# Effects of Bogies on the Wake Flow of a High-Speed Train
## 3 Dynamic Mode Decomposition  
Dynamic mode decomposition (DMD) is a data processing algorithm that extracts coherent structures with a single temporal frequency from a numerical or experimental data sequence. For the instantaneous flow field at time \(i\), the samples are obtained at \(M\) spatial locations and arranged in a certain order into a column vector \(\psi_{i}\), which is called a snapshot of the flow field. We focus on the  
Figure 5: Grid layout on the cross-section of y = 0 (upper) and on the surface of the trailing car (lower). temporal evolution processes and assume that the data are equispaced in time and the time step is \(\Delta t\). Next, we form two data matrices from \(N\)+1 snapshots:  
\[\left\{ \begin{array}{c}\mathbf{\Psi}_{0}=[\mathbf{\psi}_{0}\,\mathbf{\psi}_{1}\,\cdots\, \mathbf{\psi}_{N-1}],\\ \mathbf{\Psi}_{1}=[\mathbf{\psi}_{1}\,\mathbf{\psi}_{2}\,\cdots\,\mathbf{\psi}_{N}].\end{array} \right. \tag{1}\]  
Assume that the snapshots are generated by a discrete-time linear time-invariant system:  
\[\mathbf{\psi}_{t+1}=\mathbf{A}\mathbf{\psi}_{t},t=0,1\cdots,N-1, \tag{2}\]  
resulting in:  
\[\mathbf{\Psi}_{1}=[\mathbf{\psi}_{1}\,\mathbf{\psi}_{2}\,\cdots\,\mathbf{\psi}_{N}]\,=[ \mathbf{A}\mathbf{\psi}_{0}\,\mathbf{A}\mathbf{\psi}_{1}\,\cdots\,\mathbf{A}\mathbf{\psi}_{ N-1}]\,=\mathbf{A}\mathbf{\Psi}_{0}. \tag{3}\]  
In Equations (2) and (3), the characteristic information of the flow field is contained in matrix \(\mathbf{A}\) and hence the evolution of the flow field can be obtained through an eigen decomposition of \(\mathbf{A}\). However, in the flow field \(M\), the number of sample points is always large so that \(\mathbf{A}\)\(\in\)\(\mathbf{C}^{M\times M}\) becomes a high-dimensional matrix. The eigen decomposition of such a high-dimensional matrix is impossible to solve in practical applications. In DMD [21; 22], the characteristic information is approximated by an alternative low-dimensional matrix \(\mathbf{F}_{\text{dmd}}\)\(\in\)\(\mathbf{C}^{N\times N}\). \(N\), the number of snapshots, is always smaller than \(M\) by 1-2 orders of magnitude, which results in lower calculation costs for the eigen decomposition. Notably, all DMD calculations are performed in the complex field. First, \(\mathbf{\Psi}_{0}\) is decomposed using an economy-size singular value decomposition (SVD):  
\[\mathbf{\Psi}_{0}=\mathbf{U}\Sigma\mathbf{V}^{*}, \tag{4}\]  
where * denotes the complex conjugate transpose of the matrix. Then, the optimal low-dimensional approximation matrix \(\mathbf{F}_{\text{dmd}}\) is determined by:  
\[\mathbf{F}_{\text{dmd}}=\mathbf{U}^{*}\mathbf{\Psi}_{1}\mathbf{\
abla}\Sigma^{-1}. \tag{5}\]  
To calculate the DMD modes and corresponding eigenvalues of the flow field, the eigen decomposition of \(\mathbf{F}_{\text{dmd}}\) is required, i.e. :  
\[\mathbf{F}_{\text{dmd}}\mathbf{y}_{k}=\mu_{k}\mathbf{y}_{k},k=1,2,\cdots N. \tag{6}\]  
where \(\mathbf{y}_{k}\) is the \(k\)th eigenvector of \(\mathbf{F}_{\text{dmd}}\) and \(\mu_{k}\) is the corresponding \(k\)th eigenvalue. Thereafter, the DMD modes at \(t=0\) can be obtained by:  
\[\mathbf{\varphi}_{k}^{(0)}=\mathbf{U}\mathbf{y}_{k}. \tag{7}\]  
In Equation (7), all modes are sorted in order of descending mode energy \(\|\mathbf{\varphi}_{k}^{(0)}\|\) as:  
\[\|\mathbf{\varphi}_{1}^{(0)}\|>\|\mathbf{\varphi}_{2}^{(0)}\|>\|\mathbf{\varphi}_{3}^{(0)} \|>\cdots>\|\mathbf{\varphi}_{N}^{(0)}\|, \tag{8}\]  
where \(\|\mathbf{\varphi}_{k}^{(0)}\|\) can be evaluated by the following matrix norm [21; 23],  
\[\|\mathbf{\varphi}_{k}^{(0)}\|=\frac{1}{\text{norm}\Big{(}\mathbf{V}^{*}\Sigma^{- 1}\mathbf{y}_{k}\Big{)}}. \tag{9}\]In Equation (6), the eigenvalue \(\mu_{k}\), corresponding to the \(k\)th DMD mode, represents the characteristic information in the unsteady flow field and the corresponding mode frequency \(\omega_{k}\) and growth rate \(g_{k}\) are:  
\[\left\{ \begin{array}{l}\omega_{k}=\text{Im}\Big{[}\frac{\ln(\mu_{k})}{\Delta t }\Big{]}\\ g_{k}=\text{Re}\Big{[}\frac{\ln(\mu_{k})}{\Delta t}\Big{]}\end{array}