## 4 Results and Discussion
### Unsteady Simulation Results
#### 4.2.3 Flow Field Modal Decomposition  
The flow decomposition technique is used to distinguish different flow patterns that occur within a compressor. POD and DMD methods are commonly applied to pressure fields to study the internal flow of fluid machinery in the literature [25; 26; 27; 28; 29]. The modal decomposition methods are suitable for the study of pressure loss and flow instability caused by stall cell and impeller-diffuser interaction near the shroud side of the diffuser. With the POD method, the pressure perturbations are not divided into stochastic and deterministic but are represented by modes [30; 31; 32]. Then, the large-scale (energetic) flow structures are distinguished by filtering chaos or noise. Next, flow field reconstruction technology reveals the location and magnitude of large-scale flow structures in the flow field. The POD method is utilized to extract a basis of modal decomposition from an ensemble of snapshots to achieve this filtering. A total ensemble of 1000 (about three rotation cycles) instantaneous static pressure flow fields (0.95 span in diffuser passage of TVD and SVD) is provided for POD analysis. As the low-order POD modes characterize the energetic flow structures and dominate the flow field, the sensitivity of the ensemble size is tested by comparing low-order POD eigenvalues of each set with data subsets of various sizes. Results indicate that the eigenvalues corresponding to the low-order modes do not change significantly over more than 300 snapshots. Therefore, the ensemble size of snapshots used in this paper far exceeds the ability to provide statistically converging criteria. Finally, the DMD method is used to study the large-scale flow field structure at the special frequencies that have a great influence on the flow field. Figure 19 shows the eigenvalues corresponding to the first 300 POD modes in TVD and SVD, where the eigenvalues corresponding to each mode are normalized by the first-order eigenvalue of TVD. Note that since the POD mode exceeds 10 orders, the eigenvalue decreases sharply; the abscissa is in log scale, which provides a statistical representation of smaller-scale flow structures [32]. Figure 17: Comparison of unsteady pressure at 0.95 span of diffuser vane leading-edge (\(\varphi=0.7\ Q_{\text{d}}\)). Figure 18: Unsteady behavior of vortices (SVD, \(\varphi=0.7\ Q_{\text{d}}\)): (**a**) \(t=t_{\text{s0}}\); (**b**) \(t=t_{\text{s1}}\). As can be seen from Figure 19, the energy of POD modes in TVD is higher than that in SVD. Therefore, the POD eigenvalues indicate that large-scale flow structures of TVD possess higher energy compared to that of SVD, which is consistent with the large-scale pressure fluctuation previously observed for diffuser near-stall operating point (in Figure 17). POD eigenvector (temporal information) analysis further explains the frequency information of large-scale flow structures and distinguishes the different flow patterns more clearly. The eigenvectors corresponding to the first fourth-order POD mode were analyzed by FFT (as shown in Figure 20). For TVD, the diffuser stall frequency (86.645 Hz) and BPF appear in the first second-order mode, while the third- and fourth-order modes are also correlated with 2 BPF. However, for SVD, the third- and fourth-order modes are mainly associated with the fundamental frequency and 2 BPF, no longer affected by diffuser stall frequency. The spatial distribution of these POD modes is represented in Figure 21. Each diffuser vane is labeled with numbers from 1 to 4. Due to periodic boundary conditions, it can be considered that the left-side diffuser passage of vane 1 and the right-side diffuser passage of vane 4 are the same. The zeroth-order POD mode represents the ensemble-averaged flow-field, which includes around 91.176% of total flow energy, and its spatial pattern is shown in Figure 21a. Compared with TVD, the high static pressure zone near the SVD exit is significantly larger and more uniform. This is the reason why SVD has a higher static pressure recovery coefficient.