Results on the Prandtl-Blasius type kinetic and thermal boundary layer thicknesses in turbulent Rayleigh-Bénard convection in a broad range of Prandtl numbers are presented. By solving the laminar Prandtl-Blasius boundary layer equations, we calculate the ratio of the thermal and kinetic boundary layer thicknesses, which depends on the Prandtl number Pr only. It is approximated as 0.588Pr −1/2 for Pr Pr * and as 0.982Pr −1/3 for Pr * Pr, with Pr * ≡ 0.046. Comparison of the Prandtl-Blasius velocity boundary layer thickness with that evaluated in the direct numerical simulations by Stevens, Verzicco, and Lohse (J. Fluid Mech. 643, 495 (2010)) gives very good agreement. Based on the Prandtl-Blasius type considerations, we derive a lower-bound estimate for the minimum number of the computational mesh nodes, required to conduct accurate numerical simulations of moderately high (boundary layer dominated) turbulent Rayleigh-Bénard convection, in the thermal and kinetic boundary layers close to bottom and top plates. It is shown that the number of required nodes within each boundary layer depends on Nu and Pr and grows with the Rayleigh number Ra not slower than ∼ Ra 0.15 . This estimate agrees excellently with empirical results, which were based on the convergence of the Nusselt number in numerical simulations. arXiv:1109.6870v1 [physics.flu-dyn] 30 Sep 2011Boundary layer structure in turbulent thermal convection 2 IntroductionRayleigh-Bénard (RB) convection is the classical system to study properties of thermal convection. In this system a layer of fluid confined between two horizontal plates is heated from below and cooled from above. Thermally driven flows are of utmost importance in industrial applications and in natural phenomena. Examples include the thermal convection in the atmosphere, the ocean, in buildings, in process technology, and in metal-production processes. In the geophysical and astrophysical context one may think of convection in Earth's mantle, in Earth's outer core, and in the outer layer of the Sun. E.g., the random reversals of Earth's or the Sun's magnetic field have been connected with thermal convection.Major progress in the understanding of the Rayleigh-Bénard system has been made over the last decades, see e.g. the recent reviews [1,2]. Meanwhile it has been well established that the general heat transfer properties of the system, i. e. Nu = Nu(Ra, Pr) and Re = Re(Nu, Pr), are well described by the Grossmann-Lohse (GL) theory [3,4,5,6]. In that theory, in order to estimate the thicknesses of the kinetic and thermal boundary layers (BL) and the viscous and thermal dissipation rates, the boundary layer flow is considered to be scalingwise laminar Prandtl-Blasius flow over a plate. We use the conventional definitions: The Rayleigh number is Ra = αgH 3 ∆/νκ with the isobaric thermal expansion coefficient α, the gravitational acceleration g, the height H of the RB system, the temperature difference ∆ between the heated lower plate and the cooled upper plate, and the material constants ν, kinema...

We analyse the wind and boundary layer properties of turbulent Rayleigh–Bénard convection in a cylindrical container with aspect ratio one for Prandtl number $\mathit{Pr}= 0. 786$ and Rayleigh numbers ($\mathit{Ra}$) up to $1{0}^{9} $ by means of highly resolved direct numerical simulations. We identify time periods in which the orientation of the large-scale circulation (LSC) is nearly constant in order to perform a statistical analysis of the LSC. The analysis is then reduced to two dimensions by considering only the plane of the LSC. Within this plane the LSC is treated as a wind with thermal and viscous boundary layers developing close to the horizontal plates. Special focus is on the spatial development of the wind magnitude and the boundary layer thicknesses along the bottom plate. A method for the local analysis of the instantaneous boundary layer thicknesses is introduced which shows a dramatically changing wind magnitude along the wind path. Furthermore a linear increase of the viscous and thermal boundary layer thickness along the wind direction is observed for all $\mathit{Ra}$ considered while their ratio is spatially constant but depends weakly on $\mathit{Ra}$. A possible explanation is a strong spatial variation of the wind magnitude and fluctuations in the boundary layer region.

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.PACS numbers: 44.20.+b, 44.25.+f, 47.27.ek, 47.27.te Turbulent Rayleigh-Bénard convection (RBC) [1][2][3][4][5], consisting of a fluid confined between two horizontal plates, heated from below and cooled from above, is a system of great research interest. It is a paradigm system for studying turbulent thermal convection, which is ubiquitous in nature, occurring in the atmosphere and the mantle of the Earth as well as in stars like our Sun. Convective heat transfer is also an important problem in engineering and technological applications. The state of fluid motion in RBC is determined by the Rayleigh number Ra = αg∆H 3 /(κν) and Prandtl number Pr = ν/κ. Here α denotes the isobaric thermal expansion coefficient, ν the kinematic viscosity and κ the thermal diffusivity of the fluid, g the acceleration due to gravity, ∆ the temperature difference between the bottom and top plates, and H the distance between the plates.In turbulent RBC, there are viscous boundary layers (BLs) near all rigid walls and two thermal BLs, one above the bottom plate and one below the top plate. We denote the thicknesses of the viscous and thermal BLs by l and λ respectively. Both viscous and thermal BLs play a critical role in the turbulent heat transfer of the system and in particular λ is inversely proportional to the heat transport. Grossmann and Lohse (GL) [6], [7] developed a scaling theory of how the Reynolds number Re, determined by the mean large-scale circulation velocity U 0 above the viscous BL, and the dimensionless Nusselt number Nu, measuring the heat transport, depend on Ra and Pr for moderate Ra. The theory makes explicit use of the result l/H ∝ Re −1/2 with the proportionality constant depending only on Pr. This result follows from the assumptions that the BLs are laminar and their mean profiles, averaged over time, are described by the Prandtl-Blasius-Pohlhausen (PBP) theory [8-10] for steady-state forced convection above an infinite weakly-heated plate. Although the GL theory gives perfect agreement with the heat transport measurements, the assumption that the BLs are described by PBP theory is not fulfilled. Systematic deviations of the mean velocity and temperature profiles from the PBP predictions have been reported both in experiments and in direct numerical simulations (DNS) [11][12][13][14][15]. These deviations remain even after a dynamical rescaling procedure [16] that takes into account of the time variations of λ is used, and increase ...

For rapidly rotating turbulent Rayleigh-Bénard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one whose signature is a bimodal temperature distribution near the radial boundary. The BZF width is found to scale like Ra 1/4 Ek 2/3 where the Ekman number Ek decreases with increasing rotation rate.Turbulent fluid motion driven by buoyancy and influenced by rotation is a common phenomenon in nature and is important in many industrial applications. In the widely studied laboratory realization of turbulent convection, Rayleigh-Bénard convection (RBC) [1, 2], a fluid is confined in a convection cell with a heated bottom, cooled top, and adiabatic vertical walls. For these conditions, a large scale circulation (LSC) arises from cooperative plume motion and is an important feature of turbulent RBC [1]. The addition of rotation about a vertical axis produces a different type of convection as thermal plumes are transformed into thermal vortices, over some regions of parameter space heat transport is enhanced by Ekman pumping [3][4][5][6][7][8][9][10], and statistical measures of vorticity and temperature fluctuations in the bulk are strongly influenced [11][12][13][14][15][16][17]. A crucial aspect of rotation is to suppress, for sufficiently rapid rotation rates, the LSC of non-rotating convection [12,13,18,19], although the diameter-to-height aspect ratio Γ = D/H appears to play some role in the nature of the suppression [20].In RBC geometries with 1/2 ≤ Γ ≤ 2, the LSC usually spans the cell in a roll-like circulation of size H. For rotating convection, the intrinsic linear scale of separation of vortices is reduced with increasing rotation rate [21,22], suggesting that one might reduce the geometric aspect ratio, i.e., Γ < 1 while maintaining a large ratio of lateral cell size to linear scale [5]; such convection cells are being implemented in numerous new experiments [23]. Thus, an important question about rotating convection in slender cylindrical cells is whether there is a global circulation that substantially influences the internal state of the system and carries appreciable global heat transport. Direct numerical simulations (DNS) of rotat-ing convection [24] in cylindrical geometry with Γ = 1, inverse Rossby number 1/Ro = 2.78, Rayleigh number Ra = 10 9 and Prandtl number Pr = 6.4 (Ro, Ra and Pr defined below) revealed a cyclonic azimuthal velocity boundary-layer flow surrounding a core region of anticyclonic circulation and localized near the cylinder sidewall. The results were interpreted in the context of sidewall Stewartson layers driven by active Ekman layers at the top and bottom of the cell [25,26].Here we show through DNS and experimental measurements for a...

We report on a numerical study of the aspect-ratio dependency of Rayleigh-Bénard convection, using direct numerical simulations. The investigated domains have equal height and width while the aspect ratio Γ of depth per height is varied between 1/10 and 1. The Rayleigh numbers \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}$\end{document}Ra for this study variate between 105 and 109, while the Prandtl number is \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Pr}} = 0.786$\end{document}Pr=0.786. The main focus of the study concerns the dependency of the Nusselt number \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Nu}}$\end{document}Nu and the Reynolds number \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Re}}$\end{document}Re on \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}$\end{document}Ra and Γ. It turns out that due to Γ, differences to the cubic case (i.e., Γ = 1) in \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Nu}}$\end{document}Nu of up to 55% and in \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Re}}$\end{document}Re of up to 97% occur, which decrease for increasing \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}$\end{document}Ra. In particular for small Γ sudden drops in the \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}$\end{document}Ra-scaling of \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Nu}}$\end{document}Nu and \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Re}}$\end{document}Re appear for \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}\approx 10^6$\end{document}Ra≈106. Further analysis reveals that these correspond to the onset of unsteady motion accompanied by changes in the global flow structure. The latter is investigated by statistical analysis of the heat flux distribution on the bottom and top plates and a decomposition of the instantaneous flow fields into two-dimensional modes. For \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}$\end{document}Ra slightly above the onset of unsteady motion (i.e., \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}\approx 10^6$\end{document}Ra≈106) for all considered Γ ⩽ 1/3 a four-roll structure is present, which corresponds to thermal plumes moving vertically through the domain's center. For \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}\ge 10^7$\end{document}Ra≥107, also for small Γ, a single-roll structure is dominant, in agreement with two-dimensional simulations and experiments at larger \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Ra}}$\end{document}Ra and \documentclass[12pt]{minimal}\begin{document}$\mbox{\textit {Pr}}$\end{document}Pr.