Transition to the ultimate state of turbulent Rayleigh-Bénard convection

Ultimate Regime of Rayleigh-Bénard Turbulence: Subregimes and Their Scaling Relations for the Nusselt vs Rayleigh and Prandtl Numbers

We offer a new model for the heat transfer and the turbulence intensity in strongly driven Rayleigh-Bénard turbulence (the so-called ultimate regime), which in contrast to hitherto models is consistent with the new mathematically exact heat transfer upper bound of Choffrut et al. [Upper bounds on Nusselt number at finite Prandtl number, J. Differ. Equations 260, 3860 (2016).JDEQAK0022-039610.1016/j.jde.2015.10.051] and thus enables extrapolations of the heat transfer to geo- and astrophysical flows. The model distinguishes between four subregimes of the ultimate regime and well describes the measured heat transfer in various large-Rayleigh experiments. In this new representation, which properly accounts for the Prandtl number dependence, the onset to the ultimate regime is seen in all available large-Rayleigh datasets, though at different Rayleigh numbers, as to be expected for a non-normal-nonlinear instability.

Experimental Evidence for the Existence of the Ultimate Regime in Rapidly Rotating Turbulent Thermal Convection

What is the final state of turbulence when the driving parameter approaches infinity? For the traditional Rayleigh-Bénard convection, a possible ultimate scaling dependence of the heat transport (quantified by the Nusselt number Nu) on the Rayleigh number (Ra), which can be extrapolated to arbitrarily high Ra, is predicted by theories. The existence of the ultimate scaling has been intensively debated in the past decades. In this Letter, we adopt a novel supergravitational thermal convection experimental setup to study the possible transition to the ultimate regime. This system is characterized by the combined effects of radial-dependent centrifugal force, the Earth's gravity, and the Coriolis force. With an effective gravity up to 100 times the Earth's gravity, both Ra and shear Reynolds number can be boosted due to the increase of the buoyancy driving and the additional Coriolis forces. With over a decade of Ra range, we demonstrate the existence of ultimate regime with four direct evidences: the ultimate scaling dependence of Nu versus Ra; the appearance of the turbulent velocity boundary layer profile; the enhanced strength of the shear Reynolds number; and the new statistical properties of local temperature fluctuations. The present findings will greatly improve the understanding of the flow dynamics in geophysical and astrophysical flows.

Bayesian emulation and history matching of JUNE

We analyze JUNE: a detailed model of COVID-19 transmission with high spatial and demographic resolution, developed as part of the RAMP initiative. JUNE requires substantial computational resources to evaluate, making model calibration and general uncertainty analysis extremely challenging. We describe and employ the uncertainty quantification approaches of Bayes linear emulation and history matching to mimic JUNE and to perform a global parameter search, hence identifying regions of parameter space that produce acceptable matches to observed data, and demonstrating the capability of such methods. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.

Velocity-informed upper bounds on the convective heat transport induced by internal heat sources and sinks

Three-dimensional convection driven by internal heat sources and sinks (CISS) leads to experimental and numerical scaling laws compatible with a mixing-length-or 'ultimate'-scaling regime [Formula: see text]. However, asymptotic analytic solutions and idealized two-dimensional simulations have shown that laminar flow solutions can transport heat even more efficiently, with [Formula: see text]. The turbulent nature of the flow thus has a profound impact on its transport properties. In the present contribution, we give this statement a precise mathematical sense. We show that the Nusselt number maximized over all solutions is bounded from above by [Formula: see text], before restricting attention to 'fully turbulent branches of solutions', defined as families of solutions characterized by a finite non-zero limit of the dissipation coefficient at large driving amplitude. Maximization of [Formula: see text] over such branches of solutions yields the better upper-bound [Formula: see text]. We then provide three-dimensional numerical and experimental data of CISS compatible with a finite limiting value of the dissipation coefficient at large driving amplitude. It thus seems that CISS achieves the maximal heat transport scaling over fully turbulent solutions. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Steady thermal convection representing the ultimate scaling

Nonlinear simple invariant solutions representing the ultimate scaling have been discovered to the Navier-Stokes equations for thermal convection between horizontal no-slip permeable walls with a distance [Formula: see text] and a constant temperature difference [Formula: see text]. On the permeable walls, the vertical transpiration velocity is assumed to be proportional to the local pressure fluctuations, i.e. [Formula: see text] (Jiménez et al. 2001 J. Fluid Mech., 442, 89-117. (doi:10.1017/S0022112001004888)). Two-dimensional steady solutions bifurcating from a conduction state have been obtained using a Newton-Krylov iteration up to the Rayleigh number [Formula: see text] for the Prandtl number [Formula: see text], the horizontal period [Formula: see text] and the permeability parameter [Formula: see text]-[Formula: see text], [Formula: see text] being the buoyancy-induced terminal velocity. The wall permeability has a significant impact on the onset and the scaling properties of the found steady 'wall-bounded' thermal convection. The ultimate scaling [Formula: see text] has been observed for [Formula: see text] at high [Formula: see text], where [Formula: see text] is the Nusselt number. In the steady ultimate state, large-scale thermal plumes fully extend from one wall to the other, inducing the strong vertical velocity comparable with the terminal velocity [Formula: see text] as well as intense temperature variation of [Formula: see text] even in the bulk region. As a consequence, the wall-to-wall heat flux scales with [Formula: see text] independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Collapse of Coherent Large Scale Flow in Strongly Turbulent Liquid Metal Convection

The large-scale flow structure and the turbulent transfer of heat and momentum are directly measured in highly turbulent liquid metal convection experiments for Rayleigh numbers varied between 4×10^{5} and ≤5×10^{9} and Prandtl numbers of 0.025≤Pr≤0.033. Our measurements are performed in two cylindrical samples of aspect ratios Γ=diameter/height=0.5 and 1 filled with the eutectic alloy GaInSn. The reconstruction of the three-dimensional flow pattern by 17 ultrasound Doppler velocimetry sensors detecting the velocity profiles along their beam lines in different planes reveals a clear breakdown of coherence of the large-scale circulation for Γ=0.5. As a consequence, the scaling laws for heat and momentum transfer inherit a dependence on the aspect ratio. We show that this breakdown of coherence is accompanied with a reduction of the Reynolds number Re. The scaling exponent β of the power law Nu∝Ra^{β} crosses eventually over from β=0.221 to 0.124 when the liquid metal flow at Γ=0.5 reaches Ra≳2×10^{8} and the coherent large-scale flow is completely collapsed.

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.

Vibration-induced boundary-layer destabilization achieves massive heat-transport enhancement

Thermal turbulence is well known as a potent means to convey heat across space by a moving fluid. The existence of the boundary layers near the plates, however, bottlenecks its heat-exchange capability. Here, we conceptualize a mechanism of thermal vibrational turbulence that breaks through the boundary-layer limitation and achieves massive heat-transport enhancement. When horizontal vibration is applied to the convection cell, a strong shear is induced to the body of fluid near the conducting plates, which destabilizes thermal boundary layers, vigorously triggers the eruptions of thermal plumes, and leads to a heat-transport enhancement by up to 600%. We further reveal that such a vibration-induced shear can very efficiently disrupt the boundary layers. The present findings open a new avenue for research into heat transport and will also bring profound changes in many industrial applications where thermal flux through a fluid is involved and the mechanical vibration is usually inevitable.

Classical 1/3 scaling of convection holds up to Ra = 1015

The heat transport law in turbulent convection remains central to current research in the field. Our present knowledge of the heat transport law for Ra>1012 is inconclusive, where the Rayleigh number Ra is a measure of the strength of convection. Massively parallel simulations of the three-dimensional convection have progressed to Ra=1015 in slender cells. We resolve velocity gradients inside thin boundary layers and show that the turbulent heat transport continues to follow the classical 1/3 scaling law with no transition to the so-called “ultimate” state that is variously argued to have 1/2 scaling. Our work suggests that the boundary layers remain marginally stable and continue to act as the bottleneck for global heat transport.

The global transport of heat and momentum in turbulent convection is constrained by thin thermal and viscous boundary layers at the heated and cooled boundaries of the system. This bottleneck is thought to be lifted once the boundary layers themselves become fully turbulent at very high values of the Rayleigh number Ra—the dimensionless parameter that describes the vigor of convective turbulence. Laboratory experiments in cylindrical cells for Ra≳1012 have reported different outcomes on the putative heat transport law. Here we show, by direct numerical simulations of three-dimensional turbulent Rayleigh–Bénard convection flows in a slender cylindrical cell of aspect ratio 1/10, that the Nusselt number—the dimensionless measure of heat transport—follows the classical power law of Nu=(0.0525±0.006)×Ra0.331±0.002 up to Ra=1015. Intermittent fluctuations in the wall stress, a blueprint of turbulence in the vicinity of the boundaries, manifest at all Ra considered here, increasing with increasing Ra, and suggest that an abrupt transition of the boundary layer to turbulence does not take place.

A large thermal turbulent Taylor-Couette (THETACO) facility for investigation of turbulence induced by simultaneous action of rotation and radial temperature gradient

A thermal turbulent Taylor-Couette facility has been designed to investigate turbulent flows generated by differential rotation and radial temperature gradient. It consists of a cylindrical annulus with a rotating inner cylinder and a fixed outer cylinder. The electric heating system is installed inside the inner cylinder, and the annulus is immersed in a large cylindrical container filled with cooling fluid. Temperature regulators independently control the temperature of the inner surface of the inner cylinder and that of the cooling fluid. The facility allows us to reach values of the Reynolds number (Re ∼ 5 × 10⁵) and of the Rayleigh number (Ra ∼ 3 × 10⁶) for water as the working fluid. The facility provides torque measurements, a full optical access at the side and from the bottom for velocity measurements using particle image velocimetry (2D, stereoscopic, and tomographic). Temperature measurements in the flow can be performed by thermochromic liquid crystals or laser induced fluorescence.

Elusive transition to the ultimate regime of turbulent Rayleigh-Bénard convection

By using cryogenic ^{4}He gas as the working fluid in a cylindrical cell 0.3 m in both height and diameter, we study the influence of non-Oberbeck-Boussinesq (NOB) effects on the heat transfer in turbulent Rayleigh-Bénard convection (RBC). We show that the NOB effects increase the heat transfer efficiency when the top plate temperature closely approaches the saturation vapor curve even far away from the critical point. Viewed in this light, our analysis points to the likelihood that the claim of having observed the transition to Kraichnan's ultimate regime, under nominally similar conditions in the experiments with SF_{6} [Phys. Rev. Lett. 108, 024502 (2012)PRLTAO0031-900710.1103/PhysRevLett.108.024502], is probably an NOB effect and the important issue of the transition to the ultimate state of RBC remains open.

Rough-wall turbulent Taylor-Couette flow: The effect of the rib height

In this study, we combine experiments and direct numerical simulations to investigate the effects of the height of transverse ribs at the walls on both global and local flow properties in turbulent Taylor-Couette flow. We create rib roughness by attaching up to 6 axial obstacles to the surfaces of the cylinders over an extensive range of rib heights, up to blockages of 25% of the gap width. In the asymptotic ultimate regime, where the transport is independent of viscosity, we emperically find that the prefactor of the [Formula: see text] scaling (corresponding to the drag coefficient [Formula: see text] being constant) scales with the number of ribs [Formula: see text] and by the rib height [Formula: see text]. The physical mechanism behind this is that the dominant contribution to the torque originates from the pressure forces acting on the rib which scale with the rib height. The measured scaling relation of [Formula: see text] is slightly smaller than the expected [Formula: see text] scaling, presumably because the ribs cannot be regarded as completely isolated but interact. In the counter-rotating regime with smooth walls, the momentum transport is increased by turbulent Taylor vortices. We find that also in the presence of transverse ribs these vortices persist. In the counter-rotating regime, even for large roughness heights, the momentum transport is enhanced by these vortices.

Radiative heating achieves the ultimate regime of thermal convection

The absorption of light or radiation drives turbulent convection inside stars, supernovae, frozen lakes, and Earth's mantle. In these contexts, the goal of laboratory and numerical studies is to determine the relation between the internal temperature gradients and the heat flux transported by the turbulent flow. This is the constitutive law of turbulent convection, to be input into large-scale models of such natural flows. However, in contrast with the radiative heating of natural flows, laboratory experiments have focused on convection driven by heating and cooling plates; the heat transport is then severely restricted by boundary layers near the plates, which prevents the realization of the mixing length scaling law used in evolution models of geophysical and astrophysical flows. There is therefore an important discrepancy between the scaling laws measured in laboratory experiments and those used, e.g., in stellar evolution models. Here we provide experimental and numerical evidence that radiatively driven convection spontaneously achieves the mixing length scaling regime, also known as the "ultimate" regime of thermal convection. This constitutes a clear observation of this regime of turbulent convection. Our study therefore bridges the gap between models of natural flows and laboratory experiments. It opens an experimental avenue for a priori determinations of the constitutive laws to be implemented into models of geophysical and astrophysical flows, as opposed to empirical fits of these constitutive laws to the scarce observational data.

Transition to the Ultimate Regime in Two-Dimensional Rayleigh-Bénard Convection

The possible transition to the so-called ultimate regime, wherein both the bulk and the boundary layers are turbulent, has been an outstanding issue in thermal convection, since the seminal work by Kraichnan [Phys. Fluids 5, 1374 (1962)PFLDAS0031-917110.1063/1.1706533]. Yet, when this transition takes place and how the local flow induces it is not fully understood. Here, by performing two-dimensional simulations of Rayleigh-Bénard turbulence covering six decades in Rayleigh number Ra up to 10^{14} for Prandtl number Pr=1, for the first time in numerical simulations we find the transition to the ultimate regime, namely, at Ra^{*}=10^{13}. We reveal how the emission of thermal plumes enhances the global heat transport, leading to a steeper increase of the Nusselt number than the classical Malkus scaling Nu∼Ra^{1/3} [Proc. R. Soc. A 225, 196 (1954)PRLAAZ1364-502110.1098/rspa.1954.0197]. Beyond the transition, the mean velocity profiles are logarithmic throughout, indicating turbulent boundary layers. In contrast, the temperature profiles are only locally logarithmic, namely, within the regions where plumes are emitted, and where the local Nusselt number has an effective scaling Nu∼Ra^{0.38}, corresponding to the effective scaling in the ultimate regime.

Roughness-Facilitated Local 1/2 Scaling Does Not Imply the Onset of the Ultimate Regime of Thermal Convection

In thermal convection, roughness is often used as a means to enhance heat transport, expressed in Nusselt number. Yet there is no consensus on whether the Nusselt vs Rayleigh number scaling exponent (Nu∼Ra^{β}) increases or remains unchanged. Here we numerically investigate turbulent Rayleigh-Bénard convection over rough plates in two dimensions, up to Ra≈10^{12}. Varying the height and wavelength of the roughness elements with over 200 combinations, we reveal the existence of two universal regimes. In the first regime, the local effective scaling exponent can reach up to 1/2. However, this cannot be explained as the attainment of the so-called ultimate regime as suggested in previous studies, because a further increase in Ra leads to the second regime, in which the scaling saturates back to a value close to the smooth wall case. Counterintuitively, the transition from the first to the second regime corresponds to the competition between bulk and boundary layer flow: from the bulk-dominated regime back to the classical boundary-layer-controlled regime. Our study demonstrates that the local 1/2 scaling does not necessarily signal the onset of ultimate turbulence.

Roughness as a Route to the Ultimate Regime of Thermal Convection

We use highly resolved numerical simulations to study turbulent Rayleigh-Bénard convection in a cell with sinusoidally rough upper and lower surfaces in two dimensions for Pr=1 and Ra=[4×10^{6},3×10^{9}]. By varying the wavelength λ at a fixed amplitude, we find an optimal wavelength λ_{opt} for which the Nusselt-Rayleigh scaling relation is (Nu-1∝Ra^{0.483}), maximizing the heat flux. This is consistent with the upper bound of Goluskin and Doering [J. Fluid Mech. 804, 370 (2016)JFLSA70022-112010.1017/jfm.2016.528] who prove that Nu can grow no faster than O(Ra^{1/2}) as Ra→∞, and thus with the concept that roughness facilitates the attainment of the so-called ultimate regime. Our data nearly achieve the largest growth rate permitted by the bound. When λ≪λ_{opt} and λ≫λ_{opt}, the planar case is recovered, demonstrating how controlling the wall geometry manipulates the interaction between the boundary layers and the core flow. Finally, for each Ra, we choose the maximum Nu among all λ, thus optimizing over all λ, to find Nu_{opt}-1=0.01×Ra^{0.444}.

Dynamics of large-scale quantities in Rayleigh-Bénard convection

In this paper we estimate the relative strengths of various terms of the Rayleigh-Bénard equations. Based on these estimates and scaling analysis, we derive a general formula for the large-scale velocity U or the Péclet number that is applicable for arbitrary Rayleigh number Ra and Prandtl number Pr. Our formula fits reasonably well with the earlier simulation and experimental results. Our analysis also shows that the wall-bounded convection has enhanced viscous force compared to free turbulence. We also demonstrate how correlations deviate the Nusselt number scaling from the theoretical prediction of Ra^{1/2} to the experimentally observed scaling of nearly Ra^{0.3}.

Decay of turbulence at high reynolds numbers

Turbulent motions in a fluid decay at a certain rate once stirring has stopped. The role of the most basic parameter in fluid mechanics, the Reynolds number, in setting the decay rate is not generally known. This Letter concerns the high-Reynolds-number limit of the process. In a classical grid-turbulence wind-tunnel experiment that both reaches higher Reynolds numbers than ever before and covers a wide range of them (10^{4}<Re=UM/ν<5×10^{6}), we measure the decay rate with the unprecedented precision of about 2%. Here U is the mean speed of the flow, M is the forcing scale, and ν is the kinematic viscosity of the fluid. We observe that the decay rate is Reynolds-number independent, which contradicts some models and supports others.

Evolution of Nanofluid Rayleigh–Bénard Flows Between Two Parallel Plates: A Mesoscopic Modeling Study

The developing and developed nanofluid Rayleigh–Bénard flows between two parallel plates was simulated using the mesoscopic thermal lattice-Boltzmann method (LBM). The coupled effects of the thermal conductivity and the dynamic viscosity on the evolution of Rayleigh–Bénard flows were examined using different particle volume fractions (1–4%), while the individual effects of the thermal conductivity and the dynamic viscosity were tested using various particle sizes (11 nm, 20 nm, and 30 nm) and nanoparticle types (Al2O3, Cu, and CuO2). Two different heating modes were also considered. The results show that Rayleigh–Bénard cell in nanofluids is significantly different from that in pure fluids. The stable convection cells in nanofluids come from the expansion and shedding of an initial vortex pair, while the flow begins suddenly in pure water when the Rayleigh number reaches a critical value. Therefore, the average Nusselt number increases gradually for nanofluids but sharply for pure liquids. Uniform fully developed flow cells with fewer but larger vortex pairs are generated with the bottom heating with nanofluids than with pure liquid, with extremely tiny vortexes confined near the top heating plate for top heating. The number of vortex pairs decreases with increasing nanoparticle volume fraction and particle diameter due to the increasing of dynamic viscosity. The average Nusselt number increases with the increasing Rayleigh number, while decreases with the increasing nanoparticle diameters. The nanoparticle types have little effect on the Rayleigh–Bénard flow patterns. The Rayleigh–Bénard flows are more sensitive with the dynamic viscosity than the thermal conductivity of nanofluids.

Effect of velocity boundary conditions on the heat transfer and flow topology in two-dimensional Rayleigh-Bénard convection

The effect of various velocity boundary condition is studied in two-dimensional Rayleigh-Bénard convection. Combinations of no-slip, stress-free, and periodic boundary conditions are used on both the sidewalls and the horizontal plates. For the studied Rayleigh numbers Ra between 10(8) and 10(11) the heat transport is lower for Γ=0.33 than for Γ=1 in case of no-slip sidewalls. This is, surprisingly, the opposite for stress-free sidewalls, where the heat transport increases for a lower aspect ratio. In wider cells the aspect-ratio dependence is observed to disappear for Ra ≥ 10(10). Two distinct flow types with very different dynamics can be seen, mostly dependent on the plate velocity boundary condition, namely roll-like flow and zonal flow, which have a substantial effect on the dynamics and heat transport in the system. The predominantly horizontal zonal flow suppresses heat flux and is observed for stress-free and asymmetric plates. Low aspect-ratio periodic sidewall simulations with a no-slip boundary condition on the plates also exhibit zonal flow. In all the other cases, the flow is roll like. In two-dimensional Rayleigh-Bénard convection, the velocity boundary conditions thus have large implications on both roll-like and zonal flow that have to be taken into consideration before the boundary conditions are imposed.

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.

Scaling of heat flux and energy spectrum for very large Prandtl number convection

Under the limit of infinite Prandtl number, we derive analytical expressions for the large-scale quantities, e.g., Péclet number Pe, Nusselt number Nu, and rms value of the temperature fluctuations θ(rms). We complement the analytical work with direct numerical simulations, and show that Nu ∼ Ra(γ) with γ ≈ (0.30-0.32), Pe ∼ Ra(η) with η ≈ (0.57-0.61), and θ(rms) ∼ const. The Nusselt number is observed to be an intricate function of Pe, θ(rms), and a correlation function between the vertical velocity and temperature. Using the scaling of large-scale fields, we show that the energy spectrum E(u)(k) ∼ k(-13/3), which is in a very good agreement with our numerical results. The entropy spectrum E(θ)(k), however, exhibits dual branches consisting of k(-2) and k(0) spectra; the k(-2) branch corresponds to the Fourier modes θ[over ̂](0,0,2n), which are approximately -1/(2 nπ). The scaling relations for Prandtl number beyond 10(2) match with those for infinite Prandtl number.

Logarithmic boundary layers in strong Taylor-Couette turbulence

We provide direct measurements of the boundary layer properties in highly turbulent Taylor-Couette flow up to Re=2×106) (Ta=6.2×10(12)) using high-resolution particle image velocimetry and particle tracking velocimetry. We find that the mean azimuthal velocity profile at the inner and outer cylinder can be fitted by the von Kármán log law u+=1/κ lny+ +B. The von Kármán constant κ is found to depend on the driving strength Ta and for large Ta asymptotically approaches κ≈0.40. The variance profiles of the local azimuthal velocity have a universal peak around y+≈12 and collapse when rescaled with the driving velocity (and not with the friction velocity), displaying a log dependence of y+ as also found for channel and pipe flows.

Spatial distribution of heat flux and fluctuations in turbulent Rayleigh-Bénard convection

We numerically investigate the radial dependence of the velocity and temperature fluctuations and of the time-averaged heat flux j ¯(r) in a cylindrical Rayleigh-Bénard cell with aspect ratio Γ=1 for Rayleigh numbers Ra between 2×10^{6} and 2×10^{9} at a fixed Prandtl number Pr=5.2. The numerical results reveal that the heat flux close to the sidewall is larger than in the center and that, just as the global heat transport, it has an effective power law dependence on the Rayleigh number, j ¯(r)∝Ra{γ{j}(r)}. The scaling exponent γ{j}(r) decreases monotonically from 0.43 near the axis (r≈0) to 0.29 close to the sidewalls (r≈D/2). The effective exponents near the axis and the sidewall agree well with the measurements of Shang et al. [Phys. Rev. Lett. 100, 244503 (2008)] and the predictions of Grossmann and Lohse [Phys. Fluids 16, 1070 (2004)]. Extrapolating our results to large Rayleigh number would imply a crossover at Ra≈10^{15}, where the heat flux near the axis would begin to dominate. In addition, we find that the local heat flux is more than twice as high at the location where warm or cold plumes go up or down than in plume depleted regions.

Effect of boundary layers asymmetry on heat transfer efficiency in turbulent Rayleigh-Bénard convection at very high Rayleigh numbers [corrected]

The heat transfer efficiency in turbulent Rayleigh-Bénard convection is investigated experimentally, in a cylindrical cell of height 0.3 m, diameter 0.3 m. We show that for Rayleigh numbers 10(12) < or approximately equal to Ra < or approximately equal to 10(15) the Nusselt number closely follows Nu is proportional to Ra(1/3 if the mean temperature of the working fluid-cryogenic helium gas-is measured by small sensors directly inside the cell at about half of its height. In contrast, if the mean temperature is determined in a conventional way, as an arithmetic mean of the bottom and top plate temperatures, the Nu(Ra) is proportional to Ra(γ) displays spurious crossover to higher γ that might be misinterpreted as a transition to the ultimate Kraichnan regime.

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.

Boundary layer analysis in turbulent Rayleigh-Bénard convection in air: experiment versus simulation

We report measurements and numerical simulations of the three-dimensional velocity and temperature fields in turbulent Rayleigh-Bénard convection in air. Highly resolved velocity and temperature measurements inside and outside the boundary layers have been directly compared with equivalent data obtained in direct numerical simulations (DNSs). This comparison comprises a set of two Rayleigh numbers at Ra=3×10(9) and 3×10(10) and a fixed aspect ratio; this is the ratio between the diameter and the height of the Rayleigh-Bénard cell of Γ=1. We find that the measured velocity data are in excellent agreement with the DNS results while the temperature data slightly differ. In particular, the measured mean temperature profile does not show the linear trend as seen in the DNS data, and the measured gradients at the wall are significantly higher than those obtained from the DNS. Both viscous and thermal boundary layer thickness scale with respect to the Rayleigh number as δ(v)~Ra(-0.24) and δ(θ)~Ra(-0.24), respectively.

New perspectives in turbulent Rayleigh-Bénard convection

Recent experimental, numerical and theoretical advances in turbulent Rayleigh-Bénard convection are presented. Particular emphasis is given to the physics and structure of the thermal and velocity boundary layers which play a key role for the better understanding of the turbulent transport of heat and momentum in convection at high and very high Rayleigh numbers. We also discuss important extensions of Rayleigh-Bénard convection such as non-Oberbeck-Boussinesq effects and convection with phase changes.

Ultimate regime of high Rayleigh number convection in a porous medium

Well-resolved direct numerical simulations of 2D Rayleigh-Bénard convection in a porous medium are presented for Rayleigh numbers Ra≤4×10(4) which reveal that, contrary to previous indications, the linear classical scaling for the Nusselt number, Nu~Ra, is attained asymptotically. The flow dynamics are analyzed, and the interior of the vigorously convecting system is shown to be increasingly well described as Ra→∞ by a simple columnar "heat-exchanger" model with a single horizontal wave number k and a linear background temperature field. The numerical results are approximately fitted by k~Ra(0.4).

Thermal boundary layer profiles in turbulent Rayleigh-Bénard convection in a cylindrical sample

We numerically investigate the structures of the near-plate temperature profiles close to the bottom and top plates of turbulent Rayleigh-Bénard flow in a cylindrical sample at Rayleigh numbers Ra = 10(8) to Ra = 2 × 10(12) and Prandtl numbers Pr = 6.4 and Pr = 0.7 with the dynamical frame method [Zhou and Xia, Phys. Rev. Lett. 104, 104301 (2010)], thus extending previous results for quasi-two-dimensional systems to three-dimensional systems. The dynamical frame method shows that the measured temperature profiles in the spatially and temporally local frame are much closer to the temperature profile of a laminar, zero-pressure gradient boundary layer (BL) according to Pohlhausen than in the fixed reference frame. The deviation between the measured profiles in the dynamical reference frame and the laminar profiles increases with decreasing Pr, where the thermal BL is more exposed to the bulk fluctuations due to the thinner kinetic BL, and increasing Ra, where more plumes are passing the measurement location.

Ultimate turbulent Taylor-Couette flow

The flow structure of strongly turbulent Taylor-Couette flow with Reynolds numbers up to Re(i)=2×10(6) of the inner cylinder is experimentally examined with high-speed particle image velocimetry (PIV). The wind Reynolds numbers Re(w) of the turbulent Taylor-vortex flow is found to scale as Re(w)∝Ta(1/2), exactly as predicted by Grossmann and Lohse [Phys. Fluids 23, 045108 (2011).] for the ultimate turbulence regime, in which the boundary layers are turbulent. The dimensionless angular velocity flux has an effective scaling of Nu(ω)∝Ta(0.38), also in correspondence with turbulence in the ultimate regime. The scaling of Nu(ω) is confirmed by local angular velocity flux measurements extracted from high-speed PIV measurements: though the flux shows huge fluctuations, its spatial and temporal average nicely agrees with the result from the global torque measurements.