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The linear-theory assumption is a fundamental approach for the study of waves in fluids. The governing equations are linearized by assuming the perturbations are small so that the consequences of nonlinear terms are negligible. Nevertheless, if a wave approaches a critical level, in which the wave amplitude grows so as to create an instability of the background flow, the assumption of linearity may not hold any longer. In this case, the nonlinear terms are required to be taken into consideration.
In this thesis, two experimental setups have been proposed for the study of two scenarios, in which the nonlinear effects become significant and a traditional linear solution is no longer valid.
The first experiment focuses on an inertially oscillating rotating fluid. In the thesis, we present experimental results from a system that is simpler than classical precession experiments but still shows very similar wave interactions and a collapse to turbulence. This system consists of a partly filled rotating annulus that rotates about its symmetry axis slightly tilted with respect to the gravity vector.
In the experiments, we find a resonant collapse when the forcing frequency corresponds with a resonant frequency of the rotating tank. Two types of instability can be triggered: a parametric triadic instability, in which two free Kelvin modes arise and form a triad with the forced Kelvin mode, and a shear-type instability related to the nonlinearly excited geostrophic flow. The latter instability gives rise to a barotropic mode that interacts with the forced mode and generates secondary modes. We also observed dependency of the mode frequencies on the Ekman number, which can, at least partly, be explained by a Doppler shift due to the mean flow. Finally, we try to connect our data to a low-order dynamical system based on the weakly nonlinear theory that describes the main features of single triad interaction in precession experiments.
The second experiment concerns the study of undular bores (or tidal bores), in which the nonlinearity plays an important role. An experiment has been performed in which undular bores are produced in an open circular channel. More specifically, two different cases have been investigated: a single bore case with a rigid boundary setup and a bore colliding case with a periodic lateral boundary setup. Bores are generated by abruptly releasing a barrier that separates fluids with different surface levels. Up to our knowledge, this is the first experimental study of undular bores in a circular channel. For a setup without barriers, this geometry accomplishes in a natural way the periodic lateral boundary conditions, which is very often used in numerical simulations. The experimental results have been compared with the nonlinear numeric simulations and achieved an excellent agreement.
Resonance phenomena are ubiquitous in Nature. Resonance means that a system can accumulate large amounts of kinetic energy. In rotating flows inertial waves provide a mechanism for resonance by redistributing momentum, kinetic energy and helicity.
In order to investigate inertial waves a Taylor-Couette system was investigated which consists of a homogeneous liquid confined between two coaxial cylinders and two rigid lids. The inner cylinder is slightly conical (frustum) to break the vertical mirror symmetry. Inertial waves were excited by two different forcing configurations: the frustum in libration and the lids together with the outer cylinder in libration. Libration means that the rotation rate of the wall is modulated with a fixed amplitude and frequency of the order of the mean rotation rate. Direct numerical simulations (DNS) were conducted with a numerical solver in terrain-following coordinates.
DNS results reveal that inertial wave excitation is localised at the edges of the confinement, which is in very good agreement with recent laboratory measurements of Seelig (2014, PhD thesis, BTU Cottbus - Senftenberg). A model of the wave excitation mechanism was developed with the aid of boundary layer theory. The model suggests that a difference in the boundary layer mass flux (Ekman flux) excites the waves by driving an excess Ekman pumping velocity at the edges. The DNS results exhibit this flux difference, and the simulated kinetic energy spectra of the waves exhibit the frequency dependency predicted by the model.
However, DNS results also exhibit helical vortices at the edges which are not part of the model. Conservation properties suggest that these vortices are merely a compensating phenomenon which tends to stabilise the boundary layer flow. The details of this flow, however, appear less important for the wave excitation.
Response spectra of the kinetic energy, the dissipation rate, the helicity, and the quality factor were computed in order to assess resonance conditions. Simulated resonance peaks have a width of only 1/20th of the mean rotation rate. At these peaks, the kinetic energy was found to increase by a factor 10-50 even though viscous forces were still rather large (Ekman number of the order 1/100,000 with the length scale given by the mean radial gap width).
The flow patterns found at those resonances were investigated and found to be in very good agreement with the spatial patterns obtained by laboratory measurements and geometric ray tracing. The DNS results suggest that there are two types of resonance in rotating flows: modes and wave attractors. In contrast to a mode, a wave attractor exhibits net focusing of wave energy and occupies a finite frequency band. DNS results show that the wave attractor resonance frequency adjusts within the frequency band which suggests that wave attractor resonances complement 'classical' mode resonances and may, thus, be relevant in various applications.