## 4 Computational New Results and Analysis
### Right-Angled Isosceles Trapezoidal Cavity (Pyramid Shape)  
The different studied geometries of the pyramid-shaped right-angled isosceles trapezoidal cavities are presented in Figure 10, where the value of \(\lambda\) sweeps from \(1/17\) to \(1/3\). Figure 11 introduces the streamlines inside each geometry at Reynolds number at 1000. It is found that the changes of configuration has barely an impact on the secondary vortex located near the acute angles. Yet, the primary main vortex changes substantially as \(\lambda\) increases. It is observed that there is an infinite sequence of eddies, four distinct recirculations, adjacent to the sharp stagnant corners, as that proved in [PERSON]'s [44] investigation. For the present pyramid-shaped trapezoidal cavity, the stream lines obtained at Reynolds 7000 are presented in Figure 12. When comparing Figure 12 with Figure 11, it is observed that, as Reynolds number increases, the main vortex located at the geometry center, remains almost circular regardless of the value of \(\lambda\). Notice that in Figure 12, the main vortex is tending to have a rounded form as \(\lambda\) increases, the number of secondary vortices appearing also suffer an increase. In fact, the original number of vortices, increases to 7 whenever \(\lambda\) is \(1/3\), indicating that the number of vortices is likely to keep increasing with the increment of Reynolds number. In any case, the infinite series of eddies appearing at the sharp stagnant corners, will remain for all values of \(\lambda\) and Reynolds numbers studied, although observed at a high Reynolds numbers. In the same way, the value of \(Re^{H}\) for the pyramid-shaped trapezoid is predicted for each value of \(\lambda\). It is found that as \(\lambda\) increases, \(Re^{H}\) decreases from 23,000 to 17,000, having a huge drop to about 7400 when \(\lambda=1/3\). This observation suggests that the flow inside the pyramid-shaped trapezoid with a pointy head (smaller distance of top line) is way more stable than that with a flat (larger distance of top line) head. When \(\lambda\) increases to a certain value (\(\lambda=1/3\)), this strong stability disappears and flow recovers to a normal level of stability. A bit unstable though when compared with the classic lid-driven square cavity flow, at which \(Re^{H}\) is around 8000. Figure 11: Streamlines inside the pyramid-shaped trapezoid with various values of \(\lambda\) at \(Re=1000\). Figure 10: Different geometries of pyramid-shaped right-angled isosceles trapezoidal cavities. Figure 13 introduces the velocity profiles along the vertical and horizontal central lines for different \(\lambda\) configurations. Figure 13a shows that the main vortex turning speed increases with \(\lambda\) increase. The velocity magnitude appears to be rather constant for the two Reynolds numbers studied; this may be due to the fact that in the simulations, the different Reynolds numbers were obtained via maintaining the lid velocity and the geometry, and the fluid viscosity was the changing parameter. Compared with Figure 13a, when Reynolds number increases from 1000 to 7000, for \(\lambda<1/3\), the velocity profiles presented in Figure 13b barely changes, which is in good agreement with the observation from flow topologies presented in Figure 12a-c. However, for \(\lambda=1/3\), large oscillations of velocity profiles along the axes are observed, which is caused by the secondary vortices shown in Figure 12d. Based on Figure 13a,b, it is further proved that the flow inside a pyramid-shaped trapezoid with a pointy head is quite stable, because as can be seen from the figures, the normalized velocity profiles of \(u_{x}^{\prime}\) and \(u_{y}^{\prime}\) when \(\lambda<1/3\) nearly remain the same for both Reynolds numbers 1000 and 7000. In Table 4, which is focusing on the primary vortex location, stream function and vorticity at the vortex center are presented for different values of \(\lambda\) at steady Reynolds numbers 1000 and 7000.