scholarly journals Counter-gradient heat transport in two-dimensional turbulent Rayleigh–Bénard convection

2013 ◽  
Vol 737 ◽  
Author(s):  
Yong-Xiang Huang ◽  
Quan Zhou
Keyword(s):  
Prandtl Number ◽  
Heat Transport ◽  
Large Scale ◽  
Three Dimensional ◽  
Local Heat ◽  
Two Dimensional ◽  
Number Range ◽  
Local Heat Flux ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractWe present high-resolution numerical investigations of heat transport by two-dimensional (2D) turbulent Rayleigh–Bénard (RB) convection over the Rayleigh number range $1{0}^{8} \leqslant Ra\leqslant 1{0}^{10} $ and the Prandtl number range $0. 7\leqslant Pr\leqslant 10$. We find that there exists strong counter-gradient local heat flux with magnitude much larger than the global Nusselt number $Nu$ of the system. Two mechanisms for generating counter-gradient heat transport are identified: one is due to the bulk dynamics and the other is due to the competition between the corner-flow rolls and the large-scale circulation (LSC). While the magnitude of the former is found to increase with increasing Prandtl number, that of the latter maximizes at medium $Pr$. We further reveal that the corner–LSC competition leads to the anomalous $Nu$–$Pr$ relation in 2D RB convection, i.e. $Nu(Pr)$ minimizes, rather than maximizes as in the three-dimensional cylindrical case, at $Pr\approx 2\sim 3$ for moderate $Ra$.

A model for the emergence of thermal plumes in Rayleigh–Bénard convection at infinite Prandtl number

2000 ◽  
Vol 414 ◽  
pp. 225-250 ◽  
Author(s):  
C. LEMERY ◽  
Y. RICARD ◽  
J. SOMMERIA
Keyword(s):  
Prandtl Number ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Thermal Plumes ◽  
Bénard Convection ◽  
Benard Convection ◽  
Two Dimensional Model ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We propose a two-dimensional model of three-dimensional Rayleigh–Bénard convection in the limit of very high Prandtl number and Rayleigh number, as in the Earth's mantle. The model equation describes the evolution of the first moment of the temperature anomaly in the thermal boundary layer, which is assumed thin with respect to the scale of motion. This two-dimensional field is transported by the velocity that it induces and is amplified by surface divergence. This model explains the emergence of thermal plumes, which arise as finite-time singularities. We determine critical exponents for these singularities. Using a smoothing method we go beyond the singularity and reach a stage of developed convection. We describe a process of plume merging, leaving room for the birth of new instabilities. The heat flow at the surface predicted by our two-dimensional model is found to be in good agreement with available data.

scholarly journals Boundary layer structure in turbulent Rayleigh–Bénard convection

2012 ◽  
Vol 706 ◽  
pp. 5-33 ◽  
Author(s):  
Nan Shi ◽  
Mohammad S. Emran ◽  
Jörg Schumacher
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Prandtl Number ◽  
Boundary Layers ◽  
Large Scale ◽  
Three Dimensional ◽  
Large Scale Circulation ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractThe structure of the boundary layers in turbulent Rayleigh–Bénard convection is studied by means of three-dimensional direct numerical simulations. We consider convection in a cylindrical cell at aspect ratio one for Rayleigh numbers of $\mathit{Ra}= 3\ensuremath{\times} 1{0}^{9} $ and $3\ensuremath{\times} 1{0}^{10} $ at fixed Prandtl number $\mathit{Pr}= 0. 7$. 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 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 of 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 each dynamical phase 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.

Turbulent Rayleigh–Bénard convection for a Prandtl number of 0.67

2009 ◽  
Vol 641 ◽  
pp. 157-167 ◽  
Author(s):  
GUENTER AHLERS ◽  
EBERHARD BODENSCHATZ ◽  
DENIS FUNFSCHILLING ◽  
JAMES HOGG
Keyword(s):  
Prandtl Number ◽  
Numerical Simulations ◽  
Large Scale ◽  
Direct Numerical Simulations ◽  
Time Intervals ◽  
Number Range ◽  
Convection Roll ◽  
Helium Gas ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

For the Rayleigh-number range 107 ≲ Ra ≲ 1011 we report measurements of the Nusselt number Nu and of properties of the large-scale circulation (LSC) for cylindrical samples of helium gas (Prandtl number Pr = 0.674) that have aspect ratio Γ ≡ D/L = 0.50 (D and L are the diameter and the height respectively) and are heated from below. The results for Nu are consistent with recent direct numerical simulations. We measured the amplitude δ of the azimuthal temperature variation induced by the LSC at the sidewall, and the LSC circulation-plane orientation θ0, at three vertical positions. For the entire Ra range the LSC involves a convection roll that is coherent over the height of the system. However, this structure frequently collapses completely at irregular time intervals and then reorganizes from the incoherent flow. At small δ the probability distribution p(δ) increases linearly from zero; for Γ = 1 and Pr = 4.38 this increase is exponential. No evidence of a two-roll structure, with one above the other, was observed. This differs from recent direct numerical simulations for Γ = 0.5 and Pr = 0.7, where a one-roll LSC was found to exist only for Ra ≲ 109 to 1010, and from measurements for Γ = 0.5 and Pr ≃ 5, where one- and two-roll structures were observed with transitions between them at random time intervals.

The large-scale flow structure in turbulent rotating Rayleigh–Bénard convection

2011 ◽  
Vol 688 ◽  
pp. 461-492 ◽  
Author(s):  
Stephan Weiss ◽  
Guenter Ahlers
Keyword(s):  
Large Scale ◽  
Working Fluid ◽  
Flow Mode ◽  
Unexpected Result ◽  
Vertical Sidewall ◽  
Number Range ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractWe report on the influence of rotation about a vertical axis on the large-scale circulation (LSC) of turbulent Rayleigh–Bénard convection in a cylindrical vessel with aspect ratio $\Gamma \equiv D/ L= 0. 50$ (where $D$ is the diameter and $L$ the height of the sample). The working fluid is water at an average temperature ${T}_{av} = 40{~}^{\ensuremath{\circ} } \mathrm{C} $ with a Prandtl number $\mathit{Pr}= 4. 38$. For rotation rates $\Omega \lesssim 1~\mathrm{rad} ~{\mathrm{s} }^{\ensuremath{-} 1} $, corresponding to inverse Rossby numbers $1/ \mathit{Ro}$ between 0 and 20, we investigated the temperature distribution at the sidewall and from it deduced properties of the LSC. The work covered the Rayleigh-number range $2. 3\ensuremath{\times} 1{0}^{9} \lesssim \mathit{Ra}\lesssim 7. 2\ensuremath{\times} 1{0}^{10} $. We measured the vertical sidewall temperature gradient, the dynamics of the LSC and flow-mode transitions from single-roll states (SRSs) to double-roll states (DRSs). We found that modest rotation stabilizes the SRSs. For modest $1/ \mathit{Ro}\lesssim 1$ we found the unexpected result that the vertical LSC plane rotated in the prograde direction (i.e. faster than the sample chamber), with the rotation at the horizontal midplane faster than near the top and bottom. This differential rotation led to disruptive events called half-turns, where the plane of the top or bottom section of the LSC underwent a rotation through an angle of $2\lrm{\pi} $ relative to the main portion of the LSC. The signature of the LSC persisted even for large $1/ \mathit{Ro}$ where Ekman vortices are expected. We consider the possibility that this signature actually is generated by a two-vortex state rather than by a LSC. Whenever possible, we compare our results with those for a $\Gamma = 1$ sample by Zhong & Ahlers (J. Fluid Mech., vol. 665, 2010, pp. 300–333).

scholarly journals Azimuthal diffusion of the large-scale-circulation plane, and absence of significant non-Boussinesq effects, in turbulent convection near the ultimate-state transition

2016 ◽  
Vol 791 ◽  
Author(s):  
Xiaozhou He ◽  
Eberhard Bodenschatz ◽  
Guenter Ahlers
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Cylindrical Sample ◽  
Ultimate State ◽  
Large Scale Circulation ◽  
Classical State ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We present measurements of the orientation ${\it\theta}_{0}$ and temperature amplitude ${\it\delta}$ of the large-scale circulation in a cylindrical sample of turbulent Rayleigh–Bénard convection (RBC) with aspect ratio ${\it\Gamma}\equiv D/L=1.00$ ($D$ and $L$ are the diameter and height respectively) and for the Prandtl number $Pr\simeq 0.8$. The results for ${\it\theta}_{0}$ revealed a preferred orientation with up-flow in the west, consistent with a broken azimuthal invariance due to the Earth’s Coriolis force (see Brown & Ahlers (Phys. Fluids, vol. 18, 2006, 125108)). They yielded the azimuthal diffusivity $D_{{\it\theta}}$ and a corresponding Reynolds number $Re_{{\it\theta}}$ for Rayleigh numbers over the range $2\times 10^{12}\lesssim Ra\lesssim 1.5\times 10^{14}$. In the classical state ($Ra\lesssim 2\times 10^{13}$) the results were consistent with the measurements by Brown & Ahlers (J. Fluid Mech., vol. 568, 2006, pp. 351–386) for $Ra\lesssim 10^{11}$ and $Pr=4.38$, which gave $Re_{{\it\theta}}\propto Ra^{0.28}$, and with the Prandtl-number dependence $Re_{{\it\theta}}\propto Pr^{-1.2}$ as found previously also for the velocity-fluctuation Reynolds number $Re_{V}$ (He et al., New J. Phys., vol. 17, 2015, 063028). At larger $Ra$ the data for $Re_{{\it\theta}}(Ra)$ revealed a transition to a new state, known as the ‘ultimate’ state, which was first seen in the Nusselt number $Nu(Ra)$ and in $Re_{V}(Ra)$ at $Ra_{1}^{\ast }\simeq 2\times 10^{13}$ and $Ra_{2}^{\ast }\simeq 8\times 10^{13}$. In the ultimate state we found $Re_{{\it\theta}}\propto Ra^{0.40\pm 0.03}$. Recently, Skrbek & Urban (J. Fluid Mech., vol. 785, 2015, pp. 270–282) claimed that non-Oberbeck–Boussinesq effects on the Nusselt and Reynolds numbers of turbulent RBC may have been interpreted erroneously as a transition to a new state. We demonstrate that their reasoning is incorrect and that the transition observed in the Göttingen experiments and discussed in the present paper is indeed to a new state of RBC referred to as ‘ultimate’.

Effects of geometric confinement in quasi-2-D turbulent Rayleigh–Bénard convection

2016 ◽  
Vol 794 ◽  
pp. 639-654 ◽  
Author(s):  
Shi-Di Huang ◽  
Ke-Qing Xia
Keyword(s):  
Heat Transport ◽  
Large Scale ◽  
Lateral Aspect ◽  
Bénard Convection ◽  
Benard Convection ◽  
Reversal Frequency ◽  
Geometric Confinement ◽  
Large Scale Flow ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We report an experimental study of confinement effects in quasi-2-D turbulent Rayleigh–Bénard convection. The experiments were conducted in five rectangular cells with their height $H$ and length $L$ being the same and fixed, while the width $W$ was different for each cell to produce lateral aspect ratios (${\it\Gamma}=W/H$) of 0.6, 0.3, 0.2, 0.15 and 0.1. Direct flow field measurements reveal that the large-scale flow slows down as ${\it\Gamma}$ decreases and there are more plumes travelling through the bulk region. Moreover, the reversal frequency of the large-scale flow is found to increase drastically in smaller ${\it\Gamma}$ cells, by more than 1000-fold for the highest value of Rayleigh number reached in the experiment. The reversal frequency can be well described by a stochastic model developed by Ni et al. (J. Fluid Mech., vol. 778, 2015, R5) and the probability density functions (PDF) of the time interval between successive reversals are found to follow Poisson statistics as in the 3-D system. It is further observed that the bulk temperature fluctuation increases significantly and its PDF changes from exponential to Gaussian as ${\it\Gamma}$ decreases. The influences of geometric confinement on the global heat transport are also investigated. The measured Nu–Ra relationship suggests that, as the lateral aspect ratio decreases, the relative weight of the boundary layer contribution in the global heat transport increases compared to that from the bulk. These results demonstrate that in the quasi-2-D geometry, geometric confinement has strong effects on both the global and local properties in turbulent convective flows, which are very different from the previous findings in 3-D and true 2-D systems.

The effect of distant side-walls on the evolution and stability of finite-amplitude Rayleigh-Bénard convection

1981 ◽  
Vol 378 (1775) ◽  
pp. 539-566 ◽  
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Side Walls ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

A recent study by Cross et al . (1980) has described a class of finite-amplitude phase-winding solutions of the problem of two-dimensional Rayleigh-Bénard convection in a shallow fluid layer of aspect ratio 2 L (≫ 1) confined laterally by rigid side-walls. These solutions arise at Rayleigh numbers R = R 0 + O ( L -1 ) where R 0 is the critical Rayleigh number for the corresponding infinite layer. Nonlinear solutions of constant phase exist for Rayleigh numbers R = R 0 + O ( L -2 ) but of these only the two that bifurcate at the lowest value of R are stable to two-dimensional linearized disturbances in this range (Daniels 1978). In the present paper one set of the class of phase-winding solutions is found to be stable to two-dimensional disturbances. For certain values of the Prandtl number of the fluid and for stress-free horizontal boundaries the results predict that to preserve stability there must be a continual readjustment of the roll pattern as the Rayleigh number is raised, with a corresponding increase in wavelength proportional to R - R 0 . These solutions also exhibit hysteresis as the Rayleigh number is raised and lowered. For other values of the Prandtl number the number of rolls remains unchanged as the Rayleigh number is raised, and the wavelength remains close to its critical value. It is proposed that the complete evolution of the flow pattern from a static state must take place on a number of different time scales of which t = O(( R - R 0 ) -1 ) and t = O(( R - R 0 ) -2 ) are the most significant. When t = O(( R - R 0 ) -1 ) the amplitude of convection rises from zero to its steady-state value, but the final lateral positioning of the rolls is only completed on the much longer time scale t = O(( R - R 0 ) -2 ).

Numerical and Experimental Investigation of Phase Change Heat Transfer in the Presence of Rayleigh–Benard Convection

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
Mohammad Parsazadeh ◽  
Xili Duan
Keyword(s):  
Nusselt Number ◽  
Phase Change ◽  
Local Heat ◽  
Liquid Interface ◽  
Interface Location ◽  
Local Heat Flux ◽  
Solid Liquid Interface ◽  
Solid Liquid ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Abstract This research investigates the melting rate of a phase change material (PCM) in the presence of Rayleigh–Benard convection. A scaling analysis is conducted for the first time for such a problem, which is useful to identify the parameters affecting the phase change rate and to develop correlations for the solid–liquid interface location and the Nusselt number. The solid–liquid interface and flow patterns in the liquid region are analyzed for PCM in a rectangular enclosure heated from bottom. Numerical and experimental results both reveal that the number of Benard cells is proportional to the ratio of the length of the rectangular enclosure over the solid–liquid interface location (i.e.,, the liquified region aspect ratio). Their effect on the local heat flux is also analyzed as the local heat flux profile changes with the solid–liquid interface moving upward. The variations of average Nusselt number are obtained in terms of the Stefan number, Fourier number, and Rayleigh number. Eventually, the experimental and numerical data are used to develop correlations for the solid–liquid interface location and average Nusselt number for this type of melting problems.

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