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Stratified vortices can be found from small to large scales in geophysical and astrophysical flows. On the one hand, tornadoes and hurricanes can lead to devastation and even a large
number of casualties. On the other hand, vortices can distribute heat and momentum in the atmosphere which is important for a habitable environment on Earth. In the astrophysical context, accretion disks (from which solar systems are formed) can be seen as stratified vortices. In such systems, understanding the mechanisms that can result in an outward transport of angular momentum is a central problem. For a planet or star to be formed in a disk, angular momentum has to be carried away from its center to allow matter aggregation by gravity; otherwise, its rotation speed would be far too large, avoiding this matter aggregation (and the consequent star formation) to happen. In such gas systems, turbulence is the most likely mechanism to achieve such a large angular momentum transport. However, it was shown that the flow profile of accretion disks is stable with respect to purely shear instabilities, and the question arises about how the turbulence can be generated. Among other candidates, the strato-rotational instability (SRI) has attracted attention in recent years. The SRI is a purely hydrodynamic instability that can be modeled by a classical Taylor-Couette (TC) system with stable density stratification due to axial salinity or temperature gradients.
In this thesis, a combined experimental and high-performance computing study of new specific behaviors of the strato-Rotational Instability (SRI) is performed. The density stratification causes a change in the marginal instability transition when compared to classical non-stratified TC systems, making the flow unstable in regions where – without stratification – it would be stable. This characteristic makes the SRI a relevant phenomenon in planetary and astrophysical applications, particularly in accretion disk theory.
Despite many advances in the understanding of strato-rotational flows, the confrontation of experimental data with non-linear numerical simulations remains relevant, since it involves
linear aspects and non-linear interactions of SRI modes which still need to be better understood. These comparisons also reveal new non-linear phenomena and patterns not yet observed in the SRI, that can contribute to our understanding of geophysical flows.
We investigate the dynamics of a Newtonian liquid layer bounded on one side by a horizontal and planar substrate and on the other side by its free and deformable surface. The system is subjected to a time-periodic gravitation field in lateral or normal direction. Based on a nonlinear coordinate transformation, which maps the time-dependent surface onto a constant domain and thus eliminates the need for tracking the interface, a finite-difference method on staggered grids is presented, allowing direct numerical simulations of the full incompressible Navier-Stokes equations in two and three dimensions. Taking into account the continuity equation, a sparse linear system for the pressure is obtained from the discretized Navier-Stokes equations whose solution satisfies the conservation of momentum and mass, so that pressure corrections can be avoided. In the case of a harmonic force perpendicular to the substrate, we find in high frequency ranges the classical square patterns oscillating subharmonically with half of the driver’s frequency. For a slow excitation, hexagonal Faraday waves emerge oscillating at the forcing frequency. Vertical two-frequency excitations lead to more complex patterns — surface waves having the shape of a square superlattice are found. In the case of a lateral excitation, the formation of coarsening droplets is observed. We show that ratchet-like forces generate a nonvanishing averaged flow rate inducing a preferred direction of motion of the drops. These results correspond well with those of a simplified model based on the lubrication approximation. Our investigations also include systems in Rayleigh-Taylor configuration, where the liquid is located on the underside of the substrate. By considering rigid walls instead of periodic boundaries, wave amplification due to resonance can be studied. The corresponding numerical results are in good agreement with experimentally obtained data.
The goal of this work is to investigate pattern formation processes on the solid-liquid interface during the crystal growth of GeSi. GeSi crystals with cellular structure have great potential for applications in gamma-ray and neutron optics. The interface patterns induce small quasi-periodic distortions of the microstructure called mosaicity. Existence and properties of this mosaicity are important for the application of the crystals. The properties depend on many factors; this dependence, is currently not known even not qualitatively. A better understanding of the physics near the crystal surface is therefore required, in order to optimise the growth process. There are three main physical processes in this system: phasetransition, diffusion and melt flow. Every process is described by its own set of equations. Finite difference methods and lattice kinetic methods are taken for solving these governing equations. We have developed a modification of the kinetic methods for the advectiondiffusion and extended this method for simulations of non-linear reaction diffusion equations. The phase-field method was chosen as a tool for describing the phase-transition. There are numerous works applied for different metallic alloys. An attempt to apply the method directly to simulation GeSi crystal growth showed that this method is unstable. This instability has not been observed in previous works due to the much smaller scale of simulations. We introduced a modified phase-field scheme, which enables to simulate pattern formation with the scale observed in experiment. A flow in the melt was taken in to account in the numerical model. The developed numerical model allows us to investigate pattern formation in GeSi crystals. Modelling shows that the flow near the crystal surface has impact on the patterns. The obtained patterns reproduce qualitatively and in some cases quantitatively the experimental results.