Infinite Prandtl number limit of Rayleigh-Bénard convection

The kinetic energy balance in Rayleigh–Bénard convection is investigated by means of direct numerical simulations for the Prandtl number range $0.01\leqslant \mathit{Pr}\leqslant 150$ and for fixed Rayleigh number $\mathit{Ra}=5\times 10^{6}$. The kinetic energy balance is divided into a dissipation, a production and a flux term. We discuss the profiles of all the terms and find that the different contributions to the energy balance can be spatially separated into regions where kinetic energy is produced and where kinetic energy is dissipated. By analysing the Prandtl number dependence of the kinetic energy balance, we show that the height dependence of the mean viscous dissipation is closely related to the flux of kinetic energy. We show that the flux of kinetic energy can be divided into four additive contributions, each representing a different elementary physical process (advection, buoyancy, normal viscous stresses and viscous shear stresses). The behaviour of these individual flux contributions is found to be surprisingly rich and exhibits a pronounced Prandtl number dependence. Different flux contributions dominate the kinetic energy transport at different depths, such that a comprehensive discussion requires a decomposition of the domain into a considerable number of sublayers. On a less detailed level, our results reveal that advective kinetic energy fluxes play a key role in balancing the near-wall dissipation at low Prandtl number, whereas normal viscous stresses are particularly important at high Prandtl number. Finally, our work reveals that classical velocity boundary layers are deeply connected to the kinetic energy transport, but fail to correctly represent regions of enhanced viscous dissipation.

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

Chemical Engineering Research and Design ◽

10.1205/cherd.05064 ◽

2006 ◽

Vol 84 (1) ◽

pp. 29-37 ◽

Cited By ~ 14

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

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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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