## 2 Numerical Methods
### Integrated AITD  
As shown in Figure 2a,b, the integrated AITD, which was designed based on the original ITD of a turbofan engine, includes the aggressive intermediate-turbine duct and the integrated LPT-GV. For the AITD, its inlet aspect ratio (\(\rm{R_{hub}/R_{tip}}\)) was 0.656, and the inlet channel height (H) was 85 mm. The nondimensional duct length (L/H, ITD axial length/inlet annulus height) was 1.97, and the mean rise angle (\(\rm{\theta}\)) was 28.08\({}^{\circ}\) with an outlet-to-inlet area ratio (AR) of 1.34. Based on the above parameters, the red dot in Figure 3 shows the comparison between the design parameters of this AITD and those of other turbofan engines, where cp\({}^{\star}\) is the best AR Line of ITD for the given L/\(\rm{h_{in}}\), and cp\({}^{\star\star}\) is the best L/\(\rm{h_{in}}\) line for the given AR. The design parameters of the conventional ITD are generally located between these two lines, while the ITD (with small L/\(\rm{h_{in}}\) and large AR) studied in this paper is marked by the red dot on the left side of the cp\({}^{\star}\) line, which indicates that it is in the category of an AITD. The design parameters in Table 1 of the integrated LPT-GV refer to the prototype guide vane of the turbofan engine with its inlet and outlet metal angles unchanged. Figure 2b shows the geometric model of the integrated LPT-GV and the blade profiles for sections in the spanwise direction. To decouple the sweeping wakes from other secondary flows, such as the tip leakage flow, sweeping rods were used to simulate the wakes of HPT rotors to isolate the effects of the sweeping wakes. Previous studies have demonstrated that when the flow resistance of the rod is the same as that of the blade, the structure of the rod's wake is the same as that of the blade [34]. As shown in Figure 2, the sweeping rod was arranged 0.4H upstream of the AITD inlet, and its diameter (\(d\)) was calculated according to the measured airfoil loss (\(Y\)) of the HPT rotor by the following equation:  
\[Y = C_{d}\frac{d}{S_{b}}Z\bigg{[}1-\bigg{(}0.25C_{d}\frac{d}{S_{b}}Z-1 \bigg{)}\bigg{(}Z^{2}-1\bigg{)}\bigg{]} \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Chord**} & \multirow{2}{*}{\begin{tabular}{c} **Leading-** \\ **Edge** \\ **Radius** \\ **(mm)** \\ \end{tabular} } & \begin{tabular}{c} **Training-** \\ **Edge** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Stagger** \\ **Edge** \\ **(mm)** \\ \end{tabular} & \
\begin{tabular}{c} **Inlet Metal** \\ **(mm)** \\ \end{tabular} &
\begin{tabular}{c} **Outlet** \\ **Angle** \\ **(****)** \\ \end{tabular} & \multirow{2}{*}{**Tmax/Chord**} & \multirow{2}{*}{**Throad/Pitch**} & \multirow{2}{*}{**Solidity**} \\ \cline{1-1} \cline{6-10}  & & & & & & & & & \\ \hline Root & 74.5 & 3.5 & 2.5 & \(35^{\circ}\) & \(-6^{\circ}\) & \(62^{\circ}\) & 0.200 & 0.444 & 1.348 \\ Midspan & 94.1 & 3.5 & 2.5 & \(35^{\circ}\) & \(-6^{\circ}\) & \(62^{\circ}\) & 0.197 & 0.427 & 1.457 \\ Tip & 112.8 & 3.5 & 2.5 & \(35^{\circ}\) & \(-6^{\circ}\) & \(62^{\circ}\) & 0.194 & 0.410 & 1.566 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters of the integrated LPT-GV. Figure 3: ITD design parameters of some turbofan engines [30]. Figure 2: Cross section of the integrated AITD and measurement locations. where \(C_{d}\) represents the resistance coefficient of the rod (approximately 1.05 in the range of Re values studied in this paper), \(S_{b}\) is the rod spacing, and \(Z\) is the cosecant of the relative flow angle at the HPT outlet. Under the design conditions, the diameter of the sweeping rod is 4.6 mm (\(d/s=0.10\)) with a rotating speed of 220 RPM, which is equivalent to a reduced frequency of \(f=f_{HPT}C_{x}/u_{e}=1.07\). To study the wake effect of the HPT rotor with different trailing-edge radii, this paper also carried out unsteady simulations for \(d/s\) values of 0.056, 0.078, 0.122 and 0.143.