Session ER: Convection I

Search for the ``ultimate state" in turbulent Rayleigh-B\'enard convection

Measurements of the Nusselt number $Nu$ will be reported for turbulent Rayleigh-B\'enard convection of a cylindrical sample. They cover the Rayleigh-number range $10^{11} \alt Ra \alt 2\times 10^{15}$ using N$_2$ ($Pr = 0.72$) and SF$_6$ ($Pr = 0.79$ to 0.84) at pressures up to 19 bars and near-ambient temperatures. The sample had a height $L=2.24$m and diameter $D = 1.12$m and utilized the high-pressure vessel known as the ``Uboot of G\"ottingen" at the Max Planck Institute for Dynamics and Self-Organization in G\"ottingen, Germany. For $Ra \alt 4\times 10^{13}$ the data yielded $Nu \propto Ra^{\gamma_{eff}}$ with $\gamma_{eff} = 0.308$ and did not show the transition near $Ra = 10^{11}$ to an ``ultimate regime" that was reported by Chavanne et al. At $Ra = 4\times 10^{13}$ there is a well defined but continuous transition to a regime where $\gamma_{eff}$ is smaller than 0.30.   

Mean temperature profiles in turbulent Rayleigh-Benard convection

We report a study of mean vertical temperature profiles (TPs) in turbulent Rayleigh-Benard convection of water, $Pr=4.38$, in unit-aspect-ratio cylindrical and cubic cells for $Ra$ up to $10^9$, based on DNS. The Nusselt numbers $Nu$ computed for cylindrical cells are found to be in excellent agreement with the experimental data by Funfschilling et al. [J. Fluid Mech., vol. 536 (2005), pp. 145-154]. Based on this validation, the DNS data are used to extract TPs. In the DNS for the cylindrical geometry, reported in Shishkina \& Thess [J. Fluid Mech. (2009), in press], we find that near the heating and cooling plates the TP $\Theta(y)$ obey neither a logarithmic nor a power law. We show that the Prandtl--Blasius BL theory predicts the TP-shapes with an error $7.9\%$ within the thermal BLs alone. We further show that the profiles can be approximated by a stretched exponential approximation (SEA) of the form $\Theta(y)\approx 1-\exp(-y-0.5y^2)$ with an absolute error $<1.1\%$ within the thermal BLs and $<5.5\%$ in the whole cell. We provide also more accurate analytical approximations of the profiles involving higher-order polynomials in the SEA. Further, based on the DNS in a cube we extract TPs and estimate the accuracy of the above SEAs. Finally, we construct and analyze the quality of SEAs, which are based on local $Nu$.   

Scaling laws in turbulent Rayleigh-Benard convection with different geometry

The discovery of scaling laws in the heat flux, large-scale circulation and temperature statistics in turbulent convection has stimulated considerable experimental and theoretical efforts, aimed at understanding the universal nature of the observed scaling laws. Because of historical reasons, most of the experimental results were obtained in upright cylindrical cells with small aspect ratios. An important question one might ask is: To what extend are these scaling laws universal in that they are independent of the cell geometry? Understanding of this question has important implications to large-scale astro/geophysical convection, such as that in the atmosphere and oceans, in which boundary effects are less important. In this talk, we report an experimental study of turbulent convection in a horizontal cylinder with the bottom 1/3 (curved) surface heated and the top 1/3 surface cooled. The experiment is carried out with varying aspect ratios and Rayleigh numbers. It is found that the measured Nusselt number and Reynolds number obey the same scaling laws as those obtained in the upright cylinder. The local temperature statistics, on the other hand, change with the aspect ratio of the cell and are different from the earlier results. The experiment reveals important geometric effects of the scaling laws in turbulent convection. *Work supported by the Research Grants Council of Hong Kong SAR.   

Shallow moist Rayleigh-B\'{e}nard convection with piecewise linear equation of state

An idealized framework to study the impacts of phase transitions on atmospheric dynamics is presented. Condensation of water vapor releases a significant amount of latent heat, which directly affects the atmospheric temperature and density. Here, phase transitions are treated by assuming that air parcels are in local thermodynamic equilibrium, which implies that condensed water can only be present when the air parcel is saturated. This reduces the number of variables necessary to describe the thermodynamic state of moist air to three. It also introduces a discontinuity in the partial derivatives of the equation of state. A simplified version of the equation of state is obtained by a separate linearization for saturated and unsaturated parcels. When this equation of state is implemented in a Boussinesq system, the buoyancy can be expressed as a piecewise linear function of two buoyancy variables, $D$ and $M$, and height $z$. Numerical experiments in this setting allow then to study transitions from cumulus to stratocumulus clouds.   

Extensive Scaling of Computational Homology and Karhunen-Lo\`{e}ve Decomposition in Rayleigh-B\'{e}nard Convection Experiments

We apply two different pattern characterization techniques to large data sets of spatiotemporally chaotic flows in Rayleigh-B\'{e}nard convection (RBC) experiments. Both Computational homology (CH) and a modified Karhunen-Lo\`{e}ve decomposition (KLD) are used to analyze the data. The KLD dimension $D_{KLD}$, the number of eigenmodes required to capture a given fraction of the eigenvalue spectrum, is computed for different subsystem sizes. A similar quantity $D_{CH}$ for the same experimental data is acquired by the probability distribution of topological states constructed from the outputs of CH. We show that both $D_{CH}$ and $D_{KLD}$ scale over a large range of subsystem sizes for the state of SDC; moreover, we find the presence of boundaries leads to deviations from extensive scaling that are similar for both methodologies.   

Deterministic flow reversal in thermal convection

Two-dimensional (2D) numerical simulations of the Boussinesq equations are presented with Rayleigh numbers up to $Ra = 10^9$ and aspect ratio of about 1. They reveal a diagonal large scale convection roll (``wind of turbulence'') and smaller rolls in the two remaining corners diagonally opposing each other. These corner flow rolls play a crucial role for the mechanism of large scale wind reversal: They grow in kinetic energy and thus also in size thanks to the plume detachments from the boundary layers up to the time that they take over, leading to the breakdown of the large scale convection roll and the formation of a new one, rotating in the other direction. Based on the numerical simulations and on theoretical arguments we identify the characteristic time scale for this whole process. According to present precision it seems to grow with Ra as a power law. -- Employing PIV techniques the same deterministic flow reversal mechanism is shown to be at work for thermal convection in quasi 2D rectangular Rayleigh-Benard cells.   

Benard convection in the presence of micro particles

We study Benard convection in water containing a small volume fraction of micro particles. The investigation is motivated by recent experiments of natural convection of aqueous suspensions [1] conducted at an average temperature of 20 degrees C in which the authors report a decrease in Nusselt number compared to pure water. This effect has been attributed to density inversion in the base state taking place near the lower boundary caused by the sedimentation of the aluminum oxide particles, the density of which is greater than that of water. We attempt to elucidate these findings by carrying a stability analysis on a model of convection for a liquid suspension having a nonlinear equation of state. The model accounts for the coupled effects of Brownian motion, sedimentation and thermophoresis. The balance of the latter yields a nonlinear base profile for the concentration of particles. Density inversion occurs near either the lower or the top boundary depending on the balance between sedimentation and thermophoresis and on the size and density of the particles. Parameter range for the onset and stability of the resulting double layer convection is given and the implications the results may have on the heat transfer in nanofluids are discussed.\\[4pt] [1] B. H. Chang, A.F. Mills, E. Hernandez, Int. J. Heat Mass Transfer, 51 (2008) 1332-1341.   

Refined measurements on the structure of thermal boundary layers in turbulent Rayleigh-Benard convection

We present highly resolved temperature measurements at a large- scale Rayleigh-Benard experiment simultaneously undertaken at the top and the bottom plate using small microthermistors. For the first time the temperature measurements have been complemented by local heat flux measurements at the surface of both horizontal plates using special heat flux sensors. The experimental facility used for this purpose is an adiabatic cylinder with an inner diameter of D=7.15 m filled with air. An electrical heating plate at the bottom and a free hanging cooling plate at the top trigger the convective flow. The work reported here is limited to the H=6.30 m where the shape of the global flow is well known. We will discuss results of measurements of profiles of the mean temperature and compare the behaviour of the top and bottom boundary layers.   

Pattern Control and State Estimation in Rayleigh-B\'{e}nard Convection

We report on a new experimental approach to study instability in Rayleigh-B\'{e}nard convection. The convective fluid absorbs incident infrared laser light, thereby altering the fluid flow. Rapid scanning of the light allows nearly simultaneous actuation at many spatial locations of the pattern. This approach is used to impose reproducibly a given convection pattern. Control is demonstrated by preparing repeatedly a pattern near a straight roll instability. Selected perturbations are applied to this ensemble and decay lifetimes are measured as the system relaxes to the base state. We find that decay lifetimes increase near the instability and give a quantitative measure of distance from instability. We also create patterns that undergo the instability, giving a set of systems evolving from nearby initial conditions on both sides of the instability boundary. This set can be used to test systematically the sensitivity of state estimation, a crucial process in forecasting. Preliminary results of applying one state estimation algorithm to these diverging pattern trajectories will be discussed.   

Turbulent Rayleigh-B\'enard Convection with Conductive Plates

Despite considerable experimental, theoretical and numerical effort, for turbulent Rayleigh-B\'enard convection in the high-Ra limit the scaling of the enhanced bulk heat transport, measured by the Nusselt number Nu, with the temperature drop across the fluid, given by the Rayleigh number Ra, is still incompletely understood. While most work has assumed a fixed temperature drop across the fluid, it has recently become clear that this assumption is mathematically and experimentally inadequate, and in the quest to reconcile theory and experiment the influence of the finite conductivity of the bounding plates has been receiving increasing attention. We review recent progress in this area, and discuss in particular rigorous variational bounds on the Nu-Ra scaling for finite Prandtl number convection for general thermal boundary conditions ranging between the fixed temperature and fixed flux extremes, including the case of plates of finite conductivity. We show in particular that the usual fixed temperature assumption is a singular limit of the full problem, while in the large-Ra limit, we find a bound of Nu $\leq R^{1/3}$, where $R$ is a Rayleigh number in terms of the temperature drop across the full fluid-plate system.