## 2 Materials and Methods
### Flow 3D FEA Numerical Modelling
#### 2.5.5 Air Entrainment Model  
Air Entrainment Model in FLOW3D is based on the assumption that air entrainment in the free surface will occur. This will occur due to instabilities created by turbulence produced by flow over the complex geometry. Thus, this turbulence can overcome the stabilizing forces, caused by gravity and surface tension. As a result, air with the volume \(\delta\)V may be entrained into the fluid, which can be expressed by the Equations (8)-(10). \[\text{L}_{\text{T}}=\frac{\text{CNU}^{\frac{3}{4}}\text{k}^{\frac{3}{2}}}{ \varepsilon_{\text{T}}} \tag{8}\]  
\[\text{P}_{\text{t}}=\rho_{\text{w}}\text{k}\,;\ \text{P}_{\text{d}}=\rho_{\text {w}}\text{g}_{\text{m}}\text{L}_{\text{T}}+\frac{\sigma_{\text{sur}}}{\text{L} _{\text{T}}} \tag{9}\]  
\[\delta\text{V}=\{\text{k}_{\text{air}}\text{A}_{\text{s}}\bigg{[}\frac{2(\text {P}_{\text{t}}-\text{P}_{\text{d}})}{\rho_{\text{w}}}\bigg{]}^{\frac{1}{2}}\ \text{for}\ \text{P}_{\text{t}}>\text{P}_{\text{d}}\,,\ 0\ \text{for}\ \text{P}_{\text{t}}<\text{P}_{\text{d}}\} \tag{10}\]  
where CNU has a value of 0.09, \(\text{L}_{\text{T}}\) is turbulence scale length, and \(\varepsilon_{\text{T}}\) is turbulent dissipation. \(\rho_{\text{w}}\) is the water density \(\sigma_{\text{sur}}\) is the coefficient of surface tension. \(\text{A}_{\text{s}}\) is the surface area, \(\text{k}_{\text{air}}\) is proportionality constant. \(\delta\text{V}\) is the volume of air entrained per unit time. 2.5.6. Density Evaluation Model  
As the air is entrained air, the density of water became non-uniform. There is the formation of an air-water mixture whose density can be computed as Equation (11). \[\rho_{\rm m}=(1-\rm{C_{a}})\rho_{\rm w}+\rm{C_{a}}\;\rho_{a} \tag{11}\]  
Here, \(\rho_{\rm m}\) is the mixture density while \(\rho_{\rm a}\) is the air density. \(\rm{C_{a}}\) is the air concentration. 2.5.7. Drift Flux Model  
According to [28] phase drag and bubble diameter can be modeled using a drift flux model. According to this, model air particles are dispersed into the air over the continuous water flow. The relative velocity between the velocity of dispersed particles and continuous water flow are considered steady. Equation (12) is referred as air transport. \[\Big{(}\frac{1}{\rho_{\rm w}}-\frac{1}{\rho_{\rm a}}\Big{)}\
abla\rm{P}=\left( \frac{f\rho_{\rm w}+(1-f)\rho_{a}}{(1-f)\rho_{\rm w}\rho_{a}}\right)\times Ku_ {r} \tag{12}\]  
\(\rm{K}\) is cell drag coefficient, \(\rm{f}\) is volume fraction of water, \(\rm{u_{r}}\) shows relative or slip velocity. \(\rm{K}\) can be computed from Equation (13) given below  
\[\rm{K_{p}}=\frac{1}{2}A_{p}\rho_{w}\times\left(C_{d}U_{r}+\frac{12\mu_{w}}{ \rho_{w}\;R_{p}}\right) \tag{13}\]  
\[\rm{K}=\frac{(1-f)}{V_{p}}\big{(}K_{p}\big{)} \tag{14}\]  
\(\rm{K_{p}}\) is single drag coefficient. \(\rm{A_{p}}\) is the cross-sectional area of the bubble. \(\rm{U_{r}}\) Depending on \(\rm{u_{r}}\), \(\rm{u_{w}}\) is dynamic viscosity of water, \(\rm{C_{d}}\) is user defined drag coefficient, and \(\rm{R_{p}}\) denotes bubble diameter. 2.6. General Simulation Setup  
Both models (Models 1 and 2) were constructed in SOLIDWORKS in the form of STL (Stereolithography) 3D file. Numerical solver FLOW 3D can import 3D files from SOLIDWORKS or AUTOCAD 3D to begin the simulation. To account for the turbulence in flow, due to the presence of steps, the RNG K-\(\epsilon\) model was adopted. Different sub models (as discussed in Section 2), such as the VOF model, to capture the free surface; air entrainment model, to capture the formation of bubbles in the flow due to air; density evaluation model, to address the variation in density of the water; and drift flux model, to cope the formation of phase drag, were selected to create a virtual environment. Mesh sizes was adopted in a way to effectively capture all the features of geometry and free surface. Additionally, very fine mesh was avoided keeping in view the computing power. Same boundary conditions were adopted as discussed in Section 2. 2.7.