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Table of contents
1 Introduction
1.1 Swirling flows: introduction
1.2 Vortex breakdown and its applications
1.2.1 Definition and classification
1.2.2 Vortex breakdown in tornadoes
1.2.3 Vortex breakdown at the tip of delta wings
1.2.4 Combustion and cyclone chambers
1.3 Previous investigations and state of the art
1.3.1 Experimental investigations
1.3.2 Theoretical and numerical studies
1.4 Outline of the thesis
2 Numerical Methodology
2.1 Mathematical model
2.1.1 Flow Configuration
2.1.2 Equations of motion
2.1.3 Boundary conditions
2.2 Direct Numerical Simulation
2.2.1 Mesh
2.2.2 Boundary conditions
2.2.3 Projection method
2.2.4 Poisson equation
2.3 Steady state solutions
2.3.1 Selective Frequency Damping (SFD)
2.3.2 Recursive Projection Method (RPM)
2.3.3 Pseudo-arclength RPM continuation [97, 50]
2.4 Validation
2.4.1 Steady states
2.4.2 Temporal Dynamics
2.4.3 Influence of the Reynolds number Re
3 Viscous Effect
3.1 The Bifurcation Structure of Viscous Steady Axisymmetric Vortex Breakdown with Open Lateral Boundaries
3.1.1 Introduction
3.1.2 Mathematical Model
3.1.3 Numerical Simulations
3.1.4 The critical state of inviscid swirling flow
3.1.5 Asymptotic expansion of near-critical swirling flows in the large Reynolds number limit
3.1.6 Grabowski profile
3.1.7 Discussion and Conclusions
3.2 Convergence of the various branches in the bifurcation diagram
3.2.1 Saddle node
3.2.2 Solution with recirculation region
4 Stability of Swirling Jets
4.1 The effect of an axial pressure gradient
4.1.1 General aspects
4.1.2 Flow configuration and theoretical prediction
4.1.3 Numerical procedure
4.2 Hopf bifurcation
4.3 Spiral vortex breakdown
4.3.1 Previous studies
4.3.2 Governing equations
4.3.3 Arnoldi method used by ARPACK
4.3.4 Results
5 Conclusions and outlook
5.1 Conclusion
5.2 Outlook
6 Appendix: The pseudo-arclength continuation algorithm based on the Recursive Projection Method (RPM)
6.1 Bratu-Gelfand equation
6.2 Matlab script
6.2.1 Time step subroutine
6.2.2 Jacobian matrix-vector product and F subroutines
6.2.3 Basis increase subroutine
6.2.4 Basis decrease subroutine
6.2.5 RPM subroutine
6.2.6 Main routine
Bibliography
