| dc.description.abstracteng | This thesis is dedicated to the study of turbulent superstructures in Rayleigh-Bénard convection. Turbulent superstructures are horizontally extended large-scale flow patterns that emerge despite the presence of strong fluctuations. They persist for long time scales, evolve slowly compared to the fluctuations, and their extent increases with increasing driving. This phenomenon is observed in large-aspect-ratio experiments and simulations and is very important for various geo- and astrophysical flows. There are many open questions regarding their origin and dynamics, and we study some aspects thereof. How do small-scale fluctuations and superstructures interact? How is the emergence and stability of the large-scale flow influenced by the fluctuations? Can the effect of the fluctuations be described effectively?
In the first part, we present a numerical study of superstructures to shed light on the energetic interaction between superstructures and small-scale fluctuations. For this, we make use of a filtering approach to derive scale-resolved energy and temperature variance budgets. In those, transfer terms occur that characterize the transfer rates of energy and temperature variance between scales. At the scale of the superstructure, we find that the energy transfer rate between scales acts on average as an energy sink. Therefore, the small scales are effectively a dissipation channel for the large scales. However, if considered horizontally averaged, a more complex behaviour is revealed. Close to the wall, an inverse energy transfer rate is present at moderate Rayleigh numbers. With increasing Rayleigh number the profiles become more complex and the inverse transfer layer finally vanishes. In contrast, the transfer rate is always direct, i.e. from large to small scales in the bulk. Locally, the direction of the energy transfer is closely linked to the dynamics of the plumes. At locations of plume detaching we observe a direct energy transfer and at plume impinging an inverse transfer. Similar observations are made for the transfer rate of temperature variance. On average, the transfer is from large to small scales. In contrast, also the horizontally averaged transfer rate is almost exclusively a sink and, in addition, strongly restricted to the regions close to the wall. These results may guide future investigations of reduced three-dimensional models of superstructures.
In the second part, we study the influence of small-scale fluctuations on generic large-scale flow patterns in a phenomenological model. We base our model on the Swift-Hohenberg equation, which is a well-known reduced model for Rayleigh-Bénard convection at onset of convection. We supplement the Swift-Hohenberg equation, which leads to stationary large-scale patterns, with a random advection term to include turbulent fluctuations. Here, we use a Gaussian random velocity field, which is white in time. In combination, large-scale patterns superposed by small-scale fluctuations emerge. We show that the presence of the fluctuations shifts the onset of pattern formation to larger control parameters and increases the characteristic wavelength of the pattern. The latter effect is also observed in Rayleigh-Bénard convection, in which the length scale of the superstructures increases with increasing Rayleigh number. Therefore, our phenomenological model can qualitatively explain this observation. It may serve as the basis to derive more realistic models of superstructures, which, e.g., also capture the temporal evolution.
The last part is committed to the development of a reduced model for turbulent superstructures in Rayleigh-Bénard convection. Due to the large range of involved scales ranging from turbulent fluctuations to superstructures, the numerical and theoretical analysis of the full governing equations is very demanding. Thus, a reduced model offers the possibility of more feasible studies and, hence, more insights into the fundamental mechanisms. We propose to average the governing equations over height to reduce the dimensionality and to spatially coarse-grain to remove the fluctuations. The resulting equations contain unclosed terms that originate from the nonlinearity and the solid top and bottom wall. We propose and validate closure models based on direct numerical simulations for these terms. We show in this a priori study that rather simple closure models can reproduce the unclosed terms very well. In comparison to the previous phenomenological model, this has the advantage that it makes contact with the full governing equations and, therefore, quantitative comparisons are possible. But, in this first study of this kind, the horizontal velocity field is still introduced ad hoc as a mean flow. The proposed model provides an ideal starting point for future investigations.
In summary, in this thesis we gain insights into the energetic interaction between superstructures and turbulent fluctuations, qualitatively explain the shift of the wavelength in Rayleigh-Bénard convection, and take the first steps towards a reduced model of superstructures. | de |