Buoyancy-Driven Flow Transitions in Deep Cavities Heated From Below

A numerical investigation has been conducted of flow transitions in deep three-dimensional cavities heated from below. The first critical Rayleigh number, $RaI,$ below which the flow is at rest, and the second critical Rayleigh number, $RaII,$ for transition from steady state to oscillatory flow, have been found for cavities of aspect ratios Ar in the range 1–5. Transition to chaos has also been examined for these cases. The results show that $RaI=3583,$$2.545×104$ and $5.5×105$ and $RaII=4.07×105,$$1.65×106$ and $1.30×107$ for aspect ratios of 1, 2, and 5 respectively. The route to chaos is $PPeriodic→QP2$(Quasi-periodic with two incommensurate frequencies)→$QP3$(Quasi-periodic with three incommensurate frequencies)→$NChaotic$ for Ar=1 with the Rayleigh number varying from $4.07×105$ to $4.89×105.$ The route is $PPeriodic→P2$(Periodic doubling)→$I$(Intermittent)→$P$(Periodic)→$N$(Chaotic) for Ar=2 over a Ra range of $1.65×106$ to $1.83×106.$ The interval between periodic and chaotic flow is very short for Ar=5.