scholarly journals Bounds for Rayleigh–Bénard convection between free-slip boundaries with an imposed heat flux

2018 ◽  
Vol 837 ◽  
Author(s):  
Giovanni Fantuzzi
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Temperature Drop ◽  
Three Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We prove the first rigorous bound on the heat transfer for three-dimensional Rayleigh–Bénard convection of finite-Prandtl-number fluids between free-slip boundaries with an imposed heat flux. Using the auxiliary functional method with a quadratic functional, which is equivalent to the background method, we prove that the Nusselt number $\mathit{Nu}$ is bounded by $\mathit{Nu}\leqslant 0.5999\mathit{R}^{1/3}$ uniformly in the Prandtl number, where $\mathit{R}$ is the Rayleigh number based on the imposed heat flux. In terms of the Rayleigh number based on the mean vertical temperature drop, $\mathit{Ra}$, we obtain $\mathit{Nu}\leqslant 0.4646\mathit{Ra}^{1/2}$. The scaling with Rayleigh number is the same as that of bounds obtained with no-slip isothermal, free-slip isothermal and no-slip fixed-flux boundaries, and numerical optimisation of the bound suggests that it cannot be improved within our bounding framework. Contrary to the two-dimensional case, therefore, the $\mathit{Ra}$-dependence of rigorous upper bounds on the heat transfer obtained with the background method for three-dimensional Rayleigh–Bénard convection is insensitive to both the thermal and the velocity boundary conditions.

Bounds on Rayleigh–Bénard convection with an imposed heat flux

2002 ◽  
Vol 473 ◽  
pp. 191-199 ◽  
Author(s):  
JESSE OTERO ◽  
RALF W. WITTENBERG ◽  
RODNEY A. WORTHING ◽  
CHARLES R. DOERING
Keyword(s):  
Heat Flux ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Heat Transport ◽  
Absolute Constant ◽  
Temperature Drop ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We formulate a bounding principle for the heat transport in Rayleigh–Bénard convection with fixed heat flux through the boundaries. The heat transport, as measured by a conventional Nusselt number, is inversely proportional to the temperature drop across the layer and is bounded above according to Nu [les ] cRˆ1/3, where c < 0.42 is an absolute constant and Rˆ = αγβh4/(νκ) is the ‘effective’ Rayleigh number, the non-dimensional forcing scale set by the imposed heat flux κβ. The relation among the parameter Rˆ, the Nusselt number, and the conventional Rayleigh number defined in terms of the temperature drop across the layer, is NuRa = Rˆ, yielding the bound Nu [les ] c3/2Ra1/2.

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 ).

scholarly journals Turbulent transition in Rayleigh-Bénard convection with fluorocarbon (a)

2021 ◽  
Vol 136 (1) ◽  
pp. 10003
Author(s):  
Lucas Méthivier ◽  
Romane Braun ◽  
Francesca Chillà ◽  
Julien Salort
Keyword(s):  
Heat Transfer ◽  
Velocity Field ◽  
Rayleigh Number ◽  
Working Fluid ◽  
Turbulent Transition ◽  
Bénard Convection ◽  
Benard Convection ◽  
Image Velocimetry ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Abstract We present measurements of the global heat transfer and the velocity field in two Rayleigh-Bénard cells (aspect ratios 1 and 2). We use Fluorinert FC770 as the working fluid, up to a Rayleigh number . The velocity field is inferred from sequences of shadowgraph pattern using a Correlation Image Velocimetry (CIV) algorithm. Indeed the large number of plumes, and their small characteristic scale, make it possible to use the shadowgraph pattern produced by the thermal plumes in the same manner as particles in Particle Image Velocimetry (PIV). The method is validated in water against PIV, and yields identical wind velocity estimates. The joint heat transfer and velocity measurements allow to compute the scaling of the kinetic dissipation rate which features a transition from a laminar scaling to a turbulent Re 3 scaling. We propose that the turbulent transition in Rayleigh-Bénard convection is controlled by a threshold Péclet number rather than a threshold Rayleigh number, which may explain the apparent discrepancy in the literature regarding the “ultimate” regime of convection.

scholarly journals Bounds on Rayleigh–Bénard convection with imperfectly conducting plates

2010 ◽  
Vol 665 ◽  
pp. 158-198 ◽  
Author(s):  
RALF W. WITTENBERG
Keyword(s):  
Rayleigh Number ◽  
Biot Number ◽  
Fluid Layer ◽  
Temperature Drop ◽  
Upper Bounds ◽  
Fixed Temperature ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We investigate the influence of the thermal properties of the boundaries in turbulent Rayleigh–Bénard convection on analytical upper bounds on convective heat transport. We model imperfectly conducting bounding plates in two ways: using idealized mixed thermal boundary conditions (BCs) of constant Biot number η, continuously interpolating between the previously studied fixed temperature (η = 0) and fixed flux (η = ∞) cases; and by explicitly coupling the evolution equations in the fluid in the Boussinesq approximation through temperature and flux continuity to identical upper and lower conducting plates. In both cases, we systematically formulate a bounding principle and obtain explicit upper bounds on the Nusselt numberNuin terms of the usual Rayleigh numberRameasuring the average temperature drop across the fluid layer, using the ‘background method’ developed by Doering and Constantin. In the presence of plates, we find that the bounds depend on σ =d/λ, wheredis the ratio of plate to fluid thickness and λ is the conductivity ratio, and that the bounding problem may be mapped onto that for Biot number η = σ. In particular, for each σ > 0, for sufficiently largeRa(depending on σ) we show thatNu≤c(σ)R1/3≤CRa1/2, whereCis a σ-independent constant, and where the control parameterRis a Rayleigh number defined in terms of the full temperature drop across the entire plate–fluid–plate system. In theRa→ ∞ limit, the usual fixed temperature assumption is a singular limit of the general bounding problem, while fixed flux conditions appear to be most relevant to the asymptoticNu–Rascaling even for highly conducting plates.

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.

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