# Scale-Adaptive Simulation of Unsteady Cavitation Around a Naca66 Hydrofoil
## 4 Results and Discussion  
For a better understanding of the unsteady cavitation cycles, the total vapor volume, \(V_{\text{cav}}\), was estimated using Equation (17), which is based on studies of [PERSON] et al. [30]. The \(V_{\text{cav}}\) was divided by the volume of the total computational domain, \(V_{\text{domain}}\), due to in fact that \(V_{\text{cav}}\)./\(V_{\text{domain}}\) represents a dimensionless volume that is plotted in Figure 2(a),  
\[V_{\text{cav}.}=\sum_{q=1}^{N}\left[\alpha V\right]_{q}, \tag{17}\]  
where \({}_{q}\) is the subscript for each cell and \(N\) is the total number of cells in the computational domain. Results show three cavitation cycles from 0 to 0.7 s for both turbulence models. However, the \(V_{\text{cav}}\)./\(V_{\text{domain}}\) of ILES is twice higher than \(k-\omega\) SST SAS. This is probably due to the fact that ILES depends on the truncation error of the numerical simulation, which induces false responses in the prediction of \(V_{\text{cav}}\). Finally, the fast Fourier transform was applied to find the cavitation frequency and plotted in Figure 2(b). The highest pulse is observed at 4.01 Hz in both cases, and it matches the  
Figure 1: Geometry of NACA66 and computational domain. Figure 2: Structured grid mesh around a NACA66. frequency of the experimental result obtained by [PERSON] [22]. Moreover, ILES presented additional pulses related to SGS, which may contribute to the large computational resources demanded by ILES. The third cycle of each case from Figure 3 has been selected and plotted in Figure 4 as function of the dimensionless time, \(\xi\), and it was estimated using Equation (18) for a better comparison among cycles, which is based on previous research [4,6]. \[\xi=\frac{t-t_{\text{o}}}{t_{\text{f}}-t_{\text{o}}}, \tag{18}\]  
where \(t\) is time, \({}_{\text{o}}\) and \({}_{\text{f}}\) are subscripts for initial and final time respectively. Five instants have been selected in Figure 4 and plotted in Figure 5 to understand the unsteady cavitation cycle. Figure 5 shows a typical cavitation cycle for numerical and experimental results with the growth and detachment of the leading edge cavity from 1 to 3, the development of the re-entrant jet and the collapse of the cloud of cavities from 3 to 5. However, the detachment process of cavities is less turbulent for \(k-\omega\) SST SAS than ILES at instants 3 and 4 as indicated in Figure 5a,c, respectively. In this context, the cavitation cycle has been reproduced fairly well with \(k-\omega\) SST SAS, which is according to previous studies carried out by [PERSON] et al. [30] and [PERSON] et al. [4]. Figure 3: The total vapor variation during unsteady cavitation around a NACA6 hydrofoil with \(k-\omega\) SST scale adaptive simulation (SAS) and implicit large eddy simulation (ILES). Further analysis of the cavity detachment is indicated in Figure 6. Results of \(k-\omega\) SST SAS present a more regular detachment process of the cavity than ILES, and they match fairly well experimental results, which were based on Leroux conditions [22], and obtained by [PERSON] at National Key Laboratory on Ship Vibration and Noise of the China Ship Scientific Research Center. Moreover, it is observed that the growth of the leading edge cavity is regular without any detachment at the initial growth for \(k-\omega\) SST SAS. However, the ILES result shows several cavity detachments at the start of the growth of the leading edge cavity, which is related to the SGS and the truncation error [10; 24]. For validation of results, the pressure coefficient, \(C_{\mathrm{P}}\), has been estimated using Equation (19) for four points, which were located on the hydrofoil upper surface and plotted with bar errors as indicated in Figure 7.  
\[C_{\mathrm{P}}=\frac{p_{\mathrm{s}}-p_{\infty}}{0.5 pU_{\infty}^{2}}, \tag{19}\]  
where \(p_{\mathrm{s}}\) is the static pressure. Figure 4: Selected instants at the third cycle of both cases. Figure 5: Typical cavitation cycle and comparison among results of NACA66 with iso–contour of \(\alpha=0.1\).