Abstract
Classical fluid kinematics or Cauchy-Stokes decomposition mistreated vorticity as fluid rotation and mixed flow stretching with shearing. Classical fluid dynamics or Navier-Stokes (N-S) equations are based on the classical kinematics and treated vorticity as null in contribution of forces and mixed the stretching force with the shearing force, which is not consistent with the Galilean invariancy. N-S equations also neglect the flow rotation. It is believed that N-S equations may work for incompressible and laminar flow but are not satisfied for turbulent flow and compressible flow especially for high-speed flow. Based on Liutex, new fluid kinematics has been established by Liu in 2021, which gives a Liutex-based principal coordinate system and a new principal decomposition in that system, which has been transferred back to the original Cartesian coordinate system. The principal decomposition of velocity gradient tensor has four parts which are called rotation, stretching, anti-symmetric shearing and symmetric shearing. Four forces are derived according to the four parts of the velocity gradient tensor. According to the new fluid kinematics, it is reported in this letter that based on the principal decomposition, a new relation between the velocity gradient tensor and stress-rate tensor has been established to form a new fluid dynamics equation to govern fluid flow. The new governing equation may be applicable to both laminar flow and turbulent flow, and both incompressible flow and compressible flow including high-speed flow for reasonable results with reasonable grids. Further numerical experiment is needed to verify.
Similar content being viewed by others
References
Batchelor G. K. An introduction to fluid dynamics [M]. New York, USA: Cambridge University Press, 2000.
Jiménez J. Coherent structures in wall-bounded turbulence [J]. Journal of Fluid Mechanics, 2018, 842: P1.
Zhou Y., Clark T. T., Clark D. S. et al. Turbulent mixing and transition criteria of flows induced by hydrodynamic instabilities [J]. Physics of Plasmas, 2019, 26(8): 80901
Reynolds O. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels [J]. Philosphical Transaction of the Royal Society, 1883, 174: 935–982.
Liu C. New fluid kinematics [J]. Journal of Hydrodynamics, 2021, https://doi.org/10.1007/s42241-021-0037-5.
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(3): 35103.
Gao Y., Liu C. Rortex and comparison with eigenvalue-based vortex identification criteria [J]. Physics of Fluids, 2018, 30(8): 85107.
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, 31(2): 205–223.
Wang Y. Q., Gao Y. S., Xu H. et al. Liutex theoretical system and six core elements of vortex identification [J]. Journal of Hydrodynamics, 2020, 32(2): 197–211.
Liu C., Wang Y. Liutex and third generation of vortex definition and identification for turbulence research [M]. Berlin Heidelberg, Germany: Springer, 2021.
Liu C., Xu H., Cai X. et al. Liutex and its applications in turbulence research [M]. Cambridge, MA, USA: Academic Press, 2020.
Liu C., Gao Y. Liutex-based and Other mathematical, computational and experimental methods for turbulence structure [M]. Sharjah, United Arab Emirates: Bentham Science Publishers, 2020.
Chong M. S., Perry A. E., Cantwell B. J. A general classification of three-dimensional flow fields [J]. Physics of Fluids A: Fluid Dynamics, 1990, 2(5): 765–777.
Acknowledgments
The author thanks his students and visitors including Yi-qian Wang, Yang-rui Dong, Yi-sheng Gao, Jian-ming Liu, Pan-pan Yan, Wen-qian Xu, Yong-hua Yan, Yi-fei Yu, Sita Charkrit, Pushpa Threstha, Charles Nottage, Oscar Alverez, Vishwa Patel, Dalal Almutairi, Xuan Trieu. The author also thanks his collaborators including Hongyi Xu, Xiao-shu Cai, Hua-shu Dou. The author is grateful to Prof. Lian-di Zhou for countless discussions about the vortex definition. This work was mainly supported by the Department of Mathematics of University of Texas at Arlington as the author is the full-time professor in UTA and all students and visitors were housed by UTA. The author is grateful to Texas Advanced Computation Center (TACC) for providing computation hours.
Author information
Authors and Affiliations
Corresponding author
Additional information
Biography: Chaoqun Liu, Male, Ph. D., Professor
Rights and permissions
About this article
Cite this article
Liu, C. New ideas on governing equations of fluid dynamics. J Hydrodyn 33, 861–866 (2021). https://doi.org/10.1007/s42241-021-0050-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42241-021-0050-8