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 challenging research topics for humanity.

The purpose of this book is to encourage all experts who are doing vortex-related research around the world to pay more attention to the progress in recent vortex research.

The book has six chapters covering new vortex theories, vortex identification methods, and vortex simulation and applications.

The editor would like to thank his wife Weilan Jin, his daughter Haiyan Liu, and his son Haifeng Liu for their understanding and unconditional support.

> **Chaoqun Liu** Department of Mathematics, University of Texas at Arlington, Arlington, Texas, USA

**Chapter 1**

Method

3-dimensional torus, or <sup>3</sup>

**Abstract**

for (PNS).

**1**

elliptic, analysis

**1. Introduction**

Periodic Navier Stokes Equations

The Incompressible Navier-Stokes Equations (NSEs) are on the list of Millennium Problems, to prove their existence and uniqueness of solutions. The NSEs can be formulated in a periodic 3D domain, where they are termed the

Periodic Navier Stokes (PNS) Equations, and can be treated on a subspace spanning a

onstrates that a decaying of turbulence occurs in the 3D case for the *z* component of velocity when non-smooth initial conditions are considered for *x, y* components of velocity and that 'vorticity'sheets in the small scales of 3D turbulence dominate the flow to the extent that non-smooth temporal solutions exist for the *z* velocity for smooth initial data for the *x, y* components of velocity. Unlike the Navier-Stokes equations, which have no anti-symmetric vorticity tensor, there are new governing equations which have vorticity tensor and can be decomposed into a rotational part (Liutex), antisymmetric shear and compression and stretching. It is shown that under these recent findings, that there is a strong correlation between vorticity and vorticies

**Keywords:** periodic, Navier-stokes, blow-up, turbulence, 3-torus, Weierstrass,

This chapter gives a general model using specific periodic special functions, that is elliptic Weierstrass *P* functions. The definition of vorticity in [1], is that vorticity is a rotational part added to the sum of antisymmetric shear and compression and

stretching. Satisfying a divergence free vector field and periodic boundary conditions

periodic, then the existence of solutions which blowup in finite time can occur. On the other hand if *u*<sup>0</sup> is not smooth, then there exist globally in time solutions on *t*∈ ½ Þ 0, ∞

respectively with a general spatio-temporal forcing term *f x*!, *t*

*:* Treating the PNS Equations in <sup>3</sup>


which is smooth and

Liutex Vortex Identification

*Terry E. Moschandreou and Keith C. Afas*

for a 3D Incompressible Fluid with
