News Release

Understanding the liutex-based method for computing turbulence

The second volume of Current Developments in Mathematical Sciences explores new and exciting ways to mathematically compute vortices.

Book Announcement

Bentham Science Publishers

Liutex is a new physical quantity discovered by Chaoqun Liu et al at University of Texas at Arlington (UTA) in 2018 to describe fluid rotation, which is same important as velocity, pressure, temperature, vorticity for fluid dynamics. The discovery of Liutex probably is one of the most important breakthroughs in modern fluid dynamics especially for vortex science and turbulence research. Vortex is ubiquitous in nature and the building blocks and muscles of turbulent flow. However, vortex had no mathematical definition before, which is a bottleneck of modern fluid dynamics, causing countless significant confusions in vortex and turbulence research.

According to Liu, there are three generations of vortex identification. In 1858, Helmholtz first defined vortex as vortex tube composed by 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 are very weak. During the past three decades, many vortex identification criteria, like Q, Δ, λ_2 and λ_ci methods 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. They are all scalars and then completely dependent on so-called threshold, which is man-made and arbitrary, to show the iso-surface as the vertical structure. In addition, they are all contaminated by stretching (compression) and shearing. Liutex and the third generation of vortex definition and identification were developed by Liu and his students at UTA. Liutex is defined as a vector which uses the real eigenvector of velocity gradient tensor as its direction and twice of local fluid angular speed 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 struggle, human being first time found a physical quantity to represent fluid rotation or vortex. Liutex is defined as R ?=Rr ? and ω ??r ?>0 where:

R=ω ??r ?-?((ω ??r ? )^2-4λ_ci^2 ) ,

ω ? is the vorticity, λ_ci is the imaginary part of the eigenvalue and r ? is the real eigenvector of ?v ? .

After that, a number of vortex identification methods are developed by the UTA team including Liutex vector, Liutex vector lines, Liutex tubes, Liutex iso-surface, Liutex-Omega methods, Objective Liutex and, more recently, Liutex Core Line methods, which can more accurately visualize the vortical structure in turbulent flow, proved by countless users and 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.

In addition, the existence, uniqueness, stability, Galilean invariance of Liutex have all been proved. A new R-NR tensor decomposition is developed to replace the traditional Cauchy-Stokes (Helmholtz) decomposition. A new vorticity decomposition to Liutex and shear (RS decomposition of vorticity) is developed to reveal non-dissipative rigid rotation and dissipative shear for fluid dynamics. Liutex (rigid rotation) similarity in turbulent boundary layer is also discovered. Liutex dynamics and modified Navier-Stokes equations which can govern both laminar and turbulent flow without models are under development.

Vortex identification has six core elements including (1) absolute strength, (2) relative strength, (3) local rotational axis, (4) vortex rotation axes, (5) vortex core size, (6) vortex boundary, which are touchstones against examination of the vortex identification methods. It is confirmed with illustrative examples that only the Liutex system is able to give precise information of all six core elements in contrast to the failure of the first and second-generation methods in vortex identification.

Liutex, which is the exact quantity of fluid rotation, opens a new era of quantified vortex and turbulence research to replace traditional qualitative turbulence research which is in general based on observations, graphics, visualizations, approximations, assumptions, and hypotheses.

Liutex-based and Other Mathematical, Computational and Experimental Methods for Turbulence Structure is a new book published by Bentham Science and edited by Chaoqun Liu and Yisheng Gao. 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, (Liutex). Besides Liutex, this book also publishes a lot of efforts of analysis on turbulence structure by unobjectionable mathematics, incredible DNS computations, and marvelous experiments.

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The software of the third generation of vortex identification methods has been published online. Please visit: https://www.uta.edu/math/cnsm/public_html/cnsm/cnsm.html for free download with a short agreement for users to sign. Users can also contact the editor at cliu@uta.edu for more information.

About the Editors:

Dr. Chaoqun Liu is currently a tenured and distinguished professor and the director of Center for Numerical Simulation and Modeling at the University of Texas at Arlington, Texas, USA. He has worked on high order direct numerical simulation (DNS) and large eddy simulation (LES) for flow transition and turbulence for over 30 years, since completing his Ph.D. in applied mathematics (1989) from the University of Colorado at Denver, USA. He has worked as a principal investigator on multiple federally funded projects for NASA, US Air Force and US Navy, among other organizations. He has published 11 professional books, 120 journal papers and 145 conference papers. He is the founder and major contributor of the third generation of vortex identification methods including the Omega, Liutex/Rortex, Liutex-Omega, Modified Liutex-Omega, Liutex Core Line methods, RS vorticity decomposition and R-NR velocity gradient decomposition.

Dr. Yisheng Gao is a professor at the Department of Mathematics, Nanjing University of Aeronautics and Astronautics, China. Dr Gao received his Bachelor of Engineering degree in aircraft design and engineering (2007), Master of Engineering degree in fluid mechanics (2009) and Ph. D. degree in fluid mechanics (2016) from the same university. His research is concerned with computational fluid dynamics, including direct numerical simulation and discrete adjoint methods.

Keywords: fluid dynamics, Liutex, Vortex, vorticity, velocity gradient, Mathematical Sciences, turbulent flow


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