# Novel Fluidic Oscillator Evaluation Considering Dimensional Modifications
## 2 Governing Equations and Turbulence Model  
Under incompressible flow conditions, the continuity and momentum equations, Navier-Stokes (NS) equations, can be presented as  
\[\frac{\partial u_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{\partial x_{j}}= -\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\
u\frac{\partial^{2}u_{i}}{ \partial x_{j}\partial x_{j}} \tag{2}\]  
As in the present application the flow inside the FO is expected to be turbulent, the use of unsteady Reynolds-averaged Navier-Stokes equations (URANS) as a turbulence model approach is a good option, particularly when time or computational constraints are considered. The accuracy of the RANS and URANS turbulence models is not the same as in the LES or DNS approaches, but the computational time required to perform a simulation reduces enormously [43; 44; 45]. It is interesting to recall that the variables associated with the NS equations are the three velocity components and the pressure; therefore, the discretization of the NS equations under incompressible flow conditions and in order employs a URANS turbulence model; this requires the substitution of each variable by an average (\(\bar{\phi}\)) and a fluctuation (\(\phi^{\prime}\)) term. Calling \(\phi\) a generic variable, it can be stated that  
\[\phi=\bar{\phi}+\phi^{\prime} \tag{3}\]  
After performing the mentioned substitution and considering the fluid as incompressible, the resulting NS equations take the form  
\[\left(\frac{\partial\bar{u}_{x}}{\partial x}+\frac{\partial\bar{u}_{y}}{ \partial y}+\frac{\partial\bar{u}_{z}}{\partial z}\right)=0 \tag{4}\]  
\[ \begin{split}\frac{\partial\bar{u}_{x}}{\partial t}+\bar{u}_{x} \frac{\partial\bar{u}_{x}}{\partial x}+\bar{u}_{y}\frac{\partial\bar{u}_{x}}{ \partial y}+\bar{u}_{z}\frac{\partial\bar{u}_{x}}{\partial z}=-\frac{1}{\rho} \frac{\partial p}{\partial x}+g_{x}+\frac{1}{\rho}\frac{\partial}{\partial x }\bigg{(}2\mu\frac{\partial\bar{u}_{x}}{\partial x}-\rho(\bar{u}_{x}^{\prime })^{2}\bigg{)}\\ +\frac{1}{\rho}\frac{\partial}{\partial y}\bigg{(}\mu\bigg{(} \frac{\partial\bar{u}_{x}}{\partial y}+\frac{\partial\bar{u}_{y}}{\partial x }\bigg{)}-\rho\bar{u}_{x}^{\prime}u_{y}^{\prime}\bigg{)}+\frac{1}{\rho}\frac{ \partial}{\partial z}\bigg{(}\mu\bigg{(}\frac{\partial\bar{u}_{x}}{\partial z }+\frac{\partial\bar{u}_{z}}{\partial x}\bigg{)}-\rho\bar{u}_{x}^{\prime}u_{ z}^{\prime}\bigg{)}\end{split}  \tag{5}\]\[ \begin{split}\frac{\partial\bar{u}_{y}}{\partial t}+\bar{u}_{x} \frac{\partial\bar{u}_{y}}{\partial x}+\bar{u}_{y}\frac{\partial\bar{u}_{y}}{ \partial y}+\bar{u}_{z}\frac{\partial\bar{u}_{y}}{\partial z}=-\frac{1}{\rho} \frac{\partial p}{\partial y}+g_{y}+\frac{1}{\rho}\frac{\partial}{\partial x} \bigg{(}\mu\bigg{(}\frac{\partial\bar{u}_{y}}{\partial x}+\frac{\partial\bar{ u}_{x}}{\partial y}\bigg{)}-\rho\overline{u^{\prime}_{y}u^{\prime}_{x}} \bigg{)}\\ +\frac{1}{\rho}\frac{\partial}{\partial y}\bigg{(}2\mu\frac{ \partial u_{y}}{\partial y}-\rho(\bar{u^{\prime}_{y}})^{2}\bigg{)}+\frac{1}{ \rho}\frac{\partial}{\partial z}\bigg{(}\mu\bigg{(}\frac{\partial u_{y}}{ \partial z}+\frac{\partial\bar{u}_{z}}{\partial y}\bigg{)}-\rho\overline{u^{ \prime}_{y}u^{\prime}_{z}}\bigg{)}\end{split}  \tag{6}\] \[ \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}=-\frac{1}{\rho} \frac{\partial p}{\partial z}+g_{z}+\frac{1}{\rho}\frac{\partial}{\partial x} \bigg{(}\mu\bigg{(}\frac{\partial\bar{u}_{z}}{\partial x}+\frac{\partial\bar{ u}_{x}}{\partial z}\bigg{)}-\rho\overline{u^{\prime}_{z}u^{\prime}_{x}}\bigg{)}\\ +\frac{1}{\rho}\frac{\partial}{\partial y}\bigg{(}\mu\bigg{(} \frac{\partial\bar{u}_{z}}{\partial y}+\frac{\partial\bar{u}_{y}}{\partial z }\bigg{)}-\rho\overline{u^{\prime}_{z}u^{\prime}_{y}}\bigg{)}+\frac{1}{\rho} \frac{\partial}{\partial z}\bigg{(}2\mu\frac{\partial\bar{u}_{z}}{\partial z }-\rho(\bar{u^{\prime}_{z}})^{2}\bigg{)}\end{split}