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Table of contents
1 Introduction
1.1 Basic concepts about stratified fluid
1.1.1 Buoyancy frequency
1.1.2 Internal gravity wave
1.2 Modal instability in stratified shear flow
1.2.1 Kelvin-Helmholtz instability
1.2.2 Viscous instability (Tollmien-Schlichting wave)
1.2.3 Radiative instability
1.3 Mechanisms of modal instability in stratified fluids
1.3.1 Over-reflection phenomenon
1.3.2 Resonance phenomenon
1.4 Transient growth and optimal perturbation
1.4.1 Non-normality and finite time intervals
1.4.2 Optimal perturbation and adjoint equations
1.5 Mechanism of transient growth
1.5.1 Orr mechanism
1.5.2 Lift-up mechanism
1.5.3 Combination of Orr and lift-up mechanisms
1.6 Motivation and purpose
1.7 Summary of this thesis
2 Method
2.1 Base flow and perturbation equations
2.2 Optimal perturbations
2.3 Numerical method for eigenvalues
2.3.1 Eigenvalues for plane Poiseuille flow
2.3.2 Eigenvalues for boundary layer flow
3 Instability of a boundary layer flow on a vertical wall in a stably stratified fluid
3.1 Introduction
3.2 Temporal stability results
3.2.1 Boundary layer instability (Tollmien-Schlichting waves)
3.2.2 Radiative instability
3.2.3 Competition between radiative instability and viscous instability
3.3 Discussion
4 Instability of plane Poiseuille flow in a stably stratified fluid
4.1 Introduction
4.1.1 Modal stability of stratified Poiseuille flow
4.1.2 Non-modal stability of stratified Poiseuille flow
4.1.3 Effect of horizontal shear and vertical stratification
4.2 Modal stability analysis
4.2.1 Tollmien-Schlichting waves
4.2.2 The gravity mode in the presence of stratification
4.2.3 Resonance mechanism
4.2.4 Instability contours in wavenumber plane
4.3 Non-modal stability analysis
4.3.1 Verification for unstratified fluid
4.3.2 Eigenfunctions and optimal perturbations
4.3.3 Velocity field of transient growth
4.3.4 Stratification effects on transient growth
4.4 Discussion
5 Conclusion and perspective
