## 3 Results and Discussion
### Proper Orthogonal Decomposition (POD) Results
Indeed, for the fully confined case, when the jet was attached to one side, there were some dominated patterns that portrayed oscillating behavior and the frequency of those oscillatory patterns was a bit larger than the free external domain, because the oscillating area was reduced, as shown in Figure 18. One compelling point about the POD modes was the frequency of the third mode for the fully confined was unrelated to the first two modes, because as mentioned in Figure 15, it still had a significant percentage of energy. However, the frequency of the fourth POD mode was twice that of the first two modes, which illustrated that the jet passed through this mode twice during each cycle. This relation for the free external domain could be observed for the first three modes that the frequency of the third mode was twice of the first two modes. Figure 16: Spatial patterns of the POD modes for the both free and fully external domains at \(\mathrm{Re}=10\),730. Figure 17: Scatter distribution of time coefficients \(a_{1}(t)\) vs. \(a_{2}(t)\) at \(\mathrm{Re}=6440\). To substantiate the oscillatory pattern and large-scale coherent structures for both free and fully confined external domain, the phase identifications were calculated and depicted in Figures 19 and 20. Phase \(\theta~{}=~{}0^{\circ}\) for the free external domain described the flow when the jet inside the fluidic oscillator passed through the mixing chamber and attached to one of the converging exit's wall. Thus, the phase \(\theta~{}=~{}180^{\circ}\) represents the jet while attached to the opposite converging exit's wall and almost mirrored \(\theta~{}=~{}0^{\circ}\) around the centerline. Phases \(\theta~{}=~{}90^{\circ}\) and \(\theta~{}=~{}270^{\circ}\) represent the jet while it reached its maximum deflection. However, for the fully confined domain case, there was no symmetric behavior of the jet around the centerline, or in its oscillating domain at the small area. Indeed, from phase \(0^{\circ}\) to \(180^{\circ}\), the vortex was pushed from the wall near the exit of the fluidic oscillator, subsequently at \(\theta~{}=~{}270^{\circ}\), it shrunk, due to an increase in the pressure drop and increment in the size of the vortex at the right side of the jet. In addition, the jet almost reached maximum deflection within its oscillating area, at \(\theta~{}=~{}270^{\circ}\). Table 2 shows that the frequencies of temporal coefficients for both the free and fully confined external domains are of the same order. Since inside the fluidic oscillator was the charge of the oscillating pattern for the free external domain, we named the cycle of confined geometry, listed in the Table 2, as a \"fluidic oscillator cycle\" with high frequency. Eventually, when the jet impinged to one of the sidewalls, the left-side vortex could not be further shrunk and the pressure drop began to reduce at the left side, which led to detach the jet and push it to the opposite sidewall and this oscillating pattern was re-initiated. However, the frequency of this cycle was much smaller than the \"fluidic oscillator cycle\", subsequently, it was named as \"cavity cycle\" with low frequency. Indeed, based on the experiment, the frequency of the \"cavity cycle\" was around 0.001046 [Hz]. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Free External Domain**} & **Fully Confined** \\ \hline \(a_{1}(t)\) & 0.3968 & 0.4883 \\ \hline \(a_{2}(t)\) & 0.3968 & 0.4883 \\ \hline \(a_{3}(t)\) & 0.824 & 0.2442 \\ \hline \(a_{4}(t)\) & 0.824 & 0.9156 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Frequency [Hz] of time coefficient for both the fully confined and free external domain at \(\text{Re}=6440\). Figure 18: Corresponding POD time coefficient \(a_{i}(t)\) for both the free and fully confined external domain at \(\text{Re}=6440\); (**a**) free (**b**) fully confined.