# The Variation of Surface Shape in the Gas Jet Forming
## 2 Principles of the Gas Jet Forming and Related Inferences
### Inference of the Relationship between Jet Parameters and Mirror Blank Parameters  
According to the equilibrium theory described in Section 2.1, the surface profile of the mirror blank is closely related to the interfacial pressure and the physical properties of the photosensitive resin. The physical properties of the photosensitive resin used in this study are stable, and by changing the velocity of the gas jet and the nozzle's height from the liquid surface, the pressure on the surface of the resin pool can be altered to achieve effective control of the shape of the resulting surface. Therefore, it is important to analyse the gas flow field created by the gas jet to determine the pressure distribution on the surface of the liquid photosensitive resin. According to previous studies [16], the velocity distribution of the gas flow field on the axis of the flow stream can be divided into a potential core region and a self-similarity region, with the BOE as the transition zone, as shown in Figure 2.  
The distance of the potential core region \(S_{n}\) is determined by Equation (1) [17]. \[S_{n}=0.671\frac{r_{0}}{\alpha}, \tag{1}\]  
where \(r_{0}\) is the radius of the nozzle and \(\alpha\) is the diffusion angle; in this paper, \(\alpha\) is 0.08. As this paper is concerned with the effects of the jet velocity and the nozzle height from the liquid surface on the mirror profile, the nozzle diameter was taken as a fixed quantity. In this paper, the nozzle diameter chosen was 10 mm, and the above parameters were substituted into Equation (1) to obtain the following calculation result:  
\[S_{n}=0.671\frac{r_{0}}{\alpha}=0.671\times\frac{0.005}{0.08}=41.93\;\text{mm}, \tag{2}\]  
According to the analysis results shown in Figure 2, the gas flow field is divided by the BOE turning surface, and there is a variation trend between the potential core region and the self-similarity region. According to the calculation results obtained from Equation (2), when the self-similarity region is selected for the gas jet forming, the nozzle's height from the liquid surface is greater than 41.93 mm. At this point, it can be found through the experiment (as shown in Figure 3) that the surface formed is very shallow, and it is not easy to observe the influence of the changes in the gas jet parameters on the surface shape of the mirror blank. Figure 2: Schematic diagram of the changes in the gas flow stream. Therefore, the potential core region was chosen as the formation flow field in this study. According to Figure 2, the velocity of the flow field of the potential core region in the cross-section is the same after the gas is ejected from the nozzle, and the diameter of the core region's cross-section gradually decreases with the movement of the gas stream. In 1962, [PERSON] et al. described the main section of the jet flow field with the following equation [18]:  
\[P=\frac{1}{2}\rho_{G}V_{0}^{2}\left(\frac{d}{h}\right)\cdot e^{-c\left(\frac{h }{h}\right)^{2}}, \tag{3}\]  
where \(P\) is the pressure generated by the gas flow, \(\rho_{G}\) is the gas density, \(V_{0}\) is the gas flow rate, \(d\) is the nozzle diameter, \(h\) is the nozzle height, and \(c\) is a constant. However, the above equation can only describe the flow field distribution in the self-similarity region, but not the gas flow field in the potential core region. Comparing the self-similarity region with the flow field distribution law of the potential core region, we can see that there is a jet-preserving core region in the starting section, and there is a difference in the number of times describing the normal distribution \(e\). Therefore, it is important to develop a mechanistic model suitable for describing the flow field distribution in the potential core region and analysing the flow field pressure variations. In this study, a mechanistic model describing the flow field in the potential core region was established and modelled as follows:  
\[\left\{