Progress in the development of a new lattice Boltzmann method

Abstract

A new modeled Boltzmann equation (MBE) with four improvements made to conventional MBE is formulated. The first improvement is to include the particle internal rotational degree of freedom in the derivation of a continuous equilibrium velocity distribution function f eq; thus, rendering the MBE applicable to diatomic gas. The second improvement is made in the expansion assumed for fαeq in the lattice Boltzmann equation (LBE). This expansion is expressed in terms of the particle velocity vector (ξ) alone; hence, the LBE is no longer limited by a very low Mach number (M) assumption, and it also allows the LBE to correctly satisfy the zero divergence of the velocity field for incompressible flow. The third improvement is made to eliminate the bounce-back rule used to model no-slip wall boundary condition for fα because the rule leads to leakage at solid walls and mass conservation is compromised. The fourth improvement is carried out to render the modeled LBE truly valid for hydrodynamic flow simulation. Thus improved, the new lattice Boltzmann method (LBM) is no longer subject to the M << 1 condition and can be confidently extended to hydrodynamic simulations where the flow is truly incompressible. The fαeq expansion coefficients are found to be functions of the primitive variables, their derivatives and their products. In this LBE, fluid properties are inputs, and boundary values of fα are deduced from the primitive variables at the boundaries. Only a D2Q9 and a D3Q15 lattice model are required for 2-D and 3-D flow simulation, respectively. A finite difference splitting method is used to solve the new LBM; the scheme is labeled FDLBM, and it has been used to simulate widely different laminar flow of gas and liquid; incompressible with/without heat transfer, compressible with/without shocks, aeroacoustics with/without scattering, thermo-aeroacoustics, buoyant and double diffusive flow, and non-Newtonian flow, as well as blood flow in arteries with/without blockage. Accurate results are obtained, and they agree with other finite-difference numerical simulations of the same problems, and experimental and theoretical results whenever available. Furthermore, the aeroacoustics results are in agreement with those obtained from direct aeroacoustics simulations of the same problems.