Contents

• Abstract
• 1. Introduction
  • 1.1 The need for mesh-free Lagrangian methods
  • 1.2 Hybrid methods for diffusion
  • 1.3 Lagrangian methods for diffusion
    • 1.3.1 Random walk method
    • 1.3.2 Core expansion method
    • 1.3.3 Deterministic particle method
    • 1.3.4 Fishelov's method
    • 1.3.5 Diffusion velocity method
    • 1.3.6 Free-Lagrange method
  • 1.4 The vorticity redistribution method
  • 1.5 Organization of the thesis
  
  
• 2. Governing Equations
  • 2.1 Navier-Stokes equations
  • 2.2 Velocity field
  • 2.3 Vorticity conservation laws
  • 2.4 Summary
  
  
• 3. Vortex Methods
  • 3.1 Vortex methods for inviscid flows
  • 3.2 Vortex methods for viscous flows
  • 3.3 Summary
  
  
• 4. Vorticity Redistribution Method
  • 4.1 Mathematical formulation
  • 4.2 Physical meaning of the equations
  • 4.3 General justification
  • 4.4 Summary
  
  
• 5. Convergence Analysis
• 6. Numerical Implementation
  • 6.1 Numerical implementation of convection
  • 6.2 Numerical implementation of diffusion
    • 6.2.1 Finding the redistribution amounts
    • 6.2.2 Least maximum solution procedure
    • 6.2.3 Adding vortices
    • 6.2.4 Merging vortices
    • 6.2.5 Cut-off circulation
    • 6.2.6 Evaluation of the vorticity
  • 6.3 Boundary condition
  • 6.4 Summary
  
  
• 7. Computation of Flows in Free Space
  • 7.1 Point vortex
    • 7.1.1 Governing equations
    • 7.1.2 Vorticity field
  • 7.2 Counter-rotating vortex pair
    • 7.2.1 Governing equations
    • 7.2.2 Vorticity field
    • 7.2.3 Drift velocity
    • 7.2.4 Summary
  • 7.3 Three-dimensional Stokes flows
  
  
• 8. Computation of Flows over Solid Walls
  • 8.1 Impulsively rotated cylinder
  • 8.2 Impulsively translated cylinder
    • 8.2.1 Review of previous computations
    • 8.2.2 Streamlines
    • 8.2.3 Velocity field
    • 8.2.4 Vorticity field
    • 8.2.5 Drag force
    • 8.2.6 Effects of cut-off circulation
    • 8.2.7 Comparison with boundary layer theory
    • 8.2.8 The drag near the Van Dommelen & Shen time
    • 8.2.9 Performance compared with previous methods
    • 8.2.10 Summary
  • 8.3 Vortex-pair/cylinder interaction
  • 8.4 Summary
  
  
• 9. Discussion
  • 9.1 Resolution of short scales
  • 9.2 Automatic remeshing
  • 9.3 Computational speed
  • 9.4 Simplicity
  • 9.5 Conservation laws and positivity
  
  
• 10. Conclusions
• A. Detailed derivations
• Bibliography
• Acknowledgements
• Biographical Sketch
• List of Tables
• List of Figures