On lattice Boltzmann modeling of phase transition in an isothermal non-ideal fluid

https://doi.org/10.1016/S0029-5493(01)00435-6Get rights and content

Abstract

A new lattice Bolztmann BGK model for isothermal non-ideal fluid is introduced and formulated for an arbitrary lattice, composed of several D-dimensional sublattices. The model is a generalization of the free-energy-based lattice Bolztmann BGK model developed by Swift et al. (1996). We decompose the equilibrium distribution function in the BGK collision operator into ideal and non-ideal parts and employ second-order Chapman–Enskog expansion for treatment of both parts. Expansion coefficients for the non-ideal part are, in general, functions of macroscopic variables, designed to reproduce desired pressure tensor (thermodynamic aspects) and to eliminate the aphysical artifacts in the lattice Bolztmann model. The new model is shown to significantly improve quality of lattice Boltzmann modeling of interfacial phenomena. In the present model, the interface spurious velocity is orders of magnitude lower than that for existing LBE models of non-ideal fluids. A new numerical scheme for treatment of advection and collision operators is proposed to significantly extend stability limits, in comparison to existing solution methods of the ‘master’ lattice Bolztmann equation. Implementation of a ‘multifractional stepping’ procedure for advection operator allows to eliminate severe restriction CFL=1 in traditionally used ‘stream-and-collide’ scheme. An implicit trapezoidal discretization of the collision operator is shown to enable excellent performance of the present model in stiff high-surface-tension regime. The proposed numerical scheme is second order accurate, both in time and space.

Section snippets

Introduction and background

During the past decade, the lattice-Boltzmann Equation (LBE) method (Qian et al., 1992) has been developed extensively, and has attracted significant attention in computational physics and multiphase flow modeling communities. Lying in between microscopic molecular dynamics and conventional macroscopic fluid dynamics, the LBE simulates the macroscopic complex flow behavior by dealing with the underlying micro-world. Recently, the LBE method has been shown to provide convincing results for

Lattice geometry and symmetry

Consider the lattice composed of r sublattices in d dimenstions. Each sublattice has weight wr, which are chosen to satisfy certain symmetry requirements, which will be discussed shortly. In total, the lattice has a=0,  b links, ea.

The most important properties of the lattice are related to the symmetries of the tensors:Ei1i2…inn=aw(|ea|2)(ea)i1(ea)in,which are determined from the choice of the basic lattice directions ea.

The basic condition for standard hydrodynamic behaviour is that tensors E

Numerical treatment

In the present section, new numerical schemes for solution of lattice Boltzmann equation are introduced.

Swift et al. model

It is possible to show that the model by (Swift et al., 1995) is a particular case of the model developed in the present study, if the following coefficients of the Chapman–Enskog expansion for a correction term, Eq. (39), are chosen:β=A(c)+C(c)u2=p0*−κρ∂kkρ+(1/d−12)κ(∂kρ)2ϒ(2)ρBi(c)=0where p0*=p0ρCs2 is a ‘non-ideal part’ of the equation of state and d is a space dimension. Note, that any choice of β leads to the chosen thermodynamics (pressure tensor). We have numerically proved that

Numerical results and discussion

Computational results presented in this paper are obtained by the FlowLab code. This parallel code is written in a MPI (Message-Passing Interface) standard. Currently, there are two lattice configurations available in the FlowLab: D2Q9 and D3Q15. All results presented here are obtained for a two-dimensional D2Q9 lattice.

Conclusions

In the present paper, a new lattice Boltzmann model for isothermal non-ideal fluid is introduced. The equilibrium distribution function in the BGK collision operator is split into two parts. The first part is responsible for ideal fluid behaviour (kinetic effects), while the second part is designed to represent the desired non-ideal-fluid features. Both parts of the equilibrium distribution function are approximated by the second-order Chapman–Enskog expansions in a low-Mach-number limit.

References (27)

  • X. He et al.

    Discrete Boltzmann equation model for nonideal gases

    Physical Review Letters

    (1998)
  • Y. Kato et al.

    Amadeus project and microscopic simulation of boiling two phase flow by the Lattice-Boltzmann method

    International Journal of Modern Physics C

    (1997)
  • Landau, L.D., Lifschitz, E.M., 1988. Theoretical Physics, vol. 6, Hydrodynamics. In Russian, Nauka, Fourth Edition,...
  • Cited by (35)

    • A modified pseudopotential for a lattice Boltzmann simulation of bubbly flow

      2010, Chemical Engineering Science
      Citation Excerpt :

      There have also been many previous works for reducing the spurious velocity. Nourgaliev et al. (2002) found that the spurious velocity could be reduced with a finite difference approach in the streaming step. Wagner (2003) argued that that the spurious velocity was caused by non-compatible discretization of the driving forces, and reduced the maximum spurious velocity to O(10−16) using the potential form of the surface tension.

    View all citing articles on Scopus
    View full text