Eulerian and Lagrangian Smoothed Particle Hydrodynamics As Models for the Interaction of Fluids and Flexible Structures in Biomedical Flows
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Eulerian and Lagrangian Smoothed Particle Hydrodynamics as Models for the Interaction of Fluids and Flexible Structures in Biomedical Flows A thesis submitted to The University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2016 By Abouzied Mohamed Nasar School of Mechanical, Aerospace and Civil Engineering Contents List of Figures ................................................................................................................... 6 List of tables ...................................................................................................................... 9 List of Symbols ............................................................................................................... 10 List of Operators ............................................................................................................. 17 Abbreviations .................................................................................................................. 18 Abstract .......................................................................................................................... 20 Declaration ...................................................................................................................... 21 Copyright Statement ....................................................................................................... 21 Acknowledgements ......................................................................................................... 22 1. Introduction ................................................................................................................. 23 1.1. Background and motivation ................................................................................. 23 1.1.1. Biomedical flows ........................................................................................... 23 1.1.2. Numerical techniques for fluid structure interaction (FSI) problems ............ 24 1.2. Thesis structure ..................................................................................................... 27 2. Literature review ......................................................................................................... 29 2.1. Introduction .......................................................................................................... 29 2.2. FSI with flexible boundaries in biomedical flows ................................................ 29 2.2.1. Challenges ...................................................................................................... 31 2.2.2. State-of-the-art models for FSI in biomedical flow applications .................. 32 2.3. Meshless and Lagrangian computational methods for fluid mechanics ............... 37 2.3.1. Lagrangian particle methods .......................................................................... 37 2.4. Smoothed Particle Hydrodynamics (SPH) ........................................................... 40 2.4.1. Advantages and limitations ............................................................................ 42 2.5. FSI with Weakly Compressible Smoothed Particle Hydrodynamics ................... 47 2.6. Discrete Element Methods for granular and solid structural mechanics .............. 52 2.6.1. Overview ........................................................................................................ 52 2.6.2. Advantages and limitations ............................................................................ 54 2 2.6.3. The Vector based DEM (V-Model) for granular solids ................................. 55 2.7. Conclusions .......................................................................................................... 57 3. Review of Weakly Compressible SPH (WCSPH) ...................................................... 59 3.1. Introduction .......................................................................................................... 59 3.2. Mathematical formulation .................................................................................... 59 3.2.1. The Lagrangian description of motion........................................................... 59 3.2.2. The kernel function and the SPH volume integral ......................................... 59 3.2.3. Important considerations for SPH.................................................................. 62 3.2.4. SPH accuracy ................................................................................................. 63 3.2.5. The Navier-Stokes equations in Weakly Compressible SPH form ............... 65 3.2.6. Viscosity models ............................................................................................ 67 3.2.7. Smoothing kernel corrections ........................................................................ 70 3.2.8. Time integration ............................................................................................. 71 3.3. SPH boundary conditions ..................................................................................... 74 3.4. Treatments for spurious density fluctuations ....................................................... 77 3.4.1. Density filters ................................................................................................. 78 3.4.2.-SPH ............................................................................................................. 79 3.5. Particle shifting techniques .................................................................................. 79 3.6. Eulerian SPH ........................................................................................................ 81 3.7. Conclusions .......................................................................................................... 83 4. New boundary methods for SPH ................................................................................ 84 4.1. Introduction .......................................................................................................... 84 4.2. Investigating high-order accurate boundary models for SPH .............................. 86 4.2.1. Introduction .................................................................................................... 86 4.2.2. A high-order Mixed Finite Difference (MFD) no-slip condition .................. 88 4.2.3. Poiseuille channel flow test ........................................................................... 91 4.3. The effect of particle disorder on global accuracy ............................................... 98 4.3.1. Concluding remarks ..................................................................................... 102 3 4.4. The Immersed Boundary Model ......................................................................... 103 4.4.1. Introduction .................................................................................................. 103 4.4.2. Methodology ................................................................................................ 105 4.5. Conclusions ........................................................................................................ 110 5. The V-Model for solid elastic structures ................................................................... 112 5.1. Introduction ........................................................................................................ 112 5.2. Formulation and methodology ........................................................................... 114 5.2.1. Potential energy of a two-particle system .................................................... 114 5.2.2. Relating the material bond stiffness to the V-Model parameters ................ 118 5.2.3. The equations of motion .............................................................................. 118 5.3. Relating solid material properties to V-model bond properties ......................... 120 5.4. Quasi-static deflection of a cantilever beam under a point load ........................ 122 5.5. Test of free vibration of a cantilever beam ......................................................... 126 5.6. Conclusions ........................................................................................................ 130 6. A novel SPH-V Model for FSI with WCSPH, the V-Model and the IBM ............... 131 6.1. Introduction ........................................................................................................ 131 6.2. Impulsively started flat plate .............................................................................. 132 6.2.1. Eulerian WCSPH-IBM tests ........................................................................ 133 6.2.2. Lagrangian WCSPH-IBM tests ................................................................... 137 6.3. Oscillation of flexible plate with fluid damping using the SPH-V Model ......... 143 6.4. Investigating the effect of V-Model resolution refinement ................................ 154 6.5. On the computational solver ............................................................................... 155 6.6. Conclusions ........................................................................................................ 156 7. Application to biomedical flows ............................................................................... 158 7.1. Introduction ........................................................................................................ 158 7.2. Flow through 2-D deep leg vein