Simulation of Multiphase and Multi-component Flows by Lattice Boltzmann Method Zheng Hongwei (B. Sc., M. Sc.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 To My Parents Acknowledgements I would like to express my sincere gratitude to my advisors, Professor Shu Chang and Professor Chew Yong Tian, for their invaluable guidance and advice, and encouragement throughout the course of this thesis. Besides, I also want to take this opportunity to acknowledge and appreciate the National University of Singapore for the scholarship they have provided. In addition, I would like to give my special thanks to my parents and wife whose patient love and support enabled me to complete this work. Zheng Hongwei i Table of Contents Table of Contents Acknowledgements i Table of Contents ii Summary viii List of Tables x List of Figures xi Nomenclature xv Chapter Introduction 1.1 Background 1.2 Overview of the lattice Boltzmann method 1.3 Literature review of multiphase flows modeling 1.3.1 Traditional CFD methods 1.3.2 Lattice Boltzmann methods 14 1.4 Contributions of the dissertation 22 1.5 Scope of the dissertation 24 1.6 Organization of the dissertation 25 Chapter A General Platform for Developing a New Lattice Boltzmann Model 2.1 Introduction 28 ii Table of Contents 2.2 General framework 29 2.3 Fluid dynamics 31 2.4 A general platform 35 2.5 Generation of single speed models from the platform 40 2.6 Generation of multi-speed models from the platform 42 2.7 Numerical verification 46 2.8 Conclusions 49 Chapter Modeling Multiphase Flows by Lattice Boltzmann Method 3.1 Introduction 59 3.2 Rothman-Keller’s method 60 3.3 Shan and Chen’s method 62 3.4 Free Energy based method 65 3.4.1 Calculation of thermodynamic variables 66 3.4.2 The implementation by lattice Boltzmann method 69 3.4.3 Numerical simulation 72 3.5 Conclusion 74 Appendix. The calculation of the derivatives 76 Chapter Lattice Boltzmann Interface Capturing Method for Incompressible Multiphase Flows 4.1 Background 82 iii Table of Contents 4.2 Methodology 85 4.2.1 Development of a new lattice Boltzmann interface capturing model 85 4.2.2 Comparisons with other lattice Boltzmann interface capturing models 4.3 Results and Discussion 90 93 4.3.1 Verification 93 4.3.2 Simple translation 95 4.3.3 Solid body rotation 96 4.3.4 Shear flow 98 4.3.5 Elongation field 101 4.4 Conclusion 103 Appendix A. Derivation of the new interface capturing method 105 Appendix B. Galilean invariance of interface capturing method 108 Chapter Three-Dimensional Applications of the New Interface Capturing Method 5.1 Background 118 5.2 Methodology 120 5.2.1 Cahn-Hilliard equation by lattice Boltzmann method 120 5.2.2 The direction split flux corrected transport (FCT)’s VOF 123 5.3 Results and discussion 125 iv Table of Contents 5.3.1 Verification 126 5.3.2 Solid body rotation 127 5.3.3 Elongation 130 5.3.4 Shear flow 133 5.5 Conclusion 135 Chapter Lattice Boltzmann Model for Multiphase Flows with Large Density Ratio 6.1 Introduction 144 6.2 Governing equations 147 6.2.1 The interface evolution equation 149 6.2.2 The momentum conservation equation 150 6.3. Implementation of lattice Boltzmann method 153 6.4. Numerical validation 156 6.4.1 A bubble in the stationary flow 156 6.4.2 Capillary wave 158 6.4.3 Bubble rising under buoyancy 160 6.5 Conclusion 163 Chapter Three-Dimensional Applications of the New Lattice Boltzmann Model for Multiphase Flows 7.1 Introduction 172 v Table of Contents 7.2 Implementation of lattice Boltzmann method 173 7.3 Numerical Validation 176 7.3.1 Inteface only problem under the vortex velocity field 176 7.3.2 A 3D bubble in the stationary flow 179 7.3.3 Capillary wave 180 7.3.4 Bubble rising under buoyancy 182 7.5 Conclusion 185 Chapter Simulation of Non-sticking Multi-component Flows By LBM 8.1 Introduction 193 8.2 Method 195 8.2.1 Lattice Boltzmann method 195 8.2.2 Analysis 197 8.3 Numerical simulation 199 8.3.1 Verification 199 8.3.2 Two suspended components in static flow 200 8.3.3 Three suspension components under shear flow 202 8.4 Conclusion 203 Chapter Conclusions 9.1 Conclusions 9.1.1 Development of a general platform for LBM 213 213 vi Table of Contents 9.1.2 Development of a lattice Boltzmann interface capturing model 214 9.1.3 Development of a lattice Boltzmann model for multiphase flows with large density ratio 216 9.1.4 Development of a lattice Boltzmann model for non-sticking multicomponent flows 217 9.2 Recommendations 218 References 220 Vita vii Summary Numerical simulation of multiphase flows has drawn the attention by of many researchers due to their many practical applications in both nature and industry. When compared to the traditional computational fluid dynamics (CFD) solvers, lattice Boltzmann method (LBM) has become a promising tool due to its simplicity, since it only involves a series of collision and streaming processes. Consequently, with these advantages, it is natural to apply it to model multiphase and multi-component flows. In fact, the main aim of the thesis is to develop new lattice Boltzmann (LB) models for these complex flows. Since many theoretical derivations are involved in the LB modeling of these flows, a general platform is developed to simplify the process. It serves as a general guide for the researchers to develop their own LB models easily. The object of this platform is to answer the two questions: one is that under which conditions, a discrete velocity model can recover the Navier-Stokes equation and another question is how to construct a velocity model which will satisfy these conditions. Based on the platform, we can easily determine the equilibrium distribution functions for not only all the published models but also the new LB models according to different applications. The model of the multiphase and multi-component flows consists of interface capturing and surface tension modeling. For the interface modeling, a new interface capturing method is proposed to recover the Cahn Hilliard equation without additional terms. It can also keep the Galilean invariance. In this model, the modified LB equation (Lamura and Succi, 2002) is adopted to remove the time derivative related term. In addition, it does not require fourth order tensor of the discrete velocity model viii Chapter Conclusions and Recommendations error. After that, we also investigated the influence of curvature of the bubble. We found that the pressure jump is linear with the curvature when the surface tension coefficient is kept as a constant and the interface thickness is large enough. This linear relationship agrees well with the predicted solution by the Laplace law. For the capillary wave, it also showed that the numerical angular frequency matches the analytical solution. It further confirms that this model can produce an accurate solution. 9.1.4 Development of a lattice Boltzmann model for non-sticking multi-component flows In many multi-component flow systems such as blood flows with white cells and red cells, the components may not stick together. Currently, the factor influencing the effect of non-sticking is rarely investigated. We analyze the influence of the interaction force between different components by using the interaction model which is firstly proposed by Shan and Chen (1993). We find that the non-sticking condition could be implemented by assigning different forces to different pairs of components. Through the numerical simulation of sticking and non-sticking cases, we can find the critical point for non-sticking multi-component flows. The results shows that the numerical critical point approximately matches our analytical solution. 217 Chapter Conclusions and Recommendations 9.2 Recommendations Currently, most multiphase LB models are only suitable for structured grid. The grid is so coarse that the detailed flow structure and the interface are not well resolved. Thus, it is necessary to apply the adaptive mesh technique to the simulation of multiphase flows. It is a challenge and difficult work. One difficulty is that it is not easy to transfer the information between the finer mesh and coarser mesh. That is due to the fact that the time step of LBM is linked to the minimum spacing of the grid. Thus, the evolution step in the coarser mesh is performed after several evolution steps in the finer mesh. Besides, the neighbor sites of the pending points are not the same as those in the lattice model. Thus, another difficulty is that it is not easy to treat pending points in the adaptive grid. Besides, the present LB multiphase model is only suitable for incompressible flows. It is necessary to extend them to model the compressible multiphase flows. In many real industrial applications such as cavity problem, the compressible effect of the gas phase (bubble) must be considered. Although there are several compressible LB models, but there is still no well-accepted model that can solve compressible multiphase flows successfully. 218 Chapter Conclusions and Recommendations Furthermore, the application of multiphase flows in microscopic structure is also an interesting subject. There are many applications of Micro Electro Mechanical Systems (MEMS) such as lotus effect. Thus, in future, we may develop new models to investigate the applications in the microscopic scale. Finally, although our method has many advantages, mobility may influence the stability of our method. 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Chew, A lattice Boltzmann model for multiphase flow with large density ratio, J. Comput. Phys., 218, 1, 353-371, 2006. 229 References Zheng, H. W., C. Shu, and Y.T. Chew, Three dimensional lattice Boltzmann interface capturing method For incompressible flow, Int. J. Numer. Methods Fluids, in press. 230 Vita NAME: Zheng Hongwei DATA OF BIRTH: 18 Dec. 1977 PLACE OF BIRTH: Fujian, CHINA I was born in Fujian Prov., China in 1977. I obtained a B. Sc. Degree (1999) and M. Sc. Degree (2002) from Fudan University of China. In August of 2002, I came to Singapore and studied as a Ph. D. candidate in Department of Mechanical Engineering in National University of Singapore. The following are the publications related to my Ph. D. work: 1. Y. Peng, C. Shu, Y. T. Chew, H.W. Zheng, New lattice kinetic schemes for incompressible viscous flows, International Journal of Modern Physics C, 15 (9), 1197-1213, 2004. 2. H. W. Zheng, C. Shu, Y.T. Chew, and J., Qiu, A platform for developing new lattice Boltzmann models, International Journal of Modern Physics C, 16 (1), 61-84, 2005a. 3. H. W. Zheng, C. Shu and Y.T. Chew, Lattice Boltzmann interface capturing method for incompressible flow, Physical Review E., 72, 056705, 2005b. 4. H. W. Zheng, C. Shu and Y.T. Chew, A lattice Boltzmann model for multiphase flow with large density ratio, Journal of Computational Physics, 218, 1, 353-371, 2006. 5. H. W. Zheng, C. Shu and Y.T. Chew, Three dimensional lattice Boltzmann interface capturing Method for incompressible flow, International Journal for Numerical Methods in Fluids, in press. [...]... calculation and is not easy to be extended to multi- component flow 1.3.2 Lattice Boltzmann methods Due to the limitations in traditional CFD, that is naturally a demand to develop a lattice Boltzmann method to model multiphase flows In fact, several methods have been developed to model multiphase flows by LBM during the last twenty years (Do-Quang et al., 2000; Nourgaliev et al., 2003) They are the color method. .. solver for incompressible flows with low Reynolds numbers 1 Chapter 1 Introduction Consequently, with these advantages, it is natural to apply it to model multiphase flows 1.2 Overview of lattice Boltzmann method To understand the lattice Boltzmann method, it is necessary for us to review some basic aspects of LBM first In history, the lattice Boltzmann method comes from Lattice Gas cellular automata... non-sticking multi- component flow were presented The simulation of multiphase flows includes modeling of the interface capturing and the surface tension term This model recovers the Chan-Hilliard equation without any additional terms and reduces the spurious current by choosing the potential form of surface tension ix List of Tables Table 2.1 Comparison of vortex centers by different lattice models... transport FLAIR Flux Line-segment model for Advection and Interface Reconstruction LBE Lattice Boltzmann equation LBM Lattice Boltzmann method LGCA Lattice gas cellular automata LSM Level set method N-S Navier-Stokes ODEs Ordinary differential equations xviii TLLBM Taylor-series-expansion- and Least-square-based lattice Boltzmann method VOF Volume of fluid method xix Chapter 1 Introduction Chapter 1 Introduction... findings of Clift et al (1978) for a real bubble rising under buoyancy with density ratio of 1000 Besides simulation of multiphase flows, the non-sticking multi- component flows are also investigated In summary, four main parts are involved in this thesis A general platform for developing new LB models, a LB model for interface capturing, a LB model for multi- phase flows with large density ratio and a... Introduction by coupling with the interface description by either volume -of- fluid (Lafaurie et al., 1994) or level-set (Osher et al., 1988) methods 1.3.1.2 Interface capturing methods For interface capturing methods such as level-set (LSM) and phase field methods (PFM), the interface is implicitly captured by a contour of a particular scalar function The level set method is firstly introduced by Osher and Sethian... areas of industry Among these areas, the dynamics of multiphase flow has drawn many attentions from the scientists for a long time The term multiphase flow here is used to refer to any fluid flow consisting of more than one phase or component The dynamics of multiphase flows has many practical applications in engineering, such as oil-water flow in porous media, boiling fluids, liquid metal melting and. .. (Rothman and Keller, 1988; Gunstensen et al., 1991), the potential method (Shan et al., 1993), the free-energy based method (Swift et al., 1995 and 1996; Buick and Greated, 1998; Lamura and Gonnella, 2001; Cristea and Sofonea, 2003; Inamuro et al., 2004) and the discrete Boltzmann with the interaction force method (He et al., 1998) All these models regard the interface as a transition layer instead of a... components on either side of the interface 1.3.1.1 Interface tracking methods In interface tracking methods, the indicator (or marker or color) particles are often used to track the interfaces According to the distribution location of these virtual particles, it can be further divided into three categories: moving mesh methods, surface tracking methods, and volume tracking methods For moving-mesh methods,... , ciy Cartesian components of the i-th lattice velocity cs Speed of sound D (Spatial) Dimension DnQm Lattice notation (n = dimension, m = number of particle velocities) DnSm Single speed lattice notation (n = dimension, m = number of particle velocities) DnMm Multi- speed lattice notation (n = dimension, m = number of particle velocities) ei Vector of the i-th lattice velocity Eo Eotvos number F Body . Simulation of Multiphase and Multi-component Flows by Lattice Boltzmann Method Zheng Hongwei (B. Sc., M. Sc.) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY. Three-Dimensional Applications of the New Lattice Boltzmann Model for Multiphase Flows 7.1 Introduction 172 Table of Contents vi 7.2 Implementation of lattice Boltzmann method 173 7.3 Numerical. Chapter 8 Simulation of Non-sticking Multi-component Flows By LBM 8.1 Introduction 193 8.2 Method 195 8.2.1 Lattice Boltzmann method 195 8.2.2 Analysis 197 8.3 Numerical simulation 199