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Journal of Hydrodynamics 2022年第3期34卷 355-371页

作者： Chaoqun Liu yifei Yu yi-sheng gao Department of Mathematics University of Texas at ArlingtonArlingtonTexas76019USA College of Aerospace Engineering Nanjing University of Aeronautics and AstronauticsNanjing210016China

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.

关键词： Liutex principal coordinates principal decomposition Helmholtz Cauchy-Stokes(C-S)

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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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Journal of Hydrodynamics 2019年第3期31卷 464-474页

作者： yi-qian Wang yi-sheng gao Jian-ming Liu Chaoqun Liu School of Aerospace Engineering Tsinghua University Beijing 100084 China Department of Mathematics University of Texas at Arlington Arlington 76019 Texas USA School of Mathematics and Statistics Jiangsu Normal University Xuzhou 221116 China

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 (2c.) 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.

关键词： Vorticity Liutex Liutex-Shear(RS)decomposition explicit formula of Liutex

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A Liutex based definition and identification of vortex core center lines

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Journal of Hydrodynamics 2019年第3期31卷 445-454页

作者： yi-sheng gao Jian-ming Liu yi-fei Yu Chaoqun Liu Department of Mathematics University of Texas at Arlington Arlington Texas 76019 USA School of Mathematics and Statistics Jiangsu Normal University Xuzhou 221116 China

Six core issues for vortex definition and identification concern with (1) the absolute strength,(2) the relative strength,(3) the rotational axis,(4) the vortex core center,(5) the vortex core size, and (6) the vortex boundary (Liu C. 2019). However, most of the currently popular vortex identification methods, including the Q criterion, the criterion and the Acj criterion etc., are Eulerian local region-type vortex identification criteria and can only approximately identify the vortex boundary by somewhat arbitrary threshold. On the other hand, the existing Eulerian local line-type methods, which seek to extract line-type features such as vortex core line, are not entirely satisfactory since most of these methods are based on vorticity or pressure minimum that will fail in many cases. The key issue is the lack of a reasonable mathematical definition for vortex core center. To address this issue, a Liutex (previously named Rortex) based definition of vortex core center is proposed in this paper. The vortex core center, also called vortex rotation axis line here, is defined as a line where the Liutex magnitude gradient vector is aligned with the Liutex vector, which mathematically implies that the cross product of the Liutex magnitude gradient vector and the Liutex vector on the line is equal to zero. Based on this definition, a novel three-step method for extracting vortex rotation axis lines is presented. Two test cases, namely the Burgers vortex and hairpin vortices, are examined to justify the proposed method. The results demonstrate that the proposed method can successfully identify vortex rotation axis lines without any user-specified threshold, so that the proposed method is very straightforward, robust and efficient.

关键词： Vortex definition vortex identification flow visualization Liutex Liutex magnitude gradient vortex core lines

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Objective Omega vortex identification method

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Journal of Hydrodynamics 2019年第3期31卷 455-463页

作者： Jian-ming Liu yi-sheng gao yi-qian Wang Chaoqun Liu School of Mathematics and Statistics Jiangsu Normal University Xuzhou 221116 China Department of Mathematics University of Texas at Arlington Arlington 76019 Texas USA School of Aerospace Engineering Tsinghua University Beijing 100084 China

A new vortex identification method (Liu et al. 2016) was proposed to represent the rotation relative strength and capture and visualize the vortices in our previous study. The basic idea of the Ω method is that a ratio of the vorticity squared over the summation of the vorticity squared and the deformation squared should be used to measure the relative rotation strength. However, the vorticity tensor norm is not objective. Thus, a moving observer will observe different vortex structures in a moving reference frame, which will make confusions about the real vortex structures. In the present study, by the definitions of the net spin tensor and the net vorticity vector, an objective Ω vortex identification method is presented with examples to verify that the vortex structures retain in a moving reference frame.

关键词： Vortex identification the Ω method objectivity turbulence visualization turbulence

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Galilean invariance of Omega vortex identification method

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Journal of Hydrodynamics 2019年第2期31卷 249-255页

作者： Jian-ming Liu yi-qian Wang yi-sheng gao Chaoqun Liu School of Mathematics and Statistics Jiangsu Normal UniversityXuzhou 221116China Department of Mathematics University of Texas at ArlingtonArlington 76019TexasUSA School of Aerospace Engineering Tsinghua UniversityBeijing 100084China

In the new vortex identification method(Liu et al.2016)to represent the rotation level and capture and visualize the vortices,proposed in our previous study,the independence of the reference frame and the Galilean invariant were not proved.In the present study,the Galilean invariance of the omega vortex identification method is proved and several examples are presented to verify the conclusion.

关键词： Vortex identification Ωmethod.Galilean invariance turbulence visualization

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Mathematical foundation of turbulence generation-From symmetric to asymmetric Liutex

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Journal of Hydrodynamics 2019年第3期31卷 632-636页

作者： Jian-ming Liu Yue Deng yi-sheng gao Sita Charkrit Chaoqun Liu School of Mathematics and Statistics Jiangsu Normal University Xuzhou 221116 China Department of Mathematics University of Texas at Arlington Arlington 76019 Texas USA Department of Physics University of Texas at Arlington Arlington 76019 Texas USA

Vortices have been regarded as the building blocks and muscles of turbulence for a long time. To better describe and analyze vortices or vortical structures, recently a new physical quantity called Liutex (previously named Rortex) is introduced to present the rigid rotation part of fluid motion (Liu et al. 2018). Since turbulence is closely related to the vortex, it can be postulated that there exists no turbulence without Liutex. According to direct numerical simulations (DNS) and experiments, forest of hairpin vortices has been found in transitional and low Reynolds number turbulent flows, while one-leg vortices are predominant in full developed turbulent flows. This paper demonstrates that the hairpin vortex is unstable. The hairpin vortex will be weakened or lose one leg by the shear and Liutex interaction, based on the Liutex definition and mathematical analysis without any physical assumptions. The asymmetry of the vortex is caused by the interaction of symmetric shear and symmetric Liutex since the smaller element of a pair of vorticity elements determines the rotational strength. For a 2-D fluid rotation, if a disturbance shear effects the larger element, the rotation strength will not be changed, but if the disturbance shear effects the smaller element, the rotation strength will be immediately changed due to the definition of the Liutex strength. For a rigid rotation, if the shearing part of the vorticity and Liutex present the same directions, e.g., clockwise, the Liutex strength will not be changed. If the shearing part of the vorticity and Liutex present different directions, e.g., one clockwise and another counterclockwise, the Liutex strength will be weakened.Consequently, the hairpin vortex could lose the symmetry and even deform to a one-leg vortex. The one-leg vortex cannot keep balance, and the chaotic motion and flow fluctuation are doomed. This is considered as the mathematical foundation of turbulence formation. The DNS results of boundary layer transition are used to justify this theory.

关键词： Turbulence generation mathematical principle Liutex/Rortex asymmetry

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Liutex theoretical system and six core elements of vortex identification

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Journal of Hydrodynamics 2020年第2期32卷 197-211页

作者： yi-qian Wang yi-sheng gao Hongyi Xu Xiang-rui Dong Jian-ming Liu Wen-qian Xu Meng-long Chen Chaoqun Liu School of Mathematical Science Soochow UniversitySuzhou215006China College of Aerospace Engineering Nanjing University of Aeronautics and AstronauticsNanjing210016China Aeronautics and Astronautics Department Fudan UniversityShanghai200433China School of Energy and Power Engineering University of Shanghai for Science and TechnologyShanghai200093China School of Mathematics and Statistics Jiangsu Normal UniversityXuzhou221116China Faculty of Mechanical Engineering and Automation Zhejiang Sci-Tech UniversityHangzhou310018China SimpleHPC LTD. Shanghai200030China Department of Mathematics University of Texas at ArlingtonArlington76019USA

The third-generation vortex identification method of Liutex(previously called Rortex)was introduced by the team led by Prof.Chaoqun Liu from University of Texas at Arlington to mathematically extract the rigid rotation part from the fluid motion,and thus to define and visualize vortices.Unlike the vorticity-based first generation and the scalar-valued second generation,Q,λ2,Δandλci methods for example,the Liutex vector provides a unique,mathematical and systematic way to define vortices and visualize vortical structures from multiple perspectives without ambiguity.In this article,we summarize the recent developments of the Liutex framework and discuss the Liutex theoretical system including its existence,uniqueness,stability,Galilean invariance,locality and globality,decomposition in tensor and vector forms,Liutex similarity in turbulence,and multiple Liutex-based vortex visualization methods including Liutex lines,Liutex magnitude iso-surfaces,Liutex-Ωmethod,and Liutex core line method,etc..Thereafter,the six core elements of vortex identification,including(1)absolute strength,(2)relative strength,(3)local rotational axis,(4)vortex rotation axes,(5)vortex core size,(6)vortex boundary,are used as touchstones against which the Liutex vortex identification system is examined.It is demonstrated with illustrative examples that the Liutex system is able to give complete and precise information of all six core elements in contrast to the failure and inaccuracy of the first and second-generation methods.The important concept that vorticity cannot represent vortex and the superiority of the Liutex system over previous methods are reiterated and stated in appropriate places throughout the paper.Finally,the article concludes with future perspectives,especially the application of the Liutex system in studying turbulence mechanisms encouraged by the discovery of Liutex similarity law.As a newly defined physical quantity,Liutex may open a door for quantified vortex and turbulence research including Liutex(vortex)dynamics and

关键词： Vortex identification Liutex vector six core elements Liutex core line Liutex similarity turbulence

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Journal of Hydrodynamics 2019年第6期31卷 1259-1262页

作者： Wen-qian Xu yi-qian Wang yi-sheng gao Jian-ming Liu Hua-shu Dou Chaoqun Liu Faculty of Mechanical Engineering and Automation Zhejiang Sci-Tech UniversityHangzhou 310018China Department of Mathematics University of Texas at ArlingtonArlington 76019USA School of Mathematical Science Soochow UniversitySuzhou 215006China College of Aerospace Engineering Nanjing university of Aeronautics and AstronauticsNanjing 210016China School of Mathematics and Statistics Jiangsu Normal UniversityXuzhou 221116China

Kolmogorov's 1941 theory(K41)of similarity hypotheses and the-5/3 law for energy spectrum are considered as the most important theoretical achievement in turbulence research and the success of the modem turbulence theory.The assumptions of sufficient high Reynolds number and isotropy of turbulence that K41 based upon,however,cannot generally be met in practice,and thus discrepancy is often observed between the f law and direct numerical simulation(DNS)results of boundary layers in wall bounded turbulence,especially for moderate to low Reynolds number flows.Liutex vector is a recently defined new physical quantity which is extracted from turbulent flow to represent the rigid rotation part of fluid motion.Actually,Liutex is free from viscous dissipation and thus independent of Reynolds number,relaxing the very high Reynold number assumption of K41.Liutex similarity has been solidly demonstrated by DNS for a moderate Reynolds number turbulent boundary layer(Reθ≈1000),both the frequency and wavenumber spectrum of Liutex accurately matches the-5/3 law,which is obviously much better than the turbulence energy spectrum,while vorticity and other popular vortex identification methods,Q criterion for example,do not possess such a distinguished feature due to stretching and shearing contamination.

关键词： -5/3 law Liutex similarity K41 turbulent boundary layer moderate and low Reynolds number

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Third generation of vortex identification methods:Omega and Liutex/Rortex based systems

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Journal of Hydrodynamics 2019年第2期31卷 205-223页

作者： Chaoqun Liu yi-sheng gao Xiang-rui Dong yi-qian Wang Jian-ming Liu Yu-ning Zhang Xiao-shu Cai Nan Gui Department of Mathematics University of Texas at ArlingtonArlington 76019TexasUSA College of Energy and Power Engineering University of Shanghai for Science and TechnologyShanghai 200093China School of Aerospace Engineering Tsinghua UniversityBeijing 100084China School of Mathematics and Statistics Jiangsu Normal UniversityXuzhou 221116China Key Laboratory of Condition Monitoring and Control for Power Plant Equipment(Ministry of Education) School of EnergyPower and Mechanical EngineeringNorth China Electric Power UniversityBeijing 102206China Institute of Nuclear and New Energy Technology Collaborative Innovation Center of Advanced Nuclear Energy TechnologyKey Laboratory of Advanced Reactor Engineering and Safety of Ministry of EducationTsinghua UniversityBeijing 100084China

A vortex is intuitively recognized as the rotational/swirling motion of fluids,but a rigorous and universally-accepted definition is still not available.Vorticity tube/filament has been regarded equivalent to a vortex since Helmholtz proposed the concepts of vorticity tube/filament in 1858 and the vorticity-based methods can be categorized as the first generation of vortex identification methods.During the last three decades,a lot of vortex identification methods,including 0,A,and Aci criteria,have been proposed to overcome the problems associated with the vorticity-based methods.Most of these criteria are based on the Cauchy-Stokes decomposition and/or eigenvalues of the velocity gradient tensor and can be considered as the second generation of vortex identification methods.Starting from 2014,the Vortex and Turbulence Research Team at the University of Texas at Arlington(the UTA team)focus on the development of a new generation of vortex identification methods.The first fruit of this effort,a new Omega(/2)vortex identification method,which defined a vortex as a connected region where the vorticity overtakes the deformation,was published in 2016.In 2017 and 2018,a Liutex(previously called Rortex)vector was proposed to provide a mathematical definition of the local rigid rotation part of the fluid motion,including both the local rotational axis and the rotational strength.Liutex/Rortex is a new physical quantity with scalar,vector and tensor forms exactly representing the local rigid rotation of fluids.Meanwhile,a decomposition of the vorticity to a rotational part namely Liutex/Rortex and an anti-symmetric shear part(RS decomposition)was introduced in 2018,and a universal decomposition of the velocity gradient tensor to a rotation part(7?)and a non-rotation part(NR、was also given in 2018 as a counterpart of the traditional Cauchy-Stokes decomposition.Later in early 2019,a Liutex/Rortex based Omega method called Omega-Liutex,which combines the respective advantages of both Liutex/Rortex and Omega methods,was devel

关键词： Vortex definition vortex identification Omega Liutex/Rortex direct numerical simulation turbulence

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Influence of the Acoustic Liner in Large Eddy Simulation of Longitudinal Thermoacoustic Instability in a Model Annular Combustor

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Journal of Thermal Science 2024年第2期33卷 710-724页

作者： MENG sheng ZHANG Man gao yi AECC Commercial Aircraft Engine Co. Ltd.Shanghai200241China School of Aeronautics and Astronautics Shanghai Jiao Tong UniversityShanghai200240China

Although extensive efforts have been made to dampen the thermoacoustic instability,successfully controlling the pressure oscillations in modern gas turbines or aeroengines remains challenging.The influence of the acoustic liner on the longitudinal thermoacoustic mode in a model annular combustor is investigated by Large Eddy Simulation(LES) in this work.The result of the self-excited longitudinal thermoacoustic instability without the liner agrees well with the frequency and acoustic analysis of the pressure mode based on experimental data.Three different bias flow velocities of the liner located downstream of the combustor are then simulated.The results reveal that the existence of the liner influences not only the acoustic field but also the flow field.When the bias velocity is large,it leads to intense turbulence-induced fluctuations,and the pressure oscillation is modulated intermittently.It shows that the weak coupling between flow and pressure oscillations plays a significant role in the onset of the intermittency of a thermoacoustic system.Based on the dynamic analysis of the thermoacoustic system with the acoustic liner,this intermittency is caused by the influence of the flow field on the flame-acoustic coupling.Finally,a low-order modeling method based on Van der Pol(VdP) oscillator with additive stochastic forcing is conducted to reproduce the evolving dynamics of the thermoacoustic system.Although the numerical cases demonstrated in this work are relatively simpler than those in a practical combustion system,the results are helpful for us to understand the effect of the acoustic liner and show the attractive potential to apply this device to suppress thermoacoustic instability.

关键词： thermoacoustic instability acoustic liner Large Eddy Simulation low-order modeling

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