Three-dimensional numerical study of natural convection in vertical cylinders partially heated from the side

Three-dimensional steady and oscillatory flows are simulated in a vertical cylinder partially heated from the side. The vertical wall is heated in a zone at midheight and is insulated above and below this middle zone, while both ends of the cylinder are cooled. The cylinder aspect ratio ( A = height ∕ radius ) ranges from 2 to 8, whereas a fixed Prandtl number, Pr = 0.021 , is considered as well as a fixed length of the heated zone, equal to the cylinder radius. Three-dimensional steady and unsteady simulations as well as mode decomposition techniques and energy transfer analyses are used to characterize the flows and their transitions. The flows that develop from the steady toroidal pattern beyond the first instability threshold break the axisymmetry. At small A ( 2 ⩽ A ⩽ 2.5 ) , the flow corresponds to a two-roll rotating pattern, which is triggered by a k = 2 azimuthal mode as a result of a hydrodynamic instability. At large A ( 3 ⩽ A ⩽ 8 ) , the flow is steady and corresponds to a main one-roll pattern in the upper part of the cylinder. The flow is triggered by a k = 1 mode as a result of buoyancy effects affecting this unstably stratified upper part (Rayleigh-Bénard instability), but shear effects are involved in the instability for the smaller values of A . These steady flows then transit at a higher threshold to a standing-wave oscillatory one-roll pattern associated with the breaking of symmetry of the previous steady pattern. For intermediate values of A ( 2.7 ⩽ A ⩽ 2.9 ) , the transition is toward an oscillatory pattern, but hysteresis phenomena with multiplicity of steady and oscillatory states have been found. Comparisons with experiments performed at aspect ratios A = 4 and 8 are then considered and discussed.