## 2 Oxygen Blow and Stirring Dynamics Model
### Governing Equation and Turbulence Model  
In the steelmaking process, oxygen and molten steel are regarded as a Newtonian fluid, and the numerical solution can be obtained using the Reynolds averaged Naiver-Stokes (RANs) equation. Therefore, the continuity, momentum, and energy equation can be expressed in the following generalized form:  
\[\frac{\partial\rho\Phi}{\partial t}+div\left(\rho u\Phi\right)=div\left(\Gamma_ {\Phi}grad\Phi\right)+S_{\Phi} \tag{1}\]  
where \(\Phi\) is a generalized dependent variable; \(S_{\Phi}\) is generalized source term; and \(u\) is the velocity vector; \(\Gamma_{\Phi}\) is the generalized diffusion coefficient corresponding to \(\Phi\); when \(\Phi\) takes different variables, the specific forms of each parameter are shown in Table 2.  
The related research indicates that the supersonic gas blow is better described by the standard \(k\)-\(\omega\) model than by other models [5]. It is a two-equation eddy-viscosity turbulence model, and the eddy-viscosity-type expression of the Reynolds stress is  
\[\tau_{ij}=2\mu_{t}(S_{ij}-S_{nm}\delta_{ij}/3)-2\rho k\delta_{ij}/3 \tag{2}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Geometrical Parameters** & **Value** & **Unit** \\ \hline Nozzle throat diameter (\(d_{t}\)) & 35 & mm \\ Nozzle exit diameter (\(d_{c}\)) & 48 & mm \\ Nozzle divergent length (\(L_{\text{m}}\)) & 100 & mm \\ Nozzle inclination angle (\(\theta\)) & 13.5 & degree \\ Nozzle twist angle (\(a\)) & 0, 5, 8, 13 & degree \\ Oxygen lance height (\(H_{1}\)) & 1100\(-\)2100 & mm \\ Design Mach number (\(Ma\)) & 1.8 & 1 \\ Steel converter diameter (\(D\)) & 5000 & mm \\ Steel converter height (\(H\)) & 8000 & mm \\ Steel liquid level (\(H_{s}\)) & 2000 & mm \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometrical parameters of multi-nozzle oxygen lance and steel converter.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Equation** & \(\Gamma_{\Phi}\) & \(\Phi\) & \(S_{\Phi}\) \\ \hline Continuity & 0 & 1 & 0 \\ Momentum & \(\mu\) & \(u\) & \(-\partial p/\partial u+S_{Mu}\) \\ Energy & \(\chi\) & \(i\) & \(-p\cdot divu+\Phi+S_{i}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Generalized parameter of governing equation.  
Figure 2: Steel converter and multi-nozzle oxygen lances geometrical models: **(a)** oxygen top-blown converter steelmaking; **(b)** magnified view of area A with a conventional oxygen lance and **(c)** a nozzle-twisted lance.  
where \(\mu_{t}\) is the eddy viscosity; \(S_{ij}\) is the mean-velocity strain-rate tensor; \(\delta_{ij}\) is the Kronecker delta. Eddy viscosity can be regarded as  
\[\mu_{t}=\rho_{k}/\omega \tag{3}\]  
The turbulence kinetic energy \(k\) and its specific dissipation rate \(\omega\) can be expressed as  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial}{\partial x_{j}}\left(\rho u _{j}k-\left(\mu+\sigma^{*}\mu_{t}\right)\frac{\partial k}{\partial x_{j}} \right)=\tau_{ij}S_{ij}-\beta^{*}\rho\omega k\alpha \tag{4}\]  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial}{\partial x_{j}}\left( \rho u_{j}\omega-\left(\mu+\sigma\mu_{t}\right)\frac{\partial\omega}{\partial x _{j}}\right)=\alpha\frac{\omega}{k}\tau_{ij}S_{ij}-\beta\rho\omega^{2} \tag{5}\]  
The constants in this turbulence model are defined as \(\alpha=\frac{5}{9}\), \(\beta=\frac{3}{40}\), \(\beta^{*}=\frac{9}{100}\), \(\sigma=0.5\), \(\sigma^{*}=0.5\), and \(\text{Prt}=0.9\)[27].