Lattice Boltzmann Method For The Potential Flows

The lattice Boltzmann method (LBM) has recently become analternative method for computational fluid dynamics (CFD). Unlikeconventional methods based on macroscopic continuum equation, theLBM starts from microcosmic kinetic equations, i.e. the Boltzmannequation, to determine macroscopic fluid flows. The kinetic naturebrings certain advantages over conventional numerical methods, suchas its algorithmic simplicity, easy handing of complex boundaryconditions, and efficient hydrodynamics simulations. By using parallelcomputation scheme in a multi-processor computer, we can easilysolve a larger size of compressible flow problem with latticeBoltzmann method.Under different simple conditions, compressible fluid has alterablepredigest equations. They include the full potential equation , spansonic potential equation, subsonic and supersonic-potential equations.These solutions are difficult by eigenline method and the followingfinite analytical method. The method of velocity potential function ismore easy in solving Euler equations of gas dynamics when the fluidhas good characters, no spin and the small distribution etc.The paper uses different time scale series of equations accord tothe Boltzmann equations.2(0)(2)10( 0))()12??f αt ?τ (1 ?1τ??t+e α??xfα=?τfα (2)Considering the supposition of equilibrium distribution function,∑α f α( 0)=φ,∑α f α( 0)e αj=0, ∑α f α( 0)e αieαj=πi(j0) (3)π x( x0 )= μ[(1 ?M∞2 )]ε (τφ?12),π x( y0)=0,π y( 0y )= ε (τμ?φ12) (4)where M ∞is the mach number, and μ parameterdeduce the subsonic and supersonic-potential equation??φ t =(1 ?M∞2 )??x2φ2+??y2φ2+O(ε2) (5)By solving the Eq.(3) and Eq.(4),we get2(0)(0)(0)0()1f = φ ?πxx +πyyc,2(0)(0)(0)1)13(121f = π xx ?πyyc2(0)(0)2131f = πyyc ,2(0)(0)3131f = πyyc, (6)2(0)(0)(0)4)13(121f = π xx ?πyyc,2(0)(0)5131f = πyyc,2(0)(0)6131f = πyycTo show the effect of the model, we separately simulate the fluid oftwo dimension subsonic and supersonic-potential flow equations, anddraw respective isoline of potential function. The graphs accord withtheory results.