| Lattice Boltzmann method as a new numerical method used in the simulation of fluid flow, has been developed from Lattice Gas Automation (LGA) method. Based on the discrete approximation of Boltzmann equation, Lattice Boltzmann methed is constructed. First, the permited space of the particle velocity vectors is simplified into several vector aggregates. Second, the distribution function of the particle velocity is discretized on the regular grids in the simplified velocity space. And then the relevant discrete Boltzmann equation is constructed. That is the scheme called Lattice Boltzmann method (LB). Up to now, LB has been an alternative calculating method in the aspects of fluid flow simulation and modeling..As a new numerical method, LB has many outstanding advantages over the traditional methods, such as easy calculation, highly parallel, easy boundary processing and direct pressure calculation. So it provides a new approach to study and simulate the complex flows ,thermal and mass transport phenomena.Circulating viscous flow and Couette flow (shear flow between two planes with relatively tangential motion) as basic flows often occur in every fluid machines involving moving parts and hydrodynamics lubrication. Many striations and grooves on the surface of the mechanical parts may be caused by processing methods, production process or may be intentional. They will greatly affect the inherent flow and the works machinery systems. The background of the Couette flow with a sawtooth boundary is based on all above-mentioned. Therefore, there are theoretical and engineering significances to study them.For some particular problems of Couette flow with a sawtooth boundary, there a few numerical results reported. Some methods directly derive from the N-S equation and by simplifying or ignoring some details elements, the calculating purpose is realized.Because LB method has its peculiar advantages in the numerical simulation of flow field. This paper, in the a microscopic view, will attempt to find a way based on the LB method to calculate the Couette flow with a sawtooth boundary, as the different Re numbers and different sizes of the fringe amplitude and height. The research work in this paper are summarized as following: (1) Based on Lattice Boltzmann method, the iterative equation of single time relaxation BGK model can be expressed as: To assure the equation correctly simulate incompressible N-S equation, the equilibrium-state distribution function must satisfy the conservation laws of mass, momentum and energy. In the mean while, the corresponding macroscopic equations must agree with the dynamics equations. This paper uses the multiscale technique and Chapman-Enskog expansion and finally derives the equilibrium-state distribution function expression based on two-dimension equilateral hexagonal lattice method.(2) Using the two-dimension equilateral hexagonal lattice model to simulate the equilateral triangular cavity flow, we find the flow pattern is composed of a primary vortex and a series of secondary eddies. We also list some data of the primary vortex under the different Re numbers and analyze the motion laws of the primary vortex as the Re number increases. (3) Based on the analysis of the equilateral triangular cavity flow field, the paper numerically simulates the Couette flow with a sawtooth boundary. For each different breadth of flow, a case is selected to analyze. In the same way, we provide some data of the primary vortex under the different Re numbers and also analyze the characteristic laws of the movements of the primary vortex.(4) A derivation is also presented for the Couette flow with a sawtooth boundary. With the different sizes of the fringe amplitude and height, we analyze the flowing characters of the sawtooth-boundary Couette flow and get preferable results.(5) Utilizing the method of momentum conversion, we calculate the resistance and resistance factor CD of Couette flow with a fringe boundary, and analyze the varying laws of resistance fact...
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