ORCID Identifier(s)


Graduation Semester and Year




Document Type


Degree Name

Doctor of Philosophy in Mathematics



First Advisor

Guojun Liao

Second Advisor

Chaoqun Liu


Vortices are intuitively known as the rotational motion of fluid particles, however, unambiguous and universally accepted methods of vortex definition and identification are not available to date in the literature. First-generation vortex identification methods, also known as vorticity-based vortex criterion, were first proposed by Helmholtz. But these methods have their own problems. These methods have a shear contamination problem, and these methods did not accurately show the direction of fluid rotation. So, to overcome these problems, eigenvalues based second-generation vortex identification methods like Q, Δ, λ_(2 ), λ_(ci ), and Ω have been proposed. Most of these second-generation methods are scalar quantities based on the Cauchy-Stokes decomposition of the velocity gradient tensor. But Cauchy-Stokes’s decomposition itself is not Galilean invariant and its physical meaning is not clear. In 2017/2018, the Center for Numerical Simulation and Modeling (CNSM) at the University of Texas at Arlington proposed a Liutex vector based on eigenvector of the velocity gradient tensor to define vortex structure mathematically. Liutex method can give the local direction of fluid rotation and rotational strength. Since then, Liutex based vortex identification methods like Liutex core lines, Liutex tubes, Liutex-Omega method ( Ω_(L )), and Modified Liutex-Omega method ( Ω ̃_(L )), etc., have been proposed in fluid dynamics. In this dissertation, the Direct Numerical Simulation (DNS) data is applied to all three generations of vortex identification methods and a comparative study has been done and proposed the best method to define and visualize vortex boundary. If we are focusing on vortex direction and uniqueness of the vortex structure, then the Liutex core lines method is the appropriate method. However, if we are looking for smooth and clear iso-surface plotting of vortex structure, then we recommend the Modified Liutex-Omega method. A new coordinate system known as Principal Coordinates based on the Liutex definition is proposed. A new unique velocity gradient tensor is known as Principal Tensor has been proposed which is Galilean invariant. Then Principal Tensor is decomposed into the rotation, stretching, and shear part. Unlike traditional Cauchy-Stoke’s decomposition, Principal Tensor decomposition is unique and Galilean invariant. Also, in this dissertation, first and second-generations vortex identification methods have been redefined and each redefined vortex definition can give the degree of contamination by shear and stretching or compressing effect. These new definitions also provide the dimension of each criterion which can help us decide which criterion is close to fluid rotation and choose the better one. In the final chapter of this dissertation, proper orthogonal decomposition (POD) is used with Liutex vector as an input instead of velocity vector to extract the coherent structure of late boundary layer flow transition but with significantly lower dimension. A singular value decomposition (SVD) algorithm is used as the POD method. It is observed that given fluid motion can be modeled by few early modes as they contain a large portion of total kinetic energy and later modes can be neglected as their kinetic energy converges to zero.


Vortex, Liutex, Principal Tensor, Omega, etc.


Mathematics | Physical Sciences and Mathematics


Degree granted by The University of Texas at Arlington

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Mathematics Commons