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Vorticity generation and conservation for two-dimensional interfaces and boundaries

Published online by Cambridge University Press:  07 October 2014

M. Brøns ,
M. C. Thompson ,
T. Leweke  and
K. Hourigan
M. Brøns
Affiliation:
Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800 Lyngby, Denmark
M. C. Thompson
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, VIC 3800, Australia
T. Leweke
Affiliation:
IRPHE, UMR 7342, CNRS, Aix-Marseille Université, Centrale Marseille, 13384 Marseille, France
K. Hourigan*
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, VIC 3800, Australia Division of Biological Engineering, Monash University, Melbourne, VIC 3800, Australia
*
†Email address for correspondence: kerry.hourigan@monash.edu
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Abstract

The generation, redistribution and, importantly, conservation of vorticity and circulation is studied for incompressible Newtonian fluids in planar and axisymmetric geometries. A generalised formulation of the vorticity at the interface between two fluids for both no-slip and stress-free conditions is presented. Illustrative examples are provided for planar Couette flow, Poiseuille flow, the spin-up of a circular cylinder, and a cylinder below a free surface. For the last example, it is shown that, although large imbalances between positive and negative vorticity appear in the wake, the balance is found in the vortex sheet representing the stress-free surface.

JFM classification

Mathematical Foundations: General fluid mechanics Vortex Flows: Vortex Flows Vortex Flows: Vortex interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 758 , 10 November 2014 , pp. 63 - 93
Copyright
© 2014 Cambridge University Press 

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bozkaya, C., Kocabiyik, S., Mironova, L. A. & Gubanov, O. I. 2011 Streamwise oscillations of a cylinder beneath a free surface: free surface effects on vortex formation modes. J. Comput. Appl. Maths 235 (16), 4780–4795.CrossRefGoogle Scholar
Brøns, M. 1994 Topological fluid dynamics of interfacial flows. Phys. Fluids 6, 2730–2737.Google Scholar
Küchemann, D. 1965 Report on the I.U.T.A.M. Symposium on concentrated vortex motions in fluids. J. Fluid Mech. 21, 1–20.Google Scholar
Leweke, T., Thompson, M. C. & Hourigan, K. 2004 Vortex dynamics associated with the collision of a sphere with a wall. Phys. Fluids 16 (9), L74–L77.Google Scholar
Lighthill, M. J. 1963 Laminar Boundary Layers (ed. Rosenhead, L.), chap. 2, Oxford University Press.Google Scholar
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245 (903), 535–581.Google Scholar
Longuet-Higgins, M. S. 1992 Capillary rollers and bores. J. Fluid Mech. 240, 659–679.Google Scholar
Longuet-Higgins, M. S. 1998 Vorticity and curvature at a free surface. J. Fluid Mech. 356, 149–153.Google Scholar
Lugt, H. J. 1987 Local flow properties at a viscous free surface. Phys. Fluids 30 (12), 3647–3652.Google Scholar
Lundgren, T. & Koumoutsakos, P. 1999 On the generation of vorticity at a free surface. J. Fluid Mech. 382, 351–366.CrossRefGoogle Scholar
Mallick, D. D. 1957 Non-uniform rotation of an infinite circular cylinder in an infinite viscous liquid. Z. Angew. Math. Mech. 37, 385–392.Google Scholar
Morton, B. R. 1984 The generation and decay of vorticity. Geophys. Astrophys. Fluid Dyn. 28, 277–308.CrossRefGoogle Scholar
Ohring, S. & Lugt, H. J. 1991 Interaction of a viscous vortex pair with a free surface. J. Fluid Mech. 227, 47–70.CrossRefGoogle Scholar
Rao, A., Leontini, J. S., Thompson, M. C. & Hourigan, K. 2013a Three-dimensionality in the wake of a rotating cylinder in a uniform flow. J. Fluid Mech. 717, 1–29.Google Scholar
Rao, A., Thompson, M. C., Leweke, T. & Hourigan, K. 2013b The flow past a circular cylinder translating at different heights above a wall. J. Fluids Struct. 41, 1–21.Google Scholar
Reichl, P. J.2001 Flow past a cylinder close to a free surface. PhD thesis, Monash University, Melbourne, Australia.Google Scholar
Reichl, P., Hourigan, K. & Thompson, M. 2005 Flow past a cylinder close to a free surface. J. Fluid Mech. 533, 269–296.Google Scholar
Rood, E. P. 1994 Myths, math, and physics of free-surface vorticity. Appl. Mech. Rev. 47 (6S), S152–S156.CrossRefGoogle Scholar
Sarpkaya, T. 1996 Vorticity, free surface, and surfactants. Annu. Rev. Fluid Mech. 28, 83–128.CrossRefGoogle Scholar
Sheridan, J., Lin, J.-C. & Rockwell, D. 1997 Flow past a cylinder close to a free surface. J. Fluid Mech. 330, 1–30.Google Scholar
Stewartson, K. 1951 On the impulsive motion of a flat plate in a viscous fluid. Q. J. Mech. Appl. Maths 4, 182–198.CrossRefGoogle Scholar
Thompson, M. C., Hourigan, K., Cheung, A. & Leweke, T. 2006 Hydrodynamics of a particle impact on a wall. Appl. Math. Model. 30 (11), 1356–1369.CrossRefGoogle Scholar
Thompson, M. C., Leweke, T. & Hourigan, K. 2007 Sphere–wall collision: vortex dynamics and stability. J. Fluid Mech. 575, 121–148.CrossRefGoogle Scholar
Wu, J. Z. 1995 A theory of three-dimensional interfacial vorticity dynamics. Phys. Fluids 7 (10), 2375–2395.Google Scholar
Wu, J. Z. & Wu, J. M. 1996 Vorticity dynamics on boundaries. Adv. Appl. Mech. 32, 119–275.CrossRefGoogle Scholar
Zhang, C., Shen, L. & Yue, D. K. P. 1999 The mechanism of vortex connection at a free surface. J. Fluid Mech. 384, 207–241.CrossRefGoogle Scholar