From the continuous to the lattice Boltzmann equation: The discretization problem and thermal models

Abstract

The velocity discretization is a critical step in deriving the lattice Boltzmann (LBE) from the continuous Boltzmann equation. This problem is considered in the present paper, following an alternative approach and giving the minimal discrete velocity sets in accordance with the order of approximation that is required for the LBE with respect to the continuous Boltzmann equation and with the lattice structure. Considering to be the order of the polynomial approximation to the Maxwell-Boltzmann equilibrium distribution, it is shown that solving the discretization problem is equivalent to finding the inner product in the discrete space induced by the inner product in the continuous space that preserves the norm and the orthogonality of the Hermite polynomial tensors in the Hilbert space generated by the functions that map the velocity space onto the real numbers space. As a consequence, it is shown that, for each order N of approximation, the even-parity velocity tensors are isotropic up to rank 2N in the discrete space. The norm and the orthogonality restrictions lead to space-filling lattices with increased dimensionality when compared with presently known lattices. This problem is discussed in relation with a discretization approach based on a finite set of orthogonal functions in the discrete space. Two-dimensional square lattices intended to be used in thermal problems and their respective equilibrium distributions are presented and discussed.

High-order regularization in lattice-Boltzmann equations

“…, n b − 1, used to represent the microscopic velocity space. This formal relationship was found by Philippi and co-workers 29,30 and is based on the requirement that a discrete Hermitian representation respects the orthogonality described by Eq. (4) up to a given order, i.e.…”

Natural convection with mixed insulating and conducting boundary conditions: low- and high-Rayleigh-number regimes

“…Lattice Boltzmann Methods are well known and widely applied to a variety of single and multiphase hydrodynamic problems (Shan & Chen 1993;Shan & Doolen 1996;Succi 2005) and they have also been developed to study thermal fluids, both with the Boussinesq approximation (Benzi et al 1998;Shan 1997) and in a fully thermal regime (Scagliarini et al 2010;Zhang & Tian 2008;Biferale et al 2013;Philippi et al 2006;Prasianakis & Karlin 2007;Gonnella et. al.…”

Thermodynamic consistency in deriving lattice Boltzmann models for describing segregation in non-ideal mixtures

“…In fact, in its early development, the LBM was considered to be applicable only to flows with small Mach number and constant temperature. A key breakthrough in this area was the identification of the correspondence between the LBE and the kinetic continuous equation, and the invention of highorder LB schemes [16][17][18], leading to velocity sets that, when used in a discrete velocity kinetic scheme, ensure accurate recovery of the high-order hydrodynamic moments.…”

“…In this attempt, the thermodynamic consistency of the derived kinetic equations is thoroughly investigated in this paper. Owing to space limitations, only the kinetic equation in continuous space is presented in this paper, the LBEs themselves being considered as discrete forms of the continuous kinetic model, after using an appropriate velocity discretizaton scheme [16][17][18]. In this respect, it is important to distinguish errors due to the kinetic model used for non-isothermal segregation from errors due to the discretization procedure.…”