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

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 developed.And a latest objective Omega method,which is still under development,is also briefly introduced.These advances are classified as the third generation of vortex identification methods in the current paper.To elaborate the advantages of the third-generation methods,six core issues for vortex definition and identification have been raised,including:(1)the absolute strength,(2)the relative strength,(3)the rotational axis,(4)the vortex core center location,(5)the vortex core size,(6)the vortex boundary.The new third generation of vortex identification methods can provide reasonable answers to these questions,while other vortex identification methods fail to answer all questions except for the approximation of vortex boundaries.The purpose of the current paper is to summarize the main ideas and methods of the third generation of vortex identification methods rather than to conduct a comprehensive review on the historical development of vortex identification methods.

Analysis and correlation of fluid acceleration with vorticity and Liutex(Rortex)in swirling jets

The characteristics of the acceleration a and its correlation with the vortex core in the swirling flow are analyzed in this paper.The swirling flows of four swirl numbers(Sn=n S 0,0.10,0.20 and 0.36)are simulated by the direct numerical simulation,and the fluid mass particles from the inlet of the swirling flow are traced as a database for the current study.The correlations of the acceleration a with the vortex identification parameters of R andωare quantified,where R is the parameter of the“Liutex”-method proposed by Prof.Liu,and its early version is“Rortex”.The probability density functions of the included angles of a-R and a-ωare computed,which are used to clarify the geometrical configuration between a and R.The covariance coefficients between a and the parameters of various vortex core criteria(Q,λ2,ΩL,ΩR,ωand R)are computed and their relative levels of correlation are compared.Compared to the velocity,the expectation and the rms of the acceleration magnitudes and its projections on R and the plane perpendicular to R are computed for better explaining the correlation features between a and R.It is found that a and R are always either perpendicular or parallel to each other with the quantized included angles of∠(a,R)=kπ/2,(k=0,1,2).This confirms the role of R which works as the axis of the vortex exclusively.

Comparative assessment and analysis of Rortex vortex in swirling jets

Recently, a new definition, called Rortex, was proposed to quantify the purely rotational motion of fluids. In this work, based on the DNS data, the Rortex is used to assess and visualize the rotational motion and structure of the vortex in swirling jets in comparison with other kinds of vortex criteria, including Q , λ2,vorticity and Ω criteria. The Rortex vector,Ω, Q and criteria are found to be better than the vorticity criterion for the vortex core identification. The vector triangle formed by the Rortex R , the nonrotational shear S , and the vorticity R is analyzed to give mechanical explanations, especially of the effect of the non-rotational shear on the rotation of fluids. In addition, the probability density distributions (PDF) of the Rortex R , the nonrotational shear S, and the vorticity R are computed. The peak value of the PDF of the vorticity could be used to explain the pure rigid rotation effect and the combination effects of the rigid rotation and the non?rotational shear.

Prediction of the precessing vortex core in the Francis-99 draft tube under off-design conditions by using Liutex/Rortex method

The turbulent flow in the draft tube of a Francis turbine is very complicated while working under off-design conditions. Although the off-design conditions were widely studied, the vortex core line in the draft tube of a Francis turbine with splitter blades is not well understood, especially the vortex rope property. This letter presents a prediction of the behavior of the vortex rope in the draft tube of the Francis-99 turbine obtained by the computational fluid dynamics (CFD), where the Liutex/Rortex method, as the most recent vortex definition, is applied to analyze the periodical precession of the vortex rope in the draft tube cone. The advantage of this Liutex/Rortex method is shown by its enhanced ability to represent the vortex rope structurewith the vortex-core lines. Furthermore, since it seems to be very hard to define a sharp boundary surface for the whole vortex structure, it is advantageousfocusing only on the vortex core line,by which different vortex structures can be clearly differentiated. The evolution of the vortex core and the process of the vortex breakdown in the draft tube are revealed, which might help to comprehend the development of the turbulent flow in the draft tube.

基于开源计算流体力学软件SU2的钝体翼尖涡研究

通过开源多物理场模拟与设计集成软件平台SU2(Stanford University Unstructured),使用格点型有限体积格式,研究了来流攻角12°,雷诺数为Re=2×10^6下NACA0012半矩形翼扰流问题。通过涡向量Rortex/Liutex揭示了方形翼尖纯刚体旋转涡的形成与发展,次涡与主涡的合并,并与涡量进行了比较。结果表明:钝体矩形翼翼尖的两个锋利奇异边产生Rortex/Liutex涡,且Rortex/Liutex表示的次涡与主涡的合并与相互作用要晚于涡量;此外,除了奇异边外的翼尖侧面,虽具有较高涡量,但贡献了流体的剪切或拉伸。

Simulation of Heat Transfer Characteristics in Channel with Square Rib Under Uniform and Turbulent Inlet Conditions

The detailed flow structures and closely-related heat transfer characteristics are investigated along the wall of a cooling channel with rib tabulator by computation.Three typical Reynolds numbers defined by the rib height are set at 200,500,1300,and the Mach numbers is 0.2,respectively.Two inlet boundary conditions,including the uniform and the fully-developed turbulent conditions,are used to study the turbulence effects on the characteristics of heat transfer in the vicinity of rib and wall.Results show that the local Nusselt number increases when the Reynolds number rises from 200 to 1300.At lower Reynolds number,the turbulent inlet condition generates more tangible heat transfer enhancement.At higher Reynolds number,however,the uniform inlet condition contributes more to the convective heat transfer effects.The paper discovers that the high Nusselt number has a consistent correlation with the positive and negative sign alteration of the shear layer on the wall,which satisfactorily explains the mechanisms of heat transfer enhancement due to the flow.

ENO-AUSMPW格式对激波涡干扰的数值模拟

高阶ENO格式在复杂流场计算中具有重要的地位,它理论上可以达到任何阶的精度。本文采用二维高阶ENO插值方法,构造了高阶AUSMPW格式,并对包含复杂激波结构的激波涡干扰流场进行了数值模拟。计算结果表明了高阶ENO插值下的AUSMPW格式具有较高的激波分辨率和较低的数值耗散。

绕扁长椭球体三维分离的积分法

本文利用三维积分边界层法并结合流线法决定涡层型流动分离。由三维不可压湍流边界层方程求得边界层参数后,按流线法计算粘性流线。作为算例,研究了一带迎角的扁长椭球体在不可压缩湍流里的绕流,经与水洞里所观察到的α=15°流型的对比,二者基本一致。

第三代涡识别方法及其应用综述

根据Liu等的划分,涡识别方法的发展可以分为三代。第一代方法基于涡量,并将Cauchy-Stokes分解得到的速度梯度张量反对称部分(涡量张量)视为旋转。然而众多研究人员已发现涡量与涡的关联性并不是很高。为了克服第一代方法的缺陷,以Q、l2、D和lci等方法为代表的第二代涡识别方法应运而生。但是第二代方法在实际使用中依赖于与具体算例相关的阈值并且会被剪切影响。为了解决阈值问题,2016年发展起来的Ω方法将涡定为涡量大于变形的联通区域并以相对值表示。而2017年和2018年发展起来的Liutex向量则提供了一个系统化数学化的流体局部刚体转动定义,包括局部旋转轴和转动强度,Ω方法,Liutex向量及一系列有关方法被定义为第三代涡识别方法。该文回顾了三代涡识别方法的发展历程,系统讨论了如何精确地从流体运动中分解得到刚性旋转部分,重点介绍由美国德州大学阿灵顿分校刘超群教授及其团队主导的第三代涡识别方法所取得的突破及其应用。主要突破包括:①Ω涡识别方法克服了传统涡识别方法需要针对具体流动调整阈值的问题,并能同时捕捉到强涡和弱涡;②基于Liutex的思想,能够严格获得流体运动的局部刚性旋转部分,包括标量、矢量和张量形式;③第三代方法能够合理地回答涡的六大要素问题,即涡的绝对强度、相对强度、当地旋转轴、涡核中心、涡核大小和涡边界,而前两代方法则无法处理。该文详细介绍了第三代方法的六个具体应用,其结果表明:①在涡识别方面,第三代方法大大优于前两代方法;②第三代方法不仅是涡识别和显示的后处理方法,而且是精确表示涡的物理量;③第三代方法能够定量化地表示涡的特性(六大要素)。Liutex作为一个新生的物理量,有方向有大小。Liutex的方向就是当地涡线和涡强度(此处区别于第一代涡量线和涡量强度的定义)等值面法线的方向,其大小准确代表当地旋转角速度(涡的绝对强度),且在当地非零Liutex方向上仅有瞬时的拉压形变,流体旋转均发生在与Liutex方向垂直的平面内。在等值面缩小变为一条线的地方,它就是涡核中心。它唯一代表涡核,唯一代表湍流的涡结构,与阈值完全无关。更重要的是它准确代表一个力,代表湍流产生的原动力,直接影响湍流生成。Liutex和其他第三代涡识别方法的发现代表湍流研究的一个崭新时代的来临,它开启了湍流研究的新纪元。

A selected review of vortex identification methods with applications

In the present review, recent progress on the vortex identification methods are introduced with a focus on the newly proposed omega method（ Ω method）. The advantages of Ω method are summarized with many illustrating examples. Furthermore, comparing with other existing methods（e.g., Q criterion and λ2 criterion）, one of the characteristics of Ω method is its independence on the chosen threshold values for vortex identifications. The important parameters involved for the practical applications of Ω method are further discussed in detail together with the physical interpretation of the Ω and some suggestions of the future work. Other emerging topics（e.g., Lagrangian coherent structure and Rortex） are also introduced with comments.

Vortex identification methods in marine hydrodynamics

In this paper,several commonly used vortex identification methods for marine hydrodynamics are revisited.In order to extract and analyse the vortical structures in marine hydrodynamics,the Q,λ2-criterion and modified normalized Liutex/RortexΩR method are utilized for vortex identification for propeller open water test,ship drag test,ship propeller-rudder interaction,VIV of a marine riser and VIM of a Spar platform.The limitation of Q andλ2-criterion is discussed.The Liutex/RortexΩR method is promising for convenient and accurate vortex identification and visualization.However,care should be taken when choosing the small parameter b0 forΩR.We proposed recommended values of b0 for marine hydrodynamic problems.

Mathematical foundation of turbulence generation-From symmetric to asymmetric Liutex

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.