## 4 Numerical Method
### CFD Module Based on S-A Model  
The aerodynamic force of the membrane wing is calculated by using the CFD method based on Reynolds averaged N-S equation (RANS) and Spalart-Allmaras (S-A) turbulence model. S-A model is a single equation model [25; 26]. Compared with other turbulence models, this model has the advantages of fast calculation speed and better prediction results for inverse pressure gradient problems. Ignoring the influence of density fluctuation, the  
Figure 15: Fluid-structure coupling analysis framework. physical quantities in the N-S equation are time-homogenized, and the Reynolds time-averaged continuity equation is obtained as follows:  
\[ \begin{array}{c}\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}} \big{(}\rho u_{i}\big{)}=0\\ \frac{\partial}{\partial t}\big{(}\rho u_{i}\big{)}+\frac{\partial}{\partial x_ {j}}\big{(}\rho u_{i}\big{)}=-\frac{\partial p}{\partial x_{i}}+\frac{ \partial}{\partial x_{j}}\big{(}\mu\frac{\partial u_{i}}{\partial x_{j}}-\rho \overline{u^{\prime}}\overline{u^{\prime}}_{j}\big{)}+S_{i}\\ \frac{\partial}{\partial t}\big{(}\rho\phi\big{)}+\frac{\partial}{\partial x_ {j}}\big{(}\rho u\phi\big{)}=\frac{\partial}{\partial x_{j}}\big{(}\Gamma\frac {\partial\phi}{\partial x_{j}}-\rho\overline{u^{\prime}}_{j}\overline{u^{ \prime}}_{j}\overline{u^{\prime}}\big{)}+S\end{array}  \tag{1}\]  
where \(-\rho\overline{u^{\prime}}_{i}\overline{u^{\prime}}_{j}\) is the Reynolds stress. The Reynolds stress is connected with the turbulence average velocity gradient by the Reynolds average KBoussinesq model, which is expressed as:  
\[-\rho\overline{u^{\prime}}_{j}\overline{u^{\prime}}=\mu_{t}\Bigg{(}\frac{ \partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Bigg{)} -\frac{2}{3}\Bigg{(}\rho k+\mu_{i}\frac{\partial u_{i}}{\partial x_{i}}\Bigg{)} \delta_{ij} \tag{2}\]  
where \(\mu_{t}\) is turbulent viscosity, \(u_{i}\) is average velocity, \(\delta_{ij}\) is the symbol of \"Kroncker delta\", and \(k\) is turbulent kinetic energy. In the above equation, the purpose of calculating turbulent flow is to solve the turbulent viscosity \(\mu_{t}\). The S-A model uses a transmission equation to model \(\mu_{t}\), which is expressed as:  
\[\frac{\partial}{\partial t}\big{(}\rho\widetilde{v}\big{)}+\frac{\partial}{ \partial x_{i}}\big{(}\rho\widetilde{v}u_{i}\big{)}=G_{v}+\frac{1}{\sigma_{ \widetilde{v}}}\Bigg{\{}\frac{\partial}{\partial x_{j}}\Bigg{[}(\mu+\rho \widetilde{v})\frac{\partial\widetilde{v}}{\partial x_{j}}\Bigg{]}+C_{b2} \rho\Bigg{(}\frac{\partial\widetilde{v}}{\partial x_{j}}\Bigg{)}^{2}\Bigg{\}} -Y_{v}+S_{\widetilde{v}} \tag{3}\]  
\[\mu_{t}=\rho\widetilde{v}f_{v1} \tag{4}\]  
where \(\widetilde{v}\) is the turbulent viscosity in the area except the wall, \(G_{v}\) is the dissipative term, \(Y_{v}\) is the generating term, and \(S_{v}\) is the diffusion term. \(f_{v1}=\chi^{3}/(\chi^{3}+c_{v1}{}^{3})\), where \(\chi\) is the intermediate variable, and \(c_{v1}\) is a constant, taken as 7.1. A structured grid has higher quality and calculation accuracy compared with an unstructured grid. As displayed in Figure 16, the structured grid of the membrane wing is automatically generated in the CFD module. The chord length is \(c\), and the distance of the front and back boundaries away from the leading edge and the trailing edge is \(40c\). The up and down boundaries are \(20c\) away from the chord. In this part, the calculation accuracy of the numerical method is verified. Because the Tran-SST model can accurately reflect the laminar flow separation, transition, and reattachment behavior at a low Reynolds number, it is widely used in aerodynamic calculation of ultrahigh altitude and high-aspect ratio UAV. Therefore, this model is also added to the comparison. As shown in Figures 17-19, the test data of rigid wing in Figures 12-14 are used to verify the accuracy of S-A model and Tran-SST model.