# Analysis of a Novel Fluidic Oscillator under Several Dimensional Modifications
## 2 Governing Equations and Turbulence Model  
In Computational Fluid Dynamics (CFD) simulations and when considering the fluid as incompressible, Navier-Stokes (NS) equations take the form:  
\[\frac{\partial u_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{\partial x_{j}}=- \frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\
u\frac{\partial^{2}u_{i}}{ \partial x_{j}\partial x_{j}} \tag{2}\]  
If the fluid is considered as turbulent, several approaches to solve NS equations are possible. Ideally, Direct Numerical Simulation (DNS) should be employed; the main problem associated with DNS is the extremely small mesh cells and time steps required. Then, the Kolmogorov length and time scales should be reached to properly detect the energy dissipation range. In the vast majority of the applications, the mentioned drawbacks make it impossible to use DNS simply because the computational time is far too large. The second approach is the use of Large Eddy Simulation (LES) as a turbulence model, it is a very good option in 3D flows, yet it still requires very fine meshes and large computational times. LES requires a Subgrid-Scale Model (SGS) which is used in areas where the vortical structures are smaller than the dimension of the mesh cells. Presently, LES turbulence modeling is extensively used although the computational time required, despite the fact that supercomputers are being used, is still too large for many applications. As a result, the vast majority of the nowadays CFD applications still need to be performed using Reynolds- Averaged Navier-Stokes (RANS) or URANS (unsteady RANS) turbulence models. Their precision is not as accurate as LES or DNS models, but the computational time needed shortens drastically, which allows the performance of 2D-CFD simulations with a reasonable degree of accuracy. To discretize the NS equations under incompressible flow conditions, it needs to be kept in mind that the only variables associated with the NS equations are the pressure and the three velocity components. In order to be able to apply URANS models, each variable from the NS equations (we will call \(\phi\) any generic variable) needs to be substituted by its average \(\bar{\phi}\) and a fluctuation \(\phi^{\prime}\) term. \[\phi=\bar{\phi}+\phi^{\prime} \tag{3}\]  
Once the mentioned substitution is performed, the resulting NS equations take the form (here just the equations for two-dimensional CFD models are presented):  
\[\frac{\partial\rho}{\partial t}+\rho\bigg{(}\frac{\partial\bar{u}_{x}}{ \partial x}+\frac{\partial\bar{u}_{y}}{\partial y}\bigg{)}=0 \tag{4}\]  
\[ \begin{split}\frac{\partial\bar{u}_{x}}{\partial t}+\bar{u}_{x} \frac{\partial\bar{u}_{x}}{\partial x}+\bar{u}_{y}\frac{\partial\bar{u}_{x}}{ \partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+g_{x}+\frac{1}{\rho} \frac{\partial}{\partial x}\bigg{(}2\mu\frac{\partial\bar{u}_{x}}{\partial x }-\rho(\bar{u}_{x}^{\prime})^{2}\bigg{)}\\ +\frac{1}{\rho}\frac{\partial}{\partial y}\bigg{(}\mu\bigg{(} \frac{\partial\bar{u}_{x}}{\partial y}+\frac{\partial\bar{u}_{y}}{\partial x }\bigg{)}-\rho\bar{u}_{x}^{\prime}u_{y}^{\prime}\bigg{)}\end{split}  \tag{5}\]  
\[ \begin{split}\frac{\partial\bar{u}_{y}}{\partial t}+\bar{u}_{x} \frac{\partial\bar{u}_{y}}{\partial x}+\bar{u}_{y}\frac{\partial\bar{u}_{y}}{ \partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial y}+g_{y}+\frac{1}{\rho} \frac{\partial}{\partial x}\bigg{(}\mu\bigg{(}\frac{\partial\bar{u}_{y}}{ \partial x}+\frac{\partial\bar{u}_{x}}{\partial y}\bigg{)}-\rho\bar{u}_{y}^{ \prime}u_{x}^{\prime}\bigg{)}\\ +\frac{1}{\rho}\frac{\partial}{\partial y}\bigg{(}2\mu\frac{ \partial\bar{u}_{y}}{\partial y}-\rho(\bar{u}_{y}^{\prime})^{2}\bigg{)}\end{split}