fully inhomogeneous turbulent flow field. Theor. Comput. Fluid Dyn. 5 (4), 223242.
Michelsen, J. A.1994 Block structured multigrid solution of 2D and 3D elliptic PDE’s. Tech. Rep. AFM 94-06. Dept. of Fluid Mech., Technical University of Denmark, DTU.
Mikkelsen, R. F.2003 Actuator disc methods applied to wind turbines. PhD thesis, Dept. of Fluid Mech., Technical University of Denmark, DTU.
Okulov, V. L. & Sørensen, J. N. 2007 Stability of helical tip vortices in a rotor far wake. J. Fluid Mech. 576, 125.
Øye, S.1991 Tjæreborg wind turbine: 4. dynamic flow measurement. AFM Notat VK-204.
Prony, R. 1795 Essai expérimental et analytique. J. l’Ecole Polytech. 1 (2), 2476.
Rao, A. R., Hamed, K. H. & Chen, H. L. 2003 Nonstationarities in Hydrologic and Environmental Time Series, Water Sci. and Tech. Library, vol. 45. Springer.
Rempfer, D. & Fasel, H. F. 1994 Evolution of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid Mech. 260, 351375.
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115127.
Schlatter, P. & Örlü, R. 2012 Turbulent boundary layers at moderate Reynolds numbers. Inflow length and tripping effects. J. Fluid Mech. 710, 534.
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 528.
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. I-coherent structures. II-symmetries and transformations. III-dynamics and scaling. Q. Appl. Maths 45 (1), 561571.
Smith, G., Schlez, W., Liddell, A., Neubert, A. & Peña, A.2006 Advanced wake model for very closely spaced turbines. In Conf. Procs. EWEC Athens.
Sørensen, N. N.1995 General purpose flow solver applied to flow over hills. PhD thesis, Risø National Laboratory, Roskilde.
Sørensen, J. N. 2011 Instability of helical tip vortices in rotor wakes. J. Fluid Mech. 682, 14.
Sørensen, J. N. & van Kuik, G. A. M. 2011 General momentum theory for wind turbines at low tip speed ratios. Wind Energy 14 (7), 821839.
Sørensen, J. N. & Shen, W. Z. 2002 Numerical modeling of wind turbine wakes. Trans. ASME: J. Fluids Engng 124 (2), 393399.
Sørensen, J. N., Shen, W. Z. & Munduate, X. 1998 Analysis of wake states by a full-field actuator disc model. Wind Energy 1 (2), 7388.
Tangler, J. L., Wohlfeld, R. M. & Miley, S. J.1973 An experimental investigation of vortex stability, tip shapes, compressibility, and noise for hovering model rotors. NASA 2305.
Ta Phuoc, L.1994 Modèles de sous maille appliqués aux ecoulements instationnaires décollés. In Proceeding of the DRET Conference: Aérodynamique Instationnaire Turbulents – Aspects Numériques et Expérimentaux.
Troldborg, N.2008 Actuator line modeling of wind turbine wakes. PhD thesis, Dept of Fluid Mechanics, Technical University of Denmark, DTU.
Walther, J. H., Guenot, M., Machefaux, E., Rasmussen, J. T., Chatelain, P., Okulov, V. L., Sørensen, J. N., Bergdorf, M. & Koumoutsakos, P. 2007 A numerical study of the stabilitiy of helical vortices using vortex methods. J. Phys.: Conf. Ser. 75 (1), 012034.
Welch, P. 1967 The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust. 15 (2), 7073.
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 4, 641663.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Mutual inductance instability of the tip vortices behind a wind turbine

  • Sasan Sarmast (a1) (a2), Reza Dadfar (a1), Robert F. Mikkelsen (a2), Philipp Schlatter (a1), Stefan Ivanell (a1) (a3), Jens N. Sørensen (a2) and Dan S. Henningson (a1)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed