## 2 Aerodynamic Characteristics Analysis Model of a Cylindrical Projectile
### CFD Model  
In this section, a CFD simulation model for the flow field around a cylinder projectile is established to study its aerodynamic characteristics under different flight states, and the basic control equations for the three-dimensional unsteady compressible flow field around a projectile are given. In this study, the Reynolds-averaged Navier-Stokes (RANS) method is used to the time-averaged treatment of the continuous and momentum equations of flow domain as the following form,  
\[ \begin{split}\frac{\partial\rho}{\partial t}+\frac{\partial}{ \partial x_{i}}(\rho u_{i})=0\\ \frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x _{j}}\left(\rho u_{i}u_{j}\right)=-\frac{\partial\overline{\rho}}{\partial x_ {i}}+\frac{\partial}{\partial x_{j}}\left[\mu\left(\frac{\partial u_{i}}{ \partial x_{j}}+\frac{\partial u_{i}}{\partial x_{i}}-\frac{2}{3}\delta_{ij} \frac{\partial u_{i}}{\partial x_{j}}\right)\right]+\frac{\partial}{\partial x _{j}}\left(-\rho\overline{u^{\prime}_{i}u^{\prime}}_{j}\right)\end{split}  \tag{1}\]  
where the parameters subscripted \"\(i\)\" and \"\(j\)\" represent the values of the three axial directions of the inertial coordinate system, respectively; \(u_{i},u_{j},u_{i}\) represents the Reynolds average velocity component with the average sign omitted; \(\mu\) is the kinetic viscosity; and \(\delta_{ij}\) represents the Kronecker tensor component. \(-\rho\overline{u^{\prime}_{i}u^{\prime}}_{j}\) describes the Reynolds stress, which can be calculated using the eddy viscosity assumption proposed by [PERSON] (1965), with the following expression:  
\[-\rho\overline{u^{\prime}_{i}u^{\prime}}_{j}=\mu_{t}(\frac{\partial u_{i}}{ \partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}})-\frac{2}{3}(\rho k+ \mu_{t}\frac{\partial u_{k}}{\partial x_{k}})\delta_{ij} \tag{2}\]  
\(k\) is the turbulent kinetic energy. In order to solve the unknown quantity \(\mu_{t}\) in the above equation, a variety of computational models have been developed. The \(\omega\) transport equation in the k-\(\omega\) two-equation model can be solved by integrating through the viscous bottom layer without additional terms, which effectively reduces the requirement of the algorithm for the near-wall mesh density, improves the computational accuracy for the region of the boundary layer separation and the internal counter pressure gradient, and is more suitable for calculating the characteristic structure of the flow field under the condition of a high angle of attack. The shear stress transport (SST) k-\(\omega\) model ([PERSON] and [PERSON], 2017) additionally considers the transfer of turbulent shear stresses in the definition of turbulent viscosity, which can accurately predict the separation flow on the smooth wall. In this paper, we adopted the SST k-\(\omega\) model to simulate the turbulent flow. \[ \begin{split}\frac{\partial\rho k}{\partial t}+\frac{\partial \rho ku_{i}}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}[(\mu+\frac{\mu_{ t}}{\sigma_{k}})\frac{\partial k}{\partial x_{j}}]+G_{k}-Y_{k}\\ \frac{\partial\rho\omega}{\partial t}+\frac{\partial\rho\omega u_{ i}}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}[(\mu+\frac{\mu_{t}}{ \sigma_{\omega}})\frac{\partial\omega}{\partial x_{j}}]+G_{\omega}-Y_{\omega} \end{split}  \tag{3}\]  
\(\sigma_{k}=2\), \(\sigma_{\omega}=2\). \(G_{k}\) represents the turbulent kinetic energy due to the average velocity gradient, \(Y_{k}\) represents the dissipation of turbulent kinetic energy \(k\), \(G_{\omega}\) represents the generation of the specific dissipation rate \(\omega\), and \(Y_{\omega}\) represents the dissipation of the  
Figure 1: Basic model of the cylindrical projectile. specific dissipation rate \(\omega\). The turbulent viscosity coefficient \(\mu_{t}\) is derived from the following equation.