# Numerical Study of Partial Dam-Break Flow with Arbitrary Dam Gate Location Using VOF Method
## 2 Numerical Model  
In this work, the widespread commercial software Flow 3D is selected to deal with hydrodynamic problems. Navier Stokes equations are solved by the volume of fluid (VOF) method [33]. An assumption of stress in the fluids is the sum of a diffusing viscous term and a pressure term is used to describe the viscous flow. For the incompressible flow of a Newton fluid, the Navier Stokes equations can be expressed in three orthogonal Cartesian coordinates as:  
\[\frac{\partial\overline{u}_{i}}{\partial x_{j}}=0 \tag{1}\]The conservation of momentum:  
\[\frac{\partial\overline{u}_{i}}{\partial t}+\frac{\partial\overline{u}_{i} \overline{u}_{j}}{\partial x_{j}}=\frac{1}{\rho}\frac{\partial\overline{p}}{ \partial x_{i}}+\frac{\partial}{\partial x_{i}}\left[\frac{\mu}{\rho}\left( \frac{\partial\overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline{u}_{j }}{\partial x_{i}}\right)\right]+\frac{\partial\left(\tau_{ij}/\rho\right)}{ \partial x_{j}} \tag{2}\]  
where \(\overline{u}_{i}\); \(\overline{u}_{j}\) account for Reynolds-averaged velocities; \(x_{i}\) and \(x_{j}\) are Cartesian coordinate axes; \(\mu\) is the dynamic viscosity of the fluid; \(t\) is time; \(\rho\) is fluid density; \(\overline{p}\) is Reynolds-averaged pressure; \(\tau_{ij}=\rho\left(\overline{u_{i}u_{j}}-\overline{u}_{i}\overline{u}_{j}\right)\) represents Reynolds stress. In the presented model, computational domain can be discretized by structured rectangular mesh, which uses the FAVOR technique (Fractional Area/Volume Obstacle Representation) [33]. With improved mesh refinement, the solid boundary is better represented. The VOF is a surface-tracking technique, which is often used in tracking of liquid-gas interface. FLOW-3D uses the volume of fluid method (VOF) to determine the location of fluid within the computational mesh for problems with a variable free surface. Denoting \(A\) as the fraction of fluids, if the cell is empty the value of A is set equal to 0; if the cell is full, \(A\) is 1.0 and if the interface of two liquids or liquid-gas cuts the cell, \(A\) is greater than 0 and smaller than 1.0. According to [PERSON] and [PERSON] [34] the equation of the volume fraction can be written as:  
\[\frac{\partial A}{\partial t}+\
abla(\mathbf{V}A)=0 \tag{3}\]  
where \(\mathbf{V}\) is the vector of fluid velocity. The VOF consists of three parts: a scheme to describe the shape and location of the fluid surface; a method to track the movement of the fluid surface; and a means for applying boundary conditions at the fluid surface. In the Flow 3D model, the numerical scheme solves Navier Stokes equations on a structured grid. Fluid advection is defaulted as automatic to select the best-fit option, depending on the number of fluids and presence of a sharp interface. A first order upwind method is utilized to solve momentum advection. Explicit schemes are implemented for viscous stress, free surface pressure and advection terms. For incompressible flows, pressure forces in the momentum equation are always implicitly approximated, enforcing incompressibility and maintaining stability of the numerical solution. Viscous stresses in fluid are explicitly approximated. This produces a simple and efficient algorithm that requires a limit on time step size to maintain stability of the numerical solution [33]. The major challenge of the CFD model is to simulate turbulent flow. The reason it is a challenge is because one of the main characteristics of this flow is fluctuation in velocity distribution, which causes mixing of transported quantities, like momentum and energy. As for turbulence modeling, dam-break flow can be formulated by three turbulent models: Reynold-Averaged Navier Stokes equations (RANs), the Large Eddy Simulation model (LES) and the 2D shallow water model. The equations are filtered to separate large eddies from small ones in the LES approach [35; 36]. It is considered that the largest eddies interact strongly with the mean flow.