# Methods for direct numerical simulation of multiphase flow

If numerical simulations of three-phase flow are still a challenge in the CFD field and the purpose of lots of research groups, differents methods already exist.

Three main classes of methods for direct numerical simulations of multiphase flow exists:

**The Multiphase particle-in-cell method (mixed Eulerian-Lagrangian method)**

- All particles are assumed to be spherical. If corrections for non-spherical particles can be included in particle drag model, true interactions may therefore not be well represented.
- The size of particles must be small compared to the Eulerian grid for accurate interpolation.
- The major problem of this method is that only a low volume fraction of the dispersed second phase is acceptable. This last consideration makes the model inappropriate for the modeling of liquid-liquid mixtures, fluidized beds, or any application where the volume fraction of the second phase is not negligible.

**Moving grids methods (Lagragian methods)**

The interface is a boundary between two subdomains of the grid. The method is mainly used to follow the motion of a rising bubble, small amplitude waves or weakly deformed bubble. However, when the points move simply in a Lagrangian way, the grid may deform considerably and in case of large deformations of the interface or topology changes, the grid has to be remeshed. This complex method is only sucessful and very accurate for small interface deformation, which is not the case here.

**Fixed-grid methods (Eulerian methods)**

**The marker method**

The advantage of the method is that it allows the capture of details of interface motion on scales much smaller than the grid spacing (Sub grid resolution) and a high degree of accuracy that may be achieved by representing the interface through high-order interpolation polynomials.

**The Level Set method**

**The VOF method (Volume of Fluid)**

$$\vec{n_x}+\vec{n_y}+\vec{n_z}=\alpha$$

where n is the normal vector to the interface and $\alpha$ is a constant line.

$\alpha$ can be solved by root-finding method or analytical formulas $\alpha=\alpha(C)$ and $\vec{n}$ has several approaches such as Parker and Yong's method, Least-squarts method etc.

**Advantages of the VOF method**

The VOF's use, reliability and effectiveness are widespread: the method has been known for several decades, has gone through a continuous process of improvement and is used by many commercially available software programs.

Moreover, the volume conservation is good and no special provision is necessary to perform reconnection or breakup of the interface as the change of topology is implicit in the algorithm. The VOF method is easy to extend to 3D of space and simple to implement.

Applications of the VOF model include stratified flows or the steady or transient tracking of any liquid-gas interface, which correspond to our case.

**Limitations of the VOF method**

The simplification of nonlinear terms and the fact that high order terms are omitted after discretization can lead to less accurate solutions in some cases. Moreover, C<0 or C>1 is possible, which is not relevant with the physic. Another limit of the method is the transitional region between the phases which has to be at least equal to the grid distance.

If the interface geometry reconstruction is challenging, normal interface movements are not straightforwards. Last but not the least, the VOF method involves massive calculations and data burden, which leads to important CPU times.