Dissipation Layers in Rayleigh-B\'{e}nard Convection: A Unifying View

Boundary layers play an important role in controlling convective heat transfer. Their nature varies considerably between different application areas characterized by different boundary conditions, which hampers a uniform treatment. Here, we argue that, independent from boundary conditions, systematic dissipation measurements in Rayleigh-B\'enard convection capture the relevant near-wall structures. By means of direct numerical simulations with varying Prandtl numbers, we demonstrate that such dissipation layers share central characteristics with classical boundary layers, but, in contrast to the latter, can be extended naturally to arbitrary boundary conditions. We validate our approach by explaining differences in scaling behavior observed for no-slip and stress-free boundaries, thus paving the way to an extension of scaling theories developed for laboratory convection to a broad class of natural systems

Statistical and dynamical properties of convecting systems

Diese Arbeit analysiert die statistischen und dynamischen Eigenschaften von konvektiven Flüssigkeitsströmungen und die darin auftretenden Strukturen. Die verworrene Struktur turbulenter Konvektion verlangt eine statistische Beschreibung, da eine punktweise Vorhersage der Strömung nicht erreichbar ist, aber aufgrund des chaotischen Verhaltens auch nicht sehr aufschlussreich wäre. Nichtsdestoweniger treten in konvektiven Strömungen oft kohärente Strukturen mit einer charakteristischen Dynamik auf, die wiederum die Statistik beeinflussen und somit für ein Verständnis des Systems essentiell sind. Im ersten Teil der Arbeit werden die Temperaturfluktuationen in turbulenter Rayleigh-Bénard-Konvektion und deren Statistik mit Hilfe ihrer Verteilungsfunktion untersucht. Der zweite Teil versucht, kohärente Strukturen und deren Zusammensetzung in turbulenter Konvektion zu erfassen. Im dritten Teil werden die Statistik und die Strukturen untersucht, die sich beim Eintreten von Konvektion zeigen.This thesis analyses the statistics and the dynamics as well as the coherent structures that are found in convective fluid flows. The erratic and tangled structure of turbulent convection necessitates a statistical description, as a pointwise prediction of the flow is on the one hand out of reach, and on the other hand not very insightful due to the chaotic behavior. Nevertheless, the typical dynamics and coherent structures that are found in convective flows influence the statistics, and their behavior is crucial for a deeper understanding of the system. In the first part, the statistics of temperature fluctuations in turbulent Rayleigh-Bénard convection are examined with the help of their probability density function. The second part concentrates on the detection and composition of coherent structures in turbulent convection and their contribution to the heat transport. The third part investigates the structures and statistics that are found near the onset of convection.<br

The Lundgren-Monin-Novikov Hierarchy: Kinetic Equations for Turbulence

We present an overview of recent works on the statistical description of turbulent flows in terms of probability density functions (PDFs) in the framework of the Lundgren-Monin-Novikov (LMN) hierarchy. Within this framework, evolution equations for the PDFs are derived from the basic equations of fluid motion. The closure problem arises either in terms of a coupling to multi-point PDFs or in terms of conditional averages entering the evolution equations as unknown functions. We mainly focus on the latter case and use data from direct numerical simulations (DNS) to specify the unclosed terms. Apart from giving an introduction into the basic analytical techniques, applications to two-dimensional vorticity statistics, to the single-point velocity and vorticity statistics of three-dimensional turbulence, to the temperature statistics of Rayleigh-B\'enard convection and to Burgers turbulence are discussed.Comment: Accepted for publication in C. R. Acad. Sc

Turbulent Rayleigh-B\'enard convection described by projected dynamics in phase space

Rayleigh-B\'enard convection, i.e. the flow of a fluid between two parallel plates that is driven by a temperature gradient, is an idealised setup to study thermal convection. Of special interest are the statistics of the turbulent temperature field, which we are investigating and comparing for three different geometries, namely convection with periodic horizontal boundary conditions in three and two dimensions as well as convection in a cylindrical vessel, in order to work out similarities and differences. To this end, we derive an exact evolution equation for the temperature probability density function (PDF). Unclosed terms are expressed as conditional averages of velocities and heat diffusion, which are estimated from direct numerical simulations. This framework lets us identify the average behaviour of a fluid particle by revealing the mean evolution of fluid of different temperatures in different parts of the convection cell. We connect the statistics to the dynamics of Rayleigh-B\'enard convection, giving deeper insights into the temperature statistics and transport mechanisms. We find that the average behaviour is described by closed cycles in phase space that reconstruct the typical Rayleigh-B\'enard cycle of fluid heating up at the bottom, rising up to the top plate, cooling down and falling down again. The detailed behaviour shows subtle differences between the three cases

Investigation of temperature statistics in turbulent Rayleigh-Bénard convection using PDF methods

Rayleigh-Bénard convection in the turbulent regime is studied using statistical methods. Exact evolution equations for the probability density function of temperature and velocity are derived from first principles within the framework of the Lundgren-Monin-Novikov hierarchy. The unclosed terms arising in the form of conditional averages are estimated from direct numerical simulations. Focussing on the statistics of temperature, the theoretical framework allows to interpret the statistical results in an illustrative manner, giving deeper insight into the connection between dynamics and statistics of Rayleigh-Bénard convection

Temperature statistics in turbulent Rayleigh–Bénard convection

Rayleigh–Bénard (RB) convection in the turbulent regime is studied using statistical methods. Exact evolution equations for the probability density function of temperature and velocity are derived from first principles within the framework of the Lundgren–Monin–Novikov hierarchy known from homogeneous isotropic turbulence. The unclosed terms arising in the form of conditional averages are estimated from direct numerical simulations. Focusing on the statistics of temperature, the theoretical framework allows us to interpret the statistical results in an illustrative manner, giving deeper insights into the connection between dynamics and statistics of RB convection. The results are discussed in terms of typical flow features and the relation to the heat transfer