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A scaling for vortex formation on swept and unswept pitching wings

  • Kyohei Onoue (a1) and Kenneth S. Breuer (a1)

We examine the dynamics of the leading-edge vortex (LEV) on a rapidly pitching plate with the aim of elucidating the underlying flow physics that dictates the stability and circulation of the LEV. A wide variety of flow conditions is considered in the present study by systematically varying the leading-edge sweep angle ( $\unicode[STIX]{x1D6EC}=0^{\circ }$ , $11.3^{\circ }$ , $16.7^{\circ }$ ) and the reduced frequency ( $f^{\ast }=0.064{-}0.151$ ), while keeping the pitching amplitude and the Reynolds number fixed. Tomographic particle image velocimetry is used to characterise the three-dimensional fluid motion inside the vortex core and its relation to the LEV stability and growth. A series of control volume analyses are performed to quantify the relative importance of the vorticity transport phenomena taking place inside the LEV to the overall vortex development. We show that, near the wing apex where tip effects can be neglected, the vortex develops in a nominally two-dimensional manner, despite the presence of inherently three-dimensional vortex dynamics such as vortex stretching and compression. Furthermore, we demonstrate that the vortex formation time and circulation growth are well-described by the principles of optimal vortex formation number, and that the occurrence of vortex shedding is dictated by the relative energetics of the feeding shear layer and the resulting vortex.

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Adrian, R. J., Christensen, K. T. & Liu, Z. C 2000 Analysis and interpretation of instantaneous turbulent velocity fields. Exp. Fluids 29, 275290.
Akkala, J. M., Eslam Panah, A. & Buchholz, J. H. J. 2015 Vortex dynamics and performance of flexible and rigid plunging airfoils. J. Fluids Struct. 54, 103121.
Baik, Y. S., Aono, H., Rausch, J. M., Bernal, L. P., Shyy, W. & Ol, M. V.2010 Experimental study of a rapidly pitched flat plate at low Reynolds number. AIAA Paper 2010-4462.
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.
Beem, H., Rival, D. E. & Triantafyllou, M. S. 2012 On the stabilization of leading-edge vortices with spanwise flow. Exp. Fluids 51, 511517.
Bisplinghoff, R. L., Ashley, H. & Halfman, R. L. 1957 Aeroelasticity. Addison-Wesley.
Dabiri, J. O. 2009 Optimal vortex formation as a unifying principle in biological propulsion. Annu. Rev. Fluid Mech. 41, 1733.
Dickinson, M. H. & Gotz, K. G. 1993 Unsteady aerodynamic performance of model wings at low Reynolds numbers. J. Exp. Biol. 174, 4564.
Eslam Panah, A., Akkala, J. M. & Buchholz, J. H. J. 2015 Vorticity transport and the leading-edge vortex of a plunging airfoil. Exp. Fluids 56, 160.
Gharib, M. R., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121140.
Graftieaux, L., Michard, M. & Grosjean, N. 2001 Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows. Meas. Sci. Technol. 12, 14221429.
Granlund, K. O., Ol, M. V. & Bernal, L. P. 2013 Unsteady pitching flat plates. J. Fluid Mech. 733, R5.
Harbin, 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, 166192.
Hartloper, C. & Rival, D. E. 2013 Vortex development on pitching plates with lunate and truncate planforms. J. Fluid Mech. 732, 332344.
Jardin, T., Farcy, A. & David, L. 2012 Three-dimensional effects in hovering flapping flight. J. Fluid Mech. 702, 102125.
Jones, A. R., Pitt-Ford, C. W. & Babinsky, H. 2012 Three-dimensional effects on sliding and waving wings. J. Aircraft 48 (2), 633643.
Khalak, A. & Williamson, C. H. K. 1999 Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. J. Fluids Struct. 13, 813851.
Kim, D. & Gharib, M. 2011 Flexibility effects on vortex formation of translating plates. J. Fluid Mech. 677, 255271.
Kim, D., Strom, B., Mandre, S. & Breuer, K. 2017 Energy harvesting performance and flow structure of an oscillating hydrofoil. J. Fluids Struct. 70, 314326.
Kinzel, M., Wolf, M., Holzner, M., Lüthi, B., Tropea, C. & Kinzelbach, W. 2011 Simultaneous two-scale 3D-PTV measurements in turbulence under the influence of system rotation. Exp. Fluids 51, 7582.
Kriegseis, J., Kinzel, M. & Rival, D. E. 2013 On the persistence of memory: do initial conditions impact vortex formation? J. Fluid Mech. 736, 91106.
Kundu, P. K. & Cohen, I. M. 2008 Fluid Mechanics, 4th edn. Elsevier.
Lauder, G. V. 2015 Fish locomotion: recent advances and new directions. Annu. Rev. Mar. Sci. 7, 521545.
Lentink, D. & Dickinson, M. H. 2009 Biofluiddynamic scaling of flapping, spinning, and translating fins and wings. J. Exp. Biol. 212, 26912704.
Lisoski, D. L. A.1993 Nominally 2-dimensional flow about a normal flat plate. PhD thesis, California Institute of Technology.
O’Farrell, C. & Dabiri, J. O. 2014 Pinch-off of non-axisymmetric vortex ringss. J. Fluid Mech. 740, 6196.
Ol, M. V., Bernal, L., Kang, C.-K. & Shyy, W. 2009 Shallow and deep dynamic stall for flapping low Reynolds number airfoil. Exp. Fluids 46, 883901.
Onoue, K. & Breuer, K. 2016 Vortex formation and shedding from a cyber-physical pitching plate. J. Fluid Mech. 793, 229247.
Onoue, K., Song, A., Strom, B. & Breuer, K. 2015 Large amplitude flow-induced oscillations and energy harvesting using a cyber-physical pitching plate. J. Fluids Struct. 55, 262275.
Pozrikidis, C. 2011 Introduction to Theoretical and Computational Fluid Dynamics, 2nd edn. Oxford University Press.
Ringuette, M. J., Milano, M. & Gharib, M. 2007 Role of tip vortex in the force generation of low-aspect-ratio normal flat plates. J. Fluid Mech. 581, 453468.
Rival, D. E., Kriegseis, J., Schaub, P., Widmann, A. & Tropea, C. 2014 Characteristic length scales for vortex detachment on plunging profiles with varying leading-edge geometry. Exp. Fluids 55 (1), 18.
Rival, D. E., Prangemeier, T. & Tropea, C. 2009 The influence of airfoil kinematics on the formation of leading-edge vortices in bio-inspired flight. Exp. Fluids 46, 823833.
Sattari, P., Rival, D. E., Martinuzzi, R. J. & Tropea, C. 2012 Growth and separation of a start-up vortex from a two dimensional shear layer. Phys. Fluids 24, 107102.
Slaouti, A. & Gerrard, J. H. 1981 An experimental investigation of the end effects on the wake of a circular cylinder towed through water at low Reynolds numbers. J. Fluid Mech. 112, 297314.
Taira, K. & Colonius, T. 2009 Three-dimensional flows around low-aspect-ratio flat-plate wings at low Reynolds numbers. J. Fluid Mech. 587, 187207.
Visbal, M. R. & Garmann, D. J.2017 Numerical investigation of spanwise end effects on dynamic stall of a pitching NACA 0012 wing. AIAA Paper 2017-1481.
Wang, Z. J 2005 Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183210.
Wojcik, C. J. & Buchholz, J. H. J. 2014 Vorticity transport in the leading-edge vortex on a rotating blade. J. Fluid Mech. 743, 249261.
Wong, J. G. & Rival, D. E. 2015 Determining the relative stability of leading-edge vortices on nominally two-dimensional flapping profiles. J. Fluid Mech. 766, 611625.
Yilmaz, T. O. & Rockwell, D. 2012 Flow structures on finite-span wings due to pitch-up motion. J. Fluid Mech. 691, 518545.
Yu, H. T. & Bernal, L. P.2013 Effect of pivot point on aerodynamic force and vortical structure of pitching flat plate wings. AIAA Paper 2013-0792.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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