Three-dimensional CFD Simulation of Rayleigh–Benard Convection for Low Prandtl Number Fluids

2006 ◽  
Vol 84 (1) ◽  
pp. 29-37 ◽  
Author(s):  
S. Vedantam ◽  
M.T. Dhotre ◽  
J.B. Joshi
Keyword(s):  
Prandtl Number ◽  
Cfd Simulation ◽  
Three Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

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 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.

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