Numerical simulation of three-dimensional Bénard convection in air

Abstract

A numerical model is developed to simulate three-dimensional Bénard convection. This model is used to investigate thermal convection in air for Rayleigh numbers between 4000 and 25000. According to experiments, this range of Rayleigh numbers in air covers three regimes of thermal convection: (i) steady two-dimensional convection, (ii) time-periodic convection and (iii) aperiodic convection. Numerical solutions are obtained for each of these regimes and the results are compared with the available experimental data and theoretical predictions.At the Rayleigh number Ra = 4000 the present model is able to produce experimentally realistic wavelengths for the two-dimensional convection. The small amplitude wave disturbances at Ra = 6500 have period τ = 0·24. When they become finite amplitude travelling waves, the period is τ = 0·27. These values are in good agreement with theoretical and experimental results. A detailed study of the form of these waves and of their energetics is given in appendix A. As the Rayleigh number is increased to Ra = 9000 and 25 000, the convection manifests progressively more complex spatial and temporal variations.The vertical heat transport and other mean properties of the convection are calculated for the range of Ra considered and compared with experimental and theoretical data. A detailed comparison is also made between the mean properties of two- and three-dimensional convection at the larger values of Ra. It is found that the heat flux Nu is nearly independent of the dimensionality of the convection.

Nonlinear Bifurcations to Time-dependent Rayleigh?Benard Convection

“…Surprisingly, there have been relatively few numerical investigations in which the vertical vorticity has been incorporated. Lipps (1976) in reference to his three-dimensional model agreed with Busse (1972) that oscillatory behaviour and the generation of vertical vorticity go hand-in-hand, but said little else about the velocity structure of his solutions. In the two-dimensional roll models, the vorticity is of the form (0, cu, 0) and accordingly the vertical vorticity is zero by definition.…”

Convective circulation in littoral water due to surface cooling

“…= 9,000 and 25,000 and concluded that twodimensional wavelengths are larger than three-dimensional wavelengths. There seems to be general agreement that two-dimensional simulations predict overall Nusselt numbers (Lipps 1976) within a few percent. This observation also holds for the enclosure with differentially heated sidewalls (Mallinson and DeVahl Davis 1977;Lai and Ramsey 1987) but cannot be extrapolated to the dependence of the Nu number on the radial coordinate.…”

Numerical simulation of natural convection in a horizontal enclosure with a heat-generating conducting body

“…The above quantities are separately computed for the cold top and the hot bottom walls. The present multi-domain spectral methodology and the computer code used here have been thoroughly validated by comparing results with those of de Vahl Davis [14] and House et al [6] for the case of a vertical enclosure and with those of Lipps [2] for the case of a horizontal enclosure. For Rayleigh numbers of 10 3 , 10 4 , 10 5 and 10 6 the benchmark results on surface-averaged Nusselt number obtained by de Vahl Davis [14] are 1.118, 2.243, 4.519 and 8.800.…”

“…It has been well established that for the isothermal boundary condition the horizontal layer of fluid becomes unstable above a Rayleigh number of 1708 and convective motion sets in the form of steady convective rolls of aspect ratio (width to height) of about 2 [1]. With increasing Rayleigh number the flow undergoes a sequence of instabilities and eventually transitions to a turbulent state above a Rayleigh number of about 10 7 [2,3].…”