LOCAL DOMAIN-FREE DISCRETIZATION METHOD AND ITS COMBINATION WITH IMMERSED BOUNDARY METHOD FOR SIMULATION OF FLUID FLOWS WU YANLING NATIONAL UNIVERSITY OF SINGAPORE 2012 LOCAL DOMAIN-FREE DISCRETIZATION METHOD AND ITS COMBINATION WITH IMMERSED BOUNDARY METHOD FOR SIMULATION OF FLUID FLOWS WU YANLING (B.Eng,NUAA, M.Eng, NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2012     Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Wu Yanling 25, July,2012 I Acknowledgements  I would like to express my deepest gratitude and thank to my supervisor, Professor Shu Chang, for his invaluable guidance, constant encouragement and great patience throughout this study. I am extremely grateful to my husband, my son, and my whole family, their support and encouragement made it possible for me to complete the study. Thanks also go to the staff of Department of Mechanical Engineering for their excellent service and help. Finally, I would like to thank all my friends who have helped me in different ways during my whole period of study in NUS. Wu Yanling II  Table of Contents Declaration ………… …………………………………………… ……… .…… . I Acknowledgement … .…………………………………………… ……… .…… . II Table of Contents ………………………………………………………………… . III Summary .……………………………………………………………………… .IX List of Tables……………………………………………………………………… XI List of Figures……………………………………………………………………… XIII Nomenclature ……………………………………………………………………… XXI Chapter Introduction 1.1. Background .1 1.1.1. Analytical method .1 1.1.2. Numerical method 1.2. Domain-Free Discretization (DFD) method 1.2.1. The concept of the DFD method .8 1.2.2. The procedure of DFD method 10 1.2.3. The features of DFD method . .13 1.3. Classification of DFD method 14 III 1.3.1. Overview of Global DFD .14 1.3.2. Fundamental of Local DFD 17 1.4. Contributions andOrganization of the dissertation .17 Chapter Local Domain-Free Discretization method 2.1 Cartesian mesh methods .22 2.1.1. Saw-tooth boundary method .23 2.1.2. Immersed Boundary Method 23 2.1.3. Cut cell method .24 2.1.4. Ghost cell method .24 2.2 Comparison of LDFD method with other Cartesian mesh methods 25 2.3 LDFD on Cartesian mesh .26 2.3.1. The procedure of LDFD method .26 2.3.2. Treatment of Boundary Conditions .29 2.3.2.1Dirichlet boundary condition .30 2.3.2.2 Neumann boundary condition 30 2.3.2.3No-slip boundary condition 31 2.4 Status of mesh nodes .33 2.4.1. Classification of Status of mesh nodes 33 2.4.2. Fast algorithm of identifying the status of mesh nodes .35 2.5 Numerical application of LDFD to incompressible flow 40 2.5.1. Governing Equations 40 2.5.2. Numerical discretization .41 IV 2.5.2.1 Spatial discretization by LDFD method .41 2.5.2.2 Temporal discretization by explicit three-step formulation 42 2.5.2.3 Solving N-S equation by fractional-step method 43 2.5.3. Numerical validation: incompressible flows over a NACA0012 airfoil 45 2.6 Concluding remarks 48 Chapter Adaptive Mesh Refinement in Local DFD method 3.1 Review for Adaptive Mesh Refinement 56 3.2 Stencil Adaptive Mesh Refinement-enhanced LDFD .58 3.2.1 Two types of stencil and numerical discretization 58 3.2.2 Stencil refinement 59 3.2.3 Solution-based mesh refinement or coarsening .61 3.2.4 Stencil adaptive mesh refinement-enhanced LDFD 61 3.3 Numerical validation .64 3.4 Concluding remarks .69 Chapter LDFD-Immersed Boundary Method (LDFD-IBM) and Its Application to Simulate 2D Flows around Stationary Bodies 4.1. The Immerse Boundary Method 78 4.2. Disadvantages of conventional IBM .80 4.3. Combination of LDFD and IBM .81 4.4. Procedure of the LDFD-IBM .83 V 4.5. Numerical applications .89 4.5.1. Decaying vortex 89 4.5.2. Flows past a stationary circular cylinder 91 4.5.2.1. Steady flow over a stationary circular cylinder .93 4.5.2.2. Unsteady flow over a stationary circular cylinder .94 4.5.3. Flows over a pair of circular cylinders .96 4.5.3.1. Side-by-side arrangement 97 4.5.3.2. Tandem arrangement .99 4.5.4. Flow over three circular cylinders 100 4.5.5. Flow over four circular cylinders .103 4.6. Concluding remarks 104 Chapter Application of LDFD and LDFD-IBM to Simulate Moving Boundary Flow Problems 5.1. Status changes in moving boundary problems 127 5.2. Methodologies and procedures .129 5.2.1. LDFD for moving boundary problem .129 5.2.2. LDFD-IBM for moving boundary problem 131 5.3. Application of LDFD and LDFD-IBM to moving boundary problems 136 5.3.1. Flow around an oscillating circular cylinder .137 5.3.2. Two cylinders moving with respect to each other 139 5.4. Concluding remarks .141 VI Chapter Extension of LDFD-IBM to Simulate Three-dimensional Flows with Complex Boundary 6.1. The computational procedure for three-dimensional simulation 150 6.2. Meshing strategies for 3D case .151 6.2.1. Non-uniform mesh .151 6.2.2. Combination of stencil adaptive refinement and one-dimensional uniform mesh 154 6.3. Identification of node status in three dimension 155 6.3.1. Surface description 155 6.3.2. Fast algorithm of identifying status of mesh nodes .156 6.4. Application to three-dimensional flows around stationary boundaries 160 6.4.1. Force calculation 160 6.4.2. Numerical validation of flows around a stationary sphere .161 6.4.3. Numerical simulation for flow past a torus with small aspect ratio….163 6.4.4. Numerical simulation for 3D flow over a circular cylinders .165 6.4.4.1. Background .165 6.4.4.2. Numerical simulation for 3D flow over cylinder with periodic boundary condition .168 6.4.4.3. Numerical simulation for 3D flow over cylinder with two wall ends boundary condition .169 6.5. Concluding remarks 171 VII Chapter 7. Application of LDFD to Simulate Compressible Inviscid Flows 7.1. Euler equations and numerical discretization 184 7.2. LDFD Euler solver 187 7.2.1. Numerical discretization .187 7.2.2. Implementation of boundary condition .189 7.3. Numerical examples .194 7.3.1. Inviscid flow past the a 2D circular cylinder 194 7.3.2. Supersonic flow in a wedge channel 195 7.3.3. Compressible flow over NACA0012 airfoil 196 7.4. Concluding remarks 198 Chapter Conclusions and Recommendations 8.1 Conclusions .205 8.2 Recommendations .208 References .210 List of Journal Papers based on the thesis .224 VIII 4) Development of LDFD-IBM solver for the compressible flow In this work, the LDFD-IBM is restricted to the simulation of incompressible flows. 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Shu: 'Application of Local DFD Method to Simulate Unsteady Flows around an Oscillating Circular Cylinder’, International Journal for Numerical Methods in Fluids, 58 (11):1223-1236, 2008 3. Y.L.Wu, C.Shu, and H.Ding: 'Simulation of Incompressible Viscous Flows by Local DFD-Immersed Boundary Method’, Advances in Applied Mathematics and Mechanics, (3): 311-324,2012 224 [...]... Index of z direction L Left side of cell face R Right side of cell face Acronyms AMR Adaptive Mesh Refinement DFD Domain- Free Discretization method DQ Differential Quadrature method FD Finite Difference method FE Finite Element method FV Finite Volume method LDFD Local Domain- Free Discretization method LDFD-IBM Local Domain- Free Discretization and Immersed Boundary Method combination RBF Radial Basis Function...Summary Numerical simulation of flows with complex geometries and/ or moving boundaries is one of the most challenging problems in Computational Fluid Dynamics (CFD) In this thesis, two new non-conforming-mesh methods, Local Domain- Free Discretization (LDFD) method and hybrid LDFD and Immersed Boundary Method (LDFD-IBM), are proposed to solve this problem The concept of LDFD method is based on the... LDFD method to flexibly handle flow problems with complex geometry LDFD-IBM is a delicate combination of LDFD method and Immersed Boundary Method (IBM), and enjoys the merits of both methods For example, the penetration of streamlines into solid objects in the conventional IBM, due to inaccurate satisfaction of no-slip boundary conditions, can be avoided by using the LDFD method On the other hand,... simulations), and combination of AMR mesh and uniform mesh (for 3D simulations) are presented and they appear to work well with the two methods A variety of flow problems have been solved using the two methods, including incompressible and compressible flows with single or multiple bodies either in rest or in motion, with or without heat transfer Numerical experiments show that the LDFD method and LDFD-IBM... discretization method was 7 proposed by Shu and his co-workers (Shu and Fan 2001, Shu and Wu 2002) 1.2 Domain- Free Discretization (DFD) method 1.2.1 The concept of the Domain- Free Discretization From Section 1.1.1, the inspiration from analytical method is: ƒ the PDE and its solution domain can be treated separately; ƒ the solution obtained satisfies the PDE for both the points inside the domain and the points... Computational domain for simulation of flow around a circular cylinder 111 Figure 4.5 Local refined mesh for simulation of flow past a circular cylinder 112 Figure 4.6 Streamlines for steady flow with Re=20 and 40 .113 Figure 4.7 Instantaneous vorticity and streamlines for Re=100, 185,200 114 XIV Figure 4.8 The time-evolution of Lift and Drag coefficients for Re=100,185,200 115 Figure 4.9 Configuration of. .. 5.5 CD and CL vs time for Re=185 and Ae / D 0.2 for f e / f o =1.12 145 Figure 5.6 CD and CL vs time for Re=185 and Ae / D 0.2 for f e / f o =1.20 145 Figure 5.7 Instantaneous streamlines and vorticity contours for Re=185 and Ae/D=0.2,fe/fo=1.10 146 Figure 5.8 Geometry for flow past two cylinders moving with respect to each other 147 Figure 5.9 Comparison of CD and CL with Xu and Wang... and the points outside the domain Therefore, the basic idea of Domain Free Discretization is: (1) The governing equations can be applied to anywhere in the flow domain, no matter where the point is located inside the fluid domain or outside of the fluid domain Since the discretization is just the process of transferring governing equations into discrete form, the discrete form of the given differential... order methods (FD,FV,FE) and high order methods (Spectral method and DQ) A brief review of these methods is given below 3 ƒ Finite Difference (FD) method FD method is based on the Taylor series expansion (Strikewerda 2004) In FD method, the derivatives of the PDEs are replaced by the appropriate difference formula, giving an equation that consists solely of the values of variables at the present node and. .. are the FD method and the global method of differential quadrature (DQ) These methods discretize the derivatives in a PDE along a mesh line Thus, they require the computational domain to be regular or be a combination of regular sub-domains When a problem with complex geometry is considered, the boundary of the problem may not coincide with the mesh line To apply the finite difference and DQ methods, . LOCAL DOMAIN- FREE DISCRETIZATION METHOD AND ITS COMBINATION WITH IMMERSED BOUNDARY METHOD FOR SIMULATION OF FLUID FLOWS WU YANLING NATIONAL UNIVERSITY OF SINGAPORE 2012 LOCAL DOMAIN- FREE DISCRETIZATION. DISCRETIZATION METHOD AND ITS COMBINATION WITH IMMERSED BOUNDARY METHOD FOR SIMULATION OF FLUID FLOWS WU YANLING (B.Eng,NUAA, M.Eng, NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY. method to flexibly handle flow problems with complex geometry. LDFD-IBM is a delicate combination of LDFD method and Immersed Boundary Method (IBM), and enjoys the merits of both methods. For