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

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Abstract

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.

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Acknowledgments

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.

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Authors and Affiliations

  1. School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China

    Yi-qian Wang

  2. Department of Mathematics, University of Texas at Arlington, Arlington, Texas, 76019, USA

    Yi-sheng Gao & Chaoqun Liu

  3. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China

    Jian-ming Liu

Authors
  1. Yi-qian Wang
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  2. Yi-sheng Gao
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  3. Jian-ming Liu
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  4. Chaoqun Liu
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Corresponding author

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.

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Wang, Yq., Gao, Ys., Liu, Jm. 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

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  • DOI: https://doi.org/10.1007/s42241-019-0032-2

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