Development of mesh free methods and their applications in computational fluid dynamics

DEVELOPMENT OF MESH-FREE METHODS AND THEIR APPLICATIONS IN COMPUTATIONAL FLUID DYNAMICS DING HANG NATIONAL UNIVERSITY OF SINGAPORE 2004 DEVELOPMENT OF MESH-FREE METHODS AND THEIR APPLICATIONS IN COMPUTATIONAL FLUID DYNAMICS DING HANG (B. Eng., M. Eng.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2004 Acknowledgements I would like to express my deepest gratitude and thank to my supervisors A/P C. Shu and K. S. Yeo for their invaluable guidance, encouragement and patience throughout this study. Thanks also go to the friends and staff of the Fluid Mechanics Laboratory in NUS for their help and excellent service. My gratitude also extends to my wife and my families for their support and encouragement in all the way. Finally, I wish to thank the National University of Singapore for providing me with the research scholarship, which makes this study possible. Ding Hang I Table of Contents Acknowledgement… .…………………………………………………………… I Table of Contents ………………………………………………………………… .II List of Figures……………………………………………………………………….X List of Tables………………………………………………………………………XVI Nomenclature …………………………………………………………………XVX Summary ….…………………………………………………………………… XXIII Chapter Introduction ………………………………………….…………… 1.1 Background of computational fluid dynamics …………………………………1 1.1.1 Analytical solution of PDEs ……………………………………………… .1 1.1.2 Numerical solution of PDEs…………………………………………………1 1.2 Why mesh-free……………………………………………………………………4 1.2.1 Dynamic complexity of flow problems ……………………………… .4 1.2.2 Geometrical complexity of flow problems…………………………………4 1.2.3 Concept of mesh-free ………………………………………………………6 1.3 Literature review ……………………………………………………………… 1.3.1 Classification of mesh-free methods …………………………………… .8 1.3.2 Mesh-free methods of integral type …………………………………………9 1.3.2.1 Smoothed particle hydrodynamics (SPH) method ……… .………10 1.3.2.2 Diffuse element (DE) and Element-free Galerkin (EFG) methods .……….…………………………………………………………………… 10 1.3.2.3 Meshless local Petrov-Galerkin (MLPG) method …………………11 1.3.3 Mesh-free methods of non-integral type …………………………………12 II 1.3.3.1 Drawbacks of mesh-free methods of integral type in the flow simulations ………………………………………………… .………12 1.3.3.2 General finite difference (GFD) method ………….………….……13 1.3.3.3 Multiquadric (MQ) method ………………………………………14 1.4 Desirable mesh-free methods for fluid simulations ……………………………16 1.5 Objective of this thesis …………………………………………………………16 1.6 Organization of this thesis …….……………………………………………….17 Chapter Least Square-Based Finite Difference Method ……………20 2.1 Conventional finite difference scheme ………………………………………20 2.1.1 FD’s limitation in complex geometry ………………………… 20 2.1.2 Motivation of constructing FD-like mesh-free method .….21 2.2 Least square-based finite difference method .…………………22 2.2.1 Two-dimensional Taylor series formulation ………………………… 22 2.2.2 Local support scaling .……………….25 2.2.3 Least square technique ……………….27 2.2.4 Weighting function ……………….29 2.3 Theoretical analysis of discretization error … .…………………………………31 2.4 Numerical analysis of convergence rate ……………………………………34 2.5 Concluding remarks ……………………………………………………… 38 Chapter Local Radial Function-based Differential Quadrature (LRBFDQ) Method … 39 3.1 Radial Basis Function (RBF) and its interpolation scheme ….………………….39 3.2 Traditional RBF-based schemes and their weakness .…………………………42 III 3.3 Development of Local Radial Basis Function-based Differential Quadrature Method method ………………………………………………………………………45 3.3.1 Traditional Differential Quadrature (DQ) Method … .……………………46 3.3.2 Formulation of local MQ-DQ Method .…… ……………………………47 3.3.3 Normalization of shape parameter ……… ….……………………………51 3.4 Sample problems ………… ….…………………… ….……………………… 53 3.4.1 Poisson equation ……………….………….…………………………….…53 3.4.2 Advection-diffusion equation … ………….………………………………55 3.5 Empirical error estimate for LMQDQ method …………………………………57 3.5.1 Review of accuracy analysis of RBF-related numerical schemes …………57 3.5.2 Empirical analysis of discretization error for LMQDQ method …………58 3.5.3 Numerical Results for individual factor ………………………… .………60 3.5.3.1 Mesh size h ……………………………………………… .………60 3.5.3.2 Shape parameter c ……………………………………… ………62 3.5.3.3 Number of supporting points …………………….……… .………64 3.5.4 Relationships between numerical error and three factors …………………66 3.5.4.1 Dependence of numerical error on shape parameter and mesh size …………….….…………….….…………………………….….…………66 3.5.4.2 Relationship between numerical error, mesh size and number of supporting points ….….…………….……………………….… ………68 3.5.4.3 Relationship between numerical error, shape parameter and number of supporting points ………………………………………………………71 3.6 Concluding remarks ………………………………………………………… .73 IV Chapter Navier-Stokes Equations, Node Generation Algorithm and Solution Method …………………………………………………75 4.1 Basic equations of fluid dynamics in Eulerian form…………….….……………76 4.1.1 Compressible Euler equations ….………………………………………….76 4.1.2 Incompressible Navier-Stokes equations ……………….…………………79 4.2 Node generation algorithms …………………………………………………… 82 4.2.1 Mesh generation versus node generation ………………………………….82 4.2.2 Composite “gird” algorithm ….……………………………………………83 4.2.3 Cartesian node generator ……………………………….………………….84 4.2.4 Random node generator ……………………………………………………85 4.2.5 Locally orthogonal grid ……………………………….………………… .86 4.3 Determination of local support ………………………………………………… 89 4.4 Solution method …………………………………………………………………90 4.2.1 Steady flows ……………………………………………………………….90 4.2.2 Unsteady flows ……………….……………………………………………91 4.5 Concluding remarks ……………………………………………………………92 Chapter Applications to Steady Incompressible Flows .…………… 93 5.1 Natural convection flow in a square cavity…………………………………… 93 5.1.1 Mathematical modeling and numerical discretization ……………… .…. 93 5.1.2 Three types of node distributions and comparison of numerical results ….97 5.2 Natural convection between a square outer cylinder and a circular inner cylinder …………………….….…………….….………………………………………101 5.2.1 Mathematical modeling …………………………….….…………… .….102 5.2.2 Pressure single-value condition …………………….….…………… .….103 V 5.2.3 Definition of configuration and flow parameters ……………………… 105 5.2.3.1 Configuration parameters and “grid” size ……………………… 105 5.2.3.2 Nusselt numbers ….………….………………………………106 5.2.4 Grid-independent study ……………………………….…….……… .….107 5.2.5 Validation of numerical results … ……………………………………108 5.2.6 Global circulation ……………………………………………………….109 5.2.7 Analysis of flow and thermal fields ……………………….….….………112 5.3 Three-dimensional lid-driven cavity flow ………………………….…….…….119 5.3.1 Fractional step method ……………………………….….……….…… 120 5.3.2 Geometry configuration and physical boundary conditions …….……….123 5.3.3 Solution procedure and comparison of numerical results ……….……… 127 5.4 Concluding remarks ………………………………………………………….135 Chapter Applications to Unsteady Incompressible Flows ………136 6.1 Review of study for flow around circular cylinders ……………………………136 6.1.1 Experimental study of flow around two circular cylinders ………………137 6.1.2 Numerical study of flow around two circular cylinders ……………….…138 6.2 Scope and objective of cylinder flow study ……………………………………140 6.3 Governing equations, boundary and initial conditions …………………………141 6.3.1 Governing equations for unsteady incompressible flow …………………141 6.3.2 Boundary conditions ………………….………………………………… 142 6.3.3 Initial conditions …………………… .…………………………….…….143 6.4 Numerical solutions for flow around two circular cylinders ….….….…………143 6.4.1 Definition of flow parameters ……………………………………………144 6.4.1.1 Lift and drag coefficients ( C L & C D ) ….…………………………144 VI 6.4.1.2 Strouhal number (St) ….………….………………………………145 6.4.2 Side-by-side arrangement …………….………………………………… 145 6.4.2.1 Biased flow pattern ….……………………………………………146 6.4.2.2 Synchronized Karman vortex streets ….………….………………147 6.4.2.3 Validation of flow parameters ……………………………………148 6.4.3 Tandem arrangement …………….……… .……………………….…….149 6.4.3.1 Quasi-steady attachment ….………………………………………149 6.4.3.2 “Lock-in” phenomenon between two cylinders………….….……150 6.4.3.3 Validation of flow parameters ……………………………………151 6.4.4 Staggered arrangement …….…….……… .……………………….…….153 6.4.5 Effects of Reynolds number …….…….……… .………………….…….154 6.5 Concluding remarks………………………………… ……………….…….155 Chapter Applications to Compressible Inviscid Flows ……………172 7.1 Review of mesh-free methods fro compressible flow simulation ……………172 7.2 Mesh-free Euler solver …………………………………………………………173 7.2.1 Weakness of mesh-free method in the compressible flow simulation … 173 7.2.2 Euler equations in the conservative form ………………………………174 7.2.3 New flux G i,k ……………………………………………………………176 7.2.4 Artificial dissipation………………………………………………………177 7.2.5 Evaluation of new flux by Roe’s scheme and limiter ……………………178 7.2.6 Comparison between the upwind mesh-free scheme and finite volume method ……………………………………………………………………182 7.2.7 Boundary conditions for invicid flow ……………………………………184 7.3 Numerical examples and discussion ……………………………………………186 VII 7.3.1 Two-dimensional supersonic flow in a symmetric convergent channel …186 7.3.2 Shock tube problem ………………….………………………………… 190 7.4 Concluding remarks………………………………… ……………….…….197 Chapter Hybrid Finite Difference and Mesh-free Scheme ………199 8.1 Benefits and drawbacks of using mesh-free methods ……………….…………199 8.2 How to handle complex geometry effectively and efficiently …………………201 8.2.1 Cartesian mesh method …………………….…………….………………201 8.2.2 Overset mesh method ………………….…………………………………202 8.2.3 Motivation of using hybrid method ………………………………………202 8.3 Hybrid FD and Mesh-free Method …….….……………………………………203 8.3.1 Methodology ………………………………………….…….……………203 8.3.2 Information Exchange layer …………….……………………………… 204 8.4 Numerical examples for validation …… ….………….….……………………206 8.4.1 Flow past one isolated circular cylinder …………………….……………206 8.4.1.1 Geometry description ….…………………………………………207 8.4.1.2 Mesh design ….…….….….………………………………………208 8.4.2 Results and Discussion …………….…………………………………… 209 8.4.2.1 Effect on efficient improvement …………………………………210 8.4.2.2 Steady flow simulation at low Reynolds numbers ………….……211 8.4.2.3 Unsteady flow simulation at medium Reynolds numbers ….….…213 8.5 Concluding remarks………………………………… ……………….…….215 Chapter Conclusions and Recommendations ………………………221 9.1 Similarity and difference …………………….…………………………………221 VIII Chapter 10 Conclusions and recommendations 233 CHAPTER 10 Conclusions and Recommendations 10.1 Conclusions In this thesis, two mesh-free methods: least square-based finite difference (LSFD) method and local RBF-based differential quadrature (LRBF-DQ) method have been developed. Their abilities of dealing with the problems in fluid mechanics have also been demonstrated by applications to different types of flow problems with dynamic and geometric complexity, such as unsteady flow around two cylinders, natural convection within complex geometry, and compressible flow with shock waves. Note that both methods belong to the non-integral types of mesh-free methods. In other words, they solve the strong form of partial differential equations (PDEs), and the discretization process consists only of mesh-free derivative approximation. In this regard, they are truly mesh-free since no requirement of additional background meshes is needed. The LSFD method is based on the use of a weighted least square (WLS) technique together with a Taylor series expansion of the unknown function. The role of the weighting function is to tune the error distribution according to the relative position of the supporting points to the reference node. One important feature of the method is its ability to construct the Chapter 10 Conclusions and recommendations 234 mesh-free interpolants according to the practical node distribution in the local support. Though the least square technique is adopted to overcome the possible ill-conditioned coefficient matrix, the method does not degrade the order of accuracy. Furthermore, it holds the property of the Kronecker δ shape function, which is appreciated in the numerical schemes for fluid mechanics. From the point of view of practical implementation, LSFD method is very like the traditional finite difference method: simple and easy for programming. Several distinguishable advantages can be enjoyed from the use of LSFD method as compared with other mesh-free methods. • Truly mesh-free. In some mesh-free methods, the integration process requires background meshes. • Delta-shape function at the reference node. This circumvents the difficulty of implementation of essential boundary condition encountered in some mesh-free methods. • Simple, and easy to implement. LSFD method shares many common features with traditional finite difference method. It is easy for beginner to use this method after a short time of learning. Local RBF-DQ method, just as its name implies, gets the inspiration from the RBFs and differential quadrature (DQ) method. All previous works related to the application of radial basis functions for the numerical solution of PDEs have been based on the function approximation instead of derivative approximation. Thus, in the nonlinear partial Chapter 10 Conclusions and recommendations 235 differential problems, special numerical techniques must be used to realize the Newton iteration and convergence. This makes the solution procedure complicated and very difficult, especially for the beginners in the computational area. In the RBF-DQ method, the RBFs serve as the trial functions that are used to approximate the solution of the given differential equations. The achieved coefficients are only related to the positions of supporting points in the local support and the RBFs specified. In other words, they are problem-independent and consistently applicable to the linear and nonlinear partial differential equations. It should also be noted that both methods make no distinction regarding the dimension of the practical problems. In general, the local MQ-DQ method takes the following advantages over other RBF-based schemes: • It can be consistently applied to both linear and nonlinear problems. • It can employ a large number of nodes to solve large scale engineering problems. Some BRFs-based numerical methods can only use less than 2000 nodes. • It can preserve the valuable properties of RBFs such as high accuracy and naturally mesh-free. Successful numerical simulations of flow problems by the two mesh-free methods involving dynamic and geometric complexity prove their applicability to large scale practical applications in engineering. The simple algorithms and easy implementation of the two mesh-free methods also imply that they can be easily and extensively adopted by other end-users for the purpose of mesh-free researches/applications in their field. Chapter 10 Conclusions and recommendations 236 Despite of their attractive properties such as truly mesh-free and simple algorithm, they, however, also have the common weakness of all the mesh-free methods: poor computational efficiency. To alleviate this drawback, a hybrid scheme, which combines the mesh-free method with traditional finite difference scheme, is presented in Chapter 8. Numerical results show that compared to the pure mesh-free method, the hybrid scheme can greatly improve the computational efficiency. 10.2 Recommendations on the future work The studies conducted so far are considered as an initial investigation for the performance of mesh-free methods in the solutions of some practical flow problems. The searching for a better or more suitable mesh-free method for fluid mechanics is continuing. The high flexibility of the two mesh-free methods also presents some challenging research problems, such as looking for best radial basis functions for local RBF-DQ method, and more suitable definition of local support. From the node generation’s point of view, a more general and flexible node generator is required in order to exploit the extreme power of the mesh-free method. From the practical applications’ point of view, there is an immense opportunity for the mesh-free computational research, ranging from the small scale like micro-flow to the large one like the study of nebula. All these are considered to be the motivating factors for the future work. It is hoped that the insights presented in this thesis will help to spur more interest in and launch more investigations into the use of mesh-free methods for the numerical flow simulation. Chapter 10 Conclusions and recommendations 237 REFERNCE Atluri S.N. and Zhu T. (1998): “New Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics,” Computational Mechanics, Volume 22, Issue 2, 117-127 Babuska I. and Melenk J. (1997): “The partition of unity method,” International Journal for Numerical Methods in Engineering, Vol.40, 727-758 Bearman P.W. and Wadcock A.J. (1973): “The interaction between a pair of circular cylinders normal to a stream”, J. 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N (1999) “A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems,” Engineering Analysis with Boundary Elements, 23, 375-389 [...]... advantages of mesh- free methods as compared to the standard methods is the saving of time and human-labor on the mesh construction when complex geometry is involved Instead of mesh generation, mesh- free methods use node generation From the point of view of computational efforts, node generation is seen as an easier and faster job Another advantage of mesh- free method is ease of construction of high-order... understanding any of these methods is the study of the underlying kernel approximation and its properties Also, the study of the corresponding kernel approximation of a mesh- free method can reveal its strength and weakness, and may be of some help to further improve the method In this thesis several typical mesh- free methods are analyzed Chapter 1 Introduction 10 1.3.2 Mesh- free methods of integral... the finite point method (FPM) (Onate et al 1996), the partition of unity method (Melenk and Babuska 1996), HP-clouds methods (Duarte and Oden 1995 1996) and meshless local Petrov-Galkerkin method (MLPG) (Atluri et al 1998, Zhu et al 1999) These mesh- free methods can be categorized by their characteristics 1.3.1 Classification of mesh- free methods From the point of view of the movement of nodes, the mesh- free. .. all mesh- based methods For FVM and FEM, preliminary (pre-processing) steps are needed to establish a data base containing nodal and elemental connectivity and hierarchical mesh information The numerical results depend strongly on the mesh properties Due to their good performance, these three methods are widely used in all areas of engineering computation Chapter 1 Introduction 4 1.2 Why mesh- free? ... Successive Over-Relaxation TPS Thin Plate Splines XXII Summary The recent decade has witnessed a research boom on the mesh- free methods It is well-known that the mesh- free methods have a few clear advantages over the meshbased methods such as the requirement of node generation instead of mesh generation and easy deletion/insertion of new nodes Up to date, a lot of attentions of mesh- free researchers have been... solution of partial differential equations in the weak form As a result, many mesh- free methods can be grouped into the finite element community However, due to their dependence on the background mesh (exclusive of MLPG method) for integration, they bear the reputation of not being truly mesh- free methods One way to overcome this drawback is to develop the meshfree methods which solve the strong form of. .. unstructured meshes for FE and FV Chapter 1 Introduction 6 methods, despite recent advances in the field, is still the bottleneck in many industrial computations Another difficulty appears in the simulation of moving boundary problems With the moving of boundaries, successive re-meshing of the domain may be required to avoid the break down of the computation due to excessive mesh distortion if standard schemes... researchers showed their interests in the Chapter 1 Introduction 8 development of methods with mesh- free property As a result, a number of new meshfree methods appeared in 1990s: the diffuse element method (DEM) (Nayrole et al 1992), the element -free Galerkin method (EFGM) (Belytschko et al 1994 1995 1996), multiquadric methods (MQM) (Kansa 1990a,b), reproducing kernel particle methods (RKPM) (Liu... need of a background quadrature scheme to evaluate the integrals, which appear in the weak forms used by the Galerkin method, also impairs the efficiency and applicability of the method to the practical applications 1.3.2.3 Meshless local Petrov-Galerkin (MLPG) method In the mesh- free methods of integral type mentioned above, they all need a background mesh to evaluate the integral of weak form of governing... discretization of governing equations of flow problems does not depend on the availability of a mesh Some mesh- free methods do have a weak dependence on background meshes to carry out numerical quadrature calculation Such methods are still regarded as mesh- free because there is no fixed connection among the nodes, but not “truly” mesh- free method due to the background meshes used One of the key advantages of mesh- free . DEVELOPMENT OF MESH- FREE METHODS AND THEIR APPLICATIONS IN COMPUTATIONAL FLUID DYNAMICS DING HANG NATIONAL UNIVERSITY OF SINGAPORE. DEVELOPMENT OF MESH- FREE METHODS AND THEIR APPLICATIONS IN COMPUTATIONAL FLUID DYNAMICS DING HANG (B. Eng., M. Eng.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY. Concluding remarks………………………………… ……………….…….197 Chapter 8 Hybrid Finite Difference and Mesh- free Scheme ………199 8.1 Benefits and drawbacks of using mesh- free methods ……………….…………199 8.2 How to handle