The onset of natural convection in a cylindrical volume of fluid bounded above and below by rigid, perfectly conducting surfaces and laterally by a wall of arbitrary thermal conductivity is examined. The critical Rayleigh number (dimensionless temperature difference) is determined as a function of aspect (radius to height) ratio and wall conductivity. The first three asymmetric modes as well as the axisymmetric mode are considered. Two sets of stream functions are employed to represent a velocity field that satisfies the no-slip boundary condition on all surfaces and conservation of mass everywhere. The Galerkin method is then used to reduce the linearized perturbation equations to an eigenvalue problem. The results for perfectly insulating and conducting walls are compared with the work of Charlson and Sani[9].

Unique wavenumbers are calculated for axisymmetric Rayleigh–Bénard convection as a function of the Rayleigh number (R) up to the second critical value for several different Prandtl numbers. The analysis assumes slightly bent rolls (large radius of curvature) and that there exists a horizontal pressure gradient strong enough to force the net mean flow induced by curvature to be zero. The assumptions are satisfied for axisymmetric convection in a large aspect ratio cylinder (however, this may not be the only case). Manneville and Piquemal [Phys. Rev. A 28, 1774 (1983)] found the initial slope of the selected wavenumber with respect to Rayleigh number (using an analytic solution valid for small amplitude solutions) and our calculations agree with theirs. This initial slope is sensitive to the Prandtl number (P), but at moderate to large R the selected wavenumber is approximately independent of P when P>3. For smaller P larger wavenumbers are found, but this does not contradict any available experimental evidence.

The onset of steady natural convection in a rotating cylindrical volume of fluid completely bounded by rigid surfaces is examined for moderate Taylor numbers (Ta≤2×106) and aspect ratios (A≤2). The critical Rayleigh number for three dimensional disturbances is found to be lower than that for the radially unbounded problem by up to a factor of six. The thermal boundary condition on the lateral walls is shown to have a greater effect here than in the nonrotating case.

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The vortices near the origin of an initially laminar mixing layer have a single frequency with a well-defined phase; i.e. there is little phase jitter. Further downstream, however, the phase jitter increases suddenly. Even when the flow is forced, this same transition is observed. The forcing partially loses its influence because of the decorrelation of the phase between the forcing signal and the passing coherent structures. In the present investigation, this phenomenon is documented and the physical mechanism responsible for the phase decorrelation is identified.