## 3 PIV Experimental Apparatus
### Governing Equations  
Each phase was regarded as an interpenetrating continuum. Reynolds-averaged mass and momentum balance equations were solved for each phase. The governing equations for phase \(q\) are as follows [32]:  
Continuity equation:  
\[\frac{\partial}{\partial t}(\alpha_{i}\rho_{i})+\
abla\cdot\left(\alpha_{i} \rho_{i}\overset{\rightarrow}{u}_{i}\right)=0 \tag{1}\]  
where \(\alpha_{i}\), is the concentration of each phase, and \(\rho\) and \(\overset{\rightarrow}{u}_{i}\) are the density and velocity vector, respectively. Momentum equation:  
The momentum equation for the liquid phase is:  
\[\frac{\partial}{\partial t}\left(\alpha_{i}\rho_{i}\overset{\rightarrow}{u} _{I}\right)+\
abla\cdot\left(\alpha_{i}\rho_{i}\overset{\rightarrow}{u}_{I }\overset{\rightarrow}{u}_{I}\right)=-\alpha_{I}\
abla p+\alpha_{I}\
abla \cdot\overset{=}{\tau}_{I}+\alpha_{I}\rho_{I}\overset{\rightarrow}{g}-K \left(\overset{\rightarrow}{u}_{I}-\overset{\rightarrow}{u}_{s}\right) \tag{2}\]  
where \(p\) is the thermodynamic pressure, \(K\) the momentum transfer coefficient, \(\overset{\rightarrow}{g}\) the gravity acceleration, and \(\overset{=}{\tau}_{I}\) the viscous stress tensor of liquid phase listed in Table 3.  
The momentum equation for the solid phase is:  
\[\frac{\partial}{\partial t}\left(\alpha_{s}\rho_{s}\overset{\rightarrow}{u} _{s}\right)+\
abla\cdot\left(\alpha_{s}\rho_{s}\overset{\rightarrow}{u}_{s} \overset{\rightarrow}{u}_{s}\right)=-\alpha_{s}\
abla p-\
abla p_{s}+\
abla \cdot\overset{=}{\tau}_{s}+\alpha_{s}\rho_{s}\overset{\rightarrow}{g}+K\left( \overset{\rightarrow}{u}_{I}-\overset{\rightarrow}{u}_{s}\right) \tag{3}\]  
where \(\overset{=}{\tau}_{s}\) is the viscous stress tensor of solid phase and \(p_{s}\) is the solid pressure shown in Table 3.  
The effects of mass transfer, lift force and virtual mass force were ignored. Only the drag force, which significantly influences the simulation results, was considered in this paper [34]. The Huilin-Gidaspow model [35] was adopted to calculate the momentum exchange coefficient:  
\[K=(1-\psi)K_{E}+\psi K_{WY} \tag{4}\]  
\[\psi=\frac{\arctan[150\times 1.75(0.2-\alpha_{s})]}{\pi}+0.5 \tag{5}\]  
\[K_{E}=150\frac{\left(1-\alpha_{I}\right)^{2}\mu_{I}}{\left(\alpha_{i}d_{s} \right)^{2}}+1.75\frac{\rho_{I}(1-\alpha_{I})\left|\overset{\rightarrow}{u} _{I}-\overset{\rightarrow}{u}_{s}\right|}{\alpha_{I}d_{s}}\ \ \ \alpha_{I}\leq 0.8 \tag{6}\]  
\[K_{WY}=\frac{3}{4}C_{D}\frac{\rho_{I}(1-\alpha_{I})\left|\overset{\rightarrow }{u}_{I}-\overset{\rightarrow}{u}_{s}\right|}{d_{s}}\alpha_{I}^{-2.65}\ \ \ \alpha_{I}>0.8 \tag{7}\]  
\[C_{D}=\frac{24}{\alpha_{I}R\varepsilon_{s}}\left[1+0.15(\alpha_{I}R\varepsilon _{s})^{0.687}\right] \tag{8}\]  
\[\frac{\partial}{\partial t}\left(\alpha_{i}\rho_{i}\overset{\rightarrow}{u} _{I}\right)+\
abla\cdot\left(\alpha_{i}\rho_{i}\overset{\rightarrow}{u}_{I }\overset{\rightarrow}{u}_{I}\right)=-\alpha_{I}\
abla p+\alpha_{I}\
abla \cdot\overset{=}{\tau}_{I}+\alpha_{I}\rho_{I}\overset{\rightarrow}{g}-K \left(\overset{\rightarrow}{u}_{I}-\overset{\rightarrow}{u}_{s}\right) \tag{2}\]0where \(K\) is fluid-solid exchange coefficient, \(\psi\) is the stitching function, \(C_{D}\) is the drag function, and \(Re_{s}\) is the relative Reynolds number. [PERSON] [36] found that the mixture model is the most proper turbulence model. For this reason, both the turbulent kinetic energy and dissipation rate of the liquid phase were calculated using the mixture \(k\)-\(\varepsilon\) turbulence model where the two phases are assumed to share the same \(k\) and \(\varepsilon\).