Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T09:28:32.509Z Has data issue: false hasContentIssue false
  • English
  • Français

Influence of a wall on the three-dimensional dynamics of a vortex pair

Published online by Cambridge University Press:  20 March 2017

Daniel J. Asselin Open the ORCID record for Daniel J. Asselin [Opens in a new window]  and
C. H. K. Williamson
Daniel J. Asselin*
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853-7501, USA
C. H. K. Williamson
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853-7501, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we are interested in perturbed vortices under the influence of a wall or ground plane. Such flows have relevance to aircraft wakes in ground effect, to ship hull junction flows, to fundamental studies of turbulent structures close to a ground plane and to vortex generator flows, among others. In particular, we study the vortex dynamics of a descending vortex pair, which is unstable to a long-wave instability (Crow, AIAA J., vol. 8 (12), 1970, pp. 2172–2179), as it interacts with a horizontal ground plane. Flow separation on the wall generates opposite-sign secondary vortices which in turn induce the ‘rebound’ effect, whereby the primary vortices rise up away from the wall. Even small perturbations in the vortices can cause significant topological changes in the flow, ultimately generating an array of vortex rings which rise up from the wall in a three-dimensional ‘rebound’ effect. The resulting vortex dynamics is almost unrecognizable when compared with the classical Crow instability. If the vortices are generated below a critical height over a horizontal ground plane, the long-wave instability is inhibited by the wall. We then observe two modes of vortex–wall interaction. For small initial heights, the primary vortices are close together, enabling the secondary vortices to interact with each other, forming vertically oriented vortex rings in what we call a ‘vertical rings mode’. In the ‘horizontal rings mode’, for larger initial heights, the Crow instability develops further before wall interaction; the peak locations are farther apart and the troughs closer together upon reaching the wall. The proximity of the troughs to each other and the wall increases vorticity cancellation, leading to a strong axial pressure gradient and axial flow. Ultimately, we find a series of small horizontal vortex rings which ‘rebound’ from the wall. Both modes comprise two small vortex rings in each instability wavelength, distinct from Crow instability vortex rings, only one of which is formed per wavelength. The phenomena observed here are not limited to the above perturbed vortex pairs. For example, remarkably similar phenomena are found where vortex rings impinge obliquely with a wall.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex instability Vortex Flows: Vortex interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 817 , 25 April 2017 , pp. 339 - 373
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baker, G. R., Saffman, P. G. & Sheffield, J. S. 1976 Structure of a linear array of hollow vortices of finite cross-section. J. Fluid Mech. 74, 469–476.Google Scholar
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163.Google Scholar
Cantwell, B. J. 1981 Organized motion in turbulent flow. Annu. Rev. Fluid Mech. 13, 457–515.Google Scholar
Chu, C. C. & Falco, R. E. 1988 Vortex ring/viscous wall layer interaction model of the turbulence production process near walls. Exp. Fluids 6, 305–315.CrossRefGoogle Scholar
Crouch, J. 2005 Airplane trailing vortices and their control. C. R. Phys. 6, 487–499.Google Scholar
Crouch, J. D. 1997 Instability and transient growth for two trailing-vortex pairs. J. Fluid Mech. 350, 311–330.Google Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8 (12), 2172–2179.Google Scholar
Dee, F. W. & Nicholas, O. P.1968 Flight measurements of wing-tip vortex motion near the ground. Tech. Rep. 1065. British Aeronautical Research Council.Google Scholar
Duponcheel, M., Cottin, C., Daeninck, G., Leweke, T. & Winckelmans, G. 2007 Experimental and numerical study of counter-rotating vortex pair dynamics in ground effect. In 18ème Congrès Français de Mécanique, Association Française de Mécanique.Google Scholar
Fabre, D. & Jacquin, L. 2000 Stability of a four-vortex aircraft wake model. Phys. Fluids 12 (10), 2438–2443.Google Scholar
Georges, L., Geuzaine, P., Duponcheel, M., Bricteux, L., Lonfils, T. & Winckelmans, G.2006 LES of two-vortex system in ground effect (longitudinally uniform wakes). Tech. Rep. AST4-CT-2005-012238. 6th FAR Wake – Framework Programme for Research and Technological Development.Google Scholar
Gupta, G.2003 Generation and evolution of a viscous vortex pair. Master’s thesis, Cornell University.Google Scholar
Harris, D. M. & Williamson, C. H. K. 2012 Instability of secondary vortices generated by a vortex pair in ground effect. J. Fluid Mech. 700, 148–186.Google Scholar
Harvey, J. K. & Perry, F. J. 1971 Flowfield produced by trailing vortices in the vicinity of the ground. AIAA J. 9 (8), 1659–1660.Google Scholar
Kida, S. & Takaoka, M. 1994 Vortex reconnection. Annu. Rev. Fluid Mech. 26, 169–189.Google Scholar
Klein, R., Majda, A. J. & Damodaran, K. 1995 Simplified equations for the interaction of nearly parallel vortex filaments. J. Fluid Mech. 288, 201–248.Google Scholar
Kramer, W., Clercx, H. J. H. & van Heijst, G. J. F. 2007 Vorticity dynamics of a dipole colliding with a no-slip wall. Phys. Fluids 19, 126603.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Leweke, T., LeDizès, S. & Williamson, C. H. K. 2016 Dynamics and instabilities of vortex pairs. Annu. Rev. Fluid Mech. 48, 1–35.Google Scholar
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85–119.Google Scholar
Leweke, T. & Williamson, C. H. K. 2011 Experiments on long-wavelength instability and reconnection of a vortex pair. Phys. Fluids 23, 024101.Google Scholar
Lim, T. T. 1989 An experimental study of a vortex ring interacting with an inclined wall. Exp. Fluids 7, 453–463.Google Scholar
Luton, J. A. & Ragab, S. A. 1997 The three-dimensional interaction of a vortex pair with a wall. Phys. Fluids 9 (10), 2967–2980.CrossRefGoogle Scholar
Melander, M. V.1988 Close interactions of 3D vortex tubes. Tech. Rep. Center for Turbulence Research.Google Scholar
Melander, M. V. & Hussain, F. 1988 Cut-and-connect of two antiparallel vortex tubes. In Center for Turbulence Research Proceedings of the Summer Program, pp. 257–286. Stanford University Center for Turbulence Research.Google Scholar
Moet, H.2003 Simulation numérique du comportement des tourbillons de sillage dans l’atmosphère. PhD thesis, Institut National Polytechnique de Toulouse.Google Scholar
Moet, H., Laporte, F., Chevalier, G. & Poinsot, T. 2005 Wave propagation in vortices and vortex bursting. Phys. Fluids 17, 054109.Google Scholar
Orlandi, P. 1990 Vortex dipole rebound from a wall. Phys. Fluids 2 (8), 1429–1436.Google Scholar
Ortega, J. M., Bristol, R. L. & Savaş, Ö. 2003 Experimental study of the instability of unequal-strength counter-rotating vortex pairs. J. Fluid Mech. 474, 35–84.Google Scholar
Panton, R. L. 2001 Overview of the self-sustaining mechanisms of wall turbulence. Prog. Aerosp. Sci. 37, 341–383.CrossRefGoogle Scholar
Peace, A. J. & Riley, N. 1983 A viscous vortex pair in ground effect. J. Fluid Mech. 129, 409–426.Google Scholar
Pierrehumbert, R. T. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57 (17), 2157–2159.CrossRefGoogle Scholar
Quackenbush, T. R., Bilanin, A. J. & McKillip, R. M. 1996 Vortex wake control via smart structures technology. In Proceedings of SPIE 2721, Smart Structures and Materials 1996: Industrial and Commercial Applications of Smart Structures Technologies (ed. Robert Crowe, C.), pp. 78–92. The International Society for Optical Engineering.Google Scholar
Rennich, S. C. & Lele, S. K. 1999 Method for accelerating the destruction of aircraft wake vortices. J. Aircraft 36 (2), 398–404.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639.Google Scholar
Saffman, P. G. 1989 A model of vortex reconnection. J. Fluid Mech. 212, 395–402.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Shelley, M. J., Meiron, D. I. & Orszag, S. A. 1993 Dynamical aspects of vortex reconnection of perturbed anti-parallel vortex tubes. J. Fluid Mech. 246, 613–652.Google Scholar
Spalart, P. R. 1998 Airplane trailing vortices. Annu. Rev. Fluid Mech. 30, 107–138.Google Scholar
Stephan, A., Holzäpfel, F. & Misaka, T. 2013 Aircraft wake-vortex decay in ground proximity: physical mechanisms and artificial enhancement. J. Aircraft 50 (4), 1250–1260.Google Scholar
Swearingen, J. D., Crouch, J. D. & Handler, R. A. 1995 Dynamics and stability of a vortex ring impacting a solid boundary. J. Fluid Mech. 297, 1–28.Google Scholar
Thielicke, W.2014 The flapping flight of birds – analysis and application. PhD thesis, Rijksuniversiteit Groningen.Google Scholar
Thielicke, W. & Stamhuis, E. J. 2014a PIVlab – towards user-friendly, affordable and accurate digital particle image velocimetry in MATLAB. J. Open Res. Softw. 2 (1), e30.Google Scholar
Thielicke, W. & Stamhuis, E. J.2014b PIVlab – time-resolved digital particle image velocimetry tool for MATLAB (version: 1.40).Google Scholar
Verzicco, R. & Orlandi, P. 1994 Normal and oblique collisions of a vortex ring with a wall. Meccanica 29, 383–391.Google Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2 (1), 76–80.Google Scholar
Walker, J. D. A., Smith, C. R., Cerra, A. W. & Doligalski, T. L. 1987 The impact of a vortex ring on a wall. J. Fluid Mech. 181, 99–140.Google Scholar
Widnall, S. E., Bliss, D. B. & Tsai, C.-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66, 35–47.Google Scholar
Williamson, C. H. K., Leweke, T., Asselin, D. J. & Harris, D. M. 2014 Phenomena, dynamics and instabilities of vortex pairs. Fluid Dyn. Res. 46, 061425.Google Scholar