# Boundary layers and wind in turbulent thermal convection

by Sebastian Wagner

Date of Examination:2014-06-26

Date of issue:2014-07-24

Advisor:Prof. Dr. Dr. Andreas Dillmann

Referee:Prof. Dr. Andreas Dillmann

Referee:Prof. Dr. Andreas Tilgner

Referee:Prof. Dr. Jörg Schumacher

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## Abstract

### English

This cumulative thesis is based on seven publications and is devoted to the investigation of turbulent thermal convection by means of Direct Numerical Simulations (DNSs). Special focus is placed on the boundary layers and the global flow in the generic setup of Rayleigh–Bénard convection (RBC), i.e. a fluid, which is confined between a heated bottom and a cooled top plate as well as thermally insulating sidewalls. Under consideration of the Oberbeck-Boussinesq approximation, the RBC flow is characterized by two dimensionless parameters: the Rayleigh number and the Prandtl number. In all the publications a large amount of instantaneous temperature and velocity fields is used for a posteriori analysis. At first, the local and instantaneous boundary layer structure appearing in RBC in a cylindrical container close to the bottom plate is studied. For this purpose a method for the extraction of the large-scale circulation (wind) is introduced, to circumvent azimuthal reorientations occurring in cylindrical containers. The study reveals strong deviations from the theoretical approach of Prandtl–Blasius–Pohlhausen (PBP) for the description of laminar boundary layers. The latter approach is commonly used for the characterization of the boundary layers in RBC at moderate Rayleigh numbers and for modelling purposes. This approach is also used to estimate the Kolmogorov and Batchelor microscales at the boundary layer edge and, hence, the required spatial resolution of the boundary layers in a DNS. The theoretical PBP estimates are compared with corresponding numerical results, revealing that the estimates are not restrictive enough and therefore their improvement is desirable. This is achieved by extending the PBP approach to a non-vanishing pressure gradient parallel to the heated/cooled plates. The pressure gradient depends on the angle of attack at which the flow approaches the plates. The resulting velocity boundary layer equation is of the Falkner–Skan type and leads to a better agreement with the DNS results with respect to the ratio of the thicknesses of the thermal and viscous boundary layers. The value for the latter ratio is derived analytically for arbitrary angles of attack and infinitesimal or infinite Prandtl numbers. This leads to improved estimates for the required spatial resolution of the boundary layers in the DNS. Furthermore, the influence of the container geometry on RBC is studied. The mean heat fluxes and the global flow structures are evaluated for different Rayleigh numbers and different shapes of the container. First of all, the results obtained in a cubic enclosure are compared with the results obtained in the cylindrical container. The comparison reveals changes of the global flow, the mean heat flux and the mean kinetic energy with increasing Rayleigh number for the cubic container, which are not present in the cylinder. In addition, like in some experimental studies, quasi two-dimensional RBC is investigated, i.e. RBC in a container of equal height and length and rather short depth. The aspect ratio of depth per height is varied and the influence on the global flow structure, the mean heat flux and the mean kinetic energy are evaluated. The study reveals a strong influence of the latter quantities on the aspect ratio. For the aspect ratio of one fourth, a similar flow structure as in the experiments is obtained, even though the considered Rayleigh numbers in the DNS are much smaller. Additionally, the mean heat flux is found to be equal to that in a cylindrical container. Finally, this quasi two-dimensional geometry is extended by adding heated and cooled obstacles, which are attached to the bottom and top plates, respectively. These obstacles represent regular surface roughness, which is often used in technical applications to achieve an increased heat flux. The heat flux and the velocity magnitude for varying height and width of the obstacles is investigated. It can be well described by a presented empirical relation, reflecting the result that, for constant covering area of the surface, slender obstacles can lead to the largest heat flux increase.**Keywords:**Direct Numerical Simulation, thermal convection, boundary layers