Improved thermal lattice Boltzmann model for simulation of liquid-vapor phase change

In this paper, an improved thermal lattice Boltzmann (LB) model is proposed for simulating liquid-vapor phase change, which is aimed at improving an existing thermal LB model for liquid-vapor phase change [S. Gong and P. Cheng, Int. J. Heat Mass Transfer 55, 4923 (2012)]. First, we emphasize that the replacement of $\mathbf{\nabla }·\left(\lambda \mathbf{\nabla }T\right)/\phantom{\mathbf{\nabla }·\left(\lambda \mathbf{\nabla }T\right)\rho c_V}\rho c_V$ with $\mathbf{\nabla }·\left(\chi \mathbf{\nabla }T\right)$ is an inappropriate treatment for diffuse interface modeling of liquid-vapor phase change. Furthermore, the error terms ${\partial }_{t_0}\left(T\mathbf{v}\right)+\mathbf{\nabla }·\left(T\mathbf{vv}\right)$, which exist in the macroscopic temperature equation recovered from the previous model, are eliminated in the present model through a way that is consistent with the philosophy of the LB method. Moreover, the discrete effect of the source term is also eliminated in the present model. Numerical simulations are performed for droplet evaporation and bubble nucleation to validate the capability of the model for simulating liquid-vapor phase change. It is shown that the numerical results of the improved model agree well with those of a finite-difference scheme. Meanwhile, it is found that the replacement of $\mathbf{\nabla }·\left(\lambda \mathbf{\nabla }T\right)/\phantom{\mathbf{\nabla }·\left(\lambda \mathbf{\nabla }T\right)\rho c_V}\rho c_V$ with $\mathbf{\nabla }·\left(\chi \mathbf{\nabla }T\right)$ leads to significant numerical errors and the error terms in the recovered macroscopic temperature equation also result in considerable errors.

Figure 1

Validation of the D2 law. Snapshots of the density contours obtained by (a) the Gong-Cheng model, (b) the compromised model, and (c) the improved model. The snapshots are taken at (from left to right) $t=2\times {10}^4{\delta }_t$, $5\times {10}^4{\delta }_t$, and $1.6\times {10}^5{\delta }_t$.

Figure 2

Validation of the D2 law. Comparison of the numerical results given by the Gong-Cheng model, the compromised model, and the improved model with the data in Ref. [50], which were obtained by a finite-difference scheme.

Figure 3

Droplet evaporation on a solid surface. Snapshots of the density contours obtained by (a) the Gong-Cheng model, (b) the compromised model, and (c) the improved model. The snapshots are taken at (from left to right) $t=5\times {10}^4{\delta }_t$, $1.5\times {10}^5{\delta }_t$, and $2.5\times {10}^5{\delta }_t$. The displayed domain is $x\in \left[50,250\right]$ and $y\in \left[0,115\right]$.

Figure 4

Droplet evaporation on a solid surface. Comparison of the numerical results obtained by the Gong-Cheng model, the compromised model, the improved model, and a finite-difference scheme. Here “l.u.” denotes lattice units.

Figure 5

Simulation of bubble nucleation and departure ($g=1.5\times {10}^{-5}$). Snapshots of the density contours obtained by (a) the improved model, (b) the compromised model, and (c) the Gong-Cheng model. The snapshots are taken at (from left to right) $t=2000{\delta }_t$, $t=5000{\delta }_t$, and $t=14000{\delta }_t$.

Figure 6

Simulation of bubble nucleation and departure ($g=2.5\times {10}^{-5}$). Snapshots of the density contours obtained by (a) the improved model, (b) the compromised model, and (c) the Gong-Cheng model. The snapshots are taken at (from left to right) $t=2000{\delta }_t$, $t=5000{\delta }_t$, and $t=12000{\delta }_t$.

Figure 7

Simulation of bubble nucleation and departure. The bubble departure diameter (“l.u.” denotes lattice units) predicted by the improved model. The squares represent the numerical results obtained by the improved model and the solid line denotes the results given by $D_{\mathrm{d}}=0.209g^{-0.5}$.