Editors: Chaoqun Liu, Yisheng Gao

Series Title: Current Developments in Mathematical Sciences

Liutex-based and Other Mathematical, Computational and Experimental Methods for Turbulence Structure

Volume 2

eBook: US $99 Special Offer (PDF + Printed Copy): US $168
Printed Copy: US $119
Library License: US $396
ISSN: 2589-2711 (Print)
ISSN: 2589-272X (Online)
ISBN: 978-981-14-3758-8 (Print)
ISBN: 978-981-14-3760-1 (Online)
Year of Publication: 2020
DOI: 10.2174/97898114376011200201


The knowledge of quantitative turbulence mechanics relies heavily upon the definition of the concept of a vortex in mathematical terms. This reference work introduces the reader to Liutex, which is an accepted, accurate and mathematical definition of a vortex. The core of this book is a compilation of several papers on the subject. presented in the 13th World Congress of Computational Mechanics (WCCM2018), Symposium 704, Mathematics and Computations for Multiscale Structures of Turbulent and Other Complex Flows, New York, United States on July 27, 2018. This compilation also includes other research papers which explain the work done on the vortex definition, vortex identification and turbulence structure from different insight angles including mathematics, computational physics and experiments.

The thirteen chapters in this volume will be informative to scientists and engineers who are interested in advanced theories about fluid dynamics, vortex science and turbulence research.


Turbulence is a centuries-long world puzzle. Turbulence coherent structure really means vortex structure. However, there was no mathematical definition of vortex ever before and there was no mathematical definition for turbulence either. Therefore, there was no vortex science or vortex dynamics since vortex had no definition. Since turbulence is built up and driven by vortices, there was no serious scientific research on turbulence theory and turbulence structure because there was no definition of vortex. This book collected a lot of scientific efforts to give an accurate and mathematical definition of vortex, that is Liutex developed by Liu et al [C. Liu et al., Phys. Fluids 30, 035104 (2018)].

The core of this book is a collection of papers presented in the 13th World Congress of Computational Mechanics (WCCM2018), Symposium 704, Mathematics and Computations for Multiscale Structures of Turbulent and Other Complex Flows, New York, United States on July 27, 2018. This book also collects quite a number of other research papers working on the vortex definition, vortex identification and turbulence structure from different insight angles including mathematics, computations and experiments. Of course, the priority is dedicated to an accurate and mathematical definition for vortex, which was first named “RORTEX” in 2018 and was changed to “LIUTEX” approved unanimously by alliance of six universities and Alliance of Vortex Research in 2019. Besides Liutex, this book also publishes a lot of efforts to do analysis on turbulence structure by unobjectionable mathematics, incredible DNS computations, and marvelous experiments.

This book contains thirteen chapters which are briefly introduced in this preface.

The first chapter, Liutex – A New Mathematical Definition of Vortex and Vorticity Decomposition for Turbulence Research, is written by Chaoqun Liu, Yisheng Gao and Yifei Yu at University of Texas at Arlington. For long time, people recognize vortex as vorticity tube and measure the vortex rotation strength by vorticity magnitude. These misunderstandings have been carried out by thousands of research papers and almost all textbooks. Robinson (1989) has found the association between regions of strong vorticity and actual vortices can be rather weak. Many vortex identification criteria have been proposed. However, vortex still has no rigorous mathematical definition with direction and magnitude and the relationship between vortex and vorticity was unknown. Because we do not have definition for vortex, there was really no vortex science. Since vortex is the building block and the muscle of turbulence, lack of mathematical definition for vortex becomes a bottle neck for turbulence research. Really, there was no serious turbulence research without definition of vortex. In our recent work, a mathematical definition called Liutex (Previously called Rortex) is given to identify the rigid rotation of fluid motion. Since vortex core is near the rigid rotation, Liutex naturally represents the vortex cores. Liutex is a local mathematical vector definition with direction and magnitude of pure rotation without shear contamination, which is unique and Galilean invariant. The Liutex direction is defined as the local rotation axis and the Liutex magnitude is the local rotation. More important, we derive the accurate mathematical relation between vorticity and vortex, which is Vorticity = Liutex + Antisymmetric Shear (RS decomposition). This new discovery is an important breakthrough in modern fluid dynamics and is extremely important for turbulence research. In addition, the velocity gradient tensor has been decomposed to two parts, R (rigid rotation) and NR (non-rotation part) as a counterpart of the traditional Cauchy-Stokes decomposition (Helmholtz decomposition) which is improper since vorticity cannot represent flow rotation. Liutex is a new physical quantity like velocity, vorticity, temperature, pressure, which has been ignored by our founding fathers of fluid dynamics for centuries but is particularly important for vortex dynamics and turbulence research. Introduction of Liutex, RS decomposition of vorticity, and R-NR decomposition of velocity gradient would open a new era for vortex dynamics and new turbulence research, likely new fluid dynamics.

The second chapter, Liutex – a New Vortex definition, and its Calculation and Galilean Invariance, is written by Yiqian Wang, Yisheng Gao and Chaoqun Liu from University of Shanghai for Science and Technology in China and University of Texas at Arlington in USA. The Liutex (previously known Rortex) method introduces a vortex vector field to mathematically and systematically describe vortices in flow fields. In the present study, the calculation procedure of Liutex is revisited which includes two-step reference coordinate rotation. Then, for the first time, an explicit formula to calculate Liutex is derived and the physical intuition and efficiency improvement brought by this formula are discussed. In addition, the Galilean invariance, which has been widely accepted as a preliminary check for a successful vortex identification method is discussed for Liutex vector.

The third chapter, New Omega Vortex Identification Method Based on Determined Epsilon, is authored by Xiangrui Dong, Yisheng Gao, Chaoqun Liu from University of Shanghai for Science and Technology and University of Texas at Arlington. A new Omega method with 𝜀 determination is introduced to represent the ratio of vorticity square over the sum of vorticity square and deformation square, for the vortex identification. the advantages of the new Ω method can be summarized as follows: (1) Omega, as a ratio of the vorticity squared over the sum of the vorticity squared and deformation squared, is a normalized and case-independent function which satisfies Ω ∈ [0,1] ; (2) Compared with the other vortex visualization methods, which require a wide threshold change to capture the vortex structures, Ω can always be set as 0.52 to capture vortex for different cases and time steps; (3) 𝜀 is defined as a function without any adjustment on its coefficient in all cases; (4) The Ω method can capture both strong and weak vortices simultaneously. In addition, Ω method is quite robust with no obvious change in vortex visualization.

The fourth chapter, Stability Analysis on Shear Flow and Vortices in Late Boundary Layer Transition, is solely authored by Jie Tang from University of Texas at Arlington. Turbulence is still an unsolved scientific problem, which has been regarded as “the most important unsolved problem of classical physics”. Liu proposed a new mechanism about turbulence generation and sustenance after decades of research on turbulence and transition. One of them is the transitional flow instability. Liu believes that inside the flow field, shear (dominant in laminar) is unstable while rotation (dominant in turbulence) is relative stable. This inherent property of flow creates the trend that nonrotational vorticity must transfer to rotational vorticity and causes the flow transition. To verify this new idea, this chapter analyzed the linear stability on two-dimensional shear flow and quasirotational flow. Chebyshev collocation spectral method is applied to solve Orr–Sommerfeld equation. Several typical parallel shear flows are tested as the basic-state flows in the equation. The instability of shear flow is demonstrated by the existence of positive eigenvalues associated with disturbance modes (eigenfunctions), i.e. the growth of these linear modes. Quasi-rotation flow is considered under cylindrical coordinates. An eigenvalue perturbation equation is derived to study the stability problem with symmetric flows. Shifted Chebyshev polynomial with Gauss collocation points is used to solve the equation. To investigate the stability of vortices in flow transition, a ringlike vortex and a leg-like vortex over time from our Direct Numerical Simulation (DNS) data are tracked. The result shows that, with the development over time, both ring-like vortex and leg-like vortex become more stable as Omega becomes close to 1.

The fifth chapter, POD and DMDAnalysis in Late Flow Transition with Omega Method, is authored by Sita Charkrit and Chaoqun Liu at Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019, USA. In this chapter, the proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are applied to analyze the 3D late transitional flow on the flat plate obtained from Direct numerical simulation (DNS). POD is used to find the most persistent spatial structures while DMD is used to find single frequency modes. The omega method is applied as a vortex identification to visualize vortices with iso-surfaces Ω = 0.52. The results in POD and DMD are discussed and compared to show the same and different features such as shapes, amplitudes and time evolutions.

The sixth chapter, Comparison of Liutex and Eigenvalue-based Vortex Identification Criteria for Compressible Flows, is written by Yisheng Gao and Chaoqun Liu at Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019, USA. Most of the currently popular vortex identification methods, including the 𝐐 criterion, the Δ criterion and the 𝛌𝐜𝐢 criterion, are exclusively determined by the eigenvalues or invariants of the velocity gradient tensor and thereby can be classified as eigenvalue-based criteria. However, these criteria will suffer from several shortcomings, such as inadequacy of identifying the rotational axis and contamination by shearing. Recently, a new eigenvector-based Liutex method (previously named Rortex) was proposed to overcome the issues associated with the eigenvalue-based criteria. In this paper, the comparison of Liutex and two eigenvalue-based criteria, namely the 𝛌𝐜𝐢 criterion and a modification of the original 𝐐 criterion, are performed to assess these methods for compressible flows. According to the analysis of the deviatoric part of the local velocity gradient tensor, all the scalar, vector and tensor forms of Liutex are valid for compressible flows without any modification, while two eigenvalue-based criteria, though applicable to compressible flows, are prone to severe contamination by shearing as in incompressible flows. Vortex structures in the problem of shock-vortex interaction are examined to confirm the validity and superiority of Liutex in compressible flows.

The seventh chapter, Observation of Coherent Structures of Low Reynolds Number Turbulent Boundary Layer by DNS and Experiment, is written by Panpan Yan from Beijing Jiaotong University, Beijing, 100044, China, Chaoqun Liu from University of Texas at Arlington, Yanang Guo and Xiaoshu Cai from University of Shanghai for Science and Technology, Shanghai, 200093, China. In order to study the characteristics of coherent structures of the turbulent boundary layer, the motion single frame, and long exposure imaging (MSFLE) method is proposed and an elaborate direct numerical simulation experiment was also conducted. MSFLE method is a Lagrangian measurement method, the speed of the camera is kept the same as the speed of the coherent structure, and the particle trajectory was captured by long exposure. By calculating the trace of the points on a chosen plane of the DNS result, we can obtain the particle trajectory like MSFLE method. Multilayer of vortex structures was observed and the evolution of the vortex packets with time was recorded.The result of the DNS simulation agrees well with the experiment. The size of the vortex of the different layer is almost the same, and no vortex breakdown was observed. The formation of the small-scale vortex is caused by sweeps and ejections of the larger coherent structures rather than the breakdown process.

The eighth chapter, Direct Numerical Simulation of Incompressible Flow in a Channel with Rib Structures, is authored by Ting Yu, Duo Wang, Heng Li and Hongyi Xu from Aeronautics and Astronautics Department, Fudan University Shanghai PR China. The paper applied the state-of-the-art flow simulation method, i.e. the Direct Numerical Simulation (DNS), and strongly coupled the DNS with the heat-transfer governing equation to solve the thermal-turbulence problem in both 2- dimensional(2D) and 3-dimensional(3D) channel with rib tabulator structures. An innovative approach was applied to the simulations in one case. The surface roughness effects of the cooling vane were considered by including the roughness geometry in the DNS and the immersed-boundary method were invented to handle the geometry complexities due to the roughness. Two inlet conditions, namely the uniform flow and full-developed turbulence, were applied at the inflow surface of the channel. Half height of the channel was used as the scale length. The Prandtl number was set at Pr = 0.7. Five Reynolds number of 1000, 2500, 5000, 7500 and 1000 were calculated in the 2D cases and the Reynolds numbers of 2500 and 5000 were applied in 3D cases where a periodical condition was applied in the span-wise direction. Additionally, Reynolds number of 10000 was set in the case with roughened surface. The stream-wise velocity, turbulence intensity, the Nusselt (Nu) number were analyzed. Results in 2D cases and 3D cases presented a great difference on flow structure. At the same time, with increasing Reynolds number, the length of recirculation zone and the enhancement of heat transfer showed a decreasing trend. A vortex identification method, the newly-defined Rortex, was applied.

The ninth chapter, Vortex and Turbulent Structure Inside Hydroturbines, is written by Yuning Zhang from Key Laboratory of Condition Monitoring and Control for Power Plant Equipment (Ministry of Education), School of Energy, Power and Mechanical Engineering, North China Electric Power University, Beijing, China and Yuning Zhang from College of Mechanical and Transportation Engineering, China University of Petroleum-Beijing, Beijing China and Beijing Key Laboratory of Process Fluid Filtration and Separation, China University of Petroleum-Beijing, Beijing China. In this chapter, various kinds of vortex in the hydroturbines are briefly introduced with a focus on the swirling vortex rope in Francis turbine and the vortex in the vaneless space of the reversible pump turbine. The vortex induced pressure fluctuation and vibrations are initially demonstrated based on the on-site measurement in the power stations. Then, detailed characteristics of the vortex in the hydroturbines are demonstrated based on the plenty of examples together with the aid of the quantitative analysis.

The tenth chapter, Comparative Study of Supersonic Turbulent Channel flows between Thermally and Calorically Perfect Gases, is written by Xiaoping Chen from National-Provincial Joint Engineering Laboratory for Fluid Transmission System Technology, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, China. In this chapter, to study the effects of gas model on the turbulent statistics and flow structures, direct numerical simulations (DNSs) of supersonic turbulent channel flow for thermally perfect gas and calorically perfect gas are conducted at Mach number 3.0 and Reynolds number 4800 combined with two wall temperature of 298.15K (low temperature condition) and 596.30 K (high temperature condition). The results show that, for high temperature condition, the effects of thermally perfect gas are important because the vibrational energy excited degree exceeds 0.1. Many of turbulent statistics used to express low temperature condition for calorically perfect gas still can be generalized for high temperature condition. The gas model does not have a significant influence on the strong Reynolds analogy. Omega could capture both strong and weak vortices simultaneously for supersonic flows, even under thermally perfect gas, which is difficult to obtain by Q. Compared to the results of calorically perfect gas, the vortex structure becomes smaller, sharper and more chaotic by considering thermally perfect gas.

The eleventh chapter The Experimental Study on Vortex Structures in Low Reynolds Number Turbulent Boundary Layer, was authored by Yanang Guo, Xiaoshu Cai, Wu Zhou, Lei Zhou, Xiangrui Dong from Institute of Particle and Two-phase Flow Measurement, University of Shanghai for Science and Technology, Shanghai, China. A motion single frame and long exposure (MSFLE) imaging method, which is a Lagrangian-type measurement, is experimentally carried out to study the vortex structures in a fully developed turbulent boundary layer with a low Reynolds number on a flat plate. In order to give the process of the vortex generation and evolution, on the one hand, the measurement system moves at the substantially same velocity as the vortex structure; on the other hand, a long exposure time is selected for recording the paths of the particles. In the experiment, the vortex structure characteristics as well as the temporal-spatial development can be shown by the streamwise-normal and streamwise-spanwise images which are extracted from a fully developed turbulent boundary layer. The result shows that the interaction between high and low-speed streaks induces the generation, deformation and ‘breakdown’ of the vortex structures, and badly influences the vortex evolution.

The twelfth chapter, Experimental Studies on Coherent Structures in Jet Flows using Single-Frame- Long-Exposure (SFLE) Method is authored by Lei Zhou, Xiaoshu Cai, Wu Zhou and Yiqian Wang from Institute of Particle and Two-phase Flow Measurement, University of Shanghai for Science, China. An experimental investigation on the flow structures in jet entrainment boundary layer flows based on the Single-Frame-Long Exposure (SFLE) method is carefully performed. It is found that two entrainment mode of ‘engulfing’ and ‘nibbling’ alternatively appear in the region of 2𝑑 to 3.5𝑑 in the axial direction and 1𝑑 to 1. 25𝑑 in the radial direction with 𝑑 being the diameter of jet nozzle. The appearance probability of such a pattern and the proportion of the ‘engulfing’ mode increases with Reynolds number 𝑅𝑒 when 𝑅𝑒 ≥ 1981 (the Reynolds number is based on the nozzle diameter and jet velocity). However, the influence of Reynolds number on this flow pattern becomes weaker when 𝑅𝑒 > 2245. The main frequency of this structure is found to be between 10-19Hz with Fourier analysis. The vortical structures are further explored with the moving SFLE (MSFLE) method, and it is found that vortices always exist near the turbulent and non-turbulent interface (TNTI).

The last chapter (thirteenth), Hybrid Compact-WENO Scheme for the Interaction of Shock Wave and Boundary Layer, is co-authored by Jianming Liu from Jiangsu Normal University of China and Chaoqun Liu from Department of Mathematics, University of Texas at Arlington, Arlington, USA. In this chapter, an introduction on hybrid Weighted Essentially non-oscillatory (WENO) method is given. The hybrid techniques including both central and compact finite difference schemes are introduced. The paper reviews the driven mechanism of the high order finite scheme required for compressible flow with shock. The detailed constructing processes of the compact and WENO schemes are given and the hybrid detector.

I hope this book will be useful to scientists and engineers who are interested in fundamental fluid dynamics, vortex science and turbulence research.

In conclusion, I want to thank the numerous authors for their incredible contributions and having patience in assisting us. Furthermore, I want to acknowledge and thank the referees for their tiresome work on making this book come to fruition. Last but not the least, I would also like to thank my family including Weilan Jin (my wife), Haiyan Liu (my daughter) and Haifeng Liu (my son) for their unconditional support. The co-editor, Yisheng Gao is also grateful to his family for the strong support.

Chaoqun Liu
Department of Mathematics
University of Texas at Arlington,
Arlington, Texas 76019,