Author

Jie Tang

Graduation Semester and Year

2017

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Chaoqun Liu

Abstract

Turbulence is still an unsolved scientific problem, it has been regarded as “the most important unsolved problem of classical physics”. Dr. Liu proposed a new mechanism about turbulence generation and sustenance after decades of research on turbulence and transition. His new idea challenged the classical theorem in many aspects. One of them is the flow stability of transition. Dr. Liu believes that inside the flow field, shear (dominant in laminar) is very unstable while rotation (dominant in turbulence) is relative stable. This inherent property of flow creates the trend that non-vertical vorticity must transfer to vertical vorticity, and causes the occurrence of transition. To verify this new idea, this dissertation analyses the linear stability on two-dimensional shear flow and quasi-rotation flow. 1) Chebyshev collocation spectral method is discussed to solve Orr–Sommerfeld equation, the famous eigenvalue function describing the linear modes of disturbance. Several typical parallel shear flows are tested as the basic-state flows in the equation. The instability of shear flow is demonstrated by the existence of positive eigenvalues associated with disturbance modes (eigenfunctions), i.e. the growth of these linear modes. 2) Quasi-rotation flow is considered under Cylindrical coordinate. An eigenvalue perturbation equation is derived to study the stability problem with symmetric flows. Shifted Chebyshev polynomial with Gauss collocation points is used to solve the equation. To investigate the stability of vortices generation in real-world case, I tracked a ring-like vortex and a leg-like vortex over time from our Direct Numerical Simulation (DNS) data. The result shows that, with the generation over time, both ring-like vortex and leg-like vortex become more stable in the fact of decreasing positive eigenvalues.

Keywords

Stability, Spectral method

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

Included in

Mathematics Commons

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