# Computational Fluid Dynamics

CFD provides numerical approximation to the equations that govern fluid motion. Application of the CFD to analyze a fluid problem requires the following steps. First, the mathematical equations describing the fluid flow are written. These are usually a set of partial differential equations. These equations are then discretized to produce a numerical analogue of the equations. The domain is then divided into small grids or elements. Finally, the initial conditions and the boundary conditions of the specific problem are used to solve these equations. The solution method can be direct or iterative. In addition, certain control parameters are used to control the convergence, stability, and accuracy of the method.

### FREE SURFACE FLOWS

Free surface flows refer to flows where there is an interface between a gas and a liquid with a large density difference.Due to a low density, the inertia of the gas is usually negligible, and the only influence of the gas is its pressure acting on the interface. The liquid can thus move freely, and the locations of the free surface must be determined as part of the solution process.

One method to model a free surface is to construct a Lagrangian grid that moves with the fluid. The Lagrangian method has the advantage of tracking the free surface automatically, but fails to model flows with large amplitude surface motions due to serious deterioration of the mesh. The Arbitrary Lagrangian Eulerian (ALE) method is thus introduced with a proper regridding techniques. Still, these methods can not successfully handle complex free surface flows involving surface folding and merging. One of the early methods that can handle surface folding and merging is the Marker-and-Cell (MAC) method. The MAC method involves a Lagrangian particle movement and an Eulerian flow field calculation. The marker is used to track the centroid of the fluid element. Its velocity is obtained by averaging the Eulerian velocities in its vicinity. This method, however, suffers from an artificial creation of high or low marker number densities due to the irregularity of the flow field, such as in the converging and diverging flows and stagnation flows.

Based on a concept of a fractional volume of fluid, the volume of fluid (VOF) method is developed to follow the free surface with the volume-tracking feature of the MAC method but without its large memory and CPU costs. In this approach, only one storage word is required for each element (or cell), namely, the volume fraction, f. Consider an Eulerian structured grid and an actual curved interface cutting through it. Assume that this curve is the free surface of a liquid domain in a 2-D flow field. Therefore, one can define a volume fraction field in this mesh, f, that can take values between 1 and 0. That is, f=0 represents a cell without liquid (empty cell), f=1 a “full cell,” and 0

### VOF BASED ALGORITHMS

We have developed a series of method for the interface advection. These are as follows:

There are three basic approaches involved in FEM and Free Surface Flow. These are:
1) Fixed mesh and free boundary is tracked
2) Deformed spatial mesh using a 3-stage iterative cycle
– Stage 1: Assume shape of free boundary
– Stage 2: BVP solved after discarding 1 BC on free boundary
– Stage 3: SHape of boundary is updated using previously neglected BC
– Cycle is repeated until convergence is achieved
3) Deformed mesh and define nodes on free boundary
– Nodes give extra degrees of freedom
– Field variables and boundary position solved simultaneously using a Newton-Rapson iterative procedure We combine the FEM and VOF techniques because
– FEM solves for the field variables on a deforming boundary
– VOF used to advect the boundary interface

The advantages of this process are:
– Simulation of Large Surface Deformations (i.e. merging and breaking)
– Accurate implementation of boundary conditions
– Increase in computational efficiency