# Embedded One-Dimensional Orifice Elements for Slosh Load Calculations in Volume-Of-Fluid CFD
## 1 Introduction  
Slosh is a term that often refers to a seemingly chaotic flow of two fluids, usually a liquid and a gas, within a containment structure such as a tank. Common occurrences of slosh are within the fuel tanks of passenger vehicles [1], commercial aircraft [2; 3; 4], spacecraft [5; 6] or in tanker vessels transporting liquefied natural gas [7; 8]. The resulting slosh forces are an important consideration in the structural design of tanks as well as the dynamics and control of the vehicles carrying them [4; 9].  
Due to the inherent non-linearities involved, numerical methods are widely used to study slosh loads. These include Smoothed Particle Hydrodynamics (SPH) [10], Finite Difference Method (FDM) [11], Finite Element Method (FEM) [12] and Boundary Element Method (BEM) [13]. In this work, we focus on the Finite Volume method [14] with a Volume-Of-Fluid (VOF) [15] approach to model two-phase, immiscible fluid flow in the fuel tank slosh loads problem [3].  
Baffle structures are often found inside tanks and serve to suppress the fuel slosh loads [16], to act as structural elements [2] or both. Openings in these structures, referred to generally as orifices in this work, can significantly increase computational cost in numerical simulations due to the small geometrical scales [17; 18; 19; 20; 21; 22]. In addition to small mesh elements, orifices are often the location of high-velocity flow, as fluid is accelerated from one side of the tank to another through a constricting passage. Therefore, the computational cost is increased by both the requirement for more mesh elements and more time steps, since the allowable time step size is decreased in orifices by stability constraints such as the Courant-Friedrichs-Lewy (CFL) condition [23] or viscous diffusion in the case of explicit time integration schemes [24].  
An approach to overcome this cost is to model orifice or channel flow instead of directly discretizing and solving the governing Navier-Stokes equations. [PERSON] [25] developed a subgrid pressure jump model for use in the development of automotive catalytic converters, replacing small monolithic substrates with a subgrid model coupled to the full solution of the Navier-Stokes equations in the diffuser and outlet nozzle. [PERSON] [26] added to the work of [PERSON] by resolving the full Navier-Stokes equations within the inlet section of the monolithic substrate channels while using the pressure loss model for the downstream section (where flow is fully developed). This allows the numerical model to capture the additional losses due to developing flow in the monolithic substrate channels.  
Another approach is the coupling of 3D CFD meshes with 1D network codes [27], in which the 1D code is used to model flow in long tubes while the 3D CFD code models the remainder of the flow domain. These methods require a coupling mechanism to communicate between the network code and 3D CFD code.  
In this work, we further the methodology of [PERSON] [25] and [PERSON] [26] for constant cross-sectional orifices. In our method, we propose to embed the orifice element as part of the mesh. This allows the user to simply modify the mesh in orifice areas by drastically reducing the amount of mesh elements in these regions. A single equation is solved for momentum conservation in the entire domain with different treatment of certain terms in the bulk flow and 1D flow within the orifice. We further expand the 1D model to account for two-phase loss effects by applying existing semi-empirical correlations for flow losses. Another novel aspect of this work is giving careful attention to the VOF advection method to ensure sharp tracking of the liquid-gas interface within the 1D orifice elements. The development and validation of the above is performed in the multi-physics CFD code Elemental(r)[19; 28; 29; 30], which is developed by the Industrial CFD (InCFD) Research Group at the University of Cape Town.