Contents


Preface

Vortices are ubiquitous in the universe and include tornados, hurricanes, airplane tip vortices, polar vortices, and even star vortices in the galaxy. Vortices are also building blocks, muscles, and sinews of turbulent flows. A vortex is intuitively recognized as a rotational/swirling motion of fluids, but until recently had no rigidly mathematical definition. In 1858, Helmholtz first defined a vortex composed of so-called vortex filaments, which are infinitesimal vorticity tubes. The vorticity tube is called the first generation of vortex definition and identification, or G1. Although G1 has been accepted by the fluid dynamics community and almost all textbooks for more than a century, we can find many immediate counterexamples. For example, in the laminar boundary layer, where the vorticity (shear) is very large near the wall, but no rotation (no vortex) exists. To solve these contradictions, many vortex criteria methods have been developed during the past four decades. More popular methods are represented by the Q , ∆, λ2, and λci criteria methods. These methods have achieved partial success in vortex identification and are referred to as the second generation of vortex identification, or G2. However, G2 has several critical drawbacks. First, these methods are all scalars that have no rotation axis directions; however, a vortex is a vector. It is hard or impossible to use a scalar to represent a vector. Second, like vorticity, these criteria methods are all contaminated by shear to different degrees. Third, they are all very sensitive to threshold selections. They are also unable to show the vortex structure when both strong and weak vortices coexist. The recently developed Liutex is a third generation of vortex definition and identification, or G3, which is a uniquely defined vector. Liutex has strong potential to be applied to all fluid-related research areas.

Nowadays, the crises human beings face are mainly caused by vortices, like global climate change, polar vortices, tornados, hurricanes, environmental pollution, heart disease, and so on. Therefore, accurate definition and identification of vortices is one of the most

The purpose of this book is to encourage all experts who are doing vortex-related research

The book has six chapters covering new vortex theories, vortex identification methods,

The editor would like to thank his wife Weilan Jin, his daughter Haiyan Liu, and his son

**Chaoqun Liu**

Department of Mathematics, University of Texas at Arlington,

Arlington, Texas, USA

around the world to pay more attention to the progress in recent vortex research.

Haifeng Liu for their understanding and unconditional support.

challenging research topics for humanity.

and vortex simulation and applications.
