A New Differential Lattice Boltzmann Equation and Its Application to ...

A New Differential Lattice Boltzmann Equation 3312. STANDARD LBE AND CONVENTIONAL DIFFERENTIAL LBEThe two dimensional, standard LBE with BGK approximation can bewritten asf a (x+e ax dt, y+e ay dt, t+dt)=f a (x, y, t)+[f eqa (x, y, t) − f a (x, y, t)]/y for a=0, 1, 2,..., 8 (1)where y is the single relaxation parameter, (5) f a is the density distributionfunction, f eqa is its corresponding equilibrium state, d t is the time step ande a is the particle velocity in the a direction. The nine discrete velocitiese a =(e ax ,e ay ) are defined as0), a=0e a =˛(0,(cos[(a −1)p/2], sin[(a −1)p/2]), a=1,..., 4(cos[(a −5)p/2+p/4], sin[(a −5)p/2+p/4]) `2, a=5,..., 8(2)The equilibrium density distribution function f eqa is chosen to be in ref. 5f eqa =w a r[1+3(e ax u+e ay v)+ 9 2 (e axu+e ay v) 2 − 3 2 (u2 +v 2 )] (3)with w 0 =4/9, w a =1/9 for a=1,..., 4, and w a =1/36 for a=5,..., 8. Themacroscopic density r, the velocity u and v are obtained fromCaf a =r Caf a e ax =ru and Caf a e ay =rv (4)The speed of sound of this model is c s =1/`3 , and pressure can bedirectly computed from the equation of stateP=rc 2 s (5)The lattice-uniformity of the standard LBE (1) requires that dx=e ax · d t ,dy=e ay · d t . This feature is quite restrictive in many applications involvingexternal flows where a large gradient exists only in a small region while thedomain of interest is large. To improve the standard LBE for its generalapplication, the differential LBE can be developed by using the Taylorseries expansion. Using the following Taylor series expansionf a (x+e ax d t ,y+e ay d t ,t+d t )“f=f a (x, y, t)+e ax da (x, y, t) “ft +e“xay da (x, y, t) “ft +da (x, y, t)“yt +O(d 2 t )“t(6)