**Author details**

Yifei Yu and Chaoqun Liu\* Department of Mathematics, University of Texas at Arlington, Arlington, Texas, USA

\*Address all correspondence to: cliu@uta.edu

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Kaczorowski M, Chong K-L, Xia K-Q. Turbulent flow in the bulk of Rayleigh– Bénard convection: Aspect-ratio dependence of the small-scale properties. Journal of Fluid Mechanics. 2014;**747**:73-102

[2] Xi H-D, Zhang Y-B, Hao J-T, Xia K-Q. Higher-order flow modes in turbulent Rayleigh–Bénard convection. Journal of Fluid Mechanics. 2016;**805**:31-51

[3] Zhou Y. Rayleigh–Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. I. Physics Reports. 2017;**720**:1-136

[4] Zhou Y. Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II. Physics Reports. 2017;**723**:1-160

[5] Liu C, Gao Y-S, Dong X-R, Wang Y-Q, Liu J-M, Zhang Y-N, et al. Third generation of vortex identification methods: Omega and Liutex/Rortex based systems. Journal of Hydrodynamics. 2019;**31**:205-223. DOI: 10.1007/s42241-019-0022-4

[6] Wang Y, Yang Y, Yang G, Liu C. DNS study on vortex and vorticity in late boundary layer transition. Communications in Computational Physics. 2017;**22**:441-459. DOI: 10.4208/ cicp.OA-2016-0183

[7] Hunt JCR, Wray AA, Moin P. Eddies, stream, and convergence zones in turbulent flows. Center for Turbulent Research Report CTR-S88. 1988:193-208

[8] Chong MS, Perry AE, Cantwell BJ. A general classification of threedimensional flow fields. Physics of Fluids A: Fluid Dynamics. 1990;**2**:765-777. DOI: 10.1063/1.857730

[9] Zhou J, Adrian RJ, Balachandar S, Kendall TM. Mechanisms for generating coherent packets of hairpin vortices in channel flow. Journal of Fluid Mechanics. 1999;**387**:353-396. DOI: 10.1017/s002211209900467x

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

[11] Gao Y, Liu C. Rortex and comparison with eigenvalue-based vortex identification criteria. Physics of Fluids. 2018;**30**:085107. DOI: 10.1063/1.5040112

[12] Cuissa JC, Steiner O. Innovative and automated method for vortex identification-I. Description of the SWIRL algorithm. Astronomy & Astrophysics. 2022;**668**:A118

[13] Shen C, Yang R, Wang M, He S, Qing S. Application of vortex identification methods in vertical slit fishways. Water. 2023;**15**:2053

[14] Xu W, Wang Y, Gao Y, Liu J, Dou H-S, Liu C. Observation on Liutex similarity in the dissipation subrange of turbulent boundary layer. Computers & Fluids. 2022;**246**:105613

[15] Xu W-Q, Wang Y-Q, Gao Y-S, Liu J-M, Dou H-S, Liu C. Liutex similarity in turbulent boundary layer. Journal of Hydrodynamics. 2019;**31**:1259-1262. DOI: 10.1007/s42241-019-0094-1

[16] Yan B, Wang Y, Liu C. Liutexrepresented vortex spectrum in turbulence. Entropy. 2022;**25**:25

[17] Trieu XM, Liu J, Gao Y, Charkrit S, Liu C. Proper orthogonal decomposition analysis of coherent structure in a turbulent flow after a micro-vortex

generator. Applied Mathematical Modelling. 2022;**104**:140-162

[18] Dong X, Hao C, Liu C. Correlation between vorticity, Liutex and shear in boundary layer transition. Computers & Fluids. 2022;**238**:105371

[19] Dong X, Hao C, Dong Y, Liu C, Li Y. Investigation of vortex motion mechanism of synthetic jet in a cross flow. AIP Advances. 2022;**12**: 035045

[20] Liu C, Yu Y, Gao Y-S. Liutex based new fluid kinematics. Journal of Hydrodynamics. 2022;**34**:355-371

[21] Liu C, Liu Z. New governing equations for fluid dynamics. AIP Advances. 2021;**11**:115025. DOI: 10.1063/ 5.0074615

[22] Liu C. New ideas on governing equations of fluid dynamics. Journal of Hydrodynamics. 2021;**33**:861-866

[23] Yu Y, Shrestha P, Alvarez O, Nottage C, Liu C. Investigation of correlation between vorticity, Q, λci, λ2, Δ and Liutex. Computers & Fluids. 2021;**225**: 104977. DOI: 10.1016/j.compfluid.2021. 104977

[24] Charkrit S, Shrestha P, Liu C. Liutex core line and POD analysis on hairpin vortex formation in natural flow transition. Journal of Hydrodynamics. 2020;**32**:1109-1121

[25] Yu Y, Shrestha P, Alvarez O, Nottage C, Liu C. Correlation analysis among vorticity, Q method and Liutex. Journal of Hydrodynamics. 2020;**32**: 1207-1211. DOI: 10.1007/s42241-020- 0069-2

[26] Dong X-R, Cai X-S, Dong Y, Liu C. POD analysis on vortical structures in MVG wake by Liutex core line

identification. Journal of Hydrodynamics. 2020;**32**:497-509

[27] Wang Y-Q, Gao Y-S, Xu H, Dong X-R, Liu J-M, Xu W-Q, et al. Liutex theoretical system and six core elements of vortex identification. Journal of Hydrodynamics. 2020;**32**:197-211. DOI: 10.1007/s42241-020-0018-0

[28] Yu Y, Shrestha P, Nottage C, Liu C. Principal coordinates and principal velocity gradient tensor decomposition. Journal of Hydrodynamics. 2020;**32**: 441-453. DOI: 10.1007/s42241-020- 0035-z

[29] Tran CT, Long X-P, Ji B, Liu C. Prediction of the precessing vortex core in the Francis-99 draft tube under offdesign conditions by using Liutex/Rortex method. Journal of Hydrodynamics. 2020;**32**:623-628

[30] Wang Y, Gao Y, Liu C. Liutex and Third Generation of Vortex Identification:Workshop from Aerospace and Aeronautics World Forum 2021. Singapore: Springer Nature; 2023

[31] Liu C, Wang Y. Liutex and Third Generation of Vortex Definition and Identification: An Invited Workshop from Chaos 2020. Cham: Springer Cham; 2021

[32] Liu C, Xu H, Cai X, Gao Y. Liutex and Its Applications in Turbulence Research. London: Academic Press; 2020

[33] Liu C, Gao Y. Liutex-Based and Other Mathematical, Computational and Experimental Methods for Turbulence Structure. Singapore: Bentham Science Publishers; 2020

[34] Gao Y-S, Liu J-M, Yu Y-F, Liu C. A Liutex based definition and identification of vortex core center lines.

Journal of Hydrodynamics. 2019;**31**: 445-454. DOI: 10.1007/s42241-019- 0048-7

[35] Xu H, Cai X-S, Liu C. Liutex (vortex) core definition and automatic identification for turbulence vortex structures. Journal of Hydrodynamics. 2019;**31**:857-863. DOI: 10.1007/ s42241-019-0066-5

[36] Li H, Liu Y, Wang D, Xu H. Liutex (vortex) core and tube identification and automatic generation algorithms. Computers & Fluids. 2023;**250**:105731. DOI: 10.1016/j.compfluid.2022.105731

[37] Yan Y, Liu C. Shear layer stability analysis in later boundary layer transition and MVG controlled flow. 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. 2013

[38] Bake S, Meyer D, Rist U. Turbulence mechanism in Klebanoff transition: A quantitative comparison of experiment and direct numerical simulation. Journal of Fluid Mechanics. 2002;**459**:217-243
