## 3 Two-Phase Flow Model for CFD Calculation
### Turbulence Model  
The standard \(k\)-\(\varepsilon\) model was used as the turbulent model, and a high-y+ wall treatment (applicable to \(30\leq y+\leq 100\)) was applied for a continuous liquid phase in this work. This model is widely used in CFD calculations because it exhibits practical applicability for the two-phase flow conditions, in which bubble-induced turbulence plays a dominant role. The transport equations of the turbulent kinetic energy, \(k\), and the turbulent dissipation rate, \(\varepsilon\), for a two-phase flow are, respectively, calculated from the following two equations: \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Investigators**} & \multirow{2}{*}{**Bubble Size Correlation**} & \multicolumn{3}{c}{**Application Conditions**} \\ \cline{3-5}  & & **Flow Condition** & **Channel Geometry** & \(\mathbf{j_{g}}\) **(m/s)** & \(\mathbf{j_{f}}\) **(m/s)** \\ \hline Akita and Yoshida [20] & \(D_{b}=18\frac{\mu_{b}^{2}}{\mu^{20}\mu^{20}\mu^{20}}\left(\frac{\varepsilon}{ \sigma}\right)^{0.5}\) & upward bubble & vertical square & 0.0007–0.073 & 0 \\ Kocamustafaogullari & \(D_{b}=0.707\frac{D_{b}^{204}\mu^{20}\mu^{20}}{\left(\frac{\varepsilon}{ \sigma}\right)^{0.33}}\) & covert bubbles & horizontal pipe & 0.21–1.34 & 3.74–6.59 \\ Hibiki and Ishii [22] & \(D_{b}=12\frac{\mu^{20}\mu^{20}\mu^{20}\mu^{20}}{\left(\frac{\varepsilon}{ \sigma}\right)^{0.5}}\) & adiabatic, diabatic & vertical pipe, & 0.0029–4.87 & 0.261–6.55 \\ Hibiki and Ishii [23] & \(D_{b}=1.99\frac{D_{b}^{204}\mu^{20}}{\mu^{200}\mu^{200}}\left(\frac{\varepsilon \varepsilon}{\sigma}\right)^{0.346}\) & adiabatic, diabatic & vertical pipe, & & \\  & & bubbly flow, bubble & horizontal pipe, & 0.0007–4.87 & 0–6.55 \\  & & columns & annualus, bundle & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Existing one group bubble size correlations for the gas-liquid system. \[\frac{\partial(\rho_{c}k_{c})}{\partial t}+U_{j}\frac{\partial(\rho_{c}k_{c})}{ \partial x_{i}}=\tau_{ij}\frac{\partial(\rho_{c}U_{i})}{\partial x_{i}}-\rho_{c }\varepsilon_{c}+\frac{\partial}{\partial x_{i}}\Bigg{[}\bigg{(}\mu_{c}+\frac{ \mu_{c}^{t}}{\sigma_{\varepsilon}}\bigg{)}\frac{\partial k_{c}}{\partial x_{i }}\Bigg{]}+S_{k} \tag{2}\]  
\[\frac{\partial(\rho_{c}\varepsilon_{c})}{\partial t}+U_{j}\frac{\partial(\rho _{c}\varepsilon_{c})}{\partial x_{i}}=\frac{\varepsilon_{c}}{k_{c}}\bigg{(} \mathcal{C}_{\varepsilon 1}\tau_{ij}\frac{\partial(\rho_{c}U_{i})}{\partial x_{i}}- \rho_{c}\mathcal{C}_{\varepsilon 2}\varepsilon_{c}\bigg{)}+\frac{\partial}{ \partial x_{i}}\Bigg{[}\bigg{(}\mu_{c}+\frac{\mu_{c}^{t}}{\sigma_{ \varepsilon}}\bigg{)}\frac{\partial\varepsilon_{c}}{\partial x_{i}}\Bigg{]}+S _{\varepsilon} \tag{3}\]  
where \(\mu\) is the viscosity, \(\rho_{\mathcal{C}}\) is the density of the continuous liquid phase, \(S_{k}\) and \(S_{\varepsilon}\) are the source terms, and \(C_{\mu}\), \(C_{\varepsilon 1}\), \(C_{\varepsilon 2}\), \(\alpha_{k}\), and \(\sigma_{\varepsilon}\) are constant model coefficients. In a bubbly flow, the liquid turbulence caused by the bubbles is considered as the source term, \(S_{k}\). [PERSON] and [PERSON] [9] evaluated the bubble-induced turbulence (BIT) models proposed by [PERSON] and [PERSON] [35], [PERSON] [37], and [PERSON] et al. [38] with different time scales on the radial distribution of the void fraction, liquid velocity, and turbulent kinetic energy for a dispersed gas-liquid flow. Furthermore, the BIT model introduced by [PERSON] et al. [39], including the addition of bubble-induced eddy diffusivity, was evaluated as a reference model. All these models, except for those of [PERSON] et al., showed similar calculation results [9]. Therefore, the BIT model proposed by [PERSON] and [PERSON] was chosen for the present study.