Bulk temperature and heat transport in turbulent Rayleigh–Bénard convection of fluids with temperature-dependent properties

We critically analyse the different ways to evaluate the dependence of the Nusselt number ( $\mathit{Nu}$ ) on the Rayleigh number ( $\mathit{Ra}$ ) in measurements of the heat transport in turbulent Rayleigh–Bénard convection under general non-Oberbeck–Boussinesq conditions and show the sensitivity of this dependence to the choice of the reference temperature at which the fluid properties are evaluated. For the case when the fluid properties depend significantly on the temperature and any pressure dependence is insignificant we propose a method to estimate the centre temperature. The theoretical predictions show very good agreement with the Göttingen measurements by He et al. ( 2012, 063030). We further show too the values of the normalized heat transport $\mathit{Nu}/\mathit{Ra}^{1/3}$ are independent of whether they are evaluated in the whole convection cell or in the lower or upper part of the cell if the correct reference temperatures are used.

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title: Recent Developments in Rayleigh-Bénard Convection
abstract: ▪ Abstract This review summarizes results for Rayleigh-Benard convection that have been obtained over the past decade or so. It concentrates on convection in compressed gases and gas mixtures with Prandtl numbers near one and smaller. In addition to the classical problem of a horizontal stationary fluid layer heated from below, it also briefly covers convection in such a layer with rotation about a vertical axis, with inclination, and with modulation of the vertical acceleration.

title: New perspectives in turbulent Rayleigh-Bénard convection
abstract: 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.

title: Small-Scale Properties of Turbulent Rayleigh-Bénard Convection
abstract: The properties of the structure functions and other small-scale quantities in turbulent Rayleigh-Benard convection are reviewed, from an experimental, theoretical, and numerical point of view. In particular, we address the question of whether, and if so where in the flow, the so-called Bolgiano-Obukhov scaling exists, i.e., Sθ(r) ∼ r2/5 for the second-order temperature structure function and Su(r) ∼ r6/5 for the second-order velocity structure function. Apart from the anisotropy and inhomogeneity of the flow, insufficiently high Rayleigh numbers, and intermittency corrections (which all hinder the identification of such a potential regime), there are also reasons, as a matter of principle, why such a scaling regime may be limited to at most a decade, namely the lack of clear scale separation between the Bolgiano length scale LB and the height of the cell.

title: Non-Oberbeck–Boussinesq effects in strongly turbulent Rayleigh–Bénard convection
abstract: Non-Oberbeck–Boussinesq (NOB) effects on the Nusselt number $Nu$ and Reynolds number $\hbox{\it Re}$ 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 $\Gamma$ close to 1, were used. For each $L$, about 1.5 decades in $Ra$ were covered, together spanning the range $10^8 \,{\le}\, Ra \,{\le}\, 10^{11}$. For the largest temperature difference between the bottom and top plates, $\Delta \,{=}\, 40$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 $\chi$ of the temperature drops across the bottom and top thermal boundary layers became as small as $\chi\,{=}\,0.83$, which may be compared with the ratio $\chi \,{=}\, 1$ 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 $\hbox{\it Re}$ 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 $\Delta$. The net effects on $Nu$ are proportional to $ \Delta^2$ and thus increase only slowly and still remain minor despite drastic material-parameter 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 $\chi$ 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% of the Reynolds number, which is also consistent with the experimental data. By studying theoretically hypothetical liquids for which only one of the material parameters is temperature dependent, we are able to shed further light on the origin of NOB corrections in water: while the NOB deviation of $\chi$ from its OB value $\chi \,{=}\, 1$ mainly originates from the temperature dependence of the viscosity, the NOB correction of the Nusselt number primarily originates from the temperature dependence of the thermal diffusivity. Finally, we give predictions from our theory for the NOB corrections if glycerol were used as the operating liquid.

title: Non-oberbeck-boussinesq effects in gaseous Rayleigh-Bénard convection.
abstract: 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.

title: Non-Oberbeck-Boussinesq effects in turbulent thermal convection in ethane close to the critical point.
abstract: As shown in earlier work [Ahlers,  (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<or similar to Pr<or similar to 3.9 and 5x10(9)<or similar to Ra<or similar to 2x10(12) and are for cylindrical samples of aspect ratios 1.0 and 0.5.

title: The validity of the boussinesq approximation for liquids and gases
abstract: A new method for obtaining approximate equations for natural convection flows is presented. The systematic application of this method leads to explicit conditions for the neglect of various terms. It is shown that this method allows the specification of the conditions under which the traditional Boussinesq approximation applies to a given Newtonian liquid or gas. The method is applied to room temperature water and air.

title: On non-Oberbeck–Boussinesq effects in three-dimensional Rayleigh–Bénard convection in glycerol
abstract: Rayleigh–Bénard convection in glycerol (Prandtl number $\mathit{Pr}= 2547. 9$ ) in a cylindrical cell with an aspect ratio of $\Gamma = 1$ was studied by means of three-dimensional direct numerical simulations (DNS). For that purpose, we implemented temperature-dependent material properties into our DNS code, by prescribing polynomial functions up to seventh order for the viscosity, the heat conductivity and the density. We performed simulations with the common Oberbeck–Boussinesq (OB) approximation and with non-Oberbeck–Boussinesq (NOB) effects within a range of Rayleigh numbers of $1{0}^{5} \leq \mathit{Ra}\leq 1{0}^{9} $ . For the highest temperature differences, $\Delta = 80~\mathrm{K} $ , the viscosity at the top is ${\sim }360\hspace{0.167em} \% $ times higher than at the bottom, while the differences of the other material properties are less than $15\hspace{0.167em} \% $ . We analysed the temperature and velocity profiles and the thermal and viscous boundary-layer thicknesses. NOB effects generally lead to a breakdown of the top–bottom symmetry, typical for OB Rayleigh–Bénard convection. Under NOB conditions, the temperature in the centre of the cell ${T}_{c} $ increases with increasing $\Delta $ and can be up to $15~\mathrm{K} $ higher than under OB conditions. The comparison of our findings with several theoretical and empirical models showed that two-dimensional boundary-layer models overestimate the actual ${T}_{c} $ , while models based on the temperature or velocity scales predict ${T}_{c} $ very well with a standard deviation of $0. 4~\mathrm{K} $ . Furthermore, the obtained temperature profiles bend closer towards the cold top plate and further away from the hot bottom plate. The situation for the velocity profiles is reversed: they bend farther away from the top plate and closer towards to the bottom plate. The top boundary layers are always thicker than the bottom ones. Their ratio is up to 2.5 for the thermal and up to 4.5 for the viscous boundary layers. In addition, the Reynolds number $\mathit{Re}$ and the Nusselt number $\mathit{Nu}$ were investigated: $\mathit{Re}$ is higher and $\mathit{Nu}$ is lower under NOB conditions. The Nusselt number $\mathit{Nu}$ is influenced in a nonlinear way by NOB effects, stronger than was suggested by the two-dimensional simulations. The actual scaling of $\mathit{Nu}$ with $\mathit{Ra}$ in the NOB case is $\mathit{Nu}\propto {\mathit{Ra}}^{0. 298} $ and is in excellent agreement with the experimental data.

title: Turbulent convection at very high Rayleigh numbers
abstract: Turbulent convection occurs when the Rayleigh number (Ra)—which quantifies the relative magnitude of thermal driving to dissipative forces in the fluid motion—becomes sufficiently high. Although many theoretical and experimental studies of turbulent convection exist, the basic properties of heat transport remain unclear. One important question concerns the existence of an asymptotic regime that is supposed to occur at very high Ra. Theory predicts that in such a state the Nusselt number (Nu), representing the global heat transport, should scale as Nu ∝ Raβ with β = 1/2. Here we investigate thermal transport over eleven orders of magnitude of the Rayleigh number (106 ≤ Ra ≤ 10 17), using cryogenic helium gas as the working fluid. Our data, over the entire range of Ra, can be described to the lowest order by a single power-law with scaling exponent β close to 0.31. In particular, we find no evidence for a transition to the Ra1/2 regime. We also study the variation of internal temperature fluctuations with Ra, and probe velocity statistics indirectly.

title: Heat Transfer in Turbulent Rayleigh–Bénard Convection Below the Ultimate Regime
abstract: A Rayleigh–Bénard cell has been designed to explore the Prandtl (Pr) dependence of turbulent convection in the cross-over range 0.7<Pr<21 and for the full range of soft and hard turbulences, up to Rayleigh number Ra≃1011. The set-up benefits from the favourable characteristics of cryogenic helium-4 in fluid mechanics, in situ fluid property measurements, and special care on thermometry and calorimetric instrumentation. The cell is cylindrical with diameter/height=0.5. The effective heat transfer Nu(Ra, Pr) has been measured with unprecedented accuracy for cryogenic turbulent convection experiments in this range of Rayleigh numbers. Spin-off of this study include improved fits of helium thermodynamics and viscosity properties. Three main results were found. First the Nu(Ra) dependence exhibits a bimodality of the flow with 4–7% difference in Nu for given Ra and Pr. Second, a systematic study of the side-wall influence reveals a measurable effect on the heat transfer. Third, the Nu(Pr) dependence is very small or null: the absolute value of the average logarithmic slope (d ln Nu/d ln Pr)Ra is smaller than 0.03 in our range of Pr, which allows to discriminate between experiments with contradictory results [Ashkenazi et al.,  (1999)] [Ahlers et al.,  (2001)].

title: Non-Oberbeck-Boussinesq effects in two-dimensional Rayleigh-Bénard convection in glycerol
abstract: We numerically analyze Non-Oberbeck-Boussinesq (NOB) effects in two-dimensional Rayleigh-Benard flow in glycerol, which shows a dramatic change in the viscosity with temperature. The results are presented both as functions of the Rayleigh number Ra up to 108 (for fixed temperature difference I� between the top and bottom plates) and as functions of I� ("non-Oberbeck-Boussinesqness" or "NOBness") up to 50 K (for fixed Ra). For this large NOBness the center temperature Tc is more than 5 K larger than the arithmetic mean temperature Tm between top and bottom plate and only weakly depends on Ra. To physically account for the NOB deviations of the Nusselt numbers from its Oberbeck-Boussinesq values, we apply the decomposition of NuNOB/NuOB into the product of two effects, namely first the change in the sum of the top and bottom thermal BL thicknesses, and second the shift of the center temperature Tc as compared to Tm. While for water the origin of the Nu deviation is totally dominated by the second effect (cf. Ahlers G. et al.,  (2006) 409) for glycerol the first effect is dominating, in spite of the large increase of Tc as compared to Tm.

title: Non-Boussinesq effects in free thermal convection.
abstract: Non-Boussinesq effects in a Rayleigh-B\'enard convection system lead to a symmetry breaking between the top and the bottom boundary layers. We have found that the two layers adjust their temperature drops and the thicknesses \ensuremath{\lambda} such that their temperature scales \ensuremath{\nu}\ensuremath{\kappa}/g\ensuremath{\alpha}${\ensuremath{\lambda}}^{3}$ are equal, where \ensuremath{\alpha} is the thermal expansion coefficient, g is the gravitational acceleration, and \ensuremath{\nu} and \ensuremath{\kappa} are the kinematic viscosity and thermal diffusivity, respectively.

title: Heat-flux measurement in high-Prandtl-number turbulent Rayleigh-Bénard convection.
abstract: We report Nusselt number measurements from high Prandtl number turbulent thermal convection experiments. The experiments are conducted in four fluids with the Prandtl number Pr varying from 4 to 1350 and the Rayleigh number Ra from 2x10(7) to 3x10(10), all in a single convection cell of unity aspect ratio. We find that the measured Nusselt number decreased about 20% over the range of Pr spanned in the experiment. The measure data are also found in good agreement with the prediction of a recent theory over the extended range of Pr covered in the experiment.

title: Non-Boussinesq effect: Thermal convection with broken symmetry
abstract: We investigate large Rayleigh number (106–109) and large Prandtl number (102–103) thermal convection in glycerol in an aspect ration one cubic cell. The kinematic viscosity of the fluid strongly depends upon the temperature. The symmetry between the top and bottom boundary layers is thus broken, the so-called non-Boussinesq regime. In a previous paper Wu and Libchaber have proposed that in such a state the two thermal boundary layers adjust their length scales so that the mean hot and cold temperature fluctuations are equal in the center of the cell. We confirm this equality. A simplified two-dimensional model for the mean center temperature based on an equation for the thermal boundary layer is presented and compared with the experimental results. The conclusion is that the central temperature adjusts itself so that heat fluxes from boundaries are equal, temperature fluctuations at the center symmetrical, at a cost of very different temperature drops and Rayleigh number for each boundary.

title: High Rayleigh Number Turbulent Convection in a Gas near the Gas-Liquid Critical Point
abstract: $SF_6$ in the vicinity of its critical point was used to study turbulent convection up to exceptionally high Rayleigh numbers, $Ra$, (up to $5\cdot 10^{14}$) and to verify for the first time the generalized scaling laws for the heat transport and the large scale circulation velocity as a function of $Ra$ and the Prandtl number, $Pr$, in very wide range of these parameters. The both scaling laws obtained are consistent with theoretical predictions by B.Shraiman and E.Siggia, , 3650 (1990).

title: Scaling of hard thermal turbulence in Rayleigh-Bénard convection
abstract: An experimental study of Rayleigh-Benard convection in helium gas at roughly 5 K is performed in a cell with aspect ratio 1. Data are analysed in a ‘hard turbulence’ region (4 × 10 7 Ra 12 ) in which the Prandtl number remains between 0.65 and 1.5. The main observation is a simple scaling behaviour over this entire range of Ra . However the results are not the same as in previous theories. For example, a classical result gives the dimensionless heat flux, Nu , proportional to $Ra^{\frac{1}{3}}$ while experiment gives an index much closer to $\frac{2}{7}$ . A new scaling theory is described. This new approach suggests scaling indices very close to the observed ones. The new approach is based upon the assumption that the boundary layer remains in existence even though its Rayleigh number is considerably greater than unity and is, in fact, diverging. A stability analysis of the boundary layer is performed which indicates that the boundary layer may be stabilized by the interaction of buoyancy driven effects and a fluctuating wind.

title: Confined turbulent convection
abstract: New measurements of the Nusselt number have been made in turbulent thermal convection confined in a cylindrical container of aspect ratio unity. The apparatus is essentially the same as that used by Niemela et al. (2000), except that the height was halved. The measurement techniques were also identical but the mean temperature of the flow was held fixed for all Rayleigh numbers. The highest Rayleigh number was $2 \times 10^{15}$. Together with existing data, the new measurements are analysed with the purpose of understanding the relation between the Nusselt number and the Rayleigh number, when the latter is large. In particular, the roles played by Prandtl number, aspect ratio, mean wind, boundary layers, sidewalls, and non-Boussinesq effects are discussed. Nusselt numbers, measured at the highest Rayleigh numbers for which Boussinesq conditions hold and sidewall forcing is negligible, are shown to vary approximately as a 1/3-power of the Rayleigh number. Much of the complexity in interpreting experimental data appears to arise from aspects of the mean flow, including complex coupling of its dynamics to sidewall boundary conditions of the container. Despite the obvious practical difficulties, we conclude that the next generation of experiments will be considerably more useful if they focus on large aspect ratios.

title: Heat transfer in cryogenic helium gas by turbulent Rayleigh–Bénard convection in a cylindrical cell of aspect ratio 1
abstract: We present experimental results on the heat transfer efficiency of cryogenic turbulent Rayleigh–Benard convection (RBC) in a cylindrical cell 0.3 m in both diameter and height which has improvements with respect to various corrections connected with finite thermal conductivity of sidewalls and plates. The heat transfer efficiency described by the Nusselt number is investigated for the range of Rayleigh number , with the Prandtl number varying such that , using cryogenic He gas with well-known and in situ tunable properties as a working fluid. For our data (both corrected and uncorrected) agree with suitably corrected data from similar cryogenic experiments and are consistent with . Up to , our data could be treated as Oberbeck-Boussinesq data. For , the heat transfer efficiency becomes affected by non-Oberbeck–Boussinesq (NOB) effects, causing asymmetry of the top and bottom boundary layers. For , the Nusselt number closely follows if and are evaluated on the basis of the working fluid properties at the directly measured bulk temperature, , and suitable corrections are taken into account. In contrast, if the mean temperature is determined as an arithmetic mean of the bottom and top plate temperatures, displays spurious crossover to higher γ that might be misinterpreted as a transition to the ultimate Kraichnan regime. The second step of our analysis, reported here for the first time, is to ignore the NOB effects affecting the top half of the RBC cell. We replace it by the inverted nearly OB bottom half in order to eliminate the boundary layer asymmetry. This leads to the effective temperature difference , where denotes the bottom plate temperature, and to effective and values. The effective heat transfer efficiency obtained, showing no tendency of crossover to the ultimate regime up to in , is reported and discussed.

title: Turbulent Thermal Convection at Arbitrary Prandtl Number
abstract: The mixing‐length theory of turbulent thermal convection in a gravitationally unstable fluid is extended to yield the dependence of Nusselt number H/H0 on both Prandtl number σ and Rayleigh number Ra. The analysis assumes a layer of Boussinesq fluid contained between infinite, horizontal, perfectly conducting, rigid plates. Also obtained is the dependence of mean temperature deviation T(z), rms temperature fluctuation ψ(z), and rms velocity upon height z above the bottom plate. The theory gives H/H0 ∝ Ra1/3 (high σ), H/H0 ∝ (σ Ra)1/3 (low σ), and H/H0 ∼ 1 (very low σ). The boundaries of the several σ ranges are determined. At one intermediate Prandtl number only, the behavior of T(z) and ψ(z) reduces to that previously found by Priestly. At high σ, there is a range of z, outside the molecular conduction region, where T(z) ∝ z−1, ψ(z) ∝ z−1. The results at very low σ reduce to those of Ledoux, Schwarzschild, and Spiegel. The dynamics are found to be importantly modified at extremely large Ra because of the stirring action of small‐scale turbulence generated in shear boundary layers attached to the eddies of largest scale. The consequent corrected asymptotic law of heat transport at fixed σ is H/H0 ∝ [Ra/(In Ra)3]1/2.

title: Scale analysis and wall-layer model for the temperature profile in a turbulent thermal convection
abstract: Three sets of characteristic scales for the conduction layer, the transition layer and the convection layer are proposed to analyze the mean thermal structure in a turbulent thermal convection without mean motion. These scales are formulated based on molecular or turbulent eddy contribution to the momentum and heat transports in each layer. Using the proposed scales and a gradient matching technique at the interface between two adjacent layers, Kraichnan's (Physics Fluids5, 1374 (1962)) multi-layered structure of the mean temperature gradient profile is re-established. If the conduction scales are used to nondimensionalize mean temperature gradient data near the wall, they form a plausible correlation curve that is nearly independent of the Prandtl number and the Rayleigh number for the range of experiments. From the correlation curve, it is found that the convection layer or the similarity layer with the slope of − 4 3 begins to appear after about z+ ~ 15 and the proportionality constant of the − 4 3 power law of the mean temperature gradient is found to be about 0.6 or dΘ + /dz + = 0.6z + − 4 3 , where Θ+ and z+ are nondimensional temperature and distance scaled by the respective conduction scales. Further, a wall-layer model for the mean temperature gradient profile is formulated in accordance with the power law, dΘ+/dz+ ~ z−α+, across the layers, which is in good agreement with the data.

title: Logarithmic temperature profiles in turbulent Rayleigh-Bénard convection.
abstract: 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.

title: Logarithmic temperature profiles in the bulk of turbulent Rayleigh–Bénard convection for a Prandtl number of 12.3
abstract: We report measurements of logarithmic temperature profiles $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\varTheta (z,r) = A(r)\times \ln (z/L) + B(r)$ in the bulk of turbulent Rayleigh–Bénard convection (here $\varTheta $ is a scaled and time-averaged local temperature in the fluid, $ z$ is the vertical and $r$ the radial position, and $L$ is the sample height). Two samples had aspect ratios $\varGamma \equiv D/L = 1.00$ and 0.50 (where $D=190\ \mathrm{mm}$ is the diameter). The fluid was a fluorocarbon with a Prandtl number of $\mathit{Pr} = 12.3$ . The measurements covered the Rayleigh-number range $2\times 10^{10} \lesssim \mathit{Ra} \lesssim 2\times 10^{11}$ for $\varGamma = 1.00$ and $3\times 10^{11} \lesssim \mathit{Ra} \lesssim 2\times 10^{12}$ for $\varGamma = 0.50$ . In contradistinction to what had been found for $\varGamma = 0.50$ and $\mathit{Pr} = 0.78$ by Ahlers et al. ( 2012, art. 114501;  2014, in press), the measurements revealed no $\mathit{Ra}$ dependence of the amplitude $A(r)$ of the logarithmic term. Within the experimental resolution, the amplitude was also found to be independent of $\varGamma $ . It varied with $r$ in a manner consistent with the function $A(\xi ) = A_1/\sqrt{2\xi - \xi ^2}$ , where $\xi \equiv (R-r)/R$ with $R=D/2$ and $A_1 \simeq 0.0016$ . The results for $A(r)$ are smaller than those obtained from experiments and direct numerical simulations (Ahlers et al.,  2012, art. 114501) at similar values of $\mathit{Ra}$ for $\mathit{Pr} = 0.7$ and $\varGamma = \frac{1}{2}$ by a factor that depended slightly upon $\mathit{Ra}$ but was close to $2$ .

title: Logarithmic temperature profiles of turbulent Rayleigh–Bénard convection in the classical and ultimate state for a Prandtl number of 0.8
abstract: We report on experimental determinations of the temperature field in the interior (bulk) of turbulent Rayleigh–Bénard convection for a cylindrical sample with an aspect ratio (diameter $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}D$ over height $L$ ) equal to 0.50, in both the classical and the ultimate state. The measurements are for Rayleigh numbers $\mathit{Ra}$ from $6\times 10^{11}$ to $10^{13}$ in the classical and $7\times 10^{14}$ to $1.1\times 10^{15}$ (our maximum accessible $\mathit{Ra}$ ) in the ultimate state. The Prandtl number was close to 0.8. Although to lowest order the bulk is often assumed to be isothermal in the time average, we found a ‘logarithmic layer’ (as reported briefly by Ahlers et al.,  2012, 114501) in which the reduced temperature $\varTheta = [\langle T(z) \rangle - T_m]/\Delta T$ (with $T_m$ the mean temperature, $\Delta T$ the applied temperature difference and $\langle {\cdots } \rangle $ a time average) varies as $A \ln (z/L) + B$ or $A^{\prime } \ln (1-z/L) + B^{\prime }$ with the distance $z$ from the bottom plate of the sample. In the classical state, the amplitudes $-A$ and $A^{\prime }$ are equal within our resolution, while in the ultimate state there is a small difference, with $-A/A^{\prime } \simeq 0.95$ . For the classical state, the width of the log layer is approximately $0.1L$ , the same near the top and the bottom plate as expected for a system with reflection symmetry about its horizontal midplane. For the ultimate state, the log-layer width is larger, extending through most of the sample, and slightly asymmetric about the midplane. Both amplitudes $A$ and $A^{\prime }$ vary with radial position $r$ , and this variation can be described well by $A = A_0 [(R - r)/R]^{-0.65}$ , where $R$ is the radius of the sample. In the classical state, these results are in good agreement with direct numerical simulations (DNS) for $\mathit{Ra} = 2\times 10^{12}$ ; in the ultimate state there are as yet no DNS. The amplitudes $-A$ and $A^{\prime }$ varied as ${\mathit{Ra}}^{-\eta }$ , with $\eta \simeq 0.12$ in the classical and $\eta \simeq 0.18$ in the ultimate state. A close analogy between the temperature field in the classical state and the ‘law of the wall’ for the time-averaged downstream velocity in shear flow is discussed. A two-sublayer mean-field model of the temperature profile in the classical state was analysed and yielded a logarithmic $z$ dependence of $\varTheta $ . The $\mathit{Ra}$ dependence of the amplitude $A$ given by the model corresponds to an exponent $\eta _{th} = 0.106$ , in good agreement with the experiment. In the ultimate state the experimental result $\eta \simeq 0.18$ differs from the prediction $\eta _{th} \simeq 0.043$ by Grossmann & Lohse ( 2012, 125103).

title: Azimuthal diffusion of the large-scale-circulation plane, and absence of significant non-Boussinesq effects, in turbulent convection near the ultimate-state transition
abstract: We present measurements of the orientation ${\it\theta}_{0}$ and temperature amplitude ${\it\delta}$ of the large-scale circulation in a cylindrical sample of turbulent Rayleigh–Bénard convection (RBC) with aspect ratio ${\it\Gamma}\equiv D/L=1.00$ ( $D$ and $L$ are the diameter and height respectively) and for the Prandtl number $Pr\simeq 0.8$ . The results for ${\it\theta}_{0}$ revealed a preferred orientation with up-flow in the west, consistent with a broken azimuthal invariance due to the Earth’s Coriolis force (see Brown & Ahlers ( 2006, 125108)). They yielded the azimuthal diffusivity $D_{{\it\theta}}$ and a corresponding Reynolds number $Re_{{\it\theta}}$ for Rayleigh numbers over the range $2\times 10^{12}\lesssim Ra\lesssim 1.5\times 10^{14}$ . In the classical state ( $Ra\lesssim 2\times 10^{13}$ ) the results were consistent with the measurements by Brown & Ahlers ( 2006, ) for $Ra\lesssim 10^{11}$ and $Pr=4.38$ , which gave $Re_{{\it\theta}}\propto Ra^{0.28}$ , and with the Prandtl-number dependence $Re_{{\it\theta}}\propto Pr^{-1.2}$ as found previously also for the velocity-fluctuation Reynolds number $Re_{V}$ (He et al.,  2015, ). At larger $Ra$ the data for $Re_{{\it\theta}}(Ra)$ revealed a transition to a new state, known as the ‘ultimate’ state, which was first seen in the Nusselt number $Nu(Ra)$ and in $Re_{V}(Ra)$ at $Ra_{1}^{\ast }\simeq 2\times 10^{13}$ and $Ra_{2}^{\ast }\simeq 8\times 10^{13}$ . In the ultimate state we found $Re_{{\it\theta}}\propto Ra^{0.40\pm 0.03}$ . Recently, Skrbek & Urban ( 2015, pp. 270–282) claimed that non-Oberbeck–Boussinesq effects on the Nusselt and Reynolds numbers of turbulent RBC may have been interpreted erroneously as a transition to a new state. We demonstrate that their reasoning is incorrect and that the transition observed in the Göttingen experiments and discussed in the present paper is indeed to a new state of RBC referred to as ‘ultimate’.

title: Scaling in thermal convection: a unifying theory
abstract: A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh–Bénard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large-scale convection roll (‘wind of turbulence’) and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number Ra versus Prandtl number Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the kinetic boundary layer is thicker. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra [lsim ] 1011) the leading terms are Nu ∼ Ra1/4 Pr1/8, Re ∼ Ra1/2 Pr−3/4 for Pr [lsim ] 1 and Nu ∼ Ra1/4 Pr−1/12, Re ∼ Ra1/2 Pr−5/6 for Pr [gsim ] 1. In most measurements these laws are modified by additive corrections from the neighbouring regimes so that the impression of a slightly larger (effective) Nu vs. Ra scaling exponent can arise. The most important of the neighbouring regimes towards large Ra are a regime with scaling Nu ∼ Ra1/2 Pr1/2, Re ∼ Ra1/2 Pr−1/2 for medium Pr (‘Kraichnan regime’), a regime with scaling Nu ∼ Ra1/5 Pr1/5, Re ∼ Ra2/5 Pr−3/5 for small Pr, a regime with Nu ∼ Ra1/3, Re ∼ Ra4/9 Pr−2/3 for larger Pr, and a regime with scaling Nu ∼ Ra3/7 Pr−1/7, Re ∼ Ra4/7 Pr−6/7 for even larger Pr. In particular, a linear combination of the ¼ and the 1/3 power laws for Nu with Ra, Nu = 0.27Ra1/4 + 0.038Ra1/3 (the prefactors follow from experiment), mimics a 2/7 power-law exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through the non-normal-nonlinear transition to turbulence and another regime emerges. The theory presented is best summarized in the phase diagram figure 2 and in table 2.

title: Thermal convection for large Prandtl numbers.
abstract: The Rayleigh-Bénard theory by Grossmann and Lohse [ (2000)] is extended towards very large Prandtl numbers Pr. The Nusselt number Nu is found here to be independent of Pr. However, for fixed Rayleigh numbers Ra a maximum in the Nu(Pr) dependence is predicted. We moreover offer the full functional dependences of Nu(Ra,Pr) and Re(Ra,Pr) within this extended theory, rather than only give the limiting power laws as done in  (2000). This enables us to more realistically describe the transitions between the various scaling regimes.

title: Multiple scaling in the ultimate regime of thermal convection
abstract: Very different types of scaling of the Nusselt number Nu with the Rayleigh number Ra have experimentally been found in the very large Ra regime beyond 1011. We understand and interpret these results by extending the unifying theory of thermal convection [Grossmann and Lohse,  (2001)] to the very large Ra regime where the kinetic boundary-layer is turbulent. The central idea is that the spatial extension of this turbulent boundary-layer with a logarithmic velocity profile is comparable to the size of the cell. Depending on whether the thermal transport is plume dominated, dominated by the background thermal fluctuations, or whether also the thermal boundary-layer is fully turbulent (leading to a logarithmic temperature profile), we obtain effective scaling laws of about Nu∝Ra0.14, Nu∝Ra0.22, and Nu∝Ra0.38, respectively. Depending on the initial conditions or random fluctuations, one or the other of these states may be realized. Since the theory is for both the heat flux Nu and the ...

title: Scaling relations in large-Prandtl-number natural thermal convection
abstract: In this study we follow Grossmann and Lohse,  (2001), who derived various scalings regimes for the dependence of the Nusselt number $Nu$ and the Reynolds number $Re$ on the Rayleigh number $Ra$ and the Prandtl number $Pr$. We focus on theoretical arguments as well as on numerical simulations for the case of large-$Pr$ natural thermal convection. Based on an analysis of self-similarity of the boundary layer equations, we derive that in this case the limiting large-$Pr$ boundary-layer dominated regime is I$_\infty^<$, introduced and defined in , with the scaling relations $Nu\sim Pr^0\,Ra^{1/3}$ and $Re\sim Pr^{-1}\,Ra^{2/3}$. Our direct numerical simulations for $Ra$ from $10^4$ to $10^9$ and $Pr$ from 0.1 to 200 show that the regime I$_\infty^<$ is almost indistinguishable from the regime III$_\infty$, where the kinetic dissipation is bulk-dominated. With increasing $Ra$, the scaling relations undergo a transition to those in IV$_u$ of reference , where the thermal dissipation is determined by its bulk contribution.

title: Thermal boundary layers in turbulent Rayleigh-Bénard convection at aspect ratios between 1 and 9
abstract: We report highly resolved temperature measurements in turbulent Rayleigh–Benard convection in air at a fixed Prandtl number Pr = 0.7. Extending our previous work (du Puits et al 2007 , we carried out measurements at various aspect ratios while keeping the Rayleigh number constant. We demonstrate that the temperature field inside the convective boundary layers of both horizontal plates is virtually independent on the global flow pattern accompanying the variation in the aspect ratio. Thanks to technical upgrades of the experimental facility as well as a significant improvement of the accuracy and reliability of our temperature measurement — and unlike in our previous work — we find that the measured profiles of the time-averaged temperature field neither follow a clear power-law trend nor fit a linear or a logarithmic scaling over a significant fraction of the boundary-layer thickness. Analyzing the temperature data simultaneously acquired at both horizontal plates, various transitions in the cross-correlation and the auto-correlation function of the temperature signals are observed while varying the aspect ratio Γ. These transitions might be associated with a change in the global flow pattern from a single-roll mode at Γ = 1 toward a double- or a multi-roll mode pattern at higher aspect ratios.

title: Thermal boundary layer equation for turbulent Rayleigh-Bénard convection.
abstract: 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.

title: Mean Temperature Profiles in Turbulent Thermal Convection
abstract: To predict the mean temperature profiles in turbulent thermal convection, the thermal boundary layer (BL) equation including the effects of fluctuations has to be solved. In Shishkina et al.,  (2015), the thermal BL equation with the fluctuations taken into account as an eddy thermal diffusivity has been solved for large Prandtl-number fluids for which the eddy thermal diffusivity and the velocity field can be approximated respectively as a cubic and a linear function of the distance from the plate. In the present work we make use of the idea of Prandtl's mixing length model and relate the eddy thermal diffusivity to the stream function. With this proposed relation, we can solve the thermal BL equation and obtain a closed-form expression for the dimensionless mean temperature profile in terms of two independent parameters for fluids with a general Prandtl number. With a proper choice of the parameters, our predictions of the temperature profiles are in excellent agreement with the results of our direct numerical simulations for a wide range of Prandtl numbers from 0.01 to 2547.9 and Rayleigh numbers from 10^7 to 10^9.

title: Fluctuating Thermal Boundary Layers and Heat Transfer in Turbulent Rayleigh–Bénard Convection
abstract: We investigate the effect of fluctuations in thermal boundary layer on heat transfer in turbulent Rayleigh–Bénard convection for Prandtl number greater than one in the regime where the thermal dissipation rate is dominated by boundary layer contribution and in the presence of a large-scale circulating flow.

title: Experimental study of non-Boussinesq Rayleigh-Benard convection at high Rayleigh and Prandtl numbers
abstract: A set of experiments is performed, in which a layer of fluid is heated from below and cooled from above, in order to study convection at high Rayleigh numbers (Ra) and Prandtl numbers (Pr). The working fluid, corn syrup, has a viscosity that depends strongly on temperature. Viscosity within the fluid layer varies by a factor of 6 to 1.8×103 in the various experiments. A total of 28 experiments are performed for 104 O(107), there is no evidence for any large scale circulation, and flow consists only of plumes. Plumes have mushroom-shaped “heads” and continuous “tails” attached to their respective therma...

title: The heat transport and spectrum of thermal turbulence
abstract: In this paper a theoretical investigation is made of various properties of the steady-state inhomogeneous turbulent convection of heat in a fluid between horizontal conducting surfaces. An upper limit to the heat transport is found subject to the constraint that some minimum eddy size exists which is effective in this transport. The spectrum of convecting motions, the mean thermal gradients at each point and the eddy conductivity are then determined in terms of the minimum eddy size. The relation between the boundary conditions and eddy size is studied by an extension of the work of Pellew & Southwell using the mean thermal gradients deduced when n0 modes of motion are present to establish the Rayleigh number at which the (n0+1)th mode first becomes unstable. In a final section the spectra and mean-square values of the fluctuating velocity and temperature fields are estimated from the Boussinesq form of the hydrodynamic equations. The previously reported experimental heat transports are within 10% of those predicted. The discrete transitions are within the error limits of the observations. However, further data must be mgathered to justify the use of minimum eddy size as a defining parameter in situations of geophysical scale.

title: Thermal boundary layer profiles in turbulent Rayleigh-Bénard convection in a cylindrical sample.
abstract: 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,  (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.

title: Thermal and viscous boundary layers in turbulent Rayleigh–Bénard convection
abstract: We present the results from numerical simulations of turbulent Rayleigh–Bénard convection for an aspect ratio (diameter/height) of 1.0, Prandtl numbers of 0.4 and 0.7, and Rayleigh numbers from $1\ensuremath{\times} 1{0}^{5} $ to $1\ensuremath{\times} 1{0}^{9} $ . Detailed measurements of the thermal and viscous boundary layer profiles are made and compared to experimental and theoretical (Prandtl–Blasius) results. We find that the thermal boundary layer profiles disagree by more than 10 % when scaled with the similarity variable (boundary layer thickness) and likewise disagree with the Prandtl–Blasius results. In contrast, the viscous boundary profiles collapse well and do agree (within 10 %) with the Prandtl–Blasius profile, but with worsening agreement as the Rayleigh number increases. We have also investigated the scaling of the boundary layer thicknesses with Rayleigh number, and again compare to experiments and theory. We find that the scaling laws are very robust with respect to method of analysis and they mostly agree with the Grossmann–Lohse predictions coupled with laminar boundary layer theory within our numerical uncertainty.

title: Falkner–Skan boundary layer approximation in Rayleigh–Bénard convection
abstract: To approximate the velocity and temperature within the boundary layers in turbulent thermal convection at moderate Rayleigh numbers, we consider the Falkner–Skan ansatz, which is a generalization of the Prandtl–Blasius one to a non-zero-pressure-gradient case. This ansatz takes into account the influence of the angle of attack $\beta $ of the large-scale circulation of a fluid inside a convection cell against the heated/cooled horizontal plate. With respect to turbulent Rayleigh–Bénard convection, we derive several theoretical estimates, among them the limiting cases of the temperature profiles for all angles $\beta $ , for infinite and for infinitesimal Prandtl numbers $\mathit{Pr}$ . Dependences on $\mathit{Pr}$ and $\beta $ of the ratio of the thermal to viscous boundary layers are obtained from the numerical solutions of the boundary layers equations. For particular cases of $\beta $ , accurate approximations are developed as functions on $\mathit{Pr}$ . The theoretical results are corroborated by our direct numerical simulations for $\mathit{Pr}= 0. 786$ (air) and $\mathit{Pr}= 4. 38$ (water). The angle of attack $\beta $ is estimated based on the information on the locations within the plane of the large-scale circulation where the time-averaged wall shear stress vanishes. For the fluids considered it is found that $\beta \approx 0. 7\mathrm{\pi} $ and the theoretical predictions based on the Falkner–Skan approximation for this $\beta $ leads to better agreement with the DNS results, compared with the Prandtl–Blasius approximation for $\beta = \mathrm{\pi} $ .

title: Influence of the angle between the wind and the isothermal surfaces on the boundary layer structures in turbulent thermal convection.
abstract: We derive the asymptotes for the ratio of the thermal to viscous boundary layer thicknesses for infinite and infinitesimal Prandtl numbers Pr as functions of the angle β between the large-scale circulation and an isothermal heated or cooled surface for the case of turbulent thermal convection with laminar-like boundary layers. For this purpose, we apply the Falkner-Skan ansatz, which is a generalization of the Prandtl-Blasius one to a nonhorizontal free-stream flow above the viscous boundary layer. Based on our direct numerical simulations (DNS) of turbulent Rayleigh-Bénard convection for Pr=0.1, 1, and 10 and moderate Rayleigh numbers up to 108 we evaluate the value of β that is found to be around 0.7π for all investigated cases. Our theoretical predictions for the boundary layer thicknesses for this β and the considered Pr are in good agreement with the DNS results.