Wave Number Selection and Large-Scale-Flow Effects due to a Radial Ramp of the Spacing in Rayleigh-Bénard Convection

Abstract

We present experimental results for wave numbers q s selected in a thin horizontal fluid layer heated from below. The cylindrical sample had an interior section of uniform spacing d d 0 for radii r , r 0 (G 0 ϵ r 0 ͞d 0 43) and a ramp d͑r͒ for r . r 0 . For Rayleigh numbers R 0 . R c 1708 in the interior, straight or slightly curved rolls with an average ͗q s ͘ q c 1 ae 0 ͑e 0 ϵ R 0 ͞R c 2 1͒ and q c , q c 3.117 were selected, and q s varied on two length scales approximately equal to G 0 and to four roll wavelengths. For e & 0.03 and e * 0.18 the pattern repeatedly formed defects.

Experiments with Rayleigh-Bénard Convection

“…These patterns were, however, sidewall-induced and not intrinsic to the interior of a very large system. This was shown in an experiment where the walls were replaced by a very gentle radial ramp in the cell spacing that led to a region of pure conduction surrounding the convecting interior [7]. An example of a pattern in this system, for = 0.21, is shown in the middle of Figure 4.1.…”

“…Middle: roll pattern for a Boussinesq fluid (from [7]). Right: square pattern in binary-mixture convection (from [8]).…”

“…On the basis of the linear stability analysis of Lord Rayleigh or Sir Harold Jeffreys one knows the magnitude of the critical wavedirector, but one cannot decide whether the patterns above onset will consist of rolls, hexagons, or squares. Indeed all three patterns occur in RBC as illustrated in Figure 4.1 [6,7,8], albeit under different circumstances. Malkus and Veronis [9] predicted that the stable planform for the case of free boundaries and Boussinesq conditions [10,11] should be straight rolls rather than, for example, squares or hexagons.…”

Efficient algorithm on a nonstaggered mesh for simulating Rayleigh-Bénard convection in a box

“…(15). There is a substantial literature concerning the appropriateness and accuracy of the boundary condition Eq.…”

“…Finally, a Caltech-Duke collaboration has recently reported results [12,13] obtained with a parallel spectral element code [14] for aspect ratios up to 30. Their code can treat quantitatively most geometries and lateral boundaries used by experimentalists, including ramps [15], spoiler fins [16], and lateral walls of finite thickness and finite thermal conductivity. However, the generality of the spectral element algorithm makes it substantially more expensive to run than algorithms optimized for a simple geometry such as a box or cylinder.…”