Interdisciplinary Applied Mathematics

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tion, which admits power-law solutions for certain types of lattices.

In summary, this new entropic LB method offers many possibilities and potentially can overcome the aforementioned shortcomings of the more traditional LBM. However, its extensions to nonisothermal flows as well as complex fluid flows has not been established rigorously yet. We also note that the limit of LBM to the Navier-Stokes equations has been obtained so far only for incompressible flows but not for compressible flows. In a sense, we have a quasi-compressible formulation that can be used for both incompressible and compressible flows. In this limit the Mach number scales with the Knudsen number, and the fluctuation of density around its mean value scales with the Knudsen number squared (Boghosian et al., 2003).

15.5.1    Boundary Conditions

With regard to boundary conditions, the so-called bounce-back scheme with its origin in CA has been used to simulate wall boundary conditions. In the bounce-back scheme, when a particle distribution streams to a wall node, the particle distribution scatters back to the node it came from (Chen and Doolen, 1998). However, this approach leads to relatively low-order accuracy,    and    more    recent    work    has    attempted    to include    corrections    in

the distribution function by including velocity gradients at the wall nodes (Skordos, 1993), explicitly imposing a pressure constraint (Noble et al., 1995) or employing extrapolation techniques on staggered meshes (nodes at midpoints of lattice), similarly to the classical finite difference discretizations (Chen et    al.,    1996).    An    analysis    of the    accuracy    of    the    boundary

conditions as a function of the relaxation parameter т was performed in (Holdych et al., 2004).

A robust boundary condition was analyzed in (Wagner and Pagonabar-raga, 2002), for LBM, the so-called Lees-Edwards periodic boundary conditions. These are appropriate for simulating flows with simple boundaries, like Couette flows, subject to severe shear, which could be problematic using other types of boundary conditions. One issue is the interpolation scheme employed, since it may introduce artificial dissipation. Finally, new boundary conditions with error estimates have been formulated in (An-sumali and Karlin, 2002), and have been applied to slip flows as well as to no-slip flows.

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