## 2 Rheological Model and Numerical Algorithm  
The fundamental numerical model is Splash3D, which solves Navier-Stokes equations with large eddy simulation (LES) [43] turbulence closure. The two-step projection method [44; 45; 46] is used to solve the pressure Poisson equation and new time-step fluid velocities. The detailed description of the model is presented in the references of [PERSON] et al., [PERSON] et al., and [PERSON] et al. [47; 48]. This model considers slump and the ambient materials a volume-averaged mixture. The conservation equations of mass and momentum of the mixture are given as  
\[\frac{\partial\overline{u}_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho\overline{u}_{i})+\frac{\partial}{\partial x_ {j}}\Big{(}\rho\overline{u_{i}u_{j}}\Big{)}=-\frac{\partial\overline{P}}{ \partial x_{i}}+\frac{\partial}{\partial x_{j}}\bigg{[}\mu_{e}\bigg{(}\frac{ \partial\overline{u_{i}}}{x_{j}}+\frac{\partial\overline{u_{i}}}{x_{i}}\bigg{)} \bigg{]}+\rho g_{i} \tag{2}\]  
where the subscripts \(i,\ j=1,\ 2,\ 3\) represent the \(x,\ y,\ z\) directions respectively; \(t\) is the time, \(u\) is the velocity; \(P\) is the pressure; the over-bar represents the spatially filtered value [49]; \(g\) is the gravitational acceleration; \(\rho\) is the density; and \(\mu_{e}\) is the effective viscosity. In this study, the LES is adopted to address the turbulence effect. The effective viscosity \(\mu_{e}\) is defined as  
\[\mu_{e}=\mu\Big{(}\dot{\gamma}\Big{)}+\mu_{t}+\mu_{molecular} \tag{3}\]  
where \(\mu\Big{(}\dot{\gamma}\Big{)}\) is the rheology viscosity of the slump, \(\mu_{t}\) is the viscosity of the sub-grid scale turbulence, and \(\mu_{molecular}\) is the molecular viscosity. The Smagorinsky model [43] relates the residual stress to the rate of filtered strain. Based on the dimensional analysis, the subgrid-scale eddy viscosity is modeled as  
\[\mu_{t}=\rho l_{S}^{2}\dot{\gamma}=\rho(C_{s}\Delta)^{2}\overline{\dot{\gamma}} \tag{4}\]  
where \(l_{S}\) is the Smagorinsky length scale, a product of the Smagorinsky coefficient \(C_{s}\) and the filter width \(\Delta\); \(\dot{\gamma}\) is the second invariant of the \(\dot{\gamma}_{ij}=\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}} {\partial x_{i}}\):  
\[\dot{\gamma}=\sqrt{\frac{1}{2}\dot{\gamma}_{ij}\dot{\gamma}_{ij}} \tag{5}\]  
In general, \(C_{s}\) varies from 0.1 to 0.2 in different flows. The present 3D simulation use a value of 0.15. As for the two-dimensional simulation, \(C_{s}\)= 0.3 is suggested byLiu et al. [47]. \(\Delta\) is the filter width. Infinite volume discretization \(\Delta\) is the grid size. \[\Delta=(\Delta x_{1}\times\Delta x_{2}\times\Delta x_{3})^{\frac{1}{3}},\ for\ 3D\ simulation \tag{6}\]  
where \(\Delta x_{1}\), \(\Delta x_{2}\), and \(\Delta x_{3}\) are the three components of the grid lengths. In this study, three materials, water, air, and slump, are considered. The interfaces between them are solved by the Volume of Fluid (VOF) method [50]. The volume fraction \(f_{m}\) are used to describe the fraction of different materials in each cell, volume fraction \(f_{m}\) varies in [0, 1] and should sum to unity everywhere:  
\[\sum_{m}f_{m}=1 \tag{7}\]  
The \(f_{m}\). is calculated by the VOF equation:  
\[\frac{\partial(f_{m})}{\partial t}+\
abla\cdot(u_{i}f_{m})=0 \tag{8}\]The density used in the momentum Equation (2) is calculated from the volume average of all the materials:  
\[\rho=\sum_{m}f_{m}\rho_{m} \tag{9}\]  
where \(\rho_{m}\) represents the density of the individual material. In this study, the Bingham model is used to describe the rheological behavior of the slump. The non-linear stress-strain relation between the shear stress and strain rate is expressed by [51; 52]  
\[\tau=\tau_{y}+\mu\big{(}\dot{\gamma}\big{)}\dot{\gamma} \tag{10}\]  
\[\mu\big{(}\dot{\gamma}\big{)}=\left\{ \begin{array}{ll}\mu_{A}=\infty\;\text{ and}\;\dot{\gamma}=0,\;\text{if}\;\tau<\tau_{y}\\ \mu_{B}+\frac{\tau_{y}}{\dot{\gamma}}\;\text{and}\;\dot{\gamma}>0,\;\text{if} \;\tau\geq\tau_{y}\end{array} \right.