# Embedded One-Dimensional Orifice Elements for Slosh Load Calculations in Volume-Of-Fluid CFD
## 2 Materials and Methods
### Orifice Flow Losses  
Figure 1 shows a schematic representation of the contracting and expanding flow within a long orifice, defined as orifices in which the length \(L\) to hydraulic diameter \(D_{H}\) ratio is greater than 1.4, i.e., \(\frac{L}{D_{H}}>1.4\). The hydraulic diameter is defined as  
\[D_{H}=\frac{4A_{o}}{P_{w}}\, \tag{1}\]  
where \(A_{o}\) is the orifice cross-sectional area and \(P_{w}\) is the wetted perimeter. The point \(x_{u}\) is at the inlet to the orifice and \(x_{d}\) is at the exit to the orifice where the flow is reattached to the orifice wall, as shown. The vena contracta is the location at which the main fluid stream cross-sectional area is smallest, which is at position \(x_{c}\). Flow through orifices up to the vena contracta is nearly reversible [31]. Flow losses mainly occur downstream of the vena contracta with the majority being downstream of the orifice exit in the downstream jet. In the case of short orifices (\(\frac{L}{D_{H}}<1.4\)), the losses within the orifice are negligible [32; 33]. For long orifices (\(\frac{L}{D_{H}}>1.4\)), the flow reattaches to the orifice wall downstream of the vena contracta and, even though the majority of losses still occur downstream of the orifice, significant flow losses occur within the orifice [34].  
This work focuses on modeling the viscous flow losses and corresponding orifice mass flow rate within long orifices for two-phase flow by using one-dimensional, finite volume cells in orifice cavities. The flow outside the orifice is meshed, discretized and solved with the usual numerical discretization. Therefore, we place the onus to account for the lossesoccurring downstream of the orifice on the existing 2D/3D CFD model, as would be the case in a fully resolved simulation.  
The average velocity and pressure over the orifice cross-section (normal to the flow direction) are denoted \(\tilde{u}\) and \(\tilde{p}\), respectively. To these, we add subscripts that refer to the point where the cross-sectional average velocity or pressure is taken (upstream, downstream or at the vena contracta). Note that the orifics considered in this work all have constant cross-sectional areas; therefore, we simply refer to the orifice cross-sectional area, \(A_{o}=A_{u}=A_{d}\). \(A_{c}\) is the cross-sectional area of the vena contracta.  
The pressure loss in one-dimensional flow analyses are generally written in the form [35]  
\[\Delta\tilde{p}=C\rho\tilde{u}^{2}\,, \tag{2}\]  
where \(\rho\) is the density of the fluid and \(C\) is some loss coefficient, determined as a function of the flow conditions and boundary geometry. The effects that contribute to the vast majority of pressure losses through orifics can be placed in three categories. These are viscous losses, flow losses (venturi such as contraction and expansion) and fluid mixing (two-phase flow). These effects are discussed in the following subsections, with a general equation for the loss coefficient given in Section 2.1.4. Note that the entrance losses due to developing flow (as in the case of pipes and ducts) in long orifics can be considered negligible due to the relatively small length-to-diameter ratios [36] and are excluded in this work.