## 3 Numerical Simulation and Results Analysis
### Governing Equations  
The equations governing gas-liquid two-phase flow are provided in Equations (1) to (5):  
The continuity equation is  
\[\
abla\cdot(\alpha_{k}\rho_{k}U_{k})=0, \tag{1}\]  
The momentum equation is  
\[\
abla\cdot(\alpha_{k}\rho_{k}U_{k}U_{k}-\alpha_{k}\tau_{k})=-\alpha_{k}\
abla p +M_{k}+\alpha_{k}\rho_{k}f_{k}, \tag{2}\]  
where \(k\) means the k phase: if \(k=\) g, means gas, if \(k=\) l, means liquid. \(\alpha_{k}\) is the volume fraction of the phase. \(\rho_{k}\) denotes the density of the phase, and its unit is kg/m\({}^{3}\). \(p\) represents the pressure, and its unit is pa. \(U_{k}\) is the velocity of the phase, and its unit is m/s. \(M_{k}\) is the interaction force between phases and the unit is N-m. \(f_{k}\) denotes the mass force, and its unit is N-m.  
where \(\tau\) denotes the viscous stress tensor concerning fluid viscosity as well as turbulence viscosity. Its components are written in the following form:  
\[\tau_{k,ij}=2\mu_{k}s_{ij}-\overline{\rho_{k}w_{ki}{}^{\prime}w_{kj}{}^{\prime}}, \tag{3}\]  
where \(i,j=\) 1, 2, 3 represent the three directions of a relative Cartesian coordinate system (\(x_{1}\), \(x_{2}\), \(x_{3}\)) rotating with the impeller. \(s_{ij}\) denotes the strain tensor. \(-\overline{\rho_{k}w_{ki}{}^{\prime}w_{kj}{}^{\prime}}\) represent the turbulent Reynolds stresses due to turbulent motion. The turbulent Reynolds stress tensor is given in following form:  
\[-\overline{\rho_{k}w_{ki}{}^{\prime}w_{kj}{}^{\prime}}=2\mu_{t}s_{ij}-\frac{2} {3}\rho_{k}k\delta_{ij}, \tag{4}\]  
where the solution of \(\mu_{t}\) is based on shear stress transport (SST) \(k\)-\(w\) turbulence model. The formula of \(\mu_{t}\) is given as follows:  
\[\mu_{t}=\frac{\rho_{m}a_{1}k}{\max(a_{1}\omega,SF_{2})}, \tag{5}\]  
where \(\mu_{t}\) is the eddy viscosity--the unit is pa's--a\({}_{1}\) = 5/9, k is turbulent kinetic energy, \(\omega\) is turbulent frequency, S is the invariant of the strain rate and F\({}_{2}\) is a blending function, which restricts the limiter to the wall boundary layer [43].  
The volume fraction of each phase when performing the numerical simulation is determined according to the following equation:  
\[\alpha_{l}=\frac{Q_{l}}{Q_{l}+Q_{g}}\times 100\%, \tag{6}\]  
\[\alpha_{g}=\frac{Q_{g}}{Q_{l}+Q_{g}}\times 100\%, \tag{7}\]  
The relationship of the volume fraction between the gas and liquid is shown in Equation (8):  
\[\alpha_{l}+\alpha_{g}=1, \tag{8}\]  
where \(Q_{l}\) denotes the flow rate of the liquid phase while \(Q_{g}\) denotes the flow rate of the gas phase; the units are m3/h.  
The mixture density of the gas-liquid two-phase is defined as  
\[\rho_{m}=\alpha_{g}\rho_{g}+\alpha_{l}\rho_{l}, \tag{9}\]  
The hydraulic performance of the multiphase pump including the hydraulic head and efficiency is defined as follows:  
\[H=\frac{p_{2}-p_{1}}{\rho_{m}g}, \tag{10}\]  
where \(p_{1}\), \(p_{2}\) denote the inlet and outlet pressure of the multiphase pump, respectively.  
The efficiency of the multiphase pump can be calculated by the following equation:  
\[\eta=\frac{\rho_{m}gQH}{M\omega}, \tag{11}\]  
where \(M\) represents the torque of multiphase pump and \(\omega\) is the angular velocity of the impeller.