## 4 Results
### Global Forces  
The mean values of the lift and drag coefficients were extracted from the simulations. In addition, we carried out a Fourier analysis for the lift and drag forces on the wings to determine the unsteady characteristics of especially the sheet-cavitation shedding. The results are shown in Table 2 for all investigated cases. We observed a dominant frequency of around 30 Hz for the case with and \(\sigma=0.53\). We show an example of the time histories and corresponding frequency contents in Figure 10 obtained from DDES, and in Figure 11 obtained from the SST simulations. A frequency of 29.0 Hz for both the lift and drag coefficients was seen with coarse and medium grids when using the SST model. A dominant frequency of 29.0 Hz was observed with coarse grid DDES for both the lift and drag coefficients, while a dominant frequency of 33.4 Hz was observed for lift and 35.0 Hz was observed for the drag with medium grid DDES. Animations from fine and medium grid DDES and SST simulations \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & \(\overline{C}_{\mathcal{L}}\) & \(\overline{C}_{\mathcal{D}}\) & **Lift: \(f_{0}\) (Hz)** & **Drag: \(f_{0}\) (Hz)** \\ \hline \(U_{\infty}=8.0\) m/s, \(\sigma=0.53\) & & & & \\ \hline Left foil: coarse grid, SST & 0.078 & 0.025 & 29.0 & 29.0 \\ Right foil: coarse grid, SST & 0.078 & 0.025 & 29.0 & 29.0 \\ Left foil: coarse grid, DDES & 0.079 & 0.025 & 29.0 & 29.0 \\ Right foil: coarse grid, DDES & 0.079 & 0.025 & 29.0 & 29.0 \\ Left foil: medium grid, SST & 0.071 & 0.023 & 29.0 & 29.0 \\ Right foil: medium grid, SST & 0.071 & 0.023 & 29.0 & 29.0 \\ Left foil: medium grid, DDES & 0.075 & 0.024 & 33.4 & 35.0 \\ Right foil: medium grid, DDES & 0.075 & 0.024 & 33.4 & 35.0 \\ \hline \(U_{\infty}=8.0\) m/s, \(\sigma=1.27\) & & & & \\ \hline Left foil: coarse grid, DDES & 0.110 & 0.019 & - & - \\ Right foil: coarse grid, DDES & 0.109 & 0.019 & - & - \\ Left foil: medium grid, DDES & 0.121 & 0.017 & 160.2 & 160.2 \\ Right foil: medium grid, DDES & 0.120 & 0.017 & 160.2 & 160.2 \\ Left foil: fine grid, DDES & 0.118 & 0.016 & 159.6 & 159.9 \\ Right foil: fine grid, DDES & 0.117 & 0.016 & 159.8 & 160.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the mean lift and drag coefficients (\(C_{\mathcal{L}}\) and \(C_{\mathcal{D}}\), respectively), and their dominant frequencies (\(f_{0}\)). Figure 9: A snapshot of the simulations with \(U_{\infty}=8\) m/s and \(\sigma=1.27\). Fine grid DDES. Top frame presents an overview of the cavity extent with the transparent iso-surface of \(\alpha=0.1\) colored by blue. Lower left frame shows the pressure coefficient on the center \(z\) plane of the cavitator. Lower right frame shows the non-dimensional \(v\)-velocity on the center \(z\) plane of the cavitator. revealed a frequency of approximately 30 Hz for the sheet-cavitation shedding. Comparing the DDES and SST simulations, we see that the shedding frequency increases slightly with DDES on the medium grid. The time histories of the SST simulations appear slightly more regular, which is seen also as the sharper frequency peaks whereas these are more spread when using DDES. If we approximate the reference length of the Strouhal number as the chord length, the Strouhal numbers' range is \(St=fc/U_{\infty}=[0.22\ldots 0.26]\). If we instead take the reference velocity as \(U_{ref}=U_{\infty}\sqrt{1+\sigma}\), the Strouhal numbers would range as \(St=[0.18\ldots 0.21]\). Figure 11: Time history and frequency content of the lift and drag coefficients. Medium grid, \(\sigma=0.53\), SST simulations. Figure 10: Time history and frequency content of the lift and drag coefficients. Medium grid, \(\sigma=0.53\), DDES. For the case with \(\sigma=1.27\) a dominant frequency in between 159.6-160.2 Hz for both the lift and drag coefficients was observed with medium and fine grid DDES. As we discuss later in Section 4.6, the SST model yielded a steady-state solution for this case.