## 2 Double Distribution TLBM
### Density Distribution Function  
The D2Q9 lattice Boltzmann (LB) model has been widely used in simulations of many complex flows [33]. It consists of two parts, namely streaming and collision. A D2Q9 model with Bhatnagar-Gross-Krook approximation of the collision term is given by  
\[f_{i}(\mathbf{x}+\Delta\mathbf{x},t+\Delta t)=f_{i}(\mathbf{x},t)-\frac{1}{ \tau_{f}}[f_{i}(\mathbf{x},t)-f_{i}^{\text{eq}}(\mathbf{x},t)]+\Delta f_{i}( \mathbf{x},t), \tag{1}\]  
where \(f_{i}(\mathbf{x},t)\) is the particle distribution function in the \(i\)-th discretized velocity direction at position \(\mathbf{x}\) and time \(t\), \(\Delta x\), \(\Delta t\) and \(\tau_{f}\) are the lattice spacing, time step and relaxation time, respectively. The equilibrium distribution function is obtained by  
\[f_{i}^{\text{eq}}=\rho w_{i}\left[1+\frac{\mathbf{e}_{i}\cdot\mathbf{u}}{c_{s ^{2}}}+\frac{(\mathbf{e}_{i}\cdot\mathbf{u})^{2}-(c_{s}|\mathbf{u}|)^{2}}{2c_ {s}^{2}}\right], \tag{2}\]  
and one has  
\[\Delta f_{i}=f_{i}^{\text{eq}}(\rho,\mathbf{u}+\Delta\mathbf{u})-f_{i}^{ \text{eq}}(\rho,\mathbf{u}), \tag{3}\]  
with \(w_{0}=4/9\), \(w_{1,2,3,4}=1/9\), \(w_{5,6,7,8}=1/36\), \(c_{s}=\sqrt{3}c\) the sound speed, \(c=\Delta x/\Delta t\), \(\mathbf{u}=\sum_{i=0}^{8}f_{i}\mathbf{e}_{i}\), and the velocity set \(\mathbf{e}_{i}\) is given by  
\[\mathbf{e}_{i}=\left(e_{x}(i),\;e_{y}(i)\right)=\left\{ \begin{array}{ll}(0, 0)c&i= 0\\ (\cos[(i-1)\frac{\pi}{2}],\sin[(i-1)\frac{\pi}{2}])c&i= 1,2,3,4\\ (\sqrt{2}\cos[(2i-9)\frac{\pi}{4}],\sqrt{2}\sin[(2i-9)\frac{\pi}{4}])c&i= 5,6,7,8.\end{array} \right. \tag{4}\]  
Letting \(\mathbf{F}\) be an external force, then the macroscopic properties such as density and velocity can be obtained by  
\[\rho(\mathbf{x},t)=\sum_{i=0}^{8}f_{i}(\mathbf{x},t),\quad\mathbf{U}(\mathbf{ x},t)=\mathbf{u}(\mathbf{x},t)+\frac{1}{2}\frac{\mathbf{F}(\mathbf{x},t)}{ \rho(\mathbf{x},t)}. \tag{5}\]  
Furthermore, the kinematic viscosity \(\
u\) relates to the relaxation time by \(\
u=(\tau-0.5)c_{s}^{2}\Delta t\). The exact difference method shown as above has good performance in simulating multiphase flows due to its simplicity and accuracy [34]. Moreover, the unphysical influence of relaxation time can be avoided. In the TFE problem, the external forces \(\mathbf{F}\) include cohesive force (between fluid nodes), adhesion force (between fluid and solid nodes), and body force (e.g. gravity). They will be discussed in the following. A single component multiphase LB model is considered to investigate TFE. Numerous results on the multiphase LB model have been reported. Readers may refer to [21; 25; 35] and the references therein for more detailed information. A pseudopotential approach inspired by [32; 36] is considered in this work. The cohesive force is described as the gradient of a psudopotential function, which has the following form  
\[\phi=-G(\rho c_{s}^{2}-\beta P), \tag{6}\]  
where \(P\) is the pressure, \(G\) and \(\beta\) are two tuning parameters. As studied in [32], the parameter \(\beta\) controls the resolution of liquid-vapor interface. Note that high resolution, however, suffers fromnumerical instability. The parameter \(G\) affects the density ratio of liquid and vapor. The model reduces to the one discussed in [32] as \(G\) equals 1. It is worth mentioning that the pseudopotential function can accommodate different equation of states (EOS). In our study, the normalized Van der Waals EOS is considered, that is  
\[P=\frac{8\rho T}{3-\rho}-3\rho^{2}. \tag{7}\]  
In the above equation, the dimensionless physical variables are defined as  
\[\rho=\frac{\hat{\rho}}{\hat{\rho}_{c}},\ P=\frac{\hat{\rho}}{\hat{P}_{c}},\ T= \frac{\hat{T}}{\hat{T}_{c}}, \tag{8}\]  
where \(\hat{\rho}\), \(\hat{P}\), and \(\hat{T}\) denote the density, pressure, and temperature in physical units, respectively, while \(\hat{\rho}_{c}\), \(\hat{P}_{c}\), and \(\hat{T}_{c}\) correspond to the critical point. Throughout this paper, the normalized variables which are intrinsically dimensionless will be used in the lattice Boltzmann model.