## 5 Conclusions  
In the present paper, we performed a systematic study on the lid-driven right-angled isosceles triangular and trapezoidal cavity flow for steady solutions. We focused on the vortical structure and vortex evolution inside different geometries. Regarding the triangular chamber, steady solutions at Reynolds numbers from 100 to 8100 are presented and discussed. We found that, at low Reynolds number \(Re\leq 1000\), the domain is dominated by a large primary vortex with a tiny eddy observed at bottom corner. As Reynolds number increases, this primary vortex is squeezed by the tiny eddy located at the bottom corner and gradually elongates as the bottom eddy grows until it breaks into two secondary vortices. Meanwhile, a series of tiny eddies start to show up adjacent to the corners. Since the previous primary vortex is broken, the domain is now covered by several secondary vortices. By investigating the perturbation decay rate, \(Re^{H}\) is obtained at a Reynolds number between 7950 and 8050, at which unsteadiness shows up. Considering the trapezoidal cavity, we designed a geometric parameter \(\lambda\) to distinguish different shapes of the cavity. Simulations are performed at steady Reynolds number 1000 and 7000. For the bowl-shaped trapezoid with \(\lambda\geq 2\), as Reynolds number increases from 1000 to 7000, we found a similar trend of flow patterns observed in the triangular cavity, in which the primary vortex is squeezed and finally breaks into secondary vortices. As \(\lambda\) decreases further, it is observed that, regardless of the Reynolds number tested the enclosure is always dominated by primary vortices with large size, no tiny eddies are observed, indicating the trapezoid with smaller value of \(\lambda\) is more stable than that of larger value of \(\lambda\). The unsteadiness shows up accompanied with the appearance of tiny vortical structures. It is found that as \(\lambda\) decreases from 17 to 5, \(Re^{H}\) decreases roughly from 8300 to 8000. However, when \(\lambda\) decreases from 3 to 1.5, \(Re^{H}\) keeps increasing from roughly from 18,000 to 24,000, indicating the cavity flow tends to be incredibly stable as the enclosure is horizontally elongated. As to the pyramid-shaped trapezoid, regardless of the Reynolds number tested, the Moffatt's phenomenon is obviously observed around corners and the domain is always dominated by the primary central vortex. We found that when \(\lambda\leq 1/5\), the vortical structures remains barely the same when Reynolds number increases from 1000 to 7000, indicating the flow is quite stable. This is further observed by studying the perturbation decay rate, we discovered that the critical Reynolds number of Hopf bifurcation for \(\lambda\leq 1/5\), is around 20,000, much bigger than that of the triangular or square cavity flow. When \(\lambda\) increases to 1/3, as Reynolds number increases from 1000 to 7000, we can observe a series of secondary vortices and small scale eddies. 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.