# Effects of \"S\"-Type Bowed Guide Vanes on Unsteady Flow in 1.5-Stage Axial Compressors
## 2 Methods
### Numerical Method  
The full three-dimensional unsteady Navier-Stokes equations are used for numerical calculation, while real air is used as the working medium. The SST \(k\)-\(\omega\) turbulence model \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Guide Vane** & **Moving Vane** & **Stator Vane** \\ \hline number of blades & 54 & 36 & 45 \\ inner diameter of vane/mm & 260 & 260 & 260 \\ outer diameter of vane/mm & 447 & 445 & 442 \\ rated speed/rpm & — & 4830 & — \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structure parameters of the 1.5-stage axial compressor. Figure 1: Guide vanes with different bending coefficients. is selected for unsteady aerodynamic analysis in this study. The computational domain of aerodynamic analysis is established with the number of guide vanes, moving vanes, and stator vanes being 6, 4, and 5, respectively. The computational domain is one-ninth of the whole axial compressor model. The computational domains extend two times of the guide vane chord ahead of the guide vane and three times of the stator chord length behind the stator vane to improve the computational accuracy. Taking the \(a=0\) mm model as an example, the geometric model of the aerodynamic computational domain is shown in Figure 2.  
The numerical analysis is completed using the ANSYS CFX 16.0 software. The transient rotor stator method [29,30] is adopted for dynamic and static interfaces. The transient rotor stator method can well simulate the relative motion between the guide vane and the moving vane as well as the relative motion between the moving vane and the stator vane by using a sliding interface. The relative position of the grids on each side of the interface is updated each timestep with the rotation of the moving vane. A high accuracy difference scheme is adopted for convection terms and a second-order upwind scheme is adopted for time-term discretization. The inlet turbulence intensity is 5% for a fully developed flow. The time step equals \(7.668\times 10^{-6}\) s, which means it takes 30-time steps for a moving vane to sweep through a guide vane. In the unsteady calculation, the axial force and the tangential force of a single moving vane are detected. When there are more than 10 regular cycles, the calculation is considered to be convergent. The frequency-domain distribution of unsteady aerodynamic exciting forces is obtained using fast Fourier transform, and the frequency-domain distribution law of unsteady aerodynamic forces can be obtained, including excitation amplitude and excitation frequency, so as to further reveal the mechanism of aerodynamic excitation. In this study, the function \(f(t)\) with a period of \(2\pi\) is used to express the unsteady force of the moving vane, as shown in Equation (2) [23]. \[f(t)=\sum_{k=-\infty}^{+\infty}a_{k}e^{ikt} \tag{2}\]  
The spectrum function can be obtained using Fourier transformation of Equation (1), as shown in Equation (3) [23]. \[a_{k}=\frac{1}{2\pi}\int_{0}^{2\pi}f(t)e^{-ikt}dt \tag{3}\]  
For the overall quantities, we validated the numerical calculation method through the numerical analysis of NASA Rotor 67. [PERSON] et al. [31] gave detailed test measurement values of the blade. For the blade of Rotor 67, the number of grid nodes of the single channel numerical model is 0.921 million, and the number of elements is 0.854 million. The grid is densified at the tip clearance and the blade surface. Parameter settings for  
Figure 2: Full geometry of the investigated case. The gray parts are the vanes and the blue parts are the hubs. numerical analysis are as follows: (1) The calculation domain is set as the rotation domain, the rotational speed is 16,043 rpm, and the working medium is ideal gas.