Analysis of the typical unified lattice Boltzmann models and a comprehensive multiphase model for convection-diffusion problems in multiphase systems

The present paper analyzes the typical unified lattice Boltzmann (LB) models for different convection-diffusion (CD) problems in multiphase systems. The CD problems in multiphase systems can be roughly classified into three groups: CD problems with a continuous scalar value and a continuous flux, a discontinuous scalar value and a continuous flux, a continuous scalar value and a discontinuous flux. The characteristics of the corresponding unified LB models for the three kinds of CD problems are analyzed and the equivalence between the LB models based on different perspectives or numerical schemes is revealed. Finally, a comprehensive multiphase LB model (CMLBM) capable of solving different isotropic and anisotropic CD problems in multiphase systems is proposed. Four typical CD problems in multiphase systems are calculated to validate the CMLBM; the results show that it performs well against the typical isotropic and anisotropic CD problems in multiphase systems.

Figure 1

Schematic of heat transfer in a channel filled with two fluids.

Figure 2

Comparisons of temperature contours at Fourier numbers $F=0.025$ (a) and $F=0.05$ (b) between the calculated results (blue dashed line) and analytical solutions (red solid line).

Figure 3

Numerical error $E_2$ (a) and $E_{\infty }$ (b) analyses of the SCMLBM and the standard SRT LB model for unsteady heat convection diffusion in a channel filled with two fluids.

Figure 4

Numerical errors $E_2$ (a) and $E_{\infty }$ (b) of the SCMLBM and the standard SRT LB model for heat convection diffusion in a multiphase system at different relaxation times.

Figure 5

Schematic of the two-fluid flow between two liquid films.

Figure 6

Concentration distributions at different transient moments.

Figure 7

Numerical errors $E_2$ (a) and $E_{\infty }$ (b) of the SCMLBM and the standard SRT LB model for mass transfer in a multiphase system at different mesh sizes.

Figure 8

Numerical errors $E_2$ (a) and $E_{\infty }$ (b) of the SCMLBM and the standard SRT LB model for mass transfer in a multiphase system at different relaxation times.

Figure 9

The physical model of the solid-liquid phase change.

Figure 10

Comparisons of $T$ (a) and $f_L$ (b) between the calculated results and the analytical solutions at different times for one-dimensional one-phase melting by conduction.

Figure 11

Comparisons of $T$ (a) and $f_L$ (b) between the calculated results and the analytical solutions at different moments for one-dimensional two-phase melting by conduction.

Figure 12

The physical model of the composite anisotropic medium.

Figure 13

Temperature contours at Fourier number $F=1$ and steady state calculated by the CMLBM and FVM: (a) $F=1$, CMLBM, (b) $F=1$, FVM, (c) steady, CMLBM and (d) steady, FVM; the area in the solid red circles is the discontinuous contour lines.

Figure 14

Comparison of the transient average dimensionless temperature between the CMLBM and FVM.