# Numerical methods

Common Numerical methods:

# Moving grids

The principle of moving grid method is to relocate grid points from a fixed number of nodes to nodes concentrating in the regions of rapid variation of velocity, pressure, etc. For its application in multiphase flow, this method track the interface by grid nodes on the interface and move interface grid nodes by Lagrangian transport.

• Represent phase interface by grid nodes on the interface.

• Move interface grid nodes by Lagrangian transport:

⇒ It can result in large grid deformations which will make re-griding necessary.

• Successful for small interface deformations.

• Topology changes difficult

• Difficult normal interface movement (phase change).

# Marker Particles

The particle method is a mesh less method. Material to be simulated in form of individual particles. It is applicable for low phase fractions. For example, solid particles in air (eg. coal particle combustion) or fluid droplets in air (eg. diesel spray combustion). The limitation is that numbers of drops can be huge, which would cause large computational cost. Or drops are typically confined to relatively small regions of whole computational domain.

• It tracks phase interface by Lagrangian marker particles in a fixed grid.

• ​Phase interface can be reconstructed by polynomials through neighboring marker particles.
⇒ phase interface geometry is very accurate (normal, curvature).
⇒need to keep connectivity information of markers.
⇒topology changes are difficult.

• Normal interface movement (phase change) is challenging.

• ​The method does provides sub-grid phase interface resolution.

# VOF method

It is a surface-tracking technique applied to a fixed Eulerian mesh where the Navier Stokes equations which describe the motion of the flow have to be solved separately.

The method is based on the solution of a transport equation for variable ‘C’ (often also referred as indicator or color function) for the liquid phase.

Cij represents the portion of the area of the cell (i, j) filled with liquid phase  and the phase function χ :

We have 0 < C < 1 in cells cut by the interface S and C = 0 or 1 away from it.

The VOF method doesn't explicitly track the interface, it reconstructs the interface based on calculate the volume fraction of fluid . The Color Function also cannot be solved easily. There are already several method to approach, different approach gives different accuracy,  the most popular is PLIC (Piecewise Linear Interface Calculate). In a 3D space, the interface can be described by nx+ny+nz=a.

nx+ny+nz

# Level Sets

It is a tracking interfaces technique that makes computations on a fix grid without having to parameterize the interface. A new dimension is introduced to the case and define the interface as a level set of function G(x,y) which represent the minimum distance from each point to the interface.