# Analysis of a Novel Fluidic Oscillator under Several Dimensional Modifications
## 3 Mesh Assessment
In the present study, the flow was considered as turbulent, incompressible, and isothermal, and all simulations were two-dimensional. The fluid used was air at 15 degrees Celsius, and the dynamic viscosity and the density were \(\mu=1.813\times 10^{-5}\) kg/(m s) and \(\rho=1.225\) kg/m\({}^{3}\), respectively. The software OpenFOAM version 5.0 was employed for all simulations; the finite volume method is the approach OpenFOAM uses to discretize Navier-Stokes equations. Inlet turbulence intensity was set to 0.1% in all cases, pressure implicit with splitting operators (PISO) was used to solve the Navier-Stokes equations, the timestep being \(4\times 10^{-7}\) s, and spatial discretization was set to second order. Simulations were not considered to be converged until the residuals for the fluid velocity components, the turbulent kinetic energy, the specific dissipation rate, and the turbulent kinematic viscosity were to the order of \(10^{-6}\) or \(10^{-7}\). The residuals for the pressure were to the order of \(10^{-4}\) after all simulations. In order to further check the accuracy obtained with the different mesh densities, we decided to analyze some dynamic characteristics and introduce them in Figure 3. The FO outlet mass flow pulsating frequency is presented in Figure 3a. The instantaneous outlet mass flow and the lower feedback channel mass flow are shown in Figure 3b and \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Mesh Assessment**} \\ \hline
**2D Mesh** & **Min y+** & **Average y+** & **Max y+** & **Frequency** & **Error \%** \\
**Cells** & & & & & \\ \hline
62,143 cells & 0.007893928 & 5.255150 & 63.09918 & 24.525 Hz & 15.37\% \\
180,065 cells & 0.5630300 & 1.616285 & 16.79723 & 29.181 Hz & 0.69\% \\
367,720 cells & 0.2748158 & 0.9271403 & 8.044681 & 29.091 Hz & 0.38\% \\
718,920 cells & 0.2741867 & 0.7549317 & 8.368831 & 28.98 Hz & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results from the mesh independence study. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Boundary Conditions 2D-CFD**} \\ \cline{2-5}  & **k (m\({}^{2}\)/s\({}^{2}\))** & **Omega (s\({}^{-1}\))** & **Nut (m\({}^{2}\)/s)** & **P (Pa)** & **U (m/s)** \\ \hline inlet & \(1.5\cdot 10^{-4}\) & 0.152 & \(9.89\cdot 10^{-4}\) & zeroGradient & 10 \\ \hline outlet & zeroGradient & zeroGradient & zeroGradient & \(10^{5}\) & zeroGradient \\ \hline Top \& Bottom & empty & empty & empty & empty & empty \\ walls & \(10^{-20}\) & omegaWallFunction & nutWallFunction & zeroGradient & 0 \\  & & value \(=10^{-5}\) & value \(=10^{-7}\) & zeroGradient & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the boundary conditions for the standard case (STD), also called the baseline case. Reynolds number = 54,595. Figure 3c, respectively. The unsteady stagnation pressure measured at the mixing chamber lower converging wall is introduced in Figure 3d. Regardless of the graph chosen, it is observed that the dynamic results obtained with the meshes of 367,720 and 718,920 cells are almost identical, and clear differences are seen when using lower-resolution meshes. At this point, we can conclude that the mesh of 367,720 cells is perfectly good to perform the required simulations. At this point, and in order to evaluate the effect of the inlet turbulence intensity on the results obtained, the mass flow at the FO outlet width and at the FC lower outlet were studied for three different FO inlet turbulence intensities, 0.01%, 0.1% and 1%; the results are presented in Figure 4. The mesh used had 367,720 cells and the Reynolds number was kept constant to \(Re=54\),595. As the turbulence intensity increases, the curves slightly drift towards one side, but minor differences are observed when comparing the evaluated minimum and maximum turbulence intensities. We can conclude that using a turbulence intensity of 0.1% at the FO inlet is perfectly correct.