# Novel Fluidic Oscillator Evaluation Considering Dimensional Modifications
## 6 Results  
### Results from the 3D-CFD Simulation  
To start understanding the evolution of the jet inside the MC and at the FO outlet, it was decided to divide a full oscillation cycle into eight equidistant time steps and present the corresponding velocity and pressure fields in Figure 4. The oscillation period was chosen to start at the instant at which the pressure at the MC's upper converging wall was a maximum, as clearly observed in Figure 4b. At this instant (T = 0), the upper FC is pressurized and the jet inside the MC is being pushed downwards. At the next time step (T = 1/8), the jet inside the MC is about to reach its lower position, some reverse FC flow (flow going from the FC outlet towards the FC inlet) is observed at the lower FC, Figure 4c, and the pressure at the upper FC is slightly higher than at the lower one Figure 4d. Moving forward to T = 1/4, it is observed that the pressure at the MC's lower converging wall starts rising and keeps rising until reaching T = 1/2; see Figure 4f,h,j. This stagnation pressure increase is due to the jet impinging on the MC's lower converging wall, as observed from Figure 4e,g,i. In fact, and due to the pressure rise at the lower FC, the jet inside the MC starts moving upwards at time T = 1/4 and keeps moving upwards until T = 5/8; notice that at time T = 3/4 (Figure 4n), the pressure at the MC's upper converging wall is clearly \begin{table}
\begin{tabular}{l l l l l} \hline \hline  & \multicolumn{4}{c}{**Inlet Angles Evaluated**} \\ \hline Inlet angle \(\delta\) & \(\delta_{1}=\delta_{STD}+15^{\circ}\) & \(\delta_{2}=\delta_{STD}\) & \(\delta_{3}=\delta_{STD}-10^{\circ}\) & \(\delta_{4}=\delta_{STD}-15^{\circ}\) \\ \hline MC entrance width (m) & 0.022 & 0.0255 & 0.032 & 0.04 \\ \hline Inlet section name & Entrance 1 & \(STD\) = Entrance 2 & Entrance 3 & Entrance 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The different mixing chamber inlet angles \(\delta\) considered in the present study. Figure 3: Dimensional parametric modifications considered in the present study. Mixing chamber angle modification (**a**). Feedback channel dimensions modification (**b**). \begin{table}
\begin{tabular}{l l l l l} \hline \hline  & \multicolumn{4}{c}{**Feedback Channel Dimensions Evaluated**} \\ \hline FC modifications & \(W_{1}=(1/3)W_{STD}\) & \(W_{2}=(2/3)W_{STD}\) & \(\begin{array}{c}W_{5}=W_{STD}=\\ (K_{ext}-R_{int})\end{array}\) & \(W_{4}=(4/3)W_{STD}\) \\ \hline FC width (m) & 0.010733 & 0.021466 & 0.0322 & 0.04299 \\ \hline FC name & FC 1 & FC 2 & \(STD=FC3\) & FC 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The different feedback channel modifications considered in the present study. much higher than that at the MC's lower converging wall, which pressurizes the upper FC and pushes the jet inside the MC downwards. This process continues until reaching T = 1/4, where the jet at the MC reaches its lower position and the lower FC again begins to be pressurized. It is interesting to note that the maximum velocity exiting the FO is obtained at the instants where the pressure at the MC's upper and lower converging walls is around the respective maximum value; see Figure 4a,b; o,p; g,h and i,j. In order to further understand the FO's internal 3D vortical structure formation, evolution, and dimensions, the time-averaged Q-criterion for a value of \(2\times 10^{6}\) is computed at each of the eight equidistant time steps into which the full oscillation period is divided. The Q-criterion is conventionally defined as the second invariant of the velocity gradient tensor, and it is empirically expressed as \((1/2)(||\Omega||^{2}-||S||^{2})\), where \(\Omega\) is the rotation rate or vorticity tensor and S is the strain rate tensor. The iso-contours of the 3D structures are depicted in Figure 5. When comparing Figures 4a,c with Figure 5a,c, the FC reverse flow can now be more clearly seen. In fact, the vortical structures forming alternatively on both sides of the MC are now clearly observed in all figures thanks to the Q-criterion plots.