# The Variation of Surface Shape in the Gas Jet Forming
## 1 Introduction  
With the development of technology, aspherical mirrors are widely used in aerospace, medical testing, and information communication, and are among the core components of modern optical systems. For instance, the Giant Magellan Telescope developed by the University of Arizona and the telescopes developed by NASA all employ high sub-aspherical mirrors [1; 2]. Mirror blanks are the substrates used for optical mirror coatings, and their surface accuracy and quality determine the optical performance of the mirror after coating. Therefore, processing mirror blanks is the most critical aspect of optical mirror formation [3; 4].  
The conventional aspherical mirror processing methods are material removal formation [5], compression moulding [6], and spin-casting [7]. These methods generally have high processing costs, low flexibility, and limited surface shape. [PERSON] et al. proposed a the gas jet forming method for processing mirror blanks [8]. Under this air pressure, the pool of photosensitive resin is deformed by the impact and cured when the desired surface shape is achieved. This method allows the desired optical surface to be obtained by controlling the nozzle height and the gas flow rate.  
Currently, the studies related to the gas jet forming in the field of mirror blank processing production are still in their infancy. However, the the gas jet forming process has become one of the hot topics in the field of multiphase fluids. In 1965, [PERSON] et al. observed the liquid surface dimple phenomenon caused by cylindrical jets [9]. In 1970, [PERSON] et al. confirmed that the profile of liquid surface dimples is related to the jet velocity and nozzle diameter [10]. In 2004, [PERSON] et al. used an experimental setup of liquid metal systems to study the influence of gas flow rate and nozzle height on the depth of liquid surface dimples [11]. In 2012, [PERSON] et al. experimented with impacting liquid steel with a gas jet andestablished a depth prediction model related to the gas flow rate and nozzle height [12]. In 2017, [PERSON] et al. used the Reynolds stress model (RSM) to study the interaction process of a gas jet perpendicular to a liquid interface [13]. In the same year, [PERSON] et al. derived a nonlinear wave equation capable of quantitatively describing the surface waves caused by airflow [14]. According to the above studies, the gas jet forming process mainly consider the interaction mechanism of the gas-liquid interface, the surface wave phenomenon, and the depth of the liquid surface dimples. These studies cannot predict the diameter and principal curvature of the mirror blank. Therefore, it is necessary to establish the relationship between the gas flow rate, nozzle height, and principal curvature and surface diameter of the mirror blank for further study of the jet formation process.  
According to the research completed so far, this paper investigates the change rule of the principal curvature and surface diameter of mirror blanks in the gas jet forming. Firstly, the influence of the gas jet parameters on the pressure distribution rule in the cross-section of the gas stream is analysed, and a model of the pressure distribution in the initial section of the gas stream is established. Based on this gas pressure distribution model, an inference is drawn from the relationship between the gas jet parameters and the surface parameters. A numerical simulation is carried out to understand this inference. Finally, the inference is verified by the gas jet forming experiments. The dimensionless mathematical model of the principal curvature and surface diameter of the mirror blank with respect to the jet parameters is established by analysing the causes of the calculation deviations.