## 4 Discussion
### SETAI Behavior under Asymmetric Combustion Condition  
Table 8 presents the ratio of \(L_{\mathrm{P}}\) to \(\lambda_{\mathrm{P}}\) on each side and the measured \(P_{\mathrm{RMS}}\) for two different asymmetric combustion conditions, namely cases 66 and 67. In case 67, both OS\({}_{1}\) and OS\({}_{2}\) promote oscillation, but neither satisfies the requirement for promoting oscillation in case 66. This observation aligns with the findings illustrated in Figure 7, and it supports the notion that the occurrence of oscillation in an asymmetric condition is not determined by the overall \(\varphi\) but is dependent on the specific condition (combination of \(L_{\mathrm{P}}/\lambda_{\mathrm{P}}\) and \(\varphi\)) on each side. Under an asymmetric combustion condition where \(\varphi\) varies between the two sides, the value of \(\tau\) relies on the local \(\varphi\), rather than the global \(\varphi\). This is logical since the heat release at each flame should be controlled by the local \(\varphi\), rather than being influenced by the total \(\varphi\)[24, 26]. Consequently, the heat release oscillation differs between the two sides, even when there is an identical flow rate oscillation. This discrepancy leads to an unsynchronized effect on the pressure experienced on both sides. It can be hypothesized that altering \(\varphi\) on each side, even with a consistent total \(\varphi\), may impact the oscillation characteristic of the system, similar to adopting an asymmetric structure for air/fuel supply. To validate this hypothesis, experiments were conducted under symmetric supply structures (\(L_{1}=L_{2}=1.85\) m) but with asymmetric combustion conditions, as detailed in Table 1e. Figure 10 illustrates the observed oscillation characteristic, while Table 9 analyzes the respective effects of OS\({}_{1}\) and OS\({}_{2}\). Previous experiments have demonstrated that a \(\varphi\) range of 0.7-0.8 and 0.9-1.4, respectively, needs to be combined with an \(L_{\mathrm{P}}/\lambda_{\mathrm{P}}\) range of \((2n/4,\,(2n+1)/4)\) and \((n/4,\,2n/4)\) in order to generate a promotion effect. Among cases 68-73, where \(L_{\mathrm{P}}\) is 1.85 m, \(\varphi\) values of 0.9, 1.1, and 1.3 satisfy this requirement. Consequently, both sides have a promotion effect in cases 72 and 73, which is consistent with the observations in Figure 10 and Table 7, where strong oscillation occurs under these two cases. In cases where OS\({}_{2}\) promotes oscillation but OS\({}_{1}\) suppresses it, weak oscillation was observed in case 69, while non-oscillation occurs in cases 70 and 71. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\(\varphi_{1}=\varphi_{2}\)} & \multirow{2}{*}{\(f_{\mathrm{M}}\) (Hz)} & \multicolumn{3}{c}{**Premixed Chamber 1**} & \multicolumn{3}{c}{**Premixed Chamber 2**} & \multirow{2}{*}{\(P_{\mathrm{RMS}}\) (Pa)} \\ \cline{3-3} \cline{5-8}  & & & \(L_{1}\mu\lambda_{\mathrm{P}}\) & **Effects on Oscillation** & & \(L_{2}\mu\lambda_{\mathrm{P}}\) & **Effects on Oscillation** \\ \hline
64 & 0.9 & 75.4 & 0.377 & Promotion & 0.883 & Promotion & 2145 \\
65 & 0.7 & 71.2 & 0.616 & Promotion & 0.118 & Promotion & 2198 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Ratio of \(L_{\mathrm{P}}\) to \(\lambda_{\mathrm{P}}\) on each side and the \(P_{\mathrm{RMS}}\) under cases 64 and 65. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\(P_{1}\)} & \multirow{2}{*}{\(P_{2}\)} & \multicolumn{3}{c}{**Premixed Chamber 1**} & \multicolumn{3}{c}{**Premixed Chamber 2**} & \multirow{2}{*}{\(P_{\mathrm{RMS}}\) (Pa)} \\ \cline{3-3} \cline{5-8}  & & & \(L_{1}\mu\lambda_{\mathrm{P}}\) & & **Effects on Oscillation** & & \(L_{2}\mu\lambda_{\mathrm{P}}\) & **Effects on Oscillation** \\ \hline
64 & \(-147.5\) & 24.0 & 25.0 & 35.0 & 0.0 & 3.1 & 1.6 & 2145 \\
65 & 162.2 & \(-17.6\) & \(-18.8\) & \(-20.5\) & 0.0 & 2.5 & 1.6 & 2198 \\ \hline \hline \end{tabular}
\end{table}