Interdisciplinary Applied Mathematics

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While successful    for    simple    isothermal    fluid    flows,    the    above    LBM    for

mulation is not Galilean invariant for nonisothermal and multiphase flows; for isothermal flows it is Galilean invariant up to order O(M4), where M is the Mach number. It has also been shown to be unstable for small values of the viscosity. Indeed, in the above formulation we see that the viscosity is proportional to (r — ^), and for values of r e (^, 1] instabilities may arise. Such considerations have led to the development of the entropic lattice Boltzmann method in (Karlin et al., 1999; Boghosian et al., 2003). At the heart of    this    formulation is    the    use    of    the    H-theorem    of Boltzmann,    which

measures irreversibility. This was presented as the Boltzmann inequality in equation (15.7b). Specifically, H is the Boltzmann function based on which the entropy S is computed as S = -kBH, where

H = J f (x, v,t)ln f (x, v,t) dv dx.

Regardless of the form of the collision operator, the H-theorem states that dH/dt < 0, which in general is not satisfied in LB methods globally, although it may be satisfied locally in some versions. Specifically, it has been shown rigorously by (Yong and Luo, 2003) that the H-theorem does not exist for the lattice Boltzmann equation with polynomial equilibria.

Enforcing the H-theorem in the lattice Boltzmann formulation guarantees an asymptotically homogeneous spatial distribution of particles as t ^    ж,    which in    turn    translates    into    numerical    stability. In    the    contin

uum case, the Boltzmann inequality produces the Maxwellian distribution (used as equality in equilibrium). This is obtained as a minimizer of the H function defined above subject to the five conservation laws (as constraints) of mass, momentm (3), and energy. The local Maxwellian can also be interpreted as the zero point of the collision term, and also the zero point of the entropy production (Succi et al., 2002). In the discrete case the Maxwellian distribution is not necessarily the minimizer of the disrete H function (irrespective of the lattice), and this results in the aforementioned problems. To this end, in entropic lattice Boltzmann, use of different convex H functions is made that can be minimized relatively easily. The requirements are that Galilean invariance be preserved, as well as realizability (0 < feq < 1) and solvability. The latter condition implies that local equilibrium can be expressed in terms of the density and velocity, i.e., the continuum variables.

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