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Rayleigh-Benard Convection
The present state-of-understanding of Rayleigh-Benard convection is discussed below. In the simplest form, the flow configuration is comprised of a horizontal fluid layer confined between a pair of parallel horizontal plates. The fluid is differentially heated by maintaining the lower surface at a higher temperature compared to the top. This situation produces a top-heavy arrangement that is unstable. The dimensionless quantity that characterized the buoyancy-driven flow is the Rayleigh number defined as
When Ra is below a critical value, the gravitational potential is not sufficient to overcome the viscous forces within the fluid layer. For Rayleigh numbers above the critical value, a steady flow is established. Subsequently, flow undergoes a sequence of transitions, finally resulting in turbulence.
Transitions in Rayleigh-Benard convection depend on a Rayleigh number, a Prandtl number, and the cavity aspect ratio. Additionally, there is an effect of the geometric structure of the side walls being straight or curved [113]. The present discussion is restricted to a rectangular cavity. For a fluid layer with an infinite aspect ratio, the first transition, namely the onset of fluid motion, occurs at a Rayleigh number of 1708, irrespective of the Prandtl number. The associated flow pattern is in the form of hexagonal cells. The general effect of lowering the aspect ratio is to stabilize the flow due to the presence of the side walls and thus increase the critical Rayleigh number. All subsequent transitions are Prandtl number dependent. The present discussion is devoted to Prandtl numbers in the range 0.7-7, for which some general conclusions can be drawn.
Flow patterns in rectangular cavities can be divided into three main categories, depending on the aspect ratio. These are small  intermediate ,  and large  aspect ratio boxes. Transition and chaos in a small aspect ratio enclosure with water has been experimentally studied by Nasuno et al. Their data is in good agreement with the stability diagram of Busse and Clever. In a large aspect ratio enclosure, it has been shown that flow beyond the critical Rayleigh number is always time-dependent and non-periodic (see Ahlers and Behringer). In contrast, a large number of bifurcations have been recorded both experimentally as well as in numerical studies in small aspect ratio enclosures. Information regarding intermediate aspect ratio enclosures is sparse. The transition sequence appears to be via the formation of longitudinal rolls that are aligned with the shorter side, polygonal cells; roll-loss and displacement; and finally towards turbulence.
When the Rayleigh number is close to the critical value for the onset of convection, hexagonal convection cells have been observed both experimentally and in computation [113]. With a further increase in the Rayleigh number, stable two-dimensional longitudinal rolls have been observed. Krishnamurti [120] is one of the earliest authors to conduct an experimental study and observe roll patterns. Much later, Kessler [121] obtained steady rolls formation through a numerical simulation. With further increase in the Rayleigh number, the two-dimensional rolls were seen to bifurcate slowly to three-dimensional rolls, showing variation in shape along the roll axis. The three-dimensional rolls were found to be steady over a range of Rayleigh numbers. For further increase in the Rayleigh number, a loss-of-roll phenomena was observed. Kirchartz and Oertel [116] have shown for a box of small aspect ratio the transition from four rolls to three rolls and finally to two rolls. Simultaneously, three-dimensional rolls become unstable and a periodic motion of the roll system begins along its axis. The critical Rayleigh number for the onset of oscillatory rolls is shown to be in the range of a Rayleigh number of 30000 for air (Pr = 0.71) [121]. This critical Rayleigh number increases with the increase of the Prandtl number. The frequency of oscillation is not a strong function of the Rayleigh number, but increases slowly with increase in the Rayleigh number. Kessler [121] has also observed the exchange of mass between different rolls. This is due to a periodic motion in the location of the vertically upward and downward flow.
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