Effects of Prandtl number in quasi-two-dimensional Rayleigh–Bénard convection

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

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1981.0167 ◽

1981 ◽

Vol 378 (1775) ◽

pp. 539-566 ◽

Cited By ~ 8

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 ).

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A Numerical Study on Entropy Generation in Two-Dimensional Rayleigh-Bénard Convection at Different Prandtl Number

Entropy ◽

10.3390/e19090443 ◽

2017 ◽

Vol 19 (9) ◽

pp. 443 ◽

Cited By ~ 12

Author(s):

Yikun Wei ◽

Zhengdao Wang ◽

Yuehong Qian

Keyword(s):

Prandtl Number ◽

Entropy Generation ◽

Numerical Study ◽

Two Dimensional ◽

Bénard Convection ◽

Benard Convection ◽

Rayleigh Benard Convection ◽

Rayleigh Benard

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

Journal of Fluid Mechanics ◽

10.1017/s0022112000008582 ◽

2000 ◽

Vol 414 ◽

pp. 225-250 ◽

Cited By ~ 12

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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