## 3 Two-Phase Flow Model for CFD Calculation
### Two-Phase Interaction Models  
Drag force refers to a force acting opposite to the relative motion of any object with respect to the surrounding fluid. The drag force (\(\vec{F_{k}}^{D}\)) is given by:  
\[\vec{F_{k}}^{D}=-\frac{3}{4}C_{D}\frac{\rho_{c}\alpha}{D_{b}}|U_{r}|U_{r} \tag{18}\]  
There are several models available for the drag coefficient. In this study, the drag coefficient model proposed by [PERSON] [32], which covers a wide range of bubbly flow conditions and considers different bubble shapes via the aspect ratio, was adopted as follows:\[C_{D}=\max\left[\left(\frac{24}{\text{Re}}(1+0.15\text{Re}^{0.687})\right)\right) \frac{8 Eo}{3(Eo+4)}\right] \tag{19}\]  
\[\text{Re}=\frac{\rho_{c}|U_{I}|D_{b}}{\mu_{c}} \tag{20}\]  
\[Eo=\frac{\left|\rho_{c}-\rho_{d}\right|gD_{b}^{2}}{\sigma} \tag{21}\]  
where \(Re\) is the Reynolds number, \(Eo\) is the Eotvos number, and \(\sigma\) is the surface tension. This model accounts for the deformation effect of a spherical bubble.  
The lift force is a component perpendicular to the oncoming flow direction when a fluid flows past the surface of the body. This force was derived by [PERSON] et al. [33] as follows:  
\[\overset{\rightarrow}{F}_{k}^{L}=C_{L}\alpha\rho_{c}\left[\overset{ \rightarrow}{U}_{I}\times\left(\
abla\times\overset{\rightarrow}{U}_{c} \right)\right] \tag{22}\]  
where \(U_{c}\) is the velocity of the liquid phase and \(C_{L}\) is the lift coefficient, which has a constant value. For the bubble size distribution, the coefficient in the present study was adjusted to represent the experimental void fraction distribution.  
In two-phase flow, a significantly low void fraction is often observed in the vicinity of the wall. This phenomenon is caused by the force pushing the bubbles from the wall, which is known as the wall lubrication force. This force, as proposed by [PERSON] et al. [36], is expressed as the distance from the wall (\(y_{w}\)), as follows:  
\[\overset{\rightarrow}{F}_{k}^{WL}=-C_{WL}(y_{w})\alpha\rho_{c}\frac{\left| \overset{\rightarrow}{U}_{I}-\left(\overset{\rightarrow}{U}_{I}\cdot \overset{\rightarrow}{n}\right)\right|^{2}}{D_{b}}\overset{\rightarrow}{n} \tag{23}\]  
\[C_{WL}(y_{w})=\max\left\{-0.01+\left(\frac{0.05}{y_{w}}\right)D_{b},0\right\} \tag{24}\]  
Here, \(C_{WL}\) is a function of the inverse length and is activated only near the wall region. In Equation (23), \(\overset{\rightarrow}{n}\) is the outward-facing unit normal to the nearest point on the wall such that the force is pointing inwards to prevent bubbles from contacting the wall.  
Turbulence and lateral forces play an important role in the lateral movement of bubbles. The effect of turbulence on the redistribution of the phase concentration is modeled by a turbulent dispersion force in the phase momentum equations. When a Reynolds stress tensor is modeled for the liquid phase, the force of [PERSON] et al. [34] is expressed as:  
\[\overset{\rightarrow}{F}_{k}^{TD}=\frac{3}{2}\frac{C_{\mu}}{\text{Pr}_{t}} \frac{k_{c}}{\varepsilon_{c}}\overset{\rightarrow}{\tau_{c}} \tag{25}\]  
where \(C_{\mu}\) is the model coefficient of the standard \(k\)-\(\varepsilon\) model, \(Pr_{t}\) is the turbulent Prandtl number, \(k_{c}\) and \(\varepsilon_{c}\) are the turbulent kinetic energy and turbulent dissipation rate of the continuous phase, and \(\tau_{c}\) is the Reynolds stress tensor for the continuous phase.  
The inertia of the surrounding fluid influences the acceleration of a particle that is submerged in a flow. By including this virtual mass force, the inviscid flow theory can provide a realistic representation of the accelerating flows. This acceleration by the bubble is taken into account through the virtual mass force proposed by [PERSON] et al. [33], provided below:  
\[\overset{\rightarrow}{F}_{k}^{VM}=C_{VM}\alpha\rho_{c}\overset{\rightarrow} {a}_{r} \tag{26}\]  
Here, \(C_{VM}\) is the virtual mass coefficient depending on the bubble shape and has a value of 0.5 for the individual sperical bubbles, and \(\overset{\rightarrow}{a}_{r}\) is the relative acceleration.