next up previous contents
Next: The Conservation Equations Up: An IsotropicGalilean Invariant Previous: An IsotropicGalilean Invariant

The Equilibrium Distribution

  First we expand the equilibrium distribution tex2html_wrap_inline13963 up to second-order in the velocity tex2html_wrap_inline14035 :

equation1883

where

equation1896

The coefficients tex2html_wrap_inline14039 and tex2html_wrap_inline14041 need to be found, subject to isotropy and Galilean invariance, in order to obtain the required equilibrium distribution (to accuracy tex2html_wrap_inline14035 ). The isotropy conditions required on the second- and fourth-order tensors [47, 48] can be explicitly introduced by defining tex2html_wrap_inline14045 and tex2html_wrap_inline14047 :

  equation1908

and

  equation1915

The sum of the equilibrium distribution multiplied by an odd number of tex2html_wrap_inline13399 's is zero. Considering tex2html_wrap_inline14051 in equation (4.16) and tex2html_wrap_inline14053 in equation (4.17) gives

  equation1932

and

  equation1936

Combining these we see that

  equation1942

We now consider the Galilean invariance requirement. If tex2html_wrap_inline14055 is the jth moment,

equation1945

then we require [47, 48]

equation1953

The zeroth moment is

equation1956

Considering terms tex2html_wrap_inline14035 and using the isotropy condition, equation (4.16), we get

  equation1969

The first moment is

equation1972

Considering terms O(u) and using the isotropy condition, equation (4.16), gives

  equation1987

The second moment is

equation1990

Collecting terms of second-order in tex2html_wrap_inline12875 gives

equation2010

Considering separately the terms tex2html_wrap_inline14093 and tex2html_wrap_inline14095 we get

  equation2031

and

  equation2034

The third moment is

equation2037

Collecting terms tex2html_wrap_inline14111 gives

equation2059

This gives the final condition

  equation2067

Solving equations (4.20), (4.24), (4.26), (4.29), (4.30) and (4.33) gives tex2html_wrap_inline14113 and C = 8. The value of tex2html_wrap_inline14117 is found from equation (4.18) and tex2html_wrap_inline14119 is obtained by considering the sum of the distribution functions

equation2079

Thus we have the following equilibrium distribution for a two-dimensional lattice Boltzmann model which, by construction, must be isotropic and Galilean invariant.

equation2083

where

equation2096



James Buick
Tue Mar 17 17:29:36 GMT 1998