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.

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 Flow states in two-dimensional Rayleigh-Bénard convection as a function of aspect-ratio and Rayleigh number

2012 ◽  
Vol 24 (8) ◽  
pp. 085104 ◽  
Author(s):  
Erwin P. van der Poel ◽  
Richard J. A. M. Stevens ◽  
Kazuyasu Sugiyama ◽  
Detlef Lohse
Keyword(s):  
Rayleigh Number ◽  
Aspect Ratio ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Flow States

scholarly journals Axially homogeneous Rayleigh–Bénard convection in a cylindrical cell

2011 ◽  
Vol 691 ◽  
pp. 52-68 ◽  
Author(s):  
Laura E. Schmidt ◽  
Enrico Calzavarini ◽  
Detlef Lohse ◽  
Federico Toschi ◽  
Roberto Verzicco
Keyword(s):  
Heat Transfer ◽  
Aspect Ratio ◽  
Boundary Layers ◽  
Temperature Fields ◽  
Cylindrical Cell ◽  
Bénard Convection ◽  
Benard Convection ◽  
Velocity And Temperature Fields ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractPrevious numerical studies have shown that the ‘ultimate regime of thermal convection’ can be attained in a Rayleigh–Bénard cell when the kinetic and thermal boundary layers are eliminated by replacing both lateral and horizontal walls with periodic boundary conditions (homogeneous Rayleigh–Bénard convection). Then, the heat transfer scales like $\mathit{Nu}\ensuremath{\sim} {\mathit{Ra}}^{1/ 2} $ and turbulence intensity as $\mathit{Re}\ensuremath{\sim} {\mathit{Ra}}^{1/ 2} $, where the Rayleigh number $\mathit{Ra}$ indicates the strength of the driving force (for fixed values of $\mathit{Pr}$, which is the ratio between kinematic viscosity and thermal diffusivity). However, experiments never operate in unbounded domains and it is important to understand how confinement might alter the approach to this ultimate regime. Here we consider homogeneous Rayleigh–Bénard convection in a laterally confined geometry – a small-aspect-ratio vertical cylindrical cell – and show evidence of the ultimate regime as $\mathit{Ra}$ is increased: in spite of the lateral confinement and the resulting kinetic boundary layers, we still find $\mathit{Nu}\ensuremath{\sim} \mathit{Re}\ensuremath{\sim} {\mathit{Ra}}^{1/ 2} $ at $\mathit{Pr}= 1$. Further, it is shown that the system supports solutions composed of modes of exponentially growing vertical velocity and temperature fields, with $\mathit{Ra}$ as the critical parameter determining the properties of these modes. Counter-intuitively, in the low-$\mathit{Ra}$ regime, or for very narrow cylinders, the numerical simulations are susceptible to these solutions, which can dominate the dynamics and lead to very high and unsteady heat transfer. As $\mathit{Ra}$ is increased, interaction between modes stabilizes the system, evidenced by the increasing homogeneity and reduced fluctuations in the root-mean-square velocity and temperature fields. We also test that physical results become independent of the periodicity length of the cylinder, a purely numerical parameter, as the aspect ratio is increased.

scholarly journals Influences of aspect ratio and wall boundary condition on laminar Rayleigh–Bénard convection of Bingham fluids in rectangular enclosures

2017 ◽  
Vol 27 (2) ◽  
pp. 310-333 ◽  
Author(s):  
Sahin Yigit ◽  
Nilanjan Chakraborty
Keyword(s):  
Rayleigh Number ◽  
Aspect Ratio ◽  
Boundary Conditions ◽  
Thermal Transport ◽  
Bingham Fluids ◽  
Content Type ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Purpose This paper aims to investigate the aspect ratio (AR; ratio of enclosure height:length) dependence of steady-state Rayleigh–Bénard convection of Bingham fluids within rectangular enclosures for both constant wall temperature and constant wall heat flux boundary conditions. A nominal Rayleigh number range 103 ≤ Ra ≤ 105 (Ra defined based on the height) for a single representative value of nominal Prandtl number (i.e. Pr = 500) has been considered for 1/4 ≤ AR ≤ 4. Design/methodology/approach The bi-viscosity Bingham model is used to mimic Bingham fluids for Rayleigh–Bénard convection of Bingham fluids in rectangular enclosures. The conservation equations of mass, momentum and energy have been solved in a coupled manner using the finite volume method where a second-order central differencing scheme is used for the diffusive terms and a second-order up-wind scheme is used for the convective terms. The well-known semi-implicit method for pressure-linked equations algorithm is used for the coupling of the pressure and velocity. Findings It has been found that buoyancy-driven flow strengthens with increasing nominal Rayleigh number Ra, but the convective transport weakens with increasing Bingham number Bn, because of additional flow resistance arising from yield stress in Bingham fluids. The relative contribution of thermal conduction (advection) to the total thermal transport strengthens (diminishes) with increasing AR for a given set of values of Ra and Pr for both Newtonian and Bingham fluids for both boundary conditions, and the thermal transport takes place purely because of conduction for tall enclosures. Originality/value Correlations for the mean Nusselt number Nu ¯ have been proposed for both boundary conditions for both Newtonian and Bingham fluids using scaling arguments, and the correlations have been demonstrated to predict Nu ¯ obtained from simulation data for 1/4 ≤ AR ≤ 4, 103 ≤ Ra ≤ 105 and Pr = 500.

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.

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