# Simulation of compressible multi-phase flow phenomena using Discontinuous Galerkin methods

A broad variety of technical applications requires the numerical simulation of fluid flows. Here, the prevailing method is the so called Finite Volume Method (FVM) in which flow characteristics (like the components of the velocity vector) are represented by one constant value in each cell of the numerical grid. Even though this method serves very well in many cases, it still has several limitations concerning, for example, the extension to higher approximation orders (which greatly improves accuracy per degree of freedom) and flexibility (especially for complex or even deforming domains)

# Discontinuous Galerkin methods

In contrast to the traditional FVM approach, the Discontinuous Galerkin (DG) method introduces a local polynomial representation of characteristic flow quantities in every cell. The recovered global solution thus allows, as the name of the method implies, jumps at interfaces between neighboring cells. This extended scheme offers great opportunities in terms of

• accuracy (since the local polynomial order can be adapted according to an estimation of the error)
• parallelizability (since the amount of communication does depend on the polynomial order)
• flexibility (since neighboring cells are only coupled weakly)

The mentioned advantages come at the price of a higher overall computational effort but, especially for large scale problems, the excellent parallel scaling properties still allow the efficient solution without increasing the overall computation time significantly.

# Compressible multi-phase flows

Many fluids (in particular, gases) can significantly change their density within technically relevant flow configurations, which is why the so called compressible Navier-Stokes equations have to be considered in such cases. If the configuration additionally contains at least two immiscible fluids (like air and water or water and vapor) one speaks of a multi-phase flow.

One important example of a compressible multi-phase flow is the formation and implosion of so called cavitation bubbles. These bubbles start to form when the pressure locally falls below the vapor pressure of the fluid. When these bubbles are driven towards a region of higher pressure, they will implode violently while creating a small but very fierce water jet which can e.g. damage parts of propellers, pumps and valves.

# Outlook

The ultimate goal of our research efforts is the implementation of a robust solver for compressible multi-phase flows using a level set approach. The implementation is based on the Discontinuous Galerkin framework BoSSS which has been initiated by Florian Kummer and is under constant development at the chair of fluid dynamics (fdy). Its main features are the object-oriented design (which makes it very easy to adapt and extend its capabilities) and its wide range of applications.

# Key Research Areas

Discontinuous Galerkin methods, multi-phase flows, multi-physics, level set methods, numerical integration of discontinuous/singular integrands