# Analysis of a Novel Fluidic Oscillator under Several Dimensional Modifications
## 2 Governing Equations and Turbulence Model
 \tag{6}\]  
All RANS turbulence models are based on solving the Navier-Stokes equations by incorporating the concept of turbulence viscosity (\(\mu_{t}\)), which can be mathematically addedinto the momentum equations described above using the subsequent differential definition (via the Boussinesq hypothesis):  
\[\mu_{t}=\frac{-\rho\overline{u^{\prime}_{x}u^{\prime}_{x}}}{2\frac{\partial \overline{u}_{x}}{\partial x}}=\frac{-\rho\overline{u^{\prime}_{y}u^{\prime}_{ y}}}{2\frac{\partial\overline{u}_{y}}{\partial y}}=\frac{-\rho\overline{u^{\prime}_{x}u^{ \prime}_{y}}}{\frac{\partial\overline{u}_{x}}{\partial y}+\frac{\partial \overline{u}_{y}}{\partial x}} \tag{7}\]  
In fact, the time-averaged fluctuation terms are usually grouped in a tensor called the apparent Reynolds stress tensor (\(\tau_{app}\)), which is a \(3\times 3\) symmetric matrix. Note that in the case of two-dimensional flow, this matrix explicitly consists of four terms. The X and Y momentum terms of the NS equations can therefore be given as:  
\[\frac{\partial\overline{u}_{x}}{\partial t}+\bar{u}_{x}\frac{\partial \overline{u}_{x}}{\partial x}+\bar{u}_{y}\frac{\partial\overline{u}_{x}}{ \partial y} =g_{x}-\frac{1}{\rho}\frac{\partial p}{\partial x}+\frac{1}{\rho }\frac{\partial\overline{u}_{x}}{\partial x}\bigg{(}2(\mu+\mu_{t})\frac{ \partial\overline{u}_{x}}{\partial x}\bigg{)}+\frac{1}{\rho}\frac{\partial \overline{u}_{y}}{\partial y}\bigg{(}(\mu+\mu_{t})\bigg{(}\frac{\partial \overline{u}_{x}}{\partial y}+\frac{\partial\overline{u}_{y}}{\partial x} \bigg{)}\bigg{)} \tag{8}\] \[\frac{\partial\overline{u}_{y}}{\partial t}+\bar{u}_{x}\frac{ \partial\overline{u}_{y}}{\partial x}+\bar{u}_{y}\frac{\partial\overline{u}_{ y}}{\partial y} =g_{y}-\frac{1}{\rho}\frac{\partial p}{\partial y}+\frac{1}{\rho} \frac{\partial}{\partial x}\bigg{(}(\mu+\mu_{t})\bigg{(}\frac{\partial \overline{u}_{x}}{\partial y}+\frac{\partial\overline{u}_{y}}{\partial x} \bigg{)}\bigg{)}+\frac{1}{\rho}\frac{\partial}{\partial y}\bigg{(}2(\mu+\mu_{ t})\frac{\partial\overline{u}_{y}}{\partial y}\bigg{)} \tag{9}\]  
For the present study, we have decided to use the \(k-\omega\) SST turbulence model, which relies on using the \(k-\omega\) model near the wall, the \(k-\epsilon\) model far away from the object, and a blending function between these two. For the chosen model, the turbulence viscosity \(\mu_{t}\) can mathematically be given as:  
\[\mu_{t}=\frac{\rho k}{\omega}\quad\longrightarrow\quad \begin{cases}\rho:\text {density}\\ k:\text{turbulent kinetic energy}\\ \omega:\text{turbulent kinetic energy specific dissipation rate}\end{cases} \]  
This model employs two transport equations in order to solve \(k\) and \(\omega\). The equations used for each parameter are:  
\[\frac{\partial k}{\partial t}+u_{j}\frac{\partial k}{\partial x_{j}}=P_{k}- \beta^{*}k\omega+\frac{\partial}{\partial x_{j}}\bigg{[}(\
u+\sigma_{k}\
u_{T} )\frac{\partial k}{\partial x_{j}}\bigg{]} \tag{10}\]  
\[\frac{\partial\omega}{\partial t}+u_{j}\frac{\partial\omega}{\partial x_{j}}= \alpha S-\beta\omega^{2}+\frac{\partial}{\partial x_{j}}\bigg{[}(\
u+\sigma_{ \omega}\
u_{T})\frac{\partial\omega}{\partial x_{j}}\bigg{]}+2(1-F_{1})\sigma_{ \omega 2}\frac{1}{\omega}\frac{\partial k}{\partial x_{i}}\frac{\partial \omega}{\partial x_{i}} \tag{11}\]  
\(F_{1}\) and \(F_{2}\) are the blending functions which value depends on the distance of the cell to the wall. The first blending function takes the value of 0 far away from the wall, 1 in the cells close to the wall, and values between 0 and 1 in the transition region. The second blending function \(F_{2}\) depends on the perpendicular distance from the wall (\(d\)) and, according to [35], since the modification to the eddy viscosity has its largest impact in the wake region of the boundary layer, it is imperative that \(F_{2}\) extends further out into the boundary layer than \(F_{1}\).