Numerical Methods Based On Boltzmann Equation In Fluid Dynamics

The Boltzmann equtation is the foundational kinetic equation of microscope particles. It is a nonlinear integral and differential equation. In order to solve the equation, one must simplify it firstly. In this paper, two kinds of numerical method based on the solution of lattice Boltzmann equation are discussed. 1. Lattice Boltzmann method In recent years, the lattice Boltzmann (LB) method has attracted much attention in the computational physics and engineering communities. In this method, one solves the kinetic equation for the particle velocity distribution function. The macroscopic quantities (such as mass density ρand momentum density ρu) can then be obtained by evaluating the hydrodynamic moments of the distribution function. For two-dimensional nine-velocity (D2Q9) incompressible lattice Boltzmann model, the Boltzmann equation can be discretized in space x and time t into In the above equation, e_α(α=0,1,…8) is the particle velocity in the a direction, f_α(x,t) is the density distribution function along the a direction, f_α^eq(x,t) is its corresponding equilibrium state, x is the spatial position vector, and t is the time, τis the dimensionless relaxation time and δ_t is time step. In spite of numerous improvements in the LB method over the last several years, one important issue that has not been systematically studied is the accurate determination of the fluid dynamic force involving curved boundaries. Consider an arbitrary curved wall which separates the solid region from the fluid region. Let xw, xf and xb be the intersections of the boundary with vari-ous lattice links, the boundary node in the fluid region, and that in the solid region, respectively. Then, ? can be defined as It is well understood that the bounce-back boundary condition satisfies the no-slip boundary condition with second-order accuracy at the location of ? =1/2. But this is only true with simple boundaries of straight line parallel to the lattice grid. For a curved boundary, simply placing the boundary halfway between two nodes will alter the geometry on the grid level and degrade the accuracy of the numerical results. Equation (1) can be computed by the following two steps, collision step ^f α(x ,t)= fα(x,t)?τ1 fα[(x,t)?fα( eq )(x,t)] (3a) streaming step f α( x + eαδt ,t+δt)=^fα(x,t) (3b) where ^fαdenote the post-collision state of the distribution function. One can notice that the collision is completely local and the distribution func-tion of a lattice is only affected by neighboring ones in the streaming step. On the boundary it is important for us to define distribution function at xb in order to get the value at xf. To construct ^f α(xb,t) based upon some know information in the surrounding, Chapman-Enskog expansion for post-collision distribution function on the right-hand side of Eq. (3b) is carried out on the boundary. It has been proved that a second-order accurate no-slip boundary condition can be achieved by this method. In this paper, two approaches for force evaluation on the curved bounary in the lattice Boltzmann equation are investigated: the stress-integration method and the momentum-exchange method. The momentum-exchange method is relatively reliable, accurate, and easy to implement for both two-dimensional and three-dimensional flows. Various fluid dynamics problems including evaluation of fluid acting force on the curve boundary, four-vortex merging in spatial mixing layer, vortex merging in temporal mixing layer, incompressible viscous flow past an elliptic cylinder, flow field around a rotating circular cylinder, displacement of deformable membrane in fluid are simulated in LB method and the values about force are all evaluated by the...