The secondary flow and its stability for natural convection in a tall vertical enclosure

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Abstract
The multicellular flow between two vertical parallel plates is numerically simulated using a time-splitting pseudospectral method. The steady flow of air, and the time-periodic flow of oil (Prandtl numbers of 0.71 and 1000, respectively) are investigated and descriptions of these flows using both physical and spectral approaches are presented. The details of the time dependency of the flow and temperature fields of oil are shown, and the dynamics of the process is discussed. The spectral transfer of energy among the axial modes comprising the flow is explored. The spectra of kinetic energy and thermal variance for air are found to be smooth and viscously dominated. Similar spectra for oil are bumpier, and the dynamics of the time-dependent flow are determined to be confined to the lower end of the spectrum alone.The three-dimensional linear stability of the multicellular flow of air is parametrically studied. The domain of stable two-dimensional cellular motion was found to be constrained by the Eckhaus instability and by two types of monotone instability. The two-dimensional multicellular flow is unstable above a Grashof number of about 8550 (with the critical Grashof number for the base flow being 8037). Therefore the flow of air in a sufficiently tall vertical enclosure should be considered to be three-dimensional for most practical applications.