# Numerical Simulation of Liquid Film Characteristics during Atomization of Aluminum Alloy Powder
## 2 Atomization Model and Basic Theory
### Governing Equation  
In this paper, the atomization process involving the melting of aluminum alloy by a rotating disk is studied. At a high rotational speed, the molten metal's movement is characterized by turbulence, and the liquid molten metal flow is considered to be that of an incompressible Newtonian fluid, following the continuity equation, momentum equation, and energy equation. Combined with the actual working conditions, to better capture the characteristics of the turbulent movement process, the realizable k-\(\epsilon\) turbulence model is adopted in this paper.  
The turbulent kinetic energy \(k\) equation and the turbulent dissipation rate equation are as follows [27; 28; 29]:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial\big{(}\rho ku_{j}\big{)}}{ \partial x_{j}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{\mu_ {t}}{\sigma_{k}}\bigg{)}\frac{\partial(\rho k)}{\partial x_{j}}\Bigg{]}+\rho( P_{k}-\epsilon) \tag{1}\]  
\[\frac{\partial(\rho\epsilon)}{\partial t}+\frac{\partial\big{(}\rho\epsilon u _{j}\big{)}}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(} \mu+\frac{\mu_{t}}{\sigma_{k}}\bigg{)}\frac{\partial\epsilon}{\partial x_{j} }\Bigg{]}+\rho C_{1}E_{0}\epsilon-\rho C_{2}\frac{\epsilon^{2}}{k+\sqrt{ \epsilon\epsilon}} \tag{2}\]  
In the formula, \(C_{1}=max\Big{(}0.43,\frac{\zeta}{\zeta+5}\Big{)}\); \(C_{2}=\frac{1}{\lambda_{0}+A_{S}}U^{*}k\zeta\); \(A_{s}=\sqrt{6}cos\varphi\); \(\varphi=\frac{1}{3}arccos\Big{(}\sqrt{6}W\Big{)}\); \(W=\frac{E_{ij}E_{ij}}{\left(E_{ij}E_{ij}\right)^{2}}\); \(U^{*}=\sqrt{E_{ij}E_{ij}+\widehat{\Upsilon}_{ij}\widehat{\Upsilon}_{ij}}\); \(\widehat{\Upsilon}_{ij}=\Omega_{ij}-2\epsilon_{ijk}\omega_{k}\); \(\Omega_{ij}=\overline{\Omega}_{ij}+\epsilon_{ijk}\omega_{k}\), where \(k\) is the turbulent kinetic energy; \(\epsilon\) is the turbulent dissipation rate; \(\mu\) is the molecular viscosity coefficient; \(\mu_{t}\) is the turbulent eddy viscosity coefficient; \(x_{j}\) is the  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Atomizing** & **Atomizing** & **Nozgle** & **Tundish** & **Bottom Cone** & **Atomizing** & **Atomizing** \\
**Disk** & **Disk Diameter** & **Diameter** & **Diameter** & **Angle of** & **Tank Diameter** & **Tank Cone** \\
**Disk** & **d\({}_{0}\)/mm** & **d\({}_{1}\)/mm** & **d\({}_{2}\)/mm** & **Tundish** \(\alpha\)/\(\ell\)(\({}^{\circ}\)) & **D/mm** & **Angle \(\beta\)/(\({}^{\circ}\))** \\ \hline Flat disk & 80 & 3 & 500 & 150 & 2400 & 90 \\ Spherical disk & 80 & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Atomizing device structural parameters.  
Figure 3: Simulation geometric model and boundary conditions after treatment.  
coordinate component; \(u_{j}\) is the average relative velocity component; \(P_{k}\) is the generation term of turbulent kinetic energy; \(\mathcal{O}\) is the filter size; \(E_{ij}\), \(E_{jk}\), and \(E_{ki}\) are the time-averaged strain rates in different directions; \(\omega_{k}\) is the angular velocity; \(\Omega_{ij}\) is the average vorticity; \(\overline{\Omega}_{ij}\) is the average rotation rate; \(C_{1}\), \(C_{2}\), \(A_{0}\), \(\sigma_{\epsilon}\), \(\sigma_{k}\) are model constants; \(\epsilon_{ijk}\) is the common intermediate variable of the model; \(\zeta=\left(2E_{ij}\cdot E_{ij}\right)^{\frac{1}{2}}\frac{k}{\epsilon}\); \(E_{0}\) is the time average strain rate; \(v\) is the kinematic viscosity.