Numerical analysis of free-surface flow through rotating machines

In the context of the transformation process currently taking place in the energy production sector, energy gained from renewable power sources shall replace the present mixture, which mostly relies on fossil burnings. Therefore, in the future most of the energy shall be gained by harvesting power from sun, wind or water, geothermal heat or biomass. In case of converting energy from wind into electrical power wind turbines are used in general, while hydropower turbines are the state-of-the-art machinery to derive energy from running water. In order to convert the potential energy from running water as well water wheels pose the method of choice.
Turbines in air or water represent mechanically a two-field system, in which the structure of the turbine is surrounded by a streaming fluid. Due to the elasticity of the rotor blades the stresses of the fluid onto the structure deform the blades, which in return yield a time-dependent flow domain. Therefore turbines in a streaming fluid represent a typical example of fluid-structure interaction. Furthermore, in case of water wheels the surrounding air as third field and additional fluid phase comes into play introducing a free surface.
In this contribution the governing equations of incompressible fluid flow are presented using primal variables and discretised via the space-time finite element method [3]. The discretised model equations of the fluid are stabilised using an SUPG/PSPG approach. Shape and test functions are continuous within the space-time slabs, while across the space- time slabs the shape and test functions are continuous only in space, but discontinuous in time yielding a time-discontinuous Galerkin approach.
Due to the moving rotor blades a mesh moving technique needs to be incorporated into the computational set-up. Considering the occurring large but regular displacements of the flow boundary arising from the rotating rotor blades the shear-slip mesh update method (SSMUM) [1] as discontinuous mesh moving technique is applied.
In case of water wheels the free surface is described implicitly via a Level-Set function [2] yielding a single fluid phase with almost discontinuous density and viscosity.
The verification and validation of the developed numerical scheme is carried out with the help of computing classical benchmark problems as well as via a comparison to existing experimental data.