Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T05:24:43.139Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on the nonlinear dynamics of a typical aerofoil section in dynamic stall

Published online by Cambridge University Press:  03 February 2016

U. Galvanetto ,
J. Peirò  and
C. Chantharasenawong
U. Galvanetto
Affiliation:
Department of Aeronautics, Imperial College, London, UK
J. Peirò
Affiliation:
Department of Aeronautics, Imperial College, London, UK
C. Chantharasenawong
Affiliation:
Department of Aeronautics, Imperial College, London, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We use standard tools of the theory of dynamical systems such as phase plots, bifurcation diagrams and basins of attraction to analyse and understand the dynamic behaviour of a typical aerofoil section under dynamic stall conditions. The structural model is linear and the aerodynamic loading is represented by the Leishman-Beddoes semi-empirical dynamic stall model. The loads given by this model are nonlinear and non-smooth, therefore we have integrated the equation of motion using a Runge-Kutta-Fehlberg (RKF45) algorithm equipped with event detection. We perform simulations of the motion for a range of Mach numbers and show that the model is very sensitive to small variations. This is evidenced by the presence in the bifurcation diagram of co-existing attractors or, in other words, by the existence of more than one steady-state motion for a given Mach number. The mechanisms for the appearance and disappearance of the co-existing attractors are elucidated by analysing the evolution of their basins of attraction as the Mach number changes.

Type
Research Article
Information
The Aeronautical Journal , Volume 111 , Issue 1125 , November 2007 , pp. 731 - 739
Copyright
Copyright © Royal Aeronautical Society 2007 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. McCroskey, W.J.. Unsteady aerofoils, Annual Rev of Fluid Mech, 1982, 14, pp 285–311.Google Scholar
2. Ericsson, L.E. and Reding, J.P.. Fluid mechanics of dynamic stall part 1, Unsteady Flow Concepts. J Fluids and Structures, 1988, 2, pp 1–33.Google Scholar
3. Ericsson, L.E. and Reding, J.P.. Fluid mechanics of dynamic stall part 2, Prediction of Full Scale Characteristics, J Fluids and Structures, 2006, 2, pp 113–143.Google Scholar
4. Leishman, J.G., Principles of Helicopter Aerodynamics, 2006, Second edition, Cambridge University Press.Google Scholar
5. Johnson, W., Helicopter Theory, 1980, Princeton University Press.Google Scholar
6. Leishman, J.G.. Challenges in modelling the unsteady aerodynamics of wind turbines, Wind Energy, 2002, 5, pp 85–132.Google Scholar
7. Burton, T., Sharpe, D., Jenkins, N. and Bossanyi, E., Wind Energy Handbook, 2001, John Wiley and Sons.Google Scholar
8. Petot, D.. Modélisation du décrochage dynamique, La Recherche Aérospatiale, 1995, 5, pp 60–72, 1995.Google Scholar
9. Truong, K.V. and Costes, J.J.. Oscillatory behaviour of helicopter rotor airloads in the blade stall regime, J Aircr, 1995, 32, (5), pp 1148–1149.Google Scholar
10. Gormont, R.E.. A Mathematical model of unsteady aerodynamics and radial flow for application to helicopter rotors, May 1973, USAAVLABS, TR 72-67.Google Scholar
11. Johnson, W.. The response and airloading of helicopter rotor blades due to dynamic stall, May 1970, Massachusetts Institute of Technology, ASRL TR 130-1.Google Scholar
12. Gangwani, S.T.. Prediction of dynamic stall and unsteady airloads for rotor blades, J American Helicopter Soc, 1982, 27, pp 57–64.Google Scholar
13. Gangwani, S.T.. Synthesized aerofoil data method for prediction of dynamic stall and unsteady airloads, Vertica, 1984, 8, pp 93–118.Google Scholar
14. Leishman, J.G. and Beddoes, T.S.. A generalised model for aerofoil unsteady aerodynamic behaviour and dynamic stall using the indicial method, 1986, In Proceedings of the 42nd Annual Forum, Washington DC. American Helicopter Society.Google Scholar
15. Leishman, J.G. and Beddoes, T.S.. A semi-empirical model for dynamic stall, J American Helicopter Soc, 1989, 34, pp 3–17.Google Scholar
16. Hansen, M.H., Gaunaa, M. and Madsen, H.A.. A Beddoes-Leishman type dynamic stall model in state-space and indicial formulation, June 2004, Risø National Laboratory, Roskilde, Denmark.Google Scholar
17. Larsen, J.W., Nielsen, S.R.K. and Krenk, S.. Dynamic stall model of wind turbine aerofoils, J Fluids and Structures, 2007, 23, (7), pp 959–982.Google Scholar
18. Price, S.J. and Keleris, J.P.. Non-linear dynamics of an aerofoil forced to oscillate in dynamic stall, J Sound and Vibration, 1996, 194, (2), pp 265–283.Google Scholar
19. Li, X.G. and Fleeter, S.. Dynamic stall generated aerofoil oscillations, Int J Turbomachinery and Jet Engines, 2003, 20, pp 217–233.Google Scholar
20. Leine, R.I. and Nijmeijer, H.. Dynamics and bifurcations in non-smooth mechanical systems, 2004, Vol 18 of Lecture notes in Applied and Computational Mechanics, Springer-Verlag.Google Scholar
21. Lee, B.H.K., Price, S.J. and Wong, Y.S.. Nonlinear aeroelastic analysis of aerofoils: bifurcation and chaos, Prog in Aerospace Sci, 1998, 35, pp 205–334.Google Scholar
22. Fung, Y.C., An Introduction to the Theory of Aeroelasticity, 1993, Dover, New York.Google Scholar
23. Leishman, J.G. and Nguyen, K.Q.. State-space representation of unsteady aerofoil behaviour, AIAA J, 1988, 28, (5), pp 836–844.Google Scholar
24. Leishman, J.G. and Crouse, G.L.. State-space model for unsteady aerofoil behaviour and dynamic stall, 1989, Proceedings of the AIAA/AHS/ASME Structural Dynamics and Materials Conference, Mobile, Alabama, April 1989, AIAA paper 89-1319.Google Scholar
25. Crouse, G.L. and Leishman, J.G.. Transonic aeroelasticity analysis using state-space unsteady aerodynamic modeling, J Aircr, 1992, 29, (1), pp 153–160.Google Scholar
26. Parameswaran, V. and Baeder, J.D.. Reduction of blade vortex interaction noise using prescribed pitching, 1995, Presented at the 13th AIAA Applied Aerodynamics Conference, June 1995.Google Scholar
27. Mahajan, A.J., Kaza, K.R.V. and Dowell, E.H.. Semi-empirical model for prediction of unsteady forces on an aerofoil with application to flutter, J Fluids and Structures, 1993, 7, pp 87–103.Google Scholar
28. Shampine, L.F. and Reichelt, M.W.. The MATLAB ODE suite, SIAM J Scientific Computing, 1997, 18, (1), pp 1–22.Google Scholar
29. Galvanetto, U., Peiro’, J. and Chantharasenawong, C.. An assessment of some effects on the nonsmoothness of the Leishmann-Beddoes dynamic stall model on the nonlinear dynamics of a typical airfoil section, in press J Fluid and Structures, 2007.Google Scholar
30. Galvanetto, U.. Some remarks on the two-block symmetric Burridge-Knopoff model, Phys Lett A, 2002, 293, pp 251–259.Google Scholar