## 2 Numerical Simulation of Forced Vibration  
### Turbulence Modelling  
An excellent turbulence model can reduce the amount of numerical calculation and ensure the accuracy of the results. In the rectangular coordinate system, the N-S equation of viscous incompressible fluid after Reynolds averaging (RANS) is:  
\[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{i}\big{)}}{\partial x_{j}}=-\frac{\partial \overline{p}}{\partial x_{i}}+\mu\frac{\partial}{\partial x_{j}}\left(\frac{ \partial\overline{u}_{i}}{\partial x_{j}}-\rho\overline{u^{\prime}_{i}u^{ \prime}_{j}}\right) \tag{1}\]  
where \(\overline{u}_{i}\) and \(\overline{u}_{j}\) are the average velocity component; \(u^{\prime}_{i}\) and \(u^{\prime}_{j}\) are the pulsating velocity component; \(t\) is the time; \(\rho=1.225\) kg/m\({}^{3}\) is the air density; \(\overline{p}\) is the average pressure; \(\mu=1.7894\times 10^{-5}\) kg/m\(\cdot\)s is the dynamic viscosity coefficient.  
The new unknown quantity \(\rho\overline{u^{\prime}_{i}u^{\prime}_{j}}\) is generated by averaging the operation, so the key to making the equations closed is to introduce a model to solve it. The following assumptions are included in the eddy viscosity model proposed by [PERSON] (2016):  
\[-\rho\overline{u^{\prime}_{i}u^{\prime}_{j}}=\mu_{i}\left(\frac{\partial \overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline{u}_{j}}{\partial x_ {i}}\right)-\frac{2}{3}\rho k\delta_{ij} \tag{2}\]  
where \(k\) is turbulent kinetic energy; \(\mu_{i}\) is turbulent viscosity.  
According to the different applications of turbulence characteristic quantities in eddy viscosity model, different turbulence models have been proposed by scholars. The commonly used turbulence models can be classified into: zero-equation model, one-equation model and two-equation model according to the number of differential equations used. For general flow, the calculation accuracy is higher and the calculation amount is larger with the number of equations. Common two-equation models include standard \(k\)-\(\varepsilon\) model, RNG \(k\)-\(\varepsilon\) model, standard \(k\)-\(\omega\) model and SST \(k\)-\(\omega\) model, etc.  
SST (Shear Stress Transport) \(k\)-\(\omega\) model was proposed by [PERSON] (2007) based on \(k\)-\(\varepsilon\) model and \(k\)-\(\omega\) model. It uses the \(k\)-\(\omega\) model for the near-wall grid area and the \(k\)-\(\varepsilon\) model for the grid area further away from the wall. Because the advantages of \(k\)-\(\varepsilon\) model and \(k\)-\(\omega\) model are combined by mixed functions, the SST \(k\)-\(\omega\) model has higher reliability and accuracy, and is widely used. The expression of the SST \(k\)-\(\omega\) model is:  
\[\rho\frac{Dk}{Det}=\frac{\partial}{\partial x_{j}}\left(\mu_{eff,k}\frac{ \partial k}{\partial x_{j}}\right)+S^{k} \tag{3}\]  
\[\rho\frac{D\omega}{Det}=\frac{\partial}{\partial x_{i}}\left(\mu_{eff,\omega} \frac{\partial k}{\partial x_{i}}\right)+S^{\omega} \tag{4}\]  
where \(\mu_{eff,k}\) and \(\mu_{eff,\omega}\) is the equivalent viscosity of the turbulent term; \(S^{k}\) and \(S^{\omega}\) is the source term of the specific dissipation rate of turbulent kinetic energy.  
The reliability of numerical results using the SST \(k\)-\(\omega\) model has been confirmed by many scholars (2011, 2012, 2013, 2014). In view of this, the SST \(k\)-\(\omega\) model is used to model the turbulence viscosity in this paper. The incoming flow has been set at a turbulence intensity of 0.5%, and the ratio of turbulence viscosity to molecular viscosity is 2.