Convective pattern evolution and secondary instabilities

Abstract

Using an automated laser-Doppler scanning technique, we have performed an extensive study of pattern evolution and instabilities in a large Rayleigh-BBnard cell (20 by 30 times the layer depth) at moderate Prandtl number (2.5). This work differs from earlier experiments in that the Doppler mapping technique permits both the spatial structure and time evolution of the velocity field to be quantitatively studied, and runs lasting up to ten thousand vertical thermal diffusion times are presented. We observe and document the properties of three qualitatively different regimes above the critical Rayeigh number Rc. (i) Below 5Rc, there are a number of simple patterns in which the rolls align perpendicular to all lateral cell boundaries. The patterns are therefore curved and generally contain two boundary-related defects. A change in R induces patterns with many defects, which evolve toward the simple patterns over extraordinarily long times. These features seem to be consistent with theoretical models based on the competition between boundary, curvature and defect contributions to a Liapunov functional. However, stable states are not always reached. (ii) Above 5Rc the skewed-varicose instability of Busse & Clever causes progressively faster broadband time dependence with a spectral tail falling off approximately as the fourth power of the frequency. Doppler imaging shows that the fluctuations are caused by narrowing and pinching off of the rolls. (iii) Above 9Rc both the oscillatory and skewed-varicose instabilities cause local velocity fluctuations. However, substantial mean roll structure persists even over a 40 h period (one horizontal thermal diffusion time) at 1 5Rc. Velocity power spectra with two distinct maxima associated with the two instabilities are still resolved at 50Rc. Finally, we impose thermal inhomogeneities in order to pin the rolls, and show that the fluctuations are suppressed only if the local heat flux is a significant fraction of the convective heat transport per wavelength.

On the boundary-layer structure of patterns of convection in rectangular-planform containers

“…This illustrates the effective curvature of the grain boundary and gives a good indication of the complicated roll patterns near the corners and shorter sides of the rectangle. In general, the results are in good agreement with both the full numerical simulations of the Swift-Hohenberg equation by Greenside & Coughran (1984, figure 9) and the experiments of Pocheau & Croquette (1984) although it should be noted that the reconstruction in figure 2 excludes the L 1/2 boundary layers described in § 2.…”

“…In particular, most of the experimental visualizations indicate an outward curvature of the grain boundary which increases as the corners are approached, consistent with the present theory. In other experimental work for large-planform rectangular containers by Gollub, McCarriar & Steinman (1982), the visualization of the flow excluded the regions near the shorter sides of the rectangular planform. However, the slow time evolutions of the roll patterns exhibit many of the features observed by Greenside & Coughran (1984) in their numerical simulations of the Swift-Hohenberg equation, and the latter results provide strong evidence to support the existence of a stationary state of the type studied here.…”

“…In other experimental work for large-planform rectangular containers by Gollub, McCarriar & Steinman (1982), the visualization of the flow excluded the regions near the shorter sides of the rectangular planform. However, the slow time evolutions of the roll patterns exhibit many of the features observed by Greenside & Coughran (1984) in their numerical simulations of the Swift-Hohenberg equation, and the latter results provide strong evidence to support the existence of a stationary state of the type studied here. The numerical simulations were carried out for parameter values a = 0.668 and L = 29.2π and results for symmetric initial configurations with = 0.03 and = 0.1 led to symmetric steady states very similar to those displayed here in figures 2 and 4.…”

“…Daniels & Weinstein (1996) studied the structure of this secondary bifurcation, for the case where the pattern is restricted to rolls orthogonal to each side of the rectangle, by solving a coupled pair of amplitude equations. These were derived from the Swift-Hohenberg (1977) equation, for which extensive numerical simulations have been carried out (Greenside & Coughran 1984) but are also relevant to the full Rayleigh-Bénard system with rigid upper and lower surfaces at a particular value of the Prandtl number (Sivapragasam 1995). Numerical computations of the steady-state secondary solution branch beyond the bifurcation point showed that having initially moved inwards the grain boundaries then start to approach the walls again as the Rayleigh number increases.…”

Water surface structures observed using infrared imaging

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