Skip to main content Accessibility help
Hostname: page-component-cf9d5c678-r9vz2 Total loading time: 0.212 Render date: 2021-08-01T12:03:14.884Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Can vortex criteria be objectivized?

Published online by Cambridge University Press:  08 December 2020

George Haller
Institute for Mechanical Systems, ETH Zürich, Zürich, Switzerland
E-mail address:


Several procedures have been proposed to modify non-objective (observer-dependent) local vortex criteria so that they become objective. These modifications are only justifiable if they are equivalent to applying the original criteria after a generalized (possibly nonlinear) frame change is performed on the flow domain; otherwise, the arguments used in deriving those criteria no longer apply. To examine the feasibility of available objectivization procedures, we derive here necessary and sufficient conditions for the existence of a generalized frame change prescribed pointwise through its Jacobian field. From these conditions we conclude that, of all proposed objectivization approaches in the literature, only the replacement of the spin tensor with the spin-deviation tensor is applicable to generic fluid flows.

JFM Papers
© The Author(s), 2020. 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.)


Anghan, C., Dave, S., Saincher, S. & Banerjee, J. 2019 Direct numerical simulation of transitional and turbulent round jets: Evolution of vortical structures and turbulence budget. Phys. Fluids 31, 053606.CrossRefGoogle Scholar
Allshouse, M. R. & Peacock, T. 2015 Lagrangian-based methods for coherent structure detection. Chaos 25, 097617.CrossRefGoogle ScholarPubMed
Arnold, V. I. 1978 Ordinary Differential Equations. MIT Press.Google Scholar
Astarita, G. 1979 Objective and generally applicable criteria for flow classification. J. Non-Newtonian Fluid Mech. 6, 6976.CrossRefGoogle Scholar
Basdevant, C. & Philopovitch, T. 1994 On the validity of the “Weiss criterion” in two-dimensional turbulence. Physica D 73, 1730.CrossRefGoogle Scholar
Chakraborty, P., Balachandar, S. & Adrian, R. 2005 On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189214.CrossRefGoogle Scholar
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow field. Phys. Fluids 2, 765777.CrossRefGoogle Scholar
Drouot, R. 1976 Definition d'un transport associe un modele de fluide de deuxieme ordre. C. R. Acad. Sci. Paris A 282, 923926.Google Scholar
Drouot, R. & Lucius, M. 1976 Approximation du second ordre de la loi de comportement des fluides simples. lois classiques deduites de l'introduction d'un nouveau tenseur objectif. Arch. Mech. Stosowanej 28 (2), 189198.Google Scholar
Dubief, Y. & Delcayre, F. 2000 On coherent-vortex identification in turbulence. J. Turbulence 1, N11.CrossRefGoogle Scholar
Epps, B. 2017 Review of vortex identification methods. In AIAA SciTech Forum, 9–13 January 2017, Grapevine, Texas, 55th AIAA Aerospace Sciences Meeting, pp. 1–22. AIAA.CrossRefGoogle Scholar
Froyland, G., Santitissadeekorn, N. & Monahan, A. 2010 Transport in time-dependent dynamical systems: finite-time coherent sets. Chaos 20, 043116.CrossRefGoogle ScholarPubMed
Gao, F., Ma, W., Zambonini, G., Boudet, J., Ottavy, X., Lu, L. & Shao, L. 2015 Large-eddy simulation of 3-D corner separation in a linear compressor cascade. Phys. Fluids 27, 085105.CrossRefGoogle Scholar
Gao, Y. & Liu, C. 2018 Rortex and comparison with eigenvalue-based vortex identification criteria. Phys. Fluids 30, 085107.CrossRefGoogle Scholar
Günther, T., Gross, M. & Theisel, H. 2017 Generic objective vortices for flow visualization. ACM Trans. Graph. 36 (141), 111.CrossRefGoogle Scholar
Günther, T. & Theisel, H. 2018 The state of the art in vortex extraction. Comput. Graph. Forum 37, 149173.CrossRefGoogle Scholar
Günther, T. & Theisel, H. 2020 Hyper-objective vortices. IEEE Trans. Vis. Comput. Graphics 26, 15321547.CrossRefGoogle ScholarPubMed
Gurtin, M. E. 1981 An Introduction to Continuum Mechanics. Academic Press.Google Scholar
Gurtin, M. E., Fried, E. & Anand, L. 2010 The Mechanics and Thermodynamics of Continua. Cambridge University Press.CrossRefGoogle Scholar
Hadwiger, M., Mlejnek, M., Theussl, T. & Rautek, P. 2018 Time-dependent flow seen through approximate observer killing fields. IEEE Trans. Vis. Comput. Graphics 25, 12571266.CrossRefGoogle Scholar
Haller, G. 2000 Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos 10, 99108.CrossRefGoogle ScholarPubMed
Haller, G. 2001 Lagrangian coherent structures and the rate of strain in a partition of two-dimensional turbulence. Phys. Fluids 13, 33653385.CrossRefGoogle Scholar
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 126.CrossRefGoogle Scholar
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137162.CrossRefGoogle Scholar
Haller, G. 2016 Dynamic rotation and stretch tensors from a dynamic polar decomposition. J. Mech. Phys. Solids 86, 7093.CrossRefGoogle Scholar
Haller, G., Hadjighasem, A., Farazmand, M. & Huhn, F. 2016 Defining coherent vortices objectively from the vorticity. J. Fluid Mech. 795, 136173.CrossRefGoogle Scholar
Haller, G., Katsanoulis, S., Holzner, M., Frohnapfel, B. & Gatti, D. 2020 Objective barriers to the transport of dynamically active vector fields. J. Fluid Mech. (in press).CrossRefGoogle Scholar
Hunt, J. C. R., Wray, A. & Moin, P. 1988 Eddies, stream, and convergence zones in turbulent flows. Center for Turbulence Research Report CTR-S88, pp. 193–208.Google Scholar
Jantzen, R. T., Taira, K., Granlund, K. O. & Ol, M. V. 2014 Vortex dynamics around pitching plates. Phys. Fluids 26, 065105.CrossRefGoogle Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 6994.CrossRefGoogle Scholar
Kirwan, D. 2016 On objectivity, irreversibility and non-newtonian fluids. Fluids 1, 317.CrossRefGoogle Scholar
Lapeyre, G., Klein, P. & Hua, B. L. 1999 Does the tracer gradient vector align with the strain eigenvectors in 2-D turbulence? Phys. Fluids 11, 37293737.CrossRefGoogle Scholar
Liu, J., Gao, Y. & Liu, C. 2019 a An objective version of the rortex vector for vortex identification. Phys. Fluids 31, 065112.CrossRefGoogle Scholar
Liu, C., Gao, Y., Tian, S. & Dong, X. 2018 Rortex—a new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids 30, 035103.CrossRefGoogle Scholar
Liu, J., Gao, Y., Wang, Y. & Liu, C. 2019 b Objective omega vortex identification method. J. Hydrodyn. 31, 455463.CrossRefGoogle Scholar
Lugt, H. J. 1979 The dilemma of defining a vortex. In Recent Developments in Theoretical and Experimental Fluid Mechanics (ed. U. Muller, K. G. Riesner & B. Schmidt), vol. 13, pp. 309–321. Springer-Verlag.CrossRefGoogle Scholar
Martins, R. S., Pereira, A. S., Mompean, G., Thais, L. & Thompson, R. L. 2016 An objective perspective for classic flow classification criteria. C. R. Méc. 344, 5259.CrossRefGoogle Scholar
McMullan, W. A. & Page, G. J. 2012 Towards Large Eddy Simulation of gas turbine compressors. Prog. Aerosp. Sci. 52, 3047.CrossRefGoogle Scholar
Okubo, A. 1970 Horizontal dispersion of floatable trajectories in the vicinity of velocity singularities such as convergencies. Deep-Sea Res. 17, 445454.Google Scholar
Peacock, T., Froyland, G. & Haller, G. (Eds) 2015 Focus issue on the objective detection of coherent structures. Chaos 25, whole issue.CrossRefGoogle ScholarPubMed
Pedergnana, T., Oettinger, D. & Haller, G. 2020 Explicit unsteady Navier–Stokes solutions and their analysis via local vortex criteria. Phys. Fluids 32, 046603.CrossRefGoogle Scholar
Perry, A. E. & Chong, M. S. 1994 Topology of flow patterns in vortex motions and turbulence. Appl. Sci. Res. 53, 357374.CrossRefGoogle Scholar
Rojo, I. B. & Günther, T. 2020 Vector field topology of time-dependent flows in a steady reference frame. IEEE Trans. Vis. Comput. Graphics 26, 280290.Google Scholar
Serra, M. & Haller, G. 2016 Objective Eulerian coherent structures. Chaos 26, 053110.CrossRefGoogle ScholarPubMed
Tabor, M. & Klapper, I. 1994 Stretching and alignment in chaotic and turbulent flows. Chaos, Solitons Fractals 4, 10311055.CrossRefGoogle Scholar
Tél, T., Kadi, L., Jánosi, I. M. & Vincze, M. 2018 Experimental demonstration of the water-holding property of three-dimensional vortices. Europhys. Lett. 123, 44001.CrossRefGoogle Scholar
Tian, S., Gao, Y., Dong, X. & Liu, C. 2018 Definitions of vortex vector and vortex. J. Fluid Mech. 564, 57103.Google Scholar
Weiss, J. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48, 273294.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Can vortex criteria be objectivized?
Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Can vortex criteria be objectivized?
Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Can vortex criteria be objectivized?
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *