## 4 Computational New Results and Analysis  
### Right-Angled Isosceles Triangular Cavity  
In this section, several new results will be introduced. For each result, a numerical analysis will be followed and discussed on the behalf of fluid characteristics. Figure 3 presents the vortex structure inside a right-angled isosceles triangular cavity as steady Reynolds number increases from 100 to 7000. It is observed that as Reynolds number increases, the main vortex tends to move along the driving direction, which is from left to right. Meanwhile, as Reynolds number increases, the secondary bottom vortex keeps growing and moving toward the left corner, pushing the main vortex toward the right side of the domain. At about Reynolds number 3000, the main vortex has an elongated form, the secondary vortex has become the biggest in size and it is located on the left side of the triangular cavity. Moreover, a third vortex appears at the bottom corner and as Reynolds number keeps increasing from 3000 to 7000, this third vortex grows in size \begin{table}
\begin{tabular}{l l l l l l} \hline Equilateral triangular cavity & Data source & This paper & Ref. [13] & Ref. [11] & Ref. [15] \\  & Primary vortex & X = 0.5442 & X = 0.5401 & X = 0.5499 & X = 0.5266 \\  & & Y = 0.6455 & Y = 0.6527 & Y = 0.6539 & Y = 0.6501 \\ \hline Isosceles triangular cavity & Data source & This paper & Ref. [14] & Ref. [18] & \\  & Primary vortex & X = 0.0465 & X = 0.0448 & X = 0.045 & \\  & & Y = 0.3829 & Y = 0.3816 & Y = 0.38 & \\ \hline \end{tabular}
\end{table} 
Table 1: The position of primary vortices of the lid-driven equilateral and right-angled isosceles triangular cavity at \(Re=500\) and \(Re=100\), respectively. Figure 2: (**a**) Streamlines in the equilateral triangular cavity at \(Re=500\). (**b**) Streamlines in the right-angled isosceles triangular cavity at \(Re=100\). and moves toward the right side of the domain, tending to separate the two previously generated vortices. Accompanied by the consistent rising of Reynolds numbers, the flow is changing from steady to unsteady periodic. According to similar studies on a square cavity flow [1; 2; 3; 4; 5; 6; 7; 8], the critical Reynolds number of the transitional flow, steady to unsteady periodic, was found around 8000. The same phenomenon is obtained for the present right-angled isosceles triangular cavity. In order to determine the critical Reynolds number between steady and unsteady periodic, it is necessary to address the perturbation decay rate \(\gamma\), which is the slope of the logarithmic value of perturbation, defined as \(\epsilon=log(|u_{x}-\bar{U}|)\) for each Reynolds number, where \(\bar{U}\) is the converged value of a steady solution. Figure 4a shows the horizontal velocity history for Reynolds numbers 5000 and 7500 and measured from a probe located at geometric center. As can be seen from Figure 4a, the numerical predictions obtained via using different convergence criteria, \(10^{-6}\) and \(10^{-7}\), are exactly the same. Figure 4b,c describe the perturbation histories at Reynolds numbers 7500 and 8050, respectively. Therefore, it is concluded that the Hopf bifurcation appears at Reynolds numbers between 7500 and 8050. Figure 4d presents the phase map of \(u_{x}\) versus \(u_{y}\) obtained from the probe planted at the cavity center, which is the cross point of the horizontal and vertical central lines, at a Reynolds number 8100. It is observed that the phase trajectory at Reynolds number 8100 is a closed loop, indicating the flow to this stage in our discussion is already unsteady periodic. Table 2 shows the perturbation decay rate of each Reynolds number varying from 100 to 8050. \(t^{\prime}\) is the physical time for each Reynolds number to obtain a converged solution. \(SD_{\gamma}\) is defined as the standard deviation of perturbation decay rate. As the Reynolds number grows and approaches the value of 8000, \(\gamma\) becomes close to zero, which means the flow inside the enclosure is about to be unsteady.