## 3 Methodology
### Validation  
Here, to verify the reliability of the multi-phase framework, a dam break experiment is numerically reproduced. In the original laboratory study, the slurry was instantly poured into a rectangular flume from an upstream reservoir, and the front positions of the flow were monitored in real time [50]. A two-dimensional (2D) numerical simulation was used in the computational domain of 12 m \(\times\) 0.5 m, with the initial configuration shown in Figure 3a. For meshing, quadrilateral elements were used to discretize the entire domain. A convergence study was conducted to determine the desired mesh, and it was found that meshing with a grid cell size of 10 mm \(\times\) 10 mm is sufficient to achieve accurate results. The mixture model and the laminar model were used as the multi-phase flow model and the viscosity model, respectively. To solve the multi-phase model, a pressure-based solver with an algorithm considering the coupling of pressure and velocity and the SIMPLEC algorithm were used. A second-order implicit scheme and a second-order upwind scheme were also adopted to discretize the momentum formula and the time integration, respectively. Here, the slurry density is \(\rho_{s}=1073\) kg/m\({}^{3}\). No slip is assumed to occur along the interfaces between the base and the slurry wall. As a function of the shear strain rate, the shear stress in the run-out can be roughly predicted using the following equation [51]:  
\[s_{\text{u,B}}=s_{\text{u0,B}}+\eta\dot{\gamma} \tag{8}\]  
where \(s_{\text{u,B}}\) is the mobilized shear stress, \(s_{\text{u0,B}}\) denotes the yield stress under static loading and is selected as 42.5 Pa, \(\eta\) represents the plastic viscosity taking a value of 0.0052 Pa-s, and \(\gamma\) represents the shear strain rate. The following formula is adopted to estimate the time step \(\Delta t\) used in the simulations (ANSYS, 2011):  
\[\Delta t=\frac{\beta d_{\text{min}}}{v} \tag{9}\]  
where \(\beta\) denotes a coefficient taking a value of 0.4, \(d_{\text{min}}\) denotes the minimum mesh size and \(v\) represents the maximum velocity of the slurry.  
After the dam breaks, the front toe presents the maximum velocity, which increases to the maximum value of 2.1 m/s and then decreases to nil at deposit. The gravitational potential energy, the shear resistance along the slurry-bed and the shear resistance along slurry-wall interfaces are the controlling factors for the movement of the slurry. The final runout distance of the slurry is 9.2 m, which is very close to the value of 8.9 m from the MPM analysis [3]. The run-out profiles at 4.1 s predicted by the CFD are compared with  
Figure 3: Simulation of the dam break case with the multi-phase framework. (**a**) Initial geometry of the dam break case (not to scale); (**b**) Runout profile at 4.1 s.  
those from the MPM [52] and the laboratory observations [51]. The results show good agreement between the run-out profiles obtained through these three approaches, with a maximum difference of 10%, as shown in Figure 3b. The mixture behavior at the interface between the slurry and the air is accurately captured by the mixture model. Thus, the CFD numerical modelling approach is validated and proven to be a reliable numerical simulation tool for studying the transformation of debris flows into turbidity currents.