Deriving thermal lattice-Boltzmann models from the continuous Boltzmann equation: theoretical aspects

摘要

The particles model, the collision model, the polynomial development used forthe equilibrium distribution, the time discretization and the velocitydiscretization are factors that let the lattice Boltzmann framework (LBM) faraway from its conceptual support: the continuous Boltzmann equation (BE). Mostcollision models are based on the BGK, single parameter, relaxation-termleading to constant Prandtl numbers. The polynomial expansion used for theequilibrium distribution introduces an upper-bound in the local macroscopicspeed. Most widely used time discretization procedures give an explicitnumerical scheme with second-order time step errors. In thermal problems,quadrature did not succeed in giving discrete velocity sets able to generatemulti-speed regular lattices. All these problems, greatly, difficult thenumerical simulation of LBM based algorithms. In present work, the systematicderivation of lattice-Boltzmann models from the continuous Boltzmann equationis discussed. The collision term in the linearized Boltzmann equation ismodeled by expanding the distribution function in Hermite tensors.Thermohydrodynamic macroscopic equations are correctly retrieved with asecond-order model. Velocity discretization is the most critical step inestablishing regular-lattices framework. In the quadrature process, it is shownthat the integrating variable has an important role in defining the equilibriumdistribution and the lattice-Boltzmann model, leading, alternatively, totemperature dependent velocities (TDV) and to temperature dependent weights(TDW) lattice-Boltzmann models.