Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Explicit formula for the Liutex vector and physical meaning of vorticity based on the Liutex-Shear decomposition

  • 140 Accesses

  • 10 Citations


In the present study, the physical meaning of vorticity is revisited based on the Liutex-Shear (RS) decomposition proposed by Liu et al. in the framework of Liutex (previously called Rortex), a vortex vector field with information of both rotation axis and swirling strength (Liu et al. 2018). It is demonstrated that the vorticity in the direction of rotational axis is twice the spatial mean angular velocity in the small neighborhood around the considered point while the imaginary part of the complex eigenvalue (λci) of the velocity gradient tensor (if exist) is the pseudo-time average angular velocity of a trajectory moving circularly or spirally around the axis. In addition, an explicit expression of the Liutex vector in terms of the eigenvalues and eigenvectors of velocity gradient is obtained for the first time from above understanding, which can further, though mildly, accelerate the calculation and give more physical comprehension of the Liutex vector.

This is a preview of subscription content, log in to check access.


  1. [1]

    Wu J., Ma H., Zhou M. Vorticity and Vortices Dynamics [B]. Springer-Verlag, Berlin Heidelberg, 2006.

  2. [2]

    Dong X., Dong G., Liu C. Study on vorticity structures in late flow transition [J]. Physics of Fluids, 2018, 30:014105.

  3. [3]

    Wang Y., Yang Y., Yang G. et al. DNS study on vortex and vorticity in late boundary layer transition [J]. Communications in Computational Physics, 2017, 22: 441–459.

  4. [4]

    Robinson S. K. Coherent motion in the turbulent boundary layer [J]. Annual Review of Fluid Mechanics, 1991, 23: 601–639.

  5. [5]

    Jiménez J. Coherent structures in wall-bounded turbulence [J]. Journal of Fluid Mechanics, 2018, 842: P1.

  6. [6]

    Theodorsen T. Mechanism of turbulence [C]. in Proceedings of the Midwestern Conference on Fluid Mechanics, Columbus, Ohio, USA, 1952.

  7. [7]

    Adrian R. J. Hairpin vortex organization in wall turbulence [J]. Physics of Fluids, 2007, 19: 041301.

  8. [8]

    Wang Y., Al-Dujaly H., Yan Y. et al. Physics of multiple level hairpin vortex structures in turbulence [J]. Science China: Physics, Mechanics and Astronomy, 2016, 59: 624703.

  9. [9]

    Eitel-Amor G., órlú R., Schlatter P. et al. Hairpin vortices in turbulent boundary layers [J]. Physics of Fluids, 2015, 27: 025108.

  10. [10]

    Kasagi N., Sumitani Y., Suzuki Y. et al. Kinematics of the quasi-coherent vertical structure in near-wall turbulence [J]. International Journal of Heat and Fluid Flow, 1995, 16: 2–10.

  11. [11]

    Jeong J., Hussain F., Schoppa W. et al. Coherent structures near the wall in a turbulent channel flow [J]. Journal of Fluid Mechanics, 1997, 332: 185–214.

  12. [12]

    Iida O., Iwatsuki M., Nagano Y. Vortical turbulence structure and transport mechanism in a homogeneous shear flow [J]. Physics of Fluids, 2000, 12: 2895.

  13. [13]

    Kline S. J., Reynolds W. C., Schraub F. A., Runstadler P. W. The structure of turbulent boundary layers [J]. Journal of Fluid Mechanics, 1967, 30: 741–773.

  14. [14]

    Schoppa W., Hussain F. Coherent structure generation in near-wall turbulence [J]. Journal of Fluid Mechanics, 2002, 453: 57–108.

  15. [15]

    Chong M., Perry A., Cantwell B. A general classification of three dimensional flow fields [J]. Physics of Fluids A, 1990, 2: 765–777.

  16. [16]

    Hunt J., Wray A., Moin P. Eddies, streams, and convergence zones in turbulent flows [C]. Proceedings of the Summer Program. Center for Turbulence Research, 1988, 193–208.

  17. [17]

    Jeong J., Hussain F. On the identification of a vortex [J]. Journal of Fluid Mechanics, 1995, 285: 69–94.

  18. [18]

    Liu C., Wang Y., Yang Y. et al. New Omega vortex identification method [J]. Science China: Physics, Mechanics and Astronomy, 2016, 59:684711.

  19. [19]

    Chen H., Adrian R. J., Zhong Q. et al. Analytic solutions for three-dimensional swirling strength in compressible and incompressible flows [J]. Physics of Fluids, 2014, 26: 081701.

  20. [20]

    Chen Q., Zhong Q., Qi M. et al. Comparison of vortex identification criteria for velocity fields in wall turbulence [J]. Physics of Fluids, 2015, 27: 085101.

  21. [21]

    Zhang Y. N., Qiu X., Chen F. et al. A selected review of vortex identification methods with applications [J]. Journal of Hydrodynamics, 2018, 30(5): 767–779.

  22. [22]

    Epps B. Review of vortex identification methods [C]. 55th AIAA Aerospace Sciences Meeting, Grapevine, Texas, USA, 2017.

  23. [23]

    Liu C., Gao Y., Tian S. et al. Rortex-A new vortex vector definition and vorticity tensor and vector decompositions [J]. Physics of Fluids, 2018, 30: 034103.

  24. [24]

    Tian S., Gao Y., Dong X. et al. Definition of vortex vector and vortex [J]. Journal of Fluid Mechanics, 2018, 849: 312–339.

  25. [25]

    Gao Y., Liu C. Rortex and comparison with eigenvalue-based vortex identification criteria [J]. Physics of Fluids, 2018, 30: 085107.

  26. [26]

    Liu C., Gao Y. S., Dong X. R. et al. Third generation of vortex identification methods: Omega and Liutex/Rortex based systems [J]. Journal of Hydrodynamics, 2019, https://doi.org/10.1007/s42241-019-0022-4.

  27. [27]

    Liu J., Gao. Y., Wang Y. et al. Objective Omega vortex identification method [J]. Journal of Hydrodynamics, 2019, https://doi.org/10.1007/s42241-019-0028-y

  28. [28]

    Liu J., Wang Y., Gao Y. et al. Galilean invariance of Omega vortex identification method [J]. Journal of Hydrodynamics, 2019, https://doi.org/10.1007/s42241-019-0024-2

  29. [29]

    Lele S. K. Compact finite difference schemes with spectral-like resolution [J]. Journal of Computational Physics, 1992, 103: 16–42.

  30. [30]

    Lee C., Li R. Dominant structure for turbulent production in a transitional boundary layer [J]. Journal of Turbulence, 2007, 8: 55.

  31. [31]

    Liu C., Yan Y., Lu P. Physics of turbulence generation and sustence in a transitional boundary layer [J]. Computers and Fluids, 2014, 102: 353–384.

  32. [32]

    Liu C., Chen L. Parallel DNS for vortex structure of late stages of flow transition [J]. Computers and Fluids, 2011, 45: 129–137.

  33. [33]

    Laizet S., Lamballais E. High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy [J]. Journal of Computational Physics, 2008, 228: 5989–6015.

  34. [34]

    Laizet S., Li N. Incompact3d: A powerful tool to tackle turbulence problems with up to O(105) computational cores [J]. International Journal for Numerical Methods in Fluids, 2011, 67: 1735–1757.

Download references


The work is partly supported by China Post-Doctoral Science Foundation (Grant No. 2017M610876), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant Nos.18KJA110001), and the Visiting Scholar Scholarship of the China Scholarship Council (Grant No. 201808320079). This work is partly accomplished by using code DNSUTA developed by Dr. Chaoqun Liu at the University of Texas at Arlington.

Author information

Correspondence to Chaoqun Liu.

Additional information

Project supported by the National Nature Science Foundation of China (Grant Nos. 11702159, 91530325).

Biography: Yi-qian Wang (1987-), Male, Ph. D.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Gao, Y., Liu, J. et al. Explicit formula for the Liutex vector and physical meaning of vorticity based on the Liutex-Shear decomposition. J Hydrodyn 31, 464–474 (2019). https://doi.org/10.1007/s42241-019-0032-2

Download citation

Key words

  • Vorticity
  • Liutex
  • Liutex-Shear decomposition
  • explicit formula of Liutex