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The generation and conservation of vorticity: deforming interfaces and boundaries in two-dimensional flows

Published online by Cambridge University Press:  10 March 2020

S. J. Terrington Open the ORCID record for S. J. Terrington [Opens in a new window] ,
K. Hourigan Open the ORCID record for K. Hourigan [Opens in a new window]  and
M. C. Thompson Open the ORCID record for M. C. Thompson [Opens in a new window]
S. J. Terrington*
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, VIC 3800, Australia
K. Hourigan
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, VIC 3800, Australia
M. C. Thompson
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical and Aerospace Engineering, Monash University, Melbourne, VIC 3800, Australia
*
†Email address for correspondence: [email protected]
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Abstract

This article presents a revised formulation of the generation and transport of vorticity at generalised fluid–fluid interfaces, substantially extending the work of Brøns et al. (J. Fluid Mech., vol. 758, 2014, pp. 63–93). Importantly, the formulation is effectively expressed in terms of the conservation of vorticity, and the latter is shown to hold for arbitrary deformation and normal motion of the interface; previously, vorticity conservation had only been demonstrated for stationary interfaces. The present formulation also affords a simple physical description of the generation of vorticity in incompressible, Newtonian flows: the only mechanism by which vorticity may be generated on an interface is the inviscid relative acceleration of fluid elements on each side of the interface, due to pressure gradients or body forces. Viscous forces act to transfer circulation between the vortex sheet representing the interface slip velocity, and the fluid interior, but do not create vorticity on the interface. Several representative example flows are considered and interpreted under the proposed framework, illustrating the generation, transport and, importantly, the conservation of vorticity within these flows.

JFM classification

Vortex Flows: Vortex interactions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 890 , 10 May 2020 , A5
Copyright
© The Author(s), 2020. Published by 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., Thompson, M. C., Leweke, T. & Hourigan, K. 2014 Vorticity generation and conservation for two-dimensional interfaces and boundaries. J. Fluid Mech. 758, 63–93.CrossRefGoogle Scholar
Brown, D. L., Cortez, R. & Minion, M. L. 2001 Accurate projection methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 168 (2), 464–499.CrossRefGoogle Scholar
Buckley, M. P. & Veron, F. 2017 Airflow measurements at a wavy air–water interface using PIV and LIF. Exp. Fluids 58 (11), 161.CrossRefGoogle Scholar
Buckley, M. P. & Veron, F. 2019 The turbulent airflow over wind generated surface waves. Eur. J. Mech. (B/Fluids) 73, 132–143.CrossRefGoogle Scholar
Bulavin, P. E. & Kashcheev, V. M. 1965 Solution of the non-homogeneous heat conduction equation for multilayered bodies. Intl Chem. Engng 5, 112–115.Google Scholar
Cresswell, R. W. & Morton, B. R. 1995 Drop-formed vortex rings – the generation of vorticity. Phys. Fluids 7 (6), 1363–1370.CrossRefGoogle Scholar
Dommermuth, D. G. 1993 The laminar interactions of a pair of vortex tubes with a free surface. J. Fluid Mech. 246, 91–115.CrossRefGoogle Scholar
Donea, J., Huerta, A., Ponthot, J.-P. & Rodriguez-Ferran, A. 2004 Arbitrary Lagrangian–Eulerian methods. In Encyclopedia of Computational Mechanics (ed. Stein, E., de Borst, R. & Hughes, T. J. R.), chap. 14. Wiley.Google Scholar
Dopazo, C., Lozano, A. & Barreras, F. 2000 Vorticity constraints on a fluid/fluid interface. Phys. Fluids 12 (8), 1928–1931.CrossRefGoogle Scholar
Fenton, J. D. 1985 A fifth-order Stokes theory for steady waves. J. Waterways Port Coast. Ocean Engng 111 (2), 216–234.CrossRefGoogle Scholar
Hansen, A., Douglass, R. W. & Zardecki, A. 2005 Mesh Enhancement: Selected Elliptic Methods, Foundations and Applications. Imperial College Press.CrossRefGoogle Scholar
Küchemann, D. 1965 Report on the IUTAM symposium on concentrated vortex motions in fluids. J. Fluid Mech. 21, 1–20.CrossRefGoogle Scholar
Lighthill, M. J. 1963 Introduction. Boundary layer theory. In Laminar Boundary Layers (ed. Rosenhead, L.), chap. 2, pp. 46–109. 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. 1960 Mass transport in the boundary layer at a free oscillating surface. J. Fluid Mech. 8, 293–306.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1992 Capillary rollers and bores. J. Fluid Mech. 240, 659–679.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1998 Vorticity and curvature at a free surface. J. Fluid Mech. 356, 149–153.CrossRefGoogle Scholar
Lugt, H. J. 1987 Local flow properties at a viscous free surface. Phys. Fluids 30 (12), 3647–3652.CrossRefGoogle Scholar
Lugt, H. J. & Ohring, S. 1992 The oblique ascent of a viscous vortex pair toward a free surface. J. Fluid Mech. 236, 461–476.CrossRefGoogle Scholar
Lundgren, T. & Koumoutsakos, P. 1999 On the generation of vorticity at a free surface. J. Fluid Mech. 382, 351–366.CrossRefGoogle Scholar
Morino, L. 1986 Helmholtz decomposition revisited: vorticity generation and trailing edge condition. Comput. Mech. 1 (1), 65–90.CrossRefGoogle 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
Özışık, M. N. 1968 Boundary Value Problems of Heat Conduction. International Textbook Co.Google Scholar
Peck, B. & Sigurdson, L. 1998 On the kinetics at a free surface. IMA J. Appl. Maths 61 (1), 1–13.CrossRefGoogle Scholar
Peck, B. & Sigurdson, L. 1999 Geometry effects on free surface vorticity flux. Trans. ASME J. Fluids Engng 121 (3), 678–683.CrossRefGoogle Scholar
Reichl, P., Hourigan, K. & Thompson, M. C. 2005 Flow past a cylinder close to a free surface. J. Fluid Mech. 533, 269–296.CrossRefGoogle Scholar
Rood, E. P. 1994a Interpreting vortex interactions with a free surface. Trans. ASME J. Fluids Engng 116 (1), 91–94.CrossRefGoogle Scholar
Rood, E. P. 1994b 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
Sarpkaya, T. & Henderson, D. O.1984 Surface disturbances due to trailing vortices. Tech. Rep. NPS-69-84-00. Naval Postgraduate School, Monterey, California.Google Scholar
Sheridan, J., Lin, J.-C. & Rockwell, D. 1997 Flow past a cylinder close to a free surface. J. Fluid Mech. 330, 1–30.CrossRefGoogle Scholar
Truesdell, C. 1954 The Kinematics of Vorticity. Indiana University Press.Google Scholar
Tsuji, Y. & Nagata, Y. 1973 Stokes’ expansion of internal deep water waves to the fifth order. J. Oceanogr. Soc. Japan 29 (2), 61–69.Google Scholar
Wu, J. Z. 1995 A theory of three-dimensional interfacial vorticity dynamics. Phys. Fluids 7 (10), 2375–2395.CrossRefGoogle Scholar
Wu, J. Z. & Wu, J. M. 1993 Interactions between a solid surface and a viscous compressible flow field. J. Fluid Mech. 254, 183–211.CrossRefGoogle Scholar

Terrington et al. supplementary movie 1

Transient animation of the two-fluid Couette flow depicted in figure 8, with h1/h2 = 1, ν1/ν2 = 1, µ1/µ2 = 2 and U2 = 0.

Download Terrington et al. supplementary movie 1(Video)
Video 2.7 MB

Terrington et al. supplementary movie 2

Transient animation of the two-fluid Poiseuille flow depicted in figure 10, with h1/h2 = 1, ν1/ν2 = 4, and µ1/µ2 = 2.

Download Terrington et al. supplementary movie 2(Video)
Video 4 MB

Terrington et al. supplementary movie 3

Transient animation of the two-fluid Taylor-Couette flow depicted in figure 10, with rs/r2 = 2, r2/r1 = 3, ν1/ν2 = 1, and µ1/µ2 = 5.

Download Terrington et al. supplementary movie 3(Video)
Video 3.8 MB

Terrington et al. supplementary movie 4

Transient animation of the free-surface wave depicted in figure 14(a). The vertical scale is greatly exaggerated for clarity.

Download Terrington et al. supplementary movie 4(Video)
Video 13 MB

Terrington et al. supplementary movie 5

Transient animation of the viscous interface wave depicted in figure 14(b), with a density ratio ρ1/ρ2 = 2. The vertical scale is greatly exaggerated for clarity.

Download Terrington et al. supplementary movie 5(Video)
Video 11.5 MB

Terrington et al. supplementary movie 6

Transient animation of the interaction of a vortex pair with both a free-surface, and a viscous interface, as illustrated in figure 21. The froude number is Fr = 0.2 in both cases, while the density ratio across the viscous interface is ρ1/ρ2 = 100.

Download Terrington et al. supplementary movie 6(Video)
Video 5.8 MB