# Mathematical modelling and analysis in fluid mechanics

**Mechanical and thermodynamic laws****Turbulence models****Combustion models****Modeling of multiphase flows****Balance equations on phase interfaces and three-phase contact lines****Evaluations of the entropy principle and the symmetry method**

# Current resaerch topics

The main focus of this PhD-project is the application of new conservation laws for viscid and inviscid helical flows to turbulence. In view of the turbulence application helically symmetric turbulence is placed between the fully 3D turbulence, which determines our classical knowledge of turbulence, and plane 2D turbulence, which is rather distinct. Similar to 3D turbulence, helical turbulence admits a three-dimensional velocity field and, most important, the vortex stretching term in the vorticity transport equation responsible for the generation of small scales can be identified. Nevertheless, helical flows live on a 2D manifold, somewhat analogous to plane 2D turbulence. Employing a simulation code especially designed for simulating helical flows in the BoSSS framework we intend to answer exactly the key question in how far helical turbulence is closer to 2D or to 3D turbulence or in other words if helical flows have a preferential to transfer energy from small to large scales or vice versa.

Contact: Dominik Dierkes, M.Sc.

We investigate the problem of aerodynamic sound generation and propagation in modeled engineering shear flows. To date, neither phenomenon is fully understood, although regions with (constant) shear of velocity are ubiquitous in engineering flow systems. The definite understanding is of major importance when it comes, e.g., to the design process of turbojet engines, as noise regulations make it imperative to consider aeroacoustic properties to meet airworthiness requirements. In the acoustic realm, analyzing small amplitude perturbation dynamics yields the basic understanding. Thus, we initially analyze the linearized compressible Euler equations in the light of the breakthrough achieved by the hydrodynamic stability community in understanding shear flow phenomena in the 1990s. The methods involve Lie symmetry analysis, Kelvin mode analysis and direct numerical simulations.

Contact: Jan Niklas Hau, M.Sc.

- Development of a thermodynamic consistent model for debris flows, regarding the evolution of a dynamic pore-fluid pressure and intergranular hypoplastic frictional behavior, introduced as additional internal variables. Numerical simulations investigate the influence of the dynamic pore-fluid pressure and the intergranular friction.
- Constitutive modeling in a symbolic computational framework: The exploitation of the entropy principle in its formulation by Müller and Liu is facilitated with the aid of symbolic computation.

Contact: Julian Hess, M.Sc.

The objective of the project is to investigate the thermodynamic behavior of electrolyte solutions whose constituents react to form solids. Particularly, we are interested in modeling the thermodynamic behavior of calcium carbonate supersaturated solutions. For this purpose, we perform numerical simulations in commercial fluid dynamics softwares and compare the results with experimental information.

Contact: Dr. Martina Costa Reis

In this project a new approach to the free boundary value problem of an oscillating droplet is taken. The goal is the derivation of a new set of equations describing the surface motion. The approach is based on the recently developed mathematical technique known as “Unified Transform Method” (Fokas' Method) which allows to reformulate certain PDEs into Integro-Differential Equations (IDEs). Using this technique, it is possible to formulate a new pair of governing equations which are valid for almost arbitrary drop shapes and, additionally, takes viscous effects into account.

Contact: Dipl-Math. Dominik Plümacher