# Novel Fluidic Oscillator Evaluation Considering Dimensional Modifications
## 2 Governing Equations and Turbulence Model
 \tag{7}\]  
RANS and URANS turbulence models solve the Navier-Stokes equations via implementing the concept of turbulence viscosity (\(\mu_{t}\)), also called apparent viscosity, which has to be substituted into the three terms of the momentum equation just introduced; for this, the Boussinesq hypothesis is used, which reads  
\[ \begin{split}\mu_{t}=\frac{-\rho\overline{u^{\prime}_{x}u^{\prime }_{x}}}{2\frac{\partial\bar{u}_{x}}{\partial x}}=\frac{-\rho\overline{u^{\prime }_{y}u^{\prime}_{y}}}{2\frac{\partial\bar{u}_{y}}{\partial y}}=\frac{-\rho \overline{u^{\prime}_{z}u^{\prime}_{z}u^{\prime}_{y}}}{2\frac{\partial\bar{u} _{z}}{\partial z}}=\frac{-\rho\overline{u^{\prime}_{x}u^{\prime}_{y}}}{\frac{ \partial\bar{u}_{x}}{\partial y}+\frac{\partial\bar{u}_{y}}{\partial x}}=\frac{ -\rho\overline{u^{\prime}_{x}u^{\prime}_{z}}}{\frac{\partial\bar{u}_{y}}{ \partial z}+\frac{\partial\bar{u}_{z}}{\partial y}}=\frac{-\rho\overline{u^{ \prime}_{y}u^{\prime}_{z}}}{\frac{\partial\bar{u}_{y}}{\partial z}+\frac{ \partial\bar{u}_{z}}{\partial y}}\end{split}  \tag{8}\]  
Usually, the nine terms characterizing the time-averaged fluctuation are put together in a tensor named the apparent Reynolds stress tensor (\(\tau_{app}\)), which is a \(3\times 3\) symmetric matrix; this is why we have represented the turbulent viscosity considering six equations. After the substitution, the three components of the momentum equation take the form  
\[ \begin{split}\frac{\partial\bar{u}_{x}}{\partial t}+\bar{u}_{x} \frac{\partial\bar{u}_{x}}{\partial x}+\bar{u}_{y}\frac{\partial\bar{u}_{x}}{ \partial y}+\bar{u}_{z}\frac{\partial\bar{u}_{x}}{\partial z}=g_{x}-\frac{1}{ \rho}\frac{\partial p}{\partial x}+\frac{1}{\rho}\frac{\partial}{\partial x} \bigg{(}2(\mu+\mu_{t})\frac{\partial\bar{u}_{x}}{\partial x}\bigg{)}\\ +\frac{1}{\rho}\frac{\partial}{\partial y}\bigg{(}(\mu+\mu_{t}) \bigg{(}\frac{\partial\bar{u}_{x}}{\partial y}+\frac{\partial\bar{u}_{y}}{ \partial x}\bigg{)}\bigg{)}+\frac{1}{\rho}\frac{\partial}{\partial z}\bigg{(}( \mu+\mu_{t})\bigg{(}\frac{\partial\bar{u}_{x}}{\partial z}+\frac{\partial\bar{ u}_{z}}{\partial x}\bigg{)}\bigg{)}\end{split}  \tag{9}\]  
\[ \begin{split}\frac{\partial\bar{u}_{y}}{\partial t}+\bar{u}_{x} \frac{\partial\bar{u}_{y}}{\partial x}+\bar{u}_{y}\frac{\partial\bar{u}_{y}}{ \partial y}+\bar{u}_{z}\frac{\partial\bar{u}_{y}}{\partial z}=g_{y}-\frac{1}{ \rho}\frac{\partial p}{\partial y}+\frac{1}{\rho}\frac{\partial}{\partial x} \bigg{(}(\mu+\mu_{t})\bigg{(}\frac{\partial\bar{u}_{y}}{\partial x}+\frac{ \partial\bar{u}_{x}}{\partial y}\bigg{)}\bigg{)}\\ +\frac{1}{\rho}\frac{\partial}{\partial y}\bigg{(}2(\mu+\mu_{t}) \frac{\partial\bar{u}_{y}}{\partial y}\bigg{)}+\frac{1}{\rho}\frac{\partial}{ \partial z}\bigg{(}(\mu+\mu_{t})\bigg{(}\frac{\partial\bar{u}_{y}}{\partial z}+ \frac{\partial\bar{u}_{z}}{\partial y}\bigg{)}\bigg{)}\end{split}  \tag{10}\]  
\[ \begin{split}\frac{\partial\bar{u}_{z}}{\partial t}+\bar{u}_{x} \frac{\partial\bar{u}_{z}}{\partial x}+\bar{u}_{y}\frac{\partial\bar{u}_{z}}{ \partial y}+\bar{u}_{z}\frac{\partial\bar{u}_{z}}{\partial z}=g_{z}-\frac{1}{ \rho}\frac{\partial p}{\partial z}+\frac{1}{\rho}\frac{\partial}{\partial x} \bigg{(}(\mu+\mu_{t})\bigg{(}\frac{\partial\bar{u}_{z}}{\partial x}+\frac{ \partial\bar{u}_{x}}{\partial z}\bigg{)}\bigg{)}\\ +\frac{1}{\rho}\frac{\partial}{\partial y}\bigg{(}(\mu+\mu_{t}) \bigg{(}\frac{\partial\bar{u}_{z}}{\partial y}+\frac{\partial\bar{u}_{y}}{ \partial z}\bigg{)}\bigg{)}+\frac{1}{\rho}\frac{\partial}{\partial z}\bigg{(}2( \mu+\mu_{t})\frac{\partial\bar{u}_{z}}{\partial z}\bigg{)}\end{split}  \tag{11}\]  
The variables in the resulting momentum equations are the time-averaged pressure, the three time-averaged velocity components and the turbulence viscosity; to close the system, a new equation is therefore needed.