2.2. 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
Sn is determined by Equation (1) [
17].
where
r0 is the radius of the nozzle and
is the diffusion angle; in this paper,
α 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:
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.
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, Banks et al. described the main section of the jet flow field with the following equation [
18]:
where
P is the pressure generated by the gas flow,
ρG is the gas density,
V0 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:
where
is the core zone radius,
, and
is a constant (mm
2). Equation (4) (a) describes the pressure distribution within the core zone, while Equation (4) (b) describes the pressure distribution outside the core zone to the boundary layer.
According to
Section 2.1 of this paper, the surface shape of the mirror blank is formed by the combined action of air pressure, surface tension, and the hydrostatic pressure of the liquid. Of these, the air pressure has a positive effect on the expansion of the surface. In contrast, the surface tension and static pressure have a negative effect by inhibiting the expansion of the surface. During the the gas jet forming process, the air pressure is stronger than the combined inhibitory effect of the surface tension and static pressure, causing a tendency for the surface profile to expand outwards until the three forms of action reach equilibrium. For this reason, Equation (4) can be modified to obtain Equation (5).
According to Equation (5), the magnitude of the pressure acting on the surface of the liquid cell is related to the initial jet velocity V0. The greater the V0, the greater the resulting pressure P, resulting in a larger diameter of the surface formed by the gas jet. It can be inferred that the diameter formed by the gas jet increases as the initial gas flow rate V0 increases. In addition to velocity, the pressure is also related to the nozzle’s height from the liquid surface. Although the cross-sectional diameter of the flow stream increases as the nozzle’s height from the liquid surface increases, the greater the nozzle height, the lower the pressure within the radial cross-section, resulting in a smaller diameter formed by the gas jet.
The principal curvature is one of the key parameters describing the profile of an optical mirror, and it is not correlated with the diameter. Therefore, it is also necessary to analyse the law of change in the principal curvature of the surfaces formed in the gas jet forming. Due to the volume reduction in the potential core region and the effect of the hydrostatic pressure of the liquid, the principal curvature is determined by both the pressure in the potential core region and the pressure distribution generated by the flow streams in the vicinity of the core.
By taking the derivative of
P with respect to r in Equation (4) (b):
The second derivative of the above equation:
According to the mathematical equation, the curvature
K at each point of the airflow can be obtained from Equation (8).
Substituting Equations (6) and (7) into Equation (8) yields the curvature equation describing the pressure distribution of the gas jet acting on the fluid surface.
The focus of this paper is on the preserving core region and the pressure distribution near the preserving core region (
). At this point, the principal curvature equation of the airflow stream is shown in Equation (10).
Analysis of Equation (10) can lead to the following inference: when the nozzle diameter is determined, the principal curvature and the initial velocity of the gas stream change with the nozzle’s height from the liquid surface. In contrast, the principal curvature of the mirror blank can be inferred from the increasing initial velocity of the jet to increase with the increasing height of the nozzle from the liquid surface.