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.
Table of Contents
Chapter 1 Liutex – a New Mathematical Definition of Vortex and Vorticity Decomposition for Turbulence Research
- Chaoqun Liu, Yisheng Gao and Yifei Yu
- Introduction
- Liutex – a New Eigenvector Based Mathematical Definition for Fluid Rotation
- Four Principles
- Definition of Liutex
- Calculation Procedure for Liutex
- Velocity Decomposition
- Vortex Gradient Tensor Decomposition Based on Liutex
- Vortex Identification
- Liutex for Compressible Flow
- Conclusion
- Consent for Publication
- Conflict of Interest
- Acknowledgements
- References
Chapter 2 Liutex and Its Calculation and Galilean Invariance
- Yiqian Wang, Yisheng Gao and Chaoqun Liu
- Introduction
- Revisit of the Liutex Vector Definition
- The Physical Meaning of Liutex, Vorticity and Ci
- Derivation of the Explicit Formula to Calculate Liutex
- Derivation - Approach 1
- Derivation – Approach 2
- Galilean Invariance of Liutex Vector
- Conclusions
- List of Symbols
- Consent for Publication
- Conflict of Interest
- Acknowledgements
- References
Chapter 3 New Omega Vortex Identification Method Based On Determined Epsilon
- Xiangrui Dong, Yisheng Gao and Chaoqun Liu
- Introduction
- Definition of a Vortex by New Omega Method
- Application of Omega Method with Determined Epsilon
- Case 1: Boundary Layer Transition
- Case 2: Swblis Controlled by Mvg
- Case 3: Channel Flow with Reô=950
- Conclusions
- Consent for Publication
- Conflict of Interest
- Acknowledgements
- References
Chapter 4 Stability Analysis on Shear Flow and Vortices in Late Boundary Layer Transition
- Jie Tang
- Introduction
- A Short History Review of Research on Flow Transition and Turbulence Generation
- Liu’S New Theory on Boundary Layer Transition
- The Purpose of the Current Chapter
- Organization of This Chapter
- Chebyshev Spectral Methods
- Introduction to Spectral Methods
- Chebyshev Polynomials of First Kind
- Chebyshev Collocation Approach
- Convergence of Chebyshev Spectral Method
- Advantages of Chebyshev Nodes
- Direct Numerical Simulation (Dns) Case Set Up and Code Validation
- Case Set-Up
- Governing Equation in Generalized Curvilinear Coordinates
- Numerical Methods
- Code Validation
- DNS Visualization Method
- Instability of Two-Dimensional Shear Flow
- Linear Stability Equation
- Chebyshev Discretization of the Orr-Sommerfeld Equation
- Numerical Results for Typical Shear Flows
- Linear Stability Equation for Quasi-Rotation Flow in Cylindrical
- Coordinates
- Derivation of Linear Perturbation System
- Eigenvalue Function
- Shifted Chebyshev Polynomials and Discretization
- Shifted Chebyshev Polynomials with Linear Arg Ument
- Shifted Chebyshev Polynomials with Quadratic Arg Ument
- Equation Discretiz Ation
- DNS Observations and Numerical Results
- Comparison of Two Shifted Chebyshev Polynomials in a Hyperbolic Case
- DNS Leg-Like Vortices Cases
- DNS Observa Tions
- Numerical R Esults
- DNS Ring-Like Vortices Cases
- DNS Observa Tions
- Numerical R Esults
- Conclusion
- Consent for Publication
- Conflict of Interest
- Acknowledgements
- References
Chapter 5 Pod and Dmd Analysis in Late Flow Transition with Omega Method
- Sita Charkrit and Chaoqun Liu
- Introduction
- Case Setup and Code Validation
- The Vortex Identification and the Omega Method
- Vortex Visualization
- Modal Decomposition
- The Proper Orthogonal Decomposition (Pod)
- Pod Algorithm
- Dimension Reduction and the Matrix Reconstruction
- Linear Combination of Pod Mode
- Pod Analysis for Late Transition Flow
- Dynamic Mode D Ecomposition (Dmd)
- DMD Algorithm
- Linear Combination of Dmd Mode
- The Matrix Reconstruction
- Diagnostic from Eigenvalues and Eigenvectors of a
- Dmd Analysis for Late Transition Flow
- Discussion and Conclusion
- Consent for Publication
- Conflict of Interest
- Acknowledgement
- References
Chapter 6 Comparison of Liutex and Eigenvalue-Based Vortex Identification Criteria for Compressible Flows
- Yisheng Gao and Chaoqun Liu
- Introduction
- Review of the Definition of Liutex
- Eigenvalue-Based Vortex Identification Criteria and Compressible
- Extension
- The Applicability of Liutex to Compressible Flows
- Test Case
- Concluding Remarks
- Consent for Publication
- Conflict of Interest
- Acknowledgement
- References
Chapter 7 Observation of Coherent Structures of Low Reynolds Number Turbulent Boundary Layer by DNS and Experiment
- Panpan Yan, Chaoqun Liu, Yanang Guo and Xiaoshu Cai
- Introduction
- Case Set Up
- DNS Case Setup and Code Validation
- Numerical Me Thods
- Code Valid Ation
- Experiment Setup
- Introduction to Msfle M Ethod
- Experimental FA Cility
- Experimental Valid Ation
- Result and Discussion
- Comparison Between DNS and Experiment Results
- Analytical Relations Between Liutex, 𝑸𝑫 and 𝝀𝒄𝒊
- Discussion on Multilevel Vortex Structures
- Multilevel Vortices Ejections and Sweeps
- Conclusions
- Nomenclature
- Consent for Publication
- Conflict of Interest
- Acknowledgements
- References
Chapter 8 Direct Numerical Simulation of Incompressible Flow in A Channel with Rib Structures
- Ting Yu, Duo Wang, Heng Li and Hongyi Xu
- Introduction
- Mathematical-Physical Models and Methods
- Governing Equations
- Computation Scheme
- Vortex Identification Method
- Results of Cases with Single Rib
- Computed Cases
- Results of Validation
- Flow Structures and Heat Transfer
- Results of Case with Roughened Surface
- Structure of the Computed Case with Surface Roughness
- Results of Thermal Fields
- Results of Rortex
- Conclusions
- Consent for Publication
- Conflict of Interest
- Acknowledgements
- Reference
Chapter 9 Vortex and Flow Structure Inside Hydroturbines
- Yuning Zhang and Yuning Zhang
- A Summary of Types of Vortex in Hydroturbines
- The Effects of Vortex on Pressure Fluctuation
- The Effects of Vortex on Vibrations of Hydroturbines
- Vortex Around the Guide Plate in the Francis Turbine
- Swirling Vortex Rope in Francis Turbine
- Swirl Number Analysis
- Vortex in Vaneless Space of Reversible Pump Turbine
- Conclusions
- Consent for Publication
- Conflict of Interest
- Acknowledgement
- References
Chapter 10 a Comparative Study of Compressible Turbulent Flows Between Thermally and Calorically Perfect Gases
- Xiaoping Chen
- Introduction
- Governing Equations
- Thermally Perfect Gas (Tpg)
- Surface Roughness Geometry
- Calorically Perfect Gas (Cpg)
- Description of DNS
- DNS Results and Discussion
- Turbulent Statistics
- Strong Reynolds Analogy
- Flow Structures
- Conclusion and Outlook
- Consent for Publication
- Conflict of Interest
- Acknowledgement
- References
Chapter 11 the Experimental Study on Vortex Structures in Turbulent Boundary Layer at Low Reynolds Number
- Yanang Guo, Xiaoshu Cai, Wu Zhou, Lei Zhou and Xiangrui Dong
- Introduction
- Experimental Methods
- Motion Single Frame and Long Exposure (Msfle)
- Experiment Apparatus
- Experimental Validation
- Experimental Results and Analysis
- Measurements in the Streamwise-Normal (X-Y)-Plane
- Measurements in the Streamwise-Spanwise (X-Z)-Plane
- Conclusions
- Consent for Publication
- Conflict of Interest
- Acknowledgements
- References
Chapter 12 Experimental Studies on Coherent Structures in Jet Flows Using Single-Frame-Long-Exposure (Sfle) Method
- Lei Zhou, Xiaoshu Cai, Wu Zhou and Yiqian Wang
- Introduction
- Experimental Methods
- Single-Frame-Long-Exposure (Sfle) and Moving Sfle (Msfle)
- Experiment Apparatus
- Experimental Results and Analysis
- The Measurements of Coherent Structures in Jet Entrainment Boundary Layer
- Special Pathline Structures
- Investigation on the Coherent Structures of Jet Entrainment Boundary Layer
- Results from Msfle
- Conclusions
- Consent for Publication
- Conflict of Interest
- Acknowledgements
- References
Chapter 13 Hybrid Compact-Weno Scheme for the Interaction of Shock Wave and Boundary Layer
- Jianming Liu and Chaoqun Liu
- A Short Review on Study of High Order Finite Difference Scheme For Compressible Flows
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- Chaoqun Liu
- Yisheng Gao
