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Liutex based new fluid kinematics

  • Special Column on the 5th CMHL Symposium 2022 (Guest Editor De-Cheng Wan)
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Abstract

The traditional Cauchy-Stokes (C-S) decomposition states that the velocity gradient tensor can be decomposed into a symmetric tensor and an anti-symmetric tensor, namely the strain-rate tensor and the vorticity tensor. However, there are two problems with the C-S decomposition. One is that the anti-symmetric (vorticity) tensor cannot represent the fluid rotation or vortex. Another is that the symmetric (strain-rate) tensor cannot distinguish the stretching/compression and shear. Since vorticity cannot distinguish between the non-rotational shear and the rigid rotation, vorticity has been decomposed into a rigid rotation part called “Liutex” and an anti-symmetric shear in our previous work. A Liutex-based principal coordinate system has been proposed, and the corresponding velocity gradient tensor decomposition, called the principal decomposition, is presented in this principal coordinate system, which results in: (1) a Liutex tensor that represents rigid rotation, (2) a tensor that represents pure shear and (3) a tensor that represents stretching/compression. However, each point has its own principal coordinate system, which implies that the principal decomposition is performed in different principal coordinate systems, not the original (global) coordinate system. To address this issue, the principal decomposition in the original coordinate system is derived in this paper, and, therefore, provides a new kinematic approach to study the local rigid rotation, pure shear, and stretching/compression. The principal decomposition is unique, Galilean invariant and has clear physical meaning. The new velocity gradient tensor decomposition could become a foundation for new fluid kinematics.

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Acknowledgments

The authors are thankful for the support by the UTA Department of Mathematics which houses the UTA Vortex and Turbulence Research Team. The authors are also grateful to Texas Advanced Computing Center (TACC) for providing computation time. The DNSUTA code was released by Chaoqun Liu in 2009 and the Liutex code was released by Chaoqun Liu in 2018 which can be downloaded from the UTA web site at https://www.uta.edu/math/cnsm/public_html/cnsm/cnsm.html.

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Authors and Affiliations

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

    Chaoqun Liu & Yifei Yu

  2. College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Yi-sheng Gao

Authors
  1. Chaoqun Liu
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  2. Yifei Yu
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  3. Yi-sheng Gao
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Corresponding author

Correspondence to Chaoqun Liu.

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Biography: Chaoqun Liu, Male, Ph. D., Professor

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Liu, C., Yu, Y. & Gao, Ys. Liutex based new fluid kinematics. J Hydrodyn 34, 355–371 (2022). https://doi.org/10.1007/s42241-022-0046-z

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  • DOI: https://doi.org/10.1007/s42241-022-0046-z

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