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.

Falkner–Skan boundary layer approximation in Rayleigh–Bénard convection

2013 ◽  
Vol 730 ◽  
pp. 442-463 ◽  
Author(s):  
Olga Shishkina ◽  
Susanne Horn ◽  
Sebastian Wagner
Keyword(s):  
Boundary Layers ◽  
Large Scale ◽  
Numerical Solutions ◽  
Angle Of Attack ◽  
Convection Cell ◽  
Large Scale Circulation ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractTo 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 $\beta $ 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 $\beta $, for infinite and for infinitesimal Prandtl numbers $\mathit{Pr}$. Dependences on $\mathit{Pr}$ and $\beta $ 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 $\beta $, accurate approximations are developed as functions on $\mathit{Pr}$. The theoretical results are corroborated by our direct numerical simulations for $\mathit{Pr}= 0. 786$ (air) and $\mathit{Pr}= 4. 38$ (water). The angle of attack $\beta $ 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 $\beta \approx 0. 7\mathrm{\pi} $ and the theoretical predictions based on the Falkner–Skan approximation for this $\beta $ leads to better agreement with the DNS results, compared with the Prandtl–Blasius approximation for $\beta = \mathrm{\pi} $.

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’.

scholarly journals The origin of oscillations of the large-scale circulation of turbulent Rayleigh–Bénard convection

2009 ◽  
Vol 638 ◽  
pp. 383-400 ◽  
Author(s):  
ERIC BROWN ◽  
GUENTER AHLERS
Keyword(s):  
Large Scale ◽  
Lateral Displacement ◽  
Torsional Oscillation ◽  
Restoring Force ◽  
Large Scale Circulation ◽  
Bénard Convection ◽  
Benard Convection ◽  
Sloshing Mode ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

In agreement with a recent experimental discovery by Xi et al. (Phys. Rev. Lett., vol. 102, 2009, paper no. 044503), we also find a sloshing mode in experiments on the large-scale circulation (LSC) of turbulent Rayleigh–Bénard convection in a cylindrical sample of aspect ratio one. The sloshing mode has the same frequency as the torsional oscillation discovered by Funfschilling & Ahlers (Phys. Rev. Lett., vol. 92, 2004, paper no. 1945022004). We show that both modes can be described by an extension of a model developed previously Brown & Ahlers (Phys. Fluids, vol. 20, 2008, pp. 105105-1–105105-15; Phys. Fluids, vol. 20, 2008, pp. 075101-1–075101-16). The extension consists of permitting a lateral displacement of the LSC circulation plane away from the vertical centreline of the sample as well as a variation of the displacement with height (such displacements had been excluded in the original model). Pressure gradients produced by the sidewall of the container on average centre the plane of the LSC so that it prefers to reach its longest diameter. If the LSC is displaced away from this diameter, the walls provide a restoring force. Turbulent fluctuations drive the LSC away from the central alignment, and combined with the restoring force they lead to oscillations. These oscillations are advected along with the LSC. This model yields the correct wavenumber and phase of the oscillations, as well as estimates of the frequency, amplitude and probability distributions of the displacements.

Dynamics of the large-scale circulation in turbulent Rayleigh–Bénard convection with modulated rotation

2015 ◽  
Vol 778 ◽  
Author(s):  
Jin-Qiang Zhong ◽  
Sebastian Sterl ◽  
Hui-Min Li
Keyword(s):  
Large Scale ◽  
Modulation Frequency ◽  
Rotation Velocity ◽  
Viscous Drag ◽  
Phase Lag ◽  
Large Scale Circulation ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We present measurements of the azimuthal rotation velocity $\dot{{\it\theta}}(t)$ and thermal amplitude ${\it\delta}(t)$ of the large-scale circulation in turbulent Rayleigh–Bénard convection with modulated rotation. Both $\dot{{\it\theta}}(t)$ and ${\it\delta}(t)$ exhibit clear oscillations at the modulation frequency ${\it\omega}$. Fluid acceleration driven by oscillating Coriolis force causes an increasing phase lag in $\dot{{\it\theta}}(t)$ when ${\it\omega}$ increases. The applied modulation produces oscillatory boundary layers and the resulting time-varying viscous drag modifies ${\it\delta}(t)$ periodically. Oscillation of $\dot{{\it\theta}}(t)$ with maximum amplitude occurs at a finite modulation frequency ${\it\omega}^{\ast }$. Such a resonance-like phenomenon is interpreted as a result of optimal coupling of ${\it\delta}(t)$ to the modulated rotation velocity. We show that an extended large-scale circulation model with a relaxation time for ${\it\delta}(t)$ in response to the modulated rotation provides predictions in close agreement with the experimental results.

scholarly journals Vertical natural convection: application of the unifying theory of thermal convection

2015 ◽  
Vol 764 ◽  
pp. 349-361 ◽  
Author(s):  
Chong Shen Ng ◽  
Andrew Ooi ◽  
Detlef Lohse ◽  
Daniel Chung
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Temperature Fields ◽  
Mean Flow ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Gl Theory

AbstractResults from direct numerical simulations of vertical natural convection at Rayleigh numbers $1.0\times 10^{5}$–$1.0\times 10^{9}$ and Prandtl number $0.709$ support a generalised applicability of the Grossmann–Lohse (GL) theory, which was originally developed for horizontal natural (Rayleigh–Bénard) 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–Bénard 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–Bénard 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.

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