# Analysis of the Transonic Buffet Characteristics of Stationary and Pitching OAT15A Airfoil
## 3 Numerical Example
### Flow Field Buffet Characteristics of Pitching Airfoils
#### 3.3.2 Fluidic Modes  
The velocity snapshot samples extracted from the flow field of the pitching airfoil were decomposed by the DMD method, and the distribution of the Ritz eigenvalues in the unit circle is shown in Figure 10.  
Similar to the study of the case where the airfoil is stationary in the flow field, the analysis of the flow field with the airfoil in pitching motion started with the sorting of the DMD modal orders by calculating the modal energies, and then the loss function was calculated based on the sorted modes. The relationship between the loss function and the selected modal orders is illustrated in Figure 11.  
In Figure 11, it can be seen that in the pitching case, six modes were able to make the loss function less than 2%. The first nine modes were selected for the flow field analysis in the fixed airfoil case, and as a comparison, the same number of modes were also selected for airfoil pitching oscillation. The selected modal parameters are presented in Table 2.  
In Table 2, it can be seen that the first nine extracted DMD modes' frequencies exhibit multiples of the pitching motion frequency, which implies that the structural elastic oscillation changed the characteristics of the flow field induced by buffet (shock oscillations), and the buffet flow response vanished for airfoils oscillating at a frequency close to the buffet frequency.  
It can also be observed in Table 2 that among the nine modes selected, the growth rate and frequency of the first order mode are zero, representing the average flow in the flow field. We performed a flow velocity field contour analysis for the remaining four pairs of conjugate modes, and the corresponding contours of real parts are shown in Figure 12.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Order** & **Growth Rate** & **Frequency (Hz)** & **Loss Function** \\ \hline
1 & 0 & 0 & 0.061 \\
2–3 & \(-1.97\times 10^{-6}\) & 81.99 & 0.027 \\
4–5 & \(-1.08\times 10^{-5}\) & 163.99 & 0.023 \\
6–7 & \(-1.00\times 10^{-5}\) & 245.32 & 0.015 \\
8–9 & \(2.22\times 10^{-5}\) & 327.28 & 0.012 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Dynamical parameters of extracted DMD modes of the pitching airfoil.  
Figure 9: Snapshots of instantaneous velocity divergence in one buffet period for the pitching airfoil. (a) \(t=t_{0}\). (b) \(t=t_{0}+T/8\). (c) \(t=t_{0}+4/8T\). (d) \(t=t_{0}+5/8T\). (e) \(t=t_{0}+6/8T\). (f) \(t=t_{0}+T\).  
Figure 11: Relationship of the loss function with the numbers of DMD orders in pitching airfoil.  
Figure 10: DMD modal eigenvalues of pitching airfoil. (**a**) The distribution of all eigenvalues in the unit circle. (**b**) The distribution of first 9 modal eigenvalues extracted in unit circle.  
The shock motion and flow separation in the pitching case are given in Figure 12a,b. Figure 12b-d show that the range of shock motion is larger than that of the fixed case. In addition, the shape of the separated flow is irregular compared to the fixed case depicted in Figure 7b-d. Furthermore, there are other round-trip motions of the shock between the first and second shocks, as seen in Figure 12c. These observations demonstrate the complexity of the flow in this case, suggesting that structural elastic motions may play an important role in changing the feature of the buffet flow.  
Figure 12: The first 4 pairs of conjugate modal real part contours of streamwise velocity for the pitching airfoil. (**a**) DMD modes 2–3. (**b**) DMD modes 4–5. (**c**) DMD modes 6–7. (**d**) DMD modes 8–9.