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Using a closed set of boundary layer equations [E. S. C. Ching et al., Phys. Rev. Research 1, 033037 (2019)] for turbulent Rayleigh-Bénard convection, we derive analytical results for the dependence of the heat flux, measured by the Nusselt number (Nu), on the Reynolds (Re) and Prandtl (Pr) numbers and two parameters that measure fluctuations in the regime where the horizontal pressure gradient is negligible. This regime is expected to be reached at sufficiently high Rayleigh numbers for a fluid of any given Prandtl number. In the high-Pr limit, Nu=F1(k1)Re1/2Pr1/3 and, in the low-Pr limit, Nu tends to π−1/2Re1/2Pr1/2, where F1(k1) has a weak dependence on the parameter k1 in the eddy viscosity that measures velocity fluctuations. These theoretical results further reveal a close resemblance of the scaling dependencies of heat flux in steady forced convection and turbulent Rayleigh-Bénard convection and this finding solves a puzzle in our present understanding of heat transfer in turbulent Rayleigh-Bénard convection.

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... A notable exception is the study of Pandey, Schumacher & Sreenivasan (2021), who reported that the turbulent Prandtl number decreases steeply as a function of the molecular Prandtl number, so that Pr t ∝ Pr −1 in simulations of standard Boussinesq and variable-heat-conductivity Boussinesq convection. Similar results have recently been reported by Tai et al. (2021) and Pandey et al. (2022a). ...

... With this, if we let c ν /c χ ∝ Pr +0.25 in (3.7), instead of a fixed ratio, and note from the inset of figure 3 that the factor k u T /k T u ∝ Pr −0.25 , we would obtain from (3.7) that Pr (k− ) t becomes independent of Pr, which agrees qualitatively with our results as shown in figure 5. Therefore we conclude that the results from the k− model with a fixed value for the ratio c ν /c χ are unreliable in the present flows. We note that the strong dependence of Pr t on Pr found in Pandey et al. (2021) with the k− model has also been found using the gradient method in Tai et al. (2021) and Pandey et al. (2022a). This suggests that the behaviour of Pr t (Pr) found by these authors is not specific to the k− model in that case but reflects the properties of Boussinesq convection at constant Rayleigh number. ...

... However, given that these results differ qualitatively from those obtained from the imposed gradient method and decay experiments, the former results are considered unreliable for the system under consideration here. The fact that the gradient method yields a similar Pr dependence as the k− model for Boussinesq convection (Tai et al. 2021;Pandey et al. 2022a) suggests that in those flows the behaviour of turbulent transport can indeed be different. This is possibly connected to the strong driving of flows in the low-Pr regime when the Rayleigh number is fixed (e.g. ...

Turbulent motions enhance the diffusion of large-scale flows and temperature gradients. Such diffusion is often parameterized by coefficients of turbulent viscosity ( $\nu _{t}$ ) and turbulent thermal diffusivity ( $\chi _{t}$ ) that are analogous to their microscopic counterparts. We compute the turbulent diffusion coefficients by imposing sinusoidal large-scale velocity and temperature gradients on a turbulent flow and measuring the response of the system. We also confirm our results using experiments where the imposed gradients are allowed to decay. To achieve this, we use weakly compressible three-dimensional hydrodynamic simulations of isotropically forced homogeneous turbulence. We find that the turbulent viscosity and thermal diffusion, as well as their ratio the turbulent Prandtl number, $\textit {Pr}_{t} = \nu _{t}/\chi _{t}$ , approach asymptotic values at sufficiently high Reynolds and Péclet numbers. We also do not find a significant dependence of $\textit {Pr}_{t}$ on the microscopic Prandtl number $\textit {Pr} = \nu /\chi$ . These findings are in stark contrast to results from the $k{-}\epsilon$ model, which suggests that $\textit {Pr}_{t}$ increases monotonically with decreasing $\textit {Pr}$ . The current results are relevant for the ongoing debate on, for example, the nature of the turbulent flows in the very-low- $\textit {Pr}$ regimes of stellar convection zones.

... Bricteux et al. [13] studied a low-Pr flow through a uniformly heated channel and observed that Pr t ≈ 2 for Pr = 0.01. Recently, Tai et al. [17] studied RBC in a cylindrical cell with = 1 and observed that Pr t within the thermal boundary layer (BL) increased with decreasing Pr. ...

... where U (x) and (x) are the time-averaged velocity and temperature fields. In the literature, the turbulent viscosity is usually estimated by the flux-gradient method, according to which ν t = − u x u z /(∂U x /∂z) and the turbulent thermal diffusivity by κ t = − u z T /(∂ /∂z) [12,13,17,32,[54][55][56]. In turbulent convection, however, both ν t and κ t computed using this method become undefined at some heights. ...

... The local maxima near the plates in Fig. 6(a) are observed due to the peaks of k T (z) in Fig. 5(b). Note however that ν t and κ t in RBC have been observed to scale as z 3 in the vicinity of the plates [17,32,55,62], which suggests the constancy of Pr t (z) in the near-wall region. As we are mainly concerned here with the behavior of Pr t in the bulk region, we do not explore further the near-wall variation of the turbulent Prandtl number. ...

Convection in the Sun occurs at Rayleigh numbers, Ra, as high as 1022 and molecular Prandtl numbers, Pr, as low as 10−6, under conditions that are far from satisfying the Oberbeck-Boussinesq (OB) idealization. The effects of these extreme circumstances on turbulent heat transport are unknown, and no comparable conditions exist on Earth. Our goal is to understand how these effects scale (since we cannot yet replicate the Sun's conditions faithfully). We study thermal convection by using direct numerical simulations, and determine the variation with respect to Pr, to values as low as 10−4, of the turbulent Prandtl number, Prt, which is the ratio of turbulent viscosity to thermal diffusivity. The simulations are primarily two-dimensional but we draw upon some three-dimensional results as well. We focus on non-Oberbeck-Boussinesq (NOB) conditions of a certain type, but also study OB convection for comparison. The OB simulations are performed in a rectangular box of aspect ratio 2 by varying Pr from O(10) to 10−4 at fixed Grashof number Gr≡Ra/Pr=109. The NOB simulations are done in the same box by letting only the thermal diffusivity depend on the temperature. Here, the Rayleigh number is fixed at the top boundary while the mean Pr varies in the bulk from 0.07 to 5×10−4. The three-dimensional simulations are performed in a box of aspect ratio 25 at a fixed Rayleigh number of 105, and 0.005≤Pr≤7. The principal finding is that Prt increases with decreasing Pr in both OB and NOB convection: Prt∼Pr−0.3 for OB convection and Prt∼Pr−1 for the NOB case. The Prt dependence for the NOB case especially suggests that convective flows in the astrophysical settings behave effectively as in high-Prandtl-number turbulence.

... Bricteux et al. [12] studied a low-P r flow through a uniformly heated channel and observed that P r t ≈ 2 for P r = 0.01. Recently, Tai et al. [16] studied RBC in a cylindrical cell with Γ = 1 and observed that P r t within the thermal boundary layer (BL) increased with decreasing P r. ...

... where U (x) and Θ(x) are the time-averaged velocity and temperature fields. In the literature, the turbulent viscosity is usually estimated by the flux-gradient method, according to which ν t = − u x u z /(∂U x /∂z) and the turbulent thermal diffusivity by κ t = − u z T /(∂Θ/∂z) [11,12,16,31,[48][49][50]. In turbulent convection, however, we observe that ν t and κ t computed using this method become undefined at various heights. ...

... The local maxima near the plates in Fig. 5(a) are observed due to the peaks of k T (z) in Fig. 4(b). Note however that ν t and κ t in RBC have been observed to scale as z 3 in the vicinity of the plates [16,31,49,56], which suggests the constancy of P r t (z) in the near-wall region. As we are mainly concerned about the behavior of P r t in the bulk region, we do not explore further the near-wall variation of the turbulent Prandtl number in this work. ...

Convection in the Sun occurs at Rayleigh numbers, $Ra$, as high as $10^{22}$, molecular Prandtl number, $Pr$, as low as $10^{-6}$, and occurs under conditions that are far from satisfying the Oberbeck-Boussinesq (OB) idealization. The effects of these extreme circumstances on turbulent heat transport are unknown, and no comparable conditions exist on Earth. Our goal is to understand how these effects scale (since we cannot yet replicate the Sun's conditions faithfully). We study thermal convection by using direct numerical simulations, and determine the variation with respect to $Pr$, up to $Pr$ as low as $10^{-4}$, of the turbulent Prandtl number, $Pr_t$, which is the ratio of turbulent viscosity to thermal diffusivity. The simulations are primarily two-dimensional but we draw upon some three-dimensional results as well. We focus on non-Oberbeck-Boussinesq (NOB) conditions of a certain type, but also study OB convection for comparison. The OB simulations are performed in a rectangular box of aspect ratio 2 by varying $Pr$ from $O(10)$ to $10^{-4}$ at fixed Grashof number $Gr \equiv Ra/Pr = 10^9$. The NOB simulations are done in the same box by letting only the thermal diffusivity depend on the temperature. Here, the Rayleigh number is fixed at the top boundary while the mean $Pr$ varies in the bulk from 0.07 to $5 \times 10^{-4}$. The three-dimensional simulations are performed in a box of aspect ratio 25 at a fixed Rayleigh number of $10^5$, and $0.005 < Pr < 7$. The principal finding is that $Pr_t$ increases with decreasing $Pr$ in both OB and NOB convection: $Pr_t \sim Pr^{-0.3}$ for OB convection and $Pr_t \sim Pr^{-1}$ for the NOB case. The $Pr_t$-dependence for the NOB case especially suggests that convective flows in the astrophysical settings behave effectively as in high-Prandtl-number turbulence.

... Other few works have looked at the global probability density function of different observables [20,40,55,58] in relation to non-equilibrium statistical mechanics. Finally, a series of works have taken into account fluctuations in the determination of the structure of boundary layers [37,38,47]. Although in those works the authors took into account the existence of fluctuations in order to determine how they could affect the mean heat flux, they did not study the characteristics of the fluctuations of the heat flux and their possible scaling laws. ...

The study of the transitions among different regimes in thermal convection has been an issue of paramount importance in fluid mechanics. While the bifurcations at low Rayleigh number, when the flow is laminar or moderately chaotic, have been fully understood for a long time, transitions at higher Rayleigh number are much more difficult to be clearly identified. Here, through a numerical study of the two-dimensional Rayleigh-B\'enard convection covering four decades in Rayleigh number for two different Prandtl numbers, we find a clear-cut transition by considering the fluctuations of the heat flux through a horizontal plane, rather than its mean value. More specifically, we have found that this sharp transition is displayed by a jump of the ratio of the root-mean-square fluctuations of the heat flux to its mean value and occurs at $Ra/Pr \approx 10^9$. Above the transition, this ratio is found to be constant in all regions of the flow, while taking different values in the bulk and at the boundaries. Below the transition instead, different behaviors are observed at the boundaries and in the bulk: at the boundaries, this ratio decreases with respect to the Rayleigh number whereas it is found to be constant in the bulk for all values of the Rayleigh number. Through this numerical evidence and an analytical reasoning we confirm what was already observed in experiments, that is the decrease of the ratio of root-mean-square fluctuations of the heat flux to its mean value, observed at the boundaries below the transition, can understood in terms of the law of large numbers.

... As we have seen in this study, the lateral confinement of the convection cell can significantly influence all global response characteristics in the system and also the global structure of the convective flow. Therefore it is desired in the future to advance also the boundary-layer theory for Rayleigh-Bénard convection (see Shishkina et al. [165,166] and Ching et al. [167][168][169]) to the case of confined plates, in order to obtain accurate predictions of the profiles of the main flow characteristics in confined geometries. ...

To study turbulent thermal convection, one often chooses a Rayleigh-Bénard flow configuration, where a fluid is confined between a heated bottom plate, a cooled top plate of the same shape, and insulated vertical sidewalls. When designing a Rayleigh-Bénard setup, for specified fluid properties under Oberbeck-Boussinesq conditions, the maximal size of the plates (diameter or area), and maximal temperature difference between the plates, Δmax, one ponders: Which shape of the plates and aspect ratio Γ of the container (ratio between its horizontal and vertical extensions) would be optimal? In this article, we aim to answer this question, where under the optimal container shape, we understand such a shape, which maximizes the range between the maximal accessible Rayleigh number and the critical Rayleigh number for the onset of convection in the considered setup, Rac,Γ. First we prove that Rac,Γ∝(1+cuΓ−2)(1+cθΓ−2), for some cu>0 and cθ>0. This holds for all containers with no-slip boundaries, which have a shape of a right cylinder, whose bounding plates are convex domains, not necessarily circular. Furthermore, we derive accurate estimates of Rac,Γ, under the assumption that in the expansions (in terms of the Laplace eigenfunctions) of the velocity and reduced temperature at the onset of convection, the contributions of the constant-sign eigenfunctions vanish, both in the vertical and at least in one horizontal direction. With that we derive Rac,Γ≈(2π)4(1+cuΓ−2)(1+cθΓ−2), where cu and cθ are determined by the container shape and boundary conditions for the velocity and temperature, respectively. In particular, for circular cylindrical containers with no-slip and insulated sidewalls, we have cu=j112/π2≈1.49 and cθ=(j̃11)2/π2≈0.34, where j11 and j̃11 are the first positive roots of the Bessel function J1 of the first kind or its derivative, respectively. For parallelepiped containers with the ratios Γx and Γy, Γy≤Γx≡Γ, of the side lengths of the rectangular plates to the cell height, for no-slip and insulated sidewalls we obtain Rac,Γ≈(2π)4(1+Γx−2)(1+Γx−2/4+Γy−2/4). Our approach is essentially different to the linear stability analysis, however, both methods lead to similar results. For Γ≲4.4, the derived Rac,Γ is larger than Jeffreys' result Rac,∞J≈1708 for an unbounded layer, which was obtained with linear stability analysis of the normal modes restricted to the consideration of a single perturbation wave in the horizontal direction. In the limit Γ→∞, the difference between Rac,Γ→∞=(2π)4 for laterally confined containers and Jeffreys' Rac,∞J for an unbounded layer is about 8.8%. We further show that in Rayleigh-Bénard experiments, the optimal rectangular plates are squares, while among all convex plane domains, circles seem to match the optimal shape of the plates. The optimal Γ is independent of Δmax and of the fluid properties. For the adiabatic sidewalls, the optimal Γ is slightly smaller than 1/2 (for cylinder, about 0.46), which means that the intuitive choice of Γ=1/2 in most Rayleigh-Bénard experiments is right and justified. For the given plate diameter D and maximal temperature difference Δmax, the maximal attainable Rayleigh number range is about 3.5 orders of magnitudes smaller than the order of the Rayleigh number based on D and Δmax. Deviations from the optimal Γ lead to a reduction of the attainable range, namely, as log10(Γ) for Γ→0 and as log10(Γ−3) for Γ→∞. Our theory shows that the relevant length scale in Rayleigh-Bénard convection in containers with no-slip boundaries is ℓ∼D/Γ2+cu=H/1+cu/Γ2. This means that in the limit Γ→∞, ℓ equals the cell height H, while for Γ→0, it is rather the plate diameter D.

The study of the transitions among different regimes in thermal convection has been an issue of paramount importance in fluid mechanics. While the bifurcations at low Rayleigh number, when the flow is laminar or moderately chaotic, have been fully understood for a long time, transitions at higher Rayleigh number are much more difficult to be clearly identified. Here, through a numerical study of the two-dimensional Rayleigh-Bénard convection covering four decades in Rayleigh number for two different Prandtl numbers, we find a clear-cut transition by considering the fluctuations of the heat flux through a horizontal plane, rather than its mean value. More specifically, we have found that this sharp transition is displayed by a jump of the ratio of the root-mean-square fluctuations of the heat flux to its mean value and occurs at Ra/Pr≈109. Above the transition, this ratio is found to be constant in all regions of the flow, while taking different values in the bulk and at the boundaries. Below the transition instead, different behaviors are observed at the boundaries and in the bulk: at the boundaries, this ratio decreases with respect to the Rayleigh number whereas it is found to be constant in the bulk for all values of the Rayleigh number. Through this numerical evidence and an analytical reasoning we confirm what was already observed in experiments; that is, the decrease of the ratio of root-mean-square fluctuations of the heat flux to its mean value, observed at the boundaries below the transition, can be understood in terms of the law of large numbers.

In Rayleigh B\'enard Convection, for a range of Prandtl numbers 4.69 ≤ Pr ≤ 5.88 and Rayleigh numbers 5.52×10^5 ≤ Ra ≤ 1.21×10^9, we study the effect of shear by the inherent large-scale flow (LSF) on the local boundary layers on the hot plate. The velocity distribution in a horizontal plane within the boundary layers at each Ra, at any instant, is either (A), unimodal with a peak at around the natural convection boundary layer velocities V_bl; (B), bimodal with the first peak between V_bl and V_L, the shear velocities created by the large-scale flow close to the plate; or (C), unimodal with the peak at around V_L. Type A distributions occur more at lower Ra while type C more at higher Ra, with type B occurring more at intermediate Ra. We show that the second peak of the bimodal, type B distributions and the peak of the unimodal, type C distributions, scale as V_L scales with Ra. We then show that the areas of such regions that have velocities of the order of V_L, increase exponentially with increase in Ra and then saturate. The velocities in the remaining regions, which contribute to the first peak of the bimodal type B distributions and the single peak of type A distributions, are also affected by the shear. We show that the Reynolds number based on these velocities scale as Re_bs, the Reynolds number based on the boundary layer velocities forced externally by the shear due to the LSF, which we obtained as a perturbation solution of the scaling relations derived from integral boundary layer equations. For Pr=1 and aspect ratio Γ=1, Re_{bs} ~ Ra^{0.375} for small shear, similar to the observed flux scaling in a possible ultimate regime. The velocity at the edge of the natural convection boundary layers was found to increase with Ra as Ra^{0.35}; since V_bl~ Ra^{1/3} this suggests a possible shear domination of the boundary layers at high Ra. The effect of shear however decrease with increase in Pr and with increase in Γ, and becomes negligible for Pr ≥ 100 at Γ=1 or for Γ ≥ 20 at Pr=1, causing Re_bs~ Ra_w^{1/3}.

Natural convection arising over vertical and horizontal heated flat surfaces is one of the most ubiquitous flows at a range of spatiotemporal scales. Despite significant developments over more than a century contributing to our fundamental understanding of heat transfer in natural convection boundary layers, certain “hidden” characteristics of these flows have received far less attention. Here, we review scattered progress on less visited fundamental topics that have strong implications to heat and mass transfer control. These topics include the instability characteristics, laminar-to-turbulent transition, and spatial flow structures of vertical natural convection boundary layers and large-scale plumes, dome, and circulating flows over discretely and entirely heated horizontal surfaces. Based on the summarized advancements in fundamental research, we elaborate on the selection of perturbations and provide an outlook on the development of perturbation generators and methods of altering large-scale flow structures as a potential means for heat and mass transfer control where natural convection is dominant.

The influence of the cell inclination on the heat transport and large-scale circulation in liquid metal convection - Volume 884 - Lukas Zwirner, Ruslan Khalilov, Ilya Kolesnichenko, Andrey Mamykin, Sergei Mandrykin, Alexander Pavlinov, Alexander Shestakov, Andrei Teimurazov, Peter Frick, Olga Shishkina

In turbulent Rayleigh-Bénard convection, the boundary layers are nonsteady with fluctuations, the time-averaged large-scale circulating velocity vanishes far away from the top and bottom plates, and the motion arises from buoyancy. In this paper, we derive the full set of boundary layer equations for both the temperature and velocity fields from the Boussinesq equations for a quasi-two-dimensional flow above a heated plate, taking into account all the above effects. By solving these boundary layer equations, both the time-averaged temperature and velocity boundary layer profiles are obtained.

Computational codes for direct numerical simulations of Rayleigh–Bénard (RB) convection are compared in terms of computational cost and quality of the solution. As a benchmark case, RB convection at Ra=10⁸ and Pr=1 in a periodic domain, in cubic and cylindrical containers is considered. A dedicated second-order finite-difference code (AFID/RBFLOW) and a specialized fourth-order finite-volume code (GOLDFISH) are compared with a general purpose finite-volume approach (OPENFOAM) and a general purpose spectral-element code (NEK5000). Reassuringly, all codes provide predictions of the average heat transfer that converge to the same values. The computational costs, however, are found to differ considerably. The specialized codes AFID/RBFLOW and GOLDFISH are found to excel in efficiency, outperforming the general purpose flow solvers NEK5000 and OPENFOAM by an order of magnitude with an error on the Nusselt number Nu below 5%. However, we find that Nu alone is not sufficient to assess the quality of the numerical results: in fact, instantaneous snapshots of the temperature field from a near wall region obtained for deliberately under-resolved simulations using NEK5000 clearly indicate inadequate flow resolution even when Nu is converged. Overall, dedicated special purpose codes for RB convection are found to be more efficient than general purpose codes.

We discuss two aspects of turbulent Rayleigh-B\'{e}nard convection (RBC) on the basis of high-resolution direct numerical simulations in a unique setting; a closed cylindrical cell of aspect ratio of one. First, we present a comprehensive comparison of statistical quantities such as energy dissipation rates and boundary layer thickness scales. Data are used from three simulation run series at Prandtl numbers $Pr$ that cover two orders of magnitude. In contrast to most previous studies in RBC the focus of the present work is on convective turbulence at very low Prandtl numbers including $Pr=0.021$ for liquid mercury or gallium and $Pr=0.005$ for liquid sodium. In this parameter range of RBC, inertial effects cause a dominating turbulent momentum transport that is in line with highly intermittent fluid turbulence both in the bulk and in the boundary layers and thus should be able to trigger a transition to the fully turbulent boundary layers of the ultimate regime of convection for higher Rayleigh number. Secondly, we predict the ranges of Rayleigh numbers for which the viscous boundary layer will transition to turbulence and the flow as a whole will cross over into the ultimate regime. These transition ranges are obtained by extrapolation from our simulation data. The extrapolation methods are based on the large-scale properties of the velocity profile. Two of the three methods predict similar ranges for the transition to ultimate convection when their uncertainties are taken into account. All three extrapolation methods indicate that the range of critical Rayleigh numbers $Ra_c$ is shifted to smaller magnitudes as the Prandtl number becomes smaller.

We report simultaneous measurements of the mean temperature profile θ(z) and temperature variance profile η(z) near the lower conducting plate of a specially designed quasi-two-dimensional cell for turbulent Rayleigh-Bénard convection. The measured θ(z) is found to have a universal scaling form θ(z/δ) with varying thermal boundary layer (BL) thickness δ, and its functional form agrees well with the recently derived BL equation by Shishkina et al. [Phys. Rev. Lett. 114, 114302 (2015)]. The measured η(z), on the other hand, is found to have a scaling form η(z/δ) only in the near-wall region with z/δ≲2. Based on the experimental findings, we derive a BL equation for η(z/δ), which is in good agreement with the experimental results. These BL equations thus provide a common framework for understanding the effect of BL fluctuations.

Results from direct numerical simulations of vertical natural convection at Rayleigh numbers 1.0 x 10(5)-1.0 x 10(9) and Prandtl number 0.709 support a generalised applicability of the Grossmann-Lohse (GL) theory, which was originally developed for horizontal natural (Rayleigh-Benard) convection. In accordance with the GL theory, it is shown that the boundary-layer thicknesses of the velocity and temperature fields in vertical natural convection obey laminar-like Prandtl-Blasius-Pohlhausen scaling. Specifically, the normalised mean boundary-layer thicknesses scale with the -1/2-power of a wind-based Reynolds number, where the 'wind' of the GL theory is interpreted as the maximum mean velocity. Away from the walls, the dissipation of the turbulent fluctuations, which can be interpreted as the 'bulk' or 'background' dissipation of the GL theory, is found to obey the Kolmogorov-Obukhov-Corrsin scaling for fully developed turbulence. In contrast to Rayleigh-Benard convection, the direction of gravity in vertical natural convection is parallel to the mean flow. The orientation of this flow presents an added challenge because there no longer exists an exact relation that links the normalised global dissipations to the Nusselt, Rayleigh and Prandtl numbers. Nevertheless, we show that the unclosed term, namely the global-averaged buoyancy flux that produces the kinetic energy, also exhibits both laminar and turbulent scaling behaviours, consistent with the GL theory. The present results suggest that, similar to Rayleigh-Benard convection, a pure power-law relationship between the Nusselt, Rayleigh and Prandtl numbers is not the best description for vertical natural convection and existing empirical relationships should be recalibrated to better reflect the underlying physics.

We present high-resolution direct numerical simulation studies of turbulent
Rayleigh-Benard convection in a closed cylindrical cell with an aspect ratio of
one. The focus of our analysis is on the finest scales of convective
turbulence, in particular the statistics of the kinetic energy and thermal
dissipation rates in the bulk and the whole cell. The fluctuations of the
energy dissipation field can directly be translated into a fluctuating local
dissipation scale which is found to develop ever finer fluctuations with
increasing Rayleigh number. The range of these scales as well as the
probability of high-amplitude dissipation events decreases with increasing
Prandtl number. In addition, we examine the joint statistics of the two
dissipation fields and the consequences of high-amplitude events. We also have
investigated the convergence properties of our spectral element method and have
found that both dissipation fields are very sensitive to insufficient
resolution. We demonstrate that global transport properties, such as the
Nusselt number, and the energy balances are partly insensitive to insufficient
resolution and yield correct results even when the dissipation fields are
under-resolved. Our present numerical framework is also compared with
high-resolution simulations which use a finite difference method. For most of
the compared quantities the agreement is found to be satisfactory.

A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh-Benard 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 ≤ 1011) the leading terms are Nu ~ Ra(1/4) Pr(1/8), Re ~ Ra(1/2) Pr(-3/4) for Pr ≤ 1 and Nu ~ Ra(1/4) Pr(-1/12), Re ~ Ra(1/2) Pr(-5/6) for Pr ≥ 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 ~ Ra(1/2) Pr(1/2), Re ~ Ra(1/2) Pr(-1/2) for medium Pr ('Kraichnan regime'), a regime with scaling Nu ~ Ra(1/5) Pr(1/5), Re ~ Ra(2/5) Pr(-3/5) for small Pr, a regime with Nu ~ Ra(1/3), Re ~ Ra(4/9) Pr(-2/3) for larger Pr, and a regime with scaling Nu ~ Ra(3/7) Pr(-1/7), Re ~ Ra(4/7) Pr(-6/7) for even larger Pr. In particular, a linear combination of the 1/4 and the (1/3) power laws for Nu with Ra, Nu = 0.27Ra(1/4) + 0.038Ra(1/3) (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.

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, Phys. Rev. Lett. 86, 3316 (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 velocity amplitude Re, we can also give the scaling of the latter, namely, Re~Ra0.42, Re~Ra0.45, and Re~Ra0.50 in the respective ranges.

The progress in our understanding of several aspects of turbulent
Rayleigh-Benard convection is reviewed. The focus is on the question of how the
Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the
Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic
boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the
dynamics of the large-scale convection-roll are addressed as well. The review
ends with a list of challenges for future research on the turbulent
Rayleigh-Benard system.

The unifying theory of scaling in thermal convection (Grossmann & Lohse
(2000)) (henceforth the GL theory) suggests that there are no pure power laws
for the Nusselt and Reynolds numbers as function of the Rayleigh and Prandtl
numbers in the experimentally accessible parameter regime. In Grossmann & Lohse
(2001) the dimensionless parameters of the theory were fitted to 155
experimental data points by Ahlers & Xu (2001) in the regime $3\times 10^7 \le
Ra \le 3 \times 10^{9}$ and $4\le Pr \le 34$ and Grossmann & Lohse (2002) used
the experimental data point from Qiu & Tong (2001) and the fact that Nu(Ra,Pr)
is independent of the parameter a, which relates the dimensionless kinetic
boundary thickness with the square root of the wind Reynolds number, to fix the
Reynolds number dependence. Meanwhile the theory is on one hand well confirmed
through various new experiments and numerical simulations. On the other hand
these new data points provide the basis for an updated fit in a much larger
parameter space. Here we pick four well established (and sufficiently distant)
Nu(Ra,Pr) data points and show that the resulting Nu(Ra,Pr) function is in
agreement with almost all established experimental and numerical data up to the
ultimate regime of thermal convection, whose onset also follows from the
theory. One extra Re(Ra,Pr) data point is used to fix Re(Ra,Pr). As Re can
depend on the definition and the aspect ratio the transformation properties of
the GL equations are discussed in order to show how the GL coefficients can
easily be adapted to new Reynolds number data while keeping Nu(Ra,Pr)
unchanged.

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.

The structure of the boundary layers in turbulent Rayleigh-Benard convection
is studied by means of three-dimensional direct numerical simulations. We
consider convection in a cylindrical cell at an aspect ratio one for Rayleigh
numbers of Ra=3e+9 and 3e+10 at fixed Prandtl number Pr=0.7. Similar to the
experimental results in the same setup and for the same Prandtl number, the
structure of the laminar boundary layers of the velocity and temperature fields
is found to deviate from the prediction of the Prandtl-Blasius-Pohlhausen
theory. Deviations decrease when a dynamical rescaling of the data with an
instantaneously defined boundary layer thickness is performed and the analysis
plane is aligned with the instantaneous direction of the large-scale
circulation in the closed cell. Our numerical results demonstrate that
important assumptions which enter existing classical laminar boundary layer
theories for forced and natural convection are violated, such as the strict
two-dimensionality of the dynamics or the steadiness of the fluid motion. The
boundary layer dynamics consists of two essential local dynamical building
blocks, a plume detachment and a post-plume phase. The former is associated
with larger variations of the instantaneous thickness of velocity and
temperature boundary layer and a fully three-dimensional local flow. The
post-plume dynamics is connected with the large-scale circulation in the cell
that penetrates the boundary region from above. The mean turbulence profiles
taken in localized sections of the boundary layer for both dynamical phases are
also compared with solutions of perturbation expansions of the boundary layer
equations of forced or natural convection towards mixed convection. Our
analysis of both boundary layers shows that the near-wall dynamics combines
elements of forced Blasius-type and natural convection.

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.

Results on the Prandtl–Blasius-type kinetic and thermal boundary layer (BL) thicknesses in turbulent Rayleigh–Bénard (RB) convection in a broad range of Prandtl numbers are presented. By solving the laminar Prandtl–Blasius BL equations, we calculate the ratio between the thermal and kinetic BL thicknesses, which depends on the Prandtl number only. It is approximated as for and as for , with . Comparison of the Prandtl–Blasius velocity BL thickness with that evaluated in the direct numerical simulations by Stevens et al (2010 J. Fluid Mech. 643 495) shows very good agreement between them. Based on the Prandtl–Blasius-type considerations, we derive a lower-bound estimate for the minimum number of computational mesh nodes required to conduct accurate numerical simulations of moderately high (BL-dominated) turbulent RB convection, in the thermal and kinetic BLs close to the bottom and top plates. It is shown that the number of required nodes within each BL depends on and and grows with the Rayleigh number not slower than . This estimate is in excellent agreement with empirical results, which were based on the convergence of the Nusselt number in numerical simulations.

The Rayleigh-Bénard theory by Grossmann and Lohse [J. Fluid Mech. 407, 27 (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 J. Fluid. Mech. 407, 27 (2000). This enables us to more realistically describe the transitions between the various scaling regimes.

This paper investigates the scaling properties of the mean momentum balance (MMB) equation and the mean thermal energy balance (MHB) equation for buoyancy-driven turbulent flow and heat transfer in a differentially heated vertical channel (DHVC). Based on the characteristics of force balance, a three-layer structure is developed for the mean momentum balance equation. In Layer I, a viscous inner layer adjacent to the wall, the force balance is between the viscous force and the buoyancy force. In Layer III, the outer layer, the force balance is between the Reynolds shear force and the buoyancy force. A multiscaling analysis of the MMB equation is developed for the inner and outer layers. In the outer layer, a proper length scale is the channel half width δ, a proper velocity scale is the maximum mean streamwise velocity Umax, and a proper scale for the Reynolds shear stress is uτUmax where uτ is the friction velocity. In the viscous inner layer, a proper length scale is found to be νuτUmaxuτ, where ν is the kinematic viscosity. The structure for the MHB equation can also be divided into three layers based on the characteristics of the diffusional and turbulent heat flux. The thickness of thermal inner layer is found to be OkuτUmaxuτ, where k is the thermal diffusivity. A multiscaling analysis of the MHB equation is developed for the inner and outer layers. The outer-scaling of the MHB equation in a DHVC is similar to passive scalar transport in forced convection, where the channel half width δ is a proper length scale, friction temperature θτ is a proper temperature scale, and uτθτ is a proper scale for turbulent heat flux. The inner-scaling for the thermal inner layer in a DHVC, however, is distinctly different from that in forced convection. The thermal inner length scale in a DHVC is found to be kuτUmaxuτ. The multiscaling analysis of the MMB and MHB equations agree well with direct numerical simulation data of DHVC.

To predict the mean temperature profiles in turbulent thermal convection, the thermal boundary layer (BL) equation has to be solved. Starting from a thermal BL equation that takes into account fluctuations in terms of an eddy thermal diffusivity [Shishkina et al., Phys. Rev. Lett. 114 (2015)], 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. 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 (Pr), from Pr=0.01 to Pr=2547.9.

In this study we follow Grossmann and Lohse, Phys. Rev. Lett. 86 (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 [1], 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 [1], where the thermal dissipation is determined by its bulk contribution.

The boundary layer structure of the velocity and temperature fields in turbulent Rayleigh-Benard flows in closed cylindrical cells of unit aspect ratio is revisited from a transitional and turbulent viscous boundary layer perspective. When the Rayleigh number is large enough, the dynamics at the bottom and top plates can be separated into an impact region of downwelling plumes, an ejection region of upwelling plumes and an interior region away from the side walls. The latter is dominated by the shear of the large-scale circulation (LSC) roll which fills the whole cell and continuously varies its orientation. The working fluid is liquid mercury or gallium at a Prandtl number Pr=0.021 for Rayleigh numbers between Ra=3e+5 and 4e+8. The generated turbulent momentum transfer corresponds to macroscopic flow Reynolds numbers with values between 1800 and 46000. It is shown that the viscous boundary layers for the largest Rayleigh numbers are highly transitional and obey properties that are directly comparable to transitional channel flows at friction Reynolds numbers Re_tau slightly below 100. The transitional character of the viscous boundary layer is also underlined by the strong enhancement of the fluctuations of the wall stress components with increasing Rayleigh number. An extrapolation of our analysis data suggests that the friction Reynolds number Re_tau in the velocity boundary layer can reach values of 200 for Ra beyond 1e+11. Thus the viscous boundary layer in a liquid metal flow would become turbulent at a much lower Rayleigh number than for turbulent convection in gases and gas mixtures.

We derive the dependence of the Reynolds number Re and the Nusselt number Nu on the Rayleigh number Ra and the Prandtl number Pr in laminar vertical convection (VC), where a fluid is confined between two differently heated isothermal vertical walls. The boundary layer equations in laminar VC yield two limiting scaling regimes: Nu∼Pr1/4Ra1/4, Re∼Pr−1/2Ra1/2 for Pr≪1 and Nu∼Pr0Ra1/4, Re∼Pr−1Ra1/2 for Pr≫1. These theoretical results are in excellent agreement with direct numerical simulations for Ra from 105 to 1010 and Pr from 10−2 to 30. The transition between the regimes takes place for Pr around 10−1.

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.

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.

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 to . 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.

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 β 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 β, for infinite and for infinitesimal Prandtl numbers Pr. Dependences on Pr and β 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 β, accurate approximations are developed as functions on Pr. The theoretical results are corroborated by our direct numerical simulations for Pr=0·786 (air) and Pr=4·38 (water). The angle of attack β 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 β≈0·7π and the theoretical predictions based on the Falkner-Skan approximation for this β leads to better agreement with the DNS results, compared with the Prandtl-Blasius approximation for β=π.

We report high-resolution local-temperature measurements in the upper boundary layer of turbulent Rayleigh Bénard (RB) convection with variable Rayleigh number Ra and aspect ratio Gamma. The primary purpose of the work is to create a comprehensive data set of temperature profiles against which various phenomenological theories and numerical simulations can be tested. We performed two series of measurements for air (Pr {=} 0.7) in a cylindrical container, which cover a range from Ra {≈} 10(9) to Ra {≈} 10(12) and from Gamma {≈} 1 to Gamma {≈} 10. In the first series Gamma was varied while the temperature difference was kept constant, whereas in the second series the aspect ratio was set to its lowest possible value, Gamma {=} 1.13, and Ra was varied by changing the temperature difference. We present the profiles of the mean temperature, root-mean-square (r.m.s.) temperature fluctuation, skewness and kurtosis as functions of the vertical distance z from the cooling plate. Outside the (very short) linear part of the thermal boundary layer the non-dimensional mean temperature Theta is found to scale as Theta(z) {˜} z(alpha) , the exponent alpha {≈} 0.5 depending only weakly on Ra and Gamma. This result supports neither Prandtl's one-third law nor a logarithmic scaling law for the mean temperature. The r.m.s. temperature fluctuation sigma is found to decay with increasing distance from the cooling plate according to sigma(z) {˜} z(beta) , where the value of beta is in the range -0.30 {>} beta {>} {-}0.42 and depends on both Ra and Gamma. Priestley's beta {=} {-}1/3 law is consistent with this finding but cannot explain the variation in the scaling exponent. In addition to profiles we also present and discuss boundary-layer thicknesses, Nusselt numbers and their scaling with Ra and Gamma.

This review summarizes results for Rayleigh-Bénard 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.

In 1997, a Rayleigh-Bénard experiment evidenced a significant increase of the heat transport efficiency for Rayleigh numbers larger than Ra~1012 and interpreted this observation as the signature of Kraichnan's "Ultimate Regime" of convection. According to Kraichnan's 1962 prediction, the flow boundary layers above the cold and hot plates —in which most of the fluid temperature drop is localized— become unstable for large enough Ra and this instability boosts the heat transport compared to the other turbulent regimes. Using the same convection cell as in the 1997 experiment, we show that the reported heat transport increase is accompanied with enhanced and increasingly skewed temperature fluctuations of the bottom plate, which was heated at constant power levels. Thus, for Ra<1012, the bottom plate fluctuations can simply be accounted from those in the bulk of the flow. In particular, they share the same spectral density at low frequencies, as if the bottom plate was following the slow temperature fluctuations of the bulk, modulo a constant temperature drop across the bottom boundary layer. Conversely, to account for the plate's temperature fluctuations at higher Ra, we no-longer can ignore the fluctuations of the temperature drop across the boundary layer. These observations, consistent with a boundary layer instability, provide new evidence that the transition reported in 1997 corresponds to the triggering of the Ultimate Regime of convection.

We report measurements of the temporal fluctuations of the heat flux in Rayleigh-Bénard turbulent convection in various fluids and geometries. We observe that the rms fluctuations of the heat flux increase nearly proportionally to the temperature difference ΔT. The ratio of the rms fluctuations of the heat flux to their mean display a power law, Ra−γ on two decades in Rayleigh number (2107 < Ra < 3109). We discuss this law as well as the non-Gaussian character of the probability density function of the heat flux.

Die freie Konvektion an einer beheizten horizontalen Platte wird nach den blichen Methoden der Grenzschichttheorie untersucht. Weist die wrmebertragende Plattenseite nach oben, so ist das Problem auf dieser Basis unlsbar, wohl aber findet man eine widerspruchsfreie Lsung, wenn diese Plattenseite nach unten weist. In diesem ist die Nusselt Zahl, die den Wrmebergang kennzeichnet, proportional der 1/5-ten Potenz der Rayleigh Zahl, welche die Beheizung charakterisiert. Die bereinstimmung mit dem Experiment ist befriedigend.

We report a study of mean vertical temperature profiles (TPs) in turbulent Rayleigh-Benard convection of water, Pr=4.38, in unit-aspect-ratio cylindrical and cubic cells for Ra up to 10^9, based on DNS. The Nusselt numbers Nu computed for cylindrical cells are found to be in excellent agreement with the experimental data by Funfschilling et al. [J. Fluid Mech., vol. 536 (2005), pp. 145-154]. Based on this validation, the DNS data are used to extract TPs. In the DNS for the cylindrical geometry, reported in Shishkina & Thess [J. Fluid Mech. (2009), in press], we find that near the heating and cooling plates the TP theta(y) obey neither a logarithmic nor a power law. We show that the Prandtl--Blasius BL theory predicts the TP-shapes with an error 7.9% within the thermal BLs alone. We further show that the profiles can be approximated by a stretched exponential approximation (SEA) of the form theta(y) 1-(-y-0.5y^2) with an absolute error

The turbulent natural convection boundary layer next to a heated vertical surface is analyzed by classical scaling arguments. It is shown that the fully developed turbulent boundary layer must be treated in two parts: an outer region consisting of most of the boundary layer in which viscous and conduction terms are negligible and an inner region in which the mean convection terms are negligible. The inner layer is identified as a constant heat flux layer. A similarity analysis yields universal profiles for velocity and temperature in the outer and constant heat flux layers. An asymptotic matching of these profiles in an intermediate layer (the buoyant sublayer) as Hδ &z.tbnd gβF 0δ4 α3 → ∞ yields analytical expressions for the buoyant sublayer profiles as (T-Tw) T1 = K2( y n)- 1 3 + A(pr), U U1 = K1( y n)1 3 + B(pr), where K1, K2 are universal constants and A(Pr), B(Pr) are universal functions of Prandtl number. Asymptotic heat transfer and friction laws are obtained as Nux = C'H(Pr)H*x 1 4, τw grU21 = Cf(Pr), where C'H(Pr) is simply related to A(Pr). Finally, conductive and thermo-viscous sublayers characterized by a linear variation of velocity and temperature are shown to exist at the wall. All predictions are seen to be in excellent agreement with the abundant experimental data.

Über Flüssigkeitsbewegung bei sehr kleiner Reibung, in Verhandlungen des III

L. Prandtl, "Über Flüssigkeitsbewegung bei sehr kleiner Reibung, in Verhandlungen des III. Int. Math.
Kongr., Heidelberg, 1904 (Teubner, Leipzig, 1905), pp. 484-491.