Lattice Boltzmann Method For Boussinesq Equation

Lattice Boltzmann Method (LBM) has been 20 years since its inception, and in these 20 years, this method in the theory and application of research in the field have made a more rapid development. It is different from the traditional simulation methods, and it is based on the basis of micro- particle motion, from a molecular point of view of kinetic theory, to build a discrete velocity model. After years of effort, lattice Boltzmann method has been developed from a simple qualitative research into the accurate quantitative comparison. It can choose the equilibrium distribution function based on the physical problems need, and can even choose to accuracy. It has its own set of mathematical theory which can stand the real test of physical assumptions, It became a very effective numerical calculation method. Because of its inherent parallelism, a clear physical background, as well as handling simple boundary conditions, procedures, the advantages of easy to implement, it gradually becomes one of the hot in an international study of the related fields, which has been growing concern at home and abroad, and to be development and application in a lot of areas. Such as: micro-nano and inter- scale flow and heat transfer, porous media, multiphase flow, particle suspension flow, magnetic fluids, chemical reaction streams.Although both in the theory and applications, the lattice Boltzmann method has been achieved many remarkable achievements, and also has a number of unique advantages which conventional methods can not match, after all, it is in rapid growth stage. Compared with other numerical methods, its theoretical system is not perfect, also has a number of its own shortcomings, and in many areas of applications it is still in the exploratory stage. Therefore, the lattice Boltzmann method development and research also need more efforts and attention.In this article, which, we compare the complete summary of the lattice Boltzmann method the basic basic theory, including the basic form of lattice Boltzmann equation, the series equation derivation, the origin of the macro equation, as well as the equilibrium distribution function of the strike. Focus on this paper, a nonlinear equation (Boussinesq equation) Simulation: through the first two series of lattice Boltzmann equation, we have introduced a second order accuracy of the equation, we get the four moments of equilibrium distribution function (up to third-order moments). To 5 Bit of model basis, we obtain the corresponding equilibrium distribution function. Concrete steps are as follows:The general form of Boussinesq equations: Among them, c 02,α,βis the normal number. Order was:: We introduceωαas an additional item, and set it as: Construct lattice Boltzmann model, obtained with a source term lattice Boltzmann equation is as follows: According to Chapter II of the derivation of such, through the appropriate Chapman-Enskog start, Time multi-scale, the availability of the corresponding series of equations: Of which: It is obvious that it satisfys the following relationship: To this end, we can make: Simultaneous equations (3.8), (3.9), bitting as a model with 5,shown in Figure 3.1, we can get the appropriate solutionφα. Further, we can be the corresponding by (3.7) type:We can get a second order accuracy of the Boussinesq equation by the lattice Boltzmann series of equations: Equilibrium distribution function moment as: Bitting as a model with 5, shown in Figure 3.1, we can get equilibrium distribution function fαe q: Which M , N were:...