Two-dimensional finite-difference lattice Boltzmann method for the complete Navier-Stokes equations of binary fluids

– Based on Sirovich’s two-fluid kinetic theory and a dodecagonal discrete velocity model, a two-dimensional 61-velocity finite-difference lattice Boltzmann method for the complete Navier-Stokes equations of binary fluids is formulated. Previous constraints, in most existing lattice Boltzmann methods, on the studied systems, like isothermal and nearly incompressible, are released within the present method. This method is designed to simulate compressible and thermal binary fluid mixtures. The validity of the proposed method is verified by investigating (i) the Couette flow and (ii) the uniform relaxation process of the two components. Introduction. – Lattice Boltzmann Method (LBM) has become a viable and promising numerical scheme for simulating fluid flows. There are several options to discretize the Boltzmann equation: (i) Standard LBM (SLBM) [1]; (ii) Finite-Difference LBM (FDLBM) [1–3]; (iii) Finite-Volume LBM [1, 4]; (iv) Finite-Element LBM [1, 5]; etc. These kinds of schemes are expected to be complementary in the LBM studies. Even though various LBMs for multicomponent fluids [6–19] have been proposed and developed , (i) most existing methods belong to the SLBM [6–16], and/or based on the singlefluid theory [7–14,16,17,20]; (ii) in Ref. [6] a SLBM based on Sirovich’s two-fluid kinetic theory [21] is proposed; (iii) nearly all the studies are focused on isothermal and nearly incompressible systems. In a recent study [22], Sirovich’s kinetic theory is clarified and corresponding twofluid FDLBMs for Euler equations and isothermal Navier-Stokes equations are presented. In this letter we propose a two-fluid FDLBM for the complete Navier-Stokes equations, including the energy equation. Formulation and verification of the FDLBM. – The formulation of a FDLBM consists of three steps: (i) select or design an appropriate discrete velocity model (DVM), (ii) formulate the discrete local equilibrium distribution function, (iii) choose a finite-difference scheme. The continuous Boltzmann equation has infinite velocities, so the rotational invariance is automatically satisfied. Recovering rotational invariant macroscopic equations from a discretefinite-velocity microscopic dynamics imposes constraints on the isotropy of DVM and the