**2. Methodology**

A brief description of the POD principle for the velocity/pressure field is presented here, and more details about the theoretical derivation and practical application can be found in [19].

For moment ti, velocity or pressure components *fi* ð Þ *xm* at a monitored location m are arranged in a vector *qi* to formulate a snapshot, the total number of velocity snapshots N with sampling interval *Δt* can then be assembled in matrix *QN*. The Timeaverage field *Q* should be firstly subtracted from the instantaneous velocity or pressure field to obtain the zero-centered fluctuating matrix *Q*<sup>0</sup> *N*.

$$q\_i = \begin{bmatrix} f\_i(\mathbf{x}\_1) \\ f\_i(\mathbf{x}\_2) \\ \vdots \\ f\_i(\mathbf{x}\_m) \\ \vdots \\ \vdots \\ f\_i(\mathbf{x}\_M) \end{bmatrix} \tag{1}$$

$$Q\_N = \left[q\_1, q\_2, q\_3 \cdots q\_N\right] \tag{2}$$

$$Q'\_N = Q\_N - \overline{Q} \tag{3}$$

Where f g *x*<sup>1</sup> *x*<sup>2</sup> ⋯ *xm* signify the position of the monitored mesh points, and M is the total number of the monitored points.

The purpose of POD is to determine a set of orthogonal bases, and equivalently express *q*<sup>0</sup> *<sup>i</sup>* as the superposition of the product of the POD base and corresponding modal coefficients.

$$q\_i' = \sum\_{k=1}^{n} a\_{ki} \varphi\_k \tag{4}$$

To identify POD basis, a covariance matrix C is formulated as *C* ¼ *Q*<sup>0</sup> *N TQ*<sup>0</sup> *<sup>N</sup>*, and the eigenvalue and eigenvector can be obtained by solving the eigenfunction of matrix C as follows,

$$C\rho\_k = \lambda\_k \rho\_k \tag{5}$$

where the eigenvalue *λ<sup>k</sup>* reflects the energy contribution of each POD mode and is arrayed in descending sequence, eigenvector *φ<sup>k</sup>* represents the POD modes and any two of them are orthogonal to each other spatially.

Modal coefficients *A* ¼ *aki* ½ �ð Þ *k* ¼ 1, 2, 3⋯*n*; *i* ¼ 1, 2, 3⋯*N* are determined by projecting the original fluctuating velocity or pressure field onto the POD modes as follows,

$$A = \Phi^T Q'\_N \tag{6}$$

where the spatial POD mode Φ ¼ *φ*<sup>1</sup> *φ*<sup>2</sup> … *φ<sup>n</sup>* ½ � and the temporal modal coefficient *A* ¼ *aki* ½ �ð Þ *k* ¼ 1, 2, 3⋯*n*; *i* ¼ 1, 2, 3⋯*N* .
