# Analysis of the Transonic Buffet Characteristics of Stationary and Pitching OAT15A Airfoil
## 3 Numerical Example
### Flow Field Buffet Characteristics of the Fixed Airfoil
#### 3.2.2 Fluidic Mode  
A total of 150 velocity snapshots of the flow field were collected under the computational conditions. About 50 snapshots per shock oscillation period were sampled for a total of three buffet periods. The extracted velocity snapshot samples were decomposed according to DMD to obtain the principal modes from the simulated data. These modes were employed to analyze the flow feature. All Ritz eigenvalues can be obtained from the DMD method, and their distribution in the unit circle is shown in Figure 5a. Figure 4: Snapshots of instantaneous velocity divergence in one buffet period for the fixed airfoil. (**a**) \(t=t_{0}\). (**b**) \(t=t_{0}+T/8\). (**c**) \(t=t_{0}+T/4\). (**d**) \(t=t_{0}+3/8T\). (**e**) \(t=t_{0}+T/2\). (**f**) \(t=t_{0}+5/8T\). (**g**) \(t=t_{0}+3/4T\). (**h**) \(t=t_{0}+7/8T\). (**i**) \(t=t_{0}+T\). The modal energy criterion proposed in the literature [19; 20] (shown in Equation (9)) was used to order the modes at different frequencies, according to which the modal contributions to the overall flow field were ranked. \[I_{i}=\sum_{j=1}^{N-1}\left|a_{i}(\lambda_{i})^{j-1}\right|\|\mathbf{\varphi}_{i}\|_{ \mathrm{F}}^{2}\Delta t \tag{9}\]  
where \(I_{i}\) stands for the \(i\)th order modal energy, and \(\|\bullet\|_{\mathrm{F}}\) is Frobenius norm. The detailed derivation and meaning of this formula can be found in [19]. The loss function defined in the literature [21], representing the error between the reconstructed flow field and the real flow field by taking the first \(j\) modes, is expressed as follows,  
\[loss_{j}=\frac{\left\|\mathbf{V}_{1}^{N}-\mathbf{\Phi}(1:j)\mathbf{D}_{aj}\mathbf{\Lambda}^{i -1}\right\|_{\mathrm{F}}}{\left\|\mathbf{V}_{1}^{N}\right\|_{\mathrm{F}}} \tag{10}\]  
Figure 5: DMD modal eigenvalues of the fixed airfoil. (**a**) The distribution of all eigenvalues in unit circle. (**b**) The distribution of first 9 modal eigenvalues extracted in unit circle. where \(\mathbf{\Phi}(1:j)\) represents the first \(j\)-order modes, \(\mathbf{D}_{aj}\) represents the first \(j\)-order matrices, and \(\mathbf{\Lambda}^{i-1}{}_{j}\) represents the first \(j\) rows of \(\mathbf{\Lambda}^{i-1}\). Figure 6 shows the relationship between the modes obtained by ordering the energy criterion according to Equation (9) and the loss function defined by Equation (10). In Figure 6, we can see that the first nine modes are able to maintain the loss function below 2.5%. The distribution of the selected first nine-order modes in the unit circle is shown as the red squares in Figure 5a, and its local enlargement is shown in Figure 5b. As can be seen, all modes are basically in the vicinity of the unit circle, indicating that the system is mainly in a critical stable state under simple harmonic motion. It is also consistent with the actual physical phenomena. The growth rates and reduced frequencies of the first nine-order modes are shown in Table 1, which indicates that the growth rates of the modes are very small due to the critical stable state, and the frequency of each order mode follows the buffet frequency or its higher harmonics. The literature [18] points out that there is some phase difference between the imaginary part and the real part of the DMD mode, but the difference in the flow field characteristics is little. Based on this fact, only the real parts were analyzed for the first nine modes, as shown in Figure 5 in this paper. The modal growth rate and frequency of the first-order mode were both zero, representing the time-averaged flow field characteristics. The contours of the real parts from the remaining four pairs of conjugate modes are shown in Figure 7.  
In Figure 7a, the contour displays the range of the shock motion and shock-induced separation, which is from the root of the shock to the trailing edge of the airfoil. It reveals that there are fluid modes containing shock motion and separation in the buffet flow.