Simulation of two-phase liquid-vapor flows using a high-order compact finite-difference lattice Boltzmann method

A high-order compact finite-difference lattice Boltzmann method (CFDLBM) is extended and applied to accurately simulate two-phase liquid-vapor flows with high density ratios. Herein, the He-Shan-Doolen-type lattice Boltzmann multiphase model is used and the spatial derivatives in the resulting equations are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge-Kutta scheme to provide an accurate and efficient two-phase flow solver. A high-order spectral-type low-pass compact nonlinear filter is used to regularize the numerical solution and remove spurious waves generated by flow nonlinearities in smooth regions and at the same time to remove the numerical oscillations in the interfacial region between the two phases. Three discontinuity-detecting sensors for properly switching between a second-order and a higher-order filter are applied and assessed. It is shown that the filtering technique used can be conveniently adopted to reduce the spurious numerical effects and improve the numerical stability of the CFDLBM implemented. A sensitivity study is also conducted to evaluate the effects of grid size and the filtering procedure implemented on the accuracy and performance of the solution. The accuracy and efficiency of the proposed solution procedure based on the compact finite-difference LBM are examined by solving different two-phase systems. Five test cases considered herein for validating the results of the two-phase flows are an equilibrium state of a planar interface in a liquid-vapor system, a droplet suspended in the gaseous phase, a liquid droplet located between two parallel wettable surfaces, the coalescence of two droplets, and a phase separation in a liquid-vapor system at different conditions. Numerical results are also presented for the coexistence curve and the verification of the Laplace law. Results obtained are in good agreement with the analytical solutions and also the numerical results reported in the literature. The study shows that the present solution methodology is robust, efficient, and accurate for solving two-phase liquid-vapor flow problems even at high density ratios.

Figure 1

The D2Q9 lattice and the microscopic velocities.

Figure 2

The Maxwell construction on the isothermal $p-v$ diagram.

Figure 3

Initial condition (a) density contours and (b) sharp density profile at midplane for a 2D isothermal liquid-vapor system with a flat interface.

Figure 4

Computed (a) density field and (b) density profiles at midplane for a 2D isothermal liquid-vapor system with a flat interface using SW, SB, and IB sensors at temperatures $T=0.8$ and $T=0.7$.

Figure 5

Comparison of velocity profiles at midplane for a 2D isothermal liquid-vapor system with a flat interface using SW, SB, and IB sensors at temperatures (a) $T=0.8$ and (b) $T=0.7$.

Figure 6

Effect of value of filter coefficient $a_f$ on (a) density profile at mid-plane and (b) convergence rate of solution using SW sensor (top row), SB sensor (middle row), and IB sensor (bottom row) for a 2D isothermal liquid-vapor system with a flat interface at temperature $T=0.8$.

Figure 7

Comparison of (a) density and (b) velocity profiles at midplane for a 2D isothermal liquid-vapor system with a flat interface at $T=0.9$, $\kappa =2.5\times {10}^{-5}$ (top row) and $T=0.85$, $\kappa ={10}^{-5}$ (bottom row).

Figure 8

Coexistence curve for a 2D isothermal liquid-vapor system with a flat interface at different temperatures obtained by the present solution procedure using SW, SB, and IB sensors compared with the theory of Maxwell construction.

Figure 9

Grid refinement study on (a) density and (b) $u$-velocity profiles along midline through center of domain for 2D droplet suspended in gaseous phase at temperature $T=0.4$.

Figure 10

Comparison of computed density field of equilibrium state of 2-D droplet using SB sensor (top row) and IB sensor (bottom row) at temperatures (a) $T=0.4$, (b) $T=0.6$, and (c) $T=0.8$.

Figure 11

A schematic of cut-off zone in interfacial region of 2D droplet.

Figure 12

Comparison of density profiles at interfacial region of 2D droplet using SB and IB sensors at different temperatures.

Figure 13

Comparison of maximum value of spurious velocity for equilibrium state of 2D droplet calculated by the present solution procedure using SB and IB sensors with the standard Shan-Chen model at different temperatures.

Figure 14

Effect of value of constants $c_e$ and $c_s$ in the IB sensor on density profiles at interfacial region of 2D droplet compared with result of SB sensor at temperature $T=0.8$.

Figure 15

Effect of value of surface tension coefficient $\kappa$ on density profiles at interfacial region of 2D droplet using SB and IB sensors at temperature $T=0.5$.

Figure 16

Verification of the Laplace law for a 2D droplet investigated by the present solution procedure using SB and IB sensors at different values of surface tension coefficient $\kappa$ and temperature $T$.

Figure 17

Computed density field of a square liquid droplet between two parallel flat plates at temperature $T=0.7$ for normalized solid densities (a) $D_{\rho }=0.2$, (b) $D_{\rho }=0.5$, (c) $D_{\rho }=0.8$, and (d) $D_{\rho }=0.9$.

Figure 18

Computed wetting angle obtained by the present solution procedure for different normalized solid densities at temperature $T=0.7$.

Figure 19

Flow parameters of two droplets used for numerical simulation of coalescence phenomena.

Figure 20

Snapshots of numerical results of droplet shapes in coalescence phenomena at distances (a) $d=0.136$, (b) $d=0.116$, and (c) $d=0.076$.

Figure 21

A schematic for coalescence of two droplets and connection bridge radius $r_b$.

Figure 22

Time evolution of connection bridge radius $r_b$ for coalescence of two droplets with distances $d=0.116$ and $d=0.076$.

Figure 23

Computed density field of 2D phase separation in a liquid-vapor system using SB sensor at temperature $T=0.7$ at times $t=0.4$ (first row), $t=1.2$ (second row), $t=2.0$ (third row), and $t=5.2$ (fourth row) for surface tension coefficients (a) $\kappa ={10}^{-4}$, (b) $\kappa ={10}^{-5}$, and (c) $\kappa ={10}^{-6}$.

Figure 24

Computed density field of 2D phase separation in a liquid-vapor system using IB sensor at temperature $T=0.7$ at times $t=0.4$ (first row), $t=1.2$ (second row), $t=2.0$ (third row), and $t=5.2$ (fourth row) for surface tension coefficients (a) $\kappa ={10}^{-4}$, (b) $\kappa ={10}^{-5}$, and (c) $\kappa ={10}^{-6}$.