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On the identification of a vortex

Published online by Cambridge University Press:  26 April 2006

Jinhee Jeong
Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, USA
Fazle Hussain
Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, USA


Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena. We propose a definition of a vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor ${\bm {\cal S}}^2 + {\bm \Omega}^2$; here ${\bm {\cal S}}$ and ${\bm \Omega}$ are respectively the symmetric and antisymmetric parts of the velocity gradient tensor ${\bm \Delta}{\bm u}$. This definition captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers, unlike a pressure-minimum criterion. We compare our definition with prior schemes/definitions using exact and numerical solutions of the Euler and Navier–Stokes equations for a variety of laminar and turbulent flows. In contrast to definitions based on the positive second invariant of ${\bm \Delta}{\bm u}$ or the complex eigenvalues of ${\bm \Delta}{\bm u}$, our definition accurately identifies the vortex core in flows where the vortex geometry is intuitively clear.

Research Article
© 1995 Cambridge University Press

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Batchelor, G. K. 1967 Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bisset, D. K., Antonia, R. A. & Browne, L. W. B. 1990 Spatial organization of large structures in the turbulent far wake of a cylinder. J. Fluid Mech. 218, 439.Google Scholar
Blackwelder, R. F. 1977 On the role of phase information in conditional sampling. Phys. Fluids 20, S232.Google Scholar
Bödewadt, U. T. 1940 Die Drehströmung über festern Grund. Z. Angew. Math. Mech. 20, 141.Google Scholar
Cantwell, B. J. 1981 Organized motion in turbulent flow. Ann. Rev. Fluid Mech. 13, 457.Google Scholar
Cantwell, B. J. & Coles, D. 1983 An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder. J. Fluid Mech. 136, 321.Google Scholar
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow field. Phys. Fluids A 2, 765.Google Scholar
Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics, vol. 1. Interscience.Google Scholar
Ferré, J. A. & Giralt, F. 1989 Pattern recognition analysis of the velocity field in plane turbulent wakes. J. Fluid Mech. 198, 27.Google Scholar
Fiedler, H. E. & Mensing, P. 1985 The plane turbulent shear layer with periodic excitation. J. Fluid Mech. 150, 281.Google Scholar
Hunt, J. C. R. 1987 Vorticity and vortex dynamics in complex turbulent flows. In Proc. CANCAM, Trans. Can. Soc. Mech. Engrs 11, 21.Google Scholar
Hunt, J. C. R., Wray, A. A. & Moin, P. 1988 Eddies, stream, and convergence zones in turbulent flows. Center for Turbulence Research Report CTR-S88, p. 193.Google Scholar
Husain, H. S. & Hussain, F. 1993 Elliptic jets. Part 3. Dynamics of preferred mode coherent structure. J. Fluid Mech. 248, 315.Google Scholar
Hussain, A. K. M. F. 1980 Coherent structures and studies of perturbed and unperturbed jets. In The Role of Coherent Structures in Modelling Turbulence and Mixing (ed. J. Jimenez) Lecture Notes in Physics, vol. 136, pp. 252291. Springer.CrossRefGoogle Scholar
Hussain, F. 1986 Coherent structures and turbulence. J. Fluid Mech. 173, 303.Google Scholar
Hussain, A. K. M. F. & Hayakawa, M. 1987 Eduction of large-scale organized structure in a turbulent plane wake. J. Fluid Mech. 180, 193.Google Scholar
Hussain, F. & Melander, M. V. 1991 Understanding turbulence via vortex dynamics. In The Lumley Symposium: Studies in Turbulence, pp. 157178. Springer.CrossRefGoogle Scholar
Hussain, A. K. M. F. & Zaman, K. B. M. Q. 1980 Vortex pairing in a circular jet under controlled excitation. Part 2. Coherent structure dynamics. J. Fluid Mech. 101, 493.Google Scholar
Jeong, J. 1994 A theoretical and numerical study of coherent structures. PhD dissertation, University of Houston.Google Scholar
Jimenez, J., Moin, P., Moser, R. & Keefe, L. 1988 Ejection mechanisms in the sublayer of a turbulent channel. Phys. Fluids 31, 1311.Google Scholar
Kida, S., Takaoka, M. & Hussain, F. 1991 Collision of two vortex rings. J. Fluid Mech. 230, 583.Google Scholar
Kim, J. 1985 Turbulence structures associated with the bursting event. Phys. Fluids. 28, 52.Google Scholar
Lamb, H. 1945 Hydrodynamics. Dover.Google Scholar
Lumley, J. L. 1981 Coherent structures in turbulence. In Transition and turbulence (ed. R. E. Meyer), pp. 215242. Academic.CrossRefGoogle Scholar
Lugt, H. J. 1979 The dilemma of defining a vortex. In Recent Developments in Theoretical and Experimental Fluid Mechanics (ed. U. Müller, K. G. Roesner & B. Schmidt), pp. 309321. Springer.CrossRefGoogle Scholar
Melander, M. V. & Hussain, F. 1988 Cut-and-connect of two antiparallel vortex tubes. Center for Turbulence Research Rep. CTR-S88, pp. 257286.Google Scholar
Melander, M. & Hussain, F. 1993 Polarized vorticity dynamics on a vortex column. Phys. Fluids A 5, 1992.Google Scholar
Melander, M. V., Hussain, F. & Basu, A. 1991 Breakdown of a circular jet into turbulence. In Turbulent Shear Flows 8, Munich, pp. Scholar
Metcalfe, R. W., Hussain, F., Menon, S. & Hayakawa, M. 1985 Coherent structures in a turbulent mixing layer: a comparison between numerical simulations and experiments. In Turbulent Shear Flows 5 (ed. F. Durst, B. E. Launder, J. L. Lumley, F. W. Schmidt & J. H. Whitelaw), p. 110. Springer.CrossRefGoogle Scholar
Moffatt, H. K. 1963 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1.Google Scholar
Mumford, J. C. 1982 The structures of the large eddies in fully developed turbulent shear flows. Part 1. The plane jet. J. Fluid Mech. 118, 241.Google Scholar
Panton, R. L. 1984 Incompressible Flow. Wiley.Google Scholar
Park, K., Metcalfe, R. W. & Hussain, F. 1994 Role of coherent structures in an isothermally reacting mixing layer. Phys. Fluids 6, 885.Google Scholar
Robinson, S. K. 1991 The kinetics of turbulent boundary layer structure. PhD Dissertation, Stanford University.Google Scholar
Schoppa, W. 1994 A new mechanism of small-scale transition in a plane mixing layer: core dynamics of spanwise vortices. MS thesis, University of Houston.CrossRefGoogle Scholar
Schoppa, W., Husain, H. & Hussain, F. 1993 Nonlinear instability of free shear layers: subharmonic resonance and three-dimensional vortex dynamics. IUTAM Symp on Nonlinear Instability of Nonparallel Flows (ed. S. P. Lin et al.), 26–30 July 1993, Clarkson University, pp. 251280.Google Scholar
Shtern, V. & Hussain, F. 1993 Hysteresis in a swirling jet as a model tornado. Phys. Fluids A 5, 2183.Google Scholar
Tennekes, T. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Truesdell, C. 1953 The Kinematics of Vorticity. Indiana University.Google Scholar
Tso, j. 1983 coherent structures in a fully-developed turbulent axisymmetric jet. PhD dissertation, johns hopkins university.Google Scholar
Tso, J. & Hussain, F. 1989 Organized motions in a fully developed turbulent axisymmetric jet. J. Fluid Mech. 203, 425.Google Scholar
Virk, D., Melander, M. V. & Hussain, F. 1994 Dynamics of a polarized vortex ring. J. Fluid Mech. 260, 23.Google Scholar
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