The book introduces the fundamentals and applications of the lattice Boltzmann method (LBM) for incompressible viscous flows. It is written clearly and easy to understand for graduate students and researchers.The book is organized as follows. In Chapter 1, the SRT- and MRT-LBM schemes are derived from the discrete Boltzmann equation for lattice gases and the relation between the LBM and the Navier-Stokes equation is explained by using the asymptotic expansion (not the Chapman-Enskog expansion). Chapter 2 presents the lattice kinetic scheme (LKS) which is an extension method of the LBM and can save memory because of needlessness for storing the velocity distribution functions. In addition, an improved LKS which can stably simulate high Reynolds number flows is presented. In Chapter 3, the LBM combined with the immersed boundary method (IB-LBM) is presented. The IB-LBM is well suitable for moving boundary flows. In Chapter 4, the two-phase LBM is explained from the point of view of the difficulty in computing two-phase flows with large density ratio. Then, a two-phase LBM for large density ratios is presented. In Appendix, sample codes (available for download) are given for users.

The fluid flow and heat transfer problems encountered in industry applications span into different scales and there are different numerical methods for different scales problems. Multiscale methods are needed to solve problems involving multiple scales. In this dissertation, multiscale methods are developed by combining various single scale numerical methods, including lattice Boltzmann method (LBM), finite volume method (FVM) and Monte Carlo method. Two strategies exist in combing these numerical methods. For the first one, the whole domain is divided into multiple subdomains and different domains use various numerical methods. Message passing among subdomains decides the accuracy of this type of multiscale numerical method. For the second one, various parameters are solved with different numerical methods. These two types of multiscale methods are both discussed in this dissertation. In Chapters 3 and 4, the whole domain is divided into two subdomains and they are solved with LBM and FVM respectively. This LBM-FVM hybrid method is verified with lid driven flows and natural convections. In Chapter 5, a LBM-FVM hybrid method is proposed to take both advantages of LBM and FVM: velocity field and temperature file are solved with LBM and FVM respectively. MCM has advantages in solving radiative heat transfer, and LBM-MCM hybrid method is proposed in Chapter 6. Numerical investigation for melting problems are carried on in this dissertation. The key point in solving the melting problem is how to obtain the interface location. To overcome the disadvantages in the now existing numerical methods, an interfacial tracking method is advanced to calculate the interface location. In Chapter 7, low Prandtl fluid natural convections are solved with LBM to discuss the oscillation results. Based on these results, low Prandtl number melting problems are solved using LBM with interfacial tracking method in Chapter 8. High Prandtl number melting problems in a discrete heated enclosure are solved using FVM with interfacial tracking method in Chapter 9. To take both advantages of LBM and FVM, melting problems are solved with LBM-FVM hybrid method in chapter 10, while interfacial tracking method is advanced by porous media assumptions in fluid flow field simulation process. Problems in Chapters 3-10 are all in two-dimensional and three-dimensional problems are more general than them in the realistic applications. Double LBM-MRT model for threedimensional fluid flow and heat transfer is proposed and three types of natural convections in a cubic cavity are discussed in Chapter 11. For the first two types of cubic natural convections, the present results agreed very well with the benchmark solutions or experimental results in the literature. The results from the third type exhibited more general three-dimensional characters. Three-dimensional melting problems are solved with the proposed double LBM-MRT model with interfacial tracking method in Chapter 12. Numerical results in three conduction melting problems agree with the analytical results well. Taking Chapter 11 results in consideration, the double LBM-MRT model with interfacial tracking method is valid to solve three-dimensional conduction or convection controlled melting problems. Two convection melting problems in a cubic cavity are also solved. With a lower Rayleigh number, the convection effects are weaker; side wall effects are smaller; melting process carries on slower.

Lattice Boltzmann method (LBM) is a relatively new simulation technique for the modeling of complex fluid systems and has attracted interest from researchers in computational physics. Unlike the traditional CFD methods, which solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy) numerically, LBM models the fluid consisting of fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh.This book will cover the fundamental and practical application of LBM. The first part of the book consists of three chapters starting form the theory of LBM, basic models, initial and boundary conditions, theoretical analysis, to improved models. The second part of the book consists of six chapters, address applications of LBM in various aspects of computational fluid dynamic engineering, covering areas, such as thermo-hydrodynamics, compressible flows, multicomponent/multiphase flows, microscale flows, flows in porous media, turbulent flows, and suspensions.With these coverage LBM, the book intended to promote its applications, instead of the traditional computational fluid dynamic method.

In this work, we investigate two issues that are important to computational efficiency and reliability in fluid dynamics applications of the lattice Boltzmann equation (LBE): (1) Computational stability and accuracy of different lattice Boltzmann models and (2) the treatment of the boundary conditions on curved solid boundaries and their 3-D implementations. Three a thermal 3-D LBE models (D3Q15, D3Q27) are studied and compared in terms of efficiency, accuracy, and robustness. The boundary treatment recently developed by Filippova and Hanel and Mei et al. in 2-D is extended to and implemented for 3-D. The convergence, stability, and computational efficiency of the 3-D LBE models with the boundary treatment for curved boundaries were tested in simulations of four 3-D flows: (1) Fully developed flows in a square duct, (2) flow in a 3-D lid-driven cavity, (3) fully developed flows in a circular pipe, and (4) a uniform flow over a sphere. We found that while the fifteen-velocity 3-D (D3Q15) model is more prone to numerical instability and the D3Q27 is more computationally intensive, the D3Q19 model provides a balance between computational reliability and efficiency. Through numerical simulations, we demonstrated that the boundary treatment for 3-D arbitrary curved geometry has second-order accuracy and possesses satisfactory stability characteristics.

Abstract: We developed a three-dimensional multi-relaxation-time lattice Boltzmann method for incompressible and immiscible two-phase flow by coupling with a front-tracking technique. The flow field was simulated by using an Eulerian grid, an adaptive unstructured triangular Lagrangian grid was applied to track explicitly the motion of the two-fluid interface, and an indicator function was introduced to update accurately the fluid properties. The surface tension was computed directly on a triangular Lagrangian grid, and then the surface tension was distributed to the background Eulerian grid. Three benchmarks of two-phase flow, including the Laplace law for a stationary drop, the oscillation of a three-dimensional ellipsoidal drop, and the drop deformation in a shear flow, were simulated to validate the present model.