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Research Papers

Nonlinear Reduced-Order Models for Aerodynamic Lift of Oscillating Airfoils2

[+] Author and Article Information
Muhammad Saif Ullah Khalid

Department of Mechanical Engineering,
NUST College of Electrical and
Mechanical Engineering,
National University of Sciences and Technology,
Islamabad 44000, Pakistan
e-mail: m.saifullahkhalid@ceme.nust.edu.pk

Imran Akhtar

Department of Mechanical Engineering,
NUST College of Electrical and
Mechanical Engineering,
National University of Sciences and Technology,
Islamabad 44000, Pakistan
e-mail: akhtar@vt.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 5, 2016; final manuscript received February 27, 2017; published online May 17, 2017. Assoc. Editor: Sotirios Natsiavas.

J. Comput. Nonlinear Dynam 12(5), 051019 (May 17, 2017) (8 pages) Paper No: CND-16-1603; doi: 10.1115/1.4036346 History: Received December 05, 2016; Revised February 27, 2017

For the present study, setting Strouhal number as the control parameter, we perform numerical simulations for the flow over oscillating NACA-0012 airfoil at a Reynolds number of 1000. This study reveals that aerodynamic forces produced by oscillating airfoils are independent of the initial kinematic conditions suggesting the existence of limit cycle. Frequencies present in the oscillating lift force are composed of the fundamental harmonics and its odd harmonics. Using these numerical simulations, we analyze the shedding frequencies close to the excitation frequencies. Hence, considering it as a primary resonance case, we model the unsteady lift force with a modified van der Pol oscillator. Using the method of multiple scales and spectral analysis of the steady-state computational fluid dynamics (CFD) solutions, we estimate the frequencies and the damping terms in the reduced-order model (ROM). We show the applicability of this model to all planar motions of the airfoil; heaving, pitching, and flapping. With increasing the Strouhal number, the nonlinear damping terms for all types of motion approach similar magnitudes. Another important aspect in one of the proposed model is its ability to capture the time-averaged value of the aerodynamic lift force. We also notice that increase in the magnitude of the lift force is due to the effect of destabilizing linear damping parameter.

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References

Figures

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Fig. 1

Schematic of flow domain around the airfoil

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Fig. 2

Airfoil kinematics (solid lines: upstroke, dotted lines: downstroke), for clarity, only the mean and maximum-amplitude positions are shown here

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Fig. 3

Comparison of CL and CT for different initial positions of heaving airfoil for St = 0.10 in (a) and (c); and for St = 0.30 in (b) and (d)

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Fig. 4

Phase maps using CL and CT their time-derivatives for different initial positions and strokes of heaving airfoil at St = 0.10 in (a) and (c); and for St = 0.30 in (b) and (d)

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Fig. 5

(a) Unsteady lift and (b) amplitude-spectra for St = 0.30

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Fig. 6

Comparison of CFD solution and the proposed model for heaving airfoil at St = 0.45: (a) time histories (solid lines: CFD and circles: ROM) and (b) amplitude-spectrum (solid lines: CFD and dashed lines: ROM)

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Fig. 7

Comparison of CFD solution and the proposed model for pitching airfoil at St = 0.30 (a) time histories (solid lines: CFD and circles: ROM) (b) amplitude-spectrum (solid lines: CFD and dashed lines: ROM)

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Fig. 8

Comparison of CFD solution and the proposed model for flapping airfoil at St = 0.20 (a) time histories (solid lines: CFD and circles: ROM) (b) amplitude-spectrum (solid Lines: CFD and dashed lines: ROM)

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Fig. 9

Comparison of solutions from CFD and ROM for heaving airfoil (a) time histories (solid lines: CFD and circles: ROM) St = 0.20 (b) amplitude-spectrum (solid lines: CFD and dashed lines: ROM) St = 0.40

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Fig. 10

Plots of dynamic parameters for varying St

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Fig. 11

Comparison of CFD solution and the proposed model for pitching airfoil at St = 0.20 (a) time histories (triangles: CFD and solid line: ROM) (b) amplitude-spectrum (dashed line: CFD and solid lines: ROM)

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