# Analysis of a Novel Fluidic Oscillator under Several Dimensional Modifications
## 5 Concept of Momentum, Pressure, and Mass Flow Terms Acting on the Jet Entering the Mixing Chamber  
To be able to evaluate the lateral forces pushing the jet as it enters the mixing chamber, the momentum (M) terms on the FCs' outlets are determined. Momentum is characterized by two terms: the pressure and the mass flow (notice that the units of each term are Newtons (N)). Both terms were instantaneously evaluated at the feedback channels upper and lower outlets, see Figure 1, the equation considered takes the form:  
\[F=\dot{m}_{out}\times V_{out}+P_{f}\times S_{out}=\dot{m}_{out}^{2}/\left(S_{ out}\times\rho\right)+P_{f}\times S_{out} \tag{14}\]  
where \(\dot{m}_{out}\), \(V_{out}\), \(S_{out}\) and \(P_{f}\), respectively, are the FC outlet instantaneous mass flow defined as \(\dot{m}=\int_{s}\rho\sqrt{ds}\), the spatial- averaged fluid velocity, the FC outlet surface, and the spatial-averaged pressure instantaneously appearing at any of the FC outlets. \(\rho\) is the fluid density. The momentum considered was the one acting on the vertical direction. The net momentum acting on the jet entering the mixing chamber (MC) is obtained when considering the vertical forces defined by Equation (14) acting instantaneously on both FCs' outlets. In this study, the fluid is considered as incompressible, and fluid density is time- and spatial-independent, but fluid velocity and pressure are spatial- and temporal-dependent. Therefore, to maximize the precision of the calculations, the instantaneous momentum equation to be applied at each FC outlet was discretized and evaluated at each grid cell belonging to the corresponding surface. The resulting equation reads as the following:  
\[F=F_{massflow}+F_{pressure}=\rho\sum_{i=1}^{i=n}(S_{i}V_{i}^{2})+\sum_{i=1}^{i=n }P_{i}S_{i} \tag{15}\]  
The subindex \(i\) denotes any mesh cell belonging to the FC outlet surface. The term \(n\) defines the total number of cells corresponding to any of the two FC outlet surfaces. \(P_{i}\) and \(V_{i}\) characterize the instantaneous pressure and velocity components acting on the corresponding mesh cell in vertical direction. In the present paper, the instantaneous momentum term due to the mass flow was obtained simply by adding the elementary momentum terms of each mesh cell belonging to the chosen surface. The momentum pressure term was obtained when multiplying the instantaneous static pressure acting on each cell by the cell surface and then adding the elemental momentum pressure terms  
\begin{table}
\begin{tabular}{l l l l} \hline \hline  & \multicolumn{4}{c}{**Mixing Chamber Inlet Angles**} \\ \hline Inlet angle in \% & \(\alpha_{1}=79.61\%a_{2}\) & \(\alpha_{2}=\left(Baseline\right)\) & \(\alpha_{3}=122.55\%a_{2}\) \\ \hline Inlet angle in (degrees) & 11.9402 & 14.9968 & 18.3788 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The different mixing chamber angles considered in the present study.  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline  & \multicolumn{6}{c}{**Outlet Widths Evaluated**} \\ \hline Outlet width in \% & \(\frac{D_{2}}{75\%D_{4}}\) & \(\frac{D_{3}}{87.5\%D_{4}}\) & \(\frac{D_{4}}{\left(Baseline\right)}\) & \(\frac{D_{5}}{112.5\%D_{4}}\) & \(\frac{D_{6}}{125\%D_{4}}\) & \(\frac{D_{7}}{150\%D_{4}}\) \\ \hline Outlet width in (m) & 0.02 & 0.02335 & 0.02666 & 0.03 & 0.03333 & 0.03999 \\ \hline Outlet section name & 1 & 2 & \(STD=3\) & 4 & 5 & 6 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The different mixing chamber outlet widths considered in the present study.  
corresponding to the surface under study. Only the momentum in the vertical direction was considered in all cases.