Sebastian Wagner

Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers

We report on a numerical study of the aspect-ratio dependency of Rayleigh-Benard convection, using direct numerical simulations. The investigated domains have equal height and width while the aspect ratio Γ of depth per height is varied between 1/10 and 1. The Rayleigh numbers Ra for this study variate between 105 and 109, while the Prandtl number is Pr=0.786. The main focus of the study concerns the dependency of the Nusselt number Nu and the Reynolds number Re on Ra and Γ. It turns out that due to Γ, differences to the cubic case (i.e., Γ = 1) in Nu of up to 55% and in Re of up to 97% occur, which decrease for increasing Ra. In particular for small Γ sudden drops in the Ra-scaling of Nu and Re appear for Ra≈106. Further analysis reveals that these correspond to the onset of unsteady motion accompanied by changes in the global flow structure. The latter is investigated by statistical analysis of the heat flux distribution on the bottom and top plates and a decomposition of the instantaneous flow fields i...

Thermal boundary layer equation for turbulent Rayleigh-Bénard convection

We report a new thermal boundary layer equation for turbulent Rayleigh-Bénard convection for Prandtl number Pr>1 that takes into account the effect of turbulent fluctuations. These fluctuations are neglected in existing equations, which are based on steady-state and laminar assumptions. Using this new equation, we derive analytically the mean temperature profiles in two limits: (a) Pr≳1 and (b) Pr≫1. These two theoretical predictions are in excellent agreement with the results of our direct numerical simulations for Pr=4.38 (water) and Pr=2547.9 (glycerol), respectively.

Prandtl-number dependence of heat transport in laminar horizontal convection.

We report the Prandtl-number (Pr) and Rayleigh-number (Ra) dependencies of the Reynolds number (Re) and mean convective heat transport, measured by the Nusselt number (Nu), in horizontal convection (HC) systems, where the heat supply and removal are provided exclusively through a lower horizontal surface of a fluid layer. For laminar HC, we find that Re∼Ra^{2/5}Pr^{-4/5}, Nu∼Ra^{1/5}Pr^{1/10} with a transition to Re∼Ra^{1/2}Pr^{-1}, Nu∼Ra^{1/4}Pr^{0} for large Pr. The results are based on direct numerical simulations for Ra from 3×10^{8} to 5×10^{10} and Pr from 0.05 to 50 and are explained by applying the Grossmann-Lohse approach [J. Fluid Mech. 407, 27 (2000)] transferred from the case of Rayleigh-Bénard convection to the case of laminar HC.

Numerical Investigation of the Spatial Resolution Requirements for Turbulent Rayleigh-Bénard Convection

The key requirement for setting up a direct numerical simulation (DNS) is a sufficiently fine grid allowing to resolve locally all relevant micro-scales. In case of turbulent Rayleigh-Benard convection (RBC) this is usually done by fulfilling different analytically derived criteria for the boundary layers and the bulk flow. In order to analyse if these requirements are sufficient, DNS of turbulent RBC in a cylindrical container with aspect ratio unity and Prandtl number Pr = 0.786 have been performed for Rayleigh numbers Ra up to 109. The micro-scales obtained in the DNS as well astheir scaling with Ra are compared with the corresponding theoretical predictions. The analysis reveals that the smallest scales, occurring close to the wall, are about half of the estimated ones. Furthermore, their scaling differs slightly from the estimations while the criterion for the bulk flow fits quite well.

A Simplified Model of the Wave Generation Due to Train-Tunnel Entry

The compression wave generated when a high-speed train enters a tunnel at Mach numbers smaller than 0.4 can be described in good approximation by a linear theory of an inviscid fluid. The wave equation for the acoustic potential becomes the governing equation. It is solved by a three dimensional boundary element method in time domain which forces a vanishing normal component of the velocity at the tunnel wall. It is assumed that the elements are compact in time. This leads to a linear equation in which a special matrix-vector multiplication has to be evaluated for every time-step. The aim is to create a fast method which sets as few constraints on the geometry as possible but nevertheless gives an accurate description of the wave propagation. In a first step the elements are assumed to be rectangles and an infinitely thin cylindric tube of finite length is taken as the geometry of the tunnel. The train is modeled by a single moving mass source of monopole type. It defines a semi-infinite body whose shape slightly changes when entering the tunnel. The results of this simple model along with the comparison with analytical solutions and experimental data are shown and discussed.

DNS of Thermal Convection in Rectangular Domains with Different Depth

As a simplified model of a large class of convective processes, Rayleigh-Benard convection (RBC) enables fundamental and numerical studies of convection including Direct Numerical Simulations (DNS). Although it has been investigated for more than 100 years there are still many open questions including the influence of the geometrical characteristics of the convection cell on the flow dynamics.

Influence of the Geometry on Rayleigh-Bénard Convection

Direct numerical simulations (DNS) of Rayleigh-Benard convection in a cube and a cylinder with equal diameter and height are performed to investigate the main responses of the system, namely heat flux and motion. Differences in the latter two quantities for the two geometries suggest a transition between different flow states in the cube, which is not observed in the cylinder due to its rotational symmetry. A method is introduced to analyse the flow dynamics in the cube, which relies on the temperature distribution at the lateral walls. It reveals that above a certain Rayleigh number the global flow structure in the cube is organized in a diagonal manner and not longer parallel to the walls, which leads to differences in the heat flux and the kinetic energy in the cylindrical and the cubic sample.

Numerical Simulation of Train-Tunnel Entry Using a BEM in Time Domain

The compression wave generated when a high-speed train enters a tunnel at Mach numbers smaller than 0.4 can be described in good approximation by a linear theory of a inviscid compressible fluid. The wave equation for the acoustic potential becomes the governing equation. It is solved by a three dimensional boundary element method in time domain which forces a vanishing normal component of the velocity at the tunnel wall. It is assumed that the elements are compact in time. This leads to a linear equation in which a special matrix-vector multiplication has to be evaluated for every time-step. The aim is to create a fast method which sets as few constraints on the geometry as possible while still beeing accurate enough to be used as a first estimate of the occuring wave propagation. In a first step the elements are assumed to be rectangles and a cylinder of finite length with infinitely thin walls is taken as the geometry of the tunnel. The train is modeled by a single moving mass source of monopole type. It defines a semi-infinite body whose shape slightly changes when entering the tunnel. The results of this simple model along with the comparison with analytical solutions and experimental data are shown and discussed.