Tutorial: Multi-Species Lattice Boltzmann Models and Applications
Tutorial: Multi-Species Lattice Boltzmann Models and Applications
Tutorial: Multi-Species Lattice Boltzmann Models and Applications
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
<strong>Lattice</strong> <strong>Boltzmann</strong> solvers <strong>and</strong> applications<br />
Regular Knudsen expansion<br />
Andries-Aoki-Perthame (AAP) model<br />
Clearly the solution of the BGK equation depends on ɛ. The<br />
solution for small ɛ is investigated in the form of the asymptotic<br />
regular expansion<br />
ρ <strong>and</strong> q σi are also exp<strong>and</strong>ed:<br />
f σ = f (0)<br />
σ + ɛf (1)<br />
σ + ɛ 2 f (2)<br />
σ + · · · . (24)<br />
ρ σ = ρ (0)<br />
σ + ɛρ (1)<br />
σ + ɛ 2 ρ (2)<br />
σ + · · · , (25)<br />
q σi = ɛq (1)<br />
σi<br />
+ ɛ 2 q (2)<br />
σi<br />
+ · · · , (26)<br />
since the Mach number is O(ɛ), the perturbations of q σi starts from<br />
the order of ɛ. Consequently<br />
f σ(∗) = f (0) (1)<br />
σ(∗)<br />
+ ɛf<br />
σ(∗) + ɛ2 f (2)<br />
σ(∗) + · · · , (27)<br />
Regular expansion means ∂ α f (k)<br />
σ<br />
= O(1) <strong>and</strong> ∂ α M (k) = O(1).<br />
Pietro Asinari, PhD (Politecnico di Torino) <strong>Multi</strong>-<strong>Species</strong> LB <strong>Models</strong> Rome, Italy, on July 5-9, 2010 17 / 51
Change language