Lattice Boltzmann Model For The Euler Equations Based On Small Time Scale

Lattice Boltzmann method is developed in the past two decades in the international community as a new method for the fluid flow simulation .Its thinking ways and modeling tools have brought new ideas for hydrodynamic research, creating a new area of fluid simulation.After the 20-year development, the lattice Boltzmann method has become a very active simulation method and has a good prospect.Since the lattice Boltzmann method has been proposed, it has been concerned by many scholars. This method has a broad space for development,which involves fluid mechanics, physics, mathematics and other fields.It can simulate many problems effectively: in fluid mechanics, such as the simulation of flow round a circular cylinder, flow around baffles, pipeline flow, a two-dimensional shear flow, shock wave and so on; in other fields, this method also has a wide range of applications, such as chemical reaction and proliferation, the crystal growth , the seismic waves and so on.The lattice Boltzmann method has been trying to enhance the accuracy of the equation. One approach is to use more of the series of equations to simulate equations, but it has large computational. In this paper, we proposed a new method for the simulation of the Euler equations of second order accuracy. First of all, we introduce a small time scale assumption, and then we derivate a series of lattice Boltzmann equation based on small time scales, at last , we use the series of equations to recovery the Euler equations of second order accuracy successfully.First, we selecte the classic two-dimensional hexagonal lattice,and use this lattice discrete two-dimensional space. The lattice Boltzmann equation is:Expand distribution function using Chapman-Enskog expansionUsing multiple-dimensioned expansion to timeKnudsenε= l/L.We introduce a small time scale assumption ,take time stepΔt andε2 the same magnitude,takeΔt =ε~2 Taylor to f_α( x +Δx ,t +Δt )We let above expansions be the lattice Boltzmann equation,we obtain series of lattice Boltzmann equation based on small time scales. We obtain the Euler equations of second order accuracy using series of lattice Boltzmann equation,After computing, Equilibrium distribution function is...