# A Fluid-Structure Coupling Analysis of a Far-Field Flat Mirror for AliCPT-1 Telescope Calibration
## 3 Fluid Field Calculation  
### Theoretical Analysis  
Establishing a fluid-structure interaction model necessitates the initial determination of the fluid flow state. The fluid domain in which the FFF mirror is located is characterized as an incompressible viscous fluid flow governed by mass conservation, energy conservation, and momentum conservation [10; 11; 12]. The initial step involves determining the fluid state and considering the concept of Reynolds number (Re), which serves as an indicator to distinguish between laminar and turbulent flow regimes by evaluating the relative magnitudes of inertial and viscous forces. The formula for calculating the Reynolds number is as follows:  
\[\text{Re}=\frac{\rho\text{VL}}{\mu} \tag{2}\]  
Re signifies the Reynolds number, \(\rho\) denotes the fluid density, V represents fluid velocity, L stands for the characteristic length, and \(\mu\) represents the fluid dynamic viscosity [13; 14; 15]. When the Reynolds number is below a critical threshold (typically around 500,000), the flow is laminar, with viscous forces dominating and leading to a smooth and ordered flow. However, when the Reynolds number exceeds this threshold, turbulence occurs due to the dominance of inertial forces, resulting in irregular and chaotic flow patterns. This study demonstrates that with a basin length of 40 m, a fluid density of 1.225 kg/m, a viscosity of 1.81 \(\times\) 10\({}^{-5}\) [g/(m/s)], and a flow velocity of 17.5 m/s, the Reynolds number obtained from Equation (2) exceeds 500,000, confirming turbulent flow.  
Turbulent flow analysis primarily employs the k-\(\epsilon\) and k-\(\omega\) turbulence models. This study utilizes the realizable k-\(\epsilon\) turbulence model [16; 17]. The turbulence model and its dissipation equations are presented as follows.  
Standard model turbulence kinetic energy transport equation:  
\[\frac{\partial(\rho\text{k})}{\partial\text{t}}+\frac{\partial(\rho\text{k} \text{u}_{\text{i}})}{\partial\text{x}_{\text{i}}}=\frac{\partial}{\partial \text{x}_{\text{i}}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{\text{t}}}{\sigma_{\text{ k}}}\bigg{)}\frac{\partial\text{k}}{\partial\text{x}_{\text{j}}}\Bigg{]}+ \text{G}_{\text{k}}+\text{G}_{\text{b}}-\rho\text{\epsilon}-\text{Y}_{\text{M }}+\text{S}_{\text{k}} \tag{3}\]  
Standard model turbulence dissipation rate transport equation:  
\[\frac{\partial(\rho\text{\epsilon})}{\partial\text{t}}+\frac{\partial(\rho \text{\epsilon}\text{u}_{\text{i}})}{\partial\text{x}_{\text{i}}}=\frac{ \partial}{\partial\text{x}_{\text{j}}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{\text{t} }}{\sigma_{\text{\epsilon}}}\bigg{)}\frac{\partial\text{\epsilon}}{\partial \text{x}_{\text{j}}}\Bigg{]}+\text{C}_{1\text{\epsilon}}\frac{\text{\epsilon}} {\text{k}}(\text{G}_{\text{k}}+\text{C}_{3\text{\epsilon}}\text{G}_{\text{b} })-\text{C}_{2\text{\epsilon}}\frac{\text{\epsilon}^{2}}{\text{k}}+\text{S}_{ \text{\epsilon}} \tag{4}\]  
\(\text{G}_{\text{k}}\) represents the turbulence kinetic energy generated by mean velocity gradients, \(\text{G}_{\text{b}}\) accounts for the turbulence kinetic energy generated by buoyancy, \(\text{Y}_{\text{M}}\) signifies the influence of fluctuation expansion on turbulence dissipation rate, and \(\sigma_{\text{k}}\) and \(\sigma_{\text{\epsilon}}\) denote the turbulent Prandtl numbers for turbulence kinetic energy and turbulence dissipation rate, respectively [18; 19].  
The one-way fluid-structure coupling approach is chosen due to the rigid structure of the mirror, which has minimal influence on the fluid field. Initial fluid field calculations are conducted using the Fluent module within Workbench, providing wind pressure distribution over the mirror. These results are then used as boundary conditions for calculating mirror deformation and stress. Figure 4 presents the analysis flowchart.