# Dust Dispersion Characteristics of Open Stockpiles and the Scale of Dust Suppression Shed
## 2 Numerical Methods
### Governing Equations of Airflow and Particles  
#### 2.3.1 Governing Equations of Airflow  
In this study, we assumed that the gas phase was incompressible with constant properties. For incompressible and steady state flow, the governing equations are as follows:  
Conservation of mass equation  
\[\bigtriangledown\cdot\vec{v}=0 \tag{2}\]  
Momentum conservation equation  
\[\frac{\partial\vec{v}}{\partial t}+\vec{v}\cdot\bigtriangledown\vec{v}=-\frac {1}{\rho}\bigtriangledown\vec{p}+\bigtriangledown\cdot(\tau+\tau^{t})+\vec{f} \tag{3}\]  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Bulk Particle Properties (DPM)** & \\ \hline Material & SiO\({}_{2}\) \\ Density \(\rho_{\text{p}}\) & 2300 kg/m\({}^{3}\) \\ Particle diameter \(d_{\text{p}}\) & 8.5 m\({}^{-3}\) \\ Inject source & Surface-injection \\ \hline
**Gas Phase Properties (CFD)** & \\ \hline Material & Air \\ Density \(\rho_{\text{a}}\) & 1.25 kg/m\({}^{3}\) \\ Turbulence models & Shear-stress transport (SST) \(k-\omega\) model \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation of general setups.  
Figure 4: Distribution of velocity (velocity-inlet).  
where the mean velocity and pressure fields are \(\vec{v}\) and \(\bar{p}\), respectively, employing a data-driven closure term for the Reynolds tensor \(\tau^{t}\). The time, density, molecular stress tensor and mean momentum source term are denoted by \(t\), \(\rho\), \(\tau\) and \(\vec{\vec{f}}\), respectively.  
Analytical calculations of the flow in the computational domain showed that the flow reached a turbulent state. Turbulence closure was achieved through application of the two-equation \(k-\omega\) SST model. This is more accurate and reliable for a wider class of flow. This coincides with the computational domain of our study.  
The \(k-\omega\) SST turbulence model requires solving the turbulence kinetic energy (\(k\)) and specific dissipation rate (\(\omega\)) equations:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{ i})=\frac{\partial}{\partial x_{j}}(\Gamma_{k}\frac{\partial k}{\partial x_{j}})+ \widetilde{G_{k}}-Y_{k} \tag{4}\]  
\[\frac{\partial}{\partial t}(\rho\omega)+\frac{\partial}{\partial x_{j}}(\rho \omega u_{j})=\frac{\partial}{\partial x_{j}}(\Gamma_{\omega}\frac{\partial!}{\partial x_{j}})+G_{\omega}-Y_{\omega}+D_{\omega} \tag{5}\]  
where \(\widetilde{G_{k}}\) represents the generation of turbulence kinetic energy due to mean velocity gradients. \(G_{\omega}\) represents the generation of \(\omega\). \(\Gamma_{k}\) and \(\Gamma_{\omega}\) represent the effective diffusivity of \(k\) and \(\omega\). \(Y_{k}\) and \(Y_{\omega}\) represent the dissipation of \(k\) and \(\omega\) due to turbulence. \(D_{\omega}\) represents the cross-diffusion term.  
Symmetry boundary conditions were used for the lateral sides and the upper limits of the computational domain. The lower boundary surface as well as the pile were considered as rough walls. In accordance with the measurement, the roughness lengths on both the stockpile surface and the wall were fixed to be 0.03 mm [23]. Pressure-outlet condition was used for the right side. In addition, velocity-inlet condition was applied on the left side of the computational domain. Profiles of velocity \(u\), turbulent kinetic energy \(k\), and specific dissipation rate \(\omega\) were used to define the entry of calculation domain.  
The initial turbulence conditions at the boundary change accordingly for different inlet velocities. The inlet initial values of the turbulent kinetic energy \(k\) and the turbulent dissipation rate \(\omega\) are each given by the equation:  
\[k=\frac{3}{2}(\bar{u}I)^{2} \tag{6}\]  
where \(\bar{u}\) is the average flow velocity at the inlet, \(\bar{u}Y=\int_{0}^{Y}u_{y}ydy\), \(Y=15\). \(I\) is the turbulence intensity, \(I=0.16(Re)^{-\frac{1}{8}}\):  
\[\omega=\frac{k^{\frac{1}{2}}}{C_{\mu}l} \tag{7}\]  
where \(l\) is the turbulence length scale, \(l=\frac{0.07L}{C_{k}^{3/4}}\). \(C_{\mu}=0.09\).