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On the trajectory of leading-edge vortices under the influence of Coriolis acceleration

Published online by Cambridge University Press:  29 June 2016

Eric Limacher ,
Chris Morton  and
David Wood
Eric Limacher*
Affiliation:
Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, CanadaT2N1N4
Chris Morton
Affiliation:
Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, CanadaT2N1N4
David Wood
Affiliation:
Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, CanadaT2N1N4
*
†Email address for correspondence: [email protected]
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Abstract

Leading-edge vortices (LEVs) can form and remain attached to a rotating wing indefinitely, but the mechanisms of stable attachment are not well understood. Taking for granted that such stable structures do form, a practical question arises: what is the trajectory of the LEV core? Noting that span-wise flow exists within the LEV core, it is apparent that a mean streamline aligned with the axis of the stable LEV must exist. The present work uses the Navier–Stokes equations along such a steady, axial streamline in order to consider the accelerations that act in the streamline-normal direction to affect its local curvature. With some simplifying assumptions, a coupled system of ordinary differential equations is derived that describes the trajectory of an axial streamline through the vortex core. The model is compared to previous work, and is found to predict the trajectory of the LEV core well at span-wise locations inboard of the midspan. This result suggests that Coriolis acceleration is responsible for limiting the span-wise extent of a stable LEV by tilting it into the wake within several chord lengths from the centre of rotation. The downwash due to the tip vortex also appears to play a role, as the only significant differences between model-predicted LEV trajectories and previous results are in the plate-normal direction.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Vortex Flows: Vortex Flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 800 , 10 August 2016 , R1
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Copyright
© 2016 Cambridge University Press 

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References

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 numbers. J. Expl Biol. 207, 1063–1072.Google Scholar
Borazjani, I. & Daghooghi, M. 2013 The fish tail motion forms an attached leading edge vortex. Proc. R. Soc. Lond. B 280, 20122071.Google Scholar
Bross, M., Ozen, C. A. & Rockwell, D. 2013 Flow structure on a rotating wing: effect of steady incident flow. Phys. Fluids 25, 081901.CrossRefGoogle Scholar
Carr, Z. R., Devoria, A. C. & Ringuette, M. J. 2015 Aspect-ratio effects on rotating wings: circulation and forces. J. Fluid Mech. 767, 497–525.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, 626–630.CrossRefGoogle Scholar
Fay, J. A. 1994 Introduction to Fluid Mechanics. MIT.Google Scholar
Garmann, D. J. & Visbal, M. R. 2014 Dynamics of revolving wings for various aspect ratios. J. Fluid Mech. 748, 932–956.CrossRefGoogle Scholar
Gray, A., Abbena, E. & Salamon, S. 2006 Modern Differential Geometry of Curves and Surfaces with Mathematica® , 3rd edn. Chapman & Hall/CRC.Google Scholar
Harbig, R. R., Sheridan, J. & Thompson, M. C. 2013 Reynolds number and aspect ratio effects on the leading-edge vortex for rotating insect wing planforms. J. Fluid Mech. 717, 166–192.CrossRefGoogle Scholar
Jardin, T. & David, L. 2015 Coriolis effects enhance lift on revolving wings. Phys. Rev. E 91, 031001.Google Scholar
Kruyt, J. W., van Heijst, G. F., Altshuler, D. L. & Lentink, D. 2015 Power reduction and the radial limit of stall delay in revolving wings of different aspect ratio. J. R. Soc. Interface 12, 20150051.Google Scholar
Lee, S. J., Lee, E. J. & Sohn, M. H. 2014 Mechanism of autorotation flight of maple samaras (Acer palmatum). Exp. Fluids 55, 1718.CrossRefGoogle Scholar
Lentink, D. & Dickinson, M. H. 2009 Rotational accelerations stabilize leading edge vortices on revolving fly wings. J. Expl Biol. 212, 2705–2719.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, 1438–1440.Google Scholar
Limacher, E. & Rival, D. E. 2015 On the distribution of leading-edge vortex circulation in samara-like flight. J. Fluid Mech. 776, 316–333.Google Scholar
Maxworthy, T. 2007 The formation and maintenance of a leading-edge vortex during the forward motion of an animal wing. J. Fluid Mech. 587, 471–475.CrossRefGoogle Scholar
Murphy, D. W., Adhikari, D., Webster, D. R. & Yen, J. 2016 Underwater flight by the planktonic sea butterfly. J. Expl Biol. 219, 535–543.CrossRefGoogle ScholarPubMed
Phillips, N., Knowles, K. & Bomphrey, R. J. 2015 The effect of aspect ratio on the leading-edge vortex over an insect-like flapping wing. Bioinspir. Biomim. 10, 056020.Google Scholar
Salcedo, E., Treviño, C., Vargas, R. O. & Martínez-Suástegui, L. 2013 Stereoscopic particle image velocimetry measurements of the three-dimensional flow field of a descending autorotating mahogany seed (Swietenia macrophylla). J. Expl Biol. 216, 2017–2030.Google Scholar
Sarpkaya, T. 1971 On stationary and travelling vortex breakdowns. J. Fluid Mech. 45, 545–559.Google Scholar
Usherwood, J. R. & Ellington, C. P. 2002 The aerodynamics of revolving wings. I. Model hawkmoth wings. J. Expl Biol. 205, 1547–1564.Google Scholar
Wojcik, C. J. & Buchholz, J. H. J. 2014 Vorticity transport in the leading-edge vortex on a rotating blade. J. Fluid Mech. 743, 249–261.CrossRefGoogle Scholar
Wood, D. 2011 Small Wind Turbines: Analysis, Design and Application. Springer.Google Scholar
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