In a horizontal layer of fluid heated from below and cooled from above, cellular convection with horizontal length scale comparable to the layer depth occurs for small enough values of the Rayleigh number. As the Rayleigh number is increased, cellular flow disappears and is replaced by a random array oftransient plumes. Upon further increase, these plumes drift in one direction near the bottom and in the opposite direction near the top of the layer with the axes of plumes tilted in such a way that horizontal momentum is transported upward via the Reynolds stress. With the onset of this large-scale flow, the largest scale of motion has increased from that comparable to the layer depth to a scale comparable to the layer width. The conditions for occurrence and determination of the direction of this large-scale circulation are described. This is a report on a laboratory study of convection in a horizontal layer of fluid uniformly and steadily heated from below and cooled from above. This is the arrangement that leads to cellular convection for a range ofvalues of the Rayleigh number (the dimensionless measure of temperature difference, defined below) sufficiently small but greater than a critical value. The horizontal scale of these cells is comparable to the depth d of the layer. At successively larger values ofthis parameter, a number of transitions in the flow pattern as well as in the heat flux are observed (1-6). Most ofthese changes are within the regime of cellular flows.We shall describe here a further transition which leads to a very different scale of motion and very different transport properties. In this regime the flow is no longer cellular. It is a flow having primarily two scales of motion. The smaller scale flow, with horizontal scale comparable to the layer depth d, is best described as transient bubbles or plumes that have an organized tilt away from the vertical. The larger scale flow, having scale L, is a horizontal flow with vertical shear such that the flow is oppositely directed near the bottom and the top. L is usually the container width which, in these experiments, is an order of magnitude larger than d. The large-scale flow is apparently maintained against viscous dissipation by the Reynolds stress divergence of the tilted plume motions. APPARATUS AND PROCEDUREThe convecting fluid occupied a space ofdepth d and horizontal extent L by L such that d/L was a small number (of order 102 to 10-l). Most of the new results reported below were obtained with d = 2 and 5 cm, L = 48 cm. The fluid layer was bounded above and below by metal blocks whose thermal conductivity was several thousand times that ofthe liquids used. Even in the most extreme case within this set of experiments, when the convective heat flux was 30 times the conductive heat flux, the boundaries had conductivity 2 orders of magnitude higher than the effective conductivity ofthe fluid layer. Thus, the boundary condition was one of constant temperature along both top and bottom boundaries. On all its lateral boundaries, th...

A mathematical model of convection, obtained by truncation from the two-dimensional Boussinesq equations, is shown to exhibit a bifurcation from symmetrical cells to tilted non-symmetrical ones. A subsequent bifurcation leads to time-dependent flow with similarly tilted transient plumes and a large-scale Lagrangian mean flow. This change of symmetry is similar to that occurring with the advent of a large-scale flow and transient tilted plumes seen in laboratory experiments on turbulent convection at high Rayleigh number. Though not intended as a description of turbulent convection, the model does bring out in a theoretically tractable context the possibility of the spontaneous change of symmetry suggested by the experiments.Further bifurcations of the model lead to stable chaotic phenomena as well. These are numerically found to occur in association with heteroclinic orbits. Some mathematical results clarifying this association are also presented.

In a horizontal convecting layer of fluid, several distinct transitions occur at certain distinct Rayleigh numbers R, for a given Prandtl number Pr. The regime diagram has been extended to include the Prandtl-number range2·5 × 10−2 [les ] Pr [les ] 0·85 × 104.In particular it is found that distinct changes in the slope of the heat-flux curve occur even for Pr = 2·5 × 10−2. The flow is steady up to R = Rt = 2·4 × 103. For R > Rt, the period of oscillation is compared with the theoretical values of Busse. For Pr = 0·71 decreases as well as increases in the slope of the heat-flux curve are observed.For R just greater than Rc, the preferred orientation of rolls in various side-wall geometries is investigated. For high Prandtl number, the effect of curvature of the roll axis, forced by curved side walls, upon the second transition at RII is investigated. It is found that curvature, as well as previously discussed effects, leads to a lowering of RII. These results, along with the observed hysteresis, support the view that there are metastable states attainable by finite amplitude instability. Finally the nature of the time-dependent flow at large R and high Prandtl number is investigated in a Hele-Shaw cell. It is shown unequivocally that the observed periodicity at a fixed point is due to hot or cold plumes moving past the point.

In a horizontal convecting layer several distinct transitions occur before the flow becomes turbulent. These are studied experimentally for several Prandtl numbers from 1 to 104. Cell size plan form, transitions in plan form, transition to time-dependence, as well as the heat flux, are measured for Rayleigh numbers from 103to 105. The second transition, occurring at around 12 times the critical Rayleigh number, is one from steady two-dimensional rolls to a steady regular cellular pattern. There is associated with this a discrete change of slope of the heat flux curve, coinciding with the second transition observed by Malkus. Transitions to time-dependence will be discussed in part 2.