## 2 C-POD Method and Modal Sorting Method
### Entropy-Controlled Euclidean-to-Probability Mapping Modal Sorting Method
\[\mathbf{E}=\exp(\frac{\mathbf{Z}}{\tau})\]4 Time evolution of the waveform reconstructed using only the first mode extracted by the C-POD and POD methods, for \(c=0.1\) and \(\beta=0.2\), from \(t=0\) to \(t=2\). ## 4 Cylinder Wake Flow Case Introduction

Two-dimensional flow around a cylinder represents a classical complex flow problem in fluid dynamics, characterized by phenomena such as vortex shedding, flow separation, and turbulence. In contrast to the one-dimensional Burgers' equation, this flow configuration exhibits more intricate turbulent features, revealing a rich spectrum of physical phenomena. Consequently, it serves as an excellent benchmark for validating the efficacy of reduced-order models in capturing complex flow dynamics. ### Model Introduction

To simulate two-dimensional fluid flow around a cylinder, the computational domain was configured as a rectangular region with the cylinder positioned near the mid-left boundary. The cylinder had a radius of 5 mm, and the domain dimensions were \(L_{x}\) = 1 mm (length) and \(L_{y}\) = 40 mm (width). Inflow velocities at the left boundary were varied to adjust the Reynolds number (Re), thereby capturing a range of flow phenomena. Figure 8 illustrates the geometric configuration of the computational domain. \[\mathbf{E}=\exp(\frac{\mathbf{Z}}{\tau})\]5 Time evolution of the waveform reconstructed using only the first mode extracted by the C-POD and POD methods, for \(c\) = 2 and \(\beta\) = 2, from \(t\) = 0 to \(t\) = 2. For the two-dimensional incompressible Navier-Stokes equations, the motion of the fluid is described by the following equations:

\[\mathbf{E}=\exp(\frac{\mathbf{Z}}{\tau})\]6

\[\mathbf{E}=\exp(\frac{\mathbf{Z}}{\tau})\]7

where \(\mathbf{u}=(u,v)\) represents the velocity field components, with \(u\) being the velocity in the \(x\) direction and \(v\) is the velocity in the \(y\) direction, \(\mathbf{p}\) is the pressure field, \(\
u\) is the fluid's kinematic viscosity, and \(\mathbf{f}\) represents the body force (in this case, the body force field). For this two-dimensional flow problem, the governing equations were discretized using the finite difference method, and the implicit method combined with the alternating direction implicit (ADI) method was used for time-stepping solutions. The specific solution process is briefly described below:

1. Using the alternating direction implicit (ADI) method, the equation was split into two one-dimensional equations. The implicit solution was first performed in the \(x\) direction, followed by the implicit solution in the \(y\) direction: \[\mathbf{E}=\exp(\frac{\mathbf{Z}}{\tau})\]8 (27) \[\mathbf{E}=\exp(\frac{\mathbf{Z}}{\tau})\]9 (28)
2. The pressure field was obtained by solving the Poisson equation using the fast Fourier transform (FFT): \[\mathbf{W}=\mathbf{E}\varnothing(\mathbf{1}_{n}\mathbf{1}_{n}^{\mathrm{T}}\mathbf{E})\]0 (29)
3. We set a fixed flow velocity at the left boundary \(x=0\) and applied a no-slip boundary condition on the surface of the cylinder. 4. To quantify the rotational behavior in the flow field around the cylinder, the calculation of vorticity was introduced in the two-dimensional incompressible flow field. The vorticity is defined as: \[\mathbf{W}=\mathbf{E}\varnothing(\mathbf{1}_{n}\mathbf{1}_{n}^{\mathrm{T}}\mathbf{E})\]1 (30)

The vorticity field distribution was obtained through the differential method, as follows:

\[\mathbf{W}=\mathbf{E}\varnothing(\mathbf{1}_{n}\mathbf{1}_{n}^{\mathrm{T}}\mathbf{E})\]2

\[\mathbf{W}=\mathbf{E}\varnothing(\mathbf{1}_{n}\mathbf{1}_{n}^{\mathrm{T}}\mathbf{E})\]3 Geometric dimensions of the computational domain for the two-dimensional cylinder wake flow. ### Grid Independence and Numerical Method Validation

To validate the reliability of the numerical method and the grid used in this study, we first performed a computational model verification for a fixed single cylinder with Re = 100.