Mesoscopic Numerical Models For Nonlinear Convection-diffusion System And Microscale Fluid Flow

Fluid flow can be contributed to macroscale and microscale flow based on the char-acteristic length scale of the system. They appear to be complex and nonlinear. More-over, Microscopic flow has some distinguished appearances compared to macroscopic flow. Recently, own to their kinetic background, the lattice Boltzmann method (LBM) and Gas-kinetic method have been applied to macroscopic and microscopic fluid flow, and have achieved big success. However, it is also observed that LBM and GKS still have many problems as a kind of new numerical methods. For macroscopic convection-diffusion prob-lem, it still need to develop more work concerned on foundation of LBM including model construction and theoretical analysis. In addition, the application of LBM and GKS to the microscopic flows is just in the stage of exploration for the un-normal appearances. In sight of the above mentioned problems, some mesoscopic numerical models and analysis are de-veloped and, some new and open problems are studied for macroscopic convection-diffusion problems and microscale fluid flow. The contents of this thesis can be classified into two main aspects:(1) In term of the nonlinear convection-diffusion problem:Firstly, a lattice Boltzmann model for generalized Burgers equation is proposed. Compared to the existing LB mod-els for Burgers equation, it is more general and unified to different dimensional problems. The equilibrium distribution is easier to express and conduct; The stability analysis of the LB model is given, which is important but relatively lack for the whole works of lattice Boltzmann method. Secondly, the LB model is constructed for delay differential dynamical systems, which is rarely discussed in the LBM regime. The preserve property of Hopf bifur-cations under the LBM is studied. Thirdly, a lattice Boltzmann model is proposed to study the cross transport effects on double diffusive convection, which is usually not included in heat and mass transfer as second-order effects. These theoretical works and numerical tests are useful for LBE to investigate the field of nonlinear convection-diffusion problems deeply.(2) In term of microscale problems:Firstly, the full nonlinear high-order dynamics based on Boltzmann equation, discrete Boltzmann equation and lattice Boltmann equation are given through Chapman-Enskog expansion. Through the comparison and theoretical analysis of high-order hydrodynamics based on different mesoscopic Boltzmann models, Lattice Boltzmann method has been proved to be adequate for microscale flow. The results show the origin of the deviation from Boltzmann equation for lattice Boltzmann method in microscale flow. According to the reasons of deviation, the major ways are provided to construct LB models for microscale flow theoretically. Secondly, the stability of macroscopic equations for multiple-temperature Gas-kinetic model is discussed. Thirdly, The finite-volume GKS numerical scheme for multiple-temperature model is constructed and applied to cavity flow in the slip regime. These works provide a foundation for mesoscopic numerical models to investigate the microscale fluid flow.