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Unsteady lift for the Wagner problem in the presence of additional leading/trailing edge vortices

Published online by Cambridge University Press:  16 March 2015

Juan Li
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China
Zi-Niu Wu*
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China
Email address for correspondence:


This study amends the inviscid Wagner lift model for starting flow at relatively large angles of attack to account for the influence of additional leading edge and trailing edge vortices. Two methods are provided for starting flow of a flat plate. The first method is a modified Wagner function, which assumes a planar trajectory of the trailing edge vortex sheet accounting for a temporal offset from the original Wagner function given release of leading edge vortices and a concentrated starting point vortex at the initiation of motion. The second method idealizes the trailing edge sheet as a series of discrete vortices released sequentially. The models presented are shown to be in good agreement with high-fidelity simulations. Through the present theory, a vortex force line map is generated, which clearly indicates lift enhancing and reducing directions and, when coupled with streamlines, allows one to qualitatively interpret the effect of the sign and position of vortices on the lift and to identify the origins of lift oscillations and peaks. It is concluded that leading edge vortices close to the leading edge elevate the Wagner lift curve while a strong leading edge vortex convected to the trailing edge is detrimental to lift production by inducing a strong trailing edge vortex moving in the lift reducing direction. The vortex force line map can be employed to understand the effect of the different vortices in other situations and may be used to improve vortex control to enhance or reduce the lift.

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Anderson, J. 2010 Fundamentals of Aerodynamics (McGraw-Hill Series in Aeronautical and Aerospace Engineering). McGraw-Hill Education.Google Scholar
Ansari, S., Zbikowski, R. & Knowles, K. 2006a A nonlinear unsteady aerodynamic model for insect-like flapping hover. Part I: methodology and analysis. J. Aerosp. Engng 220, 6183.Google Scholar
Ansari, S., Zbikowski, R. & Knowles, K. 2006b Nonlinear unsteady aerodynamic model for insect-like flapping hover. Part 2: implementation and validation. J. Aerosp. Engng 220, 169186.Google Scholar
Bai, C. Y., Li, J. & Wu, Z. N. 2014 Generalized Kutta–Joukowski theorem for multi-vortex and multi-airfoil flow with vortex production – a general model. Chin. J. Aeronaut. 27, 10371050.CrossRefGoogle Scholar
Bai, C. Y. & Wu, Z. N. 2014 Generalized Kutta–Joukowski theorem for multi-vortices and multi-airfoil flow (lumped vortex model). Chin. J. Aeronaut. 27, 3439.CrossRefGoogle Scholar
Birch, J. M. & Dickinson, M. H. 2001 Spanwise flow and the attachment of the leading-edge vortex on insect wings. Nature 412, 729733.CrossRefGoogle ScholarPubMed
Birch, J. M., Dickson, W. B. & Dickinson, M. H. 2004 Force production and flow structure of the leading edge vortex on flapping wings at high and low Reynolds number. J. Expl Biol. 207, 10631072.CrossRefGoogle Scholar
Bomphrey, R. J., Lawson, N. J., Harding, N. J., Taylor, G. K. & Thomas, A. L. R. 2005 The aerodynamics of Manduca sexta: digital particle image velocimetry analysis of the leading-edge vortex. J. Expl Biol. 208, 10791094.CrossRefGoogle Scholar
Bomphrey, R. J., Taylor, G. K. & Thomas, A. L. R. 2009 Smoke visualization of free-flying bumble bees indicates independent leading-edge vortices on each wing pair. Exp. Fluids 46, 811821.CrossRefGoogle Scholar
Brown, C. E. & Michael, W. H. 1954 Effect of leading edge separation on the lift of a delta wing. J. Aeronaut. Sci. 21, 690694.CrossRefGoogle Scholar
Chow, C. Y. & Huang, M. K. 1982 The initial lift and drag of an impulsively started aerofoil of finite thickness. J. Fluid Mech. 118, 393409.CrossRefGoogle Scholar
Chow, C. Y., Huang, M. K. & Yan, C. Z. 1985 Unsteady flow about a Joukowski airfoil in the presence of moving vortices. AIAA J. 23, 657658.CrossRefGoogle Scholar
Clements, R. R. 1973 An inviscid model of two-dimensional vortex shedding. J. Fluid Mech. 57 (2), 321336.CrossRefGoogle Scholar
Crighton, D. G. 1985 The Kutta condition in unsteady flow. Annu. Rev. Fluid Mech. 17, 411445.CrossRefGoogle Scholar
Dickinson, M. H. & Gotz, K. G. 1993 Unsteady aerodynamic performance of model wings at low Reynolds numbers. J. Expl Biol. 174, 4564.Google Scholar
Eames, I., Landeryou, M. & Lore, J. B. 2008 Inviscid coupling between point symmetric bodies and singular distributions of vorticity. J. Fluid Mech. 589, 3356.Google Scholar
Ellington, C. P., van den Berg, C., Willmott, A. P. & Thomas, A. L. R. 1996 Leading-edge vortices in insect flight. Nature 384, 626630.CrossRefGoogle Scholar
Fung, Y. C. 2002 An Introduction to the Theory of Aeroelasticity. Courier Dover.Google Scholar
Graham, J. M. R. 1980 The forces on sharp-edged cylinders in oscillatory flow at low Keulegan–Carpenter numbers. J. Fluid Mech. 97, 331346.CrossRefGoogle Scholar
Graham, J. M. R. 1983 The initial lift on an aerofoil in starting flow. J. Fluid Mech. 133, 413425.CrossRefGoogle Scholar
Howe, M. S. 1995 On the force and moment on a body in an incompressible fluid, with application to rigid bodies and bubbles at high Reynolds numbers. Q. J. Mech. Appl. Maths 48, 401425.CrossRefGoogle Scholar
Huang, M. K. & Chow, C. Y. 1982 Trapping of a free vortex by Joukowski airfoils. AIAA J. 20, 292298.CrossRefGoogle Scholar
Issa, R. I. 1985 Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62 (1), 4065.CrossRefGoogle Scholar
Johansson, L. C., Engel, S., Kelber, A., Heerenbrink, M. K. & Hedenstrom, A. 2013 Multiple leading edge vortices of unexpected strength in freely flying hawkmoth. Nat. Sci. Rep. 3, 3264.CrossRefGoogle ScholarPubMed
Kanso, E. & Oskouei, B. G. 2008 Stability of a coupled body–vortex system. J. Fluid Mech. 600, 7794.CrossRefGoogle Scholar
Knowles, K., Wilkins, P., Ansari, S. & Zbikowski, R. 2007 Integrated computational and experimental studies of flapping-wing micro air vehicle aerodynamics. In Proceedings of the 3rd International Symposium on Integrating CFD and Experiments in Aerodynamics, 20–21 June 2007. U.S. Air Force Academy, Tech. Rep. No. A925515.Google Scholar
Lee, F. J. & Smith, C. A. 1991 Effect of vortex core distortion on blade–vortex interaction. AIAA J. 29, 13551362.CrossRefGoogle Scholar
Lentink, D. & Dickinson, M. H. 2009 Rotational accelerations stabilize leading edge vortices on revolving fly wings. J. Expl Biol. 212, 27052719.CrossRefGoogle ScholarPubMed
Lentink, D., Dickson, W. B., van Leeuwen, J. L. & Dickinson, M. H. 2009 Leading-edge vortices elevate lift of autorotating plant seeds. Science 324, 14381440.CrossRefGoogle ScholarPubMed
Li, J., Bai, C. Y. & Wu, Z. N. 2015 A two-dimensional multibody integral approach for forces in inviscid flow with free vortices and vortex production. Trans. ASME: J. Fluids Engng 137, 021205; Paper No. FE-13-1671.Google Scholar
Lu, Y., Shen, G. X. & Lai, G. J. 2006 Dual leading-edge vortices on flapping wings. J. Expl Biol. 209, 50055016.CrossRefGoogle ScholarPubMed
Michelin, S. & Llewellyn Smith, S. G. 2009 Linear stability analysis of coupled parallel flexible plates in an axial flow. J. Fluids Struct. 25 (7), 11361157.CrossRefGoogle Scholar
Michelin, S. & Llewellyn Smith, S. G. L. 2010 Falling cards and flapping flags: understanding fluid–solid interactions using an unsteady point vortex model. Theor. Comput. Fluid Dyn. 24 (1–4), 195200.CrossRefGoogle Scholar
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics. Macmillan Education.CrossRefGoogle Scholar
Minotti, F. O. 2002 Unsteady two-dimensional theory of a flapping wing. Phys. Rev. E 66, 051907.CrossRefGoogle ScholarPubMed
Muijres, F. T., Johansson, L. C., Barfield, R., Wolf, M., Spedding, G. R. & Hedenstrom, A. 2008 Leading-edge vortex improves lift in slow-flying bats. Science 319, 12501253.CrossRefGoogle ScholarPubMed
Muijres, F. T., Johansson, L. C. & Hedenstrom, A. 2012 Leading edge vortex in a slow-flying passerine. Biol. Lett. 8, 554557.CrossRefGoogle Scholar
Pitt Ford, C. W. & Babinsky, H. 2013 Lift and the leading-edge vortex. J. Fluid Mech. 720, 280313.CrossRefGoogle Scholar
Polhamus, E. C.1966 A concept of the vortex lift of sharp-edge delta wings based on a leading edge sunction analogy, NASA Tech. Rep. TN-D3767.Google Scholar
Pullin, D. I. 1978 The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88 (3), 401430.CrossRefGoogle Scholar
Pullin, D. I. & Wang, Z. J. 2004 Unsteady forces on an accelerating plate and application to hovering insect flight. J. Fluid Mech. 509, 121.CrossRefGoogle Scholar
Ramesh, K., Gopalarathnam, A., Edwards, J. R., OL, M. V. & Granlund, K.2011 Theoretical, computational and experimental studies of a flat plate undergoing high-amplitude pitching motion. AIAA Paper 2011-217.Google Scholar
Rossow, V. J. 1994 Aerodynamics of airfoils with vortex trapped by two spanwise fences. J. Aircraft 31, 146153.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Saffman, P. G. & Sheffield, J. S. 1977 Flow over a wing with an attached vortex. Stud. Appl. Maths 57, 107117.CrossRefGoogle Scholar
Sakajo, T. 2012 Force-enhancing vortex equilibria for two parallel plates in uniform flow. Proc. R. Soc. Lond. A 468, 11751195.CrossRefGoogle Scholar
Streitlien, K. & Triantafyllou, M. S. 1995 Force and moment on a Joukowski profile in the presence of point vortices. AIAA J. 33, 603610.CrossRefGoogle Scholar
Wagner, H. 1925 Über die Entstehung des dynamischen Auftriebs von Tragflügeln. Z. Angew. Math. Mech. 5, 1735.CrossRefGoogle Scholar
Walker, P. B.1931 Experiments on the growth of circulation about a wing. Tech. Rep. 1402. Aeronautical Research Committee.Google Scholar
Wang, X. X. & Wu, Z. N. 2010 Stroke-averaged lift forces due to vortex rings and their mutual interactions for a flapping flight model. J. Fluid Mech. 654, 453472.CrossRefGoogle Scholar
Wang, X. X. & Wu, Z. N. 2012 Lift force reduction due to body image of vortex for a hovering flight model. J. Fluid Mech. 709, 648658.CrossRefGoogle Scholar
Wojcik, C. J. & Buchholz, J. H. J. 2014 Vorticity transport in the leading-edge vortex on a rotating blade. J. Fluid Mech. 743, 249261.CrossRefGoogle Scholar
Wu, J. C. 1981 Theory for aerodynamic force and moment in viscous flows. AIAA J. 19, 432441.CrossRefGoogle Scholar
Wu, C. T., Yang, F. L. & Young, D. L. 2012 Generalized two-dimensional Lagally theorem with free vortices and its application to fluid–body interaction problems. J. Fluid Mech. 698, 7392.CrossRefGoogle Scholar
Xia, X. & Mohseni, K. 2013 Lift evaluation of a two-dimensional pitching flat plate. Phys. Fluids 25, 091901.CrossRefGoogle Scholar
Yu, Y., Tong, B. & Ma, H. 2003 Analytic approach to theoretical modeling of highly unsteady viscous flow excited by wing flapping in small insects. Acta Mechanica Sin. 19, 508516.Google Scholar
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