Aspect Ratio Dependence of Heat Transfer in a Cylindrical Rayleigh-Bénard Cell.

While the heat transfer and the flow dynamics in a cylindrical Rayleigh-Bénard (RB) cell are rather independent of the aspect ratio Γ (diameter/height) for large Γ, a small-Γ cell considerably stabilizes the flow and thus affects the heat transfer. Here, we first theoretically and numerically show that the critical Rayleigh number for the onset of convection at given Γ follows Ra_{c,Γ}∼Ra_{c,∞}(1+CΓ^{-2})^{2}, with C≲1.49 for Oberbeck-Boussinesq (OB) conditions. We then show that, in a broad aspect ratio range (1/32)≤Γ≤32, the rescaling Ra→Ra_{ℓ}≡Ra[Γ^{2}/(C+Γ^{2})]^{3/2} collapses various OB numerical and almost-OB experimental heat transport data Nu(Ra,Γ). Our findings predict the Γ dependence of the onset of the ultimate regime Ra_{u,Γ}∼[Γ^{2}/(C+Γ^{2})]^{-3/2} in the OB case. This prediction is consistent with almost-OB experimental results (which only exist for Γ=1, 1/2, and 1/3) for the transition in OB RB convection and explains why, in small-Γ cells, much larger Ra (namely, by a factor Γ^{-3}) must be achieved to observe the ultimate regime.

Boundary Zonal Flow in Rotating Turbulent Rayleigh-Bénard Convection.

For rapidly rotating turbulent Rayleigh-Bénard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one, whose signature is a bimodal temperature distribution near the radial boundary. The BZF width is found to scale like Ra^{1/4}Ek^{2/3} where the Ekman number Ek decreases with increasing rotation rate.

Heat-transport enhancement in rotating turbulent Rayleigh-Bénard convection.

We present new Nusselt-number (Nu) measurements for slowly rotating turbulent thermal convection in cylindrical samples with aspect ratio Γ=1.00 and provide a comprehensive correlation of all available data for that Γ. In the experiment compressed gasses (nitrogen and sulfur hexafluride) as well as the fluorocarbon C_{6}F_{14} (3M Fluorinert FC72) and isopropanol were used as the convecting fluids. The data span the Prandtl-number (Pr) range 0.74

Multiple transitions in rotating turbulent Rayleigh-Bénard convection.

Sometimes it is thought that sharp transitions between potentially different turbulent states should be washed out by the prevailing intense fluctuations and short coherence lengths and times. Contrary to this expectation, we found a sequence of such transitions in turbulent rotating Rayleigh-Bénard convection as the rotation rate was increased. This phenomenon was observed in cylindrical samples with aspect ratios (diameter/height) Γ=1.00 and 0.50. It became most prominent at very large Rayleigh numbers up to 2×10(12), where the fluctuations are extremely vigorous, and was manifested most clearly for Γ=1.00. It was found in the heat transport as well as in the temperature gradient near the sample center. We conjecture that the transitions are between different large-scale structures which involve changes of symmetry and thus cannot be gradual [L. Landau, Zh. Eksp. Teor. Fiz. 7, 19 (1937); L. D. Landau, Phys. Z. Sowjetunion 11, 26 (1937); L. D. Landau, in Collected Papers of L. D. Landau, (Oxford University Press, Oxford, 1965), pp. 193-216].

Logarithmic spatial variations and universal f-1 power spectra of temperature fluctuations in turbulent Rayleigh-Bénard convection.

We report measurements of the temperature variance σ(2)(z,r) and frequency power spectrum P(f,z,r) (z is the distance from the sample bottom and r the radial coordinate) in turbulent Rayleigh-Bénard convection (RBC) for Rayleigh numbers Ra = 1.6 × 10(13) and 1.1 × 10(15) and for a Prandtl number Pr ≃ 0.8 for a sample with a height L = 224 cm and aspect ratio D/L=0.50 (D is the diameter). For z/L ≲ 0.1 σ(2)(z,r) was consistent with a logarithmic dependence on z, and there was a universal (independent of Ra, r, and z) normalized spectrum which, for 0.02 ≲ fτ(0) ≲ 0.2, had the form P(fτ(0)) = P(0)(fτ(0))(-1) with P(0) = 0.208 ± 0.008 a universal constant. Here τ(0) = sqrt[2R] where R is the radius of curvature of the temperature autocorrelation function C(τ) at τ = 0. For z/L ≃ 0.5 the measurements yielded P(fτ(0))∼(fτ(0))(-α) with α in the range from 3/2 to 5/3. All the results are similar to those for velocity fluctuations in shear flows at sufficiently large Reynolds numbers, suggesting the possibility of an analogy between the flows that is yet to be determined in detail.

Logarithmic temperature profiles in turbulent Rayleigh-Bénard convection.

We report results for the temperature profiles of turbulent Rayleigh-Bénard convection (RBC) in the interior of a cylindrical sample of aspect ratio Γ≡D/L=0.50 (D and L are the diameter and height, respectively). Both in the classical and in the ultimate state of RBC we find that the temperature varies as A×ln(z/L)+B, where z is the distance from the bottom or top plate. In the classical state, the coefficient A decreases in the radial direction as the distance from the side wall increases. For the ultimate state, the radial dependence of A has not yet been determined. These findings are based on experimental measurements over the Rayleigh-number range 4×10(12)≲Ra≲10(15) for a Prandtl number Pr≃0.8 and on direct numerical simulation at Ra=2×10(12), 2×10(11), and 2×10(10), all for Pr=0.7.

Plume fragmentation by bulk interactions in turbulent Rayleigh-Bénard convection.

Using compressed gases with Prandtl numbers near 0.7, we obtained flow visualizations of turbulent Rayleigh-Bénard convection in a cylindrical sample with an aspect ratio Γ≡D/L≅10 (D is the diameter and L the height) by the shadowgraph method. Focusing on the plumes under the top plate, we found that their length had a log-normal distribution, suggesting a fragmentation process. Fragmentation events could be visually identified in the images and involved plume interactions with bulk fluctuations or upwelling domain walls. We found the mean spacing between plumes to vary with the Rayleigh number in proportion to the volume-averaged Kolmogorov length of the turbulent bulk fluctuations, providing further evidence for plume-bulk interactions.

Transition to the ultimate state of turbulent Rayleigh-Bénard convection.

Measurements of the Nusselt number Nu and of a Reynolds number Re(eff) for Rayleigh-Bénard convection (RBC) over the Rayleigh-number range 10(12)≲Ra≲10(15) and for Prandtl numbers Pr near 0.8 are presented. The aspect ratio Γ≡D/L of a cylindrical sample was 0.50. For Ra≲10(13) the data yielded Nu∝Ra(γ(eff)) with γ(eff)≃0.31 and Re(eff)∝Ra(ζ(eff)) with ζ(eff)≃0.43, consistent with classical turbulent RBC. After a transition region for 10(13)≲Ra≲5×10(14), where multistability occurred, we found γ(eff)≃0.38 and ζ(eff)=ζ≃0.50, in agreement with the results of Grossmann and Lohse for the large-Ra asymptotic state with turbulent boundary layers which was first predicted by Kraichnan.

Onset of Rayleigh-Bénard convection in cylindrical containers.

We determined the critical Rayleigh numbers Ra{c} for the onset of convection in cylindrical containers with aspect ratios 1 approximately 4, an m=1 mode was found again, but near Gamma=4 either m=1 or m=2 was observed in different runs. These results are consistent with the marginal stability curves calculated by Buell and Catton in the sense that the mode that is the first as a function of Ra to acquire a positive growth rate is the one that is observed. For Gamma approximately >4, the theoretical marginal curves for the four lowest modes lie very close together. There we found patterns near onset that corresponded to various modes, including m=2 and 4. At relatively large Gamma approximately > 6, we observed parallel straight rolls quite close to onset. Our patterns agree with several DNS investigations by others, but at some Gamma values differ from those observed experimentally by Stork and Müller. Some results for the pattern evolution with increasing Ra are reported as well.

Effect of a polymer additive on heat transport in turbulent Rayleigh-Bénard convection.

Measurements of heat transport, as expressed by the Nusselt number Nu, are reported for turbulent Rayleigh-Bénard convection of water containing up to 120 ppm by weight of poly-[ethylene oxide] with a molecular weight of 4x10{6} g/mole. Over the Rayleigh number range 5x10{9} less than or approximately Ra less than or approximately 7x10{10} Nu is smaller than it is for pure water by up to 10%.

Finite-size effects lead to supercritical bifurcations in turbulent rotating Rayleigh-Bénard convection.

In turbulent thermal convection in cylindrical samples with an aspect ratio Γ≡D/L (D is the diameter and L the height), the Nusselt number Nu is enhanced when the sample is rotated about its vertical axis because of the formation of Ekman vortices that extract additional fluid out of thermal boundary layers at the top and bottom. We show from experiments and direct numerical simulations that the enhancement occurs only above a bifurcation point at a critical inverse Rossby number 1/Ro(c), with 1/Ro(c)∝1/Γ. We present a Ginzburg-Landau-like model that explains the existence of a bifurcation at finite 1/Ro(c) as a finite-size effect. The model yields the proportionality between 1/Ro(c) and 1/Γ and is consistent with several other measured or computed system properties.

Search for the "ultimate state" in turbulent Rayleigh-Bénard convection.

Measurements of the Nusselt number Nu and of temperature variations DeltaTb in the bulk fluid are reported for turbulent Rayleigh-Bénard convection of a cylindrical sample. They cover the Rayleigh-number range 10(9) less than or similar to Ra less than or similar to 3x10(14) using He (Prandtl number Pr=0.67), N2 (Pr=0.72) and SF6 (Pr=0.79 to 0.84) at pressures up to 15 bars and near-ambient temperatures. The sample had a height L=2.24 m and diameter D=1.12 m and was located in a new High-Pressure Convection Facility (HPCF) at the Max Planck Institute for Dynamics and Self-Organization in Göttingen, Germany. The data do not show the transition to an "ultimate regime" reported by Chavanne et al. and are consistent with the measurements of Niemela et al.

Transitions between turbulent states in rotating Rayleigh-Bénard convection.

Weakly rotating turbulent Rayleigh-Bénard convection was studied experimentally and numerically. With increasing rotation and large enough Rayleigh number a supercritical bifurcation from a turbulent state with nearly rotation-independent heat transport to another with enhanced heat transfer is observed at a critical inverse Rossby number 1/Roc approximately 0.4. The strength of the large-scale convection roll is either enhanced or essentially unmodified depending on parameters for 1/Ro<1/Roc, but the strength increasingly diminishes beyond 1/Roc where it competes with Ekman vortices that cause vertical fluid transport and thus heat-transfer enhancement.

Enhanced heat transport by turbulent two-phase Rayleigh-Bénard convection.

We report measurements of turbulent heat transport in samples of ethane (C2H6) heated from below while the applied temperature difference DeltaT straddled the liquid-vapor coexistence curve T(phi)(P). When the sample top temperature T(t) decreased below T(phi), droplet condensation occurred and the latent heat of vaporization H provided an additional heat-transport mechanism. The effective conductivity lambda(eff) increased linearly with decreasing T(t), and reached a maximum value lambda(eff)(*) that was an order of magnitude larger than the single-phase lambda(eff). As P approached the critical pressure, lambda(eff)(*) increased dramatically even though H vanished. We attribute this phenomenon to an enhanced droplet-nucleation rate as the critical point is approached.

Prandtl-, Rayleigh-, and Rossby-number dependence of heat transport in turbulent rotating Rayleigh-Bénard convection.

Experimental and numerical data for the heat transfer as a function of the Rayleigh, Prandtl, and Rossby numbers in turbulent rotating Rayleigh-Bénard convection are presented. For relatively small Ra approximately 10(8) and large Pr modest rotation can enhance the heat transfer by up to 30%. At larger Ra there is less heat-transfer enhancement, and at small Pr less, similar 0.7 there is no heat-transfer enhancement at all. We suggest that the small-Pr behavior is due to the breakdown of the heat-transfer-enhancing Ekman pumping because of larger thermal diffusion.

Non-Oberbeck-Boussinesq effects in turbulent thermal convection in ethane close to the critical point.

As shown in earlier work [Ahlers, J. Fluid Mech. 569, 409 (2006)], non-Oberbeck-Boussinesq (NOB) 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 beta(T). More precisely, it is the nonlinear T dependence of the density rho(T) 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 as cited above. The theory comes to its limits if the temperature dependence of the thermal expension coefficient beta(T) is significant. The measurements reported here cover the ranges 2.1

Large-scale circulation model for turbulent Rayleigh-Bénard convection.

A model for the large-scale-circulation (LSC) of turbulent Rayleigh-Bénard convection in cylindrical samples is presented. It consists of two physically motivated stochastic ordinary differential equations, one each for the strength and the azimuthal orientation of the LSC. Stochastic forces represent phenomenologically the influence of turbulent fluctuations. Consistent with measurements, the model yields an azimuthally meandering LSC with occasional rotations, and with more rare cessations. As in experiment, cessations have a uniform distribution of LSC orientation changes.