Derivation of Zou/He boundaries

Dear all,

I chose LBM as my project for my simulation course. First of all, this site is super helpful, I appreciate it a lot.

I’ve been through the matlab sample code but have one question remaining. It concerns the derivation of the Zou/He microscopic boundary conditions for the inlet and outlet.

    fIn(2,in,col) = fIn(4,in,col) + 2/3*rho(:,in,col).*ux(:,in,col); 
    fIn(6,in,col) = fIn(8,in,col) + 1/2*(f ....

I read their paper, where it says that it was derived by

[…] use bounceback rule for the non-equilibrium part of the particle distribution normal to
the inlet to find f1 - f(eq)1= f3 - f(eq)3.]

Could somebody explain me why they can do that? To my knowledge and the way bounceback is implemented in the matlab-code, it just switches e_i and does not explicitly refer to the non-equilibrium part?!

Thank you very much