A Lattice Boltzmann Equation Method without Parasitic Currents and Its Application in Droplet Coalescence

A formulation of the intermolecular force in the nonideal gas lattice Boltzmann equation (LBE) method is examined. Discretization errors in the computation of the intermolecular force cause parasitic currents. The parasitic currents can be eliminated to round-off if the potential form of the intermolecular force is used with compact isotropic discretization. Numerical tests confirm the elimination of the parasitic currents. In order to demonstrate the applicability of the present model, inertial coalescence of droplets at high density ratio is studied. INTRODUCTION The lattice Boltzmann equation (LBE) methods for nonideal gases or binary fluids have witnessed significant progress in recent years [1–7]. The success of LBE methods can largely be attributed to their mesoscopic and kinetic nature, which enables the simulation of the macroscopic interfacial dynamics with the underlying microscopic physics. On the macroscopic level, most of these two-phase LBE methods can be considered as diffuse interface methods [8] in that the phase interface is spread on grid points and the surface tension is transformed into a volumetric force. Generally, diffuse interface methods have some advantages over sharp interface methods because computations are ∗Address all correspondence to this author. much easier for three-dimensional (3-D) flows in which topological change of the interfaces is complicated. When applied on the uniform grid, LBE methods enjoy the unit CFL (Courant, Friedrichs, and Lewy) property that eliminates any numerical errors involved in the computation of the advection operator. The inherent isotropy of the lattice guarantees isotropic discretization of the differential operators in LBE. Free from advection errors and anisotropic discretization, the LBE method can deliver much improved solutions with the same grid resolution. One undesirable feature of LBE methods as a diffuse interface method is the existence of parasitic currents. The parasitic currents are small-amplitude velocity fields caused by a slight imbalance between stresses in the interfacial region [9]. These currents increase as the surface tension force and can be reduced with large viscous dissipation, but never disappear in most cases. In the case of a 2-D liquid droplet immersed in a vapor phase, the flow tends to be organized into eight eddies with centers lying on the interface. In the diffuse interface method, the key to reducing the parasitic currents lies in the formulation of the surface tension force. Jacqmin [10] suggested that the potential form of the surface tension force was guaranteed to generate motionless equilibrium states without parasitic currents. Jamet et al. [11] later showed that the potential form ensured the correct energy transfer between the kinetic energy and the surface tension energy, eliminating parasitic currents. 1 Copyright c © 2006 by ASME Several attempts have been made to reduce the magnitude of the parasitic currents and identify their origins [12–14] in the LBE framework. Nourgaliev et al. [12] employed a finite difference approach in the streaming step of LBE and reported reduced currents compared with the previous LBE methods. Lishchuk et al. [13] noted that the parasitic currents were unwanted artifacts originating from the mesoscopic (or microscopic) nature of LBE having the interface with a finite thickness, and they tried to incorporate sharp interface kinematics into their LBE method. Cristea and Sofonea [14] argued that the directional derivative operator eα ·∇ in LBE (see Eq. (1)) generated the parasitic currents in the interfacial region, and consequently, they concluded that the parasitic currents would be reduced by the surface tension force. All these LBE schemes were able to reduce the magnitude of the parasitic currents to a certain degree but never made them entirely disappear. The discrete Boltzmann equation (DBE) proposed by He et al. [5] will be analyzed, but the analysis is equally valid for other LBE methods. We will show that the potential form of the intermolecular force in the LBE context eliminates the parasitic currents. In order to demonstrate the applicability of the present model, inertial coalescence of droplets will be examined. NOMENCLATURE fα Particle distribution function f eq α Equilibrium distribution function eα Microscopic particle velocity u Macroscopic velocity ρ Density cs Speed of sound λ Relaxation time τ Nondimensional relaxation time tα Weighting factor F Intermolecular force κ Gradient parameter p0 Thermodynamic pressure Emix Mixing energy E0 Bulk energy μ0 Chemical potential D Interface thickness σ Surface tension ν Kinematic viscosity R0 Radius of a droplet THEORY The DBE with external force F can be written as ∂ fα ∂t + eα ·∇ fα =− fα− f eq α λ + (eα−u) ·F ρcs f eq α , (1) where fα is the particle distribution function, eα is the microscopic particle velocity, u is the macroscopic velocity, ρ is the density, cs is a constant, and λ is the relaxation time. The equilibrium distribution function f eq α is given by f eq α = tαρ [ 1+ eα ·u cs + (eα ·u) 2cs − (u ·u) 2cs ]