Liutex and Third Generation of Vortex Definition and Identification: An Invited Workshop from Chaos 2020 3030702162, 9783030702168

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Chaoqun Liu Yiqian Wang   Editors

Liutex and Third Generation of Vortex Definition and Identification An Invited Workshop from Chaos 2020

Liutex and Third Generation of Vortex Definition and Identification

Chaoqun Liu  •  Yiqian Wang Editors

Liutex and Third Generation of Vortex Definition and Identification An Invited Workshop from Chaos 2020

Editors Chaoqun Liu University of Texas at Arlington Arlington, TX, USA

Yiqian Wang Soochow University Suzhou, Jiangsu, China

ISBN 978-3-030-70216-8    ISBN 978-3-030-70217-5 (eBook) https://doi.org/10.1007/978-3-030-70217-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book is a collection of papers presented in the invited workshop, “Liutex and Third Generation of Vortex Definition and Identification for Turbulence,” of CHAOS2020 conference, Florence, Italy, June 9–13, 2020. Due to the COVID-19 pandemic, the meeting was finally held online as a virtual conference. We had 33 registered speakers and 30 papers are published in this book including one invited paper. Liutex is a new physical quantity discovered by Prof. Chaoqun Liu at the University of Texas at Arlington (UTA) in 2018 (Liu et al., Physics of Fluids 30, 2018) to represent fluid rotation or vortex. The discovery of Liutex is probably one of the most important breakthroughs in modern fluid kinematics especially for vortex science and turbulence research. Vortex is ubiquitous in nature and viewed as the building blocks, the muscles, and sinews of turbulent flows (Küchemann, J. Fluid Mech., 21, 1965). However, vortex had no mathematical definition before the discovery of Liutex, which was the bottleneck of modern fluid dynamics and caused countless confusion in vortex and turbulence research. Kolář (V. Kolář and J. Šístek, Phys. Fluids 32, 091702, 2020) pointed out that “Rortex is a mathematically rigorous tool suitable for vortex characterization.” Xu (Liutex and Its Applications in Turbulence Research, ISBN-13: 978-0128190234, ISBN-10: 012819023X, Elsevier, 2020) made comments that “Liutex does lift and uncover the mask covering vortex which has puzzled our science community for so many centuries. Specifically, the Liutex core-lines limpidly bring out the skeleton of vortex structures and for the first time, vividly exhibit these structures to our visual world, which, from Xu’s experience, is so far the unique representation of vortical structures with the true, only true, nothing else but the true mathematical essences of vortex physics in entirety….” Since vortex is the building block of turbulence, without mathematical definition of vortex, turbulence research is in general limited to qualitative study which is in general relied on observation, graphics, movies, approximations, assumptions, guesses, and hypotheses. The rigorous mathematical definition of vortex or Liutex will bring turbulence research to a new era from qualitative study to quantified research. v

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According to Liu et al. (Journal of Hydrodynamics, 31(2):1–19, 2019), there are three generations of vortex identification methods in history. In 1858, Helmholtz first defined vortex as a vortex tube composed of the so-called vortex filaments, which are really infinitesimal vorticity tubes. It is classified as the first generation of vortex identification that vortex is defined as vorticity tubes. Science and engineering applications have shown that the correlation between vortex and vorticity is very weak, especially in the near-wall region. During the past four decades, many vortex identification criteria, Q, Δ, λ2 and λci methods (Hunt et al. 1988; Chong et al. 1990; Jeong et  al. 1995; Zhou et  al. 1999; Chakraborty 2005) for example, have been developed, which are classified as the second generation of vortex identification. They are all based on the eigenvalues of the velocity gradient tensor. However, they are all scalars and thus strongly dependent on the factitious and arbitrary threshold, when plotting the iso-surface to represent the vortical structures. In addition, they all lack physical meanings and obviously are contaminated by stretching (compression) and shearing. Liutex as the third generation of vortex definition and identification was developed by Liu and his students at UTA (Liu et al., Physics of Fluids 30:035103, 2018; Gao et al., Physics of Fluids, 30:085107, 2018). Liutex is defined as a vector which uses the real eigenvector of velocity gradient tensor as its direction and twice the local angular speed of the rigid rotation as its magnitude. The major idea of Liutex is to extract the rigid rotation part from fluid motion to represent vortex. After almost two hundred years of efforts, for the first time, Liutex, an accurate physical quantity to represent fluid rotation or vortex, was born in UTA. After that, a number of vortex identification methods have been developed by Liu and his UTA Team including Liutex vector, Liutex vector lines, Liutex tubes, Liutex iso-surface, Liutex-Omega methods, Objective Liutex, and, more recently, Liutex Core Line and Tube methods which can more accurately visualize the vortical structures in turbulent flows, demonstrated by countless users in research papers. Liutex Core Line, which is defined as a special Liutex line, where the gradient of R is parallel to Liutex vector, is unique and threshold-free. A so-called Principal Coordinate based on the velocity gradient tensor is defined and the fundamental vector and tensor decompositions are made in the Principal Coordinate by Liu and his students. A new vorticity decomposition to Liutex and shear, namely RS decomposition of vorticity, is proposed by Liu to separate non-­ dissipative rigid rotation from dissipative shear by decomposition of vorticity (Liu et al., Physics of Fluids 30:035103, 2018). A new Liutex-based UTA R-NR tensor decomposition is also proposed by Liu to replace the traditional Cauchy-Stokes (Helmholtz) decomposition. The Liutex-based UTA R-NR tensor decomposition is unique and Galilean invariant. The Liutex definition, Principal Coordinate, Principal Decomposition, vorticity RS decomposition, and velocity gradient tensor UTA R-NR decomposition, proposed by Liu et al., integrate the Liutex theory and pave the foundation for new fluid kinematics, new vortex science, and new turbulence research. Liutex (rigid rotation) spectrum similarity theory of -5/3 power in transitional and turbulent boundary layer is discovered by Liu and his students while Kolmogorov’s -5/3 law does not match well with DNS or experiments especially in

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low and medium Reynolds number turbulent boundary layers. Liutex dynamics and modified Navier-Stokes equations which govern both laminar and turbulent flows without models are under development by the UTA team. Prof. Liandi Zhou who is the Executive Chief Editor of the Journal of Hydrodynamics pointed out that vortex which is a group rotation has six core elements including (1) absolute strength, (2) relative strength, (3) local rotational axis, (4) vortex rotation axis, (5) vortex core size, and (6) vortex boundary, which are touchstones that the vortex identification methods should be tested against. It is confirmed by countless users that only the Liutex system is able to give precise information of all six core elements. On the other hand, the first- and second-­ generation methods in vortex identification all failed except for a rough estimate of the vortex boundaries by the second-generation methods with a so-called proper threshold, which is arbitrary and cannot be unique. The necessary condition for correctness of the real vortex structure is that they must be unique. Prof. Chaoqun Liu, the founder and major contributor of Liutex, has worked on direct numerical simulation (DNS) for flow transition and turbulence for 30 years since 1990 (Liu et al., Computers and Fluids 102, 2014). In 2015, Prof. Liu was invited to hold a short course in Beijing, China, titled “New Theory on Turbulence Generation and Sustenance,” at Tsinghua University which is the No. 1 or 2 university in China, organized by Prof. Song Fu and sponsored by 24 major universities and institutes in China. The short course received unprecedented enthusiasm from Chinese turbulence research community as 240 professionals registered for the short course. The organizers originally prepared a lecture room which can host around 200 audiences, but on the opening day, they had to remove many desks to make rooms for many additional chairs. As more people registered and came to the lecture, the organizers had to ask all local teachers and students of Tsinghua University not to enter the room and leave more space for the guest participants. There were still many people who had to stand in the room without a seat. This event gave a shock to the Chinese turbulence research community that a systematical turbulence theory was lectured by a visiting professor from the University of Texas at Arlington, Texas, USA. In 2016, Liu and Wang et al. (Sci. China Phys., Mech. Astron. 59:684711, 2016) published a paper “New Omega Vortex Identification Method.” The Omega method represents the relative strength of vortex and is insensitive to threshold selection. In addition, the Omega method can visualize both strong and weak vortices simultaneously. In that paper, Liu also proposed to decompose vorticity to a rotational part and a non-rotational part (RS decomposition of vorticity). Later, Wang and Liu found that the vorticity is smaller inside but larger outside vortex, which is an observation directly opposing the traditional and classified concept that vortex is a concentration of vorticity. The idea to extract the rigid rotation from fluid motion was given by Liu in a conference called “New Development and Key Issues of Vortex and Turbulence Research” in December 2017 at the University of Shanghai for Science and Technology (USST), Shanghai, China. Many people were excited with the new definition of vortex as they heard that “vortex” is not “vorticity” for the first time. During the discussion, several professors suggested that the new definition of

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vortex is not a traditional concept of vortex and it should be given a new name. Liu accepted one suggestion to name the new definition of vortex as “Rortex.” Unfortunately, there was a conflict that the name “Rortex” was given by a professor who was unhappy with the “name ownership,” which already caused misunderstandings. The UTA Team including many collaborators decided to rename the new vortex definition as “Liutex” in late 2018. We then had three more conferences on vortex and turbulence research. The second one was held in Zhejiang University, Hangzhou, China, with about two hundred participants and organized by Profs. Weifang Chen and Lihua Chen of Zhejiang University of China in June 2018, the third one was held in Beijing with over a hundred participants and organized by Prof. Yuning Zhang of Northern China Electric Power University in December 2018, and the fourth one was again held in USST of Shanghai with about two hundred participants and organized by Prof. Xiaoshu Cai in June 2019. Liu was the Principal Speaker for all four conferences. After the 2017 Vortex Conference, the UTA Team has published 13 papers in Physics of Fluids (PoF) and more than 20 papers in the Journal of Hydrodynamics. Liu and his collaborators also published two professional Liutex books, one by Bentham and the other by Elsevier. These new books systematically introduce Liutex theory, its mathematical definition, its theoretical foundation, and its applications in science and engineering. Liutex has been applied by countless scientists and engineers to identify vortex, visualize new vortex structure, study new physics, and, more important, uncover the turbulence mysterious veil. As more and more scientists apply Liutex and the third generation of vortex methods, people will reveal more and more secrets of turbulence. Wang made important contributions to the Omega method and explicit formula of Liutex. Liu and Wang would like to thank all UTA Team members including Yisheng Gao, Xiangrui Dong, Jianming Liu, Yifei Yu, Panpan Yan, Wenqian Xu, Guang Yang, Yonghua Yan, Yong Yang, Charles Nottage, Pushpa Shrestha, Oscar Alvarez, Xuan Trieu, Vishwa Patel, Sita Charkrit, Dalal Almutairi, and Aayush Bhattarai and collaborators including Hongyi Xu, Xiaoshu Cai, Huashu Dou, Albert Tong, and Dezhi Dai. This new Springer book is a collection of new developments in Liutex theories, methods, and applications. This book has 30 chapters. Chaps. 1–11 are provide an introduction to Liutex theories and methods, and Chaps. 12–19 report on progresses in the application of Liutex in turbulence research. Finally, Chaps. 20–30 are literatures on Liutex applications in engineering. These chapters are briefly described below: Chapter 1 is entitled “Liutex and Third Generation of Vortex Definition and Identification” written by Chaoqun Liu. Vortex is intuitively recognized as the rotational/swirling motion of fluids. However, a rigorous mathematical definition for vortex was absent for centuries. Liutex is a new physical quantity to represent fluid rotation or vortex, discovered by Liu et al. at the University of Texas at Arlington (UTA) in 2018. Liutex is thought as one of the most important breakthroughs in modern fluid dynamics. There are three generations of vortex identification methods in history. In 1858, Helmholtz first defined vortex as vorticity tubes, which is

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c­ lassified as the first generation of vortex identification. Science and engineering applications have found that the correlation between vortex and vorticity is very weak, especially in the near-wall boundary region. During the past four decades, many vortex identification criteria including Q, Δ, λ2and λci methods have been developed, which are classified as the second generation of vortex identification. They are all based on the eigenvalues of the velocity gradient tensor. However, they are all scalars and thus strongly threshold-dependent, which have problems with uniqueness and dimensions. In addition, they are all contaminated by stretching and shearing. Liutex, as the third generation of vortex definition and identification, is defined as a vector which uses the real eigenvector of velocity gradient tensor as its direction and twice the local angular speed of the rigid rotation as its magnitude. The major idea of Liutex is to extract the rigid rotation part from fluid motion to represent vortex. After that, a number of vortex identification methods have been developed by Liu and the UTA Team. The Liutex definition, Liutex-based vortex identification, Principal Coordinate, Principal Decomposition, vorticity RS decomposition, and velocity gradient tensor UTA R-NR decomposition, proposed by Liu et al., integrate the Liutex theoretic system and pave the foundation for new fluid kinematics, new vortex science, and new quantified turbulence research. Chapter 2 is entitled “Incorrectness of the Second-Generation Vortex Identification Method and Introduction to Liutex” written by Yifei Yu, Pushpa Shrestha, Oscar Alvarez, and Chaoqun Liu*. For a long time, scientists believe that vorticity reveals the essence of vortex since vorticity is a perfect indicator of rotation for rigid object although many experimental results do not support this point of view. Scientists have to develop some other vortex identification methods to improve the performance of detecting vortex. Under these situations, Q method, λ2  method, and λci method are innovated. However, these methods are all scalars and not able to locate the swirling direction, and threshold is needed to display vortex structures. To overcome these drawbacks, Liutex can correctly represent the rigid rotation of fluids. The magnitude of Liutex is proposed, which is twice the angular speed, and the direction of Liutex is the direction of swirling axis. In this chapter, the incorrectness of classical methods is explained as well as the rationality of the definition of Liutex. A correlation research is done between vorticity, Q, λci, λ2 methods, and Liutex based on a direct numerical simulation (DNS) case of boundary layer transition. The results show that the correlation between vorticity and Liutex is very small or even negative in strong shear regions, which demonstrates that using vorticity to detect vortex lacks scientific foundation and vorticity is not appropriate to represent vortex. In the last part of this chapter, a new vortex structure displaying methods including Liutex iso-surface, Liutex-Omega, and Liutex core line is introduced. Chapter 3 is entitled “Dimensional and Theoretical Analysis of Liutex and Second-Generation Vortex Identification methods,” written by Charles Nottage, Yifei Yu, Pushpa Shrestha, and Chaoqun Liu*. Many researchers have spent much effort in researching the identification of vortices for many decades. These efforts have produced many methods such as Q, λci, λ2 methods, with Q being the most popular method used in research and industry. Although these methods can visualize the vortical structure, they have apparent faults. In this chapter, a dimensional

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analysis is conducted to examine and compare the dimensions of each method with the dimension of angular velocity α. The results show that λci is the only method other than Liutex that has the correct dimension relative to the angular velocity α. Chapter 4 is entitled “Mathematical Study on Local Fluid Rotation Axis— Vorticity is Not the Rotation Axis” authored by Charles Nottage, Yifei Yu, and Chaoqun Liu*. A vortex can be intuitively recognized as the rotational/swirling motion of the fluids. The fascination of the phenomenon brought about many years of research to classify and identify the vortical structure. Vorticity was one of the first theories developed to identify the vortex. Although the vorticity theory was mathematically sound, it did not line up with experimental results. This setback brought forth new eigenvalue-based methods such as Q, λci, λ2 criteria. However, since these eigenvalue-based methods are scalar-valued, many researchers and textbooks still accept that vorticity is vortex. In recent years, a new vortex identification method called Liutex is developed. Liutex is a vector quantity with a clear physical meaning that overcomes the drawbacks of the previous methods. The physics behind a vortex reveals that there exists a local fluid rotation axis. In this chapter, a mathematical study on five local fluid rotation axis candidates is conducted, including, namely, the symmetrical tensor’s three eigenvectors, the vorticity vector, and the Liutex vector. The results show that the vorticity vector satisfied the local fluid rotation axis requirements only in particular cases while the Liutex directional vector unconditionally satisfied the requirements to be considered the local fluid rotation axis. Chapter 5 is entitled “No Vortex in Flows with Straight Streamlines—Some Comments on Real Schur Forms of Velocity Gradient ∇v” authored by Xiangyang Xu, Zhiwen Xu, Changxin Tang, Xiaohang Zhang, and Wennan Zou*. Velocity gradient is the basis of many vortex recognition methods, such as Q criterion, Δ criterion, λ2 criterion, λci criterion, and Ω criterion. Except the λci criterion, all these criteria recognize vortices by designing various invariants, based on the Helmholtz decomposition that decomposes velocity gradient into strain rate and spin. In recent years, the intuition of “no vortex in straight flows” has promoted people to analyze the vortex state directly from the velocity gradient, in which vortex can be distinguished from the situation that the velocity gradient has a couple complex eigenvalues. A specious viewpoint to adopt the simple shear as an independent flow mode was emphasized by many authors: among them, Kolář (2004) proposed the triple decomposition of motion by extracting a so-called effective pure shearing motion; Li et al. (2014) introduced the so-called quaternion decomposition of velocity gradient and proposed the concept of eigen rotation; Liu et al. (2019) further mined the characteristic information of velocity gradient and put forward an effective algorithm of Liutex (namely eigen rotation), and then developed the vortex recognition method. However, there is another explanation for the increasingly clear representation of velocity gradient, which is the local streamline pattern based on critical point theory. In this chapter, the tensorial expressions of the right/left real Schur forms of velocity gradient are clarified from the characteristic problem of ∇v. The relations between the involved parameters are derived and numerically verified. Comparing with the geometrical features of local streamline pattern, we confirm that the

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p­ arameters in the right eigen-representation based on the right real Schur form of velocity gradient have good meanings to reveal the local streamline pattern. Some illustrative examples from the DNS data are presented. Chapter 6 is entitled “Mathematical Definition of Vortex Boundary and Boundary Classification Based on Topological Type” written by Xiang Li*, Qun Zheng, and Bin Jiang. 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 is explained through geometric relations in this chapter, 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. Chapter 7 is entitled “A Comparison of Liutex with Other Vortex Identification Methods on the Multiphase Flow Past a Cylinder Using LBM on GPU” authored by Pengxin Cheng, Nan Gui*, Xingtuan Yang, Jiyuan Tu, and Shengyao Jiang. Multiphase flow past obstacles is extensively applied in engineering and industries. The interaction of two different phases associated with impact of solid results in complex flow characteristics and vortex field. In this chapter, the GPU-­accelerated Lattice Boltzmann method is used to study the process of multiphase flow past a 2-D cylinder. The drag force components induced by continuous and dispersed phase respectively as well as the total force are illustrated and the underlying mechanisms are interpreted. The vortex identification methods based on traditional approaches and Liutex are compared. The relationship between the extremums of disparate vortex identification variables and bubble deformation process is investigated. Chapter 8 is entitled “On the Comparison of Liutex Method with Other Vortex Identification Methods in a Confined Tip-Leakage Cavitating Flow” by Xiaorui Bai, Huaiyu Cheng*, and Bin Ji. In the current chapter, Large Eddy Simulation on the confined tip leakage cavitating flow generated by a straight NACA0009 hydrofoil with a clearance size gap=10mm is performed. TLV cavitation and TSV cavitation are reasonably captured. Different vortex identification methods of all three generations, including vorticity, Q criterion, Ω method, and Liutex method, are compared in detail with the simulated tip leakage cavitating results. It is observed that the Ω method and the Liutex method show advantages in visualizing vortical structures in the currently studied flow regime with a strong capability in filtering out shearing

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contaminations. Both Liutex vector field and Liutex iso-surface show potentials in investigating vortical cavitating flow. Differences of cavitating and non-cavitating tip vortices mainly lie in their sizes and the generation of small-scale vortices around the TSV, which indicates the effect of cavitation. Chapter 9 is entitled “Lagrangian Liutex” written by Yiqian Wang. A Lagrangian version and an objective Lagrangian version of the Liutex, which has been introduced to identify and visualize vortices, are developed in this study. The capability of the new methods to educe vortices in geophysical Bickley jet flow is discussed in detail. It is found the Lagrangian version surpasses vorticity-based methods as it is free from shear contamination and also reveals the Lagrangian nature of vortices. The objective version is capable of visualizing vortices independent of selected reference frame. However, selecting the volume to apply the objective Lagrangian Liutex method differently leads to different results. A general principle of selecting symmetry flow field with zero mean vorticity is proposed to overcome this issue. Chapter 10 is entitled “Visualizing Liutex Core Using Liutex Line and Liutex Tubes” authored by Oscar Alvarez, Yifei Yu, Pushpa Shrestha, and Chaoqun Liu*. Liutex is a vortex identification method which can be used to determine the absolute and relative strength of a vortex, the local rotation axis of a vortex, the vortex core center, the size of the vortex core, and the vortex boundary. A vortex core is defined as a concentration of Liutex vectors. It would be nice if one could visualize the Liutex core to gain a better understanding. In this chapter, the Liutex core lines and Liutex tubes are applied to show the vortex structure by using pre-processed DNS data, which can help researchers of fluid dynamics understand the turbulence physics. There is no reason to believe that iso-surfaces can represent vortex structure, but it is more reasonable to demonstrate the vortex structures by using the Liutex core lines and Liutex tubes as Liutex is a new vector quantity to represent flow rotation or vortex. Chapter 11 is entitled “Analysis of Difference Between Liutex and λci” written by Yisheng Gao, Yiqian Wang, and Chaoqun Liu*. The λci criterion is an extension of the ∆ criterion and has been widely used for vortex identification and visualization. However, one obvious drawback of the λci criterion is the contamination by shear. In this paper, the theoretic analysis of difference between Liutex and λci is presented, from the perspective of the eigensystem of the velocity gradient tensor. It is found that λci is actually a measurement of angular speed in the non-orthonormal coordinate system, and thus is contaminated by shear, even for two-dimensional case. In contrast, Liutex is determined in a special orthonormal coordinate system which is called the principal coordinate and eliminates the shear contamination and thus can reasonably measure the actual rigid rotation part of local fluid motion. Chapter 12 is entitled “Hairpin Vortex Formation Mechanisms Based on LXCLiutex Core Line Method” authored by Heng Li, Duo Wang, and Hongyi Xu*. Based on the direct numerical simulation (DNS) data, the generation mechanisms of hairpin vortices are studied in this chapter. The momentum thickness Reynolds number range is 250  0 is used to keep  the definition unique and consistent when the fluid motion is pure rotation. ω = ∇ ×v is vorticity and λci is the imaginary part of the conjugate complex eigenvalues of ∇v.

2.2  Principal Coordinate and Principal Decomposition [37] Principal Coordinate According to Chong and Perry [8], in the vortex area, the velocity gradient tensor must have one real eigenvalue and two conjugate complex eigenvalues. coordinate (X, Y, Z) has Z-axis aligned with the real Definition 2  A principal  eigenvector of ∇v and two diagonal elements equal to each other (Fig.  1.1). In π addition, the X-Y coordinate rotation angle θ < . In the principal coordinate: 2



 ∂U  ∂X   ∂V ∇V =   ∂X   ∂W  ∂X

∂U ∂Y ∂V ∂Y ∂W ∂Y

  ∂U 0   ∂X   ∂ V 0 =   ∂X   ∂W   ∂W ∂Z   ∂X

∂U ∂Y ∂V ∂Y ∂W ∂Y

 0  0   λr  

(1.2)

1  Liutex and Third Generation of Vortex Identification Methods

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Fig. 1.1 Principal coordinates



  λcr   1 ∇Vθ =  R +  2   ξ 

1 − R 2

λcr η

 0  0 ,   λr  

(1.3)

where R is the Liutex magnitude, λr is the real eigenvalue, λcr is the real part of the conjugated complex eigenvalues, ξ, η, ϵ are shears. The principal coordinate is  uniquely defined based on ∇v. Principal Tensor Decomposition The velocity gradient tensor decomposition can be written as





 λcr   ∇V =  R / 2 +   ξ

−R / 2 0  λcr 0  = R + NR η λr 

−R / 2 0  0 λcr 0 0  and NR =  s R =  R / 2  0  ξ 0 0 

0 λcr η

0 0  , λr 

(1.4)



which is called “UTA R-NR decomposition”, where R stands for the rotational part of the local fluid motion, which is called the tensor version of Liutex, and NR the non-rotational part. It is clear that NR has three real eigenvalues, so NR itself implies no local rotation. The UTA R-NR decomposition of velocity gradient tensor is extremely important in fluid mechanics, which is unique, Galilean invariant and well representing physics.

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Fig. 1.2  Illustration of vorticity vector decomposition at Point A

Vorticity RS Decomposition Vorticity cannot be applied to represent flow rotation. Otherwise, we would not have vortex identification methods like Q, λci, λ2, Δ. Vorticity vector must be decomposed to a rotational part (Liutex) and a non-rotational part (shear). The vorticity RS decomposition can be obtained by



0  η      ω = ∇ × V = R + S =  0  +  −ξ   R   s 

(1.5)

    where S = ω − R can be considered as shearing vector since the components of S indicate the strengths of the simple shear along different axes (Fig. 1.2).

2.3  Liutex Methods for Vortex Identification Liutex Iso-surface Since Liutex is a rigid rotational part extracted from fluid motion, Liutex vector, Liutex vector lines, Liutex tubes, Liutex iso-surface can all be applied to display vortex structure (Figs.  1.3 and 1.4). The advantage of the Liutex method is that Liutex is a vector unlike others which are all scalar. The other advantages of Liutex are that Liutex represents a pure rotation without contamination by shears while all other second generation vortex identification methods are contaminated by shears. Of course, Liutex-iso-surface still needs thresholds like other scalar methods, but it is the pure rotation strength. A number of Liutex-based vortex identification methods have been developed including Modified Liutex-Omega methods, Objective Liutex and, more recently, Liutex Core Line methods [1–3, 13, 20, 22, 23, 25, 34, 38] which can more accurately visualize the vortical structures in turbulent flows, proved by countless users

1  Liutex and Third Generation of Vortex Identification Methods

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Fig. 1.3  Liutex iso-surface and Liutex lines (color represents the rotation strength)

Fig. 1.4  Liutex iso-surface for the vortex structure in early transition (R = 0.1)

and research papers. A brief introduction is given below for modified Liutex-Omega method and Liutex-Core Line method. Liutex-Omega Method Ω Method Definition 3  The Ω method is given by Liu et al. [38] Ω=

BF AF + BF + ε



(1.6)

where A and B are the symmetric and antisymmetric part from Cauchy-Stokes 2 decomposition, ⋅ F represents the Frobenius norm, ε is a small positive number introduced to avoid division by zero. Dong et al. [16] suggests that ε could be determined at each time step by

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ε = 0.001 ( BF − AF )max





(1.7)

The advantages of Ω method include: (1) easy to perform, (2) physical meaning is clear, (3) normalized from 0 to 1, (4) no need to adjust threshold by setting Ω = 0.52, (5) able to capture strong and weak vortices simultaneously. Liutex-Omega Method [20] Definition 4  Liutex-Omega method combines the idea of Liutex and Omega methods to define: ΩR = 2

where α =

β2 , α + β2 +ε 2

(1.8)

2

1  ∂V ∂U   ∂V ∂U  1  ∂V ∂U  ,β =  . − + −   + 2  ∂Y ∂X   ∂X ∂Y  2  ∂X ∂Y 

Modified Liutex-Omega Method [23] Liutex-Omega method has some bulging phenomenon on the iso-surfaces. In order to overcome such a problem, a modified Liutex-Omega method is proposed by  = Ω R

β2 1 β 2 + α 2 + λcr2 + λr2 +  2

(1.9)

which is equivalent to  = Ω R



(ω · r )

2

 2 2 (ω · r ) − 2λci2 + 2λcr2 + λr2  + ε  

(1.10)

The modified Liutex-Omega method is normalized, not contaminated by shear, insensitive to threshold selection, which is now quite popular among users (Fig. 1.5). Liutex Core Line Method [13, 34] All iso-surface methods are threshold-dependent. Liutex core line is defined as the rotation axis of each vortex. Liutex Core Line, where the gradient of R is parallel to Liutex vector, is unique and threshold-free.

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1  Liutex and Third Generation of Vortex Identification Methods

 Fig. 1.5  Iso-surfaces of hairpin vortex structures. (a) Old ΩR method (b) New  R method

Definition 5  The vortex core line is defined as a special Liutex line which passes the points satisfying the condition of

  ∇R × r = 0, r ≠ 0

(1.11)

 where r represents the direction of the Liutex vector. The Liutex (vortex) rotation core lines in the flow field which is uniquely defined without any threshold requirement (Fig.  1.6). Therefore, the Liutex core rotation axis lines with the Liutex strength are derived uniquely and are believed the only entity that is capable of cleanly and unambiguously representing the vortex structures.

3  First and Second Generations of Vortex Identification 3.1  F  irst Generation: Vorticity-Based Vortex Identification Methods Since the concepts of vorticity tube and filament were proposed by Helmholtz [46] in 1858, it is generally believed that vortices consist of vorticity  tubes and vortex strength is measured by the magnitude of vorticity, i.e. ∇ × v [47]. Although the vorticity is widely adopted to detect vortices, one immediate counter-example is that in the laminar boundary layer the average shear force generated by the no-slip wall is so strong that an extremely large amount of vorticity exists but no rotation motions are observed in the near-wall regions, which implies that vortex cannot be represented by vorticity. Robinson [41–42] has found that “the association between regions of strong vorticity and actual vortices can be rather weak in the turbulent boundary layer, especially in the near wall region vortices.

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Fig. 1.6  Vortex structure in flow transition displayed by Liutex core line methods (color represents Liutex strength) (a) Liutex iso-surface and core lines, (b) Liutex core lines, (c) Liutex core lines for flow transition

As shown by Wang et al. [27], the magnitude of vorticity can be substantially reduced along vorticity lines entering the vortex core region near the solid wall in a flat plate boundary layer (Fig. 1.7). For a transitional flow over a flat plate, Fig. 1.7 clearly indicates that in the near-wall region of the boundary layer, the local vorticity vector can deviate from the direction of vortical structures and a vortex can appear in the area where the vorticity is smaller than the surrounding area where the vorticity is larger than the vorticity inside the vortex. These results demonstrate that vorticity cannot be used to represent vortex. Vortex is a natural phenomenon, but vorticity is a mathematical definition. There is no reason to say that vortex is vorticity. Instead, “vortex is not vorticity”, which has been proved by rigorous mathematical proof [1, 2] and countless engineering applications. They are two different concepts. In fact, if vortex cannot be ended inside the flow field, how turbulence can be generated by “vortex breakdown”. Vortex can break down, which means vortex is not vorticity tubes. The correlation analysis between vorticity and vortex (fluid rotation) shows there exist no correlations between vortex and vorticity in general, especially near the wall. Although both vortex and vorticity are vectors, they are different vectors without correlation.

1  Liutex and Third Generation of Vortex Identification Methods

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Fig. 1.7 (a) Vortex appears in the area where vorticity is relatively smaller, (b) Vorticity line is not aligned with vortex, (c) Vorticity is rather smaller than surrounding

3.2  Second Generation of Vortex Identification During the past four decades, a number of vortex identification methods have been proposed. Q- Criterion [7]



Q=

1 2 BF − AF2 2

(

)

(1.12)

where A and B are the symmetric and antisymmetric parts of the velocity gradient 2 tensor respectively and ⋅ F represents the Frobenius norm. Theoretically, Q > 0 can identify the vortex boundary but, in practice, a threshold Qthreshold must be specified to define the regions where Q > Qthreshold. Q-criterion shows the symmetric tensor has roles to balance the anti-symmetric tensor. λci Criterion λci method [10] defines the strength of vortex as the imaginary part λci of the complex eigenvalues of the velocity gradient tensor ∇V. λ2 Method λ2 method [9] defines the strength of vortex by using the second largest eigenvalue λ2 of A2 + B2, where A and B are the symmetric and anti-symmetric parts of the velocity gradient tensor. By assuming the fluid is incompressible, steady and non-viscous, the Navier-­ Stokes equation is converted to A2 + B2 =  −  ∇ (∇p)/ρ, where p and ρ represent pressure and density respectively. Jeong and Hussain define the rotational area by the existence of two negative eigenvalues of the symmetric tensor A2 + B2.

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Three Problems: Threshold, Rotation Direction and Strength of Vortex Without exception, the aforementioned criteria require user-specified thresholds. Since different thresholds will indicate different vortical structures [48, 49], it is critical to determine an appropriate threshold. When a large threshold for the λ2 criterion is used, “vortex breakdown” (see Fig.  1.8) can be observed in the late boundary layer transition. But if a small threshold is applied, no “vortex breakdown” (see Fig. 1.8) will be exposed, which means that an appropriate threshold is vital to these vortex identification methods. Many computational results have revealed that the threshold is case-related, empirical, sensitive, time step-related, and hard to adjust. Furthermore, no one knows whether the specified threshold is proper or improper. Actually there may be no single proper threshold especially if there are strong and weak vortices co-exist. If the threshold is too small, weak vortices may be captured, but strong vortices could be smeared and become vague. If the threshold is too large, weak vortices will be wiped out. The other disadvantage of these vortex identification methods is that these vortex identification methods can only provide iso-surface without any information about rotation axis or vortex direction. A more serious question is raised that why the iso-surface can be used to represent the rotation strength? The answer is no, since they are different to each other (not unique), contaminated by shear and stretch in different degrees, and failed to represent the rigid rotation of fluid motion. Mathematical Misunderstandings of First and Second Generations The major mathematical misunderstanding is the consideration of vorticity as the fluid rotation axis and vorticity is the measurement of vortex strength.

Fig. 1.8  Vortex breakdown with a large threshold of λ2 = −0.017 and no vortex breakdown with a small threshold of λ2 = −0.003 (same DNS data set)

1  Liutex and Third Generation of Vortex Identification Methods

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Vorticity is Not Fluid Rotation Axis Classical theory [47] considers vorticity vector as the rotation axis and vorticity as the strength of the vortex, which is really a misunderstanding for over a century. Definition 6  At a moment, a local fluid rotation axis is defined as a vector which can only have stretching (compression) along its length.      It is a basic math that the increment of v in the direction of dr is dv = ∇v.dr. Theorem 1  Liutex is the local fluid rotation axis.    Proof  In the Liutex direction, which is the real eigenvector,  dv = ∇v.r = λr r . According to Definition 6, Liutex is the Local rotation axis as R = Rr . Theorem 2  Vorticity is in general not local fluid rotation axis.           dv = ∇v ⋅ ω = A ⋅ ω + B ⋅ ω = A ⋅ a1 r1 + a2 r2 + a3 r3 + ( ∇ × v ) × ω Proof         = a1λ1 r1 + a2 λ2 r2 + a3 λ3 r3 + 0 ≠ λ a1 r1 + a2 r2 + a3 r3 = λω

(

)

(

)

(1.13)

unless λ1 = λ2 = λ3 = λ The velocity increment has stretches in three directions which violates Definition    6, where ω=( ∇× v ) , A and B are are symmetric and anti-symmetric tensors of ∇v , λ1 , λ2 , λ3 , r1 , r2 , r3 are real eigenvalues and corresponding eigenvectors of symmetric matrix A. A rotation axis cannot be stretched in three directions but only in its own direction. The other common misunderstanding is consideration of λci as the fluid swirling/ rotation strength Theorem 3  Liutex magnitude is the local rotation strength. Proof  Liutex is the rigid rotation part extracted from the fluid motion and twice the local angular speed. Theorem 4  λci is not the local rotation strength. Proof  According to Zhou et al. [10]



λr      ∇v = [ vr vcr vci ]  0  0

0 λcr −λci

0  λci  [ vr λcr 

 vcr

  −1 vci ] = T ∇VT −1

(1.14)

 ∇ V The transformation is similar butnot orthogonal since T is not orthogonal.  has same eigenvalues as original ∇v has, but different rotation speed from ∇v as  ∇ v vorticity of is not equal to λci − (−λci) = 2λci. This contradicts to the Galilean invariance of vorticity.

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 ∇V is exactly same as the original In the Liutex-based Principal Coordinate,  ∇v in the XYZ-frame since Q (transformation matrix) is the orthogonal matrix. Of course, other second generation methods cannot be used as the fluid rotation strength either like Q, Δ, or λ2 which are all contaminated by shear and even have different dimensions from the fluid angular speed. Contaminations of First and Second Generations of Vortex Identification Liutex is a physical quantity which exactly represents the rigid rotation or vortex and all other methods are contaminated [50]. Contamination of Vorticity In a principal coordinate:

ω = (η , − ξ ,R + ε )

T



(1.15)



and its magnitude is

ω = η2 +ξ 2 + (R + ε )



2

(1.16)



From the Eqs. (1.15) and (1.16), it can be concluded that a vorticity vector does not only represent rotation but also claims shear to be a part of the vortical structure. Contamination of the Q Method In a principal coordinate:



1 1    ε − R 0   λcr  λcr 2 2     1 1 λcr ∇V =  R +ε 0  =  ε λcr 2  2    1 η λr   1  ξ ξ η    2 2 1 1 1   0 − R− ε − ξ  2 2 2   1 1 1   R+ ε − η = AQ + BQ 0 2 2 2    1  1ξ η 0   2  2

1  ξ 2   1  η + 2   λr  

(1.17)



1  Liutex and Third Generation of Vortex Identification Methods

(

)

19

2 2 1 R ε  ξ  η   2 +  + 2  + 2    F F 2   2 2  2  2   2 2 2 1 ε  ξ  η   − 2 λcr 2 + λr 2 + 2   + 2   + 2    2  2 2  2   2 1 R 1 =   + R ⋅ ε − λcr 2 − λr 2 2 2 2  

Q=



1 BQ 2

2

− AQ

2

=

(1.18)

Therefore, the value of Q is undoubtedly contaminated by shear and stretching. In addition, Q contains an R2 term, indicating dimensional inconsistence with a fluid rotation. Contamination of λci Criterion In a principal coordinate, we have RR  + ε  = λci2  22 



(1.19)

Thus,

λci =

RR  +ε   22 

(1.20)



The expression of λci has ε, which is a component of the shear part and thus is contaminated by shear. Numerical Contamination Analysis [51] The velocity gradient tensor in the Principal Coordinate is:



 ∂u  ∂x   ∂v ∇V =   ∂x  ∂w   ∂x

∂u ∂y ∂v ∂y ∂w ∂y

∂u   λ ∂z   cr  ∂v   1 R +ε = ∂z   2  ∂w   ξ  ∂z  

1 − R 2

λcr η

 0  0   λr  

(1.21)



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In order to demonstrate the shear and stretching contaminations of different criteria numerically, shear and stretching components are separately added and calculated to see how different criteria respond. Adding Shear Components A matrix corresponding to shear is added to the original matrix:



0 0 ∇Vshear = ε a 0 ξ a η a

0 0  0 

(1.22)

The new velocity gradient tensor in the Principal Coordinates after adding shear is



1  λcr − R  2  1 λcr ∇V1 =  R + ε + ε a 2  η + ηa  ξ + ξa 

 0  0   λr  

(1.23)

Under the Principal Coordinate, the local rotation axis is the Z-axis, so ξa (value ∂w ∂w change of component) and ηa (value change of component) are not in the ∂y ∂x rotation plane. Thus, these components will not influence the rotation strength. However, εa is in the rotation plane, thus affecting rotation strength. By the Liutex ∂v  ∂u ∂v  ∂u , .  and definition, the magnitude of rotation strength is min θ ∂ y y x ∂ ∂ ∂ xθ  θ  θ  θ ∂u ∂v are the minimum absolute values of ∂y and respectively, when we rotate the ∂x π π coordinate θ angle anti-clockwise along the z-axis ( − < θ ≤ ) . The rotation 2 2 1 1 strength does not change if εa ≥  − ε since R + ε + ε a ≥ − R , which shows that 2 2 1 1 the Liutex magnitude is still R. If εa   l > lDI, the energy spectrum Ek(f)of the 2

−

5

3 3 turbulence must be of the form of Ek ( f ) = Cε 0 f where f is the wave number and ε0is the turbulence energy dissipation, which is a famous −5/3 law. The similarity is based on Kolmogorov’s third hypothesis that assumes in the inertial subrange the

29

1  Liutex and Third Generation of Vortex Identification Methods Table 1.3  Correlation between Liutex and vorticity, Q, λci and λ2 at x = 815.5 ρ(R, Vor) ρ(R, Q) ρ(R, λci) ρ(R, −λ2)

z1 0.10 0.90 0.91 0.90

z2 0.03 0.86 0.90 0.87

z3 0.08 0.70 0.92 0.80

z4 0.30 0.69 0.88 0.75

z5 0.55 0.83 0.90 0.89

z6 0.73 0.79 0.94 0.88

z7 0.41 0.69 0.90 0.78

z8 0.81 0.85 0.96 0.89

z9 0.71 0.84 0.95 0.87

z10 0.47 0.68 0.93 0.74

Fig. 1.12  Correlation between Liutex and vorticity, Q, λci and λ2 at x = 815.5

turbulence energy spectrum is solely determined by energy dissipation and the wave a b number, independent of viscosity. This will give Ek ( f ) = Cε 0 f where f is the wave number and its dimension is 1/m. The power coefficients a and b are easily obtained by dimensional analysis. However, Kolmogorov’s law requires homogeneous incompressible flow with very high Reynolds number and is for the inertial subrange. The −5/3 law is hard to match DNS and experiment in a turbulent boundary layer [54]. However the Liutex similarity is immediate found in a low Reynolds number turbulent boundary layer (Fig. 1.13). For so thrilling discovery of the Liutex similarity in many different boundary layers, the Liutex similarity cannot be an accident but very meaningful for finding turbulence structure and turbulence modeling. There is no similarity for vorticity and Q-criterion (Fig. 1.14)

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Fig. 1.13 Liutex −5/3 similarity and Kolmogorov K41 Similarity

Fig. 1.14  Vorticity and Q-criterion have no similarity

4.3  C  orrelation Between Shock Fluctuation and Liutex Spectrum A high order large eddy simulation (LES) of flow separation induced by shock and turbulent boundary layer interaction (SBLI) at Mach number 2.4 in a compression corner is conducted. A low frequency of pressure oscillation caused by SBLI has been found for decades, which is one of the major hurdles of the supersonic commercial aircraft design. The correlation of the pressure oscillation and Liutex spectrum are very closely correlated, over 0.9 at most points shown in Fig.  1.15 and Table  1.4, which could help find the mechanism of the low frequency noise

31

1  Liutex and Third Generation of Vortex Identification Methods

Fig. 1.15  Correlation of spectrum of pressure fluctuation and Liutex. (a) Shock boundary layer Interaction (b) Seven sample points (c) Spectrum of pressure and Liutex at point 5 Table 1.4  Correlation between pressure fluctuation and Liutex spectrum Point 1 0.9324

Point 2 0.9259

Point 3 0.9553

Point 4 0.9374

Point 5 0.9166

generation by SBLI and ways to reduce or removal of the low frequency noise. It cannot be an accident for such a close correlation. The correlation between pressure fluctuation and Liutex spectrum is very large (see Table 1.4)

5  Conclusions and Future Work Following conclusions can be made based on the aforementioned analyses 1. Liutex is a new physical quantity to represent flow rotation or vortex with direction as the fluid rotation axis and magnitude as twice the fluid angular speed. 2. There is a principal coordinate based on Liutex which is parallel to the Z-axis. 3. There is a principal decomposition of velocity gradient tensor or UTA R-NR decomposition which is unique and Galilean invariant while the Cauchy-Stokes (Helmholtz) decomposition is not unique and Galilean invariant. In addition, the anti-symmetric tensor cannot represent the flow rotation

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4. Vorticity is not vortex and cannot represent fluid Vorticity must be   rotation. decomposed to Liutex and shear, or ∇ × v = R + S (RS decomposition of vorticity) 5. The first and second generations of vortex identification methods are all contaminated by shearing and some of them are contaminated by stretching. 6. The second generation of vortex identification methods is scalar and then strongly threshold-dependent. They have problems with non-uniqueness and wrong dimension. 7. The Omega method is insensitive to threshold change and capable to capture both strong and weak vortices. 8. The modified Liutex-Omega method is not contaminated by shearing and insensitive to threshold change and capable to capture both strong and weak vortices. This method is particularly important for complex vortex structure as a mixture of shear, weak vortices and strong vortices in addition to advantages of insensitive feature to threshold change. 9. The Liutex core line method is unique and threshold-free, which can give a unique and accurate vortex structure of the fluid field. 10. If shear is large like areas near the solid wall, vorticity cannot represent vortex and the correlation between vorticity and Liutex is very small which shows they are not correlated. In the near wall region, the correlation between Liutex and second generation of vortex identification methods like Q, λ2, λci is also small because these methods are all contaminated by shear. In the area away from the solid wall and in the upper boundary layer, shear becomes weak, the correlation between Liutex and first and second generations becomes large or close to 1. However, turbulence is likely generated by the near wall boundary layers where the first and second generations of vortex identification methods are in general inappropriate to represent vortex structure. In general vorticity is worse than the second generation. 11. Hairpin vortex in turbulence is generated by K-H instability (shear layer instability). 12. Liutex has similarity of −5/3 law in the dissipation subrange of turbulence boundary layer. 13. The low frequency noise generated by shock and turbulent boundary layer interaction (SBLI) is dominated by the Liutex spectrum. Therefore, control of Liutex spectrum is the key of SBLI control. The future work would include: 1. Unique and threshold-free vortex structure Liutex is a mathematical definition for fluid rotation and vortex, but iso-­ surface of Liutex is still threshold-dependent for vortex identification. The modified Liutex-Omega method is insensitive to threshold change and a nice tool for vortex identification. However, it is still not unique. The Liutex core line and tube method is the only candidate to give a unique vortex structure by defining a   special Liutex line which satisfies: ∇R × r = 0 , r ↑ 0 . However, finding of Liutex core lines and tubes are mainly conducted by manual methods so far,

1  Liutex and Third Generation of Vortex Identification Methods

33

which is not realistic for sophisticated vortex structure in turbulent flow. The key issue is how to find the local maxima of Liutex in a 3-D flow field. 2. Similarity of Liutex and subgrid model Although K41 is the most well recognized and most important discovery in turbulence research, K41 is not coincided with experiment or DNS very well in a turbulent boundary layer. The similarity of Liutex should be a thrilling discovery after K41, which almost exactly matches the −5/3 law. However, it is still unknown why Liutex spectrum meets the −5/3 law and how to use the Liutex −5/3 law for turbulence modeling, which is still an unanswered question, but extremely important to theoretic turbulence research and development of turbulence models. 3. Liutex dynamics As addressed above, vorticity cannot represent fluid rotation or vortex. The existing “Vortex Dynamics” [55] is really vorticity dynamics. Liutex is the mathematical definition of fluid rotation or vortex. We must establish the transport equation for Liutex to replace the vorticity transport equation, which could be used for vortex science and turbulence research. 4. Physics of Turbulence generation and sustenance Vortex structure is still a mystery even for a very simple question if there are or are not hairpin vortices. Most people believe yes [54], but some say no and recent researches show yes for low Reynolds number but no for high Reynolds number [48, 49]. We understand this was caused by that we have no vortex definition and iso-surface of existing vortex identification methods are all threshold-­ dependent but not unique. After Liutex was discovered, the unique and threshold-free vortex structure must be obtained. Although Richardson [40] and Kolmogorov [39] gave the possible turbulence generation hypotheses, but no one can prove by DNS or experiment that turbulence is generated by large vortex breakdown to small pieces. After the unique vortex structure is found, visualization will provide some hints that how large vortices are generated, how smaller vortices are generated and how they become non-symmetric and chaotic. The Liutex dynamics will be applied to quantified research for turbulence generation and sustenance. 5. Modified Navier-Stokes equations for both laminar and turbulent flows without model Navier-Stokes (NS) equations satisfies conservation of mass, momentum and energy and works well for laminar flow which has no mass transfer between fluid layers or has no Liutex in a laminar boundary layer. Turbulent flow is a fluid rotation dominant or Liutex dominant flow. In such a case, we must consider the Liutex role which can be achieved by the conservation of moment of momentum and centripetal force by adding some additional force terms. We call this the Liutex-­ based modified NS equations. In general, we need a turbulence model for Reynolds averaged NS equation (RANS) or a subgrid model for large eddy simulation (LES). In DNS, we do not need model as all vortices are resolved, which is too costly for engineering applications. In the modified NS equations, there is no turbulence model needed and the governing equation should work for both

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Laminar and turbulent flow. These Liutex-based terms will automatically disappear when the flow is laminar where Liutex is zero or the flow field is fully resolved where Liutex terms are zero. All aforementioned intellectual merit will produce significant impact to fundamental fluid mechanics by clarifying misunderstandings on vortex and turbulence in textbooks and thousands of research papers, which have dominated the community for centuries. New fluid kinematics will be established by using Liutex theoretical system to replace old fluid kinematics and then the new Liutex dynamics will be established. It will let us enter a new era from qualitative turbulence research to quantified turbulence research. Turbulence is filled with vortices, how to conduct quantified research if we did not have mathematical definition for vortex? As vortex exists everywhere in the universe, a mathematical definition of vortex or Liutex will play a critical role in scientific research. There is almost no place without vortex in fluid dynamics. As a projection, the Liutex theory will play an important role to the investigations of the vortex dynamics applying to hydrodynamics, aerodynamics, thermodynamics, oceanography, meteorology, metallurgy, civil engineering, astronomy, biology, etc. and to the researches on the generation, sustenance, modelling and controlling of turbulence. Acknowledgement  The author would like to thank the Department of Mathematics of University of Texas at Arlington where the UTA Team is housed and where the birthplace of Liutex is. The authors are grateful to Texas Advanced Computational Center (TACC) for providing computation hours. The author would like to thank his students and visitors, especially Y.  Wang, Y.  Gao, X. Dong, J. Liu, P. Yan, W. Xu, Y. Yu, P. Shrestha.

References 1. C. Liu, Y. Gao, S. Tian, X. Dong, Rortex—A new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids 30, 035103 (2018). https://doi.org/10.1063/1.5023001 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 3. 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. Hydrodyn. 31(2), 1–19 (2019). https://doi.org/10.1007/s42241-­019-­0022-­4 4. D. Küchemann, Report on the IUTAM symposium on concentrated vortex motions in fluids. J. Fluid Mech. 21, 1–20 (1965) 5. V. Kolář, J. Šístek, Consequences of the close relation between Rortex and swirling strength. Phys. Fluids 32, 091702 (2020). https://doi.org/10.1063/5.0023732 6. C. Liu, H. Xu, X. Cai, Y. Gao, Liutex and Its Applications in Turbulence Research, ISBN-13: 978-0128190234, ISBN-10: 012819023X (Elsevier, 2020 7. J. Hunt, A. Wray P. Moin, Eddies, streams, and convergence zones in turbulent flows, in Center for Turbulence Research Proceedings of the Summer Program, vol. 193 (1988) 8. M.  Chong, A.  Perry, B.  Cantwell, A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765–777 (1990) 9. J. Jeong, F. Hussain, On the identification of a vortices. J. Fluid Mech. 285, 69–94 (1995) 10. J. Zhou, R. Adrian, S. Balachandar, T. Kendall, Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396 (1999)

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11. P. Chakraborty, S. Balachandar, R.J. Adrian, On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189–214 (2005) 12. C.  Liu Y.  Gao, Liutex-based and Other Mathematical, Computational and Experimental Methods for Turbulence Structure, vol. 2, ISSN: 2589-2711, eISSN: 2589-272X (Online), ISBN: 978-981-14-3758-8, eISBN: 978-981-14-3760-1 (Online) (Bethman, 2020) 13. Y. Gao, J. Liu, Y. Yu, C. Liu, A Liutex based definition and identification of vortex core center lines. J. Hydrodyn. 31(2), 774–781 (2019) 14. Y. Gao, Y. Yu, J. Liu, C. Liu, Explicit expressions for Rortex tensor and velocity gradient tensor decomposition. Phys. Fluids 31, 081704 (2019) 15. Y.  Gao, C.  Liu, Rortex based velocity gradient tensor decomposition. Phys. Fluids 31(1), 011704 (2019) 16. X. Dong, Y. Wang, X. Chen, Y. Zhang, C. Liu, Determination of epsilon for Omega vortex identification method. J. Hydrodyn. 30(4), 541–548 (2018) 17. 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 18. X. Dong, S. Tian, C. Liu, Correlation analysis on volume vorticity and vortex in late boundary layer transition. Phys. Fluids 30, 014105 (2018) 19. X. Dong, G. Dong, C. Liu, Study on vorticity structures in late flow transition. Phys. Fluids 30, 104108 (2018) 20. X. Dong, Y. Gao, C. Liu, New normalized Rortex/vortex identification method. Phys. Fluids 31, 011701 (2019). https://doi.org/10.1063/1.5066016 21. X. Dong, X. Cai, Y. Dong, C. Liu, POD analysis on vortical structures in MVG wake by Liutex core line identification. J. Hydrodyn. 32, 497–509 (2020) 22. J. Liu, Y. Gao, C. Liu, An objective version of the Rortex vector for vortex identification. Phys. Fluids 31(6), 065112 (2019). https://doi.org/10.1063/1.5095624 23. J. Liu, C. Liu, Modified normalized Rortex/vortex identification method. Phys. Fluids 31(6), 061704 (2019). https://doi.org/10.1063/1.5109437 24. J. Liu, Y. Gao, Y. Wang, C. Liu, Galilean invariance of Omega vortex identification method. J. Hydrodyn. (2019). https://doi.org/10.1007/s42241-­019-­0024-­2 25. J. Liu, Y. Gao, Y. Wang, C. Liu, Objective Omega vortex identification method. J. Hydrodyn. (2019). https://doi.org/10.1007/s42241-­019-­0028-­y 26. J. Liu, Y. Deng, Y. Gao, S. Charkrit, C. Liu, Mathematical foundation of turbulence generation from symmetric to asymmetric Liutex. J. Hydrodyn. 31(3), 632–636 (2019) 27. Y. Wang, Y. Yang, G. Yang, C. Liu, DNS study on vortex and vorticity in late boundary layer transition. Comm. Comp. Phys. 22, 441–459 (2017) 28. Y.  Wang, Y.  Gao, C.  Liu, Galilean invariance of Rortex. Phys. Fluids 30, 111701 (2018). https://doi.org/10.1063/1.5058939 29. 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.  Hydrodyn. (2019). https://doi. org/10.1007/s42241-­019-­0032-­2 30. Y.  Wang, Y.  Gao, C.  Liu, Letter: Galilean invariance of Rortex. Phys. Fluids 30(11), 111701 (2019) 31. Y. Wang, Y. Gao, H. Xu, X. Dong, J. Liu, W. Xu, M. Chen, C. Liu, Liutex theoretical system and six core elements of vortex identification. J. Hydrodyn. 32, 197–221 (2020) 32. W. Xu, Y. Gao, Y. Deng, J. Liu, C. Liu, An explicit expression for the calculation of the Rortex vector. Phys. Fluids 31, 095102 (2019). https://doi.org/10.1063/1.5116374 33. W. Xu, Y. Wang, Y. Gao, J. Liu, H. Dou, C. Liu, Liutex similarity in turbulent boundary layer. J. Hydrodyn. 31(6), 1259–1262 (2019) 34. H. Xu, X. Cai, C. Liu, Liutex core definition and automatic identification for turbulence structures. J. Hydrodyn. 31(5), 857–863 (2019) 35. Y. Zhang, X. Qiu, F. Chen, K. Liu, Y. Zhang, X. Dong, C. Liu, A selected review of vortex identification methods with applications. J. Hydrodyn. 30(5) (2018). https://doi.org/10.1007/ s42241-­018-­0112-­8

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36. Y.  Zhang, X.  Wang, Y.  Zhang, C.  Liu, Comparisons and analyses of vortex identification between Omega method and Q criterion. J. Hydrodyn. 31(2), 224–230 (2019) 37. Y. Yu, P. Shrestha, C. Nottage, C. Liu, Principal coordinates and principal velocity gradient tensor decomposition. J. Hydrodyn. 32, 441–453 (2020) 38. C. Liu, Y. Wang, Y. Yang, Z. Duan, New Omega vortex identification method. Sci. China Phys. Mech. Astron. 684711, 59 (2016) 39. A.N.  Kolmogorov, Local structure of turbulence in an incompressible fluid at very high Reynolds numbers. Dokl. Akad. Nauk SSSR 26, 115–118 (1941) 40. L.F.  Richardson, Weather Prediction by Numerical Process (Cambridge University Press, Cambridge, UK, 1922) 41. S. Robinson, S. Kline P. Spalart, A review of quasi-coherent structures in a numerically simulated turbulent boundary layer, Tech. rep., NASA TM-102191 (1989) 42. S. Robinson, A review of vortex structures and associated coherent motions in turbulent boundary layers, in Structure of Turbulence and Drag Reduction (Springer, Berlin Heidelberg, 1990) 43. R. Adrain, Hairpin vortex organization in wall turbulence. Phys. Fluids 19, 041301 (2007) 44. C. Liu, Y. Yan, P. Lu, Physics of turbulence generation and sustenance in a boundary layer. Comput. Fluids 102, 353–384 (2014) 45. Y. Zhang, K. Liu, H. Xian, X. Du, A review of methods for vortex identification in hydroturbines. Renew. Sust. Energ. Rev. 81, 1269–1285 (2017) 46. H.  Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. Journal für die reine und angewandte Mathematik 55, 25–55 (1858) 47. C. Truesdell, The Kinematics of Vorticity (Indiana University Publications Science Seres Nr. 14.) XVII + 232 S (Indiana University Press, Bloomington, 1954) 48. I. Marusic, B.J. McKeon, P.A. Monkewitz, H.M. Nagib, A.J. Smits, K.R. Sreenivasan, Wall-­ bounded turbulent flows at high Reynolds numbers: recent advances and key issues. Phys. Fluids 22, 065103 (2010) 49. J. Jiménez, Coherent structures in wall-bounded turbulence. J. Fluid Mech. 842, P1 (2018). https://doi.org/10.1017/jfm.2018.144 50. V. Kolář, J. Sistek, Stretching response of Rortex and other vortex identification schemes. AIP Adv. 9, 105025 (2019) 51. P.  Shrestha, C.  Nottage, Y.  Yu, O.  Alvarez, C.  Liu, Stretching and shearing contamination analysis for Liutex/Rortex and other vortex identification methods. arXiv:06779 (2020) 52. Y. Yu, P. Shrestha, O. Alvarez, C. Nottage, C. Liu, Correlation analysis among vorticity, Q method and Liutex. J. Hydrodyn. arXiv:2008.06781 (2020) 53. S. Charkrit, C. Liu, Vortex core and POD analysis on hairpin vortex formation in flow transition. J. Hydrodyn. arXiv:1912.01975 (2020) 54. X.  Wu, P.  Moin, Direct numerical simulation of turbulence in a nominally zero-pressure-­ gradient flat-plate boundary layer. J. Fluid Mech. 630(10), 5–41 (2009) 55. J. Wu, H. Ma, M. Zhou, Vorticity and Vortices Dynamics (Springer, Berlin Heidelberg, 2006) 56. B. Epps, Review of Vortex Identification Methods, AIAA 2017-0989 (2017)

Chapter 2

Incorrectness of the Second-Generation Vortex Identification Method and Introduction to Liutex Yifei Yu, Pushpa Shrestha, Oscar Alvarez, Charles Nottage, and Chaoqun Liu

Abstract  For a long time, scientists believe that vorticity reveal the essence of vortex since vorticity is a perfect indicator of rotation for rigid object although many experimental results do not support this point of view. Even though scientists do not doubt the correctness of vorticity, they have to develop some other vortex identification methods to improve the performance of detecting vortex. Under this situation, Q method, λci method and λ2 method are innovated. However, these methods are all scalars and not able to locate the swirling direction, and threshold is needed to be used to display vortex structures. To overcome these drawbacks, Liu proposed Liutex which can correctly represent the rigid rotation of fluids. The magnitude of Liutex represents the twice angular speed and the direction of Liutex is the direction of swirling axis. In this paper, the incorrectness of the classical methods is explained as well as the rationality of the definition of Liutex. A correlation research is done between vorticity, Q, λci, λ2 methods and Liutex based on a direct numerical simulation (DNS) case of boundary layer transition. The results show that the correlation between vorticity and Liutex is very small or even negative in strong shear regions, which demonstrates that using vorticity to detect vortex lacks scientific foundation and vorticity is not appropriate to represent vortex. In the last part of this paper, new vortex structure displaying methods including Liutex iso-surface, Liutex-Omega and Liutex core line are introduced. Keywords  Correlation · Liutex · Vorticity · Liutex-Omega · Liutex core line · Q method · λcimethod · λ2 method

Y. Yu · P. Shrestha · O. Alvarez · C. Nottage · C. Liu (*) Department of Mathematics, University of Texas at Arlington, Arlington, TX, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_2

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1  Introduction Scientists have been finding a proper indicator of fluid rotation for a long time. At the very beginning, vorticity was used to represent fluid rotation since the magnitude of vorticity is exactly twice of the angular speed and the direction of vorticity is the direction of the swirling axis for rigid body mechanics. In 1858, Helmholtz [1] advised to use a vorticity tube/filament to display vortex structure. However, it has been found that the predictions of vortex with this assumption conflict with experimental results in many cases. For example, 2D Couette flow [2] is a type of straight-line laminar flow which implies there is no vortex in the domain. However, vorticity can be found in the 2D Couette flow. Some direct numerical simulations (DNS) show the same situation as well. A DNS research done by Wang et al. [3] shows the magnitude of vorticity is small at some places where the fluid rotation is strong. This evidence clearly enunciates vorticity and vortex are not equivalent. All the counterexamples presented above prompted scientists to look for more appropriate methods to represent vortex. In the past several decades, some vortex identification methods have been proposed, including Q criterion [4], ∆ criterion [5], λci method [6] and λ2 method [7]. Although, compared with vorticity, these methods perform better in terms of predict vortex strength. However, there is a common drawback that none of these methods can solve. This drawback is all of these methods are scalar methods which implies the only way to display the structure of the vortex is using iso-surfaces dependent on threshold whose selection is generally empirical and some kind arbitrary. There is not a uniform criterion of threshold selection and as a result different people may choose different thresholds, creating the difficulty of vortex structures comparison among different scientists. Additionally, all these methods are not able to indicate the direction of the swirling axis because they can only produce the strengths of the vortex by scalar values. Liu [8] classified vorticity-based methods as the first generation methods, Q criterion, ∆ criterion, λci method as the second generation methods because they are all based on eigenvalues of the velocity gradient tensor, and to overcome the drawbacks of second generation methods, Liu developed a new physical quantity—Liutex [9, 10] categorized as the third generation method. Liutex is a vector quantity whose magnitude indicates the strength of rotation and whose direction represents the swirling axis. The Liutex concept comes from the idea that the rotation axis can be only stretched or compressed and the magnitude should correspond with the rigid rotation of vortex. Detailed information of Liutex is discussed later in the paper. After the innovation of Liutex, it has been proved to be unique and Galilean invariant [11], and many theories based on Liutex are gradually proposed and developed such as Liutex similarity [12], Liutex core line [13, 14], Liutex-Omega method [15, 16], Objective Liutex [17], Principal Coordinate and Principal Decomposition [18]. Even though a lot of counterexamples and experiments can be provided to disprove vorticity is vortex, many text-books [19] still consider vorticity as the fundamental vortex identification method since it looks perfect in math and works very well for rigid objects. So, it is of significant necessity to clarify vorticity is not

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vortex. In this paper, the incorrectness of the classical methods is explained as well as the rationality of the definition of Liutex, and furthermore an evidence from correlation perspective is provided as well to convince people that vorticity and vortex are different. Correlation coefficient is a statistics concept which reveals the extent that two groups of data are related. If vorticity is vortex, their correlation coefficient should be close to 100%. However, from our research, only in the region where shear is weak, can the correlation be around 100% and in some areas where the shear is strong, the correlation can even become negative. Apparently, vorticity cannot be vortex. The paper is organized as follows. In Sect. 2, we briefly review some vortex identification methods and explain the reason why they are not accurate. Theoretical correlation analysis can be found in Sect. 3, followed by the numerical analysis based on DNS research data in Sect. 4. In Sect. 5, some vortex identification methods based on Liutex are introduced.

2  Incorrectness of Classical Vortex Identification Method In this section, it is explained why classical vortex identification methods are not correct.  Definition 1  ω is called vorticity if     v (2.1)  v is the velocity vector. Admittedly, vorticity is an ideal indicator of rotation for rigid objects, it has problem to deal with fluids. To clarify the criterion, what is a local rotation axis is defined firstly. The fundamental physics of a rotation axis is that the rotation axis can only be stretched (or compressed) but cannot rotate itself. In other words, along the direction axis,  the change of the velocity must be in the same direction, i.e.  of swirling  dv   dr , where r is the direction of swirling axis. Based on this idea, the definition of rotation axis is derived as follows.  Definition 2  The local rotation axis r [9] is the normalized vector that satisfies:

  dv   dr    r  0

(2.2)

(2.3)    ω refers to vorticity, dv is the increment of velocity, dr is the increment of where   r , and α is a constant. Eq. (2.3) is included in the definition to further uniquify r since a straight line has positive and negative directions.

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 Theorem 1  The r in Def. 2 is the normalized eigenvector of velocity gradient tensor [9].        Proof  Since dv  v  dr , then Def. 2 implies that v  dr   dr . Therefore, dr is the eigenvector corresponding to the real eigenvalue of ∇v . With the Def. 2, it can be proved that direction of vorticity is not rotation axis. Theorem 2  Vorticity is in general not local fluid rotation axis.      Proof  dv  v    A    B   (2.4) where  B are symmetric and antisymmetric part of the velocity gradient tensor  A and ∇v , and ω represents vorticity. Since A is a symmetric matrix, it must have three real eigenvalues andthree  core1 , e2 , e3 be responding eigenvectors. Let λ1, λ2, λ3 be the three eigenvalues of A and   e ,e ,e the three corresponding eigenvectors.   1 2 3 are a set of basis of the space and there exists a1, a2, a3 such that   a1e1  a2 e2  a3 e3 . Therefore,

   dv  A    B  



      A   a1e1  a2 e2  a3 e3      v   



    a11e1  a2 2 e2  a3 3 e3  0





    a11e1  a2 2 e2  a3 3 e3        a11e1  a2 2 e2  a3 3 e3    a1e1  a2 e2  a3 e3   

(2.5)

It is obvious that unless λ1 = λ2 = λ3 = λ. Thus, vorticity does not satisfy the definition of the rotation axis in general. Another issue of vortex is its strength. It is known that the velocity gradient tensor of rigid rotation in 2D is u  y   0  R   v   R 0  y 

(2.6)



 u  x  v    v  x 

u  y  0 2 a   v  0 0  y 

(2.7)



 u  x  v    v  x 

u v u v    . R represents the angular speed of the rigid rota 0 and i.e. x y y x ∂u v tion. A problem arises that how to choose the value of R if is not exactly  . ∂y  x Take the following velocity gradient tensor as an example.



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Fig. 2.1  2D Couette flow

1 The magnitude of vorticity of Eq. (2.7) is ω = 2a, and 2   a is considered as the angular speed according to the classical theory. It can be seen from this example ∂u ∂v u v ,   . But, is that vorticity takes average of as angular speed when ∂y ∂x y x u v   taking average the correct way to deal with ? In fact, Eq. (2.7) is the y x velocity gradient tensor corresponding to 2D Couette flow. The figure of the velocity distribution is shown in Fig. 2.1. Apparently, this is a straight-line laminar flow where there is no vortex, however, 1 u v   a  0 . So taking average is not the correct way to deal with  issue. 2 y x After several years’ research, Liu [9] pointed out that taking minimum should be the correct way to evaluate the vortex strength in the “Liutex” concept which is innovated recently. Taking minimum is more reasonable than taking average because v  0 in the original velocity gradient tensor, but vorticity artificially create a in x ∂v the entry, and that is reason why the non-existing vortex appears. On the con∂x trary, taking minimum correctly reflects the physical phenomena of this 2D Couette flow. In many recent papers [20], it has been shown that vorticity consists of    rotation   RS , and shear. There is a so-called R-S decomposition of vorticity, namely,      where R means rotation and S represents shear. Note that all ω , R and S are 1  vectors. Therefore, ω representing angular speed is only a special case in solid 2 mechanics with the condition that there is no shear. After the discussion of what should be the rotation axis and vortex strength, the definition of “Liutex” becomes natural. Liutex is a vector first defined by Liu such that its magnitude represents twice angular speed and its direction is the swirling axis. Definition 3  Liutex [9] is a vector method defined as

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  R = Rr (2.8)  where R is the magnitude of Liutex and r is the local rotation axis. Wang et al. [21] and Xu et al. [22] proposed an explicit expression of R i.e.





  R   r 

 

  r 

2

 4ci2



(2.9)

   ∇v where ω is vorticity, r is the local rotation axis, and λci is the imaginary   part of  complex eigenvalue. In order to let r be unique, it is required that   r  0 .

Definition 4  Principal Coordinate [18] is the coordinate that satisfies    1 . Its Z-axis is parallel to the r (direction of Liutex) and   r  0 . 2. The velocity gradient tensor under this coordinate is in the form of:



  cr    V V   X   W  X

U Y

cr W Y

 0  0   r  

(2.10)

where λr and λcr are the real eigenvalue and real part of the conjugate complex eigenvalue pair of the velocity gradient tensor, respectively, for rotation points. U V U  0 and  3. . Y Y X 4. The rotational angle of the X-Y coordinates around the Z-axis must be smaller than 90° or −90°  0, there exists one real and two conjugate complex eigenvalues. The latter means that the point is inside a vortex region. Although the Δ-method can capture the vortex region successfully, it is susceptible to the threshold value, which is man-made and arbitrary in general.

2.2  Q Method Proposed by Hunt et al. [5], the Q method is one of the most popular methods used to visualize the vortex structure. 𝑄 is defined as the difference between the squared Frobenius norms of the vorticity and strain-rate tensors. i.e., Q



1 2 BF  AF2 2





(3.7)

A and B are the symmetric (strain-rate tensor) and anti-symmetric (vorticity tensor) part of the velocity gradient tensor.

A



1    v  v T 2





 u  x   1  v u       2  x y    1  w  u   2  x z  

1  u v     2  y x  v y 1  w v     2  y z 

1  u w     2  z x   1  v w      2  z y    w   z 

(3.8)



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B

1    v  v T 2







 0    1  v u       2  x y    1  w  u   2  x z  

1  u v     2  y x  0 1  w v     2  y z 

1  u w     2  z x   1  v w      2  z y     0  

(3.9)

The Q method considers that a vortex occurs in the region where Q > 0. Using the Q method, it is easy to track the vortical structure by iso-surface plotting. However, Q is scalar-valued, and a proper threshold is required to visualize the vortex region.

2.3  λ2 Criterion The λ2 criterion  is calculated based on the observation that pressure tends to be the lowest on the axis of a swirling motion of fluid particles in a vortical region. This occurs because the centrifugal force is balanced by the pressure force (the cyclostrophic balance). This method is valid only in a steady inviscid planar flow [6]. However, this assumption fails to accurately identify vortices under strong, unsteady, and viscous conditions. By neglecting these unsteady and viscous effects, the symmetric part S of the gradient of the incompressible Navier–Stokes equation can be expressed as: S  A2  B2  

  p 





(3.10)

where p is the pressure and Eq. (3.10) is a representation of the pressure Hessian ∂2 p matrix ((∇(∇p))ij= . ∂xi ∂yi To capture the region of local pressure minimum in a plane perpendicular to the vortex core line, Jeong and Hussain [6] defined the vortex core as a connected region with two positive eigenvalues of the pressure Hessian matrix, i.e., a connected region with two negative eigenvalues of the symmetric tensor S. If λ1, λ2 & λ3 are three real eigenvalues of the symmetric tensor S, then by reordering them as λ1 ≥ λ2 ≥ λ3, there must be λ2  0 , where ω is the vorticity vector [9]. The result of a Direct Numerical Simulation is taken as an example to show the incorrectness mentioned above of the candidates expect Liutex. The point selected to analyze is shown in Fig. 4.1. The velocity gradient tensor at this point is



0.2818661 0.2621670   0.0533380  ∇V =  −0.0139413 0.0003662 0.1193656   −0.0055126 −0.0798357 −0.0548821

(4.29)

And the corresponding Liutex, vorticity can be expressed as

 T R = [ −0.1292245 0.0197261 −0.0100723]

Fig. 4.1  The selected point

(4.30)

4  Mathematical Study on Local Fluid Rotation Axis: Vorticity is Not the Rotation Axis



 T ω = [ −0.1992013 0.2676797 −0.2958074 ]

81

(4.31)

The direction of the three eigenvectors of symmetric matrix D is:

 T d1 = [ −0.6333458 0.4286110 0.6443335]  T d2 = [ 0.1600420 −0.7420691 0.6509377]  T d3 = [ 0.7571391 0.5153891 0.4013906 ]

(4.32) (4.33) (4.34)

Figure 4.2 shows the directions of Liutex, velocity, and vorticity. Figure  4.3 shows the directions of the three eigenvectors of the symmetric matrix. It can be seen easily that the direction of Liutex is inside and generally parallel to the isosurface shape while the other vectors are not. Besides the intuitive feeling from the graph, some mathematical analysis will be done. Recall that the rotation axis can only be stretched or compressed in one of the physical rotation instances. The increase of velocity along Liutex, vorticity, and the eigenvectors are described as follows:

   T dVR = ∇V ⋅ R = [ −0.0039731 6.0649258 −3.0968137]    T dVω = ∇V ⋅ ω = [ 0.0127261 −0.0324341 −0.0040377]    T dVd1 = ∇V ⋅ d1 = [ 0.2559524 0.0858979 −0.0660894 ]

Fig. 4.2  Directions of Liutex, vorticity, and velocity

(4.35) (4.36) (4.37)

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Fig. 4.3  Directions of the three eigenvectors of the symmetric matrix



   T dVd 2 = ∇V ⋅ d2 = [ −0.0299734 0.0751966 0.0226365]    T dVd 3 = ∇V ⋅ d3 = [ 0.2908864 0.0375455 −0.0673495]

(4.38) (4.39)

The cross product result can be used to test if the directions of two vectors are parallel. All vectors are normalized before taking the cross product to avoid the influence of magnitudes. 



( dV )

 T × Rnormed =  −3.5 × 10 −18 2.5 × 10 −17 1.1 × 10 −16    T dVω × ωnormed = [ 0.68 −0.19 −0.63] normed   T dVd1 × d1normed = [ 0.08 −0.12 0.16 ] normed   T dVd 2 × d2normed = [ 0.07 0.02 0.01] normed   T dVd 3 × d3normed = [ 0.05 −0.17 0.12 ] normed

R normed



(



(



(

)

(

)



)

)

(4.40) (4.41) (4.42) (4.43) (4.44)

The definition of Liutex guarantees that its direction is parallel to the direction of velocity increase, and in our numerical analysis, the cross product is almost zero. However, for the other four vectors, the cross product results are far from zero, i.e., they do not satisfy the first condition of the swirling axis. The second condition for an axis to be the swirling axis is that it can not rotate itself. In classical theory, the vorticity tensor is misunderstood to represent the

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83

rotation part; however, the real rotation part should be described by Liutex. The rotation matrix A is  0   1 A= R 2  1 − R  2

−

1

1 z

R

2 −

0

z



R

2

1 2

1 y

2

R

x

0

y

R

x

  0    =  −0.0050362     −0.0098630  

0.0050362 0

0.0098630 

0.0646122 

−0.0646122

0

(4.45)

 

All vectors are also normalized to avoid the influence of magnitudes.  T A ⋅ Rnormed = [ 0 0 0 ]



 T A ⋅ ωnormed = [ −0.0035 −0.0406 −0.0344 ]  T A ⋅ d1normed = [ 0.0085 0.0448 −0.0214 ]  T A ⋅ d2normed = [ 0.0027 0.0413 0.0464 ]  T A ⋅ d3normed = [ 0.0066 0.0221 −0.0408]

(4.46) (4.47) (4.48) (4.49) (4.50)

Obviously, except Liutex, all other vectors rotate themselves.

3  Conclusions 1. The vorticity tensor can be decomposed into a rigid rotational tensor and anti-­ symmetric shear tensor, which implies that vorticity is not strictly rotation. 2. The vorticity vector contains shear and rotational factors. 3. The rotational axis cannot be affected by rotation on itself, but it can have some stretching or compression. 4. There were five vector candidates to represent the rotational axis: Three eigenvectors of the symmetrical tensor, the vorticity vector from the vorticity tensor (null space), and the Liutex vector from the velocity gradient tensor (vortex area). 5. The symmetrical tensor’s three eigenvectors could not satisfy the conditions to be the rotational axis. 6. The vorticity vector satisfied the conditions in two specific cases: (a) no rotation (R = 0) with λcr = λr. (b) No Shear in the X and Y directions (ξ = η = 0).  7. The Liutex directional vector r , is the only one that satisfies the conditions to be the rotational axis unconditionally.

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References 1. S.K. Robinson, Coherent motion in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639 (1991) 2. B. Epps, Review of vortex identification methods, in 55th AIAA Aerospace Sciences Meeting, Grapevine, Texas, USA (2017 https://doi.org/10.2514/6.2017-­0989 3. C. Liu, Y. Gao, X. Dong, Y. Wang, J. Liu, Y. Zhang, X. Cai, N. Gui, Third generation of vortex identification methods: Omega and Liutex/Rortex based systems. J.  Hydrodyn. 31(2), 205–223 (2019) 4. J.C.R. Hunt, A.A. Wray, P. Moin, Eddies, stream, and convergence zones in turbulent flows, Center for turbulence research report CTR-S88, 193 (1988) 5. J. Zhou, R.J. Adrian, S. Balachandar, T.M. Kendall, Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353 (1999) 6. J. Jeong, F. Hussain, On the identification of a vortex. J. Fluid Mech. 285(1), 69 (1995) 7. Y. Yu, P. Shrestha, C. Nottage, C. Liu, Principal coordinates and principal velocity gradient tensor decomposition. J. Hydrodyn. 32, 441–453 (2020) 8. Y.  Gao, C.  Liu, Rortex and comparison with eigenvalue-based vortex identification criteria. Phys. Fluids 30(8), 085107 (2018) 9. Y. Gao, J. Liu, Y. Yu, C. Liu, A Liutex based definition and identification of vortex core center lines. J. Hydrodyn. 31(3), 445–454 (2019)

Chapter 5

No Vortex in Straight Flows: On the Eigen-­ Representations of Velocity Gradient ∇v Xiangyang Xu, Zhiwen Xu, Changxin Tang, Xiaohang Zhang, and Wennan Zou

Abstract  Velocity gradient is the basis of many vortex recognition methods, such as Q criterion, Δ criterion, λ2 criterion, λci criterion and Ω criterion, etc. Except the λci criterion, all these criterions recognize vortices by designing various invariants, based on the Helmholtz decomposition that decomposes velocity gradient into strain rate and spin. In recent years, the intuition of “no vortex in straight flows” has promoted people to analyze the vortex state directly from the velocity gradient, in which vortex can be distinguished from the situation that the velocity gradient has a couple complex eigenvalues. A specious viewpoint to adopt the simple shear as an independent flow mode was emphasized by many authors, among them, Kolář proposed the triple decomposition of motion by extracting a so-called ‘effective’ pure shearing motion; Li et  al. introduced the so-called quaternion decomposition of velocity gradient and proposed the concept of eigen rotation; Liu et al. further mined the characteristic information of velocity gradient and put forward an effective algorithm of Liutex (namely eigen rotation), and then developed the vortex recognition method. However, there is another explanation for the increasingly clear representation of velocity gradient, that is the local streamline pattern based on the critical-­ point theory. In this paper, the tensorial expressions of the right/left real Schur forms of velocity gradient are clarified from the characteristic problem of ∇v. The relations between the involved parameters are derived and numerically verified. X. Xu Institute of Engineering Mechanics, Nanchang University, Nanchang, China e-mail: [email protected] Z. Xu · W. Zou (*) Institute for Advanced Study, Nanchang University, Nanchang, China e-mail: [email protected] C. Tang Institute of Photovoltaics, Nanchang University, Nanchang, China e-mail: [email protected] X. Zhang School of Science, Nanchang Institute of Technology, Nanchang, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_5

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Comparing with the geometrical features of local streamline pattern, we confirm that the parameters in the right eigen-representation based on the right real Schur form of velocity gradient have good meanings to reveal the local streamline pattern. Some illustrative examples from the DNS data are presented. Keywords  Velocity gradient · Left/right real Schur forms · Right/left eigen-­ representations · Local streamline pattern · Vortex recognition

1  Introduction The intuition of no vortex in straight flows becomes strong in the understanding of vortex identification. Lugt [1] presented that a vortex is the rotating motion of a multitude of material particles around a common center, while Robinson [2] proposed that a vortex exists when instantaneous streamlines mapped onto a plane normal to the vortex core exhibit a roughly circular or spiral pattern, when viewed from a reference frame moving with the center of the vortex core. Therefore, the fact—vorticity cannot distinguish between pure shearing motions and the actual swirling motion of a vortex [3–5]—has won support among the people to develop other ways instead of vorticity to identify a vortex. In the framework of classical flow theory, the vorticity, indicating an average angular velocity of fluid elements, appears as one of the unique natural choices for a vortex-identification criterial measure. The fatal flaw of using vorticity to indicate the vortex is that we have to admit the vortex in straight flows. She et al. [6] found the tube-like feature of strong vortices in the DNS turbulence, Saffman [7] added that “we shall use this term to denote any finite volume of vorticity immersed in irrotational fluid”, and recently Wu and Yang [8] proposed that “vortex is a specific region of vorticity field with tubular structure”, and tried to consider the dynamic mechanism within the definition of vortex. In order to consider the effect of strain rate in addition to vorticity, several methods were developed to analyze the vortex under the requirement of the Galilean invariance [9, 10]. Most of them are derived from the velocity gradient d =  ∇v, where the methods based on the scalar invariants (eigenvalues) of velocity gradient (or its sum decomposition) include Q criterion [11–14], λ2 criterion [3] and λci criterion [15, 16], among them only the λci criterion has nothing to do with the Helmholtz decomposition. After a lot of theoretical and practical exploration, the application of complex measures derived from d has already revealed its importance in the analysis of vortical structures in complicated flows. But the flow mechanism other than the Helmholtz decomposition seems to be difficult to construct. Kolář [17, 18] proposed a triple decomposition by extracting of a so-called ‘effective’ pure shearing motion, limited to planar flows. Li et  al. [19] presented the quadruple decomposition of velocity gradient, namely dilatation, axial deformation along the principal axes of

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87

the strain-range sensor, planar motion, and pure shearing. Liu and his coworkers [20, 21] realized such a decomposition is actually based on the real Schur form (Golub and van Loan [22]) of the velocity gradient, and constructed a systemic criterion called Liutex/Rortex. But the abandon of the Helmholtz decomposition means the mechanism of viscous interaction coming from the strain rate must be modified. An alternative idea making use of velocity gradient to catch flow patterns stems from the critical point theory [23–25]. Since Perry and Chong [25] pointed out the usefulness of critical-point concepts in the understanding of flow patterns, Chong et  al. [26] proposed the use of the region where a couple of complex eigenvalue implies the appearance of a vortex, Zhou et al. [15] used the imaginary part of the complex eigenvalue of velocity gradient, and presented the local streamline pattern (LSP) to visualize a vortex, Wang et al. [27] also investigated the imaginary part λci of the complex eigenvalue of the velocity gradient as the pseudo-time average angular velocity of a trajectory moving circularly or spirally around the axis. In practice, many people thought it necessary to combine this methodology with the concentration of vorticity magnitude, namely dividing the vorticity into a rotation part and a shear part, to obtain a reasonable shape for the vortex. In this paper, we will focus on the vortex identification methods derived from the real Schur form of velocity gradient. In Sect. 2, three studies starting from the complex eigenvalues of velocity gradient are summarized and compared with each other. In Sect. 3, we focus on the tensorial representations and the relationship between different representations. The LSPs are classified in Sect. 4, and the correspondence between the geometrical features and the parameters in the eigen-­ representations is investigated. Discussion and Case study are presented in Sect. 5, while a brief conclusion is given in Sect. 6.

2  V  ortex Recognition Methods From Velocity Gradient with Complex Eigenvalues For simplicity, the fluid is uniform and incompressible in this paper, that means, the divergence of velocity vanishes everywhere. For a three-dimensional flow, the matrix form of velocity gradient is written as



 ∂u   ∂x  ∂u ∇v =  ∂y   ∂u   ∂z

∂v ∂x ∂v ∂y ∂v ∂z

∂w   ∂x   d 11 ∂w   = d 21 ∂y     d31 ∂w   ∂z 

d12 d22 d32

d13  d23  ≡ d. d33 

(5.1)



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When the velocity gradient has only one real eigenvalue, Zhou et al. [15] wrote the velocity gradient by1

d T = [ vcr



vci

 1  − 2 λr  vr ]  −λi    0 

 0  0  [ vcr   λr  

λi 1 − λr 2 0

vci

vr ] , −1

(5.2)



where λr is the real eigenvalue with vr as its eigenvector, while the two conjugate 1 complex eigenvalues − λr ± iλi have corresponding eigenvectors vcr ± ivci. In the 2 local affine coordinate system {y1, y2, y3} defined by bases {vcr, vci, vr}, the local streamline starting at point dr field v = = d T ⋅ r as dt

( x ,x ,x )



0 2

0 3

can be solved from the velocity

y3 ( t ) = x30 eλr t ,



0 1

y1 ( t ) = e

1 − λr t 2

y2 ( t ) = e

1 − λr t 2



(5.3a)

 x10 cos ( λi t ) + x20 sin ( λi t )  ,

(5.3b)

 x20 cos ( λi t ) − x10 sin ( λi t )  .

(5.3c)

Zhou et al. [15] pointed out that “the local flow is either stretched or compressed along the axis vr, while on the plane spanned by the vectors vcr and vci, the flow is swirling”, as shown in Fig. 5.2 in [15]. That means the normal of plane (vcr, vci), or the right eigenvector of the real eigenvalue of d, is referred to as the rotation direction of local vortex, and the left eigenvector vr indicates the direction we will define as the extension direction of local vortex. In addition, Zhou et  al. [15] used the imaginary part of the complex eigenvalue pair as the local swirling strength of the vortex. Unlike the two-dimensional flow [17, 18], in which the rotation and extension directions of the vortex are the same and definitely perpendicular to the plane, in the three-dimensional flow both rotation and extension directions of the vortex are to be determined. According to the Schur theorem [22], there are real Schur forms for a real matrix with the real eigenvalue λr (the only one if not specified), say

1  According to the matrix notation, say the Eq. (5.1) in Chong et al. [26], the velocity gradient used by Zhou et al. [15], is actually v∇, so we denote it by dT in this paper. We also rearrange the order of eigenvalues and eigenvectors.

5  No Vortex in Straight Flows: On the Eigen-Representations of Velocity…

d = P ∗T DR P ∗ = P T DL P



89

(5.4a)

with



 D11 DL =  D21  0

D12 D22 0

 D11∗ D13   ∗ D23  , DR =  D21 ∗  D31 D33  

D12∗ ∗ D22 ∗ D32

0   0 , ∗  D33 

(5.4b)

∗ = D33 = λr , n3 ∙ d = λrn3, d ∙ m3 = λrm3. It is easy to testify that n3 is equivaand D33 lent to vr while m3 indicates the axis of (vcr, vci). After Kolář’s proposition and practice for two-dimensional flows, Li et al. [19] first introduced an orthonormal frame {n1, n2, n3} (right hand if not specified) to express DL as in Table 5.1, and called ψ the proper rotation. Years later, based on the same left real Schur form (see Table 5.1), Liu et al. [20, 21] proposed the rigid-body rotation vector by combining the parameter ϕ with the direction n3, and presented an effective algorithm for calculating the involved parameters and achieved a lot of applications through cooperation. As listed in Table 5.1, Zhou, Li and Liu started from the same (left) characteristic problem of velocity gradient tensor, Li and Liu introduced the same orthonormal frame, used the same parameter to characterize the strength of rotation, but both of them paid no attention to the extension direction.

3  T  hree Forms of Tensorial Representation for the Velocity Gradient Dividing the fluid domain into the regions with or without vortex is the primary objective in vortex identification, which can be worked out by the feature that the characteristic polynomial of velocity gradient tensor has complex roots or not.

3.1  S  pectral Representation Under the Affine Frame in the Vortex Region For the velocity gradient  d, the characteristic equations of its left characteristic problem N ∙ d = λN and its right characteristic problem d ∙ N = λN are the same.  1  Assume that the characteristic polynomial has eigenvalues − λ3 ± ιβ ,λ3  with 2   the unit imaginary number ι = −1 , and the corresponding right eigenvectors are {N1 ± ιN2, N3} satisfying

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Table 5.1  Three representations of velocity gradient tensor ∇v with one real eigenvalue λr and its left eigenvector vr or n3 (1) Zhou et al. (λci) Representation

− 1 λ  2   −λ    0  i



λi

r

−

1 2

(2) Li et al. (Proper rotation)

   0   λ  

− 1 λ  2   −ψ    0 

0

λr

0

r

r

ψ +γ

−

1 2 0

λr

(3) Liu et al. (Liutex) − 1 λ  2   −φ    0 

   α   λ   β

r

Rotation axis

the axis of plane (vcr, n3 vci)

n3

Rotation strength

λi

ψ

ϕ

Extension direction

Stretch/compress direction n3

−

−

r

φ+s

−

1 2 0

λr

   η   λ   ξ

Remarks (2) and (3) are orthonormal frames, (1) is affine frame

r

(1) is rotation plane, (2) and (3) are the same (2) and (3) are the same No definition in (2) and (3)

 1  d ⋅ N3 = λ3 N3 , d ⋅ ( N1 ± ι N2 ) =  − λ3 ± ιβ  ( N1 ± ι N2 ) ;  2 

(5.5)

and the corresponding left eigenvectors are {N1 ∓ ιN2, N3} satisfying



 1  N 3 ⋅ d = λ3 N 3 , N 1  ι N 2 ⋅ d =  − λ3 ± ιβ  N 1  ι N 2 .  2 

(

)

(

)

(5.6)

From (5.5) and (5.6), the spectral representation of velocity gradient tensor



 1   d = λ3 N3 ⊗ N 3 + Re  − λ3 + ιβ  ( N1 + ι N2 ) ⊗ N 1 − ι N 2  , 2   

(

)

(5.7a)

with the combination of two right-handed frames {N1, N2, N3} and {N1, N2, N3} is form-invariant by requiring the dual relations N ⋅ N = δ .2 The dual relations show that the two right-handed frames {N1, N2, N3} and {N1, N2, N3} are uniquely interdependent in the spectral representation. An equivalent expression of (5.7a) is i

i

j

j

 The dual conditions can be solved explicitly. Assume that {N1, N2, N3} with the triple product g >0 is known, the corresponding {N1, N2, N3} can be expressed −1 1 i ijk by N = g  N j × Nk , and vice versa. 2 2

[ N1,N2 ,N3 ] = ( N1 × N2 ) ⋅ N3 =

5  No Vortex in Straight Flows: On the Eigen-Representations of Velocity…

d = λ3 N 3 ⊗ N − 3



1 2

(

λ3 N1 ⊗ N + N 2 ⊗ N 1

2

) + β (N

⊗ N − N2 ⊗ N 2

1

91

1

),



(5.7b)

which coincides with the matrix expression given by Zhou et al. [15]. It is obvious in the spectral representation (5.7a) and (5.7b) that the real eigenvector, N3 or N3, can be determined to be only one non-zero real factor difference as long as the tensor product N3  ⊗  N3 is constant, while the complex eigenvector, N1 + ιN2 or N1 + ιN2, can be determined to be only one non-zero complex factor difference as long as the tensor product (N1 + ιN2) ⊗ (N1 − ιN2) is invariant. Let alone the size changes of N1, N2, N3, consider the basic transforms including the sign changes of N1, N2, N3 and exchanges of N1 and N2, there are 16 possible frames where half of them are right-handed. Among the 8 right-handed frames, four of them {N1, N2, N3}, {−N1, −N2, N3}, {N2, −N1, N3} and {−N2, N1, N3} result in the same spectral representation as (5.7a) and (5.7b), and if N3 changes to its opposite −N3, the remaining four frames {N1, −N2, −N3}, {−N1, N2, −N3}, {N2, N1, −N3} and {−N2, −N1, −N3} will yield the same form of representation but β changes to −β. Therefore, if we limit β to a positive number, the directions of real eigenvectors N3 and N3 are uniquely determined, and there are four equivalent right-handed frames for the spectral representation (5.7a) and (5.7b).

3.2  E  igen-Representations Under the Orthonormal Frames in the Vortex Region In this subsection, we will denote the orthonormal frame based on N3 by the left orthonormal frame, and refer to the tensorial form of the matrix expression of velocity gradient under this frame [19, 21] as the left eigen-representation. We propose the right eigen-representation under the right orthonormal frame based on N3, and derive the relation between it and the spectral representation (5.7a) and (5.7b). Theorem 1  For the case that the characteristic polynomial of velocity gradient tensor has complex roots, there exists a unique right-handed orthonormal frame {m1, m2, m3}, and the right eigen-representation d = λ m ⊗ m + ( R + τ ) m ⊗ m − Rm ⊗ m + m ⊗ (τ m + τ m )  3 3 3 3 1 2 2 1 3 1 1 2 2

(5.8)

under {m1, m2, m3}, with R > 0, τ3 > 0 and τ1 > 0 (or τ2 > 0 when τ1 = 0), where the symmetric traceless base ⌊m3  ⊗  m3⌋ is defined by 1 1 m3 ⊗ m3 ≡ m3 ⊗ m3 − m1 ⊗ m1 − m2 ⊗ m2 . If the coefficient of m2 ⊗ m1 is neg2 2 ative, m3 cannot change to −m3; If τ1 = τ2 = 0, {−m1, −m2, m3} is an equivalent frame; if τ3 = 0, {−m2, m1, m3} also becomes an equivalent frame. In order to guarantee the uniqueness of correspondence between (5.7a), (5.7b) and (5.8), we need to regulate the eigenvectors in (5.7a) and (5.7b). As mentioned, the four right-handed frames {N1, N2, N3}, {−N1, −N2, N3}, {N2, −N1, N3} and

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{−N2, N1, N3} are indistinguishable according to the basic eigen-parameters λ3 and β. The requirement of an obtuse angle between N1 and N2 to distinguish {N1, N2, N3} and {N2, −N1, N3}, and the sign of m1 ∙ N1 or m3 ∙ d ∙ m1 to distinguish {N1, N2, N3} and {−N1, −N2, N3}. Because of the dual relations, the treatment to the right-handed frames {N1, N2, N3} is also workable to the right-handed frames {N1, N2, N3}. Now we start from (5.8) to achieve the spectral representation (5.7a) and (5.7b). Since the bases {m1, m2, m3} and the parameters λ3, R, τ1, τ2, τ3 are known, we first have N3 = m3 and β = R ( R + τ 3 ) . To make N1 and N2 unique, we regulate them to have the same size, and the included angle between N1 and m1 to be acute. Because both N1 and N2 are in the plane normal to N3, they can be expressed by unit vectors m1 and m2, while their sizes are further required to meet the requirement N1 × N2 = m3. From (5.8) and the characteristic relation we obtain N1 =

(N

1

  1 − ι N 2 ⋅ d =  − λ3 + ιβ  N 1 − ι N 2 ,  2 

R β  m1 m2  2 −   ,N = R  2  β

)

R β  m1 m2  +  , R  2  β

(

)

(5.9a)

or N 1 − ι N 2 = (1 − ι )

m  R β  m1 −ι 2 .  2  β R 

(5.9b)

Assume that N3 =  m3 +  C1m1 +  C2m2, combining with N3 ∙ d = λ3N3 and (5.8) 3  2 λ3 results in   β2 −  R



 R  C  τ    1  =  1  , which can be solved to get 3   C2   τ 2  λ3  2  β2 3 3 λ3τ 1 − Rτ 2 λ3τ 2 + τ 1 R m. N 3 = m3 + 2 m1 + 2 2 9 2 9 2 2 λ3 + β λ3 + β 2 4 4

(5.10)

Finally, making use of the orthogonality N1 ∙ N1 = N2 ∙ N2 = 1 and N1 ∙ N2 = N2 ∙  N  = (N1 + ιN2) ∙ N3 = 0, we have 1

N1 + ι N2 = (1 + ι )

   βτ 1 + ι Rτ 2 1  m3  .  β m1 + ι Rm2 − 3 2 Rβ   λ3 − ιβ   2 

(5.11)

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93

The above results for eigenvectors can be testified by substituting them into the spectral representation (5.7a) and (5.7b), which will yield the right eigen-­ representation (5.8) of velocity gradient under the orthonormal frame {m1, m2, m3}. The inverse relations can be obtained as follows. When the eigenvalues  1  − λ3 ± ιβ ,λ3  are given, the right eigenvectors {N1 ± ιN2, N3} are determined to  2  some extent. We further choose the right-handed frame {N1, N2, N3} by making the coefficient of N1 ⊗ N2 in (5.7b) positive (If not so, use the equivalent frame {N1, −N2, −N3} instead) and an obtuse angle between N1 and N2 (If not so, use the equivalent frame {−N2, N1, N3} instead); then normalize N3 to a unit vector and isomorphize N1 and N2 in advance through a rotation of θ around the axis N3 so that the real part and the imaginary part of (N1 + ιN2)e−ιθ have the same size. Now, if all these requirements are met, namely (a) {N1, N2, N3} is right-handed, (b) N3 is a unit vector, (c) N1 and N2 are isomorphized and their included angle is obtuse, it is easy to calculate the orthonormal frame {m1, m2, m3} and the parameters λ3, β, R, τ1, τ2, τ3 in the right eigen-decomposition (5.8): 1. Set m3 being the normalized N3; 2. Calculate R and m1, 2 by R = β

N1 + N 2 N −N 1

2

≤ β , m1,2 =

from N 2 ± N1 N1 ± N 2

the

isomorphized

N1,

N2

;

3. Calculate other parameters as τ3 = (β2 − R2)/R, τ1 = m3 ∙ d ∙ m1, τ2 = m3 ∙ d ∙ m2; 4. Finally, if τ1  0 2

(

)

(8.14)

where A and B are the symmetric part and antisymmetric part of velocity gradient tensor respectively, and ‖⋅‖F denotes the Frobenius norm. Liu et al. [9] introduced the Ω method as a measure of local vorticity density, which is computed by a ratio as the relative vortex strength: Ω=

BF2 A + BF2 + ε 2 F

ε = 0.001( BF2 − AF2 )

(8.15)

max



(8.16)

A threshold of 0.52 is suggested by the authors, which has been tested effective by several flows, indicating robust and reliable features of the method.

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By using a Liutex-­shear decomposition instead of the Cauchy-­Stokes decomposition, the Liutex vector is deduced by Gao and Liu [38] to define the physical rigid-­­ rotation part of the flow, which is expressed as:   R = Rr (8.17)





2 ( β − α ) if α 2 − β 2 0, β 0  R = 2 ( β + α ) if α 2 − β 2 < 0, β < 0 0, if α 2 − β 2 ≥ 0, 

(8.18)

2

α=

1  ∂V ∂U   ∂V ∂U  − + + 2  ∂Y ∂X   ∂X ∂Y 

(8.19)

1  ∂V ∂U  β=  + (8.20) 2  ∂X ∂Y   where R denotes the rigid rotation strength and r represents the rotation axis of the local fluid. Capital letters in the equations represent velocity component in a new  XYZ-­frame with the Z-­axis parallel to r .

3  Numerical Setup The discussed TLV cavitating flow here is generated by a truncated NACA0009 hydrofoil, whose chord C is 100  mm, incidence angle is 10°, and the distance between foil tip and the gapwall is 10 mm. The flow field and boundary conditions

Fig. 8.1  Schematic of the computational domain

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Table 8.1  The parameters of the numerical meshes employed Mesh Mesh 1 Mesh 2 Mesh 3

Number of total nodes 7.36 million 11.46 million 15.59 million

Number of refined nodes 3.81 million 6.07 million 10.17 million

Refined edge length L 0.6 mm 0.5 mm 0.4 mm

are schematically shown in Fig. 8.1. The inlet total pressure is fixed at 153,000 Pa, and the outlet velocity is 10 m/s to keep consistent with the referenced experiments [32, 39]. To have a good prediction of the vortices generated from the boundary layer around the foil and the gapwall, no-­slip wall condition is applied to these solid boundaries. Three sets of meshes were generated with the Cartesian cut-­cell method by refining regions around the foil surface and the tip gap, details of which are listed in Table 8.1.

4  Results and Discussion 4.1  Sensitivity Study of Mesh Resolution Figure 8.2 shows vortical structures predicted with LES by Mesh1–3 under non-­­ cavitating condition. The hair-­pin vortices on the foil surface and tip vortices, including TLV, TSV and second induced vortices are captured by all meshes. More turbulent structures are provided by the refined meshes of finer resolutions as expected. It is noted that TLV and TSV break into small-­scale turbulent vortices after the intersection point. Figure 8.3 shows the simulated time-­averaged cavities with the iso-­surface of αv = 0.1, as compared with the expeimental picture [39], in which partial cavity, TLV cavity and TSV cavity are in good agreement with the experimental results in both cavity size and trajectory. Therefore, to strike a balance between the computational resources and the simulation accuracy, Mesh 2 is used for further analysis.

4.2  Comparison of Vortex Identification Methods A set of initial thresholds for different identification methods showing similar sizes of vortical structures in the current flow, denoted as Vinitial, is determined according to their maximum values in the flow field, i.e. 1500  s−1 for vorticity method, 150,000 s−2 for Q-­criterion and 550 s−1 for Liutex method. In addition, the initial threshold for Ω method is set to be 0.52 as suggested because the parameter is non-­­ dimensional. Cavitating vortices at a typical instant are displayed with thresholds ranging from 0.75Vinitial to 1.5Vinitial for vorticity method, Q-­criterion and Liutex method, and from 0.51 to 0.6 for Ω method in the following discussion.

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Fig. 8.2  Vortical structures predicted by different meshes (iso-­surface of Q = 100,000 s−2)

In Fig. 8.4, significant shear contaminations attributed to the wall boundary layer are displayed by the vorticity method around the foil surface and gapwall. Sizes of foil trailing edge vortices (FTEVs) and tip vortices (TVs) decrease significantly with increase of the threshold, while the wall boundary layers and the foil leading edge vortex (FLEV) vary little. Iso-­surface of vorticity  =  3Vinitial is provided in Fig. 8.5a to make it clearer, showing that TVs and FTEVs nearly vanish, whereas the boundary layer over the gapwall and the foil, as well as the FLEV are still clearly visible. In terms of FLEV, great velocity gradients exist mainly due to large pressure and velocity differences at the cavity interface as shown in Fig. 8.5b, c. It

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Fig. 8.3  Time-­averaged cavities for three meshes as compared with the experiment picture [39]

Fig. 8.4  Cavitating vortices showing with vorticity method

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Fig. 8.5  Cavitating vortices showing by vorticity method with the threshold of 3Vinitial and flow field around the foil leading edge

is illustrated that structures in the tip leakage cavitating flow visualized with the vorticity method are heavily influenced by velocity gradients in both shearing and vortical regions, which makes it completely fail to filter shear layers. Figure 8.6 shows cavitating vortices visualized with Q-­criterion, in which little variation appears for different thresholds, and strips of shear layer contaminations also exist around solid boudaries. The reason why Q structures didn’t change much is supposed that the dimension of Q is the square of vorticity, causing a maximum five orders of magnitude greater than that of the vorticity method and the Liutex method. Therefore, visible distinctions are supposed to appear for thresholds differing by orders of magnitude. Isosurface of ten times the initial Q value is shown in Fig.  8.7b, in which TVs and FTEVs are significantly reduced as compared with Fig. 8.7a.

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Fig. 8.6  Cavitating vortices showing with Q-­criterion

Fig. 8.7  Illustration of the effect of maximum magnitude order in the flow field on vortices displayed by Q-­criterion

Defined as a non-­dimensional variable, the Ω method is suggested to be applied with the value 0.52 [17], which is confirmed to be effective in the current flow as presented in Fig. 8.8. Shear contaminations can be filtered out successfully with all four thresholds ranging from 0.51 to 0.6 for the Ω method and from 0.75Vinitial to 1.5Vinitial for the Liutex method as shown in Fig. 8.9. Weaker vortices downstream the foil are captured by both Ω and Liutex methods as marked with dashed red circles, and disappear as thresholds increase. It is indicated that the 0.52 threshold of Ω method is robust and the newly proposed Liutex method possesses a good capability in capturing vortical structures in the tip leakage cavitating flow. Therefore, considering that flow in the current study is dominated by both strong vortices, including TVs and FTEVs, as well as large shear layers existing over the gapwall and the foil surface simultaneously, ability in shearing motion filtering of the Ω

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Fig. 8.8  Cavitating vortices showing with Ω method

Fig. 8.9  Cavitating vortices showing with Liutex method

method and the Liutex method makes these two methods more appropriate for the application to vortices investigation here.

4.3  P  erformance of Liutex Method in Investigating Tip Vortices Considering that tip vortices are of various directions, between two superior methods based on the above discussions, which are the Ω method and the Liutex method, the latter one is preferred in the following analysis of characteristics of non-­­ cavitating and cavitating tip vortices because it is a vector quantity.

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Vector field of the Liutex method is checked in Fig. 8.10 for wake vortices downstream the foil trailing edge, including the hairpin vortex and the stretched FTEVs. It is shown that the Liutex vectors are tangential to the iso-­surface of R = 550 for both these two kinds of vortices as described by Gao and Liu [38] and Liu et al. [10], indicating the potential of Liutex method in providing both magnitude and direction information of the visualized vortex. Then the Liutex vector field is used to investigate the non-­cavitating and cavitating TVs which are of typical directions as compared to other vortices in the current flow field.

Fig. 8.10  Liutex vector fields for wake vortices downstream the foil trailing edge (iso-­surface of R = 550)

Fig. 8.11  Liutex vector field for non-­cavitating tip vortices (iso-­surface of R = 550)

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Fig. 8.12  Liutex vector field for cavitating tip vortices (iso-­surface of R = 550)

The Liutex vector field for tip vortices without cavitation is displayed in Fig. 8.11, where TLV, TSV1 and TSV2 are of the same direction, opposite to the main stream direction. A secondary vortex is found at the foil tip suction side corner with the same direction as that of the main stream. Moreover, as marked by black dashed circles, surfaces of TSV1 and TSV2 becomes rough as it moves closer to TLV, and small-­scale vortices are observed around TSV1 with the direction almost orthogonal to it. The small-­scale structures are possibly attributed to elliptic instability of co-­rotating vortex pair formed by TLV and TSV [40]. Directions of TLV and TSV are revealed to be same with and without cavitation. In addition, it is observed that TLV and TSV tends to merge quickly after cavitation happens as shown in Fig. 8.12, resulting in a larger cavitating TLV without small-­scale vortices, which reflects the significant effect of cavitation in changing vortical field.

5  Conclusions In the present investigation, Large eddy simulation of the tip leakage cavitating flow is presented with a good agreement for both vortical and cavitating structures between the numerical results and referenced experiment [32, 39]. Applicability of all three generations of vortex identification methods, including the vorticity method, the Q-­criterion, the Ω method and the Liutex method is compared thoroughly. With the superior identification method, characteristics of TLV and TSV, as

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well as the effect of cavitation on TVs are discussed briefly. The main conclusions are summarized as follows: 1. The 0.52 threshold of Ω method is tested to be robust in the tip leakage cavitating flow. The Liutex method possesses a good capability in capturing vortical structures in the investigated flow as compared with methods categorized as the first and second generations, which fails to filter out strong shear layers around the foil surface and the gapwall. 2. It is checked that the Liutex vectors are tangential to the iso-­­surface of Liutex in the currently investigated flow, which shows the potential of Liutex method in visualizing vortices in cavitating flow by providing both magnitude and direction information. 3. Under non-­­cavitating condition, TSV tends to show instability with rough surface and small-­­scale structures around TSV, which is possibly due to the formation of co-­­rotating vortex pair by TLV and TSV.  And it is not the case in cavitating flow. Acknowledgements  This work was financially supported by the National Natural Science Foundation of China (Project nos. 51822903, 11772239, and 11772305). The numerical calculations in this paper were done on the supercomputing system in the Supercomputing Center of Wuhan University.

References 1. X.  Long et  al., Large eddy simulation and Euler-Lagrangian coupling investigation of the transient cavitating turbulent flow around a twisted hydrofoil. Int. J.  Multiphase Flow 100, 41–56 (2018) 2. X.-R. Bai et al., Large eddy simulation of tip leakage cavitating flow focusing on cavitation-­ vortex interaction with Cartesian cut-cell mesh method. J. Hydrodynam. 30(4), 750–753 (2018) 3. C.-C.  Wang et  al., Cavitation vortex dynamics of unsteady sheet/cloud cavitating flows with shock wave using different vortex identification methods. J.  Hydrodynam. 31(3), 475–494 (2019) 4. R.E.A. Arndt, Cavitation in fluid machinery and hydraulic structures. Annu. Rev. Fluid Mech. 13, 273–328 (1981) 5. J. Katz, Cavitation phenomena within regions of flow separation. J. Fluid Mech. 140(MAR), 397–436 (1984) 6. Q. Le, J.P. Franc, J.M. Michel, Partial cavities - global behavior and MEAN pressure distribution. J. Fluids Eng. Trans. Asme 115(2), 243–248 (1993) 7. R.E.A. Arndt, Cavitation in vortical flows. Annu. Rev. Fluid Mech. 34(1), 143–175 (2002) 8. Y.N.  Zhang et  al., A selected review of vortex identification methods with applications. J. Hydrodynam. 30(5), 767–779 (2018) 9. C.Q.  Liu et  al., New omega vortex identification method. Sci. China Phys. Mech. Astron. 59(8), 9 (2016) 10. C.Q. Liu et al., Rortex-a new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids 30(3), 12 (2018)

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11. J. Liu, Y. Gao, C. Liu, An objective version of the Rortex vector for vortex identification. Phys. Fluids 31(6) (2019) 12. J.  Liu, C.  Liu, Modified normalized Rortex/vortex identification method. Phys. Fluids 31(6) (2019) 13. J.-M.  Liu et  al., Objective Omega vortex identification method. J.  Hydrodynam. 31(3), 455–463 (2019) 14. H. Xu, X.-s. Cai, C. Liu, Liutex (vortex) core definition and automatic identification for turbulence vortex structures. J. Hydrodynam. 31(5), 857–863 (2019) 15. Y.-S.  Gao et  al., A Liutex based definition and identification of vortex core center lines. J. Hydrodynam. 31(3), 445–454 (2019) 16. C.Q. Liu et al., Third generation of vortex identification methods: Omega and Liutex/Rortex based systems. J. Hydrodynam. 31(2), 205–223 (2019) 17. Y.-Q.  Wang et  al., Liutex theoretical system and six core elements of vortex identification. J. Hydrodynam. 32(2), 197–211 (2020) 18. X.-D.  Bai et  al., The visualization of turbulent coherent structure in open channel flow. J. Hydrodynam. 31(2), 266–273 (2019) 19. N. Gui et al., Analysis and correlation of fluid acceleration with vorticity and Liutex (Rortex) in swirling jets. J. Hydrodynam. 31(5), 864–872 (2019) 20. L. Wang et al., Extension omega and omega-Liutex methods applied to identify vortex structures in viscoelastic turbulent flow. J. Hydrodynam. 31(5), 911–921 (2019) 21. X.R. Dong et al., POD analysis on vortical structures in MVG wake by Liutex core line identification. J. Hydrodynam. 32(3), 497–509 (2020) 22. Y.-F. Wang et al., The applicability of vortex identification methods for complex vortex structures in axial turbine rotor passages. J. Hydrodynam. 31(4), 700–707 (2019) 23. X.  Bai et  al., Comparative study of different vortex identification methods in a tip-leakage cavitating flow. Ocean Eng. 207 (2020) 24. J. Chen et al., Numerical investigation of the cavitating flow structure with special emphasis on the vortex identification method. Mod. Phys. Lett. B 34(4), 19 (2020) 25. R.E.A. Arndt, V.H. Arakeri, H. Higuchi, Some observations of tip-vortex cavitation. J. Fluid Mech. 229, 269–289 (1991) 26. D.H.  You et  al., Study of tip-clearance flow in turbomachines using large-eddy simulation. Comput. Sci. Eng. 6(6), 38–46 (2004) 27. B. Lakshminarayana, M. Pouagare, R. Davino, Three-dimensional flow field in the tip region of a compressor rotor passage—Part I: Mean velocity profiles and Annulus Wall boundary layer. J. Eng. Power 104(4), 760–771 (1982) 28. K. Roussopoulos, P.A. Monkewitz, Measurements of tip Vortex characteristics and the effect of an anti-cavitation lip on a model Kaplan Turbine Blade. Flow. Turb. Combust. 64(2), 119 (2000) 29. D.  Bertetta et  al., CPP propeller cavitation and noise optimization at different pitches with panel code and validation by cavitation tunnel measurements. Ocean Eng. 53, 177–195 (2012) 30. A.  Asnaghi, U.  Svennberg, R.E.  Bensow, Large Eddy simulations of cavitating tip vortex flows. Ocean Eng. 195, 26 (2020) 31. H.Y. Cheng et al., Large eddy simulation of the tip-leakage cavitating flow with an insight on how cavitation influences vorticity and turbulence. App. Math. Model. 77, 788–809 (2020) 32. M. Dreyer et al., Mind the gap: a new insight into the tip leakage vortex using stereo-PIV. Exp. Fluids 55(11) (2014) 33. Q.  Guo et  al., Numerical simulation for the tip leakage vortex cavitation. Ocean Eng. 151, 71–81 (2018) 34. H. Cheng et al., A new Euler-Lagrangian cavitation model for tip-vortex cavitation with the effect of non-condensable gas. Int. J. Multiphase Flow, 103441 (2020) 35. F. Nicoud, F. Ducros, Subgrid-scale stress Modelling based on the square of the velocity gradient tensor. Flow. Turb. Combust. 62(3), 183–200 (1999)

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Chapter 9

Lagrangian Liutex Yiqian Wang

Abstract  A Lagrangian version and an objective Lagrangian version of the Liutex, which has been introduced to identify and visualize vortices, are developed in this study. The capability of the new methods to educe vortices in geophysical Bickley jet flow is discussed in detail. It is found the Lagrangian version surpasses vorticity-­based methods as it is free from shear contamination, and also reveals the Lagrangian nature of vortices. The objective version is capable of visualizing vortices independent of selected reference frame. However, selecting the volume to apply the objective Lagrangian Liutex method differently leads to different results. A general principle of selecting symmetry flow field with zero mean vorticity is proposed to overcome this issue. Keywords  Liutex · Lagrangian based vortex identification

1  Introduction Shear is one of the very first introduced concepts in fluid mechanics. Based on Cauchy-­Stokes decomposition, however, a pure shear, say ∂u/∂y = 2a in two dimensions which integrate to u = 2ay and v = 0 if we ignore additional constants, with u and v as the velocity components in the direction of x and y coordinates respectively and a a positive constant, is decomposed into a deformation part and a rotational part represented by A and B as 0 a  1 ∇V + ∇V T =   2 a 0 

(9.1)

 0 a 1 ∇V − ∇V T =   2  −a 0 

(9.2)

A= B=

(

(

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Y. Wang (*) School of Mathematical Science, Soochow University, Suzhou, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_9

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∇V is the velocity gradient tensor and the superscript T represents matrix transpose. The corresponding velocity vector on the axes and streamlines of ∇V, A and B for the considered two-­dimensional pure shear are shown in Fig. 9.1. It is thus argued that a pure shear whose streamlines (or pathlines) are all straight lines has been decomposed into a rigid rotation part B and an additional deformation part A which seems to cancel the rotation in B. This mathematical correct decomposition may not be well suited to reveal the physics. These issues prompted Liu et al. [1] to introduce a new decomposition of velocity gradients in which the pure shear is viewed as a basic unit that cannot be further decomposed, and in this spirit the Liutex vector is introduced to represent the rigid rotation part of fluid motion with its direction as the local rotational axis and magnitude as twice the angular speed of rigid rotation. The Liutex based decomposition of velocity gradient tensor extract rigid rotation with residual stretching or compression and pure shear [1, 2]. An important result based on the Liutex vector is that the vorticity is further decomposed into Liutex and residual pure shear as

ω = R+S

(9.3)

where ω is the vorticity vector, R is the Liutex vector and S is the residual pure shear. This relation confirms that vorticity is contaminated by shear and thus cannot

Fig. 9.1  Cauchy-­Stokes decomposition of two-­dimensional pure shear

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represent vortex in high shear flows, for example, the near-­wall region of wall bounded flows. Compared with previous vortex identification methods, the Liutex system has two distinctive features. First, the directional information is included by introducing  a vector quantity R with its direction as the local rotation axis. Intuitively, any rotational motion should have an axis to rotate about which, however, has been generally ignored. Second, the magnitude of the Liutex vector represents twice the local angular speed of the rigid rotation part of the fluid motion and thus free from shear contamination. Six critical aspects of vortex identification including absolute strength, relative strength, local rotational axis, global rotation axis, vortex core size and vortex boundary can be extracted form flow field based on the Liutex system. Multiple vortex identification strategies have been introduced based on the Liutex vector, for example, Liutex magnitude iso-­ surface, objective Liutex, Liutex-­ Ω method, Liutex core line method, etc. [3–7], and those methods have been proven to be effective in capturing vortices in various flows [8, 9]. The phenomena of vortices is Lagrangian, i.e., a process of fluid particles rotating. Previous Lagrangian methods have been developed based on vorticity. However, as stated above, vorticity is contaminated by shear and thus is not suited to represent fluid rotation, i.e., vortices, especially in wall-­bounded flows. In this study, we will develop a Lagrangian method based on the Liutex vector, and thus name it as Lagrangian Liutex. Several examples will be used to illustrate the performance of Lagrangian Liutex in identifying vortices.

2  Lagrangian Liutex The Lagrangian Liutex can be defined as t



LLtt0 ( x0 ) = ∫ R ( x ( s;x0 ) ,s ) ds t0



(9.4)

which represents the rotated angle from time t0 to t. For this Lagrangian method, the focus is not put on the instantaneous angular velocity any more like Eulerian methods. We ask the question that for a given time period of t − t0, the rotation angle of any selected particles (not air molecule particles) at t0, and use this rotation angle to quantify the strength of the particle rotating. For two dimensional flows, the Liutex vector will always point to the third direction, and thus can be viewed as a scalar. Correspondingly, the two-­ dimensional version of Lagrangian Liutex can be written as t



LLtt0 ( x0 ) = ∫ R3 ( x ( s;x0 ) ,s ) ds t0



(9.5)

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We apply the two-­dimensional Lagrangian Liutex to geophysical Bickley jet flow. The stream function is given by

ψ ( x,y,t ) = ψ 0 ( y ) + ψ 1 ( x,y,t )





(9.6)

where ψ0(y) is the steady background flow given as y ψ 0 ( y ) = UL tanh   L



(9.7)

And the perturbation ψ1(x, y, t) is obtained by



 y  3 ψ 1 ( x,y,t ) = ULsech 2   Re  ∑ fn ( t ) exp ( ikn x )   L   n=1 

(9.8)

U and L are characteristic velocity and length scales. For our case, we select U  =  62.66 ms−1, L  =  1770 km, kn  =  2n/r0 and r0  =  6371  km. In addition, fn(t) = ϵn exp (−ikncnt) with kn as the wave number and cn as corresponding travelling speed. The magnitudes ϵn are chosen to be ϵ1 = 0.0075, ϵ2 = 0.15 and ϵ3 = 0.3 and the wave speed cn are given as c2  =  0.205U, c3 = 0.461U and c1 = c3 + 5 − 1 / 2 ⋅ k2 / k1 ⋅ ( c2 − c3 ) . The spatial domain we calculate is [0, 6.371π] × [−3, 3] (million meters), and the timespan we considered is [0,40 days]. The results of Lagrangian Liutex is shown in Fig. 9.2 and as comparison the Lagrangian-­averaged vorticity deviation (LAVD) method developed by Haller et al. [10] applied to the same flow is shown in Fig. 9.3. As shown in the Figs. 9.2 and 9.3, the vortices identified by Lagrangian Liutex are more concentrated in isolated islands while those of LAVD are surrounded by areas with substantial magnitude regions as shown by the green and light blue (around LAVD = 60) areas in Fig. 9.3. The reason is that vorticity is highly contaminated by shear, which exists among those vortices. However, vorticity-­based methods mistreat shear as rotation, and thus identify a smoother “vortex strength” field. Intuitively, vortex is generally isolated phenomenon and from this perspective, this

(

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Fig. 9.2  Lagrangian Liutex applied to geophysical Bickley jet

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Fig. 9.3  LAVD applied to geophysical Bickley jet

Fig. 9.4  Instantaneous Liutex distribution at t0

new Lagrangian Liutex has better capability to visualize vortices as isolated areas. At the vortex core area, we can see that magnitude of LAVD is larger than that of Lagrangian Liutex (LL), which also comes from the fact that LL is free from shear contamination. Finally, a silent zone exists around the center between the upper vortices and lower vortices, in which area in fact the flow is free from both rotation and shear. In this silent zone, the velocity gradient is small, and behaves like inviscid flow. The existence of such silent zone states the fact that when the flow is void of shear, LL and LAVD would be the same, or Liutex would be identical to vorticity as Eq. (9.1) given S = 0. However, fluid is not rigid body, and flow is always a combination of rotation and shear, i.e., where there is rotation (vortex), there will always be residual shear. Figure 9.4 shows the instantaneous Liutex distribution at t0, i.e., twice the instantaneous angle velocity of the rotational part of the flow motion. Different from LL, in which the value is always positive, instant positive Liutex means the rotation is counter-­clockwise while negative value means the rotation is clockwise. Besides that, the instantaneous swirling strength of the vortices is rather uneven distributed as the vortices close to x  =  0 and x  =  20 have higher Liutex magnitude than the

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Fig. 9.5  Distribution of Liutex magnitude of particle trajectories at day 10

vortices near x = 10. On the other hand, the vortices identified by LL as shown in Fig. 9.2 have almost equal magnitude, which comes from the integration in the definition of LL as some kind of average. In Bickley jet, a vortex would experience both fast-­rotating and slow-­rotating states, and LL smoothed every vortex by averaging. An advantage of LL is that the vortices it identifies are circular which is in accordance with intuition of human on vortices. On the other hand, the instantaneous Liutex vortices have shapes like triangles. Therefore, identification of vortex center with instantaneous Liutex would not be as easy as with LL. Figures 9.5, 9.6, 9.7 and 9.8 show the distribution of Liutex of particle trajectories at different time steps. For example, a red point in Fig. 9.5 means that for the particle on this point at t0, 10 days later this Lagrangian particle will have the Liutex magnitude represented by the red color, which means it is counter-­clockwise rotating. As time evolves, we can see that originally counter-­rotating particles will always be counter-­rotating and vice versa. In addition, particles originally non-­­ rotating could be entrained into vortical regions and has a spiral shape as shown in Figs. 9.5, 9.6, 9.7 and 9.8. The swirling strength of the particles will also alternate between high and low magnitude as time evolves.

3  Objective Lagrangian Liutex The objectivity of defining vortices requires that the identified structures should remain the same with moving observers at different states, which is especially important in geophysical research. According to Haller, vorticity is not objective and thus methods directly based on vorticity would not be objective either. Objectivity means that the vector to represent vortex should be invariant under moving reference frame. Generally, the reference frame transformation can be represented by

9  Lagrangian Liutex

Fig. 9.6  Distribution of Liutex magnitude of particle trajectories at day 20

Fig. 9.7  Distribution of Liutex magnitude of particle trajectories at day 30

Fig. 9.8  Distribution of Liutex magnitude of particle trajectories at day 40

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x X  Y  = Q t  y  + b t ( )  ( )    z   Z 

(9.9)

where x, y, z and X, Y, Z denote the coordinates in the initial and new reference frame respectively. Q is an orthogonal rotation matrix which satisfies QQT = QTQ = I, and b represents translation. The time-­dependent feature is represented by the independent variable t. According to Liu et al. [3], objective vector and objective tensor can be defined as follows. Definition 1  An objective vector in the original reference frame represented by l and in the moving reference frame represented by L, should satisfy the following relationship, L = QT ( t ) l



(9.10)



Definition 2  An objective tensor in the original reference frame represented by p and in the moving reference frame represented by P, should satisfy the following relationship,

P = QT ( t ) pQ ( t )

(9.10)



Note that both Q(t) and b(t) are time-­dependent. If Q is independent of time and b(t)  =  ct, with c as constant velocity vector, the reference frame transformation becomes Galilean transformation. It has been shown that the Liutex vector itself is invariant under Galilean transformation, but not objective. It has been proposed by Liu et al. [3] that an objective version of Liutex can be obtained by introducing an objective net velocity gradient tensor as

v = ∇v − B ∇

(9.11)

where B is an instantaneously spatial averaged antisymmetric tensor defined by resorting to the spatially averaged vorticity ω .

ω (t ) =

1 ∫ ω ( x,t ) dV Vol ( t )

(9.12)

while B is then defined by satisfying



1 Be = ω × e, ∀e ∈  3 2

(9.13)

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 is objective, and It then can be verified that the net velocity gradient tensor ∇v thus based on the definition of this new net velocity gradient tensor, an objective version of Liutex Rˆ is introduced by Liu et al. [3] An objective Lagrangian Liutex vector can be defined as t



 tt ( x ) = Rˆ ( x ( s;x ) ,s ) ds LL 0 0 0 ∫ t0



(9.14)

Using the same example of geophysical Bickley jet, the vortices can be visualized with the new objective Lagrangian Liutex method as shown in Fig. 9.9. It is not surprising that Fig. 9.9 and Fig. 9.2 are basically the same, which is not surprising since there is certain symmetry of the instantaneous vorticity for the considered spatial domain. As shown in Fig. 9.10 for t = 0, the mean vorticity would almost be zero since the negative and positive values of vorticity cancel each other at all times, and thus the net velocity gradient tensor would be almost the same as v = ∇v . Accordingly, the Liutex vector would be the original vorticity tensor, i.e., ∇ the same and so on to Lagrangian Liutex. Although the objective version makes the visualization no longer depending on particular reference frame, it also introduces certain ambiguity by subtracting the velocity gradient tensor by a spatial average tensor corresponding to spatial vorticity. The process of spatial average relies on the selection of a spatial range to average over. If we select the domain shown in Fig. 9.9 to do average, it is found than the mean vorticity almost equal zero, and the objective version of Lagrangian Liutex happens to capture vortices almost the same as the Lagrangian Liutex method. However, if other strategies to select averaging domain are adopted, the resulting visualization could be significantly changed. We apply the objective Lagrangian Liutex separately to the upper half and lower half domain, both the spatial vorticity average of which would certainly not be zero, and the results are shown in Fig. 9.11. First, the locations of the vortices are basically correctly captured. However, the magnitude decreases especially in the vortex

Fig. 9.9  Objective Lagrangian Liutex applied to geophysical Bickley jet

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Fig. 9.10  The vorticity distribution at t = 0

Fig. 9.11  Objective Lagrangian Liutex applied to the upper half and lower half domain

centers. Second, the silent zone between the upper vortex layer and lower vortex layer is now filled with vortices, which is different from our perception of this particular flow. It is thus believed that the selection of spatial domain plays a critical role in successfully using the objective Lagrangian Liutex method, and as a general principle, it would be better if we can use some symmetry of the considered physical problem to select areas with zero mean vorticity as the visualization domain.

4  Conclusions Vortices are rotating motion of fluids, which certainly has Lagrangian nature. Based on the newly introduced Liutex vector, whose direction represents the local rotation axis and magnitude represents twice the angular velocity of the rigid rotation part, we introduced the Lagrangian Liutex and objective Lagrangian Liutex. For the Lagrangian Liutex, it is better than vorticity based Lagrangian methods in that it is

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free from shear and it surpass the instantaneous Liutex method because it encompasses the Lagrangian nature of vortices. The objective version is particularly important when the observer or reference frame are moving irregularly. However, ambiguity is introduced when we consider how to select the volume to do vorticity average. A general principle of using symmetry to select zero mean vorticity areas has been proposed and illustrated with the example of geophysical Bickley jet flow.

References 1. C. Liu, Y. Gao, X. Dong, et al., Third generation of Vortex identification methods: Omega and Liutex/Rortex based systems. J. Hydrodynam. 31, 205–223 (2019) 2. Y. Wang, Y. Gao, H. Xu, et al., Liutex theoretical system and six core elements of Vortex identification. J. Hydrodynam. 32, 197–211 (2020) 3. J. Liu, Y. Gao, C. Liu, An objective version of the Rortex vector for vortex identification. Phys. Fluids 31, 065112 (2019) 4. X. Dong, Y. Gao, C. Liu, New normalized Rortex/vortex identification method. Phys. Fluids 31, 011701 (2019) 5. J.  Liu, C.  Liu, Modified normalized Rortex/vortex identification method. Phys. Fluids 31, 061704 (2019) 6. Y. Gao, J. Liu, Y. Yu, C. Liu, A Liutex based definition and identification of vortex core center lines. J. Hydrodynam. 31(3), 445–454 (2019) 7. H. Xu, X. Cai, C. Liu, Liutex (vortex) core definition and automatic identification for turbulence vortex structures. J. Hydrodynam. 31(5), 857–863 (2019) 8. N. Gui, L. Ge, P. Cheng, et al., Comparative assessment and analysis of Rortex vortex in swirling jets. J. Hydrodynam. 31(3), 495–503 (2019) 9. Y. Wang, W. Zhang, X. Cao, et al., The applicability of vortex identification methods for complex vortex structures in axial turbine rotor passages. J. Hydrodynam. 31, 700–707 (2019) 10. G. Haller, A. Hadjighasem, M. Farazmand, et al., Defining coherent vortices objectively from the vorticity. J. Fluid Mech. 795, 136–173 (2016)

Chapter 10

Visualizing Liutex Core Using Liutex Lines and Tubes Oscar Alvarez, Yifei Yu, Pushpa Shrestha, Dalal Almutairi, and Chaoqun Liu

Abstract  Liutex is a vortex identification method which can be used to determine the absolute and relative strength of a vortex, the local rotation axis of a vortex, the vortex core center, the size of the vortex core, and the vortex boundary. A vortex core is defined as a concentration of Liutex vectors. It would be nice if we could visualize the Liutex core to gain a better understanding. In this paper, we will use the Liutex core lines and Liutex tubes to show the vortex structure by using pre-­ processed DNS data, which can help researchers studying fluid dynamics understand the turbulence physics. There is no reason to believe iso-surfaces can represent vortex structure, but it is more reasonable to demonstrate the vortex structures by using the Liutex core lines and Liutex tubes as Liutex is a new vector quantity to represent flow rotation or vortex. Keywords  Liutex · Vortex · Vortex core · Liutex core lines · Liutex tube · Liutex core tube

1  Introduction Modern computing has made it possible to create and analyze large and complicated fluid structures. These fluid structures have made it possible to improve current technologies and to create new more efficient ones. Over the past decades, we have developed methods that have allowed us to determine and observe specific properties of fluids. Some methods can be grouped together into a single category where their purpose is to define one critical/fundamental part of all fluids—fluid rotation. Determining fluid rotation for the sake of finding turbulent regions within a fluid has been an integral part of fluid mechanics since near the beginning. Many methods

O. Alvarez · Y. Yu · P. Shrestha · D. Almutairi · C. Liu (*) Department of Mathematics, University of Texas at Arlington, Arlington, TX, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_10

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(methods such as Q criterion [1], δ criterion [2], λci criterion [3], and λ2 criterion [4]) have been created to try and describe fluid rotation (or a vortex). This second generation [5] of vortex identification methods have many disadvantages though. One disadvantage is that each case requires an independent threshold which makes these methods sensitive to threshold change. Some cases may not even have any suitable threshold [5]. Another big disadvantage is that they are based solely on a scalar quantity. Having only a scalar quantity does not allow us to determine the rotation axis of the vortex (among other things) making it difficult to analyze some structures. Most importantly, these previous generational methods are contaminated by shearing [6] in their identification process so they do not and cannot define or show pure fluid rotation. It has not been up until recently (2018) when Dr. Chaoqun Liu et al. [7] of the University of Texas at Arlington had found a way to rigorously mathematically describe a vortex in an fluid—Liutex. This new method Liutex (previously known as Rortex), isolates fluid rotation apart from other extraneous properties such as shearing, etc. This isolation allows us to find the pure rotation of an fluid which leads to more accurate interpretations of a fluid’s behavior. Liutex also benefits us as the result of rotation in an fluid returns to us in the form of a tensor (as opposed to a scalar which is the result of previous generations of vortex identification methods). The resulting Liutex tensor provides us with enough information to resolve the following six issues [5]: • • • • • •

What is the absolute strength of a vortex? What is the relative strength of a vortex? What is the rotation axis? Where is the vortex core center? What is the size of the vortex core? Where is the vortex core boundary?

In this chapter, we will be answering all these questions with the introduction of the Liutex Core Tube. We will show the method in manually finding the Liutex Core Tube and introduce some definitions that will make this possible.

2  Liutex (Rortex) It has been previously established that if the velocity gradient tensor (∇v) has one real eigenvalue and two complex conjugate eigenvalues, then there exists an instantaneous local swirling motion in the direction of the real eigenvector. Or,

∇v ⋅ r = λr r

where λr is the real eigenvalue of ∇v. Since the normalized eigenvector is unique up to a ±, a second condition is introduced

10  Visualizing Liutex Core Using Liutex Lines and Tubes



171

ω ⋅r > 0

where ω represents the vorticity vector. The magnitude of the Liutex vector r represents the local rotational strength of the fluid. The magnitude is determined on the plane perpendicular to r, so c­ oordinate rotation is required. Though recently, Wang et al. [8] introduced an explicit formula to find the strength of Liutex without having to the expensive coordinate rotation. Here is the explicit formula:

R =ω ⋅r −

(ω ⋅ r )

2

− 4λci2



where λci is the imaginary part of the complex eigenvalue of the vorticity gradient tensor. Using this formula, we can uniquely define the Liutex vector R by

R = Rr.

We can use the magnitude of the Liutex vector to analyze regions of fluids that have a specific strength. In Fig. 10.1, we see DNS data of a 3D boundary layer during early transition on a flat plate where the iso-surfaces have been created using Liutex and drawn where the Liutex magnitude is 0.07.

Fig. 10.1  DNS of flat plate boundary layer with iso-surfaces drawn using Liutex Magnitude of 0.07 in the early transition phase

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3  Liutex Core Line A great benefit of Liutex being a vector instead of a scalar is that it allows us to find the center axis of rotation of a vortex within an fluid. Finding the center of rotation can be extremely useful when determining the structure of turbulence within the fluid. We will find the Liutex core line by defining the gradient of the Liutex magnitude vector,



 ∂R     ∂x   ∂R  ∇R =   ∂y    ∂R   ∂z   

Through integration of the Liutex magnitude vector, we can obtain the Liutex gradient lines. The integration of the Liutex gradient vector can produce discontinuous Liutex gradient lines, so, we can combine the Liutex lines with the Liutex gradient lines to extract the vortex rotation axis lines. We can define the vortex core center (or the vortex core axis line) as a line consisting of points which satisfy the condition

∇R × r = 0,

R >0

where r represents the direction of the Liutex vector. Understanding of the Liutex core line can be improved through visualization. We will use the same DNS early transition data of a flat plate to construct the Liutex core line. First, we consider a slice that intersects with a vortex region within our data. In Fig.  10.2, we take a slice on the x-axis that intersects with the leg of the vortex. Then in Fig. 10.3, we take a sample of Liutex gradient lines near the intersection within the vortex region and the x-slice. We can see from Fig. 10.3 that the Liutex gradient lines point towards the center line (core line) of the vortex and intersects the x-slice where the center line is located. We can now pinpoint the exact location of the Liutex core line. We notice that the intersection of the Liutex gradient lines and the x-slice is located at a local maximum Liutex magnitude of the slice. It is to say that there is a Liutex core line at every local maximum (seed point) Liutex magnitude of the slice. We will now draw the Liutex core line [9]. It is made from the intersection point of the Liutex gradient lines with the reference x-slice. We will create a Liutex line (streamtrace) at that point of intersection as shown in Figs. 10.4 and 10.5.

10  Visualizing Liutex Core Using Liutex Lines and Tubes

Fig. 10.2  x-slice at x = 412.333

Fig. 10.3  Liutex gradient lines drawn near intersection to locate Liutex Core Line

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Fig. 10.4  Liutex line drawn at the intersection of the x-slice and Liutex gradient lines creating the unique Liutex Core Line

Fig. 10.5  Segment of the Liutex Core Line found at the local maximum Liutex Magnitude value of 0.166685

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We have now found the rotation axis of our vortex. But, because all DNS data is limited by resolution, we see a break or discontinuity in the formation of the Liutex core line. This can be mitigated by creating Liutex core lines at some or all local maximums (seed points) on the reference x-slice. For the purpose of this example, we will only use this segment of the Liutex core line.

4  Liutex Core Tube We will now answer the question of finding the vortex core size by defining the strength of the core tube with the Liutex Core Tube Magnitude Vector. Definition 1  The Liutex Core Tube Magnitude Vector can be defined as a vector that lies on a plane intersection with its initial point at the Liutex core line and its terminal point at an arbitrary percentage between 0 and 1 (non-inclusive) of the local maximum at its initial point. What this means is that we can decide the strength and size of the core tube by establishing a percentage of the local maximum value that we found when creating the Liutex core line. In Fig. 10.6, we can say that point p is an arbitrary percentage of the local maximum Liutex magnitude value c, or

Fig. 10.6  Liutex Core Tube Magnitude Vector p

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p = ac

where we have the condition that 0   0, there are one real and two complex eigenvalues, which indicates a closed or spiraling streamline pattern. For this reason, Chong et al. [21] suggest the Δ criterion which defines a vortex core to be the region where the velocity gradient tensor has complex eigenvalues. But the physical meaning of the discriminant Δ is not clear. The λci criterion can be regarded as an extension of the Δ criterion and identical to the Δ criterion when zero threshold is applied. Hence, λci only exists when the velocity gradient tensor has complex eigenvalues. To derive the λci criterion, Eq. (11.1) is expressed in the eigenvector basis, which can be written as  dc  = V −1GVc dt

(11.13) T  c = [ c1 c2 c3 ] is the position vector based on the eigenvector basis and has the following relation with the position vector y   Vc = y



(11.14)

In this case, the velocity gradient tensor can be decomposed in a non-standard way as

∗

∗

( )

G =V Λ V

∗

−1

 = [ vcr

 vci

 λcr   vr ]  −λci  0

λci λcr 0

0  0  [ vcr λr 

 vci

 −1 vr ] (11.15)

λcr, λci, λr are the real part, the imaginary part of the complex eigenvalue and the real  eigenvalue, respectively. vcr ± ivci represents the complex eigenvectors correspond ing to the eigenvalue λcr ± iλci and vr the real eigenvector. Accordingly, the ODEs given by Eq. (11.13) is



 dc1   dt     λcr  dc2  =  −λ  dt   ci    0  dc3   dt 

λci λcr 0

0   c1  0  c2  λr   c3 

(11.16)



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The solution of Eq. (11.16) is

c1 ( t ) = c1 ( 0 ) cos ( λci t ) + c2 ( 0 ) sin ( λci t )  eλcr t

(11.17a)



c2 ( t ) = c1 ( 0 ) cos ( λci t ) − c2 ( 0 ) sin ( λci t )  eλcr t

(11.17b)



c3 ( t ) = c3 ( 0 ) eλr t

(11.17c)

 c1 ( 0 )   y1 ( 0 )     −1  c2 ( 0 )  = V  y2 ( 0 )   c3 ( 0 )   y3 ( 0 )     

(11.18)

with



Zhou et al. [2] suggest that λci can measure the swirling strength since λci plays a role of angular speed and the orbit time of the trajectory is 2π/λci. A larger λci seems to indicate a stronger vortex from the idea of λci. Figure 11.1 shows eigenvectors and the trajectory around a point O selected from the following boundary layer transition case. The velocity gradient tensor at the point O is



 0.0068 0.0796 0.4804  G =  −0.0074 0.0040 −0.0461  −0.0103 −0.0323 −0.0108 

The eigenvalues are λr  =  0.0314, λcr  ±  λci  =    −  0.0157  ±  0.0685i and the T  corresponding eigenvectors are vr = [ 0.8910 − 0.4383 0.1183] , T  T  vcr = [ 0.9852 − 0.0557 0.0370 ] , vci = [ 0 0.0972 0.1234 ] . Obviously, the instantaneous trajectory is a swirling streamline around the direction of the real eigen vector vr . The concept of Liutex (previously called Rortex) is first proposed by Liu et al. [10]. The direction of the Liutex vector, representing the local rotational axis, is  defined by the direction of the real eigenvector vr which fulfills

  G ⋅ vr = λr vr

(11.19)

The justification of this definition is that according to Eq. (11.17), all trajectories of neighboring particles swirl or rotate around the direction of the real eigenvector, as shown in Fig. 11.1. The magnitude of Liutex is obtained in a special coordinate system called the principal coordinate [22]. In the principal coordinate, the velocity gradient tensor can be written as



λcr R/2 0   G pc =  − ( R / 2 +  ) λcr 0   ξ η λr 

(11.20)

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Here, R is the magnitude of Liutex, and ϵ, ξ, η represent the strength of simple shear along different axes. R/2 is the actual rotation strength and equal to angular speed 1 for rigid body rotation. A constant is applied to be consistent with the vorticity 2 and vorticity decomposition [10]. This special coordinate can be realized by two successive coordinate rotations. The first rotation makes the z-axis along the direction of the real eigenvector and the second planar rotation makes the leading two diagonal terms of the velocity gradient tensor equal to the real part of the complex eigenvalues. The justification of the magnitude definition has been discussed in [23]. Actually, from Eqs. (11.19) and (11.20), an explicit expression of the Liutex vector can be derived [11, 18].



   R = (ω ⋅ vr ) − 

 

(ω ⋅ vr )

2

 − 4λci 2  vr 

(11.21)

  where ω is the vorticity vector and the direction of vr is determined by the con  dition ω ⋅ vr > 0 .

3  Analytical Relation Between Liutex and λci Since the coordinate rotation is an orthogonal transformation which does not change the eigenvalues, Eq. (11.20) and Eq. (11.15) have the identical eigenvalues. From the relation between roots and coefficients, we have





R R  +   = λci2  22 

λci =

R R  +  2  2 

(11.22)

(11.23)

Equation (11.23) clearly shows that λci is not a precise measurement of rotation due to the shear effect given by ϵ. If the shear effect ϵ is large enough, the λci criterion would improperly identify a region with large shear as a strong vortex. It has been discussed in [9]. On the other hand, consider the eigendecomposition given by Eq. (11.15). λci appears anti-symmetrically in the matrix Λ∗ and seems to be a representation of rotation. However, Eq. (11.15) is only a similarity transformation but not an orthogonal transformation. It means that Λ∗ in Eq. (11.15) has the same eigenvalues as G, but it is a matrix representation of the velocity gradient tensor G in a non-­orthonormal basis, so it cannot be regarded as the same tensor in the orthonormal coordinate system. Therefore, λci is not a direct measurement of the rotational/swirling strength in an orthonormal coordinate system due to the non-orthonormality of the eigenvectors. Even if in two-dimensional cases, the eigenvectors in the non-standard form

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Table 11.1  Comparison between λci and Liutex Matrix form (ignore diagonal terms for stretching/compression) λci

 0  −λ  ci  0

λci 0 0

Coordinate system Non-orthonormal system

0 0  0 

Contamination by shear Yes

Principal coordinate No

Liutex  0 R / 2 0 − R / 2 0 0  ( )   0 0 0 

are usually orthogonal but not orthonormal, which implies that λci is still a “contaminated” measurement of rotation due to the non-normal basis. Table 11.1 summaries the difference between λci and Liutex. To explore the contamination due to the non-orthonormality, Gram-Schmidt orthogonalization is applied to Eq. (11.15) to obtain the expression of R in terms of λci and eigenvectors as

(

R=

  2     2    vci ⋅ vci − ( vr ⋅ vci )2  vci ⋅ vci − ( vr ⋅ vci ) − ( vcr ⋅ vcr ) + ( vr ⋅ vcr )     −     2    + ( v ⋅ v ) − ( v ⋅ v )2  cr cr r cr +4 ( vcr ⋅ vci − ( vr ⋅ vcr ) ( vr ⋅ vci ) )  

( )

2 det V ∗

)

2

λci (11.24)

The detailed derivation of Eq. (11.24) is provided in [24]. Obviously, if the eigenvectors constitute the orthonormal basis, the matrix V∗ is a proper orthogonal matrix. In this case, there is no shear along three axes, so λci is equal to R and can represent the rotation strength. Nevertheless, in general, the three eigenvectors are not mutually orthogonal or have different lengths. Denote

    2 A = vci ⋅ vci − ( vr ⋅ vci ) ,



    2 B = ( vcr ⋅ vcr ) − ( vr ⋅ vcr ) ,



      C = vcr ⋅ vci − ( vr ⋅ vcr ) ( vr ⋅ vci ) . We have R=



( A + B) − ( A − B)

2

+ 4C 2

( )

2 det V ∗

λci

(11.25)

According to [24],

( )

det V ∗ = AB − C 2



(11.26)

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  and A + B > 0 since ω ⋅ vr > 0 . We obtain R=



( A + B) − ( A − B)

2

+ 4C 2

2 AB − C 2

λci

(11.27)

After some simple algebraic manipulations, it can be found that 0 < ( A + B) −



( A − B)

2

+ 4C 2 < 2 AB − C 2

(11.28)

which implies 2 R ( A + B ) − ( A − B ) + 4C = λci < λci 2 4 AB − C 2 2



(11.29)

Equation (11.29) indicates that the rotation strength R/2 is always less than λci and only part of λci contributes to rotation and the remaining part contributes to shear. Combined with Eq. (11.23), we have =

8 AB − 10C 2 − A2 − B2 + ( A + B ) 2 AB − C 2 ( A + B ) − 

( A − B)

( A − B)

2

2

+ 4C 2

+ 4C 2  

λci

(11.30)

4  Examples 1. 2D Rigid rotation superposed with a prograde shear motion As a model problem, a 2D Rigid rotation superposed with a prograde shear motion is examined. The velocity of rigid rotation is  u = ( R / 2 ) y ⋅ ( R > 0)  v = − ( R / 2 ) x



The velocity of the prograde shear motion used here is u=0  v = − x,  > 0

The velocity will be

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Fig. 11.2  Velocity of 2D Rigid rotation superposed with a prograde shear motion

 u = ( R / 2 ) y  v = − ( R / 2 +  ) x



as shown in Fig. 11.2. In this case, the velocity gradient tensor can be written as 0 R / 2  G= 0  − ( R / 2 +  )

The eigenvalues are

λcr = 0





λci =

R R  +  2  2 

and the corresponding eigenvectors are



 R  vcr =  R +    0

   

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0    vci =  R + 2 −  R +



   

The eigen decomposition is  R  R + G=   0 

  0   R + 2   −λci −  R + 

 R +  0   λci   R   0  R +  −  0  R + 2     From the eigenvectors, it can be found that although vcr and vci are orthogonal in the non-standard form, but they are not unit vectors. So λci is actually a measurement of the rotation in an orthogonal but not orthonormal coordinate system. Figure 11.3 demonstrates streamlines across though the point (1,0) if R is fixed and ϵ = R/2, 4.5R 0

Fig. 11.3  Streamlines across though the point (1,0) with different ϵ

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and 24.5R. It is clear that the shear becomes stronger, but the rotation strength remains the same. In the limit, if ϵ trends to infinity, the motion trends to strong simple shear with weak rotation. However, λci will also trend to infinity, which would mistakenly identify this motion as strong rotation. 2. Boundary layer transition The direct numerical simulation (DNS) of boundary layer transition on a flat plate is used to analyze Liutex and λci for the realistic flow. The details of DNS can be found in [25]. A point on the iso-surface R  =  0.05 is examined, as shown in Fig. 11.4. The velocity gradient tensor at the point is



0.7677   0.0275 0.1656 G =  −0.0112 0.0136 −0.0124   −0.0105 −0.0406 −0.0424 



The eigenvalues are λr  =  0.0388, λcr  ±  λci  =    −  0.0201  ±  0.0957i and the corT  resonding eigenvectors are vr = [ 0.8874 − 0.4477 0.1096 ] , T T   vcr = [ 0.9874 0.0217 − 0.0659] , vci = [ 0 0.0993 0.1017] . Using coordinate rotation, the velocity gradient tensor in the principal coordinate will be



−0.0201 λcr R/2 0  0.025 0      G pc =  − ( R / 2 +  ) λcr 0  =  − ( 0.025 + 0.3429 ) −0.0201 0   ξ η λr   0.6909 0.0772 0.0388 

Fig. 11.4  Liutex iso-surface for vortex leg (R = 0.05)

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Fig. 11.5  Directions of principal coordinate, eigenvectors and instantaneous trajectory

The direction of the x-axis of the principal coordinate is T  x pc = [ 0.0425 0.3162 0.9478] . The direction of the y-axis of the principal coor T dinate is y pc = [ −0.4590 −0.8364 0.2996 ] . It can be easily validated that x pc ,  y pc and vr are mutually orthogonal to each other, but the eigenvectors are not. The directions and an instantaneous trajectory are illustrated in Fig. 11.5. According to Gpc, the shear ϵ is relatively large, leading to a large λci. Therefore, λci would be severely contaminated by shear.

5  Conclusions In this paper, the analytical relation between Liutex and λci is discussed based on the eigensystem of the velocity gradient tensor. It is found that λci is not a direct measurement of the rotational/swirling strength in an orthonormal coordinate system due to the non-orthonormality of the eigenvectors. Even if in two-dimensional

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cases, the eigenvectors in the non-standard form can be orthogonal but not orthonormal. So λci would be contaminated by shear. Liutex can eliminate the shear contamination and thus can reasonably measure the actual rigid rotation part of local fluid motion. Acknowledgement  The first author is supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). The authors are also grateful to Texas Advanced Computational Center (TACC) for providing computation hours.

References 1. J.  Hunt, A.  Wray, P.  Moin, Eddies, streams, and convergence zones in turbulent flows, in Center for Turbulence Research Proceedings of the Summer Program, (1988), p. 193 2. J. Zhou, R. Adrian, S. Balachandar, T. Kendall, Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396 (1999) 3. P. Chakraborty, S. Balachandar, R.J. Adrian, On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189–214 (2005) 4. J. Jeong, F. Hussain, On the identification of a vortices. J. Fluid Mech. 285, 69–94 (1995) 5. B. Epps, Review of vortex identification methods. AIAA 2017-0989 (2017) 6. C.  Liu, Y.  Gao, X.  Dong, Y.  Wang, J.  Liu, Y.  Zhang, X.  Cai, N.  Gui, Third generation of vortex identification methods: Omega and Liutex/Rortex based systems. J.  Hydrodyn. 31, 205–223 (2019) 7. G. Haller, Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137–162 (2015) 8. Y. Maciel, M. Robitaille, S. Rahgozar, A method for characterizing cross-sections of vortices in turbulent flows. Int. J. Heat Fluid Flow 37, 177–188 (2012) 9. Y. Gao, C. Liu, Rortex and comparison with eigenvalue-based vortex identification criteria. Phys. Fluids 30, 085107 (2018) 10. C. Liu, Y. Gao, S. Tian, X. Dong, Rortex—A new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids 30, 035103 (2018) 11. 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. Hydrodyn. 31(3), 464–474 (2019) 12. Y. Gao, Y. Yu, J. Liu, C. Liu, Explicit expressions for Rortex tensor and velocity gradient tensor decomposition. Phys. Fluids 31, 081704 (2019) 13. X. Dong, Y. Gao, C. Liu, New normalized Rortex/vortex identification method. Phys. Fluids 31, 011701 (2019) 14. J.  Liu, C.  Liu, Modified normalized Rortex/vortex identification method. Phys. Fluids 31, 061704 (2019) 15. Y. Wang, Y. Gao, C. Liu, Galilean invariance of Rortex. Phys. Fluids 30, 111701 (2018) 16. J. Liu, Y. Gao, C. Liu, An objective version of the Rortex vector for vortex identification. Phys. Fluids 31, 065112 (2019) 17. Y.  Gao, C.  Liu, Rortex based velocity gradient tensor decomposition. Phys. Fluids 31, 011704 (2019) 18. W. Xu, Y. Gao, Y. Deng, J. Liu, C. Liu, An explicit expression for the calculation of the Rortex vector. Phys. Fluids 31, 095102 (2019) 19. Y. Gao, J. Liu, Y. Yu, C. Liu, A Liutex based definition and identification of vortex core center lines. J. Hydrodyn. 31(3), 445–454 (2019) 20. W. Xu, Y. Wang, Y. Gao, J. Liu, H. Dou, C. Liu, Liutex similarity in turbulent boundary layer. J. Hydrodyn. 31(6), 1259–1262 (2019)

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21. M.  Chong, A.  Perry, B.  Cantwell, A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765–777 (1990) 22. Y. Yu, P. Shrestha, C. Nottage, C. Liu, Principal coordinates and principal velocity gradient tensor decomposition. J. Hydrodyn. 32, 441–453 (2020) 23. J. Liu, Y. Deng, Y. Gao, S. Charkrit, C. Liu, Mathematical foundation of turbulence generation-­ symmetric to asymmetric Liutex/Rortex. J. Hydrodyn. 31, 632–636 (2019) 24. C. Liu, Y. Gao, Liutex-based and other mathematical, computational and experimental methods for turbulence structure (Bentham Science, Sharjah, 2020) 25. C. Liu, Y. Yan, P. Lu, Physics of turbulence generation and sustenance in a boundary layer. Comput. Fluids 102, 353–384 (2014)

Part II

Liutex Applications for Turbulence Research

Chapter 12

Hairpin Vortex Formation Mechanisms Based on LXC-Liutex Core Line Method Heng Li, Duo Wang, and Hongyi Xu

Abstract  Based on the direct numerical simulation (DNS) data, the generation mechanisms of hairpin vortices are studied in the research. The momentum thickness Reynolds number range is 250  0, r represents the direction of the Liutex vector. In another words, Liutex core line is the line where all the Liutex gradient lines align with direction of Liutex vector. In fact, it is the local maximum of magnitude of Liutex vector in a plane perpendicular to the local axis of fluid rotation.

3  Application to DNS Data The DNS data is for a compressible, zero-pressure-gradient flat-plate boundary layer as in [1, 2]. The case set up for this research is presented in Sect. 2 above. We have applied Liutex based methods and other traditional vortex identification schemes to DNS data and compared them. The iso-surface presented below are for early transition of flat plate boundary layer data.

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Fig. 19.3 (a–c) Iso-surface plotting of Δ at various thresholds. (a) Δ  =  0. (b) Δ  =  0.005. (c) Δ = 0.03

3.1  ∆Criterion Iso-surface of ∆ at various thresholds are presented in the Fig.  19.3 below. Figure 19.3a shows the iso-surface with ∆ = 0 which seems thick and smeared vortex structure can be seen. Also, spanwise vortex in the beginning of the flat plate appears clearly. But when we increase the threshold to 0.005, ∧ and hairpin vortices become sharp and spanwise vortices disappear. When we further increase the threshold to 0.03, ∧ as well as hairpin vortices disappear, and the vortex structure looks broken like a collection of debris. This is largely because of the threshold selection. Selecting small threshold results in vortex structure being smeared and vague while bigger threshold causes vortex to break down theoretically into several pieces. This creates the confusion in determining the proper threshold if we do not have prior knowledge about the structure of fluid flow in the boundary layer. This method does not give us the proper guideline to construct the flow structure as flow structures keep changing with the change in threshold.

3.2  Q Criterion Iso-surface of Q at various thresholds are presented in the Fig.  19.4 below. Figure 19.4a shows the iso-surface with Q = 0.001, where spanwise vortices, lambda vortices and hairpin vortices are overcoated and unclear due to the capture of the very weak vortices because of selecting small threshold. As in Δ method, as we keep increasing threshold uniformly, the flow structure becomes thinner and sharp, and ultimately it starts to disappear, making it very hard to select the proper threshold.

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Fig. 19.4 (a–c) Iso-surface plotting of Q at various thresholds. (a) Q = 0.001. (b) Q = 0.003. (c) Q = 0.03

Fig. 19.5 (a–c) Iso-surface plotting of λci at various thresholds. (a) λci = 0.00005. (b) λci = 0.05. (c) λci = 0.5

3.3  λci Criterion As in ∧ and Q methods, streamwise vortex can be seen in Fig. 19.5a with small threshold. When threshold is increased a thin and sharp lambda vortex can be seen as in Fig. 19.5b. But hairpin vortex in this method is not as clear as ∧and Q method. As we further decrease the threshold, the head of hairpin vortex starts disappearing which shows that head of the hairpin vortex has weaker vortex strength compared to legs.

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Fig. 19.6 (a–c) Iso-surface plotting of λ2 at various thresholds. (a) λ2 = −0.000. (b) λ2 = −0.1. (c) λ2 = −0.7

3.4  λ2 Criterion Iso-surface of Q at various thresholds are presented in the Fig.  19.6 below. Figure 19.6a shows the iso-surface plot of λ2 =  − 0.0001 which thick and smeared. As in previous method, the more we decrease the threshold (in negative values), the more the vortex structure starts to break down.

3.5  Omega Criterion Since Omega method gives the relative strength of vorticity in fluid flow coupled with deformation, we set threshold a little bigger than 50%. Figure 19.7a–c represents the iso-surface of flow structure of transitional boundary layer at various thresholds. But there is not much change in the vortex structure, providing the ample evident that this method is only mildly affected by threshold change. This method can capture the both weak and strong vortex structure simultaneously as it represents the relative strength of vorticity and is expressed between o and 1. So, it is evident that Omega method is a useful method if we want to get rid from the headache of selecting proper threshold.

3.6  Modified Liutex-Omega Method This method was proposed by Liu and Liu [14] to improve the Liutex-Omega method, which is a combination of Liutex and Omega method. Although Liutex-­ Omega method robustly able to capture vortex structures, the Iso-surface plotting

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Fig. 19.7 (a–c) Iso-surface plotting of Ω at various thresholds. (a) Ω = 0.52. (b) Ω = 0.57. (c) Ω = 0.62

Fig. 19.8 (a–c) Iso-surface plotting of  L at various thresholds. (a)  L =0.53. (b)  L =0.60. (c)  L =0.65

was not smooth and had bulging phenomena. So, Modified Liutex-Omega method was proposed to resolve these problems. Like Omega method, this method also calculates the relative strength of vorticity in the rotated frame XYZ where Z-axis is parallel to the real eigen vector r of velocity gradient tensor. This method is also a relative method, so both weak and strong vortex can be captured simultaneously. Furthermore, this method is minimally affected by threshold changes. So, this is the most effective method if we want to get rid of threshold selection dilemma (Fig. 19.8).

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3.7  Liutex Core Lines Method Figure 19.9 represents the flow structure of early transition of boundary layer by Liutex core lines. Liutex core lines are local maximum of Liutex magnitude in the planes perpendicular to the axis of rotation. Since this is a line, not the iso-surface, so all the annoyances that come with selection of threshold for iso-surface plotting is not the case anymore. Therefore, not only we are exempted from nuisance of choosing a proper threshold, Liutex core lines method is also unique that can display the vortex structure with ease. Moreover, this method can clearly depict the strength of vortex cores at vortex positions like lambda vortex, hairpin vortex, etc. In the following figure, different colors represent different strength of vortex core lines.

Fig. 19.9 (a–c) Evolution of Liutex core lines in early transition boundary

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4  Conclusions With the application of different schemes on DNS data of transitional boundary layer, we can draw the following conclusions: 1. Q, Δ, λ2, and λcimethods are threshold sensitive. They all give different vortex structure with different threshold, making it hard to choose the proper one. A prior knowledge about the flow structure is needed if one is to get the vortex structure from these methods. 2. Omega method is moderately sensitive to threshold change. A prior knowledge about the flow structure is not needed to draw iso-surface of vortex structure as any number little greater than 0.50 gives the consistent figure of iso-surface. 3. Omega method can capture both weak and strong vortices simultaneously. 4. Modified Liutex-Omega method can capture both weak and strong vortices simultaneously and is moderately sensitive to threshold changes. 5. Q, Δ, λ2, ,λciand Ω method present the similar figures if proper threshold is selected. However, we prefer omega method over others for two reasons. First, it has less effect of threshold change. Second, it has a clear physical meaning. i.e. the proportion of vorticity in the fluid flow should be more than 50%. 6. Out of all the vortex identification schemes, Liutex core line method is unique and there is no need to select threshold as it is a line passing through the points with greatest rotational strength.

References 1. X. Wu, P. Moin, Direct numerical simulation of turbulence in a nominally zero-pressure gradient flat-plate boundary layer. JFM 630, 5–41 (2009) 2. Y. Yan, C. Chen, F. Huankun, C. Liu, DNS study on Λ-vortex and vortex ring formation in the flow transition at Mach number 0.5. J. Turbul. 15(1), 1–21 (2014) 3. C. Liu, Y. Yan, P. Lu, Physics of turbulence generation and sustenance in a boundary layer. Comput. Fluids 102, 353–384 (2014) 4. A.M. Perry, M.S. Chong, A description of eddying motions and flow patterns using critical-­ point concepts. Annu. Rev. Fluid Mech. 19, 125–155 (1987) 5. M.S. Chong, A.E. Perry, A general classification of three-dimensional flow fields. Phys. Fluids A 2(5), 765–777 (1990) 6. J. Jeong, F. Hussain, On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995). https://doi.org/10.1017/S0022112095000462 7. J. Zhou, R. Adrian, S. Balachandar, T.M. Kendall, Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396 (1999) 8. C.  Liu, Y.  Gao, X.  Dong, Y.  Wang, J.  Liu, Y.  Zhang, X.  Cai, N.  Gui, Third generation of vortex identification methods: Omega and Liutex/Rortex based systems. J. Hydrodyn. 31(2), 205–223 (2019) 9. C. Liu, Y. Wang, Y. Yang, Z. Duan, New omega vortex identification method. Sci. China Phys. Mech. Astron. 59(8), 1–9 (2016) 10. C. Liu, Y. Gao, S. Tian, X. Dong, Rortex- A new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids 30, 035103 (2018)

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11. Y.  Gao, C.  Liu, Rortex and comparison with eigenvalue-based identification criteria. Phys. Fluid 30, 085107 (2018) 12. Y.  Yu, P.  Shrestha, C.  Nottage, C.  Liu, P.  Coordinates, Principal velocity gradient tensor decomposition. J. Hydrodyn. (2020). https://doi.org/10.1007/s42241-­020-­0035-­z 13. X. Dong, Y. Gao, C. Liu, New normalized Rortex/vortex identification method. Phys. Fluids 31(1), 011701 (2019) 14. J. Liu, C. Liu, Modified normalized Rortex/vortex identification method. Phys. Fluids 31(6), 061704 (2019) 15. Y. Gao, J. Liu, Y. Yu, C. Liu, A Liutex based definition and identification of vortex core center lines, J. Hydrodynamics. 31(3) (2019) 16. Y. Yan, C. Chen, F. Huankun, C. Liu, DNS study on Λ-vortex and vortex ring formation in flow transition at Mach number 0.5. J. Turbul. 15(1), 1–21 (2014) 17. C. Liu, Y. Yan, P. Lu, Physics of turbulence generation and sustenance in a boundary layer. Comput. Fluids 102, 353–384 (2014) 18. X.R. Dong, Y. Wang, X. Chen, Y. Dong, Y. Zhang, C. Liu, Determination of epsilon for omega vortex identification method. J. Hydrodyn. 30(4), 541–546 (2018)

Part III

Liutex Applications in Engineering

Chapter 20

Investigation of Flow Structures Around Cylinders with High Reynolds Number by Liutex Vortex Identification Methods Jiawei He, Songtao Chen, Weiwen Zhao, and Decheng Wan

Abstract  The computational results of flow around cylinders using PANS method based on Menter shear stress transport (SST) turbulence model were presented in this investigation. Partially averaged Navier-Stokes (PANS) method derived from the traditional Reynolds averaged Navier-Stokes (RANS) methods by introducing controlling parameter into original RANS equations to modify these equations, is one kind of hybrid methods which can effectively simulate the separated turbulent flows. The Liutex was used to identify vortex structures in late boundary layer transition on flow around cylinders. The program of Liutex is based on Professor Liu’s theory, compiled with C++ language, which was embedded in OpenFOAM for post-­ processing. Several comparative studies on simple examples and realistic flows are studied to confirm the superiority of Liutex. Keywords  Partially averaged Navier-Stokes (PANS) · Reynolds number · Flow around cylinder · Liutex vortex identification method

J. He · S. Chen · W. Zhao Computational Marine Hydrodynamics Lab (CMHL), School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China D. Wan (*) Computational Marine Hydrodynamics Lab (CMHL), School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China Ocean College, Zhejiang University, Zhoushan, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_20

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1  Introduction The development of vortex identification method can be classified into three generations. The first-generation methods are based on the vorticity. The second generation of vortex identification methods, including Q, λ2, Δ and λci criteria. Liutex was novelly obtained as the third-generation of the vortex definition and identification methods. Recent studies have given an improved physical understanding of the Liutex method. Tian et  al. proposed a new vector quantity that is called vortex vector, which accurately describe the local fluid rotation and clearly display vortical structures [1]. Liu et al. [2] novelly obtained Rortex identifies both the precise swirling strength and the rotational axis, and thus it can reasonably represent the local fluid rotation and provide a new powerful tool for vortex dynamics and turbulence research. Gao and Liu [3] pointed out that the Rortex eliminates the shear contamination and thus can accurately quantify the local rotational strength. What’s more, with continued study they found that the Rortex vector field and Rortex lines can also be used to visualize and investigate vortical structures, not just the iso-surface of Rortex. Wang et al. [4] obtained an explicit expression of the Rortex vector in terms of the eigenvalues and eigenvectors of velocity gradient for the first time. Dong et al. [5] pointed out that the third generation of Rortex method ΩR is quite robust and can be always set as 0.52 to capture vortex structures in different cases and at different time steps. Furthermore, Liu et  al. [6] have been raised six core issues for vortex definition and identification to elaborate the advantages of the third-generation methods. Xu et al. [7] used an example of fully-developed turbulence in a square annular duct to demonstrate the vortex structure represented by the vortex-core lines. According to Xu’s finding, the third generation vortex identification approach based on the vortex-core lines is capable of profoundly uncovering the vortex natures. Liu et al. [8] presented a modified normalized Rortex identifica tion method named  R to improve the original ΩR method and resolve the bulging phenomenon on the isosurfaces, which is caused by the original ΩR method. The primary aim of the research described in this paper is to study numerically the flow around a circular cylinder. The emphasis is on clarifying the flow field in the wake of the cylinder by using Liutex vortex identification method.

2  Computational Domain and Grids The simulations are performed on an O-grid, with a computational domain extending 25D in the length,10D in the width and πD in the depth, where D is the diameter of the cylinder. The full computational domain is shown in Fig. 20.1. The total grid number for the simulation is around 4.2 million. The time step Δt is set to maintain a Courant number of approximately 0.5, which Δt=0.001s. The end of simulation time is set to t= D/U0 = 25. The parallel simulation are scheduled on the HPC cluster

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Fig. 20.1  Computational domain and boundary conditions. (a) Computational domain. (b) Boundary conditions

in our lab with 28 physical cores (Intel Xeon E5-4627, 3.4 GHz), consumed 20 h in total.  The boundary conditions used for the numerical simulations can be defined as follows. Velocity will be assigned as fixed value ux = U0, uy = uz = 0, in the inlet. On the cylinder boundary no slip condition for the velocity will be applied. The pressure boundary condition in inlet is of a type zero gradient on the cylinder. At the outlet boundary, the pressure gradient is set equal to 0. The rest of the boundaries is

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Fig. 20.2  Details of the refinement mesh. (a) Mesh around the cylinder. (b) The value of y + 

defined as symmetry boundary for the reason of assuming that the height of the cylinder is infinite. Details of the refinement mesh around the cylinder is shown in Fig. 20.2. The wall y + values range from about 1 to around 5, indicating sufficient resolution in the near boundary layer for the simulation.

3  Discretization Format In the present study, the governing equations are discretized using a finite volume method for solving the incompressible Navier–Stokes equations. The time discretization is done using second order implicit Euler Scheme. A second order Gauss integration is used for spatial gradient calculations. The convection operator is discretized using a total variation diminishing (TVD) scheme. The merged PISOSIMPLE (PIMPLE) algorithm is used for solving the coupled pressure–velocity equations. The governing equations are discretized using a finite volume method for solving the incompressible Navier–Stokes equations using a self-­developed solver naoe-FOAM-SJTU, with a newly implemented SST-PANS turbulence model. The suitability of the present solver has been clarified by Zhao et al. for solving flow past two circular cylinders in tandem [9] and simulating of vortex-­induced motions of a semi-submersible [10].

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4  Turbulence Model The zonal hybrid RANS–LES turbulence model of SST-PANS has been used for current simulation. The details of SST-PANS model has been presented and evaluated for flow around cylinder by Elmiligui et al., and the model gives good agreement with the experiment results [11, 12]. ∂ ( ρ ku ) ∂t



+

∂ ( ρUi ku ) ∂xi

∂ ( ρωu )

∂ku  ∂  = Pku − β ∗ ρ kuωu +  ( µ + σ ku µu ) ∂xi  ∂xi 

∂ ( ρUiωu )

γ = Pku − β ′ρ kuωu2 + ∂t ∂xi vu ∂ωu  ρσ ω 2 u ∂ku ∂ωu ∂  ( µ + σ ωu µu )  + 2 (1 − F1 ) ∂xi  ∂xi  ωu ∂xi ∂xi



+

(20.1)

(20.2)

where,

β ′ = γβ ∗ −

f f γβ ∗ β + ;σ ku = σ k ω ;σ ωu = σ ω ω fω fω fk fk



(20.3)

where β∗=0.09, γ1 = 5/9, γ2 = 0.44, β1 = 0.075, β2 = 0.0828, σk1 = 0.85, σk2 = 1.0, σω1 = 0.5, σω2 = 0.0828. fk =

ku f ω ; fω = u = ε k fk ω



(20.4)

where k is the total turbulent kinetic energy and ω is the total specific dissipation rate, respectively. The expressions for the two blending functions in the case of SST-PANS are given by:

(20.5)



4     ku 500 µ  4 ρσ k     ω 2u u  } F1 = tanh min  max  ∗ , ,  β ωu d d 2 ρωu  CDkω d 2            

(20.6)



 2 ρσ ω 2 u ∂ku ∂ωu −10  CDkω = max  ,10   ωu ∂xi ∂xi 

(20.7)



2   2 ku 500 µ     F2 = tanh  max  ∗ ,   β ωu d d 2 ρωu         

The unresolved eddy viscosity μu is defined as:

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 ρk ρa k  µu = min  u , 1 u   ωu SF2 

(20.8)

5  Results 5.1  Force Coefficient The non-dimensional drag forces coefficient is given by:



Cd ( t ) =

2 Fx ( t ) ρ HDU 2

(20.9)

where Fx(t) is the force in the in-line direction; U is the velocity of flow; ρ is the density of water; D is the characteristic diameter of cylinder. The non-dimensional lift forces coefficient is given by:



Cl ( t ) =

2 Fy ( t )

ρ HDU 2

(20.10)

where Fy(t) is the force in the transverse direction; U is the velocity of flow; ρ is the density of water; D is characteristic diameter of cylinder.

Fig. 20.3  The time histories of the present lift (Cl) and drag (Cd) coefficients

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Table 20.1  Overall flow parameters of the flow past a circular cylinder, Reynolds number(Re) Re = 140,000 Data source Experiment [13] Present SST-PANS Present SST-IDDES

Cd 1.24 1.31 0.93

Cpb 1.21 1.29 0.96

St 0.179 0.20 0.21

Fig. 20.4  Mean pressure coefficient on the surface of the cylinder with the experimental data and the numerical simulation

Figure 20.3 demonstrates the time histories of the present lift (Cl) and drag (Cd) coefficients for a period of 35 vortex shedding cycles. Furthermore, Summary of the global flow quantities for circular cylinder flow is listed in Table 20.1. Based on the results listed in Table 20.1 and the comparisons of Fig. 20.4, it was concluded that the current PANS compared well with Cantwell’s experimental data. The values obtained by the SST-PANS and SST-IDDES are in good agreement with the experimental data. Figure 20.4 show the mean surface pressure coefficient distribution of the cylinder in the horizontal plane from numerical calculations and the experiment. Experimental measurements taken from Cantwell and Coles [13] (Re  =  140,000, unfilled symbols). There is a good agreement between the present simulation results and the experiments with respect of flow getting to the angle from separation point. Comparing the mean surface pressure coefficient of SST-PANS and SST-IDDES models with experimental results, the SST-PANS models appear to be in better agreement with experimental results than the SST-IDDES turbulent model.

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5.2  Instantaneous Vorticity Q Criterion The definition of Q is given by: Q=



(

)

1 2 2 Ω −S 2

(20.11)

where S is invariant measure of the strain rate tensor, define as S = 2 Sij Sij , and 1  ∂u ∂u  Sij =  i + j  , Ω is invariant measure of vorticity tensor. 2  ∂x j ∂x j  Omega-Liutex Ω Liu In 2018 [1], a Liutex vector (previously named Rortex) was proposed to provide a mathematical and systematical definition of the local rigid rotation part of the fluid motion, including both the local rotational axis and the rotational strength. As a new physical quantity raised by Liu et al. [2], Liutex has direction and magnitude. Its direction represents the local vortex line and is parallel to the normal vector of the vortex iso-surface. Its magnitude is exactly the local angular speed, and Liutex represents a force which is the driving force of turbulence generation. According to Liu et al. [6] the definition of Omega-liutex ΩLiu is given by: Ω Liu =

2

where

α=

β β+α +ε 2

1  ∂U ∂V   ∂V ∂U  − + + , 2  ∂X ∂Y   ∂X ∂Y 

(20.12) 1  ∂V ∂U  β=  − , 2  ∂X ∂Y 

and

ε = 0.001 × (β2 − α2)max. Figure 20.5 shows the instantaneous isosurface of Omega-liutex ΩLiu criterion and Q-criterion, which have been colored with velocity magnitude. According to different vortex identification methods, it can be observed for the wake vortex structure that powerful vortices with high vorticity magnitude mostly appear in the vicinity of the tail. The vortex structures obtained by Q criterion and Omega-liutex ΩLiu are very similar. The results show that compared with Q criterion, the accuracy of vortex recognition results of Omega-liutex method is not affected by the selection of subjective thresholds, and it can identify strong and weak vortex structures at the same time, with better recognition results. The comparison in Fig. 20.5 clearly illustrates the effect of thresholds for Q criterion. Figure 20.6 show the Streamwise vortex cores at the horizontal plane. The flow in the wake field predicted by different vortex identification methods shows significant discrepancy. Overall, the distribution of vorticity is clearer than the Liutex.

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Fig. 20.5  Instantaneous isosurface of Omega-liutex criterion and Q-criterion (dye visualization with velocity magnitude). (a) Omega-liutex criterion ΩLiu  =  0.52. (b) Q-criterion(Q  =  10). (c) Q-criterion (Q = 25). (d) Q-criterion (Q = 50). (e) Q-criterion (Q = 100). (f) Q-criterion (Q = 200)

Lots of vorticity appears near the wall of cylinder, while the Liutex is close to zero. As shown in Fig. 20.6, the formation of a vortex is an important physical process in local flow, so that the vortex boundary does not appear. The increase in the proportion of vorticity means that the shear effect is decreasing. Although the strength of the vortex core is significantly reduced, the dissipation at this time is low and the rotational motion has a dominant effect. This can affect the tail vortex structure. Shear deformation play significant roles in forming those energetic vortices, due to

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Fig. 20.6  Streamwise vortex cores at the horizontal plane, Liutex (left) and Vorticity (right). (a) Liutex Magnitude. (b) Vorticity Magnitude. (c) Liutex X. (d) Vorticity X. (e) Liutex Y. (f) Vorticity Y. (g) Liutex Z. (h) Vorticity Z

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boundary layers separated from the streamlined tail. And turbulent eddies have to be rotated and strained by the strong shears, thus resulting in prominent turbulent characteristics.

6  Conclusions A flow field around a circular cylinder at subcritical Reynolds numbers has been numerically calculated by the SST-PANS model simulation, and the vortex structure in the near wake detailedly discussed as a focus. The results show that compared with Q criterion, the accuracy of vortex recognition results of Omega-Liutex method is not affected by the selection of subjective thresholds, and it can identify strong and weak vortex structures at the same time, with better recognition results. For further work, simulations for the higher Reynolds numbers with the SST-PANS model and Liutex vortex identification method should be performed. Acknowledgements  This work is supported by the National Natural Science Foundation of China (51879159), The National Key Research and Development Program of China (2019YFB1704200, 2019YFC0312400), Chang Jiang Scholars Program (T2014099), and Innovative Special Project of Numerical Tank of Ministry of Industry and Information Technology of China (2016-23/09), to which the authors are most grateful.

References 1. S. Tian, Y. Gao, X. Dong, C. Liu, Definitions of vortex vector and vortex. J. Fluid Mech. 849, 312–339 (2018). https://doi.org/10.1017/jfm.2018.406 2. C. Liu, Y. Gao, S. Tian, X. Dong, Rortex—A new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids 30(3), 035103 (2018). https://doi.org/10.1063/1.5023001. 3. Y. Gao, C. Liu, Rortex and comparison with eigenvalue-based vortex identification criteria. Phys. Fluids 30(8), 085107 (2018). https://doi.org/10.1063/1.5040112. 4. 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. Hydrodyn. 31(3), 464–474 (2019). https://doi.org/10.1007/s42241-­019-­0032-­2 5. X. Dong, Y. Gao, C. Liu, New normalized Rortex/vortex identification method. Phys. Fluids 31(1), 011701 (2019). https://doi.org/10.1063/1.5066016. 6. C. Liu et al., Third generation of vortex identification methods: Omega and Liutex/Rortex based systems. J. Hydrodyn. 31(2), 205–223 (2019). https://doi.org/10.1007/s42241-­019-­0022-­4. 7. H.  Xu, X.  Cai, C.  Liu, Liutex (vortex) core definition and automatic identification for turbulence vortex structures. J.  Hydrodyn. 31(5), 857–863 (2019). https://doi.org/10.1007/ s42241-­019-­0066-­5 8. J. Liu, C. Liu, Modified normalized Rortex/vortex identification method. Phys. Fluids 31(6), 061704 (2019). https://doi.org/10.1063/1.5109437. 9. W. Zhao, D. Wan, Benchmark for detached-eddy simulation of flow past tandem cylinders, Elev. Asian Computational Fluid Dynamics Conference, Sept 16–20, 2016, Dalian, China (2016), pp. 397–401.

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10. W.  Zhao, L.  Zou, D.  Wan, Z.  Hu, Numerical investigation of vortex-induced motions of a paired-column semi-submersible in currents. Ocean Eng. 164, 272–283 (2018). https://doi. org/10.1016/j.oceaneng.2018.06.023 11. A. Elmiligui, K. Abdol-Hamid, S. Massey, S. Pao, Numerical Study of Flow Past a Circular Cylinder Using RANS, Hybrid RANS /LES and PANS Formulations (2004). https://doi. org/10.2514/6.2004-­4959 12. F.S.S. Pereira, G. Vaz, L. Eça, S.S.S. Girimaji, Simulation of the flow around a circular cylinder at Re=3900 with Partially-Averaged Navier-Stokes equations. Int. J. Heat Fluid Flow 69, 234–246 (2018). https://doi.org/10.1016/j.ijheatfluidflow.2017.11.001 13. B. Cantwell, D. Coles, An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder. J. Fluid Mech. 136(1), 321 (1983). https://doi.org/10.1017/ S0022112083002189.

Chapter 21

Vortex Identification Study of Flow Past Stationary or Oscillating Cylinder Weiwen Zhao and Decheng Wan

Abstract  In this paper, vortex identifications for turbulent flow past a stationary and oscillating cylinder is performed on the three-dimensional velocity field obtained by delayed detached-eddy simulation. The Reynolds number of the flow based on the cylinder diameter is 41,750. For the oscillating case, the moving boundary and the motion of the cylinder is archived by Arbitrary Lagrangian-­ Eulerian method. Third generation vortex identification methods, namely Liutex vector and the Omega-Liutex method are presented to understand the coherent turbulent flow structures. Quantitative flow variables such as drag and lift coefficients, pressure on the cylinder surface are also presented. Keywords  Vortex identification · Cylinder flow · VIV · Rortex · Liutex

1  Introduction Cables and risers in ocean engineering can subject to strong ocean currents. The vibrations induced by vortex are critical issues for these flexible slender bodies due to the potential accelerations of fatigue damage. For the sake of simplicity, the complex physical phenomena can be reduced to a physical model of flow past a circular cylinder. During the past several decades, flow over a cylinder has been extensively studied both experimentally and numerically. Comprehensive reviews can be referred to Williamson’s group work [1, 2]. For laminar flow at a few hundred Reynolds number without flow separation, the physics have been well studied and W. Zhao Computational Marine Hydrodynamics Lab (CMHL), School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China D. Wan (*) Computational Marine Hydrodynamics Lab (CMHL), School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China Ocean College, Zhejiang University, Zhoushan, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_21

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understood. At high Reynolds number, however, there still requires much efforts to understand the flow instability, shear layer, vortex generation and shedding established. The Re = 3900 case is very representative for subcritical cylinder flow. It has been a classical test case for different turbulence modeling methods. Ma et al. [3] performed direct numerical simulation (DNS) of cylinder flow at Re = 3900. The mean velocity profiles and power spectra in both near wake and far downstream regions are in good agreement with experimental measurements. Studies based on large eddy simulations (LES) were also carried out by Moin et  al. [4–6] and Parnaudeau et al. [7]. Moin’s group is pioneer of LES studies for cylinder flow at Re = 3900. In their numerical studies, the mean velocity and Reynolds stress profiles are in good agreement with experimental data in the far wake region. They discussed the numerical dissipation of upwind and central differencing discretization schemes and found out the low-dissipation central differencing scheme gives more accurate power spectra of velocity at high wave numbers. However, their results show large discrepancy of velocity profile shapes in the near wake regions. This is believed to be attributed to the experimental error of the original data. In the paper published by Parnaudeau et al. [7], they not only performed LES simulations, but also conducted experimental investigations. The near wake region velocity profiles between numerical simulations and experimental measurements show no differences. Both are in good agreement with previous LES results of Kravochenko and Moin [6] and DNS results of Ma et al. [3]. Recent studies [8, 9] also increase the confidence level of the experimental data from Parnaudeau et al. [7] Higher Reynolds cylinder flows have also been investigated despite it’s more challengeable. Breuer [10] investigated cylinder flow at Re = 140,000 to evaluate the applicability of LES for practically relevant high-Re flows. He also discussed the influence of subgrid scale modeling and grid resolution on the quality of the predicted results. However, the grid refinement does not show improvement of the result compare to experimental data. Travin et  al. [11] performed detached eddy simulation (DES) for cylinder flow at Re = 50,000, 140,000 and 3 × 106, respectively. The objective is to assess the applicability of DES methods to the unsteady massively separated flows. Similar to Breuer’s [10] observations, they found out that grid refinement does not always improve the results. For cylinder subject to vortex-induced vibrations (VIV), a comprehensive overview can be found in the publications of Sarpkaya [12, 13], Bearman [14, 15], Williamson and Govardhan [2], Wu et  al. [16]. Blackburn et  al. [17] presented results from low Reynolds number experimental data of VIV and compare the results with 2D and 3D DNS. They concluded that 3D simulations are essential in order to capture the correct structural response. In recent years, the flexible cylinder subject to VIV gained more and more attentions [18, 19] especially in offshore oil industry. Tognarelli et al. [20] performed experiments of a long flexible riser (with aspect ratio 481.5) in uniform and linearly sheared currents to study the VIV features in different flow conditions. Their results showed that the vibration mode appears as standing wave for uniform current, while it becomes a much more complex motion which behaves as the combination of standing and travelling waves. As for numerical

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simulations, fully 3D numerical model can accurately predict the detailed flow field along the spanwise direction of the long cylinder [21, 22]. However, this method requires that all the spanwise should be covered by computational mesh. For large aspect ratio flexible cylinders, the computational cost is unaffordable. To reduce computational cost, the strip model was proposed [23, 24]. In this model, the flow is solved on a set of domains along the spanwise directions. At each spanwise location, the flow is solved in 2D domain instead of 3D. In this paper, we focus on the cylinder flow at Re = 41,750. Flow past stationary and oscillating cylinder are investigated numerically. The paper is organized as follows. First, the numerical methods and models are introduced. Then the case conditions and numerical results are presented and discussed. Finally we drawn our conclusion.

2  Numerical Models 2.1  Turbulent Flow Modeling Considering the incompressible turbulent flow past a rigid circular cylinder. The governing equations of the flow in Arbitrary Lagrange-Eulerian (ALE) frame can be written as:

∇·u = 0

(21.1)



∂u 1 + ∇· u ( u − u g ) = − ∇p + ∇·( 2ν eff S ) ∂t ρ

(21.2)

(

)

where, u is velocity vector field, ug is grid velocity. p is pressure field. ρ is flow density. νeff = ν + νt is the effective viscosity, with ν the molecular viscosity and νt the eddy viscosity. S is the mean strain rate tensor. We employ the k − ω SST delayed detached eddy simulation (DDES) for the modeling of turbulent flow [25]. This is an hybrid RANS-LES model, which behaves like a LES subgrid scale (SGS) model in the separated flow region, and acts as an k − ω SST model in the near wall and other regions. With such a modeling strategy, the grid number in boundary layer is reduced and at the same time the accuracy is maintained. The switching between RANS and LES is controlled by a modified turbulent length scale, which is defined as the minimum of the calculated length scale from RANS model and the local grid scale [25] lDDES = lRANS − fd max ( 0,lRANS − C DES ∆ )

(21.3) in which, fd is a “delay” function to protect the boundary from earlier separation and CDES is an empirical coefficient called DES constant. Details about the formula of the functions and the values of the constant coefficients can be referred to Gritskevich et al. [26]



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2.2  Dynamic Deforming Mesh In the convection term of the momentum equations, the grid point velocity ug represent the deforming rate of the grid. To obtain the grid velocity, we solve the following Laplacian equations

γ∇x g = 0





(21.4)

1 where γ = is the inverse of distance to the moving boundary, xg is the grid point r displacement. After solving the Laplacian equations, the grid velocity is then obtained by dividing grid point displacement by time.

2.3  Vortex Identification Methods To study the behavior of vortices, the vortex structures should first be identified and extracted from the flow field. In this paper, we adopts the vortex identification methods proposed by Prof. Liu’s group at the University of Texas at Arlington [27– 31]. According to Liu et al. [32], the vorticity of fluid can be divided into vortical and non-vortical parts. Based on this concept, they proposed several methods, such as Omega method [32], Liutex/Rortex method [27, 29], normalized Liutex/Rortex method [30] and modified normalized Liutex/Rortex method [31]. In this paper, we use the modified normalized Liutex/Rortex method, defined as

( ω·r )

˜

ΩR =

2

2 ( ω·r ) − 2λci2 + 2λcr2 + λr2  + ε   2

(21.5)

The above equation defines a parameter in the regions where the velocity gradient tensor has two complex and one real eigenvalues. λci λcr is the imaginary and real part of the complex eigenvalues. λr is the real eigenvalue. ω is the vorticity vector and r is the Liutex vector. ε = b0 max (β2 − α2) is a small parameter to avoid division by zero and b0 is a small positive number around 0.001–0.002.

3  Flow Case Conditions We first introduce the geometric parameters of the circular cylinder. The diameter of the cylinder is 0.125 m. The aspect ratio, or the length over diameter ratio, is 4. Two configurations are considered. In the first configuration, the cylinder is stationary. While in the second one, the cylinder is oscillating in the cross flow direction. The oscillation can be expressed by

21  Vortex Identification Study of Flow Past Stationary or Oscillating Cylinder



y = Y0 sin ( 2π f0 t )



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(21.6)

The current velocity is 0.334  m/s, corresponding to a Reynolds number of 41,750. For the oscillating case, the amplitude is Y0 = 0.3D and the frequency is f0U/D = 0.2. The averaged/filtered continuity and momentum equations are solved on a collocated cell-centered unstructured grid. We choose different discretization schemes for different terms in different equations. For temporal term, we employ a threelevel second-order backward Euler scheme for discretization. The convection terms in momentum equations are discretized by linear upwind scheme with stabilization (LUST). The convection terms in turbulent quantity transport equations are discretized using a limited linear scheme. The coupled velocity-pressure is solved by the standard PISO algorithm with three corrector loops in each time steps. The computational domain extends from -10D at inlet to 20D at outlet in the inline direction, and from -6D to 6D in the cross-flow direction. The cylinder is placed at the origin of the Cartesian coordinate. Figure 21.1 shows the top view and side view of the computational domain and mesh. We use an octree cutting method to refine the grid. The initial mesh consists of uniform spacing hexahedral cells. Up to four levels of cell splitting is achieved in the vicinity and behind the cylinder. In order to resolve accurate velocity profile in the inner boundary layer, eight layers of boundary mesh are added to the cylinder surface. The non-dimensioned wall distances of the first layer cells satisfy y+   ϵ, w

1 i , j ,k − 2

=w

1 i , j ,k − 2

− ∆t ⋅ patm

α i , j ,k − α i , j ,k −1 ∆z

⋅

(22.9)

An 2-D example is given to illustrate the efficiency of the above approach in generating breaking waves. In this case, A1/L = 0.02, L = 2π, Tf = 4π, Uw =  − 1, k1  =  1,  w1  =  1. By implementing above discretization method, the resultant patm curves and wave profiles can be obtained, as shown in Fig. 22.2. Apparently, present

Fig. 22.2 (a) patm curves at different time instants during Tf, (b) corresponding wave profiles induced by patm (A1/L = 0.02)

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method gives a smooth evolution of a steep wave. Noted for all of the 2-D simulations in this work, initial random disturbance is out of our consideration. The Froude number is defined by Fr = U ′ gL ′ , where U′, L′ represent the characteristic velocity and length, respectively. Wave breaking phenomenon involves a broad range of relevant length and time scales, Therefore it is challengable to be fully resolved by traditional simulators. In this study, a parallel solver based on block structured adaptive mesh refinement (BAMR) [10] is applied to study the highly complex hydrodynamic phenomenon in surf zone, by which the breaking of complex fractioning, coalescence of bubbles and droplets can be captured numerically. To avoid the non-physical velocity arising from the high-density-ratio flows, a mass-momentum consistent scheme based on sharp interface model and CLSVOF method is developed. Time discretisation of the momentum equation is explicit and an second order Adams-Bashforth scheme is used. The velocity–pressure coupling by the incompressible flow constraint is decoupled with the fractional-step pressure correction method [11]. The equations are discretised on a staggered grid by means of the conservative finite difference scheme. The space derivatives of the convective term are discretised by a fifth-order WENO (weighted essentially non oscillation) scheme, whereas the viscous term is approximated by a second order centred scheme. An algebraic multigrid method is adopted for solving Poisson equation of pressure iteratively.

3  Result and Discussion In this section, a series of wave breaking cases are simulated with numerical methods mentioned above. For all the simulations in this study, Froude number Fr = 1 with U′ = 1, L′ = 1 is considered. The length of the computational domain is L = 2π, water depth and the height of the air are equal to π, the width of the domain is 0.5π. For all of the 2-D/3-D cases, adaptive mesh with 2–7 refinement levels are used, corresponding to a minimal grid resolution Δh = L/512. All the simulation cases are performed until dimensionless time t = 40 with a constant CFL = 0.2 is used. The parameters of the atmospheric pressure patm in (22.6) are Uw  =    −  1, k1  =  1, w1 = 1, Tf = 4π, and ε = 0.1A1. The periodic boundary condition in the wave propagation direction is enforced for all the simulation. According to Brucker et al. [8], four types of wave breaking are found, the weak plunging (WP), plunging (P), strong plunging (SP), and very strong plunging wave. Therefore, five simulation cases with initial setup of A1/L  =  0.017, 0.018, 0.019, 0.02, 0.022 are performed. According to our numerical tests, for A1/L = 0.017, no plunging breaking wave is generated, in other cases (A1/L = 0.018~0.022) plunging waves, from WP to SP waves are obtained. First, preliminary 2-D simulations are performed to study the vortex detachment phenomenon around the free surface. Figure  22.3 shows the vorticity field with wave profiles at three times of the plunging breaking cases (A1/L = 0.2). Each of the

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Fig. 22.3  Instantaneous vorticity distribution of the air phase at the initial stage of wave breaking events (A1/L = 0.02). (a) t = 5. (b) t = 10. (c) t = 15

frames has been given to correspond to the following stages in the breaking events. Figure 22.3a: In the initial stage, a very thin vorticity layer is generated attached to the free surface. The vorticity keeps invariable until t = 7. Figure 22.3b: the boundary layer at the rear part (down wind side) of the wave peak starts to become unstable and produces vertical dipole structures. Figure 22.3c: More vortices are released and roll up subsequently and detach continuously from the free surface. In terms of the water phase, the six frames in Fig. 22.4 correspond to the vortex evolution stages in the breaking events are summarized as follows. Figure 22.4a: Initially, vorticity magnitude is too low to be observed. Figure 22.4b: Similar vortex dipole are shown with much lower magnitude than in air phase. Figure 22.4c: Vortex pairs attach to the free surface are extruded gradually. Figure  22.4d–f: After the wave breaking occurs, complex vortex interactions with high vorticity magnitude caused by the turbulent two-phase mixing are shown. Finally, to demonstrate the performance of the BAMR solver for 3D cases, a 3D case considering A1/L = 0.02 is carried out. The breaking process can be summarized as following: First, a jet is generated along the wave propagating direction, then it impacts the water surface due to gravity. At the initial stage, wave breaking exhibit 2-D flow dynamics pattern, as in Fig. 22.4a–c. After jet-impacting occurs, the entrapment of air cavity brings large number of bubbles, fully 3-D turbulent pattern is formed. Figure 22.5a–d shows the free surface visualized by the α = 0.5 of the volume fraction at four time instants for the plunging breaking cases. It is noted that the bubbles stay for a longer duration than the appearance of droplets, even the free surface recovers stable state, the entered air-cavity still exists. For A1/L = 0.02 case, the volume of entrapped air increase sharply after the jet impaction, and reach the peak at t ≈ 1.43Tf. Afterwards, bubbles driven by buoyancy emerge from the water surface gradually, corresponding air volume decrease to zero with t ≈ 3.9Tf. The vortex filament generated by pluging is one of the essential breaking-wave phenomenon available for direct observation. This vortex structure was obtained

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Fig. 22.4  Instantaneous vorticity distribution of the water phase during the whole wave breaking events (A1/L = 0.02). (a) t = 5. (b) t = 10. (c) t = 15. (d) t = 20. (e) t = 30. (f) t = 40

from simulation by Lubin and Glockner [12] and discussed. In present work, a third generation of vortex identification methods: a noval Liutex approach [13] is adopted for visualization of the vortex structure in breaking wave. As shown in Fig. 22.6a–c, vortex structure of A1/L = 0.02 pluging wave in three instants is given. At the initial stage of breaking events, aligned vortex filament is created by the entrainment of air cavity (Fig.  22.6a). Subsequently, strong turbulent mixing of bubbles and water disturb the regular vortex structure (Fig. 22.6b). From Fig 22.6b–c, it is observed that the turbulent flow region exhibits aggregation and reducing slowly with the stability of flows. HPC cluster in our lab with 28 physical cores (Intel Xeon E5-4627, 3.4 GHz) are in use for present parallel computation. For A1/L = 0.02 wave breaking case (3-D), 31 h are consumed for a 5-wave period simulation.

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Fig. 22.5  Evolution of the free surface at the middle and later stage of wave breaking events (A1/L = 0.02, side view). (a) t = 10. (b) t = 20. (c) t = 30. (d) t = 50

4  Conclusions In present study, plunging wave in deep water is under preliminary 2-D and 3-D simulations. From the numerical results, following conclusions are drawn. First, the water phase adjacent to interface region produces dipole vortex structures with about 1/10 vorticity magnitude compared with the air phase. Second, after jet plunging, entrapped air experiences a sharp increment and stays in water for a long duration until t ≈ 3.9Tf for a A1/L = 0.02 breaking wave. Many characteristic variables during wave breaking, including the void fraction, bubble size distribution are expected to be presented laterly. The fast algorithm for two-phase flow simulations, which can greatly improve the computational efficiency, will be considered in future. Acknowledgement  This work is supported by the National Natural Science Foundation of China (11902199, 51979160, 51879159), Shanghai Pujiang Talent Program (19PJ1406100), The National Key Research and Development Program of China (2019YFB1704204, 2019YFC0312400), Chang Jiang Scholars Program (T2014099), Shanghai Excellent Academic Leaders Program (17XD1402300), and Innovative Special Project of Numerical Tank of Ministry of Industry and Information Technology of China (2016-23/09), to which the authors are most grateful.

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Fig. 22.6 Instantaneous free surface and vortex structure generated by Liutex approach (A1/L = 0.02, bottom view, vortex tubes are colored by velocity magnitude). (a) t = 10. (b) t = 20. (c) t = 30

References 1. F. Veron, C. Hopkins, E.L. Harrison, et al., Sea spray spume droplet production in high wind speeds. Geophys. Res. Lett. 39, 16 (2012) 2. Z. Tian, M. Perlin, W. Choi, Evaluation of a deep-water wave breaking criterion. Phys. Fluids 20, 6 (2008) 3. G.B.  Deane, M.D.  Stokes, Scale dependence of bubble creation mechanisms in breaking waves. Nature 418(6900), 839–844 (2002) 4. A. Saket, W.L. Peirson, M.L. Banner, et al., Wave breaking onset of two-dimensional deep-­ water wave groups in the presence and absence of wind. J. Fluid Mech. 811, 642–658 (2015) 5. J. Song, M.L. Banner, On determining the onset and strength of breaking for deep water waves, Part 1: Unforced irrotational wave groups. J. Phys. Oceanogr. 32, 2541–2558 (2002) 6. M.L. Banner, W.L. Peirson, Wave breaking onset and strength for two-dimensional deep-water wave groups. J. Fluid Mech. 585, 93–115 (2007) 7. P.  Lubin, S.  Vincent, S.  Abadie, et  al., Three-dimensional Large Eddy Simulation of air entrainment under plunging breaking waves. Coast. Eng. 53(8), 631–655 (2006) 8. K.A.  Brucker, T.T.  O’Shea, D.G.  Dommermuth, P.  Adams, Three-dimensional simulations of deep-water breaking waves, in Proceedings of the 28th Symposium on Naval Hydrodynamics (2010)

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9. C.Q.  Liu, Y.Q.  Wang, Y.  Yang, et  al., New omega vortex identification method. Sci. China Phys. Mechan. Astron. 59(8), 684–711 (2016) 10. C.  Liu, C.H.  Hu, An adaptive multi-moment FVM approach for incompressible flows. J. Comput. Phys. (2018) 11. C.  Liu, C.H.  Hu, An efficient immersed boundary treatment for complex moving object. J. Comput. Phys. 274, 654–680 (2014) 12. P. Lubin, S. Glockner, Numerical simulations of three-dimensional plunging breaking waves: generation and evolution of aerated vortex filaments. J. Fluid Mech. 767, 364–393 (2015) 13. C.Q.  Liu, Y.S.  Gao, X.R.  Dong, et  al., Third generation of vortex identification methods: Omega and Liutex/Rortex based systems. J. Hydrodynam. Ser. B 31, 205–223 (2019)

Chapter 23

Numerical Investigation of Complex Flow Field in Ship Self-Propulsion and Zigzag Maneuverability Zhen Ren, Weiwen Zhao, and Decheng Wan

Abstract  During the last decades, there are many proposed vortex identification methods, including Q, Δ, λ2 and λci criteria. However, these methods are based on the Cauchy-Stokes decomposition and/or eigenvalues of the velocity gradient. And these methods are not able to capture both strong and weak vortices simultaneously. In the present study, the third generation of vortex identification, liutex, is adopted to capture the vortex structure in the flow field of ship self-propulsion. The method can capture both strong vortices and weak vortices, and the six core issues for vortex definition and identification has been raised. At the same time, the second generation of vortex identification method, Q criterion, is applicated, and the results are compared with that obtained by using the liutex method. In the numerical simulations, the self-propulsion and zigzag manoeuvrability of KCS with a Lpp length of 6.0702  m are carried out, couping with KP505 propeller and the rudder with NACA0018 section. And the Froude number is 0.26. The vortex structure in both cases are present. Firstly, the vortex structure in the self-propulsion is analyzed. Both resluts obtained by liutex method and Q criterion are compared. And the same comparison of calcluted results is carried out in zz10/10 manoeuvrability. The results show that the liutex method better presents the size and direction of the vortex structure.

Z. Ren Computational Marine Hydrodynamics Lab (CMHL), School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China W. Zhao Ocean College, Zhejiang University, Zhoushan, China D. Wan (*) Computational Marine Hydrodynamics Lab (CMHL), School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China Ocean College, Zhejiang University, Zhoushan, China e-mail: [email protected]; https://dcwan.sjtu.edu.cn/ © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_23

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Keywords  Liutex · Q · Vortex identification · Self-propulsion · Zigzag manoeuvrability

1  Introduction Vortex structures are widely found in naval architecture and ocean engineering. There are a large number of vortex structures with different scales and intensities around the hull, propeller and rudder. Accurate identification of vortex structures is important for understanding the flow mechanism of the flow field around hull-­ propeller-­rudder. The first generation of vorticity-based vortex identification method is completely mathematically correct. But the physical meaning is not very clear [1, 2]. Vortex structure cannot be accurately captured based on vorticity theory. Later, researchers [3–6] have proposed the second generation of vortex identification methods represented by Q, λ2, Δ, and λci. But there are still many problems to identify the vortex structure. In response to the problems of the first and second generation vortex recognition methods, Liu et  al. [7] proposed the third generation of vortex identification method. The first and second generation of vortex identification methods are currently used in various fields. Guilmineau et  al. [8] used the DES and RANS models to analyze the open water performance of propeller, INSEAN E779A model. Q criterion and λ2 methods are adopted to display the vortex structure. Posa and Balaras [9] presented the vortex structure around the DARPA SUBOFF model by using LES method. In their study, the quasi-streamwise vortices are captured with 3.5 billion cells in the numerical simulations. Ren [10] and Wang [11] utilized the vorticity-­ based vortex identification and Q criterion to analyze the bow wave breaking and the flow mechanism of the overturning and breaking of the bow wave of KCS model. Olivieri et al. [12] analyzed the vorticity during the bow wave breaking of DTMB 5415. The predicted results are in good agreement with the experiments. To study the flow mechanism around hull-propeller-rudder, vortex structures are always presented in the numerical simulations. Shen et  al. [13, 14] displaied the vortex structure around the hull-propeller-rudder in the numerical calculations of self-propulsion and zigzag maneuver of KCS model. In their studies, Q criterion and vorticity-based vortex identification method are also used. Wang et al. [15, 16] also used the first and second generation of vortex identification method to show the vortex structure around the ONRT ship model with twin rudder and twin screw. The flow mechanism of complex wake field was revealed further. In the present study, the numerical simulation of self-propulsion and standard zigzag maneuver of KCS are carried out. The paper is organized as follows. Numerical methods include the basic numerical schemes and the brief introduction of three generation of vortex identification methods. Then the geometry model of the hull, propeller and rudder and grid generation are presented. The vortex struc-

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tures of self-propulsion and zigzag maneuver simulation are analyzed and discussed. Finally, a brief conclusion is drawn.

2  Numerical Method 2.1  Basic Numerical Scheme In the present study, the CFD solver, naoe-FOAM-SJTU, is used, which is deve loped based on the open source platform OpenFOAM. The self-propulsion and zigzag maneuver can be carried out by this solver because that the dynamice overset mesh and 6DOF motion are also introduced. In the present numerical calculations, RANS equations are solved by coupling with shear stress transport tubulence model, SST k-ω model. The volume of fluid (VOF) approach is adopted to capture the free surface.

2.2  Vortex Identification Methods First Generation of Vorticity-Based Vortex Identification Methods In 1858, the concepts of vorticity was proposed by Helmholtz. The first generatioin of vortex identification method is based on the definition of vorticity. The vorticity represents the mathematical definition of velocity curl, ▽ × ν. But it is not very clear in terms of its physical meaning. Generally speaking, it is considered to be twice the rotational angular velocity of the rigid body rotation angular velocity of the fluid mass based on the motion analysis of fluid mass and Cauchy-Stokes decomposition. The Cauchy-Stokes decomposition [2] is expresses as:



1 V ( r + dr ) = V ( r ) + Adr + ω × dr 2

(23.1)

According to the decomposition, the movement of fluid mass can be decomposed into three parts: translation, the deformation represented by the symmetrical tensor A and rigid body rotation represented by vorticity ω. Although the last equation is completely mathematically correct, it is obviously problematic to simply understanding the vorticity ω as the rigid body rotating part of the fluid mass movement.

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Second Generation of Eigenvalue-Based Vortex Identification Methods Obviously, vorticity and vortex are two different concepts, so the first generation of vorticity-based vortex identification was not satisfactory. In order to identify vortex structures more effectively, some methods such as Q, λ2, Δ, and λci were proposed. Although their theories were different, the Q and λ2 methods was the modifications of the Cauchy-Stokes decomposition and the Δ and λci methods were based on the effect of the velocity gradient tensor ▽V on the local instantaneous streamlines. Here is a brief introduction of the Q criterion. The Q criterion proposed by Hunt et al. is a very popular vortex identification method. It is one of the invariants of the velocity gradient tensor ▽V and can be expressed as [2]. Q=



1 2 BF − AF2 2

(

)

(23.2)

where A and B represent the symmetric and antisymmetric parts of the velocity gradient tensor, resprectively. Generally, an artificially given threshold is required in the second generation of vortex identification method. Obviously, this threshold plays an important influence on the display of the vortex structure. Third Generation of Vortex Identification Methods In order to overcome the problems of the second generation of vortex identification methods, Liu et  al. proposed the third vortex identification method: Omega and Liutex/Rortex. The Omega method was also derived from Cauchy-Stokes decomposition, and the vorticity ω is further decomposed into a rotating part and a non-rotating part. The Ω [2] can be expressed as Ω=

BF2 AF2 + BF2 + ε

(23.3)

Obviously, Ω is between 0 and 1, which can be understood as the concentration of vorticity. More specifically, it represents the rigidity of the fluid motion. When Ω = 1, it means that the fluid rotates as a rigid body.Ω > 0.5 indicates that the anti-­ symmetric tensor B is dominant over the symmetric tensor A. So Ω = 0.51 or 0.52 are always selected as a fixed threshold to identify the vortex structure. In the Omega vortex identification method, ε is a small positive number used to avoid the division by zero. Dong et al. proposed a method for selecting ε, which defined as a function of BF2 − AF2 [2].

(



)

max

ε = 0.001 × ( BF2 − AF2 )

max



(23.4)

The Omega vortex identification method advoids the problem to artificially adjust the threshold.

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According to the Cauchy-Stokes decomposition, the antisymmetric tensor B cannot represent the rigid body part of fluid motion. The Liutex/Rortex vector ­systematically solves the problem of how to obtain a rigid rotating part from the fluid motion. The Liutex vector is defined as

(

)

R = ω ,r − ω ,r 2 − 4λci2 r



(23.5)

where ω is the vorticity vector. Note that, being different from all other vortex identification methods, to define Liutex vector, not only the eigenvalues of the velocity gradient tensor ▽V, but the eigenvectors of ▽V and the local vorticity are also used; Liutex vector not only represents the rotation intensity, also the local rotation axis; Compared with Cauchy-­ Stokes decomposition, Liutex vector provides accurate motion decomposition to capture the rigid rotation part in the flow field.

3  Geometry and Grid Generation 3.1  Geometry Model In the present simulations, the hull of KCS is the 6.0702 m with the KP505 propeller and rudder. Figure 23.1 shows the geometry of the hull, propeller and rudder. And the advance speed of the ship is 2.006  m/s. In the present study, the self-­ propulsion and 10/10 zigzag maneuver are performed. And extensive experiments have been carried out by MARIN.

3.2  Grid Generation The unstructured grid adopted in the numerical calculations are generated by the commercial software, HEXPRESS. are presented in Fig. 23.2 shows the computational domain and boundary condition. Due to overset grid method, the overlap

Fig. 23.1  Geometry model of KCS

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Fig. 23.2  Computational domain and boundary condition

domains of hull, propeller and rudder are selected, as shown in Fig. 23.3. The grids of each domain are generated, respectively. The total grid number of the ­computational domain is about 6.07 million. Figure  23.4 presents the local grid distribution of hull, propeller and rudder.

4  Results and Analysis 4.1  Presentation of Vortex in Self-Propulsion In order to compare different vortex identification methods, Fig.  23.5 shows the vortex structures obtained by using four vortex identification methods in the simulation of self-propulsion. As we can see, the vortex structure obtained by the first vortex identification methods is worse than the other results, with the magnitude of vorticity is 75. The hull, propeller and rudder are covered fullly by the vortex structure. Compared with the vortex structure obtained by vorticity-based vortex identification methods, more complex vortex structure is captured by Q criterion, which the threshold is set as 50 artificially. The vortex structure is more regular and longer. On the other hand, the hull and rudder are covered partially by the vortex structure. The vortex structure obtained by OmegaR is similar to the results captured by Q criterion. But the threshold is not selected artificially. The results presented by Liutex are different from the others. A lot of breaking vortex are captured by this third generation of vortex identification method. Since the vortex structure varies drastically around the propeller and rudder, Fig. 23.6 shows the development and evolution of the vertical vorticity and Liutex on the horizontal plane passing through the propeller shaft. The vorticity presented by first generation is very regular, clear and continuous. The distribution of maxi-

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Fig. 23.3  Boundary of overlap domain

mum and minimum of vorticity is very concentrated. Compared with the results from the first generation, the regularity and continuity of Liutex are weaker. Figures 23.7 and 23.8 show the Liutex and vorticity distribution on the plane at x/L = 0.9774 and x/L = 1.02, respectively. The plane at x/L = 0.9774 is in front of the propeller disk. And the plane at x/L = 1.02 is after the rudder. Overall, the distribution of vorticity is more regular and clearer than the Liutex. The distribution of Liutex is more disordered and random. In front of the propeller disk, the distribution of axial Liutex seems to be opposite to that of axial vorticity. The negative Liutex mainly occurs on the portside and the positive Liutex is concentrated on the starboard. While the positive axial vorticity mainly appears on the portside and the negative is concentrated on the starboard. The lateral and vertical Liutex is more random than the axial results. The same disorder and random of the distribution of Liutex appeare on the plane after the rudder. The continuity of Liutex is much worse than the vorticity. But generally, it can be found that where a positive vorticity appears, a negative Liutex appears.

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Fig. 23.4  Local grid distribution

4.2  Presentation of Vortex in Zigzag Maneuvering In order to compare different vortex identification methods better, the vortex structures in the 10/10 zigzag maneuver are analyzed in the next. Figure 23.9 shows the time history of rudder angle and yaw angle for this case. A represents the second overshoot angle. The results presented in the next are at this point in time. Figure 23.10 shows the vortex structures presented by using the three generation vortex identification methods in the simulation of 10/10 zigzag maneuver. As we can see, the vortex structure obtained by the first vortex identification methods is longer than the results in the self-propulsion. But the hull, propeller and rudder are also covered fully by the vortex structure. The vortex structures obtained by Q

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Fig. 23.5  Vortex structure of self-propulsion identified by different vortex identification methods

Fig. 23.7  Liutex (top) and vorticity (bottom) distribution at x/L = 0.9774

Fig. 23.6  Vertical Liutex (third generation, left) and vorticity (first generation, right) distribution on the horizontal plane through the shaft

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Fig. 23.8  Liutex (top) and vorticity (bottom) distribution at x/L = 1.02

Fig. 23.9  Time history of rudder angle and yaw angle for 10/10 zigzag maneuver

c­ riterion and OmegeR are very similar. The vortex structure presented by Liutex are also more break and irregular than the others. Due to the steering, the vortex structure is longer than that in the self-propulsion. Due to the influence of rudder angle, the vortex structure varies. Figure 23.11 shows the development and evolution of the vertical vorticity and Liutex on the horizontal plane passing through the propeller shaft for the case zigzag maneuver. The vorticity presented by first generation is not symmetrical. The larger vorticity occurs on the portside of the rudder. And the vorticity develops downstream continuously. The vertical Liutex on the portside of rudder also changes, which is effected by the rudder angle. While the Liutex is negative and not continuous. Here

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Fig. 23.10  Vortex structure of 10/10 zigzag maneuver identified by different vortex identification methods

Fig. 23.11  Vertical Liutex (left) and vorticity (right) distribution identified by first and third generation of vortex identification methods on the horizontal plane through the shaft

it is more clearly shown that the region where the negative Liutex appears corresponds to the positive vorticity. Figures 23.12 and 23.13 show the Liutex and vorticity distribution on the plane at x/L = 0.9774 and x/L = 1.02, respectively. Overall, the distribution of vorticity is also more regular and clearer than the Liutex. Lots of vorticity also appears near the hull, while the Liutex is close to zero. Due to the posture of the hull, the distribution of Liutex and vorticity are oblique. In front of the propeller disk, the region where negative axial Liutex appears corresponds to positive axial vorticity. Since the flow field is effected by the rudder angle, numerous Liutex and vorticity appears on the starboard of rudder. At the root of rudder, a small region with the drastic ­vorticity/ Liutex appears. But generally, it can be found that where a positive vorticity appears, a negative Liutex appears.

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Fig. 23.12  Liutex (top) and vorticity (bottom) distribution at x/L = 0.9774

Fig. 23.13  Liutex (top) and vorticity (bottom) distribution at x/L = 1.02

5  Conclusions In the present numerical simulations, the RANS model is used to solve the flow field around the hull-propeller-rudder of KCS.  Dynamic overset method is applied to simulate the 6DOF motion of the hull, propeller and rudder. Four vortex identification methods, including vorticity-based (first generation), Q criterion (eigenvalue-­ based, second generation) and OmegaR and Liutex/Rortex (third generation) are adopted to present the vortex structure. By comparison, it is found that the vortex structure can be more conveniently and accurately display by using the third ­generation of vortex identification method. Using the OmegaR method does not require manual selection of thresholds. Tinier and break vortex structure are captured by Liutex method. The distribution of Liutex is more disordered and random by comparing with the results of vorticity. Generally, where a positive vorticity

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appears, a negative Liutex appears. In the future work, more vortex identification methods should be adopted to present the vortex structure. Acknowledgements  This work is supported by the National Natural Science Foundation of China (51879159), The National Key Research and Development Program of China (2019YFB1704200, 2019YFC0312400), Chang Jiang Scholars Program (T2014099), and Innovative Special Project of Numerical Tank of Ministry of Industry and Information Technology of China (2016-23/09), to which the authors are most grateful.

References 1. C. Liu, Y. Gao, X. Dong, et al., Third generation of vortex identification methods: Omega and Liutex/Rortex based systems. J. Hydrodyn. Ser. B 31(2), 205–223 (2019) 2. Y. Wang, N. Gui, A review of the third-generation vortex identification method and its applications. Chinese J. Hydrodyn. 34, 4 (2019) 3. J. Hunt, A.Wray, P. Moin, Eddies, Stream and Convergence Zones in Turbulent Flows. Center for Turbulent Research Report CTR-S88 (1988) 4. J. Jeong, F. Hussain, On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995) 5. M. Chong, A. Perry, A general classification of three- dimensional flow fields. Phys. Fluids A 2(5), 765–777 (1990) 6. J.  Zhou, R.  Adrian, S.  Balachandar, et  al., Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396 (1999) 7. C. Liu, Y. Gao, S. Tian, et al., Rortex-a new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids 30, 035103 (2018) 8. E. Guilmineau, G.B. Deng, A. Leroyer, et al., in Influence of the Turbulence Closures for the Wake Prediction of a Marine Propeller. Fourth International Symposium on Marine Propulsors smp’15, Austin, Texas, USA, June 2015 9. Antonio Posa, Elias Balaras, in Large-Eddy Simulations of the DARPA SUBOFF Model in Towed and Propelled Configurations. 31st Symposium on Naval Hydrodynamics Monterey, CA, 11–16 September 2016 10. Z. Ren, J. Wang, D. Wan, in Numerical Simulations of Ship Bow and Shoulder Wave Breaking in Different Advancing Speeds. The ASME 2018 37th International Conference on Ocean, Offshore and Arctic Engineering, Madrid, Spain, 17–22 June 2018 (2018) 11. J. Wang, Z. Ren, D. Wan, Study of a container ship with breaking waves at high Froude number using URANS and DDES methods. J. Ship Res. (2020) 12. A. Olivieri, F. Pistani, R. Wilson, et al., Scars and vortices induced by ship bow and shoulder wave breaking. J. Fluids Eng. 129, 1445–1459 (2007) 13. Z. Shen, D. Wan, P. Carrica, Dynamic overset grids in OpenFOAM with application to KCS self-propulsion and maneuvering. Ocean Eng. 108, 287–306 (2015) 14. Z. Shen, P. Carrica, D. Wan, in Ship Motions of KCS in Head Waves with Rotating Propeller Using Overset Grid Method. Proceedings of the 33rd International Conference on Ocean, Offshore and Arctic Engineering, San Francisco, CA, USA, July 8–13, 2014 15. J. Wang, L. Zou, D. Wan, CFD simulations of free running ship under course keeping control. Ocean Eng 141, 450–464 (2017) 16. J. Wang, L. Zou, D. Wan, Numerical simulations of zigzag maneuver of free running ship in waves by RANS-overset grid method. Ocean Eng. 162, 55–79 (2018)

Chapter 24

Application of Liutex for Analysis of Complex Wake Flows Characteristics of the Wind Turbine Yang Huang, Liushuai Cao, and Decheng Wan

Abstract  To further understand the complicated wake characteristics of the wind turbine under various conditions, the Liutex vector is applied to identify the vortex in the turbine wake. The actuator line model is used to predict the aerodynamic performance of wind turbine. By combining the actuator line model and finite element method, the aeroelastic responses of wind turbine are obtained. Moreover, the actuator line model is embedded into in-house CFD code naoe-FOAM-SJTU to predict the coupled aero-hydrodynamic performance of the FOWT. Different vortex identification methods including the Vorticity method, Q method and Liutex method are adopted to visualize the wake characteristics. The wake characteristics of wind turbine calculated from different simulations are discussed in detail. Several conclusions can be drawn from the simulation results and discussions. Compared with the Vorticity method and Q method, the Liutex vector can better describe the complex wake characteristic of wind turbine in various conditions. The wake vorticity distribution behind the wind turbine is found to be asymmetric. In addition, the wake characteristics of the FOWT become more uneven when the platform motions are taken into consideration, and significant wake expansion phenomenon is observed. Keywords  Wind turbine · Wake characteristics · Actuator line model · Liutex vector

Y. Huang · L. Cao Computational Marine Hydrodynamics Lab (CMHL), School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China D. Wan (*) Computational Marine Hydrodynamics Lab (CMHL), School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China Ocean College, Zhejiang University, Zhoushan, China e-mail: [email protected]; https://dcwan.sjtu.edu.cn/ © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_24

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1  Introduction Wind energy is believed to be one of the most promising renewable energy resources. With the rapid development of wind power industry, the wind turbine is developing in the direction of large capacity. The diameter of rotor sharply increases with the generator power of wind turbine [1, 2]. This leads to large structural deformation of the wind turbine blades and further change the aerodynamic performance of wind turbine. Moreover, the blade deformation complexes the interactions between the rotating blades and the turbine wake, which alters the wake characteristics and changes the inflow condition of the downstream wind turbine [3]. Thus, the aeroelastic responses of the wind turbine, especially the wake characteristics, should be carefully investigated. A number of aeroelastic models have been applied to study the coupled aeroelastic performance of the wind turbine. Meng et al. [4] proposed an elastic actuator line model (EAL) by combing the actuator line model (ALM) and the finite element method (FEM) for the modelling of aeroelastic responses of wind turbine. Ma et al. [5] developed an aeroelastic analysis tool ALFEM to study the wake characteristics of wind turbine with considering the structural deformation. To obtain detailed wake characteristics induced by the blade deformation, the computational fluid dynamic (CFD) method coupled with multi-body dynamics (MBD) model is also applied to perform aeroelastic simulations for the wind turbine [6]. It should be noted that the CFD method requires a lot of computational resources. The body force method, such as ALM mentioned above, needs fewer computational costs to obtain relatively accurate results. FEM method is widely applied to calculate the structural deformation of wind turbine. To simply the structural calculation, equivalent beam theory is usually adopted in the simulation [7]. The wind turbines are usually clustered in wind farm to convert wind power into electric energy. Limited by the space, wake interactions between wind turbines in the wind farm cannot be avoided. The aerodynamic performance of downstream wind turbine is significantly affected by the wake of upstream wind turbine. The wake interactions further enhance the instability of the power output of wind turbine. The wake development and the complicated wake characteristic in the wake farm show great effects on the annual power output of the wind farm. Therefore, it is essential to study the wake interactions between the wind turbines. The wake development is also important for the wind farm layout optimization. Many researches are conducted to focus on the complicated wake characteristics in the wind farm. Choi et al. [8, 9] investigated the wake interactions between three 2-MW wind turbines with tandem layout. The dynamic responses of the wind turbine including the aerodynamic loads and the wake velocity field were analyzed. Moreover, the influence of inter-turbine spacing on the wake interactions were also studied. Fletcher and Brown [24] studied the wake interactions between two wind turbines with different layouts using vorticity transport model. The influence of stream-wise and transverse distances between wind turbine on wake characteristics were discussed in detail. Moreover, Mikkelsen et  al. [25] used the actuator line model to detect the wake interaction phenomena between wind turbines. Wake

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development behind the wind turbine was observed. The effects of wake interactions on the rotor power and thrust was discussed in detail. In the wind farm, the wake interactions between wind turbines are complicated. It is changeable to capture detailed wake characteristics. Compared with onshore wind resources, the offshore wind is stronger and smoother. With the wind farm heading for the deep water, the floating offshore wind turbine (FOWT) is believed to be the development trend of wind turbine [10]. The FOWT is a rather complicated system, which is composed of a wind turbine, a floating platform and a mooring system. Interference effects between the wind turbine and the floating platform show significant effects on the coupled performance of the FOWT [11]. Due to the platform motion, there are complicated interactions between the rotating blades and the vortex structure, which further make the wake characteristics of the FOWT complex. In addition, the platform motions are also greatly altered by the aerodynamic loads. A number of researches have been conducted to investigate the complicated wake characteristics of the FOWT. Leble and Barakos [12] studied the wake development of DTU 10-MW wind turbine with pitching and yawing motions. The influence of platform motions on the aerodynamic loads were also discussed. The wake interactions between FOWTs become more complicated compared onshore wind turbines. The platform motions and the wake interactions both affect the wake development in downstream. To better capture the wake characteristics of the FOWT, model tests are conducted in the wind tunnel. The Particle Image Velocimetry (PIV) technique is applied to visualize the wake behavior [13]. Above all, the wake characteristics of wind turbine are rather complex and need to be further investigated. In order to better understand the vortex generated by the rotating blades and the wake behavior of wind turbine in different conditions, the third generation of vortex identification method Liutex [14] is utilized to capture the vortex structure and illustrate the flow characteristics. The Liutex vortex has been successfully applied in the analysis of various vortices [15, 16]. In the present work, the wake characteristics and the wake vortex of the onshore wind turbine and the FOWT are analyzed with various vortex identification methods. Moreover, the results obtained from different vortex identification methods are also compared and discussed to further understand the complex wake characteristics of wind turbine.

2  Numerical Method 2.1  Elastic Actuator Line Model In order to model the aeroelastic responses of wind turbine, aerodynamic loads and structural deformation both need to be calculated. In the present work, the actuator line model [17] is applied to calculate the aerodynamic loads of wind turbine, the finite method is used to predict the structural deformation of wind turbine blades. Moreover, the ALM is combined with the FEM to establish a coupled aeroelastic

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analysis model named elastic actuator line model (EAL) for the wind turbine. The coupling between the aerodynamic loads and structural deformation is achieved by data exchange. Actuator line technique is a simplified method to predict the aerodynamic loads and capture the three-dimensional wake characteristics of the wind turbine. Actual wind turbine blades are replaced with virtual actuator lines withstanding body forces. Thus, the boundary of wind turbine blades can be ignored in the simulation, and the computational resources can be greatly reduced. To obtain the body forces distributed on the virtual actuator lines, the local relative wind speed and local attack angle should be firstly calculated. Figure 24.1 shows the velocity distribution at the blade section, where the coordinate system is defined at (θ, z) plane. To take the influence of structural deformation into consideration, an additional velocity Us induced by the blade deformation is considered in the calculation of local attack angle and local relative wind speed. The relative wind speed Urel can be obtained by the following equation:

Urel = Uθ − Ω ⋅ r + U z + U s

(24.1)



where Uθ and Uz are projections of inflow wind speed in the inertial frame, respectively. Ω is the rotational speed. r represents the distance between the blade section and the hub center. The attack angel α is defined by the following equation:

α = φ − θt

(24.2)

where ϕ represents the inflow angle. θt is the local twist angle. Based on a tabulated two-dimensional airfoil data, the lift and drag coefficients can be obtained by linearly interpolating. Then the body force can be expressed as:



ρ cN b Urel f= 2rdrdz

2

(CL eL + CD eD )



(24.3)

where ρ is the air density. c represents the chord length at the blade section. Nb is the blade number of wind turbine. CL and CD are the lift coefficient and drag coefficient, respectively. eL and eD are the unit vectors along lift and drag forces, respectively. It

Fig. 24.1  Velocity distribution at the blade section in EAL

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is noted that the body forces need to be smoothed by a regularization kernel function in order to avoid singular behavior in the calculation. Finally, the body forces distributed in the flow field can be written as: N

fε ( x,y,z,t ) = ∑ f ( xi ,yi ,zi ,t ) i =1



  d 2  1 exp −    ε 3π 3 / 2   ε  

(24.4)

where d represents the distance between the actuator point and the gird point in flow field. ε is a constant parameter which is used to adjust the strength of regularization function. Three-dimensional Reynolds-averaged Navier-Stocks (RANS) equations with the k–ω SST turbulence model are adopted as the governing equations for the aerodynamic calculations of wind turbine.

∇⋅U = 0

(24.5)



∂U 1 + (U ⋅ ∇ ) U = − ∇p + ν∇ 2U + fε ∂t ρ

(24.6)

where U represents the velocity field. ρ donates the air density. ν is the kinematic viscosity coefficient. One-dimensional FEM is adopted to calculate the structural deformation of wind turbine blades. A two-node, four degree-of-freedom (DOF) beam element is used to discretize the blade. As shown in Fig. 24.2, the wind turbine blade is discretized into a series of elements. It is noted that only the deformation along flap-wise and edge-­ wise directions are taken into consideration. The structural governing equations are second-order ordinary differential equations:

[ M ]  x  + [C ][ x ] + [ K x ][ x ] = [ Fx ] ¨



 



[ M ]  y  + [C ][ y ] +  K y  [ y ] =  Fy 

(24.7)

¨



 

Fig. 24.2  Structural discretization model



(24.8)

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where [M] is the mass matrix; [C] is the damping matrix; [K] is the stiffness matrix. [x] and [y] represent the deformation along flap-wise and edge-wise direction, respectively. [Fx] and [Fy] represents the external forces including the aerodynamic forces, gravity forces and centrifugal forces. Moreover, Newmark-beta method is applied to solve the MCK equations. A two-way wake coupling method is applied to achieve the interface effects between the aerodynamic loads and structural deformation. The aerodynamic forces calculated by the actuator line model are considered in the MCK equations, and the calculation of aerodynamic loads take into account the structural deformation velocity.

2.2  Unsteady Actuator Line Model As introduced above, the ALM can obtain relatively accurate aerodynamic loads with fewer computational resources compared with CFD method. Thus, this model is selected in the present work to predict the unsteady aerodynamic performance of the FOWT. Based on original ALM, some modifications are made to consider the influence of platform motions on the aerodynamic loads. Figure 24.3 shows a cross-­ sectional element at radius r, where the velocity components are shown in the (θ, z) plane. Compared with actuator line model, the unsteady actuator line model (UALM) considers the additional velocity (UM) induced by the platform motions [18]. Then the relative wind speed can be calculated by the following equation:

Urel = Uθ − Ω ⋅ r + U z + U M



(24.9)

The calculation procedure of body forces in UALM is the same with that in the aerodynamic part of EAL, which has been introduced above. To achieve the coupled aero-hydrodynamic simualtions for the FOWT, the aerodynamics of wind turbine and the hydrodynamics of floating platform both need to be modelled. Based on the UALM, the unsteady aerodynamic peformacne of wind turbine can be predictd. Moreover, a two-phase CFD code naoe-FOAM-SJTU [19] is applied to sovle the hydrodynamic resposnes of the flaoting platform. Volume of fluid (VOF) method with bounded compression technique is utilized to capture the

Fig. 24.3  Cross-sectional velocity components in UALM

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free surface. By combining the UALM and the naoe-FOAM-SJTU, a coupled analysis tool FOWT-UALM-SJTU [20] is establised for the simulations of FOWT under wind and wave loads. The three-dimensional Reynolds-averaged Navier-Stocks (RANS) equations with the k-ω SST turbulence model are chosen as governing equations for the complex two-phase flow problem. ∇·U = 0

∂ ( ρU )



(

(24.10)

)

+ ∇· ρ ( U − U g ) U = −∇pd − g· x∇ρ + ∂t ∇·( µeff ∇U ) + ( ∇U )·∇µeff + fσ + fs + fε

(24.11)

where U is the velocity in flow field. Ug donates the grid node speed. pd is the dynamic pressure. g is the gravitational acceleration vector. ρ is the mixture density with two phases. μeff represents effective dynamic viscosity. fσ is the surface tension term in two phases model. fs is the source term for sponge layer. fε donates the body forces calculated from the UALM. Figure 24.4 shows the solving procedure of coupled aero-hydrodynamic calculations for the FOWT. It is seen that the coupling between the wind turbine and the floating platform is achieved. Moreover, a wake coupling way is selected in the present simulations. The platform motion obtained from the last time-step is considered in the UALM, and the body forces calculated by the UALM are added into the 6DOF motion equations and momentum equations in this time step.

Fig. 24.4  Solving procedure of coupled aero-hydrodynamic simulations for the FOWT

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3  Results and Discussions With the help of the third generation of vortex identification methods, Linux and Omega, complicated wake characteristics in the aeroelastic responses of wind turbine, the wake interactions between wind turbines and the wake of floating offshore wind turbine are analyzed in detail.

3.1  Aeroelastic Responses of Wind Turbine The NREL 5-MW wind turbine [21], which is a typical multi-megawatt offshore wind turbine, is chosen as the analysis object to investigate the aeroelastic responses of wind turbine under uniform wind inflow condition. Main parameters are summarized in Table 24.1. The computational domain is shown in Fig. 24.5, where the main sizes of the computational domain are marked out. The mesh in computational domain in Fig. 24.6. In order to better capture the wake characteristics, the grids behind the wind turbine are refined. The total grid number is 3.2 million. A constant wind speed of 11.4 m/s is adopted in the simulation. The corresponding rotational speed of the wind turbine is 12.1 rpm. In order to better understand the complicated wake characteristics of the NREL 5-MW wind turbine, the wake structure is visualized by different vortex identification methods. Figure 24.7 shows the vortex structure generated from the blade tip of wind turbine. Three different vortex identification methods, including the first generation (Vorticity method), the second generation (Q method) and the third generation (Liutex method), are applied to visualize the blade tip vortex structure. Moreover, the vorticity of wake flow in horizontal plane at hub height (z = 90 m) is also presented, as shown in Fig. 24.8. It is seen from Fig. 24.7 that clearly spiral vortex is generated from the blade tip of wind turbine. The vortex structure can be well predicted by the third vortex identification method. However, it is seen that the vortex structure of wind turbine visualized with different methods is obviously dependent the threshold that is used to determine the vortex boundary. When the selected threshold is small, the vortex Table 24.1  Main parameters of the NREL 5-MW wind turbine [21]

Rotor orientation, configuration Rotor, hub diameter Hub height Cut-in, rated, cut-out wind speed Cut-in, rated rotor speed Overhang, shaft tilt, pre-cone angle Rotor mass Nacelle mass Tower mass

Upwind, three blades 126 m, 3 m 90 m 3 m/s, 11.4 m/s, 25 m/s 6.9 rpm, 12.1 rpm 5 m, 5°, 2.5° 110,000 kg 240,000 kg 347,460 kg

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Fig. 24.5  Computational domain for aeroelastic modelling of NREL 5-MW wind turbine

Fig. 24.6  Grid distribution in the computational domain

structures visualized by different methods have small difference. For different methods, the magnitude of proper threshold is different. As shown in Fig.  24.7, Q = 0.001 may be the appropriate threshold, while the proper threshold for R that represents the magnitude of Liutex vector is about 0.5. It should be noted that the wake vortex will be ignored when the selected threshold is large. To avoid influence of threshold on the wake structure, Omega-R method developed by Liu et al. [14] is also used to the visualized the vortex structure. Relative vortex strength is used to visualize the vortex in this method, as shown in Fig. 24.8. The strong vortex and the wake vortex can both be clearly observed. Moreover, the vorticity in the wake field is also visualized with Vorticity method and Liutex method. Comparing the Fig. 24.8a and the Fig. 24.8d, the vorticity in the wake field predicted by different methods shows significant discrepancy. Vortex-shedding phenomenon can be

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Fig. 24.7  Vortex structure of the wind turbine visualized by different vortex identification methods

clearly captured by the Liutex method, while it is not obvious in the wake visualized by the Vorticity method. The wake vorticity along z direction also show significant discrepancy between the Vorticity method and the Liutex method. It is seen that the wake vortices at the same position predicted by different methods are just opposite.

3.2  Wake Interactions Between Two Wind Turbines Wake interactions in the wind farm have significant effects on the aerodynamic performance of the wind turbine. In the present work, wake interactions between two wind turbines with tandem layout are investigated. The NREL 5-MW wind

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Fig. 24.8  Wake vorticity in horizontal plane at hub height (z = 90 m)

Fig. 24.9  Arrangement of wind turbines with tandem layout

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Fig. 24.10  Grid distribution in the computational domain

turbine is also selected as the analysis object, and main parameters are summarized in Table 24.1. The arrangement of wind turbines is presented in Fig. 24.9. The distance between two wind turbines along stream-wise direction is 3D (D = 126 m is the diameter of rotor). Grid distribution is illustrated in Fig. 24.10. Considering the expansion of turbine wake, refined girds are generated in the wake region behind the wind turbine. The minimum gird size in the wake region is 1 m × 1 m × 1 m, and total grid number is 8.56 million. A low wind speed of 5 m/s is chosen in the simulations of two in-line wind turbines, and the corresponding rotational speed is 7.5 rpm. The vortex structure of two wind turbines under tandem layout is presented in Fig. 24.11. The distance between the vortex rings visualized by Vorticity method and Q method is small when a small threshold is adopted. Integrate vortex structure is captured by the Vorticity method and Q method, while broken wake structure is generated with the Liutex method. The main reason is that the absolute vortex strength is used in the vortex identification. In addition, Omega-R method is also applied to visualize the wake structure of two in-line wind turbines. It is shown that the clear vortex structure is generated from the blade tip of upstream wind turbine, and it transfers to the downstream. Moreover, the vortex rings gradually evenish with the distance from the upstream wind turbine. Compared with the vortex structure of upstream wind turbine, the distance between the vortex ring of the downstream wind turbine is much smaller. The main reason is that the inflow wind speed for the downstream wind turbine is relatively low due to the existing of upstream wind turbine. Thus, the transmission speed of the vortex ring along downstream is small, which further leads to the fusion of the vortex rings. Besides, the vorticity in the wake field along different directions is also investigated, as shown in Fig. 24.12. Compared with the Vorticity method, the Liutex method can better describe the wake vorticity in the flow field. It is observed that the vorticity distribution become much more uneven in the wake of downstream wind turbine. Moreover, the vorticity distribution in the wake field is asymmetric.

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Fig. 24.11  Vortex structure of the two wind turbines with tandem layout

Fig. 24.12  Wake vorticity in horizontal plane at hub height (z = 90 m)

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3.3  Complex Wake Characteristics of FOWT There are strong interactions between the wind turbine and the floating platform. Due to the platform motions, the inflow condition for the FOWT becomes complex. Moreover, the turbine wake interacts with the rotating blades, which further complex the wake characteristics of the FOWT.  In this study, coupled aero-hydrodynamic simulations are conducted for a spar-type FOWT under combined wind-wave loads. The spar-type FOWT is composed of the NREL 5-MW wind turbine, the OC3-Hywind spar platform [22] and a mooring system [23]. Main properties of the floating platform and mooring system are summarized in Table 24.2 and Table 24.3. Wake field of the FOWT is investigated in detail. Figure 24.13 shows the arrangement of the computational domain. The schematic diagram of the FOWT is presented in Fig.  24.14. Different grid resolutions are adopted in the computational domain in order to reduce the total grid number, as shown in Fig. 24.15. Moreover, the grids near the water surface and the platform is refined. To capture detailed wake characteristics behind the FOWT, the mesh in the turbine wake is also refined. The 1st order Stokes wave with a typical wave period of 10s and a wave height of 4 m is adopted as the incident wave condition. In addition, the exponential model is selected to describe the high-dependent wind speed, which is defined by the following equation:



 z  U z = U0   H

0.143



(24.12)

where Uz represents the wind speed at the height of z. H = 90 m is the hub height. U0 = 11.4 m/s donates the inflow wind speed. The corresponding rotational speed of the wind turbine is 12.1 rpm. It is noted that control strategies for the rotor speed and the blade pitch angle is not taken into consideration. Affected by the platform motions, the wake structure of the FOWT becomes more complicated. As shown in Fig.  24.16, the vortex structure of the FOWT is Table 24.2  Main properties of the OC3-Hywind spar platform [22]

Depth to platform base below SWL Elevation to platform top above SWL Depth to top of taper below SWL Depth to bottom of taper below SWL Platform diameter above taper Platform diameter below taper Platform mass, including ballast CML location below SWL Platform roll inertia about CM Platform pitch inertia about CM Platform yaw inertia about platform centerline

120 m 10 m 4 m 12 m 6.5 m 9.4 m 7,466,330 kg 89.9155 m 4,229,230,000 kg m2 4,229,230,000 kg m2 164,230,000 kg m2

24  Application of Liutex for Analysis of Complex Wake Flows Characteristics… Table 24.3  Gross parameters of the mooring system [23]

Number of mooring lines Angle between adjacent lines Depth to anchors below SWL Depth to fairleads below SWL Radius to anchors from platform centerline Radius to fairleads from platform centerline Unstretched mooring line length Mooring line diameter Equivalent mooring line mass density

367

3 120° 320 m 70.0 m 853.87 m 5.2 m 902.2 m 0.09 m 77.7066 kg/m

Fig. 24.13 Spar-­ type FOWT

visualized by different vortex identification methods. It is seen that the vortex structure obviously leans backward due to the relatively large platform pitch angle. Moreover, wake expansion phenomenon can be clearly observed. The diameter of vortex ring significant increases with the distance from the FOWT. Due to the platform motion, the distance between the vortex rings also change with time, which

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Fig. 24.14  Computational domain

Fig. 24.15  Grid distribution in the computational domain

can also be clearly observed in the vorticity distribution in the wake field. Figure 24.17 shows the vorticity along different directions in the horizontal plane. It is noted that the vorticity distributed in the horizontal plane is obviously asymmetric. The vorticity in the left half plane has a larger strength than that in the right half plane. This phenomenon is observed when the vorticity is calculated by Liutex method. It means that the Liutex method can better capture the wake characteristics in complex wake field. Irregular vortex shedding phenomenon is also captured by Liutex method when the platform motion is taken into consideration.

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Fig. 24.16  Vortex structure of the FOWT under combined wind-wave loads

4  Conclusions In this study, complicated wake characteristics in the wake field of NREL 5-MW wind turbine under various conditions are investigated by different vortex identification methods. The finite element method is combined with the actuator line model to predict the aeroelastic responses of wind turbine. Moreover, the unsteady actuator line model that considers the influence of platform motion on aerodynamic loads is embedded into the in-house CFD code naoe-FOAM-SJTU to perform coupled aero-hydrodynamic simulations for the spar-type FOWT. The vortex structure and wake vorticity in the turbine wake are visualized by Vorticity method, Q method and Liutex method. By comparing the wake characteristics obtained from different vortex identification methods, several conclusions can be drawn from the simulation results and discussions. Compared with the Vorticity method and Q method, the Liutex vector can better describe the complex wake characteristic of wind turbine in various conditions. The wake vorticity distribution behind the wind turbine is found to be asymmetric. Due to the wake interactions between the wind turbines, the vortex rings of the upstream wind turbine transfer to the downstream and merge with the vortices generated from the blade tip of downstream wind turbine. In addition, the wake characteristics of the FOWT become more uneven when the platform motions are taken into consideration, and significant wake expansion phenomenon is observed. In the future work, the Liutex vector will be applied to further investigate the complicated flow phenomena in the turbine wake.

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Fig. 24.17  Vortex structure of the FOWT under combined wind-wave loads Acknowledgements  This work is supported by the National Natural Science Foundation of China (51879159), The National Key Research and Development Program of China (2019YFB1704200, 2019YFC0312400), Chang Jiang Scholars Program (T2014099), and Innovative Special Project of Numerical Tank of Ministry of Industry and Information Technology of China (2016-23/09), to which the authors are most grateful.

References 1. Ö. Ceyhan, F. Grasso, Investigation of wind turbine rotor concepts for offshore wind farms. J. Phys.: Conf. Ser. 524(1), 012032 (2014) 2. N. Ederer, The right size matters: Investigating the offshore wind turbine market equilibrium. Energy 68, 910–921 (2014) 3. L.  Wang, X.  Liu, A.  Kolios, State of the art in the aeroelasticity of wind turbine blades: Aeroelastic modelling. Renew. Sust. Energ. Rev. 64, 195–210 (2016)

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4. H. Meng, F.S. Lien, L. Li, Elastic actuator line modelling for wake-induced fatigue analysis of horizontal axis wind turbine blade. Renew. Energy 116, 423–437 (2018) 5. Z. Ma, P. Zeng, L.P. Lei, Analysis of the coupled aeroelastic wake behavior of wind turbine. J. Fluids Struct. 84, 466–484 (2019) 6. M.C. Hsu, Y. Bazilevs, Fluid–structure interaction modeling of wind turbines: Simulating the full machine. Comput. Mech. 50(6), 821–833 (2012) 7. L.  Wang, X.  Liu, A.  Kolios, State of the art in the aeroelasticity of wind turbine blades: Aeroelastic modelling. Renew. Sust. Energ. Rev. 64, 195–210 (2016) 8. N.J.  Choi, H.N.  Sang, J.H.  Jeong, K.C.  Kim, CFD study on aerodynamic power output changes with inter-turbine spacing variation for a 6 mw offshore wind farm. Energies 7(11), 7483–7498 (2014) 9. N.J. Choi, H.N. Sang, J.H. Jeong, K.C. Kim, Numerical study on the horizontal axis turbines arrangement in a wind farm: effect of separation distance on the turbine aerodynamic power output. J. Wind Eng. Ind. Aerodyn. 117(117), 11–17 (2013) 10. X. Sun, D. Huang, G. Wu, The current state of offshore wind energy technology development. Energy 41(1), 298–312 (2012) 11. Y. Huang, D. Wan, Investigation of interference effects between wind turbine and spar-type floating platform under combined wind-wave excitation. Sustainability 12(1), 246 (2020) 12. V. Leble, G. Barakos, 10-MW wind turbine performance under pitching and yawing motion. J. Sol. Energy Eng. 139(4), 041003 (2017) 13. H.  Hu, M.  Khosravi, P.  Sarkar, in An Experimental Investigation on the Aeromechanic Performance and Wake Characteristics of a Wind Turbine Model Subjected to Pitch Motions. 34th Wind Energy Symposium (2016) 14. C. Liu, Y.S. Gao, X.R. Dong, Y.Q. Wang, J.M. Liu, Y.N. Zhang, Third generation of vortex identification methods: omega and liutex/rortex based systems. J.  Hydrodyn. Ser. B. 31, 205–223 (2019) 15. Y.N. Zhang, K.H. LIU, J.W. Li, Analysis of the vortices in the inner flow of reversible pump turbine with the new omega vortex identification method. J. Hydrodyn. 30(3), 463–469 (2018) 16. N. Gui, H.B. Qi, L. Ge, Analysis and correlation of fluid acceleration with vorticity and Liutex (Rortex) in swirling jets. J. Hydrodyn. (2019) 17. J.N. Sørensen, W.Z. Shen, Numerical modeling of wind turbine wakes. J. Fluids Eng. 124(2), 393–399 (2002) 18. P. Li, P. Cheng, D. Wan, Q. Xiao, in Numerical Simulations of Wake Flows of Floating Offshore Wind Turbines by Unsteady Actuator Line Model. The Ninth International Workshop on Ship and Marine Hydrodynamics, Glasgow, UK, August (2015) 19. Z.R.  Shen, H.J.  Cao. D.C.  Wan, in Manual of CFD solver for ship and ocean engineering flows: NAOE-FOAM-SJTU. Technical Report for Solver Manual, Shanghai Jiao Tong University (2012) 20. P. Cheng, Y. Huang, D. Wan, A numerical model for fully coupled aero-hydrodynamic analysis of floating offshore wind turbine. Ocean Eng. 173, 183–196 (2019) 21. J. Jonkman, Definition of the Floating System for Phase IV of OC3 (No. NREL/TP-500-47535) (National Renewable Energy Lab (NREL), Golden, CO (United States), 2010) 22. J. Jonkman, S. Butterfield, W. Musial, G. Scott, Definition of a 5-MW Reference wind Turbine for Offshore System Development (No. NREL/TP-500-38060) (National Renewable Energy Lab (NREL), Golden, CO (United States), 2009) 23. J. Jonkman, W. Musial, Offshore Code Comparison Collaboration (OC3) for IEA Wind Task 23 Offshore wind Technology and Deployment (No. NREL/TP-5000-48191) (National Renewable Energy Lab (NREL), Golden, CO (United States), 2010) 24. T. M. Fletcher, R. E. Brown, Simulation of wind turbine wake interaction using the vorticity transport model. Wind Energy, 13(7), 587–602 (2010) 25. R. Mikkelsen, J. N. Sørensen, S. Øye, N. Troldborg, Analysis of Power Enhancement for a Row of Wind Turbines Using the Actuator Line Technique. J. Phys: Conf. Ser. 75, 012044 (2007)

Chapter 25

Application of Omega-Liutex Identification Method in the Cavitating Flows Around a Three-Dimensional Bullet Rundi Qiu, Renfang Huang, Yiwei Wang, and Chenguang Huang

Abstract  The objectives of this paper are (1) to provide a better insight into the dynamics and structures of transient cavitation, and (2) to better understand the interaction between unsteady cavitating flow and vortex dynamics using different vortex-identification methods. Numerical simulations of unsteady cavitating flows are conducted around a three-dimensional bullet using Large Eddy Simulation (LES) method via the open-source code, OpenFOAM. The results reliably predict unsteady cavitation patterns, including cavity initiation, growth, and collapse in a short time. To clarify the mechanism of cavitation-vortex interaction, the Omega-­ Liutex method is used to visualize vortex structure. The results show strong correlations between a cavity and vortex structure. Keywords  Transient cavitation · OpenFOAM · Vortex identification · Omega-Liutex

R. Qiu Key Laboratory for Mechanics in Fluid Solid Coupling Systems, Institute of Mechanics, Chinese Academy of Sciences, Beijing, China School of Future Technology, University of Chinese Academy of Sciences, Beijing, China e-mail: [email protected] R. Huang (*) Key Laboratory for Mechanics in Fluid Solid Coupling Systems, Institute of Mechanics, Chinese Academy of Sciences, Beijing, China e-mail: [email protected] Y. Wang · C. Huang Key Laboratory for Mechanics in Fluid Solid Coupling Systems, Institute of Mechanics, Chinese Academy of Sciences, Beijing, China School of Future Technology, University of Chinese Academy of Sciences, Beijing, China School of Engineering Science, University of Chinese Academy of Sciences, Beijing, China e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_25

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1  Introduction Cavitation is a common hydrodynamic phenomenon which usually occurs where the speed of water is high enough. When local pressure is lower than saturated vapor pressure, water turns into vapor and copious bubbles are generated. The collapse of bubbles increases the fluctuation of the pressure field in a short time, which can greatly reduce hydraulic mechanical performance. Compared with general hydrodynamic phenomena, cavitation has unique characteristics, including a shorter characteristic time and a free surface, neither of which can be predicted. These characteristics result in difficulties in obtaining a theoretical solution about cavitation. Thus, numerical simulation has become an effective method in exploring the properties of cavitation. The presence of unsteady cavitating flow is accompanied by complex vortex formation and elimination. How to show the process of vortex motion is a vital issue in a numerical simulation of cavitation. Initially, such a simulation was based on the Reynolds averaged Navier–Stokes equations (RANS). Coutier-Delgosha et al. [1] compared the influences on the numerical results of four turbulence models based on RANS. The result showed that standard k − ε and standard k − ω models are unable to predict cavity shedding. After these models were corrected by including turbulent viscosity, the simulation results agreed well with experiment. With increases in computing speed, large eddy simulations (LES) are gaining increased attention. Compared with RANS, LES can resolve vortex structure more accurately. Therefore, LES are more suitable for cavitating flows with complex vortex motion. Dittakavi et al. [2] used LES to calculate cavitating flow in a Venturi tube. They pointed out that LES can describe vortex structure more clearly than RANS, and they derived the law of vorticity transport. Roohi et al. [3] applied LES to a two-­ dimensional Clark-Y hydrofoil cavitating flow. They indicated that LES predicted the shape of cavitation more accurately than RANS, even if the mesh did not strictly comprise two-dimensional grids. LES performed equally well in three dimensions. Nowadays, it is common to use the LES method for high-fidelity simulations of cavitating flows. The interactions between vortices have a significant effect on the dynamics of cavitation flow. For the unsteady cavitation of a hydrofoil, cavity instability originates from the mechanism of the re-entrant jet [4]. At the cavitation-shedding stage, the re-entrant jet creates numerous vortices of different sizes in a fluid. How to identify and analyze vortex structure is helpful in exploring cavitation characteristics. An understanding of vortices broadly derives from a strict mathematical definition. Helmholtz proposed the concepts of vorticity tubes/filaments [5]. Based on the definition of a vortex, three vortex theorems can be derived, which provides a basis for detailed analysis of vortex structures. However, vorticity cannot distinguish between the rotation area and the shear area. Although the mathematics of vorticity are clear and easy to use, they are not suitable for cavitating flow. To clarify the physical meaning of a vortex, researchers have constructed a second-generation vortex definition method based on eigenvalues, including the Q criterion [6], the

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Δ  criterion [7], and the λ2  criterion [8]. These criteria have appropriate physical meaning and can distinguish rotation from shear in a flow field. Nevertheless, due to its very definition, a vortex structure selected using various criteria is subjective. Different parameters in the criteria lead to different results and various vortex structures, and there is no general agreement on vortex identification. To solve these problems, Liu et  al. have developed a new identification method for vortices to overcome the disadvantages inherent to traditional methods. They proposed the Omega vortex-identification method [9–11]. Based on this method, Dong et al. created a new method named Omega-Liutex [12]. By normalizing the value of a criterion to a value between 0 and 1, Omega and Omega-Liutex reflect the shape of the vortex structure objectively. They claim that this is a third-generation vortex-­ identification method, which performs well in different cases; and that it is a reliable tool for analyzing the characteristics of a vortex. Wang et al. [13] investigate the turbulent drag reduction mechanism by analyzing the characteristic of vortex structure in the viscoelastic turbulent flows, and they compared various vortex identification method. Researchers improve Omega-Liutex method and obtain Rortex method, which is helpful to identify the vortex line and vortex core [14–16]. The vortex-identification method has wide application prospects in the field of cavitating flow, because the cavitation dynamics is closely related to the evolution of cavitation vortex structures. Wang et al. [17] compare different vortex identification method in cavitation vortex dynamics and show the performances of various methods in identifying cavitation vortex structures. Studies on water pumps [18] and turbine [19] have been conducted in recent years, but related research in hydrodynamics is still to be developed. The focus of this work is on investigating the interaction between a vortex and transient cavitation. We simulate a three-­ dimensional cavitating flow around a bullet using LES turbulence and a Schnerr– Sauer cavitation model. The volume fraction of the water phase is used to analyze cavitation-shape evolution over time. The wake flow of the bullet is used to observe the characteristics of fluid motion. To obtain a clear vortex structure, the Omega-­ Liutex identification method is utilized to analyze the flow field around the bullet. We hope our work will provide a reference for the application of Omega-Liutex in three-dimensional problems in actual engineering structures.

2  Numerical Methods and Computation Setting In this paper, the flow field around the bullet is obtained by solving a discrete Navier–Stokes equation. Multiphase flow cavitation theory can be used to calculate the interface between the vapor and liquid. In general, the cavitation-flow regime is regarded as a homogeneous bubbly flow. The continuity equation and momentum equation are simplified for a single-phase flow scenario. According to the above assumptions and conditions, the expression for the phase transition is achieved using a cavitation model of the transport-equation type. The phase transition

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between gas and liquid follows the rule of mass transfer. So, the mass conservation equation can be obtained: ∂ρ + ∇ ⋅ ( ρ U ) = m + − m − ∂t



(25.1)

α, ρ, U, m + , m − represent the volume fraction, effective mixture density, velocity vector, evaporation rate, and the condensation rate, respectively. The effective mixture density in homogeneous bubbly flow is defined as:

ρ = α l ρl + α v ρ v



(25.2)

where the subscripts l and v represent the vapor phase and the liquid phase, respectively. Different expressions for m + and m − are derived for different cavitation models. In this paper, the Schnerr–Sauer cavitation model is proposed to simulate the process of evaporation and condensation. m + and m − are defined as follows: m + = Ce m − = Cc

3 ρl ρ vα l (1 − α l ) 2 max ( psat − p,0 ) 3 ρ Rb ρl



3 ρl ρ vα l (1 − α l ) 2 max ( p − psat ,0 ) 3 ρ Rb ρl



(25.3)

(25.4)

1

 1 − αl 3 3 Rb =    α l 4π n 



(25.5)

psat is the saturated vapor pressure, Rb is the radius of a bubble, and n is the nuclei density per unit volume of liquid. Using momentum conservation, the following equation is derived: ∂ ( ρU)

∂t

+ ∇ ⋅ ( ρ UU ) − ∇ ⋅ τ = −∇p + ρ g + F



(25.6)

τ,g,F indicate viscous stress, gravity, and force, respectively. The 3D model of the bullet is presented in Fig. 25.1, and the computation domain of the bullet is shown in Fig. 25.2a, including four types of boundaries: inlet, outlet, bullet, and symmetry plane. This case is based on a bullet in OpenFOAM, which is in the tutorial directory. The original case used laminar flow as a turbulence model. In this paper, the turbulence model is modified as k − μ one equation LES model. The radius of the bullet, d, is equal to 0.01 m. The distance between the bullet and inlet is 5d. The distance from the outlet to the bullet is 10d and the distance from bullet to the symmetry plane is 5d, as shown in Fig. 25.2b. Fig. 25.3a and Fig. 25.3b show the mesh generated by snappyHexMesh, a Cartesian meshing tool in

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Fig. 25.1  3D view of the bullet

Fig. 25.2  Computational domain in (a) full view (b) y–z plane

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Fig. 25.3  Grid generated by snappyHexMesh (a) full view (b) close-up view

OpenFOAM, in full view and close-up view, respectively. The number of cells in this grid is 1.44 × 106 (Fig. 25.3). The cavitation number is set to 0.4885, in terms of the following equation:

σ=

p∞ − psat 1 ρ U2 2 ∞ ∞

(25.7)

ρ∞  is the water density and U∞ is the flow velocity in the inlet, which is set to 20 m/s. The Reynolds number is 2 × 105. In order to observe and analyze the evolution of the vorticity field behind the bullet, we use the Omega-Liutex identification method to describe vortex structure. The Omega-Liutex method is enhanced based on the Omega method and the Omega-R method [20–24]; the value of Omega-R is calculated using Eq. (25.8):

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ΩR =



β2 α + β2 +ε

379

(25.8)

2

ΩR can be chosen as 0.52 to visualize vortex structure. ε is empirically defined as a function about α and β:

ε = b×(β 2 −α 2 )



max

(25.9)



where b is a small positive number. α and β can be written as: 2

α=

2

1  ∂V ∂U   ∂V ∂U  − + + 2  ∂Y ∂X   ∂X ∂Y  1  ∂V ∂U  β=  − 2  ∂X ∂Y 



(25.10)

(25.11)

U, V, W represent the velocity components in the x-, y-, and z-directions, respectively. The derivative  of each velocity component can be obtained using the velocity gradient tensor ∇V :  ∂U ∂U   ∂X , ∂Y , 0      ∂V ∂V  , ,0 ∇V =  ∂X ∂Y     ∂W , ∂W , ∂W   ∂X ∂Y ∂Z 



(25.12)

Liu et al. improved Eq. (25.8) and obtained the improved Omega-Liutex identifying method: ˜

ΩR =

 

(ω ⋅ r )

2

  2 2 (ω ⋅ r ) − 2λci2 + 2λcr2 + λr2  + ε  

(25.13)

 where λ represents the eigenvalues of the velocity gradient tensor ∇V . The subscripts ci, cr, and r represent the imaginary part of the complex eigenvalue, the real   part of the complex eigenvalue, and the real eigenvalue, respectively. ω ,r represent  the vorticity and the direction of Liutex. r has the same direction as the real eigenvector of the velocity gradient tensor ∇V .

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3  Results and Discussion The simulation calculates a total of 60-time steps, from 0 s to 0.012 s. The data are written every 0.0002 s. The results show that the whole procedure can be divided into two stages. The first stage is from 0 s to 0.0032 s, during which transient behaviors are significant. The second stage is from 0.0034  s to 0.012  s. At the second stage, the border of cavitation in the flow field gradually stabilizes, and the wake flow behind the bullet is turbulent. In order to elucidate the attributes of the flow field during the two different stages, the velocity field and water volume fraction field are shown in Figs. 25.4 and 25.7, respectively. Figure 25.4 shows the formation and progression of the re-entrant jet from 0.002 s to 0.0044 s. The contour of the water volume fraction and the vector diagram at the wake region of bullet are presented in Fig. 25.5. At the beginning of the first step, due to the transient behavior of flow, an inception cavitation occurs at the top of the bullet and expands rapidly. In a few time steps, the cavity occupies the entire bullet and extends behind the bullet. At 0.002 s, the growth of the cavity stops and begins to shrink from the tail of the cavity, while the re-entrant jet begins to generate at the end of cavity. At 0.0026 s, the flow at the end of the cavity invades the center of the cavities traveling in the opposite direction to the incoming flow, forming a re-entrant jet structure, as shown in Fig.  25.5. Experimental research reveals that cavitation collapse is mostly caused by re-entrant jet structure. As the re-entrant jet touches and covers the surface of the bullet, the free interface of the cavities begins to become unstable due to interference from the re-entrant jet; the cavity quickly shrinks, which makes the pressure on the surface of the bullet oscillate significantly. After 0.0032s, the attached cavity breaks into small-scale cloud cavity and move downstream. Since the shape of the re-entrant jet is symmetric about the central axis, it can be predicted that the jet and the velocity field are also roughly symmetric about the central axis, as shown in Fig. 25.4. In order to quantitatively analyze the impact of the reversed flow on the bullet, the drag coefficient curve is calculated and shown in Fig. 25.6. The drag coefficient, Cd, is calculated thus: Cd =

Fdrag 1 ρU ∞2 A 2

(25.14)

According to the initial settings for the bullet, the cross-section area of bullet is set to 7.854 × 10−5m2, and U∞ is set to 20 m/s. The generation of the re-entrant jet occurs in a very short time, and the instantaneous impact between re-entrant jet and the bullet is much greater than the fluctuation in drag force caused by vortex shedding. Comparable phenomena can be perceived for other variables, so we choose the drag coefficient as a typical variable and show it in Fig. 25.6. At about 0.0031 s, the drag coefficient always becomes abruptly much greater than the coefficient at any other time. The huge variation in the drag coefficient mainly takes place between

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Fig. 25.4  Flow fields from 0.0002 s to 0.0044 s in (a) the z-direction velocity field. The re-entrant jet was generated behind the bullet and quickly penetrated forward until it reached the tail of the bullet. (b) Volume fraction of water. The boundary of the cavity expanded rapidly; when it reached its maximum volume, a re-entrant jet was generated in a short time, breaking through the cavity. The cavities at the tail then detached from the bullet and oscillated. A slice in the y–z plane is chosen

0.003 s and 0.0032 s. By comparing Figs. 25.5 and 25.6, it is clear that the fluctuation in drag coefficient is not stimulated by the re-entrant jet contacting the tail of the bullet, but by the collapse of the cavity at the tailing edge of bullet. The cavity is cut off from the center of the re-­entrant jet, which produces a strong shock. Then the shock propagates to the bullet. After the cavity has collapsed, the drag coefficient quickly returns to a quasi-stable state, and thereafter, it mainly reflects the vortex shedding effect at the tail of the bullet. After the re-entrant jet disappears, the cavitation detaches from the bullet and repeats the formation-expansion-collapse cycle at the rear of the bullet for some time. When cavitation no longer appears in about 0.01  s, the cavitation near the

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Fig. 25.5  Evolution of re-entrant jet in 0.001 s. The re-entrant jet originates from a point, which is distanced from about three times the diameter of the bullet and is directed towards the tail of the bullet. A strong fluctuation in velocity occurs after the jet has reached the tail. A slice in the y–z plane is chosen

Fig. 25.6  Drag coefficient curve from 0.0026 s to 0.0038 s. The cavity is cut off from the middle of the cavity, causing a huge fluctuation in drag coefficient

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Fig. 25.7  Physical field from 0.006 s to 0.012 s: (a) the z-direction velocity field. The wake-flow regime remained turbulent during the whole process. (b) Volume fraction of water. The cavity generated in the former stage collapses over time. The thick layer attached around bullet gradually becomes steady. A slice in the y–z plane is chosen

leading edge of bullet becomes stable. There is no cavitation behind the bullet, and the top of the bullet is covered with a thin layer of cavities. This phenomenon can be clearly seen in Fig. 25.7a, b, which present the features of cavitation in the second stage. Figure 25.7a shows that the velocity field of the bullet after cavitation dissipation is similar to the velocity field for non-cavitation. Due to the use of the LES turbulence model, the wake region of the flow field is turbulent. Figure 25.7b shows the cavity shape in 0.012 s. The reason why the thick cavitation layer around the bullet maintains its stability is because of the cavitation number. The top of the bullet is spherical, which makes it less likely to induce super cavitation. In this case, the cavitation number is not low enough to generate super cavitation, which can wrap around the whole bullet. In this case, the quasi-periodic shedding of cavities does not occur because of the shape of the bullet.

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In this paper, we mainly focus on vortex generation when a re-entrant jet develops in a short time. The Omega-Liutex program is applied to describe the evolution of vortices during the formation and elimination of the jet. To understand the relationship between a vortex and cavitation, the volume fraction is plotted in the same figure with an iso-surface obtained using Omega-Liutex in Fig. 25.8. The iso-­surface  = 0.52 . In 0.0022 s, the cavity reaches its maximum of Omega-Liutex is set to Ω R volume. Since the jet direction is opposite to the direction of the external flow field, a small annular vortex appears at the tip of the jet. The vortex follows the jet and touches the wall of the bullet, and the fluid is forced to change direction, which induces vortex formation at the tail of the bullet. Meanwhile, the vortex that escapes from the bullet draws the fluid on both sides of liquids into the inside of the cavity, cutting the cavity from the center. Tail vortex shedding is induced by the groove near the rear of the bullet. It can be inferred that the groove at the rear end of the bullet has an important influence on the shape of transient cavitation. The groove-­ induced shedding vortex entrains the fluid into the cavitation, causing the cavitation to break from the center. Compared with the bullet without grooves, the bubbles behind the grooved bullet accelerate the elimination of cavitation. After 0.003 s, the cavity is surrounded by vorticity, which indicates strong vortex motion inside and outside the cavity. To explore the ability of different methods to extract the vortex structure in cavitating flow, we make a simple comparison of the iso-surface maps obtained using the three different vortex methods for the flow from 0.0022 s to 0.0032 s in Fig. 25.9.

Fig. 25.8  Iso-surface according to the Omega-Liutex criterion and the contours of the water volume fraction in (a) 0.0022 s, (b) 0.0024 s, (c) 0.0026 s, (d) 0.0028 s, (e) 0.003 s, and (f) 0.0032 s.  = 0.52 The contour is plotted in the x-plane. The iso-surface of Omega-Liutex is set to Ω R

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 = 0.52 Fig. 25.9  Different results about various vortex identification using (a) Omega-Liutex Ω R , (b) the λ2-criterion λ2 =  − 5 × 106, and (c) the Q-criterion Q = 1 × 107, from 0.0022 s to 0.0032 s

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The criterion for adjusting Q and λ2 is to make the shape of the shedding vortex at the tail of the bullet approximately the same. The most obvious advantage of Omega-Liutex is that the shape of the vortex can be obtained without modifying the value of the criterion. The λ2-criterion and Q-criterion many adjustments to make the vortex structure as clear as possible. Using these criteria (Fig. 25.9), the vortex structure calculated using Omega-Liutex is clearer and the boundary of vortex is more distinct. The iso-surfaces obtained using the λ2-criterion and Q-criterion have different degrees of adhesion at the tail of the bullet, meaning the boundary of the vortex structure given by Omega-Liutex is clearer, and the main vortex structure can be better distinguished from the complex flow area. Changing the size of the Omega-­ Liutex criterion can be done to distinguish between a strong vortex and a weak vortex.

4  Conclusions In this paper, a 10-cm diameter bullet under water is calculated using OpenFOAM. Based on the simulation results, the process is divided into two stages. The transient behavior of the flow field in the first stage is dominant. At the beginning, a large cavity appears and wraps around the whole bullet. As time progresses, the re-entrant jet at the tail of the cavitation is generated and migrates forward, breaking through the cavity and causing the cavity to shrink in a short time. The whole process takes about 0.01 s. The second stage is considered to comprise quasi-­ steady flow, which is marked by the attachment between the jet and the bullet. After the re-entrant jet has reached the tail of the bullet, the cavity is divided into two parts. In the first part, cavity fluctuates slowly and eventually stabilizes in front of the bullet, forming a thin layer. The second part sheds from the bullet, and the volume of the cavity is quickly attenuated. Soon afterwards, the wake flow of the bullet becomes turbulent. To explore vortex structure during its initial stages, the Omega-­ Liutex method is utilized to describe a vortex in a wake flow. A comparison between different method is applied and the result shows that Omega-Liutex method has unique advantage in vortex identification. The goal of this paper is to study the phenomenon of a three-dimensional bullet when it is suddenly launched in deep water, and to analyze the vortex structure of the flow problem using Omega-Liutex. The problems similar to this bullet are common in high-speed hydrodynamics. Solving this case about bullet can provide a reference for the modeling and analysis of actual engineering problems. Omega-Liutex provides accurate vortex-structure criteria, providing a reliable tool to better understand cavitating flow fields. This paper studies cavitation around the bullet when it is far away from a free surface. At different depths, the cavitation shape changes, caused by the interaction between the free surface and the bullet; hence this is also worthy of observation and analysis. Acknowledgement  This work is accomplished using the code Omega-LiutexUTA, released by Chaoqun Liu at the University of Texas at Arlington. The project is supported by the National Key R&D Program (Grant Nos. 2016YFC0300800, 2016YFC0300802), National Natural Science

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Foundation of China (Nos. 11772340 and 11672315), and the Science and Technology on Water Jet Propulsion Laboratory (Grant No. 6142223190101).

References 1. O. Coutier-Delgosha, R. Fortes-Patella, J.L. Reboud, Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation. J. Fluid Eng.: Trans. ASME 125(1), 38–45 (2003) 2. N. Dittakavi, A. Chunekar, S. Frankel, Large Eddy simulation of turbulent-cavitation interactions in a Venturi nozzle. J. Fluid Eng.: Trans. ASME 132(12), 11 (2010) 3. E. Roohi, A.P. Zahiri, M. Passandideh-Fard, Numerical simulation of cavitation around a two-­ dimensional hydrofoil using VOF method and LES turbulence model. Appl. Math. Model. 37(9), 6469–6488 (2013) 4. M. Callenaere, J.P. Franc, J.M. Michel, M. Riondet, The cavitation instability induced by the development of a re-entrant jet. J. Fluid Mech. 444, 223–256 (2001) 5. C.Q.  Liu, Y.Q.  Wang, Y.  Yang, Z.W.  Duan, New omega vortex identification method. Sci. China: Phys. Mech. Astron. 59(8), 9 (2016) 6. P.-H. Alfredsson, A. Johansson, J. Kim, in Turbulence Production Near Walls: The Role of Flow Structures with Spanwise Asymmetry. Stanford Univ., Studying Turbulence Using Numerical Simulation Databases, 2. Proceedings of the 1988 Summer Program, pp. 131–141 (1988) 7. M.S. Chong, A.E. Perry, B.J. Cantwell, A general classification of 3-dimensional flow-fields. Phys. Fluids A: Fluid Dyn. 2(5), 765–777 (1990) 8. J. Jeong, F. Hussain, On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995) 9. X.R. Dong, Y.Q. Wang, X.P. Chen, Y. Dong, Y.N. Zhang, C.Q. Liu, Determination of epsilon for Omega vortex identification method. J. Hydrodyn. 30(4), 541–548 (2018) 10. Y.N. Zhang, X. Qiu, F.P. Chen, K.H. Liu, Y.N. Zhang, X.R. Dong, C.Q. Liu, A selected review of vortex identification methods with applications. J. Hydrodyn. 30(5), 767–779 (2018) 11. J.M.  Liu, Y.S.  Gao, Y.Q.  Wang, C.Q.  Liu, Objective Omega vortex identification method. J. Hydrodyn. 31(3), 455–463 (2019) 12. C.Q. Liu, Y.S. Gao, X.R. Dong, Y.Q. Wang, J.M. Liu, Y.N. Zhang, X.S. Cai, N. Gui, Third generation of vortex identification methods: Omega and Liutex/Rortex based systems. J. Hydrodyn. 31(2), 205–223 (2019) 13. L. Wang, Z.Y. Zheng, W.H. Cai, W.Y. Li, Extension Omega and Omega-Liutex methods applied to identify vortex structures in viscoelastic turbulent flow. J. Hydrodyn. 31(5), 911–921 (2019) 14. W. Xu, Y. Gao, Y. Deng, J. Liu, C. Liu, An explicit expression for the calculation of the Rortex vector. Phys. Fluids 31(9) (2019) 15. X. Dong, Y. Gao, C. Liu, New normalized Rortex/vortex identification method. Phys. Fluids 31(1) (2019) 16. J.M. Liu, Y.S. Gao, C.Q. Liu, An objective version of the Rortex vector for vortex identification. Phys. Fluids 31(6), 11 (2019) 17. C.C. Wang, Y. Liu, J. Chen, F.Y. Zhang, B. Huang, G.Y. Wang, Cavitation vortex dynamics of unsteady sheet/cloud cavitating flows with shock wave using different vortex identification methods. J. Hydrodyn. 31(3), 475–494 (2019) 18. S. Xu, X.P. Long, B. Ji, G.B. Li, T. Song, Vortex dynamic characteristics of unsteady tip clearance cavitation in a waterjet pump determined with different vortex identification methods. J. Mech. Sci. Technol. 33(12), 5901–5912 (2019) 19. Y.F.  Wang, W.H.  Zhang, X.  Cao, H.K.  Yang, The applicability of vortex identification methods for complex vortex structures in axial turbine rotor passages. J.  Hydrodyn. 31(4), 700–707 (2019)

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20. Y.S. Gao, C.Q. Liu, Rortex and comparison with eigenvalue-based vortex identification criteria. Phys. Fluids 30(8), 18 (2018) 21. Y.Q. Wang, Y.S. Gao, C.Q. Liu, Letter: Galilean invariance of Rortex. Phys. Fluids 30(11), 6 (2018) 22. C.Q. Liu, Y.S. Gao, S.L. Tian, X.R. Dong, Rortex—A new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids 30(3), 12 (2018) 23. X.R. Dong, Y.S. Gao, C.Q. Liu, New normalized Rortex/vortex identification method. Phys. Fluids 31(1), 5 (2019) 24. X.R.  Dong, G.  Dong, C.Q.  Liu, Study on vorticity structures in late flow transition. Phys. Fluids 30(10), 8 (2018)

Chapter 26

Analysis of Vortex Evolution in Turbine Rotor Tip Region Based on Liutex Method Yufan Wang and Weihao Zhang

Abstract  In turbine rotor tip region, vortex structures are complex and aerodynamic loss is large. The double-cavity tip is expected to reduce the aerodynamic loss in this region. In this paper, influence of the double-cavity tip on the flow field in the tip region is numerically studied. The evolution characteristics of vortex structures are analyzed by using the Liutex method, and control effects of different double-­ cavity tip geometries on tip-leakage flow and leakage loss are discussed. The results show that the double-cavity tip can form multiple aero-labyrinth liked sealing effect and effectively control the leakage flow. The double-cavity tip with small pressure side cavity has better control effect, which can improve the turbine stage efficiency by more than 0.16%. Increasing the cavity depth of the double-cavity tip will enhance the squealer rib corner vortex, but the comprehensive control effect on leakage flow is not obvious. The inclined pressure side rib of double-cavity tip can provide larger space for scraping vortex by inhibiting the pressure side squealer rib corner vortex, which is beneficial to controlling the leakage flow. However, the inclination of the middle rib will cause the open separation at the top of the pressure side rib, which has a negative impact on the control of the leakage flow. Keywords  Turbine rotor · Double-cavity tip · Tip-leakage flow · Liutex method · Vortex evolution

Y. Wang · W. Zhang (*) National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy & Power Engineering, Beihang University, Beijing, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_26

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1  Introduction Due to the existence of tip-leakage flow, the flow field in the turbine rotor tip region is extremely complex. There are multi-scale and multi-intensity vortex structures, such as tip leakage vortex, upper passage vortex and tip gap vortices. These vortices have strong interaction with each other, and they are strongly mixed with the mainstream and tip-leakage flow, which leads to great aerodynamic loss. Denton [1] pointed out that the loss caused by tip-leakage flow in the blade tip region accounts for more than 30% of the total aerodynamic loss in the rotor passage. For an accurate estimate of tip leakage loss, Denton [1], Yaras et al. [2] and Yang et al. [3] have established good leakage loss models, respectively. In order to improve the ­aerodynamic performance of the turbine, it is necessary to control the tip-leakage flow and reduce the mixing loss between the main-flow, tip-leakage flow and vortex structures. In order to analyse the evolution of vortex structures in the flow field, the vortex identification methods, such as Q criterion and ∆ criterion, are often used to identify vortices. However, the results identified by these methods are often contaminated by shear, which is not convenient to accurately understand the evolution of vortex structures in the tip region. In 2018, Liu et al. [4] proposed Liutex method, which is a systematic vortex identification method. The advantages of Liutex were illustrated by Gao and Liu [5]. Shortly afterwards, Liu and Liu [6] proposed modified normalized Liutex vortex identification method, and Xu et al. [7] proposed the definition of Liutex core, which marks the maturity of Liutex method. Since that, Wu et al. [8], Wang et al. [9], Zhang et al. [10], Wang et al. [11] have tried to apply the Liutex method to the study of energy dissipation analysis, boundary layer transition, pump and turbine, and have proved the advantages of Liutex method in robustness, threshold selection, simultaneous identification of strong and weak vortices, and elimination of shear interference. Therefore, this paper will use the Liutex method to accurately identify the vortex structures in the tip region and analyze the interaction between vortices. In order to improve the aerodynamic performance of the turbine, tip-leakage flow needs to be effectively controlled. Therefore, a variety of tip-leakage flow control methods, including squealer tip, winglet tip and tip injection, have been proposed. Dey et al. [12] found that the suction side rib has good blocking effect on tip-leakage flow, while pressure side rib has little effect on restraining tip-leakage flow. Inspired by the function of wing blades, Coull et al. [13] proposed a method to control tip-leakage flow by adding winglets at rotor tip, and tried various winglet geometries such as full winglet, pressure side winglet and suction side winglet. Based on the fact that both winglets and squealer ribs can effectively improve the aerodynamic performance of turbines, Schabowski et al. [14] proposed to combine the winglets and squealer ribs to obtain greater aerodynamic performance benefits. For the first time, Pougare and Weinhold [15] put forward the experiment of controlling tip-leakage flow by blade tip injection, and found that tip injection has the potential to control tip-leakage flow and reduce tip-leakage loss.

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The squealer tip has great control effect on tip-leakage flow, and the flow field in the gap and the control mechanism of squealer tip have been studied in many studies. Mischo et al. [16] found that there are three vortex structures in the tip cavity, namely, the scraping vortex, the pressure side squealer rib corner vortex and the suction side squealer rib corner vortex. Yang and Feng [17] found that the cavity tip can significantly reduce the tip-leakage flow rate and momentum by enhancing the mixing inside the gap. After analyzing many vortex structures in the cavity, Zou et  al. [18] considered that the scraping vortex generated inside the cavity is the dominant flow structure affecting turbine aerodynamic performance. The scraping vortex plays the role of aero-labyrinth and blocks the leakage flow, thus changing the key aerodynamic parameters such as leakage flow rate and momentum. Based on this conclusion, the possibility of forming multiple aero-labyrinth liked sealing effect at the blade tip is considered in this paper. Based on the evolution law of the scraping vortex in the clearance, a variety of double-cavity tip geometries are constructed. For further improvement of turbine aerodynamic performance, geometric optimization of squealer tip has been investigated in several studies. Prakash et al. [19] found that the inclined pressure side squealer rib can increase the blocked area through a larger flow separation in the pressure side squealer tip gap, enhancing leakage flow control. Senel et al. [20] found that the increase of rib height and the decrease of rib width lead to the decrease of tip-leakage flow. Zeng et al. [21] discussed the impacts of squealer geometric, such as the inclined pressure side rib and squealer rib width, on the vortex structures in the gap and tip-leakage loss. In this paper, taking a transonic single-stage high-pressure turbine as the research object, six kinds of double-cavity tips were constructed, and the evolution of vortices in the tip region are analyzed by using the Liutex method. The control mechanism of the double-cavity tip on the tip-leakage flow are analyzed. Based on the above analysis, the effects of cavity depth and inclined ribs on vortices, tip-leakage flow aerodynamic parameters’ distributions and tip-leakage loss distribution are further studied, and the influence rules of double-cavity tip geometry parameters on tip-leakage flow is summarized.

2  Numerical Methods 2.1  Objects and Case Setup The turbine geometry in this study is a transonic single stage high-pressure turbine. The main geometric parameters and the operation conditions of the turbine stage are listed in Table 26.1, where τ is clearance height, π is the total-to-total pressure ratio, Mar, exit is relative Mach number at the exit, and Reexit is Reynolds Number at the exit. In order to study the effects of double-cavity tip geometry on tip-leakage flow, 6 different double-cavity tip geometries were constructed based on single-cavity tip

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Table 26.1  Turbine parameters and boundary conditions Stator Rotor Blade passing frequency (HZ) 17,610

N 46 72 π 4.08

Aspect ratio 0.6 1.15 Mar, exit 1.11

τ/%span – 1.16% Reexit 1.12 × 106

Fig. 26.1  Blade tip geometries of seven cases and sketch of double-cavity tip

Table 26.2  The geometric characteristics of case2–7 Case case2 case3 case4 case5 case6 case7 Cavity height 1.5τ 1.5τ 1.5τ 2.5τ 2.5τ 2.5τ Position of inclined squealer rib – – – – Pressure side and middle Middle

(case1). Geometric factors such as the position of middle rib, cavity depth and inclined rib were considered. The blade tip geometry of each case is shown in Fig.  26.1a, and Fig.  26.1b is the schematic diagram of geometric parameters of double-cavity tip. The geometric characters of case2-case6 are shown in Table 26.2. According to the position of the middle rib in the double-cavity tip, the geometries of the six double-cavity tips can be divided into two types. Type 1 is the double-­ cavity tip with small suction side cavity (such as case2), and Type 2 is the double-­ cavity tip with small pressure side cavity (such as case3–case7).

2.2  Computational Setup The numerical simulation in this study uses the software ANSYS CFX 19.0 to solve steady viscous Reynolds Averaged Navier-Stokes equations with the Shear Stress Transport (SST) turbulence model to close equations. The computational domain consists of one stator passage and one rotor passage. Total temperature, total pres-

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sure and turbulence intensity are used as the inlet boundary conditions, and inflow direction is normal to boundary condition. As the outlet boundary condition, static pressure can be adjusted to make sure certain total-to-total pressure ratio. Both surfaces at the circumferential side of the stator and the rotor passages are set as periodic boundary conditions. And surface between stator and rotor is set as mixing-plane. All the wall faces are set as adiabatic with no slip.

2.3  Grid Independence Validation The commercial software NUMECA Autogrid 5 was used to generate the stator domain computational grid, and the ICEM was used to generate the rotor domain computational grids. The mesh elements in both domains are hexahedrons. Local encryption is carried out near the shroud, the hub and the rotor blade. The wall distance of first mesh cell is set to 0.001 mm, and the average value of calculated y + is about 1. To eliminate the influence of different grid topology on the numerical results, all the cases use the same stator domain computational grid and the same topological structure is used in the rotor domains of all the cases. The grids are shown in Fig. 26.2. Considering that the grid density may affect the accuracy of the numerical results, in order to prevent the grid density from affecting the turbine aerodynamic performance, the grid independence validation was performed and the appropriate grid number was determined. The radial grid number in the rotor tip is analyzed and determined. The six cases with different gap grid densities were calculated in the turbine with flat tip. The grid distribution of each case is shown in Table 26.3. A comparison of tip-leakage flow rate and the turbine isentropic efficiency of six cases is shown in Fig. 26.3. As the pictures show, with the increase of grid number, the differences of tip-leakage flow rate and turbine isentropic efficiency between

Fig. 26.2  Sketch of computational grid

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Table 26.3  Cases of varied mesh allocation

Grid1 Grid2 Grid3 Grid4 Grid5 Grid6

Axial × Tangential × Radial Stator Rotor (gap) 156 × 40 × 56 214 × 43 × 90(13) 214 × 43 × 96(19) 214 × 43 × 102(25) 214 × 43 × 108(31) 214 × 43 × 114(37) 214 × 43 × 120(43)

Total number /×106 2.39 2.57 2.74 2.92 3.10 3.28

Fig. 26.3  Comparison of aerodynamic parameters

two adjacent cases decreases gradually. When the number of radial grids in the gap is more than 31, the difference of tip-leakage flow rate is less than 0.06%, and the difference of turbine isentropic efficiency is less than 0.01%, which can be ignored. Therefore, the radial grid number in the gap was determined as 31, which can ensure the accuracy of the numerical results and save the computation cost as much as possible.

3  Results and Discussion 3.1  V  ortex Evolution in Tip Region of Rotor with Double-­Cavity Tip The turbine stage efficiency and dimensionless tip-leakage flow rate of case1-case4 are shown in Fig. 26.4. Obviously, the double-cavity tip can significantly improve the turbine stage efficiency. The difference is that compared with the single-cavity tip (case1), the turbine stage efficiency of ‘Type 1’ with small suction side cavity is increased by about 0.04%, while that of ‘Type 2’ with small pressure side cavity is increased by more than 0.16%. According to Fig. 26.4b, ‘Type 1’ results in a slight

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Fig. 26.4  Comparison of turbine stage efficiency and leakage flow rate

Fig. 26.5  Vortex structures in the tip region (Identified by Liutex = 5 × 104 iso-surface and colored with streamwise vorticity; SV scraping vortex)

increase in tip-leakage flow rate relative to the single-cavity tip, indicating that ‘Type 1’ doesn’t have more efficient control effect on tip-leakage flow than case1. Zou et  al. [18] pointed out that the scraping vortex, which can form an aero-­ labyrinth liked sealing effect in the cavity, is the key vortex structure for controlling the tip-leakage flow. The vortex structures in the clearance identified by Liutex iso-­ surface are shown in Fig. 26.5, and the vortices are colored with streamwise vortic-

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ity. Obviously, in cases with double-cavity tip, there are scraping vortices in each cavity, that is to say, multiple aero-labyrinth liked sealing effect is formed in case2-­ case4. However, the chordwise length of the scraping vortex in the suction side cavity of case 2 is small, and the ‘Location A’ in the Fig. 26.5 represents the position of the scraping vortex flowing out of the clearance in case2. In addition, ‘Location B’ in the figure represents the position of the scraping vortex flowing out of the clearance in case3 and case4. The difference of the scraping vortex evolution will inevitably lead to the weaker blocking effect of case2 on tip-leakage flow than case3 and case4 in the region between ‘Location A’ and ‘Location B’. Moreover, in case2, the scraping vortex in the suction side cavity flows out of the clearance too early, which will cause the enhancement of tip leakage vortex in the passage. It is worth mentioning that, according to the distribution of streamwise vorticity, case2 can enhance the entrainment effect of the scraping vortex in the pressure side cavity on tip-leakage flow. Moreover, compared with case3 and case4, the scraping vortex in pressure side cavity of case2 have larger chordwise length, which causes that the control effect of case2 on tip-leakage flow downstream of “Location B” is much stronger. Figure 26.6 shows the leakage flow rate distributions and normal momentum difference distributions of four cases at the gap outlet, which is helpful to directly evaluate the blocking effect of the scraping vortex structure on the leakage flow. The dotted lines ‘A’ and ‘B’ in the figure correspond to ‘Location A’ and ‘Location B’ where scraping vortices flows out of the clearance, respectively. Scraping vortices flows out of the clearance, which results in the sharp decrease of the control effect on the tip-leakage flow and the sharp increase of the leakage flow rate and momentum at the gap outlet. It can be seen from Fig. 26.6a that the leakage flow rate of case2 is similar to that of case1 over a large axial range, and case2 can’t reduce the leakage flow rate more effectively. The leakage flow rate and momentum of case3 and case4 are significantly less than those of case1 in 40–70% axial chord region, which indicates that the position of effective control on the tip-leakage flow is just the region where have

Fig. 26.6  Comparison of aerodynamic parameters’ distributions at the gap exit along the streamwise direction

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two scraping vortex in the circumferential direction. While the tip-leakage flow rate and normal momentum of case3 and case4 are significantly greater than those of case1 in 75–85% axial chord region, which is due to the fact that there is no scraping vortex at this position in case3 and case4, but there are scraping vortex in pressure side cavity of case1 and case2. In order to obtain the aerodynamic loss distribution in the blade tip region, the calculation method of local entropy production proposed by Kock and Herwit [22] is used to measure the aerodynamic loss, and it can be calculated as follows: k T T  t k T T 2  sij sij     Sg’’’,local  2 T T T xi xi  T 2 xi xi



(26.1)

where S is local entropy generation rate, k is the thermal conductivity, αt and α are the turbulent thermal diffusivity and thermal diffusivity respectively, μ is dynamic viscosity, and ε is turbulence eddy dissipation. Figure 26.7 shows the distribution of aerodynamic loss in tip region. And the Liutex iso-line with a value of 5 × 104 are also given in this figure, as the black lines show, which is convenient to identify the vortex structures. Obviously, the energy dissipation in the tip region of case2 and case1 are similar. Comparing the loss ’’’ g ,local

Fig. 26.7  Comparison of local entropy production distribution in the tip region

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d­ istribution at ‘Zone 1’ in case1 and case4, it can be seen that case4 effectively reduces the energy dissipation in the pressure side cavity and the mixing loss caused by tip-­leakage flow and main-flow in the passage. The loss distribution of ‘Zone 2’ in the figure shows that the double-cavity tip with small pressure side cavity has a large loss near the trailing edge. From the above, the double-cavity tip with small pressure side cavity can control the tip-leakage flow and leakage loss more effectively. But there are still some problems in this kind of double-cavity tip, which has the possibility of further ­optimization. And these problems are mainly reflected in the suppression of the development of the scraping vortex by the larger SRCV and the large loss near the trailing edge.

3.2  Geometric Optimization of Double-Cavity Tip According to the above analysis of vortex structure and loss of case1-case4, based on the case4 with the best aerodynamic performance, case5 is obtained by increasing the cavity depth; Case6 and case7 are obtained by changing the straight rib to the inclined rib. Figure  26.8 shows the turbine stage efficiency of case4–case7. Obviously, the increase of cavity depth and inclined ribs can improve the aerodynamic performance of the turbine, and the effect of rib inclination on turbine stage efficiency is more obvious than that of cavity depth. The streamlines in tip region and the distribution of Liutex in different slices of case4 and case5 are shown in Fig. 26.9. Comparing the evolution of SRCV in case4 and case5, it can be seen that the increase of cavity height causes that the size of SRCV increases. In the pressure side cavity, due to the small cavity width, the size

Fig. 26.8  Comparison of turbine stage efficiency

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Fig. 26.9  Distribution of Liutex and streamline of scraping vortex inside the suction side cavity (SRCV squealer rib corner vortex)

Fig. 26.10  Comparison of loss distribution in the tip region

of SRCV increases, which inhibits the development space of the scraping vortex, and causes that the scraping vortex at ‘Zone A’ in the figure is closer to the middle rib. In the suction side cavity, the cavity width is larger, and the strength of the scraping vortex at ‘Zone B’ in case4 is weak. And the increase of the size of SRCV leads to the obvious increase of the scraping vortex strength. The above two changes of the scraping vortex are beneficial to enhancing the entrainment effect of the scraping vortex on the tip-leakage flow and blocking the leakage flow better.

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However, the increase of SRCV results in the decrease of the chordwise length of the scraping vortex, as shown in the region at ‘Zone C’ of figure. Obviously, ­cavity will lose the aero-labyrinth liked sealing effect at the places where have no scraping vortex, which is not conducive to the control of leakage flow. Figure 26.10 shows the loss distribution in the tip region of case4 and case5. It can be seen that, the increase of cavity depth has little effect on the mixing loss in the passage and the loss in the middle and front section of the gap, mainly reducing the loss near the trailing edge of the gap. In order to solve the problem of large SRCV in case4, this paper attempts to use inclined ribs to weaken the size of SRCV. In case6, the pressure side rib and the middle rib are inclined, while in case7, only the ribs on the pressure side are inclined. Figure 26.11 shows the Liutex distribution in the normal slices of the cambers. The scraping vortex in the pressure side cavity of case6 is restrained, while that of case7 is enhanced, which indicates that the inclined middle rib has a strong inhibitory effect on the development of scraping vortex in the pressure side cavity, and is not conducive to controlling tip-leakage flow. In order to better understand the flow field structure, ‘Slice A’ in the figure is selected for analysis, and the distributions of Mach number and velocity vector in the ‘Slice A’ are shown in Fig. 26.12. It can be seen from Fig. 26.12 that, if the middle rib is inclined, the leakage will flow into the clearance more horizontally, which results in larger SRCV and smaller scraping vortex. According to the distribution of Mach number at the top of pressure side squealer rib, there is obvious open separation at the top of pressure side squealer rib of case6, which is the reason why the leakage flow flows into the clearance parallel to the blade tip. It can be seen that the inclined middle rib leads to the open separation at the top of the pressure side rib, which increases the size of SRCV, thus

Fig. 26.11  Comparison of Liutex distribution in the tip region (SV scraping vortex)

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Fig. 26.12  The distributions of Mach Number and velocity vector in the ‘Slice A’ (SV scraping vortex, SRCV squealer rib corner vortex)

inhibiting the development of the scraping vortex, and causing adverse effects on blocking tip-leakage flow.

4  Conclusions In this paper, a transonic single-stage high-pressure turbine was taken as the research object to study the evolution characteristics of vortices in the clearance of various double-cavity tips. Combined with the aerodynamic parameter distributions of tip-­ leakage flow, the control mechanism and some problems of double-cavity tip are summarized. In order to solve these problems, this paper studies the influence of cavity geometry factors such as cavity depth and inclined ribs on tip-leakage flow, and further improves the turbine aerodynamic performance. The main conclusions are as follows: 1. Scraping vortices are formed in each cavity of the double-cavity tip, thus forming multiple aero-labyrinth liked sealing effect, effectively controlling the leakage flow, and obviously improving the aerodynamic performance of the turbine. 2. The double-cavity tip with small pressure side cavity can control the tip-leakage flow and leakage loss more effectively, and can increase the turbine stage efficiency by more than 0.16%. 3. With the increase of the cavity height in double-cavity tip, the size of SRCV increases, and the intensity of SV increases, which is conducive to blocking the leakage flow; however, the increase of SRCV also causes that SV flows out of the clearance early, which is not conducive to controlling the leakage flow. These two different effects comprehensively control the influence of the cavity height on the leakage flow. 4. In the double-cavity tip, the inclined pressure side squealer rib is helpful to control the leakage flow; while the inclined middle squealer rib results in the open

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separation at the top of pressure side squealer ribs, which will restrain the development of the SV and cause adverse effects on blocking tip-leakage flow. Acknowledgement  The authors would like to acknowledge the support of the National Natural Science Foundation of China (No. 51406003).

References 1. J. Denton, Loss mechanisms in turbomachines. J. Turbomach. 115(4), 621–656 (1993) 2. M. Yaras, S. Sjolander, Prediction of tip-leakage losses in axial turbines. J. Turbomach. 114(1), 204–210 (1990) 3. H. Yang, W. Zhang, Z. Zou, et al., The development and applications of a loading distribution based tip leakage loss model for unshrouded gas turbines. J. Turbomach. 142(7), 071005 (2020) 4. C. Liu, Y. Gao, S. Tian, et al., Rortex—A new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids 30(3), 035103 (2018) 5. Y. Gao, C. Liu, Rortex and comparison with eigenvalue-based vortex identification criteria. Phys. Fluids 30(8), 085107 (2018) 6. C. Liu, J. Liu, Modified normalized Rortex/vortex identification method. Phys. Fluids 31(6), 061704 (2019) 7. H. Xu, X. Cai, C. Liu, Liutex (vortex) core definition and automatic identification for turbulence vortex structures. J. Hydrodyn. 2 (2019) 8. Y. Wu, W. Zhang, Y. Wang, et al., Energy dissipation analysis based on velocity gradient tensor decomposition. Phys. Fluids 32(3), 035114 (2020) 9. Y. Wang, Y. Yang, G. Yang, et al., DNS study on vortex and vorticity in late boundary layer transition. Commun. Comput. Phys. 22(02), 441–459 (2017) 10. Y. Zhang, K. Liu, J. Li, et al., Analysis of the vortices in the inner flow of reversible pump turbine with the new omega vortex identification method. J. Hydrodyn. 30(3), 463–469 (2018) 11. Y. Wang, W. Zhang, X. Cao, et al., A discussion on the applicability of vortex identification methods for complex vortex structures in axial turbine rotor passages. J.  Hydrodyn. 31(4), 700–707 (2019) 12. D. Dey, L. Kavurmacioglu, C. Camci, Tip desensitization of an axial turbine rotor using partial squealer rims. Turbine Blade Tip Design and Tip Clearance Treatment, VKI Lecture Series 2, 19–23 (2004) 13. J. Coull, N.R. Atkins, H.P. Hodson, Winglets for improved aerothermal performance of high pressure turbines. J. Turbomach. 136(9), 091007 (2014) 14. Z.  Schabowski, H.  Hodson, D.  Giacche, et  al., Aeromechanical optimization of a winglet-­ squealer tip for an axial turbine. J. Turbomach. 136(7), 071004 (2014) 15. M. Pouagare, W. Weihold, in Tip Leakage Reduction Through Tip Injection in Turbomachines. 22nd Joint Propulsion Conference. 1986: 1746 (2012) 16. B. Mischo, T. Behr, R.S. Abhari, Flow physics and profiling of recessed blade tips: Impact on performance and heat load. J. Turbomach. 130(2), 021008 (2008) 17. D. Yang, Z. Feng, Study on tip leakage flow and heat transfer for a squealer tip blade. J. Eng. Thermophys. 28(6), 936–938 (2007) (in Chinese) 18. Z. Zou, F. Shao, Y. Li, et al., Dominant flow structure in the squealer tip gap and its impact on turbine aerodynamic performance. Energy 138, 167–184 (2017) 19. C.  Prakash, C.P.  Lee, D.G.  Cherry, et  al., Analysis of some improved blade tip concepts. J. Turbomach. 128(4), 639–642 (2006) 20. C. Senel, H. Maral, L. Kavurmacioglu, et al., An aerothermal study of the influence of squealer width and height near a HP turbine blade. Int. J. Heat Mass Transfer 120, 18–32 (2018)

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21. F. Zeng, W. Zhang, Y. Wang, et al., Effects of squealer geometry of turbine blade tip on the tip-leakage flow and loss. J. Therm. Sci. (2020) (in press) 22. F. Kock, H. Herwit, Entropy production calculation for turbulent shear flows and their implementation in CFD codes. Int. J. Heat Mass Transfer 26(4), 672–680 (2005)

Chapter 27

Numerical Simulation of Leakage Flow Inside Shroud and Its Interaction with Main Flow in an Axial Turbine Dongming Huang and Weihao Zhang

Abstract  In a turbine, the leakage loss is an important source of turbine aerodynamic loss. In this paper, steady numerical simulations were performed to analyze the flow characteristics in the shroud of a 1.5-stage axial turbine and the interaction between the leakage flow and the mainstream. A spatial loss audit inside shroud was undertaken and the loss in the shroud was assessed. Based on the Liutex method, the vortices in the shroud were identified. The results show that the leakage flow in the shroud could be regarded as a jet process with several parts which are free jet and wall attached jet. The rest part of the shroud was occupied by the vortices and separated flows. There is both inflow and outflow on the interface of outlet cavity and mainstream, accompanied by a series of vortices in the outlet cavity. The loss in the inlet cavity is affected by the leakage flow intensity. The loss in the outlet cavity accounts for the largest part of the loss in the shroud. Keywords  Flow characteristics · Vortex · Turbine · Shroud · Leakage loss · Liutex method

1  Introduction Leakage flow impacts turbine performance through reducing extracted work in rotor blades. Apart from this, Wallis et al. [1] identified four other loss mechanisms that the shroud leakage flows affect turbine performance. They are mixing in the inlet shroud, mixing through labyrinth seal, mixing loss in the exit cavity because of the difference between the leakage flow and the main flow, and the effect of non-ideal incidence in the downstream blade row. Rosic and Denton [2] analysed the flow in D. Huang · W. Zhang (*) National Key Laboratory of Science and Technology on Aero-Engine Aero-thermodynamics, School of Energy & Power Engineering, Beihang University, Beijing, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_27

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a 1.5 stage turbine and separated the contribution of each leakage loss mechanism. For the shrouded turbine investigated, the efficiency drop due to the mixing loss in the inlet cavity and exit cavity is 0.20% and 0.25%, respectively. The re-entry mixing loss and downstream negative incidence losses causes 0.40% efficiency drop. The loss due to the reduced work extraction can be estimated by the mass fraction of the leakage flow which is directly proportional to the former. Based on Denton’s ­previous work, Yang [3] proposed a strategy to calculate the mixing loss between the tip leakage flow and mainstream flow by a control volume analysis, and this developed method could be referred to calculate the re-entry mixing loss. Reducing the re-entry mixing loss has been recognized as a promising way to reduce aerodynamic loss and further improve turbine performance. This perspective has been verified by many researchers. Denton and Johnson [4] stated that the shroud leakage flow remains roughly unchanged as the flow pass through the shroud. The mixing loss due to the difference in tangential velocity between the mainstream and the leakage flow is significant. Hunter and Manwaring [5] researched on the interaction of the cavity flow and the main flow. They found that the disparity in circumferential momentum of the two flows led to increased loss. Gier et al. [6] showed that the re-entry mixing loss accounted for up to a half of the losses related to cavity flow. Based on this, several groups have used turning elements expecting to reduce the circumferential momentum disparity at the cavity exit. Wallis et al. [1], Rosic and Denton [2] incorporated turning devices onto the shroud surface and at the cavity exit respectively in order to redirect the cavity exit flow. Contradictory results have been obtained. Rosic and Denton measured an improvement of 0.4% in overall aerodynamic efficiency while Wallis got a reduction in efficiency of 3.5% due to the increased loss from main flow ingress in the outlet cavity. Gao et al. [7] took a similar approach after Rosic and Denton, using stationary vanes fixed to the casing but between radial seals. Their results demonstrate that the turning elements can reduce mixing losses, but the overall isentropic efficiency may decrease due to the large separation at the trailing edge of the turning vanes, which depends on the shape and number of turning vanes. It’s unachievable to control the loss as expected without acquaintance with the flow in the shroud, which resent researches have cast new light on. Wang [8] revealed the detailed flow field in the shroud cavities, and built the vortex models of some local flow. Palmer et al. [9] assessed the toroidal vortex system in loss generation and regulation of mass flow, and the correlation between the vorticity of the vortex and its associated viscous dissipation and the recirculating mass flow rate was identified. With the advance of vortex identification method, the vortex structures in the shroud could be captured more accurately. In 2018, Liu et al. [10] proposed a new systematic vortex identification method named Liutex method, whose advantages were then illustrated by Gao and Liu [11]. Modified normalized Liutex vortex identification method was soon proposed by Liu and Liu [12]. The definition of Liutex core proposed by Xu et al. [13] marks the maturity of Liutex method. Since then, the effectiveness and advantages of Liutex method in robustness, threshold selection

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and so on have been demonstrated by many researchers [14–17]. Therefore, Liutex method will be used in this paper to capture the vortices in the shroud. In this paper, a numerical simulation was performed to explore the flow characteristics and loss mechanisms in the shroud. Apart from quantitative analysis of the loss on the different regions in the shroud, a spatial loss audit inside shroud was undertaken and the loss in the shroud was assessed.

2  Numerical Methods The turbine in this research, whose primary geometric and aerodynamic parameters at the design point are listed in Table 27.1, contains a stator passage, a rotor passage and an inter-turbine duct, which is not shown in Fig. 27.1. The rotor blade is furnished with stepped labyrinth seal shroud. The meridional view of the shroud is shown in Fig. 27.2. The numerical simulation was performed by Finite Volume Solver (ANSYS CFX 19.0). Reynolds-averaged Navier-Stokes equations were solved by using a finite volume mothed. SST turbulence model was used for turbulence closure, and variable specific heat calculation was applied in the simulation. All the wall faces were set as no slip and adiabatic wall faces and the rotor hub wall rotated with the rotor blade. Mass flow-averaged total temperature and total pressure, inflow angle was imposed in the inlet plane while static pressure was applied as the outflow condition. Gird independence is checked and totally three million nodes are used in the simulation. The shroud domain and each blade passage domain consists of about one million grid nodes, respectively. The meshes on the stationary domains are coupled with the rotational domains by mixing plane approach. However, the mesh on the interface between the shroud and mainstream is matched strictly as shown in Fig. 27.3.

3  Flow Characteristics in the Shroud As shown in the Fig. 27.4, the meridional velocity distribution in the meridional plane of the shroud indicates that the leakage flow can be treated as a jet process with several different parts which are free jet and wall attached jet. At first, the leakage flow was turned because of the blocking of the shroud. The leakage flow thereTable 27.1  Aerodynamic and geometric parameters of the LP at the design point Inlet total temperature (K) Inlet total pressure (Pa) Exit static pressure (Pa)

1183.4 655,527 280,000

Rotating speed (r/min) Number of blades Solidity

10,736 53/94/7 1.58/1.41/0.65

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Fig. 27.1  Mesh in the stator passage and the rotor passage

Fig. 27.2  Meridional view of the shroud

fore is concentrated around the vicinity of the leading edge of the shroud, forming a free jet that flows into the inlet cavity. Next, the leakage jet made another turn and flows as a wall attached jet towards the first seal. The leakage jet encounters large acceleration after flowing past the seals. Finally, the leakage jet is turned third time and flows out the outlet cavity, joining the mainstream. A large separation area is also formed at the downstream casing wall of the outlet cavity. Figure 27.5 indicates the vortex structures in the shroud, which are captured by iso-surface |Liutex| = 5 × 104 and colored by vorticity in y direction. Figure 27.5 shows the space outside the leakage jet in the shroud is occupied by vortices and separated flows, which are mainly induced by the leakage flow. In Fig. 27.5b, the vortices in the inlet cavity are cut by a circumferential plane. Chons et al. [18] state

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Fig. 27.3  Mesh in the rotor passage and shroud

Fig. 27.4  Meridional velocity distribution in the meridional plane of the shroud

that the fluid particles within a vortex should exhibit as a swirling motion, implying that the closed orbits or spirals formed by the streamlines in the circumferential plane should be considered as vortices. However, it can be found in Fig. 27.6 that the spirals in the plane were less than the regions with high Liutex values, indicating that there are more vortices than that the streamline displays. In fact, the center of the obits or spirals do not correspond to the vortex core because the flow in these areas carries low momentum. For example, the vortex V1 and V2(marked in Fig. 27.6) have same rotation direction, namely anti-clockwise. The two vortices

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Fig. 27.5  Iso-surfaces of |Liutex| = 5 × 104, colored by Vorticity-y: (a) shroud; (b) inlet cavity with a circumferential plane; (c) outlet cavity

Fig. 27.6  Liutex distribution and streamlines in the circumferential plane shown in Fig. 27.5b

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interact with each other, resulting in the low velocity between them. The high Liutex value exists in the area where the streamline curves. The vortices in the outlet cavity seems simpler than that in the inlet cavity. In the outlet cavity, the mainstream flow periodically ingress into the cavity due to the blade to blade pressure gradient. The incoming flow rolls up and forms the cavity vortex. With the help of vortex identification by Liutex, the cavity vortex could be clearly captured, which is marked in red circle in Fig.  27.5. Therefore, it can be found that the cavity vortex starts at position marked P in Fig. 27.5 and then advances circumferentially. It will be shown to play an important role in the entropy generation of the outlet cavity in the later section.

4  Loss Mechanisms in the Shroud Denton’s canonical review paper [19] states that the rational measure of loss in an adiabatic machine is entropy creation. Basic thermodynamics tells us that entropy creation occurs due to viscous friction in either boundary layers or free layers and heat transfer across finite temperature differences. Therefore, the entropy generation per unit volume can be described simply as follows: k ∂T ∂T 2 µ sij sij ′′′ ′′′ + Svisc = 2 + Sg′′′,local = Sthemal T ∂xi ∂xi T





′′′ g ,local

S denotes the local entropy generation per unit volume, called entropy generation rate and sij represents shear strain rate. In fact, some parameters in the equation cannot be solved directly until the direct numerical simulations (DNS) are employed [20]. To calculate the viscous part and thermal part directly from RANS results, the equation should be transformed into another form based on Reynolds average studied by Herwig [21] and Kramer-Bevan [22]:



k ∂T ∂T α t k ∂T ∂T 2 µ sij sij ερ + + + Sg′′′,local = 2 T T T ∂xi ∂xi α T 2 ∂xi ∂xi



where “  ̅” means the numerical Reynolds averaged parameters. Altogether four groups of entropy production terms in turbulent flows could be identified. Next, entropy production rate calculated in this way would be used to analyse the loss in different regions of the shroud. The flow is non-uniform in the circumferential direction. Figure 27.7 shows the entropy generation rate in four different circumferential planes whose location is shown in Fig. 27.8. It can be found that the high entropy generation rate area locates at the proximity of the leakage jet and the wall. Apart from the loss at the wall caused by the viscous friction in boundary layers, there are two jet-like area with high entropy production rate (marked A1 and A2 in the C3 plane), where the loss is

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Fig. 27.7  Entropy generation rate distribution in four circumferential planes

Fig. 27.8  The location of the four circumferential planes in the inlet cavity

mainly caused by the interaction between the leakage flow and low momentum fluid in the inlet cavity. The leakage intensity is different in varied planes, which is manifested as the change of loss in the two areas. When the leakage jet is more intense, the high loss production rate area at the proximity of the interface between inlet cavity and the mainstream (marked A1) is shorter while the area marked A2 is longer, deeper inside the inlet cavity. While the distribution of the loss changes with the flow intensity, the total loss in different planes of the inlet cavity increases with flow intensity. The distribution of entropy generation rate and axial velocity in a circumferential plane of the cavity between seals is shown in the Fig. 27.9. In the cavity between

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Fig. 27.9  Entropy generation rate and axial velocity distribution in a circumferential plane of the cavity between seals

Fig. 27.10  Entropy generation rate distribution in a circumferential plane of the seals

seals, the momentum of leakage jet decays with distance from the first seal because of the wall dissipation, and a thin layer of high entropy generation rate area is formed at the casing wall, suggesting high loss is caused. After passing through the step, the leakage jet mixes with the vortex in the corner, and loss is generated as well. The loss on the tip of each seal is probably caused by the open separation. Compare the entropy generation rate distribution of the two seals shown in Fig. 27.10, it can be found that the loss on the tip of the second seal is larger than the first seal. That is because the leakage flow passing through the second seal encounters higher pressure drop, forming a more intense jet. Therefore, the open separation region is larger, and more separated flow mixes with the jet. The flow in the outlet cavity is more complex. The entropy generation rate distribution and streamline in the circumferential planes indicates that the high intensity jet eject from the seal mixes with the low momentum fluid near the shroud, leaving a long and wide area with high entropy production rate. The mainstream flow ingresses periodically into the outlet cavity and forms the cavity vortex, as shown in C6 and C7 plane. Large entropy production is generated at the corner and the wall of the trailing edge of the shroud due to the mixing process with the cavity flow and the friction of the wall, respectively.

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In C5 and C6 plane where no mainstream flow ingresses into the outlet cavity, the outflow divides into two streams, in which one stream of fluid flows around the cavity vortex, then mixes with the other stream of fluid, generating entropy production. The entropy production is affected by the velocity of the two streams. With the evolution of the cavity vortex induced by the ingress flow, the flow pattern in the outlet cavity changes, as well as the loss (Figs. 27.11 and 27.12). In conclusion, the loss inside the shroud is related to three objects: wall, leakage jet and vortices. Comparing the distribution of entropy production rate with Liutex, it can be found that high loss does not appear at the core of vortices (Fig. 27.13). Fig. 27.11  The location of the four circumferential planes in the outlet cavity

Fig. 27.12  Entropy generation rate distribution and streamline in four circumferential palnes

Fig. 27.13  Entropy generation rate and Liutex distribution in a circumferential plane

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Fig. 27.14  Local entropy generation rate (left axis) and loss accumulation (right axis) in the shroud

Because the Liutex represents the strength of the vortex and the flow in the core region of vortices is rotation. Pure rotation does not cause loss [14]. Altogether some qualitative concepts about the loss in the shroud have been developed. Next, some quantitative calculation was made. Figure 27.14 gives the local entropy generation rate on different axial location and loss accumulation in the shroud. The horizontal axis of the chart stands for normalized axial location, which is calculated by x’ = x/L. The maximum value of entropy generation rate in the inlet and outlet cavity corresponds to the loss caused by leakage jet and wall dissipation (marked in red boxes in Fig. 27.13), respectively. Besides, it shows that more loss was generated in the outlet cavity. Table 27.2 gives the loss of each region. The total pressure loss is given as reference. It shows that the entropy production and total pressure loss in the outlet cavity account for the largest part of the loss in the shroud. Nearly a half of loss in the shroud is generated in the outlet cavity. This fact marks the control of the loss in the outlet cavity as a promising way for reducing aerodynamic losses.

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Table 27.2  Entropy production and Total pressure loss in different regions of the shroud Region Inlet cavity First seal Cavity Second seal Outlet cavity Total

Entropy production Value (W/K) 0.1279 0.0074 0.1523 0.0142 0.2753 0.5771

Proportion 22.16% 1.27% 26.40% 2.47% 47.71% 100%

Total pressure loss Value (Pa) Proportion 25,660 18.21% 4143 2.94% 38,364 27.23% 7527 5.34% 65,192 46.27% 140,886 100%

5  Conclusions In this study, the flow characteristics and the loss mechanisms in the shroud was analyzed by performing steady numerical simulations on a 1.5-stage axial shrouded turbine. The following conclusion can be drawn. Firstly, the flow in the shroud shows that the leakage flow can be treated as a jet process with several different parts which are free jet and wall attached jet. The other spaces in the shroud are occupied by vortices and separated flows. Secondly, the entropy production rate distribution in the shroud shows that the high entropy generation rate area in the inlet cavity is affected by the leakage flow intensity. The mainstream flow ingress periodically into the outlet cavity, changing the flow pattern and affecting the loss in the outlet cavity. Finally, qualitative calculation clearly shows that more than 45% of the loss in the shroud is generated in the outlet cavity, which accounts for the largest part. It indicates that the control of the leakage loss in the outlet cavity is a promising way for reducing aerodynamic losses. Acknowledgments  The authors would like to acknowledge the support of the National Natural Science Foundation of China (No. 51406003).

Nomenclature k Thermal conductivity L Length of the cavity T Static temperature x Axial coordinate αt Turbulent thermal conductivity μ Fluid dynamic viscosity

References 1. A.M.  Wallis, J.D.  Denton, A.A.J.  Demargne, in The Control of Shroud Leakage Flows to Reduce Aerodynamic Losses in a Low Aspect Ratio, Shrouded Axial Flow Turbine. ASME Paper 2000-GT-475 (2000) 2. B. Rosic, J.D. Denton, The control of shroud leakage loss by reducing circumferential mixing. J. Turbomach. 130, 021010(1–7) (2008)

27  Numerical Simulation of Leakage Flow Inside Shroud and Its Interaction with Main… 417 3. H. Yang, W. Zhang, Z. Zou, et al., The development and applications of a loading distribution based tip leakage loss model for unshrouded gas turbines. J. Turbomach. 142(7), 071005 (2020) 4. J.D. Denton, C.G. Johnson, An Experimental Study of the Tip Leakage Flow Around Shrouded Turbine Blades. CEGB Report No R/M/N848, Marchwood Engineering Laboratories (1976) 5. S.D. Hunter, P.O. Orkwis, Endwall Cavity Flow Effects on Gaspath Aerodynamics in an Axial Flow Turbine: Part II—Source Term Model Development (Power for Land, Sea, & Air, ASME Turbo Expo, 2000) 6. J. Gier, K. Engel, B. Stubert, R. Wittmaack, Modelling and analysis of main flow-shroud leakage flow interaction in LP turbines. Proc. ASME TurboExpo 2006 (2006) 7. J. Gao, Q. Zheng, G. Yue, L. Sun, Control of shroud leakage flows to reduce mixing losses in a shrouded axial turbine. Proc. IMechE Part C: J. Mech. Eng. Sci. 226, 1263–1277 (2012) 8. Z. Zou, J. Liu, W. Zhang, et al., Shroud leakage flow models and a multi-dimensional coupling CFD (computational fluid dynamics) method for shrouded turbines. Energy 103, 410–429 (2016) 9. T.R. Palmer, C.S. Tan, H. Zuniga, D. Little, M. Montgomery, A. Malandra, Quantifying loss mechanisms in turbine tip shroud cavity flows. Proc. ASME TurboExpo 2014(2014) ASME GT2014-25783 10. C. Liu, Y. Gao, S. Tian, et al., Rortex—A new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids 30(3), 035103 (2018) 11. Y. Gao, C. Liu, Rortex and comparison with eigenvalue-based vortex identification criteria. Phys. Fluids 30(8), 085107 (2018) 12. C. Liu, J. Liu, Modified normalized Rortex/vortex identification method. Phys. Fluids 31(6), 061704 (2019) 13. H. Xu, X. Cai, C. Liu, Liutex (vortex) core definition and automatic identification for turbulence vortex structures. J. Hydrodyn. 2 (2019) 14. Y. Wu, W. Zhang, Y. Wang, et al., Energy dissipation analysis based on velocity gradient tensor decomposition. Phys. Fluids 32(3), 035114 (2020) 15. Y. Wang, Y. Yang, G. Yang, et al., DNS study on vortex and vorticity in late boundary layer transition. Commun. Comput. Phys. 22(02), 441–459 (2017) 16. Y. Zhang, K. Liu, J. Li, et al., Analysis of the vortices in the inner flow of reversible pump turbine with the new omega vortex identification method. J. Hydrodyn. 30(3), 463–469 (2018) 17. Y. Wang, W. Zhang, X. Cao, et al., The applicability of vortex identification methods for complex vortex structures in axial turbine rotor passages. J. Hydrodyn. 31(4), 700–707 (2019) 18. M.S.  Chong, A.E.  Perry, B.J.  Cantwell, A general classification of three-dimensional flow fields. Phys. Fluids A: Fluid Dyn. 2(5), 765–777 (1990) 19. J.D. Denton, Loss mechanisms in turbomachines. J. Turbomach. 115(4), 621–656 (1993) 20. Z.Y. Li, Numerical Simulation of Axial Compressor and Construction of Entropy Generation Loss Model Based on the Hybrid RANS/LES Method (University of Chinese Academy of Sciences, Beijing, China, 2009) (in Chinese) 21. F. Kock, H. Herwit, Entropy production calculation for turbulent shear flows and their implementation in CFD codes. Int. J. Heat Mass Transf. 26(4), 672–680 (2005) 22. J.S. Kramer-Bevan, A Tool for Analyzing Fluid Flow Losses (University of Waterloo, Waterloo, Canada, 1992)

Chapter 28

The Identification of Tip Leakage Vortex of an Axial Flow Waterjet Pump by Using Omega Method and Liutex Xiaoyang Zhao, Jie Chen, Biao Huang, and Guoyu Wang

Abstract  The tip leakage vortex (TLV) is a common phenomenon in the field of rotating machinery. The formation and evolution of TLV can lead to some adverse effects, such as the interfering with the main flow, the occurrence of vortex cavitation, etc. This paper investigates the structure of TLV in an axial flow waterjet pump by using two vortex identification methods, namely the omega method and the Liutex. These two criterions can capture the primary characteristics of the vortex structures with different vorticity strength in the flow passage and the tip gap region. As the cavitation number decrease, the tip leakage cavitation and the sheet cavitation on the suction surface of the rotor blade present evident interactions with the TLV. The accuracy and capability of these two methods present different effects. The Liutex method can not only capture more detailed vortex characteristics in the unstable regions but also some weak vortex structures alongside with the TLV core. Keywords  Omega method · Liutex method · Tip leakage vortex · Axial flow waterjet pump

1  Introduction The axial flow waterjet pump has been widely used in high speed boats, amphibious vehicles and submarines. Due to the clearance between the blade tip and the casing, the tip leakage flow (TLV) is one of the typical flow structures in the rotor passage. The TLV may disturb the main flow and cause instability. The TLV core is often considered as the low pressure region, where the vortex cavitation occurs. The evolution of TLV and the mechanisms of the interaction between TLV and other cavitation structures have drawn many researchers’ attention. Miorini et al. investigate the X. Zhao · J. Chen · B. Huang (*) · G. Wang School of Mechanical Engineering, Beijing Institute of Technology, Beijing, China e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_28

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inner structures of the tip flow and the evolution of the TLV by using PIV measurements [1]. They found that the leakage flow can lead to the flow separation by entraining the casing boundary layer. Wu et al. study the cavitation in the tip region by PIV measurements [2]. They found that the cavitation bubbles accumulate along the tip on the pressure side firstly, then the TLV will rolls up these bubbles and transports them pass across the tip region. Apart from these experimental works, many researchers have applied CFD method and revealed the mechanisms about the evolution of the TLV and tip leakage cavitation. Hah and Katz combined RANS and LES method to simulate the TLV and presented that the periodic TLV, which cross the tip gap of the adjacent blade, will produce vortex ropes and lead to the pressure changes in the tip gap region [3]. Zhang et al. simulated the formation of 3-D TLV cavitation cloud and the breakdown of the suction-side-perpendicular cavitating vortices (SSPCV) [4]. They found that the SSPCV is produced due to the shedding cavitation cloud in the suction surface and will further evolve under the influence of the TLV and the tip leakage flow. To better understand the vortex structures, it is necessary to realize the vortex visualization by introducing some vortex identification criterions. The well-defined notion of the vortex has been a long-standing issue. In the early days, the Helmholtz’s theorems were widely accepted and the concept of vorticity was used to describe the vortices [5, 6]. However, Robinson et al. proposed that the relationship between the actual vortex structures and the high vorticity regions could be weak, especially for the turbulent boundary layer near wall [7]. Since then, many vortex identification methods (i.e. Q criterion [8], Δ criterion [9], λ2 criterion [10], λci criterion [11]) have been developed by many researchers. Those vortex identification methods still have some shortcomings, such as the dependence of the threshold and the misidentification of the strong shear region. In recent years, Liu et al. proposed the new omega method [12] and the Liutex method [13], which improves the accuracy and practicability of the vortex identification. On that basis, Liu et al. developed the objective omega method to ensure that the vortex structures keep constant for a moving observer [14]. Dong et al. proposed a new normalized vortex identification method named Ω-Liutex method, which originates from omega method and Liutex method [15]. Zhang et al. applied the new omega method to analyze the vortices in different working conditions for a revisable pump turbine [16]. They found that both the strong and the weak vortices can be identified by the omega method at the same threshold. Wang et al. investigated the complex vortices in the turbine rotor passage by using for different kinds of identification methods [17]. Their results show that the Liutex method is capable of analyzing the vortex structures and their evolution in the rotor passage. Xu et  al. applied different vortex identification methods to study the vortex dynamic characteristics in the waterjet pump [18]. Their results show that the omega method and the Liutex method can predict both the strong and the weak vortex structures in the cavitating flow.

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2  Numerical Model 2.1  Conservation of Mass and Momentum The incompressible and unsteady Reynolds-averaged Navier-Stokes (URANS) equations can be shown as follows in cartesian coordinates. ∂ρ m ∂ ( ρ m u j ) + =0 ∂t ∂x j

∂ ( ρ m ui )

∂t

+

∂ ( ρ m ui u j ) ∂x j

=−

∂p ∂ + ∂xi ∂x j

(28.1)

  ∂u ∂u j 2 ∂uk   δ ij   (28.2) − ( µ m + µ t )  i +   ∂x   j ∂xi 3 ∂xk  



∂ρlα l ∂ ( ρlα l u j ) = m + + m − + ∂t ∂x j



ρ m = ρlα l + ρ vα v

(28.4)



µm = µlα l + µvα v

(28.5)

(28.3)

Where u is the velocity, p is the pressure, ρm is the mixture density, ρl is the liquid density, ρv is the vapor density, μm is the mixture laminar viscosity, μl and μv are respectively the liquid and vapor dynamic viscosity, μt is the turbulent viscosity, m + and m − are the source term and sink term, which respectively represent the condensation rate and evaporation rate.

2.2  Turbulence Model The SST model can produce the reasonable simulation results for the flow regions away from the wall or near the wall. The SST k–ω turbulence model, being used to solve the URANS equations in this study, are shown as follows in Einstein notation. ∂ ( ρm k ) ∂t



∂ ( ρ mω ) ∂t

+

+

∂ ( ρmu j k ) ∂x j

∂ ( ρmu jω ) ∂x j

= Pk − Dk +

µ ∂    µm + t σk ∂x j  

 ∂k   ∂x j

  

µ  ∂ω  ∂     µm + t  σ k  ∂x j  ∂x j   1 ∂k ∂ω + 2 ρ m (1 − F1 ) σ ω 2 ω ∂x j ∂x j

(28.6)

= Cω Pω − βω ρ mω 2 +

(28.7)

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The viscosity is defined as: vt =

ρ a1k max ( a1ω ,SF2 )

(28.8)

Where vt = μt/ρ, F1 and F2 are the blending functions, S is an invariant measure of the strain rate.

2.3  Cavitation Model for Simulation The Zwart cavitation model has been applied in this study. The cavitating flow is governed by the mass transfer equation (i.e. Eq. 28.3) and the condensation rate m + and the evaporation rate m − are defined as follows: 1

m + = C prod

3α v ρ v  2 p − pv  2   , p > pv RB  3 ρl 

(28.9)

1

3α (1 − α v ) ρ v  2 pv − p  2 m = −Cdest nuc   , p < pv RB  3 ρl  −



(28.10)

where pv is the saturated vapor pressure, p is the local fluid pressure, αnuc is the nuclei volume fraction, RB is the bubble radius, Cdest is the constant generation rate of vapor when p  pv.

3  Vortex Identification Methods 3.1  Omega Method The basic principle of the new omega method is to introduce a parameter Ω to represent the ratio of the vertical vorticity over the whole vorticity, and the Ω can be defined as follow:

Ω = b / (a + b + ε )



a = tr AT A Σ 3 i = 1Σ 3 j = 1( Aij )



b = tr BT B Σ 3 i = 1Σ 3 j = 1( Bij )

(28.11)



(

)

2

(

)

2



(28.12) (28.13)

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Where a and b are the square of the Frobenius norm of A and B, respectively, ε is a small positive number used to avoid division by zero.

3.2  Liutex Method Liutex is a newborn physical quantity, which has direction and magnitude. The explicit expression of the magnitude of liutex can be defined as follow:   R = Rr



1  ∂v ∂u   ∂v ∂u  + −   +  2  ∂Y ∂X   ∂X ∂Y  2

α=



(28.14)

1  ∂v ∂u  β=  − 2  ∂X ∂Y 

2

(28.15)

(28.16)

And if α2-β2  0, then R = 2(β-α); if α2-β2  0.5 represents that the antisymmetric tensor B is dominant over the symmetric tensor A, so Ω slightly larger than 0.5 can be used as the criterion for vortex identification. Liutex represents the rigid rotating part of fluid motion. In order to obtain the size R of Liutex, first, use a Q rotation to rotate the initial coordinate system xyz1 to xQyQzQ, so that the rotated zQ is in the same direction as the rotation axis r, and the speed gradient after rotation Tensor ∇VQ becomes:



 ∂uQ   ∂xQ  ∂v ∇VQ = Q∇VQ T =  Q  ∂xQ   ∂wQ   ∂xQ

∂uQ ∂yQ ∂vQ ∂yQ ∂wQ ∂yQ

 0    0    ∂wQ  ∂zQ 

(29.9)



Among them, Q is the coordinate rotation matrix, (uQ, vQ, wQ) is the velocity component in the xQyQzQ coordinate system after rotation. Here, the Rodrigues rotation formula is used to solve Q, and the specific expressions can be found in Gao and Liu [18] and Liu et al. [19]

4  Results and Analysis From Fig. 29.5, it is found that the temperature and high temperature are mainly concentrated at 0.4–0.8 m, and there is no obvious high temperature zone behind the furnace, which indicates that the gas mixture is reduced later. The gas is mainly mixed in the front half of the furnace. In Fig. 29.6 it is found that the recirculation of flue gas is mainly concentrated at 0–0.8 m of the furnace, indicating that the main combustion zone of the furnace is in the first half of the furnace, which is consistent with the phenomenon in Fig. 3.1. Asymmetric nozzles are used, and different speeds are used to transport gas fuels. The gas collides in the furnace. Due to the different speeds of the gas, gas recirculation will be formed after the collision, resulting in a swirl structure. In Figs. 29.7, 29.8 and 29.9, the vortex structure in the furnace is found to be concentrated at the position of 0–0.8 m in the furnace. It shows that the temperature combustion in the furnace is mainly concentrated at the position of 0–0.8 m, which can just coincide with the previous temperature cloud diagram and streamline diagram. Explain that the gas mixture in the furnace can be judged by the vortex structure diagram. However, comparing Figs. 29.7, 29.8 and 29.9, we find that the three identification methods have different recognition effects on the vortex structure. The vortex structure in the furnace can be identified by the Liutex vortex structure identification method, while the Q criterion can only identify the vortex structure dia-

436

Fig. 29.5  Temperature cloud chart when Z = 0

Fig. 29.6  Streamline cloud diagram when Z = 0

Fig. 29.7  Liutex vortex structure recognition map

Fig. 29.8  Q criterion vortex structure identification diagram

Fig. 29.9  Omega criterion vortex structure recognition diagram

C. Guo et al.

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gram in the front half of the furnace, and the vortex structure in the back half of the furnace cannot be accurately identified by the Q criterion. In combination with the streamline diagram, the front half of the furnace is the main mixing zone for gas flow, and the second half of the furnace is the secondary mixing zone. However, the Q criterion cannot well identify the vortex structure in the second half of the furnace, indicating that the gas mixture cannot be Q criterion accurate judgment. From Fig. 29.9, it is found that the vortex structure is distributed throughout the furnace. The vortex structure is stronger in the front half of the ­furnace and weaker in the second half. Combining Figs. 29.7 and 29.9, the gas mixture in the furnace can be well reflected, but the combination of Figs. 29.8 and 29.9 can not make the gas mixing in the furnace reflect well, because in Fig. 29.8 in the second half of the furnace Part of the vortex structure cannot be displayed well, which contradicts the result in Fig. 29.9. Regarding the regulation of the threshold, the threshold range of the Q criterion is too large and difficult to control. The threshold range of the Liutex vortex recognition method is significantly smaller than the Q criterion, which is more convenient to use. Therefore, in the identification of vortex structure, the Liutex vortex structure identification method has more advantages than Q criterion. So using the Liutex vortex identification method can better reflect the gas mixture in the furnace.

5  Conclusions Through direct numerical simulation of the oxygen heating furnace, the simulation data is imported into tecplot for analysis, three eddy structure recognition methods are used to analyze the data, and the temperature cloud diagram and streamline diagram are verified, and the following conclusions are obtained: 1. The gas mixing in the furnace is mainly concentrated in the front part of the furnace, which can be clearly identified by the vortex structure identification method. 2. The Liutex vortex recognition method is obviously better than the Q criterion in vortex recognition, and the gas mixing situation in the furnace can be better reflected by the Liutex vortex recognition method. 3. The small vortex structure in the furnace can be well recognized by the Ω vortex recognition. It can be combined with the Liutex method to clearly verify that the asymmetric nozzle is easier to achieve MILD combustion. Acknowledgments  This study was financially supported by National Natural Science Foundation of China (Grant Nos. 51964039, 51464041, 51164025) and National Natural Science Foundation of Inner Mongolia (Grant No. 2018MS05009).

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References 1. P.-f. Li, Experimental and Numerical Investigation of the MILD Combustion of Gaseous Fuels in a Laboratory-Scale Furnace (Peking University, Beijing, 2013) 2. M. Katsuki, T. Hasegawa, The science and technology of combustion in highly preheated air. Proc. Combust. Inst. 27, 3135–3146 (1998) 3. A.K. Gupta, in Flame Characteristics and Challenges with High Temperature Air Combustion. Proceedings of 2000 International Joint Power Generation Conference, Miami Beach, Florida, 2000, vol 2, no 36, pp. 1–18 4. H.  Tsuji, A.  Gupta, D.G.  Lilley, et  al., High Temperature Air Combustion: From Energy Conservation to Pollution Reduction (CRC Press, FL, 2003) 5. J.A.  Wünning, J.G.  Wünning, Flameless oxidation to reduce thermal no-formation. Prog. Energy Combust. Sci. 23(1), 81–94 (1997) 6. C. Rottier, C. Lacour, G. Godard, B. Taupin, A.M. Boukhaifa, et al., in An Aerodynamic Way to Reach Mild Combustion Regime in a Laboratory-Scale Furnace. Third European Combustion Meeting ECM (2007) 7. C. Liu, Y. Yan, P. Lu, Physics of turbulence generation and sustenance in a boundary layer. Comput. Fluids 102, 353–384 (2014) 8. J.M. Wallace, Highlights from 50 years of turbulent boundary layer research. J. Turbul. 13(53), 1–70 (2013) 9. J. C. R. Hunt, A. A. Wray, P. Moin, Eddies, streams, and convergence zones in turbulent flows. Proceedings of the 1988 CTR Summer Program, pp. 193–208 (1988) 10. C. Liu, Y. Wang, Y. Yang, et al., New omega vortex identification method. Sci. China Phys. Mech. Astron. 59(8), 6–9 (2016) 11. C. Liu, Y. Wang, Y. Yang, et al., New omega vortex identification method. Sci. China Phys. Mech. Astron. 59(8), 684711 (2016) 12. Y.N. Zhang, K.H. Liu, J.W. Li, et al., Analysis of the vortices in the inner flow of reversible pump turbine with the new omega vortex identification method. J. Hydrodyn. 30(3), 463–469 (2018) 13. Y.N. Zhang, X. Qiu, F.P. Chen, et al., A selected review of vortex identification methods with applications. J. Hydrodyn. 30(5), 767–779 (2018) 14. C. Liu, Y. Gao, S. Tian, et al., Rortex—A new vortex vector definition and vorticity tensor and vector decompositions. Phys. Fluids 30, 035103 (2018) 15. Q. Ouyang, L.M. Zhao, L.Y. Wen, C.G. Bai, Simulation study on radiative imaging of pulverised coal combustion in blast furnace raceway. Ironmak. Steelmak. 38(3), 181–118 (2014) 16. J.C.  Hunt, A.A.  Wray, P.  Moin, Eddies, stream, and convergence zones in turbulent flows. Center Turbul. Res. 25, 193–208 (1988) 17. X.R. Dong, Y.Q. Wang, F.P. Chen, Y.L. Dong, Y.N. Zhang, C.Y. Liu, Determination of epsilon for Omega vortex identification method. J. Hydrodyn. 30(04), 5–12 (2018) 18. Y. Gao, C. Liu, Rortex and comparison with eigenvalue- based vortex identification criteria. Phys. Fluids 30, 085107 (2018) 19. C. Liu, Y. Gao, X.R. Dong, et al., Third generation of vortex identification methods: Omega and Liutex/Rortex based systems. J. Hydrodyn. 31(2), 205–223 (2019)

Chapter 30

Numerical Investigation of the Cavitation-­Vortex Interaction Around a Twisted Hydrofoil with Emphasis on the Vortex Identification Method Jie Chen, Biao Huang, Qin Wu, and Guoyu Wang

Abstract  Large eddy simulations combined with Zwart–Gerber–Belamri cavitation model is applied to investigate the physical interaction of the cavitation-vortex dynamics around a 3D twisted hydrofoil. The vortex structure of the hydrofoil surface at a typical instant is divided into three regions by the iso-surfaces of Ω method, i.e., the attach region-A, the shedding region-B and C. The results show that each region presents the different characteristic vortex structure, the attach region-A is primary and the secondary U-type vortex, the shedding region-B is O-shape vortex and 3D waves of the vortex structure, and the shedding region-C is the large-scale vortex structures. Then, the different vortex identification methods, namely, ω criterion, Q criterion, Ω method and Liutex method, is employed to capture vortex structures of unsteady cavitating flow. The 3D waves of the vortex structure and large-scale vortex structures are accurately captured with other vortex identification method, except for the ω criterion. It is noticeable that Liutex method can not only clearly display the U-type vortex structure, but also accurately describe its direction. The distribution of Liutex inside the primary U-type vortex structure is symmetric, and the head magnitude of U-type vortex is smaller than that of legs. The average magnitude of Liutex at the core is 1394.46. For the secondary U-type vortex structure, the magnitude of votex structure near the center of the hydrofoil is greater than that of the other side, and the average value is 738.1. Keywords  Large eddy simulations · Cavitation-vortex structure · Liutex · U-type vortex

J. Chen · B. Huang (*) · Q. Wu · G. Wang School of Mechanical Engineering, Beijing Institute of Technology, Beijing, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 C. Liu, Y. Wang (eds.), Liutex and Third Generation of Vortex Definition and Identification, https://doi.org/10.1007/978-3-030-70217-5_30

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1  Introduction Cavitation commonly occurs in fluid machinery when the local fluid pressure is lower than the vapor pressure, such as valves, pumps, and turbines. The occurrence of it significant influence on the hydrodynamic performance and generates noise, vibration, and surface erosion [1–3]. The unsteadiness of cavitating flow and its shedding dynamics involves the complex interactions between phase-change and vortex dynamics [4]. Gopalan and Katz [5] applied PIV and high-speed photography to investigate the flow structure around the closure region and the downstream of sheet cavitation in the nozzle. The results demonstrated that the collapse of cavity was a primary source of large-scale hairpin vortices and massive vorticity. Foeth [6, 7] conducted a series of experiments to study the cavitating flows around the Delft twisted hydrofoil. He found that the highly unsteady cavitating flow experiences quasi-periodic primary and secondary cavity shedding. Huang et al. [8] and Ji et al. [9] analyzed the effect of cavitation on different terms of vorticity transport equation. The results demonstrated that the terms of dilatation and baroclinic of the cavity surface increased with the process of mass transfer, which is responsible for cavitation vortex generation. Ji et al. [9, 10] simulated the evolution of cavitation structure around the Delft Twist hydrofoil. The horse-shoe vortex in cavitating flow was observed, which was closely associated with the cavity shedding. Peng et al. [11] suggested that the interactions between the circulating flow and the shedding vapor cloud may be responsible for the formation of the U-type cloud cavity structures. Many interesting works have been conducted to understand the complex physical interactions between cavitation and vortex structure. However, the vortex identification method is crucial to accurately identify and analyze the vortex structures of the cavitation process. Long et al. [12] investigated the cavitation-vortex interaction around a Clark-Y hydrofoil with Q criterion. They pointed out that the periodic vortex shedding, the steady growth of the length of vortex structure and large-scale vortex with a three-dimensional turbulent structure transported into the hydrofoil wake. Hossain et al. [13] applied Q criteria, Δ criteria [14], λ2 criteria [15], λci criteria [16], and vorticity to investigate the large-scale vortex structures and the instantaneous three-dimensional coherent structures inside the cavity. The results revealed that the different vortex identification methods with suitable threshold can captured the same vortex structures. Recently, a new vortex identification criterion—Ω method with insensitive threshold was proposed by Liu et al. [17, 18]. Compared to the other vortex identification methods, Ω method release the definite physical meaning. Chen et al. [19] employed ω criterion, Q criterion and Ω method to investigate cavitation-vortex dynamics interaction around 2D Clark-Y hydrofoil. The results indicated that the Ω method can capture both strong and weak vortical structures, especially for the weak vortices of the boundary of the attached cavity and the back region of the re-entrant jet. Wang et al. [20] applied different vortex identification methods involves vorticity, Q criterion, Ω method and λ2 method to investigate the cavitation vortex dynamics. The result showed that the Ω method can identify

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the vortices outside the cavity, at the pressure side and at wake region of hydrofoil, which testified that this method can simultaneously identify both the strong and weak vortices. The correlation between the cavitation and vortex dynamic is still not inadequate understood due to lack of the precise mathematical definition of vortex. Recently, Liu et  al. [21, 22] proposed a new vortex identification method named Liutex. The Liutex vector simultaneously represents the magnitude of local rotation intensity and the direction of local rotation axis [22, 23]. Xu et al. [24] investigated the tip leakage vortex (TLV) of waterjet pumps using different vortex identification methods, including vorticity, Q criterion, λci, λ2 criterion, Ω method and Liutex/ Rortex. The result indicated that the Ω and Liutex can effectively captured the TLV core, and only the Ω and Liutex can predict weak vortices in cavitation flow [20]. In the present paper, the physical interaction of the cavitation-vortex dynamics around the twisted hydrofoil is investigated by large eddy simulations (LES) combined with Zwart-Gerber-Belamri cavitation model. The vortex structure of the hydrofoil surface at a typical instant is divided into three regions by the iso-surfaces of Ω method, i.e., the attach region-A, the shedding region-B and C.  The vortex structures of unsteady cavitation around hydrofoil are analyzed by using different vortex identification methods, i.e. ω criterion, Q criterion, Ω method and Liutex method. To present the vortex details, the magnitude of Liutex inside the primary and secondary U-type vortex structure is captured.

2  Mathematical Formulations and Numerical Method 2.1  Cavitation and Turbulence Model The large eddy simulation (LES) method is firstly proposed by Smagorinsky [25] to simulate atmospheric air currents. The main idea of LES is to separate large-scale vortex structure from small-scale vortex structure by the low-pass filter. Then, the large-scale vortex structure is solved directly and the small-scale vortex structure is modeled by sub-grid scale (SGS) model. Compared to Direct Numerical Simulation (DNS), LES reduces the requirement in computational resource. This model can remarkably improve the accuracy of prediction in cavitating cases [26, 27]. In present paper, LES method is used to solve the unsteady cavitating flow around a Delft twisted hydrofoil. In the cavitating flow, the mixture model of liquid/vapor two-phase flow is assumed to be homogeneous, thus the multiphase fluid components are considered to share the same velocity and pressure. The continuity and momentum equations for the mixture flow are



∂ρm ∂ ( ρm u j ) + =0 ∂t ∂x j

(30.1)

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∂ ( ρm ui ) ∂t



+

∂ ( ρm ui u j ) ∂x j

=−

∂u  ∂p ∂  +  µm i  ∂xi ∂x j  ∂x j 

(30.2)

where ui is the velocity component in the i direction, p is pressure. The density ρm and laminar viscosity μm of mixture is defined as

ρm = ρlα l + ρvα v



(30.3)



µm = µlα l + µvα v

(30.4)

where α is the volume fraction of one component. The subscripts v and l refer to the vapor and liquid components. The LES equation can be obtained by applying a Favre-filtering operation to Eqs. (30.1) and (30.2),

(



(

∂ ρm u i

)

∂ρm ∂ ρm u j + =0 ∂t ∂x j

∂t

) + ∂ ( ρ u u ) = − ∂p + m

i

j

∂x j

∂x i

(30.5)

∂u ∂   µ m i ∂x j  ∂x j

 ∂τij  −  ∂x j

(30.6)

where the over-bars denote filtered quantities. An extra non-linear term is observed in Eq. (30.6), called the Sub-Grid Scale (SGS) stresses, which resulting from the filtering operation are unknown and require modeling, are defined as

(

τij = ρm u i u j − u i u j

)

(30.7)

To model the SGS stresses, one commonly used SGS model is the eddy-viscosity model, which assumes that the SGS stress is proportional to the modulus of the strain rate tensor, Sij,



1 τij − τkk δ ij = −2µ t Sij 3

(30.8)

where τkk is the isotropic part of the SGS stresses, Sij is the rate-of-strain tensor for 1  ∂u ∂u j   , and μt is the sub-grid the resolved scale, which defined by Sij =  i + 2  ∂x j ∂xi  scale turbulent viscosity. In order to close the equation, the Wall-Adapting Local Eddy-Viscosity (WALE) model [28] is adopted is adopted to the sub-grid scale turbulent viscosity,

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µt = ρ L

2 s

(S S ) (S S ) + (S S ) d ij

5/ 2

ij

Sijd =



gij =

d ij

3/ 2

d ij

ij

d ij

(30.9)

5/ 4



1 2 1 gij + g 2ji − δ ij gkk2 2 3

∂ui , ∂x j

(

)

(

443

Ls = min kd ,CsV 1/ 3

(30.10)

)

(30.11)

where Ls is the mixing length for the SGS, κ is von Karman’s constant, d is the distance to the closest wall, V is the volume of the computational cell and Cs is the WALE constant having the value of 0.325.

2.2  Numerical Setup In order to accordance with the experiment [6], the 3D twist hydrofoil with chord length c is 0.15 m and the attack angle of the entire foil equals to −2° is applied in the present study, the detaile of the geometry structure is shown in Fig. 30.1. The numerical simulation is conducted based on the commercial CFD package ANSYS FLUENT 17.1. The computational domain and boundary conditions are consistent with the experiment [6], as shown in Fig. 30.2. The height and length of computational domain is 2c and 7c, respectively. In order to reduce the requirement on computational resources, only half of the test section is used in simulations due to its geometric symmetry. The free slip walls are exerted on the top, bottom and side of the channel, the inlet flow velocity is set as U∞ = 6.97 m/s, and the outlet static pressure is 29,159 Pa according to the cavitation number, defined as σ = (pout-­ pv)/(0.5ρlU∞2) = 1.07, where pout is the outlet static pressure. The hydrofoil surface is set as no-slip boundary condition and the midplane is set to symmetry. The

Fig. 30.1  Three-dimensional Delft Twist-11 hydrofoil: (a) 3D view; (b) attach angle distribution across the span

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Fig. 30.2  Computational domain and boundary conditions

c­ omputational domain with an O-H type grid is sufficient refined near the foil surface due to the requirement of the wall-normal grid distance in LES, and a typical configuration of the computational meshes around the three dimensional hydrofoil surface with 80 nodes in the spanwise direction as shown in Fig. 30.3. Excessive iteration in each time step will cost too much computational resource, and insufficient iterations lead to insufficient accuracy. Finally, 40 iterations in each time step is selected to keep balance between computational accuracy and efficiency. Meanwhile, the RMS residuals target is set to 10−4 and time steps is 1 × 10−5 s, which is sufficient for unsteady flows with cavitation [12]. The cavitation model is first switched off to compute the steady non-cavitating flow with k-ω SST turbulence model until a fully developed turbulence is achieved. Then, the cavitation model and LES are turned on for the unsteady cavitation flow simulation.

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Fig. 30.3  Typical 3D hydrofoil surface with 80 nodes in the spanwise direction of the computational domain mesh

Fig. 30.4  Vortex structures on the suction side in non-cavitating flow (Q = 100 s−2)

2.3  Validation of the Numerical Results LES requires substantially finer mesh to capture the intrinsic turbulence structures. Validation of mesh resolution for LES cannot rely on mesh convergence analysis. Figure 30.4 shows the vortex structures of the suction side in non-cavitating flow by iso-surface of Q criterion, defined as,

446



J. Chen et al. 2 1  ∂u i  ∂u i ∂u j    Q=   − 2  ∂x t  ∂x j ∂x i   

(30.12)

The results of simulation suggest that the vortex structure on the surface of hydrofoil is accurately presented, especially near the trailing edge. Moreover, the more details of hairpin vortex (red dashed arrow) are captured with this mesh resolution. The dimensionless wall distance y +