## 4 Flow around the 30P30N Three-Element High-Lift Airfoil Configuration
### Simulation Result by Wall-Modelled Navier-Stokes Simulation  
The isosurface of the second invariant of the velocity gradient tensor (Q-criterion) around the 30P30N high-lift airfoil configuration at the angle of attack \(\theta=5.5^{\circ}\) is shown in Figure 6.  
Vortices are shed from the trailing edge of the slat (Figure 6a), the main element, and the flap. The intermittent vortices of the outer layer of the turbulent boundary layer are generated on the suction side of the main element (Figure 6b) and flap. Figure 7 shows the distribution of the velocity component \(u_{x}\) at the nearest cell from the airfoil surface and an \(xy\)-plane cross-section of the fine grid simulation, providing a close-up view at  
Figure 5: Computational grid around a 30P30N high-lift airfoil configuration. (**a**) Grid in all the computational domain; (**b**) Computational grid around the trailing edge of the slat and the leading edge of the main element. Figure 6: Q-criterion around a 30P30N high-lift airfoil configuration when \(\theta=5.5^{\circ}\). (**a**) Wake of the slat; (**b**) Vortices of the outer layer of the turbulent boundary layer on the suction side of the main element. \(x/c\sim 0.15\). The dark blue area in Figure 7 shows the negative value of the velocity component, which indicates that the leading-edge separation bubble forms at \(x/c\sim 0.15\). The location of the separation bubble is consistent with the experiment [24]. This separation bubble formed in the fine grid simulation but not in the medium and coarse grid simulations. Figures 7(a) and 9 show the instantaneous and time-averaged distribution of the velocity component \(u_{x}\), respectively, around the flap when \(\theta=5.5^{\circ}\). Flow separation occurs on the suction side of the upstream side of the trailing edge of the flap in the instantaneous flow field as shown in Figure 7(a). It does not reattach before the trailing edge of the flap. In contrast with Figure 7(b), the flow does not separate in the time-averaged flow field (Figure 9). This indicates that the separation near the trailing edge of the flap occurs intermittently. The lift coefficients in this study and the \(\theta\)-sweep of that in the experiment in Reference [23] are shown in Figure 10.  
Figure 8: Instantaneous distribution of velocity component \(u_{x}\) around the flap when \(\theta=5.5^{\circ}\). Figure 7: The leading edge separation bubble around \(x/c=0.15\) in the fine grid simulation; color shows velocity component \(u_{x}\) at the nearest cell from the airfoil surface and an \(x,y\) cross-section. The Cartesian cut-cell simulation with the wall-stress model accurately predicts the lift coefficient. The lift coefficient of the Cartesian cut-cell simulation with the wall-stress model, the boundary-fitted mesh simulation [25], the hybrid mesh simulation [26], and the experiment [23] are shown in Table 4. The hybrid mesh in Reference [26] is composed of the boundary fit mesh in the near-field region and the quadtree-based Cartesian mesh in the far-field region. Compared with the experiment, the Cartesian cut-cell simulation with the wall-stress model underestimates the lift coefficient. The difference from the experiment decreases monotonically with increasing grid resolution. However, the difference decreases slower than expected accuracy in the formulation. Figure 10: Lift coefficient of the 30P30N high-lift airfoil configuration. Figure 9: Time-averaged distribution of velocity component \(u_{x}\) by the wall-modelled Navier–Stokes simulation around the flap when \(\theta=5.5^{\circ}\). The pressure coefficient in the case of \(\theta=5.5^{\circ}\) is shown in Figure 11.  
The surface pressure distribution at the pressure side of this study coincides with the experiment and referenced simulations. It indicates that the surface pressure distribution at the pressure side is independent of the grid type and the near-wall treatment.