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Lift coefficient estimation for a rapidly pitching airfoil

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Abstract

We develop a method for estimating the instantaneous lift coefficient on a rapidly pitching airfoil that uses a small number of pressure sensors and a measurement of the angle of attack. The approach assimilates four surface pressure measurements with a modified nonlinear state space model (Goman–Khrabrov model) through a Kalman filter. The error of lift coefficient estimates based only on a weighted-sum of the measured pressures are found to be noisy and biased, which leads to inaccurate estimates. The estimate is improved by including the predictive model in an conventional Kalman filter. The Goman–Khrabrov model is shown to be a linear parameter-varying system and can therefore be used in the Kalman filter without the need for linearization. Additional improvement is realized by modifying the algorithm to provide more accurate estimate of the lift coefficient. The improved Kalman filtering approach results in a bias-free lift coefficient estimate that is more precise than either the pressure-based estimate or the Goman–Khrabrov model on their own. The new method will enable performance enhancements in aerodynamic systems whose performance relies on lift.

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References

  • An X, Williams DR, da Silva AF, Colonius T, Eldredge J (2017) Response of the separated flow over an airfoil to a short-time actuator burst. In: 47th AIAA fluid dynamics conference, p 3315

  • Bansmer S, Radespiel R (2012) Flapping flight: high thrust and propulsive efficiency due to forward gliding oscillations. AIAA J 50:2937–2942. https://doi.org/10.2514/1.J051749

    Article  Google Scholar 

  • Bartholomay S, Wester TTB, Perez-Becker S, Konze S, Menzel C, Hölling M, Spickenheuer A, Peinke J, Nayeri CN, Paschereit OP, Oberleithner K (2020) Pressure based lift estimation and its application to feedforward load control employing trailing edge flaps. Wind Energy Sci Discuss 1–39 (2020). https://doi.org/10.5194/wes-2020-91. https://wes.copernicus.org/preprints/wes-2020-91/

  • Brunton SL, Dawson ST, Rowley CW (2014) State-space model identification and feedback control of unsteady aerodynamic forces. J Fluids Struct 50:253–270

    Article  Google Scholar 

  • Darakananda D, da Silva AFDC, Colonius T, Eldredge JD (2018) Data-assimilated low-order vortex modeling of separated flows. Phys Rev Fluids 3(12), 124701

  • Dawson ST, Schiavone NK, Rowley CW, Williams DR (2015) A data-driven modeling framework for predicting forces and pressures on a rapidly pitching airfoil. In: 45th AIAA fluid dynamics conference, pp 1–14

  • Goman M, Khrabrov A (1994) State-space representation of aerodynamic characteristics of an aircraft at high angles of attack. J Aircraft 31(5):1109–1115

    Article  Google Scholar 

  • Grimau L (2014) Energy savings for uav flight in unsteady gusting conditions through trajectory optimization. Master’s thesis, Illinois Institute of Technology, Chicago, USA

  • Hemati MS, Dawson ST, Rowley CW (2016) Parameter-varying aerodynamics models for aggressive pitching-response prediction. AIAA J

  • Kalman RE et al (1960) A new approach to linear filtering and prediction problems. J Basic Eng 82(1):35–45

    Article  MathSciNet  Google Scholar 

  • Le Provost M, Williams DR, Brunton S (2018) Sindy analysis of disturbance and plant model superposition on a rolling delta wing. In: 2018 Flow control conference, p 3068

  • Peters DA, Karunamoorthy S, Cao WM (1995) Finite state induced flow models. Journal of aircraft 32(2):313–322

    Article  Google Scholar 

  • Platzer M, Jones K, Young J, Lai J (2008) Flapping wing aerodynamics: progress and challenges. AIAA J 46:2136–2149. https://doi.org/10.2514/1.29263

    Article  Google Scholar 

  • Reißner FA (2015) Hysteresis modeling and comparison of controller effectiveness on a pitching airfoil. Master’s thesis, Technical University Berlin, Berlin

  • Shamma JS, Athans M (1992) Gain scheduling: Potential hazards and possible remedies. IEEE Control Syst Mag 12(3):101–107

    Article  Google Scholar 

  • Theodorsen T (1979) General theory of aerodynamic instability and the mechanism of flutter

  • Wagner H (1925) Über die entstehung des dynamischen auftriebes von tragflügeln. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 5(1):17–35

    Article  Google Scholar 

  • Welch G, Bishop G, et al (1995) An introduction to the kalman filter

  • Williams DR, An X, Iliev S, King R, Reißner F (2015) Dynamic hysteresis control of lift on a pitching wing. Exp Fluids 56(5):112

    Article  Google Scholar 

  • Williams DR, King R (2018) Alleviating unsteady aerodynamic loads with closed-loop flow control. AIAA J 56(6):2194–2207

    Article  Google Scholar 

  • Williams DR, Reißner F, Greenblatt D, Müller-Vahl H, Strangfeld C (2016) Modeling lift hysteresis on pitching airfoils with a modified Goman–Khrabrov model. AIAA J

Download references

Acknowledgements

The support for this work by the US Air Force Office of Scientific Research FA9550-18-1-0440 with program manager Gregg Abate is gratefully acknowledged. Supported under the US Air Force Office of Scientific Research FA9550-14-1-0328 with program manager Douglas Smith is also gratefully acknowledged. The first author would like to thank the supported under ONR MURI Grant N00014-14-1-0533 with program manager Robert Brizzolara. The insightful suggestions from Professor C. Rowley and S. Otto at Princeton University are sincerely appreciated.

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Authors and Affiliations

  1. MAE Department, Princeton University, Princeton, NJ, 08544, USA

    Xuanhong An

  2. MMAE Department, Illinois Institute of Technology, Chicago, IL, 60616, USA

    David R. Williams

  3. MAE Department, University of California Los Angeles, Los Angeles, CA, 90095, USA

    Jeff D. Eldredge

  4. MCE Department, California Institute of Technology, Pasadena, CA, 91125, USA

    Tim Colonius

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  1. Xuanhong An
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  2. David R. Williams
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  4. Tim Colonius
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Correspondence to Xuanhong An.

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An, X., Williams, D.R., Eldredge, J.D. et al. Lift coefficient estimation for a rapidly pitching airfoil. Exp Fluids 62, 11 (2021). https://doi.org/10.1007/s00348-020-03105-3

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