Hostname: page-component-7dd5485656-glrdx Total loading time: 0 Render date: 2025-10-30T17:15:18.741Z Has data issue: false hasContentIssue false
  • English
  • Français

The generation and diffusion of vorticity in three-dimensional flows: Lyman's flux

Published online by Cambridge University Press:  25 March 2021

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: stephen.terrington@monash.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine Lyman's (Appl. Mech. Rev., vol. 43, issue 8, 1990, pp. 157–158) proposed definition of the boundary vorticity flux, as an alternative to the traditional definition provided by Lighthill (Introduction: boundary layer theory. In Laminar Boundary Layers (ed. L. Rosenhead), chap. 2, 1963, pp. 46–109. Oxford University Press). While either definition may be used to describe the generation and diffusion of vorticity, Lyman's definition offers several conceptual benefits. First, Lyman's definition can be interpreted as the transfer of circulation across a boundary, due to the acceleration of that boundary, and is therefore closely tied to the dynamics of linear momentum. Second, Lyman's definition allows the vorticity creation process on a solid boundary to be considered essentially inviscid, effectively extending Morton's (Geophys. Astrophys. Fluid Dyn., vol. 28, 1984, pp. 277–308) two-dimensional description to three-dimensional flows. Third, Lyman's definition describes the fluxes of circulation acting in any two-dimensional reference surface, enabling a control-surface analysis of three-dimensional vortical flows. Finally, Lyman's definition more clearly illustrates how the kinematic condition that vortex lines do not end in the fluid is maintained, providing an elegant description of viscous processes such as vortex reconnection. The flow over a sphere, in either translational or rotational motion, is examined using Lyman's definition of the vorticity flux, demonstrating the benefits of the proposed framework in understanding the dynamics of vortical flows.

JFM classification

Vortex Flows: Vortex dynamics

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A106
Check for updates
Copyright
© The Author(s), 2021. Published by 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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Beardsell, G., Dufresne, L. & Dumas, G. 2016 Investigation of the viscous reconnection phenomenon of two vortex tubes through spectral simulations. Phys. Fluids 28 (9), 095103.CrossRefGoogle Scholar
Bernard, P.S. 2011 The hairpin vortex illusion. J. Phys.: Conf. Ser. 318, 062004.Google Scholar
Bernard, P.S. 2019 On the inherent bias of swirling strength in defining vortical structure. Phys. Fluids 31 (3), 035107.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
Calabretto, S.A.W., Denier, J.P & Levy, B. 2019 An experimental and computational study of the post-collisional flow induced by an impulsively rotated sphere. J. Fluid Mech. 881, 772–793.CrossRefGoogle Scholar
Calabretto, S.A.W., Levy, B., Denier, J.P. & Mattner, T.W. 2015 The unsteady flow due to an impulsively rotated sphere. Proc. R. Soc. A 471, 20150299.CrossRefGoogle Scholar
Chorin, A.J. 1973 Numerical study of slightly viscous flow. J. Fluid Mech. 57 (4), 785–796.CrossRefGoogle Scholar
Eyink, G.L. 2008 Turbulent flow in pipes and channels as cross-stream “inverse cascades” of vorticity. Phys. Fluids 20 (12), 125101.CrossRefGoogle Scholar
Eyink, G.L., Gupta, A. & Zaki, T.A. 2020 Stochastic Lagrangian dynamics of vorticity. Part 1. General theory for viscous, incompressible fluids. J. Fluid Mech. 901, A2.CrossRefGoogle Scholar
Fuentes, O.U.V. 2007 On the topology of vortex lines and tubes. J. Fluid Mech. 584, 147–156.CrossRefGoogle Scholar
Greene, J.M. 1993 Reconnection of vorticity lines and magnetic lines. Phys. Fluids B 5 (7), 2355–2362.CrossRefGoogle Scholar
Gresho, P.M. 1991 Incompressible fluid dynamics: some fundamental formulation issues. Annu. Rev. Fluid Mech. 23 (1), 413–453.CrossRefGoogle Scholar
Gushchin, V.A., Kostomarov, A.V., Matyushin, P.V. & Pavlyukova, E.R. 2002 Direct numerical simulation of the transitional separated fluid flows around a sphere and a circular cylinder. J. Wind Engng Ind. Aerodyn. 90 (4–5), 341–358.CrossRefGoogle Scholar
Gushchin, V.A. & Matyushin, R.V. 2006 Vortex formation mechanisms in the wake behind a sphere for $200 < Re < 380$. Fluid Dyn. 41 (5), 795–809.CrossRefGoogle Scholar
Huggins, E.R. 1971 Dynamical theory and probability interpretation of the vorticity field. Phys. Rev. Lett. 26 (21), 1291–1294.CrossRefGoogle Scholar
Huggins, E.R. 1994 Vortex currents in turbulent superfluid and classical fluid channel flow, the Magnus effect, and Goldstone boson fields. J. Low Temp. Phys. 96 (5–6), 317–346.CrossRefGoogle Scholar
Huggins, E.R. & Bacon, D.P. 1980 Vortex currents and hydrodynamic instability in Taylor cells. Phys. Rev. A 21 (4), 1327–1330.CrossRefGoogle Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.CrossRefGoogle Scholar
Johnson, T.A. & Patel, V.C. 1999 Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 19–70.CrossRefGoogle Scholar
Kida, S. & Takaoka, M. 1991 Breakdown of frozen motion of vorticity field and vorticity reconnection. J. Phys. Soc. Japan 60 (7), 2184–2196.CrossRefGoogle Scholar
Kida, S. & Takaoka, M. 1994 Vortex reconnection. Annu. Rev. Fluid Mech. 26 (1), 169–177.CrossRefGoogle Scholar
Kolár, V. 2003 On the Lyman problem. Cent. Eur. J. Phys. 1 (2), 258–267.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.CrossRefGoogle Scholar
Kustepeli, A. 2016 On the Helmholtz theorem and its generalization for multi-layers. Electromagnetics 36 (3), 135–148.CrossRefGoogle Scholar
Leonard, A. 1980 Vortex methods for flow simulation. J. Comput. Phys. 37 (3), 289–335.CrossRefGoogle Scholar
Lighthill, M.J. 1963 Introduction: boundary layer theory. In Laminar Boundary Layers (ed. L. Rosenhead), chap. 2, pp. 46–109. Oxford University Press.Google Scholar
Lyman, F.A. 1990 Vorticity production at a solid boundary. Appl. Mech. Rev. 43 (8), 157–158.Google Scholar
McGavin, P. & Pontin, D.I. 2018 Vortex line topology during vortex tube reconnection. Phys. Rev. Fluids 3 (5), 054701.CrossRefGoogle Scholar
Melander, M.V. & Hussain, F. 1989 Cross-linking of two antiparallel vortex tubes. Phys. Fluids A 1 (4), 633–636.CrossRefGoogle Scholar
Melander, M.V. & Hussain, F. 1990 Topological aspects of vortex reconnection. In Topological Fluid Mechanics (ed. H.K. Moffatt & A. Tsinober), pp. 485–499. Cambridge University Press.Google 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
Panton, R.L. 1984 Incompressible Flow. John Wiley and Sons.Google Scholar
Payne, R.B. 1956 A numerical method for calculating the starting, and perturbation of a two-dimensional jet at low Reynolds number. Tech. Rep. 3047. Aero. Res. Counc. (Lond.), Rep. & Mem.Google Scholar
Payne, R.B. 1958 Calculations of unsteady viscous flow past a circular cylinder. J. Fluid Mech. 4 (1), 81–86.CrossRefGoogle Scholar
Saffman, P.G. 1990 A model of vortex reconnection. J. Fluid Mech. 212, 395–402.CrossRefGoogle Scholar
Stanaway, S., Shariff, K. & Hussain, F. 1988 Head-on collision of viscous vortex rings. In Studying Turbulence Using Numerical Simulation Databases 2: Proceedings of the 1988 CTR Summer Program, pp. 257–286. Stanford University.Google Scholar
Takaoka, M. 1996 Helicity generation and vorticity dynamics in helically symmetric flow. J. Fluid Mech. 319, 125–149.CrossRefGoogle Scholar
Terrington, S.J., Hourigan, K. & Thompson, M.C. 2020 The generation and conservation of vorticity: deforming interfaces and boundaries in two-dimensional flows. J. Fluid Mech. 890, A5.CrossRefGoogle Scholar
Thompson, M.C., Leweke, T. & Provansal, M. 2001 Kinematics and dynamics of sphere wake transition. J. Fluids Struct. 15 (3–4), 575–585.CrossRefGoogle Scholar
Tiwari, S.S., Pal, E., Bale, S., Minocha, N., Patwardhan, A.W., Nandakumar, K. & Joshi, J.B. 2020 Flow past a single stationary sphere, 2. Regime mapping and effect of external disturbances. Powder Technol. 365, 215–243.CrossRefGoogle Scholar
Truesdell, C. 1954 The Kinematics of Vorticity. Indiana University Press.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
Wu, J.Z. & Wu, J.M. 1996 Vorticity dynamics on boundaries. Adv. Appl. Mech. 32, 119–275.CrossRefGoogle Scholar
Wu, J.Z. & Wu, J.M. 1998 Boundary vorticity dynamics since Lighthill's 1963 article: review and development. J. Theor. Comput. Fluid Dyn. 10 (1–4), 459–474.CrossRefGoogle Scholar

Terrington et al. supplementary movie 1

See pdf file for movie caption

Download Terrington et al. supplementary movie 1(Video)
Video 8 MB

Terrington et al. supplementary movie 2

See pdf file for movie caption

Download Terrington et al. supplementary movie 2(Video)
Video 16.7 MB

Terrington et al. supplementary movie 3

See pdf file for movie captions

Download Terrington et al. supplementary movie 3(Video)
Video 21.1 MB

Terrington et al. supplementary movie 4

See pdf file for movie captions

Download Terrington et al. supplementary movie 4(Video)
Video 5.5 MB
Supplementary material: PDF

Terrington et al. supplementary material

Captions for movies 1-4

Download Terrington et al. supplementary material(PDF)
PDF 15.7 KB