Abstract
Recently, new progress has been made in vortex recognition, and the definition of Liutex (Rortex) based on eigenvectors has established a relation between rotation axis and eigenvectors of velocity gradient tensor. Based on this relation, the geometric and physical meanings of Liutex magnitude and the concept of rigid-rotation of vortex are revisited. Considering the simple shear, geometric meaning of Liutex can be explained through geometric relations in this paper, believing that the dominant quantity rotation described by Liutex implies rotational strength without non-circular symmetry. Moreover, The mathematical condition of vortex boundary is given: the set of points with multiple roots in the characteristic equation of velocity gradient tensor in a flow field. In this way, the topological structure of critical point theory is applied to vortex boundary. According to whether the velocity gradient tensor can be diagonalized, there are shear boundary and non-shear boundary, while according to the positive, negative and zero of the double root, there would be stable boundary, unstable boundary and degenerate boundary. Under different decompositions, we can analyze the superposition of fluid deformation behavior more clearly by mapping image space, which can be used to compare Helmholtz decomposition with Liutex velocity gradient decomposition.
National Science and Technology Major Project (2017-II-0006-0019, 2017-I-0009-0010).
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References
J. Wu, Y. Yang, Thoughts on vortex definition. Acta Aerodyn. Sin. 1, 1–8 (2020)
C. Liu, Y. Gao, S. Tian, et al., Rortex a new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids 30(3) (2018)
W. Xu, Y. Gao, Y. Deng, et al., An explicit expression for the calculation of the Rortex vector. Phys. Fluids 31(9), 095102 (2019)
Y. Wang, Y. Gao, C. Liu, Galilean invariance of Rortex. Phys. Fluids 30, 111701 (2018). https://doi.org/10.1063/1.5058939
V. Kolář, J. Å Ãstek, Stretching response of Rortex and other vortex-identification schemes. AIP Adv. 9(10), 105025 (2019)
Y. Gao, J. Liu, Y. Yu, C. Liu, A Liutex based definition and identification of vortex core center lines. J. Hydrodynam. 31(2), 774–781 (2019)
C. Liu, Y. Gao, X. Dong, J. Liu, Y. Zhang, X. Cai, N. Gui, Third generation of vortex identification methods: Omega and Liutex/Rortex based systems. J. Hydrodynam. 31(2), 1–19 (2019). https://doi.org/10.1007/s42241-019-0022-4
L. Zhen, Z. Xi-Wen, F. He, Evaluation of vortex criteria by virtue of the quadruple decomposition of velocity gradient tensor. Acta Phys. Sin. 05, 249–255 (2014)
W. Xu, Y. Gao, Y. Deng, J. Liu, C. Liu, An explicit expression for the calculation of the Rortex vector. Phys. Fluids (2019)
Y. Gao, Y. Yu, J. Liu, C. Liu, Explicit expressions for Rortex tensor and velocity gradient tensor decomposition. Phys. Fluids (2019)
J. Liu, C. Liu, Modified normalized Rortex/vortex identification method. Phys. Fluids 31, 061704 (2019). https://doi.org/10.1063/1.5109437
J. Liu, Y. Gao, C. Liu, An objective version of the Rortex vector for vortex identification. Phys. Fluids 31, 065112 (2019). https://doi.org/10.1063/1.5095624
Y. Wang, Y. Gao, J. Liu, C. Liu, Explicit formula for the Liutex vector and physical meaning of vorticity based on the Liutex-Shear decomposition. J. Hydrodynam. (2019). https://doi.org/10.1007/s42241-019-0032-2
Y. Gao, C. Liu, Rortex and comparison with eigenvalue-based vortex identification criteria. Phys. Fluids 30, 085107 (2018). https://doi.org/10.1063/1.5040112
Y. Gao, C. Liu, Letter: Rortex based velocity gradient tensor decomposition. Phys. Fluids 31, 011704 (2019). https://doi.org/10.1063/1.5084739
X. Dong, Y. Gao, C. Liu, New normalized Rortex/vortex identification method. Phys. Fluids 31, 011701 (2019). https://doi.org/10.1063/1.5066016
X. Dong, Y. Gao, C. Liu, Study on vorticity structure in late flow transition. Phys. Fluids 30, 104108 (2018)
S. Tian, Y. Gao, X. Dong, C. Liu, A definition of vortex vector and vortex. J. Fluid Mech. 849, 312–339 (2018). https://doi.org/10.1017/jfm.2018.406
X. Dong, Y. Yan, Y. Yang, G. Dong, C. Liu, Spectrum study on unsteadiness of shock wave-vortex ring interaction. Phys. Fluids 30, 056101 (2018). https://doi.org/10.1063/1.5027299
X. Dong, S. Tian, C. Liu, Correlation analysis on volume vorticity and vortex in late boundary layer transition. Phys. Fluids 30, 014105 (2018)
H. Xu, X. Cai, C. Liu, Liutex core definition and automatic identification for turbulence structures. J. Hydrodynam. (2019)
J. Liu, Y. Deng, Y. Gao, S. Charkrit, C. Liu, Letter: Mathematical foundation of turbulence generation-symmetric to asymmetric Liutex/Rortex. J. Hydrodynam. (2019)
J. Liu, Y. Gao, Y. Wang, C. Liu, Galilean invariance of Omega vortex identification method. J. Hydrodynam. (2019). https://doi.org/10.1007/s42241-019-0024-2
J. Liu, Y. Gao, Y. Wang, C. Liu, Objective Omega vortex identification method. J. Hydrodynam. (2019). https://doi.org/10.1007/s42241-019-0028-y
Y. Zhang, X. Qiu, F. Chen, K. Liu, Y.-N. Zhang, X.-R. Dong, C. Liu, A selected review of vortex identification methods with applications. Int. J. Hydrodynam. 30(5) (2018). https://doi.org/10.1007/s42241-018-0112-8
X. Dong, Y. Wang, X. Chen, Y. Zhang, C. Liu, Determination of epsilon for Omega vortex identification method. Int. J. Hydrodynam. 30(4), 541–548 (2018)
Z. Zhifen, Qualitative Theory of Differential Equations (Science Press, Beijing, 1985)
J. Wu, Introduction to Vortex Dynamics (Higher Education Press, Beijing, 1993)
M.S. Chong, A.E. Perry, B.J. Cantwell, A general classification of three-dimensional flow fields. Phys. Fluids A Fluid Dynam. 2(5), 765–777 (1990)
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Li, X., Zheng, Q., Jiang, B. (2021). Mathematical Definition of Vortex Boundary and Boundary Classification Based on Topological Type. In: Liu, C., Wang, Y. (eds) Liutex and Third Generation of Vortex Definition and Identification. Springer, Cham. https://doi.org/10.1007/978-3-030-70217-5_6
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DOI: https://doi.org/10.1007/978-3-030-70217-5_6
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