Non-Oberbeck-Boussinesq (NOB) effects on the Nusselt number Nu and Reynolds number Re in strongly turbulent Rayleigh-Bénard (RB) convection in liquids were investigated both experimentally and theoretically. In the experiments the heat current, the temperature difference, and the temperature at the horizontal midplane were measured. Three cells of different heights L, all filled with water and all with aspect ratio Γ close to 1, were used. For each L, about 1.5 decades in Ra were covered, together spanning the range 10 8 6 Ra 6 10 11 . For the largest temperature difference between the bottom and top plates, ∆ = 40 K, the kinematic viscosity and the thermal expansion coefficient, owing to their temperature dependence, varied by more than a factor of 2. The Oberbeck-Boussinesq (OB) approximation of temperature-independent material parameters thus was no longer valid. The ratio χ of the temperature drops across the bottom and top thermal boundary layers became as small as χ = 0.83, which may be compared with the ratio χ = 1 in the OB case. Nevertheless, the Nusselt number Nu was found to be only slightly smaller (by at most 1.4%) than in the next larger cell with the same Rayleigh number, where the material parameters were still nearly height independent. The Reynolds numbers in the OB and NOB case agreed with each other within the experimental resolution of about 2%, showing that NOB effects for this parameter were small as well. Thus Nu and Re are rather insensitive against even significant deviations from OB conditions. Theoretically, we first account for the robustness of Nu with respect to NOB corrections: the NOB effects in the top boundary layer cancel those which arise in the bottom boundary layer as long as they are linear in the temperature difference ∆. The net effects on Nu are proportional to ∆ 2 and thus increase only slowly and still remain minor despite drastic materialparameter changes. We then extend the Prandtl-Blasius boundary-layer theory to NOB Rayleigh-Bénard flow with temperature-dependent viscosity and thermal diffusivity. This allows calculation of the shift in the bulk temperature, the temperature drops across the boundary layers, and the ratio χ without the introduction of any fitting parameter. The calculated quantities are in very good agreement with experiment. When in addition we use the experimental finding that for water the sum of the top and bottom thermal boundary-layer widths (based on the slopes of the temperature profiles at the plates) remains unchanged under NOB effects within the experimental resolution, the theory also gives the measured small Nusselt-number reduction for the NOB case. In addition, it predicts an increase by about 0.5%

Non-Oberbeck-Boussinesq (NOB) effects are measured experimentally and calculated theoretically for strongly turbulent Rayleigh-Bénard convection of ethane gas under pressure where the material properties strongly depend on the temperature. Relative to the Oberbeck-Boussinesq case we find a decrease of the central temperature as compared to the arithmetic mean of the top-and bottom-plate temperature and an increase of the Nusselt number. Both effects are of opposite sign and greater magnitude than those for NOB convection in liquids like water. DOI: 10.1103/PhysRevLett.98.054501 PACS numbers: 47.27.ÿi, 47.20.Bp Turbulent convection in a fluid heated from below and cooled from above (Rayleigh-Bénard convection) is an important model system in fluid dynamics [1]. The induced temperature difference across a sample is represented by the Rayleigh number Ra gL 3 = ( is the thermal expansion coefficient, g the acceleration of gravity, L the sample height, the thermal diffusivity, and the kinematic viscosity). The problem usually is analyzed within an approximation due to Oberbeck [2] and Boussinesq [3] (OB), where it is assumed that all fluid properties are constant within the entire sample except for the density where it induces the buoyancy force. Here we address, both experimentally and theoretically, the nature of deviations from this approximation.A central aspect of this system is an understanding of the boundary layers (BLs) near the top and bottom plates. At modest Ra they remain laminar while the fluid interior is turbulent, and their instabilities impact the Nusselt number Nu (the effective thermal conductivity eff normalized by the diffusive thermal conductivity ) [4,5]. At much larger Ra it was suggested [6 -8] that the BLs become turbulent as well, that they then no longer influence Nu, and that an asymptotic (''ultimate'' or ''Kraichnan'') regime is achieved where Nu Ra 1=2 . In the present Letter we show, by comparing theoretical calculations with new experimental measurements, that the laminar BLs are also significantly influenced by non-Oberbeck-Boussinesq (NOB) effects. These effects can be described well by an extension of the Prandtl-Blasius boundary-layer theory [9,10].Though NOB effects in turbulent Rayleigh-Bénard convection were measured already 15 years ago [11,12], a quantitative comparison between OB and NOB convection was only done recently [13]. However, such study was restricted to NOB effects in liquids like water and glycerol. It found a decrease of Nu and an increase of the center temperature T c as compared to the OB case. The latter could be explained quantitatively by an extension of the Prandtl-Blasius BL theory [13]. Extending this theory to gases is considerably more challenging, as then also the density depends on temperature (beyond the OB dependence), leading to a density boundary layer. Moreover, all material properties such as , , the shear viscosity , and the specific heat c P depend on both temperature and density. Here we shall show that nevertheless an extension of th...

The phenomenon of irregular cessation and subsequent reversal of the large-scale circulation in turbulent Rayleigh-Bénard convection is theoretically analysed. The force and thermal balance on a single plume detached from the thermal boundary layer yields a set of coupled nonlinear equations, whose dynamics is related to the Lorenz equations. For Prandtl and Rayleigh numbers in the range 10 −2 ≤ Pr ≤ 10 3 and 10 7 ≤ Ra ≤ 10 12 , the model has the following features: (i) chaotic reversals may be exhibited at Ra ≥ 10 7 ; (ii) the Reynolds number based on the root mean square velocity scales as Rerms ∼ Ra One important issue in turbulent Rayleigh-Bénard convection is the interplay between the large-scale circulation (the so-called wind) [1] and the dynamics of plumes detached from the thermal boundary layers [2]. In particular, such interplay seems to be relevant in the process of circulation reversals, which occur in an irregular time sequence [3,4,5,6,7,8]. Remarkably, similar reversals are also observed in the wind direction of the atmosphere [9] and in the magnetic polarity of the earth [10].In principle, two reversal scenarios are possible: Reversal through cessation of the convection roll, and reversal through its azimuthal rotation. With two temperature sensors placed close to each other near the sidewall [4,5], one can detect roll reversals, but not distinguish between the two scenarios. With several sensors placed along the azimuth of the cell, Cioni et al.[6] succeeded to detect reversal through azimuthal rotation of the roll. Reversal through rotation was also detected in refs. [7,8]. However, with an ingenious multi-probe setup, Brown, Nikolaenko, and Ahlers [8] were able to distinguish between the rotation and cessation scenarios, and many reversals through cessation were detected. Reversal through cessation was also observed in two-dimensional numerical simulations of the Boussinesq equations (see fig. 8 of ref.[11] and fig. 12 of ref. [12]), where the rotation scenario is of course impossible.Since reversal through cessation is a more surprising scenario, the aim of the present work is to reveal its physical mechanism. Qualitatively, the picture is as follows [13]: If an uprising hot plume gets too fast because of a temperature surplus, it fails to cool down sufficiently when passing the top plate. It then is still warmer than the ambient fluid when advected down along the sidewall. By buoyancy it therefore looses speed and counteracts the large-scale circulation. Indeed, the downward wind may be counteracted so strongly that it stops or even reverses its direction. This mechanism can be effective only for sufficiently strong wind, i.e., for sufficiently large Reynolds number, because for slow motion the thermal diffusivity κ has enough time to reduce the temperature surplus of the originally warmer plume relative to its neighbourhood. Then its power to reverse the circulation by buoyancy is gone.The model : In order to quantify the cessation mechanism discussed above, let us first characterize the s...

As shown in earlier work ͓Ahlers et al., J. Fluid Mech. 569, 409 ͑2006͔͒, non-Oberbeck-Boussinesq ͑NOB͒ corrections to the center temperature in turbulent Rayleigh-Bénard convection in water and also in glycerol are governed by the temperature dependences of the kinematic viscosity and the thermal diffusion coefficient. If the working fluid is ethane close to the critical point, the origin of non-Oberbeck-Boussinesq corrections is very different, as will be shown in the present paper. Namely, the main origin of NOB corrections then lies in the strong temperature dependence of the isobaric thermal expansion coefficient ␤͑T͒. More precisely, it is the nonlinear T dependence of the density ͑T͒ in the buoyancy force that causes another type of NOB effect. We demonstrate this through a combination of experimental, numerical, and theoretical work, the last in the framework of the extended Prandtl-Blasius boundary-layer theory developed by Ahlers et al. as cited above. The theory comes to its limits if the temperature dependence of the thermal expension coefficient ␤͑T͒ is significant. The measurements reported here cover the ranges 2.1Շ PrՇ 3.9 and 5 ϫ 10 9 Շ RaՇ 2 ϫ 10 12 and are for cylindrical samples of aspect ratios 1.0 and 0.5.

The large scale “wind of turbulence” of thermally driven flow is analyzed for very large Rayleigh numbers between 4∙1011 and 7∙1011 and Prandtl number of 0.71 (air at 40°C) and aspect ratios order of one. The wind direction near the upper plate is found to horizontally oscillate with a typical time scale very similar to the large eddy turnover time. The temporal autocorrelation of the wind direction reveals an extremely long memory of the system for the direction. We then apply and extend the dynamical model of Gledzer, Dolzhansky, and Obukhov to the flow, which is based on the Boussinesq equations in the bulk and which can be solved analytically in the inviscid and unforced limit, but which completely ignores the boundary layer and plume dynamics. Nevertheless, the model correctly reproduces both the oscillations of the horizontal wind direction and its very long memory. It is therefore concluded that the boundary layers and the plumes are not necessary to account for the oscillations of the wind direction. The oscillations rather occur as intrinsic precession of the bulk flow.