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On the aerodynamic forces on heaving and pitching airfoils at low Reynolds number

  • M. Moriche (a1), O. Flores (a1) and M. García-Villalba (a1)

The influence that the kinematics of pitching and heaving 2D airfoils has on the aerodynamic forces is investigated using direct numerical simulations and a force decomposition algorithm. Large-amplitude motions are considered (of the order of one chord), with moderate Reynolds numbers and reduced frequencies of order $O(1)$ , varying the mean pitch angle and the phase shift between the pitching and heaving motions. Our results show that the surface vorticity contribution (viscous effect) to the aerodynamic force is negligible compared with the contributions from the body motion (fluid inertia) and the vorticity within the flow (circulation). For the range of parameters considered here, the latter tends to be instantaneously oriented in the direction normal to the chord of the airfoil. Based on the results discussed in this paper, a reduced-order model for the instantaneous aerodynamic force is proposed, taking advantage of the force decomposition and the chord-normal orientation of the contribution from vorticity within the flow to the total aerodynamic force. The predictions of the proposed model are compared with those of a similar model from the literature, showing a noticeable improvement in the prediction of the mean thrust, and a smaller improvement in the prediction of the mean lift and the instantaneous force coefficients.

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Andersen, A., Pesavento, U. & Wang, Z. J. 2005 Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid Mech. 541, 6590.
Anderson, J. M., Streitlien, K., Barret, K. S. & Triantafyllou, M. S. 1998 Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 4172.
Ansari, R., Zbikowski, R. & Knowles, K. 2006 Aerodynamic modelling of insect-like flapping flight for micro air vehicles. Prog. Aerosp. Sci. 42, 129172.
Ashraf, M. A., Young, J. & Lai, J. C. S. 2011 Reynolds number, thickness and camber effects on flapping airfoil propulsion. J. Fluids Struct. 27, 145160.
Baik, Y. S., Bernal, L. P., Granlund, K. & Ol, M. V. 2012 Unsteady force generation and vortex dynamics of pitching and plunging aerofoils. J. Fluid Mech. 709, 3768.
Carr, L. W. 1988 Progress in analysis and prediction of dynamic stall. J. Aircraft 25 (1), 617.
Chang, C.-C. 1992 Potential flow and forces for incompressible viscous flow. Proc. R. Soc. Lond. A 437, 517525.
Choi, J., Colonius, T. & Williams, D. R. 2015 Surging and plunging oscillations of an airfoil at low Reynolds number. J. Fluid Mech. 763, 237253.
Dickinson, M. H., Lehmann, F.-O. & Sane, S. P. 1999 Wing rotation and the aerodynamic basis of insect flight. Science 284 (5422), 19541960.
von Ellenrieder, K. D., Parker, K. & Soria, J. 2008 Fluid mechanics of flapping wings. Exp. Therm. Fluid Sci. 32, 15781589.
Ellington, C. P., Van Den Berg, C., Willmott, A. P. & Thomas, A. L. R. 1996 Leading-edge vortices in insect flight. Nature 384, 626630.
Fung, Y. C. 2002 An Introduction to the Theory of Aeroelasticity. Courier Corporation.
Jones, K. D. & Platzer, M. F.1997 Numerical computation of flapping-wing propulsion and power extraction. AIAA Paper AIAA-97-0826.
Lewin, G. C. & Haj-Hariri, H. 2003 Modelling thrust generation of a two-dimensional heaving airfoil in a viscous flow. J. Fluid Mech. 492, 339362.
Long, L. N. & Fritz, T. E. 2004 Object-oriented unsteady vortex lattice method for flapping flight. J. Aircraft 41 (6), 12751290.
Lua, K. B., Lim, T. T., Yeo, K. S. & Oo, G. Y. 2007 Wake-structure formation of a heaving two-dimensional elliptic airfoil. AIAA J. 45, 15711583.
Martín-Alcántara, A., Fernandez-Feria, R. & Sanmiguel-Rojas, E. 2015 Vortex flow structures and interactions for the optimum thrust efficiency of a heaving airfoil at different mean angles of attack. Phys. Fluids 27 (7), 073602.
Mittal, R., Dong, H., Bozkurttas, M., Najjar, F. M., Vargas, A. & Von Loebbecke, A. 2008 A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys. 227 (10), 48254852.
Moriche, M.2017 A numerical study on the aerodynamic forces and the wake stability of flapping flight at low Reynolds number. PhD thesis, Universidad Carlos III de Madrid.
Moriche, M., Flores, O. & García-Villalba, M. 2015 Generation of thrust and lift with airfoils in plunging and pitching motion. J. Phys.: Conf. Ser. 574, 012163.
Moriche, M., Flores, O. & García-Villalba, M. 2016 Three-dimensional instabilities in the wake of a flapping wing at low Reynolds number. Intl J. Heat Fluid Flow 62A, 4455.
Noca, F., Shiels, D. & Jeon, D. 1999 A comparison of methods for evaluating time-dependent fluid dynamic forces on bodies, using only velocity fields and their derivatives. J. Fluids Struct. 13 (5), 551578.
Pesavento, U. & Wang, Z. J. 2004 Falling paper: Navier–Stokes solutions, model of fluid forces, and center of mass elevation. Phys. Rev. Lett. 93 (14), 144501.
Platzer, M. F., Jones, K. D., Young, J. & Lai, J. C. S. 2008 Flapping-wing aerodynamics: progress and challenges. AIAA J. 46 (9), 21362149.
Ramamurti, R. & Sandberg, W. 2001 Simulation of flow about flapping airfoils using finite element incompressible flow solver. AIAA J. 39 (2), 253260.
Read, D. A., Hover, F. S. & Triantafyllou, M. S. 2003 Forces on oscillating foils for propulsion and maneuvering. J. Fluids Struct. 17 (1), 163183.
Rozhdestvensky, K. V. & Ryzhov, V. A. 2003 Aerohydrodynamics of flapping-wing propulsors. Prog. Aerosp. Sci. 39, 585633.
Sedov, L. I. 1965 Two-dimensional Problems in Hydrodynamics and Aerodynamics. Interscience Publishers.
Shyy, W., Aono, H., Chimakurthi, S. K., Trizila, P., Kang, C.-K., Cesnik, C. E. S. & Liu, H. 2010 Recent progress in flapping wing aerodynamics and aeroelasticity. Prog. Aerosp. Sci. 46, 284327.
Shyy, W., Aono, H., Kang, C.-K. & Liu, H. 2013 An Introduction to Flapping Wing Aerodynamics. Cambridge University Press.
Taha, H. E., Hajj, M. R. & Beran, P. S. 2014 State-space representation of the unsteady aerodynamics of flapping flight. Aerosp. Sci. Technol. 34, 111.
Taha, H. E., Hajj, M. R. & Nayfeh, A. H. 2012 Flight dynamics and control of flapping-wing MAVs: a review. Nonlinear Dyn. 70 (2), 907939.
Theodorsen, T.1949 General theory of aerodynamic instability and the mechanism of flutter. NACA Tech. Rep.
Uhlmann, M. 2005 An immersed boundary method with direct forcing for the simulation of particulate flows. J. Comput. Phys. 209 (2), 448476.
Wagner, H. 1925 Über die Entstehung des dynamischen Auftriebes von Tragflügeln. Z. Angew. Math. Mech. 5 (1), 1735.
Wang, S., Zhang, X., He, G. & Liu, T. 2015 Evaluation of lift formulas applied to low-Reynolds-number unsteady flows. AIAA J. 53 (1), 161175.
Wang, Z. J. 2000 Vortex shedding and frequency selection in flapping flight. J. Fluid Mech. 410, 323341.
Wei, Z. & Zheng, Z. C. 2014 Mechanisms of wake deflection angle change behind a heaving airfoil. J. Fluids Struct. 48, 113.
Widmann, A. & Tropea, C. 2015 Parameters influencing vortex growth and detachment on unsteady aerodynamic profiles. J. Fluid Mech. 773, 432459.
Wu, J. Z., Pan, Z. L. & Lu, X. Y. 2005 Unsteady fluid-dynamic force solely in terms of control-surface integral. Phys. Fluids 17 (9), 098102.
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Moriche et al. supplementary movie
Contours of spanwise vorticity (left), thrust density (center) and lift density (right) of case B090. Same as Figure 6 in the full paper.

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