# The Variation of Surface Shape in the Gas Jet Forming
## 3 Numerical Simulation
### Control Equations and Discrete Methods  
The VOF interface tracking model in the Reynolds-averaged Navier-Stokes framework was used in the simulation to predict the the gas jet forming process. [PERSON] and [PERSON] first proposed the VOF method in 1975 to capture free surface calculations through a computational grid [PERSON] and [PERSON] (1981). The fluid model was refined by these two researchers in 1981 and is still in use today [PERSON] and [PERSON] (1981).  
The numerical simulation of the the gas jet forming method is a multinomial fluid model. During the solution process, the gas and liquid phases fill the calculation domain, and in this model both the gas and liquid phases are continuous, satisfying the Navier-Stokes equations. Due to the low influence of temperature on the experiment, the energy conservation factor is ignored in the calculations. The mass equation is shown in Equation (11), and the momentum equation is shown in Equation (12) [PERSON] and [PERSON] (1981).  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho U)=0, \tag{11}\]  
\[\frac{\partial\rho U}{\partial t}+\
abla\cdot(\rho UU)-\
abla\cdot\tau=- \
abla p+\rho g+F, \tag{12}\]  
Figure 4: The model diagram of the gas jet forming.  
where \(t\) is the time, \(U\) is the gas flow rate, \(p\) is the pressure, \(g\) is the gravity vector, \(\mu\) is the viscosity, \(\rho\) is the density, and \(F\) is the source term, which is provided in the experiments of the jet spray moulding method, including surface tension.  
According to Section 2.1 above, surface tension is an important influencing factor for the prediction of the mirror blank's surface shape. The surface tension model in this paper is the continuous surface force model proposed by [PERSON]. As shown in Equation (13), in this model, the surface tension is added to the source term of the momentum equation [22].  
\[F_{\text{CSS}}=\
abla\!\cdot\!T, \tag{13}\]  
Expressing the stress surface tensor \(T\)[22]:  
\[T=\sigma(|\
abla\alpha|I-\frac{\
abla\alpha\otimes\
abla\alpha}{|\
abla\alpha |}), \tag{14}\]  
where \(I\) is the stress tensor, \(\alpha\) is the volume fraction, \(\sigma\) is the surface tension coefficient, \(\otimes\) is the tensor product of the two vectors, and \(n\) is the volume fraction gradient.  
Within the grid of the VOF method, the two phases share a common set of momentum equations. Therefore, the rate properties and the material properties within the grid are jointly determined by the physical properties of the two phases, along with the volume fraction [23].  
The formula for the mixing density \(\rho_{m}\) is as follows [23]:  
\[\rho_{m}=\alpha_{c}\rho_{c}+\alpha_{d}\rho_{d}, \tag{15}\]  
where \(\alpha_{c}\) is the volume fraction of the gas phase, \(\rho_{c}\) is the physical density of the gas phase, \(\alpha_{d}\) is the volume fraction of the liquid phase, and \(\rho_{d}\) is the physical density of the liquid phase.  
The formula for the mixing viscosity \(\mu_{m}\) is as follows [23]:  
\[\mu_{m}=\alpha_{c}\mu_{c}+\alpha_{d}\mu_{d}, \tag{16}\]  
where \(\mu_{c}\) is the physical viscosity of the gas phase, \(\mu_{d}\) is the physical viscosity of the liquid phase. The grid rate \(\mathbf{U_{m}}\) property has the following formula [23]:  
\[\mathbf{U_{m}}=\frac{\alpha_{c}\rho_{c}\mathbf{U_{c}}+\alpha_{d}\rho_{d} \mathbf{U_{d}}}{\rho_{m}}, \tag{17}\]  
where \(\mathbf{U_{C}}\) is the gas phase rate property at the last time step of the grid, while \(\mathbf{U_{d}}\) is the liquid phase rate property at the last time step of the grid.  
In order to ensure the better stability of the VOF, an explicit algorithm is used in the time domain, the transient solution is used for the numerical simulation process, and the PISO algorithm is applied to solve the pressure-rate field. The first-order upwind algorithm is used to discretise the momentum equation, the PRESTO algorithm is used to discretise the pressure space, and the least squares cell-based algorithm is used to discretise the gradient space.