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# Heat flux in turbulent Rayleigh-Bénard convection: Predictions derived from a boundary layer theory

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## Abstract

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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... 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. ...
... 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. ...
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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.
... 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. ...
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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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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 {&ap;} 10(9) to Ra {&ap;} 10(12) and from Gamma {&ap;} 1 to Gamma {&ap;} 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 {&ap;} 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.
Article
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.
Article
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.
Article
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.
Article
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.
Conference Paper
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
Article
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.