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Artificial boundary conditions in the Lattice Boltzmann method / vorgelegt von Daniel Heubes, geboren in Haan. Wuppertal, Februar 2017
Content
Contents
List of figures
Abbreviations
Notation
List of symbols
1 Introduction
1.1 Outline of this thesis
1.2 Related scientific works
2 Fluid dynamics and the lattice Boltzmann method
2.1 Modeling fluid flows
2.1.1 Microscopic formulation
2.1.2 Macroscopic formulation
2.1.3 Mesoscopic formulation
2.2 Boltzmann equation and its macroscopic limit
2.2.1 Moments of the single particle distribution
2.2.2 Moments of the Boltzmann equation
2.2.3 Alternative collision model
2.3 Lattice Boltzmann method
2.3.1 Characteristic curves
2.3.2 Integration of the kinetic model along its characteristics
2.3.3 Analytical derivation of the D3Q19 velocity set
2.3.4 Lattice Boltzmann equation
2.4 Lattice Boltzmann beyond Navier-Stokes equations
2.4.1 Advection equation with the D1Q2 model
2.4.2 Thermodynamical flows and other applications
2.5 Problem formulation
3 Non-reflecting boundary conditions
3.1 An ideal transparent boundary condition
3.2 Artificial boundary conditions
3.2.1 Perfectly matched layers
3.2.2 Impedance boundary condition
3.3 Characteristic boundary conditions
3.3.1 Basic systems of characteristic boundary conditions
3.3.2 Characteristic analysis
3.3.3 Solution of the CBC system
3.3.4 Determination of boundary populations
3.4 Enhanced characteristic boundary conditions
3.4.1 Incorporation of viscous terms
3.4.2 Inlet and outlet characteristic boundary conditions
4 Exact discrete artificial boundary condition for linear collision model
4.1 Evolution represented with digraphs
4.1.1 Domain of dependence
4.1.2 Lattice Boltzmann equation with reversed perspective
4.1.3 Visualization by digraphs
4.2 Weighted digraphs for linear collision models
4.3 Construction principle of the exact discrete artificial boundary condition
4.4 Exact discrete artificial boundary condition
4.4.1 Node weights for contributing fictitious nodes
4.4.2 Node weights for past boundary nodes
4.4.3 Exact discrete artificial boundary condition
5 Discrete artificial boundary conditions (DABCs)
5.1 One-dimensional DABCs for a linear collision model
5.1.1 Decreasing node weights
5.1.2 Approximate discrete artificial boundary condition
5.1.3 Proof of node weights' general decrease
5.2 Interpretation as subproblems
5.3 One-dimensional DABCs for a nonlinear collision model
5.4 General discrete artificial boundary conditions in any dimension
5.4.1 Computational grid
5.4.2 Solving the subproblem
5.5 Analysis of the discrete artificial boundary conditions
5.5.1 Error sources
5.5.2 On history depth and initialization
5.5.3 Implementation
5.5.4 Computational costs
6 Numerical results
6.1 Terminology
6.2 Approximate DABCs for linear collision models
6.3 One-dimensional pressure wave
6.4 Visual interpretation of discrete artificial boundary conditions
6.5 An isolated vorticity wave
6.6 Plane waves with different angles of incidence
6.7 3D square duct flow past an obstacle
7 Conclusions and outlook
A Alternative discrete velocity models
B Matrices for characteristic boundary conditions
B.1 Matrices of the basic system
B.2 Coefficient matrices of the characteristic system
C Completion of the proof of Lemma 4
D Optimal selection for expanding the domain of convergence
E Additional numerical results
References