## 4 Model Description
### Generation of Turbulent Inflow, Introduction into the Computational Domain and Boundary Conditions  
In the homogeneous case, the use of the Mann method requires adjusting two parameters to produce the turbulence with the demanded characteristics: the lengthscale \(L\) and TI. The latter is normally controlled by means of varying the coefficients \(\alpha\varepsilon^{2/3}\) (\(\alpha\) is the Kolmogorov constant) of the von Karman energy spectrum [22] until the desired TI is achieved. As it is not straightforward to give an exact relationship between \(\varepsilon\) and the generated TI, the procedure suggested by [46] is used. Instead of changing \(\varepsilon\), a scaling factor \(SF=\sqrt{\sigma_{target}^{2}/\sigma^{2}}\), is used, where \(\sigma_{target}^{2}\) is the desired average variance of the turbulence and \(\sigma^{2}\) is the variance of the turbulence field in each direction. In this way, the desired TI can be obtained by multiplying \(SF\) by each velocity component of the turbulence box. It is expected that when the HIT field is convected at a uniform velocity, the TI will decay monotonically in the streamwise direction. To estimate the turbulence intensity value that the box (TI\({}_{\text{B}}\)) should have in order to attain the desired TI at the given position, empirical relations obtained from fits to experimental results can be used [37; 39; 40]. However, such relations depend on fitting parameters that are reported to vary within a certain margin. This fact and the expected numerical dissipation cause that TI\({}_{\text{B}}\) cannot be estimated beforehand with precision. Furthermore, the averaged value of TI at the position where turbulence is introduced in the computational domain does not correspond to TI\({}_{\text{B}}\) (due to the interpolations between turbulence planes and the adjustment of the synthetic velocity field to the LES conditions, to be discussed later on). In consequence, some testing was necessary to find the right value. Likewise, empirical relations for the development of \(L_{1}\)[37] are not sufficient to predict the value of \(L_{1,\text{B}}\), so tests were necessary to determine its magnitude. The values found for TI\({}_{\text{B}}\) and \(L_{1,\text{B}}\) are presented in the results section. Boundary conditions are set to replicate the conditions in the wind tunnel. Thus, slip conditions are used for the lateral boundaries, whereas the outlet is set to zero gradient. For the time derivative, a second-order backward scheme is employed. When the disks are introduced, these are located at \(x_{D}\), this is \(x=0.5\) m from the inlet, at the centre of the \(y\)-\(z\) plane. Assuming the [PERSON] hypothesis of equivalent spatial and temporal correlations, the streamwise axis of the turbulence box is assumed equivalent to time. To introduce the synthetic turbulence into the computational domain, cross-sectional planes of the turbulence box are taken for every available longitudinal position and their velocity values are mapped onto the inlet of the computational domain. The procedure is illustrated in Figure 2, where the planes extracted from the synthetic turbulence field are separated by \(\Delta x_{\text{B}}\). As the crosswise locations of the cell centres of the synthetic turbulence do not exactly correspond to those of the computational domain, linear interpolations are used to evaluate the velocity values at \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**LES Domain Size**} & \multicolumn{2}{c}{**20D \(\times\) 5D \(\times\) 5D**} \\ \hline  & Layout & Uniform region \\  & & \(20D\times 3.6D\times 3.6D\) \\ \hline  & LES domain grid & \(1000\times 208\times 208\) cells \\ Case Ti3 & Turbulence box & \(40D\times 5D\times 5D\) \\  & Box grid & \(2048\times 256\times 256\) cells \\ \hline  & LES domain grid & \(500\times 104\times 104\) cells \\ Case Ti12 & Turbulence box & \(80D\times 5D\times 5D\) \\  & Box grid & \(2048\times 128\times 128\) cells \\ \hline \hline \end{tabular}
\end{table}