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Definitions of vortex vector and vortex

Published online by Cambridge University Press:  18 June 2018

Shuling Tian Open the ORCID record for Shuling Tian [Opens in a new window] ,
Yisheng Gao ,
Xiangrui Dong  and
Chaoqun Liu Open the ORCID record for Chaoqun Liu [Opens in a new window]
Shuling Tian
Affiliation:
College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, USA
Yisheng Gao
Affiliation:
Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, USA
Xiangrui Dong
Affiliation:
Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, USA National Key Laboratory of Transient Physics, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China
Chaoqun Liu*
Affiliation:
Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, USA
*
†Email address for correspondence: cliu@uta.edu
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Abstract

Although the vortex is ubiquitous in nature, its definition is somewhat ambiguous in the field of fluid dynamics. In this absence of a rigorous mathematical definition, considerable confusion appears to exist in visualizing and understanding the coherent vortical structures in turbulence. Cited in the previous studies, a vortex cannot be fully described by vorticity, and vorticity should be further decomposed into a rotational and a non-rotational part to represent the rotation and the shear, respectively. In this paper, we introduce several new concepts, including local fluid rotation at a point and the direction of the local fluid rotation axis. The direction and the strength of local fluid rotation are examined by investigating the kinematics of the fluid element in two- and three-dimensional flows. A new vector quantity, which is called the vortex vector in this paper, is defined to describe the local fluid rotation and it is the rotational part of the vorticity. This can be understood as that the direction of the vortex vector is equivalent to the direction of the local fluid rotation axis, and the magnitude of vortex vector is the strength of the location fluid rotation. With these new revelations, a vortex is defined as a connected region where the vortex vector is not zero. In addition, through direct numerical simulation (DNS) and large eddy simulation (LES) examples, it is demonstrated that the newly defined vortex vector can fully describe the complex vertical structures of turbulence.

JFM classification

Vortex Flows: Contour dynamics Mathematical Foundations: Topological fluid dynamics Vortex Flows: Vortex Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 849 , 25 August 2018 , pp. 312 - 339
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Copyright
© 2018 Cambridge University Press 

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References

Adrian, R. J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19, 041301.Google Scholar
Berdahl, C. & Thompson, D. 1993 Eduction of swirling structure using the velocity gradient tensor. AIAA J. 31, 97–103.Google Scholar
Chakraborty, P., Balachandar, S. & Adrian, R. J. 2005 On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189–214.Google Scholar
Chashechkin, Y. D. 1993 Visualization and identification of vortex structures in stratified wakes. In Eddy Structure Identification in Free Turbulent Shear Flows (ed. Bonnet, J. P. & Glauser, M. N.), Fluid Mechanics and its Applications, vol. 21, pp. 393–403. Springer.Google Scholar
Chong, M., Perry, A. & Cantwell, B. 1990 A general classification of three dimensional flow fields. Phys. Fluids A 2, 765–777.Google Scholar
Dallmann, U.1983 Topological structures of three-dimensional flow separation. AIAA Paper 1983-1735.Google Scholar
Elsas, J. H. & Moriconi, L. 2017 Vortex identification from local properties of the vorticity field. Phys. Fluids 29, 015101.Google Scholar
Epps, B. P.2017 Review of vortex identification methods. AIAA Paper 2017-0989.Google Scholar
Gentle, J. E. 1998 Numerical Linear Algebra for Applications in Statistics. Springer.Google Scholar
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 1–26.Google Scholar
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137–162.Google Scholar
Haller, G. 2016 Dynamically consistent rotation and stretch tensors from a dynamic polar decomposition. J. Mech. Phys. Solids 86, 70–93.Google Scholar
Haller, G. & Beron-Vera, F. J. 2013 Coherent lagrangian vortices: the black holes of turbulence. J. Fluid Mech. 731, R4.Google Scholar
Haller, G., Hadjighasem, A., Farazmand, M. & Huhn, F. 2016 Defining coherent vortices objectively from the vorticity. J. Fluid Mech. 795, 136–173.Google Scholar
Helmholtz, H. 1867 About integrals of hydrodynamic equations related with vortical motions. Phil. Mag. 33, 485–512.Google Scholar
Huang, Y. & Green, M. A. 2015 Detection and tracking of vortex phenomena using lagrangian coherent structures. Exp. Fluids 56, 147.Google Scholar
Hunt, J. C., Wray, A. A. & Moin, P. 1988 Eddies, streams, and convergence zones in turbulent flows. In Center for Turbulence Research Proceedings of the Summer Program, vol. CTR‐S88, pp. 193–208.Google Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.Google Scholar
Karrasch, D., Huhn, F. & Haller, G. 2014 Automated detection of coherent lagrangian vortices in two-dimensional unsteady flows. Proc. R. Soc. Lond. 471, 20140639.Google Scholar
Katz, A. 1996 Computational Rigid Vehicle Dynamics. Krieger Publishing Co.Google Scholar
Kolář, V. 2007 Vortex identification: New requirements and limitations. Intl J. Heat Fluid Flow 28, 638–652.Google Scholar
Kolář, V. & Šístek, J. 2014 Recent progress in explicit shear-eliminating vortex identification. In Proceedings of 19th Australasian Fluid Mechanics Conference, Melbourne, Australia, December 8–11 (ed. Chowdhury, H. & Alam, F.), RMIT University, Article no. 274.Google Scholar
Kolář, V. & Šístek, J. 2015 Corotational and compressibility aspects leading to a modification of the vortex-identification q-criterion. AIAA J. 53, 2406–2410.Google Scholar
Kolář, V., Šístek, J., Cirak, F. & Moses, P. 2013 Average corotation of line segments near a point and vortex identification. AIAA J. 51, 2678–2694.Google Scholar
Küchemann, D. & Weber, J. 1965 Report on the iutam symposium on concentrated vortex motions in fluids. J. Fluid Mech. 21, 1–20.Google Scholar
Liu, C., Wang, Y., Yang, Y. & Duan, Z. 2016 New omega vortex identification method. Sci. China Phys., Mech. Astron. 59, 684711.Google Scholar
Liu, C., Yan, Y. & Lu, P. 2014 Physics of turbulence generation and sustenance in a boundary layer. Comput. Fluids 102, 353–384.Google Scholar
Lugt, H. J. 1979 The dilemma of defining a vortex. In Recent Developments in Theoretical and Experimental Fluid Mechanics (ed. Roesner, K. G., Müller, U. & Schmidt, B.), pp. 309–321. Springer.Google Scholar
Murman, S. M., Aftosmis, M. J. & Berger, M. J.2003 Simulations of 6-dof motion with a cartesian method. AIAA Paper 2003-1246.Google Scholar
Robinson, S. K. 1990 A review of vortex structures and associated coherent motions in turbulent boundary layers. In Structure of Turbulence and Drag Reduction. Springer.Google Scholar
Robinson, S. K., Kline, S. J. & Spalart, P. R.1989 A review of quasi-coherent structures in a numerically simulated turbulent boundary layer. NASA Tech. Mem. 102191.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Sayadi, T., Hamman, C. W. & Moin, P. 2013 Direct numerical simulation of complete h-type and k-type transitions with implications for the dynamics of turbulent boundary layers. J. Fluid Mech. 724, 480–509.Google Scholar
Vollmers, H., Kreplin, H. P. & Meier, H. U. 1983 Separation and vortical-type flow around a prolate spheroid – evaluation of relevant parameters. In Proceedings of the AGARD Symposium on Aerodynamics of Vortical Type Flows in Three Dimensions, AGARD-CP-342.Google Scholar
Wang, Y., Yang, Y., Yang, G. & Liu, C. 2017 Dns study on vortex and vorticity in late boundary layer transition. Commun. Comput. Phys. 22, 441–459.Google Scholar
Wu, J., Xiong, A. & Yang, Y. 2005 Axial stretching and vortex definition. Phys. Fluids 17, 038108.Google Scholar
Wu, J., Yan, Y. & Zhou, M. 2006 Vorticity and Vortex Dynamics. Springer.Google Scholar
Yang, Y., Yan, Y. & Liu, C. 2016 Iles for mechanism of ramp-type mvg reducing shock induced flow separation. Sci. China-Phys. Mech. Astron. 59, 124711.Google Scholar
Zhang, Y., Liu, K., Xian, H. & Du, X. 2018 A review of methods for vortex identification in hydroturbines. Renew. Sustain. Energy Rev. 81, 1269–1285.Google Scholar
Zhou, J., Adrian, R. J., Balachandar, S. & Kendall, T. M. 1999 Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396.Google Scholar