distribution and equilibruim distribution function

Lattice Boltzmann Method LBM: Theory
#1

Hi everybody,

According to conservation laws for D2Q9 and for LBGK model;

sum(0-8) [f] = sum(0-8) [f_eq] = rho
sum(0-8) [ef] = sum(0-8) [ef_eq] = rho*u

f: distribution function
f_eq: equilibruim distribution function.
rho: density
u: fluid velocity
0 -8: lattice numbers (form 0 to 8)

In Multi-Chapman Enskog Analysis.
f= f_0 + eps*f_1

f_0: Equilibruim distribution function (Zeroth order distribution function)
f_1: First order distribution function

Therefore;

sum(0-8) [f_1] =0 (Eq*)
sum(0-8) [e*f_1] = 0 (Eq **)

My Question is; why do the Eq* and Eq** like this. Where do they come form?

Thank you,
Daracan

#2

Hi all,

I found what is the reason, all the sum of ’ f_1 ’ and ’ e*f_1 ’ is zero.

A global conservation of macroscopic quantity is expressed locally by collisional invariant. In other words, conservation mass in a fluid is enforced by local conservation mass during the collisional between particles.

Col: Colision process
In LBGK=> Col= -omega (f-f_eq)

For mass;
Sum(Col)=0
=> sum (-omega (f-f_eq))=0
=> sum(f)=sum(f_eq)

For momentum;
Sum (eCol)=0
=> sum (e(-omega (f-f_eq)))=0
=> sum(ef)=sum(ef_eq)

Therefore;
sum(f_1)=0
sum(e*f_1)=0

Best Regards
Daralcan