scholarly journals Prandtl–Blasius temperature and velocity boundary-layer profiles in turbulent Rayleigh–Bénard convection

2010 ◽  
Vol 664 ◽  
pp. 297-312 ◽  
Author(s):  
QUAN ZHOU ◽  
RICHARD J. A. M. STEVENS ◽  
KAZUYASU SUGIYAMA ◽  
SIEGFRIED GROSSMANN ◽  
DETLEF LOHSE ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Rayleigh Number ◽  
Reference Frames ◽  
Temperature Profiles ◽  
Number Range ◽  
Velocity Boundary Layer ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

The shapes of the velocity and temperature profiles near the horizontal conducting plates' centre regions in turbulent Rayleigh–Bénard convection are studied numerically and experimentally over the Rayleigh number range 108 ≲ Ra ≲ 3 × 1011 and the Prandtl number range 0.7 ≲ Pr ≲ 5.4. The results show that both the temperature and velocity profiles agree well with the classical Prandtl–Blasius (PB) laminar boundary-layer profiles, if they are re-sampled in the respective dynamical reference frames that fluctuate with the instantaneous thermal and velocity boundary-layer thicknesses. The study further shows that the PB boundary layer in turbulent thermal convection not only holds in a time-averaged sense, but is most of the time also valid in an instantaneous sense.

scholarly journals Aspect ratio dependence of heat transport by turbulent Rayleigh–Bénard convection in rectangular cells

2012 ◽  
Vol 710 ◽  
pp. 260-276 ◽  
Author(s):  
Quan Zhou ◽  
Bo-Fang Liu ◽  
Chun-Mei Li ◽  
Bao-Chang Zhong
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Finite Conductivity ◽  
Turbulent Heat ◽  
Number Range ◽  
Aspect Ratios ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractWe report high-precision measurements of the Nusselt number $Nu$ as a function of the Rayleigh number $Ra$ in water-filled rectangular Rayleigh–Bénard convection cells. The horizontal length $L$ and width $W$ of the cells are 50.0 and 15.0 cm, respectively, and the heights $H= 49. 9$, 25.0, 12.5, 6.9, 3.5, and 2.4 cm, corresponding to the aspect ratios $({\Gamma }_{x} \equiv L/ H, {\Gamma }_{y} \equiv W/ H)= (1, 0. 3)$, $(2, 0. 6)$, $(4, 1. 2)$, $(7. 3, 2. 2)$, $(14. 3, 4. 3)$, and $(20. 8, 6. 3)$. The measurements were carried out over the Rayleigh number range $6\ensuremath{\times} 1{0}^{5} \lesssim Ra\lesssim 1{0}^{11} $ and the Prandtl number range $5. 2\lesssim Pr\lesssim 7$. Our results show that for rectangular geometry turbulent heat transport is independent of the cells’ aspect ratios and hence is insensitive to the nature and structures of the large-scale mean flows of the system. This is slightly different from the observations in cylindrical cells where $Nu$ is found to be in general a decreasing function of $\Gamma $, at least for $\Gamma = 1$ and larger. Such a difference is probably a manifestation of the finite plate conductivity effect. Corrections for the influence of the finite conductivity of the top and bottom plates are made to obtain the estimates of $N{u}_{\infty } $ for plates with perfect conductivity. The local scaling exponents ${\ensuremath{\beta} }_{l} $ of $N{u}_{\infty } \ensuremath{\sim} R{a}^{{\ensuremath{\beta} }_{l} } $ are calculated and found to increase from 0.243 at $Ra\simeq 9\ensuremath{\times} 1{0}^{5} $ to 0.327 at $Ra\simeq 4\ensuremath{\times} 1{0}^{10} $.

scholarly journals Local boundary layer scales in turbulent Rayleigh–Bénard convection

2014 ◽  
Vol 758 ◽  
pp. 344-373 ◽  
Author(s):  
Janet D. Scheel ◽  
Jörg Schumacher
Keyword(s):  
Boundary Layer ◽  
Rayleigh Number ◽  
Aspect Ratio ◽  
Layer Thickness ◽  
Boundary Layers ◽  
Local Boundary ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractWe compute fully local boundary layer scales in three-dimensional turbulent Rayleigh–Bénard convection. These scales are directly connected to the highly intermittent fluctuations of the fluxes of momentum and heat at the isothermal top and bottom walls and are statistically distributed around the corresponding mean thickness scales. The local boundary layer scales also reflect the strong spatial inhomogeneities of both boundary layers due to the large-scale, but complex and intermittent, circulation that builds up in closed convection cells. Similar to turbulent boundary layers, we define inner scales based on local shear stress that can be consistently extended to the classical viscous scales in bulk turbulence, e.g. the Kolmogorov scale, and outer scales based on slopes at the wall. We discuss the consequences of our generalization, in particular the scaling of our inner and outer boundary layer thicknesses and the resulting shear Reynolds number with respect to the Rayleigh number. The mean outer thickness scale for the temperature field is close to the standard definition of a thermal boundary layer thickness. In the case of the velocity field, under certain conditions the outer scale follows a scaling similar to that of the Prandtl–Blasius type definition with respect to the Rayleigh number, but differs quantitatively. The friction coefficient $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}c_{\epsilon }$ scaling is found to fall right between the laminar and turbulent limits, which indicates that the boundary layer exhibits transitional behaviour. Additionally, we conduct an analysis of the recently suggested dissipation layer thickness scales versus the Rayleigh number and find a transition in the scaling. All our investigations are based on highly accurate spectral element simulations that reproduce gradients and their fluctuations reliably. The study is done for a Prandtl number of $\mathit{Pr}=0.7$ and for Rayleigh numbers that extend over almost five orders of magnitude, $3\times 10^5\le \mathit{Ra} \le 10^{10}$, in cells with an aspect ratio of one. We also performed one study with an aspect ratio equal to three in the case of $\mathit{Ra}=10^8$. For both aspect ratios, we find that the scale distributions depend on the position at the plates where the analysis is conducted.

How surface roughness reduces heat transport for small roughness heights in turbulent Rayleigh–Bénard convection

2017 ◽  
Vol 836 ◽  
Author(s):  
Yi-Zhao Zhang ◽  
Chao Sun ◽  
Yun Bao ◽  
Quan Zhou
Keyword(s):  
Heat Transfer ◽  
Rayleigh Number ◽  
Heat Transport ◽  
Roughness Height ◽  
Physical Reason ◽  
Number Range ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Rough surfaces have been widely used as an efficient way to enhance the heat-transfer efficiency in turbulent thermal convection. In this paper, however, we show that roughness does not always mean a heat-transfer enhancement, but in some cases it can also reduce the overall heat transport through the system. To reveal this, we carry out numerical investigations of turbulent Rayleigh–Bénard convection over rough conducting plates. Our study includes two-dimensional (2D) simulations over the Rayleigh number range $10^{7}\leqslant Ra\leqslant 10^{11}$ and three-dimensional (3D) simulations at $Ra=10^{8}$. The Prandtl number is fixed to $Pr=0.7$ for both the 2D and the 3D cases. At a fixed Rayleigh number $Ra$, reduction of the Nusselt number $Nu$ is observed for small roughness height $h$, whereas heat-transport enhancement occurs for large $h$. The crossover between the two regimes yields a critical roughness height $h_{c}$, which is found to decrease with increasing $Ra$ as $h_{c}\sim Ra^{-0.6}$. Through dimensional analysis, we provide a physical explanation for this dependence. The physical reason for the $Nu$ reduction is that the hot/cold fluid is trapped and accumulated inside the cavity regions between the rough elements, leading to a much thicker thermal boundary layer and thus impeding the overall heat flux through the system.

Thermal and viscous boundary layers in turbulent Rayleigh–Bénard convection

2012 ◽  
Vol 711 ◽  
pp. 281-305 ◽  
Author(s):  
J. D. Scheel ◽  
E. Kim ◽  
K. R. White
Keyword(s):  
Boundary Layer ◽  
Rayleigh Number ◽  
Scaling Laws ◽  
Viscous Boundary Layer ◽  
Variable Boundary ◽  
Bénard Convection ◽  
Benard Convection ◽  
Viscous Boundary ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractWe 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.

Torsional oscillations of the large-scale circulation in turbulent Rayleigh–Bénard convection

2008 ◽  
Vol 607 ◽  
pp. 119-139 ◽  
Author(s):  
DENIS FUNFSCHILLING ◽  
ERIC BROWN ◽  
GUENTER AHLERS
Keyword(s):  
Rayleigh Number ◽  
Aspect Ratio ◽  
Large Scale ◽  
Small Scale ◽  
Large Scale Circulation ◽  
Number Range ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Measurements over the Rayleigh-number range 108 ≲ R ≲ 1011 and Prandtl-number range 4.4≲σ≲29 that determine the torsional nature and amplitude of the oscillatory mode of the large-scale circulation (LSC) of turbulent Rayleigh–Bénard convection are presented. For cylindrical samples of aspect ratio Γ=1 the mode consists of an azimuthal twist of the near-vertical LSC circulation plane, with the top and bottom halves of the plane oscillating out of phase by half a cycle. The data for Γ=1 and σ=4.4 showed that the oscillation amplitude varied irregularly in time, yielding a Gaussian probability distribution centred at zero for the displacement angle. This result can be described well by the equation of motion of a stochastically driven damped harmonic oscillator. It suggests that the existence of the oscillations is a consequence of the stochastic driving by the small-scale turbulent background fluctuations of the system, rather than a consequence of a Hopf bifurcation of the deterministic system. The power spectrum of the LSC orientation had a peak at finite frequency with a quality factor Q≃5, nearly independent of R. For samples with Γ≥2 we did not find this mode, but there remained a characteristic periodic signal that was detectable in the area density ρp of the plumes above the bottom-plate centre. Measurements of ρp revealed a strong dependence on the Rayleigh number R, and on the aspect ratio Γ that could be represented by ρp ~ Γ2.7±0.3. Movies are available with the online version of the paper.

Direct numerical simulations of Rayleigh–Bénard convection in water with non-Oberbeck–Boussinesq effects

2019 ◽  
Vol 881 ◽  
pp. 1073-1096 ◽  
Author(s):  
Andreas D. Demou ◽  
Dimokratis G. E. Grigoriadis
Keyword(s):  
Numerical Simulations ◽  
Two Dimensions ◽  
Direct Numerical Simulations ◽  
Wall Region ◽  
Number Range ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Near Wall

Rayleigh–Bénard convection in water is studied by means of direct numerical simulations, taking into account the variation of properties. The simulations considered a three-dimensional (3-D) cavity with a square cross-section and its two-dimensional (2-D) equivalent, covering a Rayleigh number range of $10^{6}\leqslant Ra\leqslant 10^{9}$ and using temperature differences up to 60 K. The main objectives of this study are (i) to investigate and report differences obtained by 2-D and 3-D simulations and (ii) to provide a first appreciation of the non-Oberbeck–Boussinesq (NOB) effects on the near-wall time-averaged and root-mean-squared (r.m.s.) temperature fields. The Nusselt number and the thermal boundary layer thickness exhibit the most pronounced differences when calculated in two dimensions and three dimensions, even though the $Ra$ scaling exponents are similar. These differences are closely related to the modification of the large-scale circulation pattern and become less pronounced when the NOB values are normalised with the respective Oberbeck–Boussinesq (OB) values. It is also demonstrated that NOB effects modify the near-wall temperature statistics, promoting the breaking of the top–bottom symmetry which characterises the OB approximation. The most prominent NOB effect in the near-wall region is the modification of the maximum r.m.s. values of temperature, which are found to increase at the top and decrease at the bottom of the cavity.

scholarly journals Heat transfer in rotating Rayleigh–Bénard convection with rough plates

2017 ◽  
Vol 830 ◽  
Author(s):  
Pranav Joshi ◽  
Hadi Rajaei ◽  
Rudie P. J. Kunnen ◽  
Herman J. H. Clercx
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Ekman Layer ◽  
Ekman Pumping ◽  
Ekman Boundary Layer ◽  
Bénard Convection ◽  
Roughness Elements ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

This experimental study focuses on the effect of horizontal boundaries with pyramid-shaped roughness elements on the heat transfer in rotating Rayleigh–Bénard convection. It is shown that the Ekman pumping mechanism, which is responsible for the heat transfer enhancement under rotation in the case of smooth top and bottom surfaces, is unaffected by the roughness as long as the Ekman layer thickness $\unicode[STIX]{x1D6FF}_{E}$ is significantly larger than the roughness height $k$. As the rotation rate increases, and thus $\unicode[STIX]{x1D6FF}_{E}$ decreases, the roughness elements penetrate the radially inward flow in the interior of the Ekman boundary layer that feeds the columnar Ekman vortices. This perturbation generates additional thermal disturbances which are found to increase the heat transfer efficiency even further. However, when $\unicode[STIX]{x1D6FF}_{E}\approx k$, the Ekman boundary layer is strongly perturbed by the roughness elements and the Ekman pumping mechanism is suppressed. The results suggest that the Ekman pumping is re-established for $\unicode[STIX]{x1D6FF}_{E}\ll k$ as the faces of the pyramidal roughness elements then act locally as a sloping boundary on which an Ekman layer can be formed.

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