## 3 Results and Discussion
### Proper Orthogonal Decomposition (POD) Results  
To study the dynamics of the unsteady flow structures, POD was applied to extract major coherent flow structures from an unsteady flow field by extracting the most energetic eigenmodes. The velocity field could be written as:  
\[\mathrm{u}(\mathrm{x},\mathrm{t})\ =\ \overline{\mathrm{u}}+\ \mathrm{u}^{\prime}( \mathrm{x},\mathrm{t}) \tag{4}\]  
\[\mathrm{u}^{\prime}(\mathrm{x},\mathrm{t})\ =\ \sum\
olimits_{\mathrm{i}\ =\ 1}^{ \mathrm{N}}\mathrm{a}_{\mathrm{i}}(\mathrm{t})\phi_{\mathrm{i}}(\mathrm{x}) \tag{5}\]  
where \(\mathrm{u}(\mathrm{x},\mathrm{t})\) is the instantaneous velocity, \(\overline{\mathrm{u}}\) is the time-averaged velocity, and \(\ \mathrm{u}^{\prime}(\mathrm{x},\mathrm{t})\) is the velocity fluctuation that can be shown by the linear combination of the products of POD modes (\(\phi_{\mathrm{i}}(\mathrm{x})\)) and corresponding modal coefficients (\(\mathrm{a}_{\mathrm{i}}(\mathrm{t})\)) for each snapshot. The phase angle for each snapshot was calculated by corresponding time coefficients \(\mathrm{a}_{\mathrm{1}}(\mathrm{t})\) and \(\mathrm{a}_{\mathrm{2}}(\mathrm{t})\) and for obtaining the phase-averaged fields, the phase angle of \(\pm 3^{\circ}\) was achieved by dividing the entire cycle into 120 phase intervals. \[\theta\ =\ \mathrm{archtan}(\frac{\mathrm{a}_{\mathrm{1}}(\mathrm{t})}{ \mathrm{a}_{\mathrm{2}}(\mathrm{t})}) \tag{6}\]  
The POD analysis included 12,000 velocity field snapshots, and the eigenvalues of POD modes converged to stable constants. The convergence method was also used to find the number of POD modes, as [PERSON] et al. [39] proposed. Figure 13 shows the percentage of energy contained in the first 20 modes. The third mode was still energetic at \(\mathrm{Re}=6440\) with around 10% of the energy. In contrast, for the other cases, the first two modes contained a dominant percentage of the total energy and the third mode had less than 5%. On the other hand, for the sweeping jet in the free external domain, as the previous study [25; 27] suggested, the first two modes had a predominant percentage of the energy. Figure 12: (**a**) Momentum flux for both the free and fully confined domains at different Reynolds number, entrainment of fluid for (**b**) free and (**c**) fully confined external domain. For the fully confined geometry, at \(\mathrm{Re}=2680\), the jet was close to the centerline. By increasing the Re number, the values of the non-dimensional velocity components were higher than that of the free external domain. For \(\mathrm{Re}=6440\) and 10,730, as was expected, the high fluctuation areas were on the left and right sides of the cavity, as the jet was attached to those areas and was prevented from periodically oscillating. Figures 17 and 18 show the time coefficients for the free and fully confined cases at \(\mathrm{Re}=6440\). Figure 17 depicts the circle shape for the amplitude relations of the time coefficients. Thus, the first two modes for both external domains had a phase shift \(\pi/2\) and illustrated switching behavior at the outside region. FFT analysis on the temporal time coefficients is listed in Table 2 and showed that the characteristic frequency of time coefficients was identical to the frequency of the sweeping jet for the free external domain (Figure 5), which represents that the first two modes could be aggregated into a pair. Additionally, the distribution of time coefficients revealed that for the fully confined geometry, although the jet could not oscillate as the free external domain case, it oscillated in a small region, while it was inclined toward the sidewall due to the oscillatory behavior at the inside of the fluidic oscillator.