## 2 Numerical Analysis
### Modeling of the Inflow Generator  
We investigated a synthetic inflow generator having a variety of length scales, which was based on [PERSON] and [PERSON]'s method ([PERSON] and [PERSON] [26]) (XC). In this section, we present a brief review of the XC model. The XC model imposes correlations using an exponential function to satisfy the prescribed space and time correlations. The instantaneous value of the velocity \(u_{i}\) may be written as:  
\[u_{i}=\overline{u}_{i}+a_{ij}\psi_{j} \tag{7}\]where \(\overline{u}_{i}\) is the mean and \(a_{ij}\psi_{j}\) is the fluctuating part of \(u_{i}\), which is called Reynolds decomposition. The fluctuating part can be separated into two parts, \(a_{ij}\), which is the amplitude tensor, and \(\psi_{j}\), which is the unscaled fluctuation with zero mean (\(\overline{\psi}_{j}=0\)), a unit variance (\(\overline{\psi_{j}\psi_{j}}=1\)), and zero covariance with the other distributions (\(\overline{\psi_{j}\psi_{j}}=0,i\
eq j\)). The term \(a_{ij}\) using Reynolds stress \(R_{ij}\) is the Cholesky decomposition of the Reynolds stress by [PERSON] and [PERSON] (1965) as follows,  
\[a_{ij}=\begin{pmatrix}\sqrt{R_{11}}&0&0\\ R_{21}&\sqrt{R_{22}-a_{21}^{2}}&0\\ R_{31}/a_{11}&(R_{32}-a_{21}a_{31}/a_{22}&\sqrt{R_{33}-a_{31}^{2}-a_{32}^{2}} \end{pmatrix} \tag{8}\]  
where \(R_{ij}\) is the Reynolds stress tensor obtained from the experimental or previously calculated numerical simulation data. If the boundary layer is homogeneous in the lateral direction, \(R_{31}=R_{32}=0\); therefore, \(a_{31}\) and \(a_{32}\) can be removed.  
The synthetic inflow generation technique using digital filtering aims to find the unscaled fluctuation (\(\psi_{j}\)), which is a one-dimensional number sequence with zero mean, unit variance, and spatial correlation. The correlations in the most turbulent shear flow have similar features. The advantage of the correlation function is that it approximates the relation of two different points to the form of an exponential function. The exponential function is \(\exp\left(-\pi r/\beta L\right)\), where \(\beta=2\). However, in the equation, \(\beta\) is not clearly defined, and its function remains unclear. In this study, the exponential function was carefully tested and subsequently modified. The modified exponential function is expressed as follows  
for the application of the plane velocity field data for the next time step. Based on [PERSON]'s hypothesis, the Lagrangian time scale can be expressed as follows:  
\[T=L/\overline{u} \tag{13}\]  
where \(L\) is the length scale and \(\overline{u}\) is the local mean velocity. In addition, the Eulerian time scale, \(T\), can be obtained from experimental or previous numerical analyses. The velocity of the next time step can be calculated by Equation (14) using the velocity at the previous time step.  
\[\psi_{i}(t+\triangle t,y,z)=\psi_{i}(t,y,z)\exp\left(-\frac{\pi\triangle t}{ 4T}\right)+\phi_{i}(t,y,z)\left[1-\exp\left(-\frac{\pi\triangle t}{2T}\right) \right]^{0.5} \tag{14}\]  
where \(\phi_{i}\) is obtained using a new set of random data; however, \(\psi_{i}\) is the plane velocity field data of the current or next time step and \(T\) is a constant value for the entire inlet plane representing the Eulerian time scale. Note that \(\phi_{i}\) is fully random in time, its variance is unity, and it is an independent variable from \(\psi_{i}(t,y,z)\).