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Bifurcation characteristics of a two-dimensional structurally non-linear airfoil in turbulent flow

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Abstract

The dynamics of a structurally non-linear two-dimensional airfoil in turbulent flow is investigated numerically using a Monte Carlo approach. Both the longitudinal and vertical components of turbulence, corresponding to parametric (multiplicative) and external (additive) excitation, respectively, are modelled. The properties of the airfoil are chosen such that the underlying non-excited, deterministic system exhibits binary flutter; the loss of stability of the equilibrium point due to flutter then leads to a limit cycle oscillation (LCO) via a supercritical Hopf bifurcation. For the random system, the results are examined in terms of the probability structure of the response and the largest Lyapunov exponent. The airfoil response is interpreted from the point of view of the concepts of D- and P-bifurcations, as defined in random bifurcation theory. It is found that the bifurcation is characterized by a change in shape of the response probability structure, while no discontinuity in the variation of the largest Lyapunov exponent with airspeed is observed. In this sense, the trivial bifurcation obtained for the deterministic airfoil, where the D- and P-bifurcations coincide, appears only as a P-bifurcation for the random case. At low levels of turbulence intensity, the Gaussian-like bell-shaped bi-dimensional PDF bifurcates into a crater shape; this is interpreted as a random fixed point bifurcating into a random LCO. At higher levels of turbulence intensity, the post-bifurcation PDF loses its underlying deterministic LCO structure. The crater is transformed into a two-peaked shape, with a saddle at the origin. From a more universal point of view, the robustness of the random bifurcation scenario is critiqued in light of the relative importance of the two components of turbulent excitation.

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References

  1. Lee, B.H.K., Price, S.J., Wong, Y.S.: Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos. Prog. Aerosp. Sci. 35, 205–334 (1999)

    Google Scholar 

  2. Alighanbari, H.: Flutter analysis and chaotic response of an airfoil accounting for structural nonlinearities. PhD Thesis, McGill University, Montreal, Canada (1995)

  3. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences. Springer-Verlag, New York (1983)

  4. Fujimori, Y., Lin, Y.K., Ariaratnam, S.T.: Rotor blade stability in turbulent flows: Part II. AIAA J. 17(7), 673–678 (1979)

    Google Scholar 

  5. Prussing, J.E., Lin, Y.K.: A closed-form analysis of rotor blade flap-lag stability in hover and low-speed flight in turbulent flow. J. Am. Helicopter Soc. 28(3), 42–46 (1983)

    Google Scholar 

  6. Bucher, C.G., Lin, Y.K.: Stochastic stability of bridges considering coupled modes: II. J. Eng. Mech. (ASCE) 115(2), 384–400 (1989)

    Google Scholar 

  7. Lin, Y.K.: Stochastic stability of wind-excited long-span bridges. Probab. Eng. Mech. 11, 257–261 (1996)

    Google Scholar 

  8. Náprstek, J., Pospíšil, S.: Flutter onset influenced by combined multiplicative and additive random noise. In: Proceedings of the 5th International Colloquium on Bluff Body Aerodynamics and Applications, pp. 289–292. NRCC/University of Ottawa, Ottawa, Canada (2004)

  9. Horsthemke, W., Lefever, R.: Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology. Springer−Verlag, Berlin, Germany (1984)

  10. Arnold, L.: Random Dynamical Systems. Springer-Verlag, Berlin, Germany (1998)

  11. Ariaratnam, S.: Some illustrative examples of stochastic bifurcations. In: Nonlinearity and Chaos in Engineering Dynamics, Thompson, J., Bishop, S. (eds.), pp. 267–274. Wiley, Chichester, UK (1994)

  12. Arnold, L.: Random Dynamical Systems. Lectures given at the 2nd Session of the CIME on Dynamical Systems, pp. 1–43. Springer-Verlag, Berlin, Germany (1995)

  13. Fung, Y.C.: An Introduction to the Theory of Aeroelasticity. Wiley, New York (1955)

  14. Poirel, D., Price, S.J.: Response probability structure of a structurally nonlinear fluttering airfoil in turbulent flow. Probab. Eng. Mech. 18(2), 185–202 (2003)

    Google Scholar 

  15. Poirel, D., Price, S.J.: Random binary (coalescence) flutter of a two-dimensional linear airfoil. J. Fluids Struct. 18, 23–42 (2003)

    Google Scholar 

  16. Van der Wall, B.G., Leishman, J.G.: On the influence of time-varying flow velocity on unsteady aerodynamics. J. Am. Helicopter Soc. 39(4), 25–36 (1994)

    Google Scholar 

  17. Houbolt, J., Steiner, R., Pratt, K.: Dynamic response of airplanes to atmospheric turbulence including flight data on input and response. NASA Tech. Rep. R-199 (1964)

  18. Zentner, I., Poirion, F.: Stability of fluid-structure coupled systems in the presence of multiplicative and additive noise. Application to airfoil flutter. In: Proceedings of the 7th International Conference on Computational Structures Technology. Lisbon, Portugal (2004)

  19. Poirel, D., Price, S.J.: Structurally nonlinear fluttering airfoil in turbulent flow. AIAA J. 39(10), 1960–1968 (2001)

    Google Scholar 

  20. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin (1992)

  21. Schreiber, T.: Interdisciplinary application of nonlinear time series methods. Phys. Rep. 308, 1–64 (1999)

    Google Scholar 

  22. Argyris, J., Faust, G., Haase, M.: An exploration of chaos – an introduction for natural scientists and engineers. In: Texts on Computational Mechanics, vol. VII. North-Holland, Amsterdam, The Netherlands (1994)

  23. Argyris, J., Mlejnek, H.-P.: Dynamics of structures, texts on computational mechanics, vol. V. North-Holland, Amsterdam, The Netherlands (1991)

  24. Leng, G., Sri Namachchivaya, N., Talwar, S.: Robustness of nonlinear systems perturbed by external random excitation. J. Appl. Mech. 59, 1015–1022 (1992)

  25. Poirel, D., Price, S.J.: Post-instability behaviour of a structurally nonlinear airfoil in longitudinal turbulence. J. Aircraft 34(5), 619–626 (1997)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Royal Military College of Canada, P.O. Box 17000, Stn Forces, Kingston, Ontario, K7K7B4, Canada

    Dominique Poirel

  2. Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montréal, Québec, H3A 2K6, Canada

    Stuart J. Price

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  1. Dominique Poirel
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  2. Stuart J. Price
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Correspondence to Dominique Poirel.

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Poirel, D., Price, S.J. Bifurcation characteristics of a two-dimensional structurally non-linear airfoil in turbulent flow. Nonlinear Dyn 48, 423–435 (2007). https://doi.org/10.1007/s11071-006-9096-y

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