## 4 Computational New Results and Analysis
### Right-Angled Isosceles Trapezoidal Cavity (Bowl Shape)  
In this subsection, a series of right-angled isosceles trapezoidal enclosures (bowl shape) with different top-line/base-line ratios defined as \(\lambda=L_{top}/L_{base}\) are shown in Figure 5. Figure 6 shows the velocity profiles along the vertical and horizontal central lines obtained from different grid resolutions at a steady Reynolds number 1000. The normalized velocity profiles \(u_{x}^{\prime}=u_{x}/U_{lid}\) and \(u_{y}^{\prime}=u_{y}/U_{lid}\) along the vertical and horizontal central lines, respectively, are introduced. It is found that whenever the grid spacing is smaller than 0.00390625, the normalized velocity profiles remain constant, suggesting that a grid spacing of \(\Delta x=0.001667\) is appropriate for steady solutions. Figure 7 illustrates the streamlines and vortical structures for different geometries of the right-angled isosceles trapezoidal cavities at Reynolds number 1000. It is interesting to realize that the following trend appears in all cavities presented in Figure 7. As \(\lambda\) decreases, the main primary vortex keeps moving toward the positive x axis. Meanwhile, it is noticed that the bottom secondary vortex is compressed gradually whenever the length of the base line is outstretched. When the base line extends to a certain value with \(\lambda=5\), the secondary bottom vortex breaks up, generating two subvortices, located near the right and left corner, respectively. With the further extension of the base line \(\lambda=5/3\), these two subvortices keep separating and finally disappear. From Figure 8, it is observed that for each value of \(\lambda\), at \(Re=7000\), three main vortices, two negative and one positive, are almost covering the entire flow domain, in all these cases, a fourth much smaller positive vortex appears on the upper left hand side corner. As soon as \(\lambda\) reaches a value of \(5/3\), the previous existing four vortices collapse and generate two elongated main vortices, a positive and a negative one, which respectively separate the fluid into an upper and lower domain. It is important to realize when comparing Figure 8 with Figure 7 that an increment of Reynolds number promotes the flow complexity inside the domain; this is due to the higher kinetic energy embedded in the fluid. The approximate  
Figure 4: Velocity component \(u_{x}\) versus iteration steps \(t\). (**a**) Steady solutions at \(Re=5000\) and \(Re=7500\). (**b**) Logarithmic value of perturbation at \(Re=7500\). (**c**) Logarithmic value of perturbation at \(Re=8050\). (**d**) Phase map of \(u_{x}\) versus \(u_{y}\) at a periodic Reynolds number 8100.  
value of \(Re^{H}\) for the bowl-shaped trapezoid is roughly predicted for each value of \(\lambda\) through a similar systematic study of perturbation decay rate presented in Table 2. It is found that when \(\lambda\geq 5\), as \(\lambda\) decreases, \(Re^{H}\) decreases. Specifically, \(Re^{H}\) is around 8300, 8200, and 8000 when \(\lambda=17\), 9 and 5, respectively. However, when \(\lambda\) decreases from 3 to 1.5, \(Re^{H}\) keeps increasing from roughly from 18,000 to 24,000. Notice that this phenomenon is similar to authors' previous research on the transitions inside a lid-driven shallow square cavity, where when the aspect ratio (H/L) was 0.1, \(Re^{H}\) was found to be around 19000, indicating the cavity flow tends to be stable as the enclosure is horizontally elongated. Figure 5: Different geometries of right-angled isosceles trapezoidal cavity with the long topline driven. Figure 6: Velocity profiles along the vertical (**a**) and horizontal (**b**) central lines with different resolutions. Figure 9 shows the normalized velocity profiles at Reynolds numbers 1000 and 7000 along the vertical and horizontal central lines for all the geometries studied in this case. In Figure 9a, it is introduced the horizontal velocity distribution along the vertical central line.