## 3 Numerical Simulation  
Steady-state numerical simulations were done using ANSYS FLUENT 18.2 software based on finite volume discretization. The boundary conditions and numerical method for the basic and modified gas torches are shown in Table 3. The standard k-\(\varepsilon\) turbulence model was used in this study. The SIMPLE algorithm was adopted for coupling the pressure and velocity in the simulation. For more accuracy, the second-order upwind scheme was selected for the spatial discretization. The convergence criterion was set up as \(10^{-4}\) for the momentum, continuity, and turbulence equations. The numerical simulation was conducted to investigate the velocity profile inside and at the outlets of the gas torches. The meshes and boundary conditions of the basic model are shown in Figure 4. The inlet boundary condition was set as a pressure inlet of the fuel/air mixture. An ambient pressure outlet boundary condition was set at the outlet of the gas torch. A no-slip wall boundary condition was set for the other sides of the torch. \begin{table}
\begin{tabular}{c c c c} \hline  & **Model 1** & **Model 2** & **Model 3** \\ \hline Guide vane holes diameters (\(d_{h}\)) & 3–4.5 mm & 3 mm & 3–3.5 mm \\ Guide vane length (\(l\)) & 480 mm & 30 mm & 100 mm \\ Guide vane width (\(w\)) & 25 mm & 25 mm & 25 mm \\ Guide vane angle (\(\alpha\)) & 160 degrees & 160 degrees & 160 degrees \\ Outer shape height (\(l_{p}\), \(h_{t}\)) & - & 63 mm & 63 mm \\ Outer shape angle (\(\alpha_{p}\), \(\alpha_{t}\)) & - & 45 degrees & 45 degrees \\ Y-branch pipes diameter (\(d_{p}\)) & - & 15 mm & - \\ \hline \end{tabular}
\end{table} 
Table 2: The dimensions of inner and outer shapes of the three new models. Figure 3: The inner and outer shapes of three new gas torch models: (**a**) model 1; (**b**) model 2; (**c**) model 3. The governing equations for the continuity, momentum, kinetic energy, and dissipation energy of the turbulence model are the following:  
Continuity equation:  
\[\
abla.\left(\vec{\rho u}\right)=0 \tag{1}\]  
Momentum equation:  
\[\
abla.\left(\vec{\rho u}\overrightarrow{u}\right)=-\
abla p+\
abla.\left(\mu+ \mu_{t}\right)\left[\left(\
abla\overrightarrow{u}+\
abla\overrightarrow{u} ^{T}\right)-\frac{2}{3}\
abla.\overrightarrow{u}\overrightarrow{l}\right] \tag{2}\]  
The turbulence kinetic energy \(k\) and the rate of its dissipation \(\varepsilon\) are obtained from the following transport equations:  
\[\frac{\partial}{\partial x_{i}}(\rho ku_{i})=\frac{\partial}{\partial x_{j}} \left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)\frac{\partial k}{\partial x_ {j}}\right]+G_{k}-\rho\varepsilon \tag{3}\]  
\[\frac{\partial}{\partial x_{i}}(\rho\varepsilon u_{i})=\frac{\partial}{ \partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}}\right) \frac{\partial\varepsilon}{\partial x_{j}}\right]+C_{1\varepsilon}\frac{ \varepsilon}{k}G_{k}-C_{2\varepsilon}\rho\frac{\varepsilon^{2}}{k} \tag{4}\]  
The turbulent viscosity is modeled as follows:  
\[\mu_{t}=\rho C_{t}\frac{k^{2}}{\varepsilon} \tag{5}\]  
The production of turbulence kinetic energy is defined as follows:  
\[G_{k}=-\rho\overrightarrow{u^{\prime}_{i}u^{\prime}}\frac{\partial u_{j}}{ \partial x_{i}} \tag{6}\]  
\[-\rho\overrightarrow{u^{\prime}_{i}u^{\prime}}_{j}=\mu_{t}\left(\frac{ \partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right) -\frac{2}{3}\left(\rho k+\mu_{t}\frac{\partial u_{k}}{\partial x_{k}}\right) \delta_{j} \tag{7}\]  
Figure 4: Meshes and boundary conditions of a three-dimensional gas torch. \begin{table}
\begin{tabular}{c c} \hline Gauge pressure inlet & 100–500 Pa \\ Gauge pressure outlet & 0 Pa \\ Walls & Non-slip wall \\ Turbulent model & Standard \(k\)-\(\varepsilon\) \\ Convergence criterion & \(10^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Boundary conditions and numerical method.