TU CE GSC CE People at the Graduate School CE Students Johannes Kromer Research Project of Johannes Kromer

Modeling and simulation of nonlinear vibrations in fluid particles

In contrast to solid objects, fluid particles such as liquid drops or gas bubbles may strongly and dynamically deform while they move through ambient media. The shape deformation is mainly induced by pressures differences resulting from local flow caused by the buoyancy-driven particle motion.

Of course, other mechanisms of excitation may occur as well. Depending on the nondimensional flow parameters, such as the *Reynolds* and *Eötvös* number, the respective trajectories of the particle motion display linear, spherical and helical behavior as well as irregular curves. Simultaneously - especially for liquid drops and gas bubbles - the shape of the fluid particle oscillates due to the feedback of the ambient flow, which crucially re-affects the dynamics of motion. While restrictive assumptions concerning the dynamics allow for an analytical treatment, more realistic formulations of the respective equations of motion exhibit highly nonlinear characteristics, producing nonlinear oscillations of both the path and shape of the fluid particle. The physical effects behind the dynamics exhibit complex bifurcations as well as instabilities.

There is a wide variety of technical applications for which the motion of fluid particles in ambient media is decisive, in particular in the chemical industry, but also reaching from atmospherics science to human medicine. Despite their industrial importance and application, there still is little understanding of the phenomena mentioned above, especially regarding the characteristics of the local nonlinear oscillations.

Hence, to shed more light on the underlying dynamics, the focus of this thesis shall be the \textit{direct numerical simulation} of the two-phase *Navier-Stokes* equations in its one-field formulation, applying the *volume-of-fluid* method. The institute *Mathematical Modeling and Analysis* resorts to and appropriately develops the software FS3D, which was developed specifically for the direct numerical simulation of incompressible two-phase flows and therefore provides several beneficial characteristics: from numerous applications and articles the inhouse code is well validated. It allows to incorporate artificial boundary conditions (ABC) as well as surface tension forces (balanced CSF). Furthermore, an implemented windowing technique provides traceability of the moving disperse phase, which considerably reduces the required simulation domains

and, thus, computational costs. In order to exploit the benefits of high-performance machines, a parallelization via MPI and OpenMPI has been implemented.

While the examination of Newtonian fluids allows first investigations of the underlying dynamics, the main focus of this thesis will cover non-Newtonian fluids. From the wide range of complex rheological material models, the *Oldroyd* model and variations thereof are of key interest, since they incorporate viscoelasticity.

Besides the numerical examination, a semi-analytical treatment of the *Navier-Stokes* equations shall be conducted for selected 2D- and 3D-flows. From these results, criteria will developed to condense nonlinear vibration characteristics. These criteria are validated by application on a broad basis of numerical computations.