## 2 C-POD Method and Modal Sorting Method
### Entropy-Controlled Euclidean-to-Probability Mapping Modal Sorting Method
As illustrated in Figure 10, the average Cr value of C-POD was higher at lower modal orders; however, as the mode number increased, the average Cr value of C-POD decreased rapidly and eventually stabilized at values consistently lower than those observed for POD. At higher modal orders, C-POD maintained a lower average Cr value, indicating improved reconstruction accuracy across multiple modes relative to POD. Furthermore, as shown in Figure



dimensionality reduction model accuracy--both in the lower bound of capability (maximum Cr) and the overall average accuracy (mean Cr)--thereby confirming its enhanced dimensionality reduction capability over that of POD. #### 4.4.3 Comparison of Modal Decomposition Results

Based on Figure 9, the first ten modes were selected, and the POD and C-POD methods were used to decompose the vorticity field of the cylinder wake at \(t=236\) s and \(t=240\) s under the condition of Re = 150, when the flow transitioned into a stable vortex street. The vorticity field dominated by the first six modes is shown in Figures 12 and 13. From the comparison of Figures 12 and 13, it can be observed that the vorticity field dominated by the first six modes extracted by C-POD mainly contained four types of vortical structures, as shown in Figure 14. These are named as:

Figure 11: Variation in the lower bound of dimensionality reduction and reconstruction capability of POD and C-POD methods at different modal orders. Figure 10: Variation in average dimensionality reduction and reconstruction capability of POD and C-POD methods at different modal orders. 1. Type 1 vortex: Symmetrical side vortices extending downstream from both sides of the cylinder into the flow region. 2. Type 2 vortex: Counter-rotating pair of vortices formed at the rear of the cylinder due to recirculation, resembling a \"horseshoe\" shape. 3. Type 3 vortex: Developing horseshoe-shaped vortices. 4. Type 4 vortex: Asymmetric side vortices on both sides of the cylinder. Figure 12: Distribution of the vorticity field dominated by the first six modes extracted by C-POD and POD at \(t=236\) s.

Figure 13: Distribution of the vorticity field dominated by the first six modes extracted by C-POD and POD at \(t\) = 240 s.

Between \(t=236\) s and \(t=240\) s, vortex formation in the vorticity field was characterized by changes in the 1 st, 2 nd, and 5 th modes, with all vortices exhibiting a contraction trend toward the rear of the cylinder. Concurrently, the rotational direction of vortices in the 1 st, 3 rd, 5 th, and 6 th modes reversed. Lower-order POD modes corresponded to the primary flow structures and were thus more intuitively interpretable. However, as the modal order increased, modes such as the 2 nd and 3 rd captured more complex flow characteristics, including primary vortices, secondary vortices, and localized vortices. Despite this, the physical interpretation of vortex evolution--particularly the temporal dynamics of higher-order modes--remains a challenging endeavor. To further validate the reliability of the ECEPM method in ranking the significance of the basis for C-POD method clustering, the total number of modes was set to 10. The ECEPM method selected the first \(n\) modes, reconstructing the vorticity field distributions at \(t=236\) s and \(t=240\) s while computing the corresponding Cr values. These results were then compared with those obtained using the POD method, as shown in Figure 15. Figure 15 illustrates that when the modal order was less than 3, the reconstruction accuracy of the C-POD method increased rapidly with additional modes, exhibiting a significantly greater improvement than the POD method. This finding suggests that the first \(n\) modes selected by the ECEPM method promptly captured the system's primary features, and the C-POD method demonstrated robust reconstruction capabilities at lower modal orders. However, as the modal order increased further, the reconstruction accuracies of both methods tended to converge. Figure 14: Various vortex patterns in the vorticity field dominated by C-POD modes.