## 2 C-POD Method and Modal Sorting Method
### Entropy-Controlled Euclidean-to-Probability Mapping Modal Sorting Method
The effects of the total grid number and dimensionless time step on the results were considered, and the results were compared with the Stanton number and Lift coefficient values obtained from the literature [28; 29; 30]. The comparison results are presented in \[\mathbf{W}=\mathbf{E}\varnothing(\mathbf{1}_{n}\mathbf{1}_{n}^{\mathrm{T}}\mathbf{E})\]4 The dimensionless time step \(\Delta t*\) is defined as shown in Equation (32):

\[\mathbf{W}=\mathbf{E}\varnothing(\mathbf{1}_{n}\mathbf{1}_{n}^{\mathrm{T}}\mathbf{E})\]5

where \(\Delta t\) represents the time step, \(U_{0}\) is the upstream inlet velocity of the cylinder, and \(D_{c}\) is the diameter of the cylinder. From Table 1, it can be observed that the simulation results remained consistent as the grid number increased from 400 \(\times\) 160 to 800 \(\times\) 320. When the dimensionless time step was less than or equal to 0.01, the results were in close agreement with those in the literature. Therefore, in this study, we chose the A\({}_{3}\) condition, setting the total grid number to 400 \(\times\) 160 and the dimensionless time step to 0.01. ### Presentation of Computational Results

Figure 9 presents the vorticity distribution corresponding to several typical flow state transitions in the two-dimensional flow around a cylinder at Re = 150. The specific flow states are detailed as follows:

1. From \(t\) = 1 to 9 s, a pair of side vortices formed on either side of the cylinder, gradually creating a pair of vortices with opposite rotational directions behind the cylinder. 2. From \(t\) = 9 to 33 s, the counter-rotating vortices behind the cylinder gradually moved and diffused in the direction of the flow. 3. From \(t\) = 33 to 110 s, the counter-rotating vortices behind the cylinder disappeared, and the flow became more stable. 4. For times exceeding 110 s, the flow pattern underwent notable changes, with vortices detaching periodically to form a stable vortex street. Multiple regions of alternating positive and negative vorticity emerged, indicating pronounced shear and recirculation phenomena within the flow field--hallmarks of the classic Karman vortex street. \begin{table}
\begin{tabular}{c c c c c} \hline
**Operating Condition Name** & **Total Number of Grids** & \(\Delta t^{*}\) & **CD** & **St** \\ \hline A1 & 100 \(\times\) 40 & 0.01 & 0.215 & 0.16 \\ A2 & 200 \(\times\) 80 & 0.01 & 0.225 & 0.164 \\ A3 & 400 \(\times\) 160 & 0.01 & 0.229 & 0.165 \\ A4 & 600 \(\times\) 240 & 0.01 & 0.229 & 0.165 \\ A5 & 800 \(\times\) 320 & 0.01 & 0.229 & 0.165 \\ A6 & 400 \(\times\) 160 & 0.005 & 0.229 & 0.165 \\ A7 & 400 \(\times\) 160 & 0.02 & 0.231 & 0.167 \\ Experiment from [28] & & & & 0.165 \\ CFD from [29] & & & & 0.226 & 0.165 \\ CFD from [30] & & & & 0.233 & 0.166 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of numerical solutions with literature results for varying total grid numbers and dimensionless time steps. Cylinder Wake Flow Field Dimensionality Reduction and Reconstruction Accuracy, and Modal Decomposition

#### 4.4.1 Dataset and Test Set Construction

To test the accuracy variations of POD and C-POD in the reduced-order modeling and reconstruction of vorticity fields in flow around a cylinder, as well as their modal decomposition capabilities for the flow behavior of the cylinder, a database was constructed over the entire evolution of the cylinder's flow pattern at Re = 150. Specifically, the time range was set from t = 0 to 145 s, with data collected every 0.1 s. The vorticity field distribution data were then written into the dataset, resulting in a total database size of 64,000 \(\times\) 1450. #### 4.4.2 Dimensionality Reduction and Reconstruction Accuracy of Cylinder Wake Flow Field

This section assesses the accuracy of the POD and C-POD methods in both the dimensionality reduction and reconstruction of the cylinder wake vorticity field, as well as their modal decomposition capabilities for characterizing the cylinder's wake flow behavior.