# The Variation of Surface Shape in the Gas Jet Forming
## 3 Numerical Simulation
### The Setting of the Numerical Simulation Parameters and Analysis of the Results  
According to Section 2.2, it can be seen that when the nozzle is too high from the liquid surface, it will make the formation difficult to observe macroscopically. In contrast, a low nozzle distance from the liquid surface will cause waves at the gas-liquid interface. Therefore, the nozzle height was set between 20 mm and 40 mm, and the gas jet velocity was between 9.5 m/s and 11 m/s. The jet parameters for the numerical simulation and the subsequent experiments were used in 4 \(\times\) 5 orthogonal experiments, and the gas jet parameters were set as shown in Table 1.  
The material used to form the mirror blanks was photosensitive resin, and the gas emitted from the nozzle was air. The relevant properties of the photosensitive resin were tested using a DA-130N densitometer, a VT-06 viscometer, and a SITA surface tension meter, and the results are shown in Table 2.  
The final mirror blank profile curves for the different gas jet parameters were obtained by numerical simulation, as shown in Figure 7, where the yellow part is the air and the red part is the photosensitive resin liquid pool. By collecting the calculated data from the numerical simulation, the relationship between the gas flow rate and the diameter of the forming surface was determined, as shown in Figure 8a, and the relationship between the distance from the nozzle to the liquid surface and the diameter of the formed surface is shown in Figure 8b. As shown in Figure 8, the diameter of the surface formed by the gas jet increased as the gas flow rate increased when the nozzle's height from the liquid surface was fixed. When the gas flow rate was fixed, the diameter of the surface formed by the gas flow rate decreased as the nozzle height increased. When the gas flow rate was between 9.5 m/s and 11 m/s and the nozzle's height from the liquid surface was between 20 mm and 30 mm, the minimum diameter of the formed surface was 18.6904 mm and the maximum was 30.8936 mm in the simulations. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Numerical Value** \\ \hline Gas flow rate (m/s) & 9.5, 10, 10.5, 11 \\ Nozzle height (mm) & 20, 22.5, 25, 27.5, 31 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Gas jet parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Air** & **Photosensitive Resin** \\ \hline Density/kg\(\cdot\)m\({}^{-3}\) & 1.225 & 977 \\ Viscosity/cps & 1.7894 \(\times\) 10\({}^{-5}\) & 2000 \\ Surface tension/mN\(\cdot\)m\({}^{-1}\) & & 37.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties. Figure 7: Numerical simulation results of the the gas jet forming. The principal curvature of the mirror blank is challenging to determine directly from the profile information obtained from the numerical simulation, unlike the diameter of the surface formed by the gas jet. Therefore, using the surface profile data calculated from the simulation, the principal curvature was calculated using a data-fitting method. The shape equation for fitting the data is given in Equation (19) [27]. \[z=\frac{Cr^{2}}{1+\sqrt{1-(K+1)C^{2}r^{2}}}+a\tau^{4}+br^{6}+\cdots-h_{m}, \tag{19}\]  
where \(C\) is the principal curvature of the mirror blanks, \(K\) is the cone coefficient, \(a\), \(b\)... is a multinomial coefficient, and \(h_{m}\) is the lowest point of the mirror blank. The principal curvature of the formed surface, simulated for different gas flow rates and nozzle heights from the liquid surface, is shown in Figure 9.  
As shown in Figure 9, both the gas flow rate and the nozzle height from the liquid surface play a significant role in the changes in the principal curvature of the mirror blank. As the nozzle height increases, the principal curvature increases. As the gas flow rate increases, the principal curvature increases. Within the range of jet parameters set in this  
Figure 8: The diameter of the formed surface with different gas flow rates and nozzle heights in simulations.