Interdisciplinary Applied Mathematics

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P = 53 fi and pv = 53 ci fi.


In simulating incompressible isothermal flows, the relaxation time t is taken constant in    the    BGK    mode,    but    as    we    have    discussed    already    in    Section

15.4, this is not valid for nonisothermal flows or flows with variable density, as in gas microflows. To this end, it has been proposed by Chen and collaborators (Nie et al., 1998) to modify the relaxation parameter as follows:

,    1 АО    1

T=2 + 7lT“2

where p0 = 1. The viscosity v and mean free path are defined based on the relaxation parameter from

v = с?вАЬ(т’ — 1/2) and A=—(r’— 1/2).    (15.50)


In order to arrive at the Navier-Stokes equation from the LBM we also need to employ the limit of long wavelength and low Mach number, and using the Chapman-Enskog multiscale expansion the resulting equations are

The pressure is given by P = csp. If the density variations are small, we recover the more familiar form of Navier-Stokes from the above equations.

The BGK version of LBM is typically more accurate for values of т < 1. A systematic truncation error analysis of LBM is performed in (Holdych et al., 2004). The method is second-order accurate in time for fixed lattice spacing, but is first-order accurate in time and second-order accurate in space for a constant value of т. If the lattice spacing is reduced for a constant ratio Ax/At, then LBM does not converge. It is also recognized that LBM has the same type of inconsistency as the classical Dufort-Frankel scheme for diffusion, and thus consistency is satisfied only for At/Ax2 remaining constant during refinement. For constant values of т, LBM is consistent in the classical sense.

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