We present experimental results for pattern formation in a thin fluid layer heated time periodically from below. They were obtained with computer-enhanced shadowgraph flow visualization and with heat-flux measurements. The experimental cell was cylindrical, with a radius-to-height ratio of 11.0. The temperature of the top plate was held constant while that of the bottom plate was modulated sinusoidally so that the reduced Rayleigh number \ensuremath{\epsilon}\ensuremath{\equiv}\ensuremath{\Delta}T/\ensuremath{\Delta}${\mathit{T}}_{\mathit{c}}$-1 had the form \ensuremath{\epsilon}(t)=${\mathrm{\ensuremath{\epsilon}}}_{0}$+\ensuremath{\delta} sin(\ensuremath{\omega}t). Here the time t and frequency \ensuremath{\omega} are scaled by the vertical thermal diffusion time. Experiments were performed within the ranges 8.0\ensuremath{\le}\ensuremath{\omega}\ensuremath{\le}18.0, 0.4\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}3.3, and -0.2\ensuremath{\le}${\mathrm{\ensuremath{\epsilon}}}_{0}$\ensuremath{\le}0.6. Measurements of the convective threshold shift ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{c}}$(\ensuremath{\delta},\ensuremath{\omega}) were in good agreement with theoretical predictions. Comparisons were made with theoretical predictions of a range ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{A}}$(\ensuremath{\delta},\ensuremath{\omega})\ensuremath{\le}${\mathrm{\ensuremath{\epsilon}}}_{0}$${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$(\ensuremath{\delta},\ensuremath{\omega}) (${\mathrm{\ensuremath{\epsilon}}}_{\mathit{A}}$${\mathrm{\ensuremath{\epsilon}}}_{\mathit{c}}$,${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$\ensuremath{\gtrsim}${\mathrm{\ensuremath{\epsilon}}}_{\mathit{c}}$) where only a hexagonal pattern with downflow at the cell centers is predicted to be stable, a range ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$\ensuremath{\le}${\mathrm{\ensuremath{\epsilon}}}_{0}$${\mathrm{\ensuremath{\epsilon}}}_{\mathit{B}}$(\ensuremath{\delta},\ensuremath{\omega}) where both hexagonal and roll patterns are expected to be stable, and a range ${\mathrm{\ensuremath{\epsilon}}}_{0}$\ensuremath{\ge}${\mathrm{\ensuremath{\epsilon}}}_{\mathit{B}}$ where only a roll pattern should be stable. At low modulation amplitudes (\ensuremath{\delta}\ensuremath{\lesssim}1.2 for \ensuremath{\omega}=15) only rolls were observed over the range of ${\mathrm{\ensuremath{\epsilon}}}_{0}$ studied, although the rolls appeared perturbed for ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{A}}$\ensuremath{\le}${\mathrm{\ensuremath{\epsilon}}}_{0}$${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$. At moderately high amplitudes (1.2\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}2.3 for \ensuremath{\omega}=15), a cellular pattern with local sixfold symmetry and downflow at the cell centers, which was reproducible from one cycle to the next, was observed over the range ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{A}}$\ensuremath{\lesssim}${\mathrm{\ensuremath{\epsilon}}}_{0}$\ensuremath{\lesssim}${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$. Over the range ${\mathrm{\ensuremath{\epsilon}}}_{0}$\ensuremath{\gtrsim}${\mathrm{\ensuremath{\epsilon}}}_{\mathit{B}}$ roll-like patterns were observed. Over the range ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$\ensuremath{\lesssim}${\mathrm{\ensuremath{\epsilon}}}_{0}$\ensuremath{\lesssim}${\mathrm{\ensuremath{\epsilon}}}_{\mathit{B}}$, where theory predicts bistability of rolls and hexagons, a coexistence between the two patterns was found. At high values of \ensuremath{\delta} (\ensuremath{\delta}\ensuremath{\gtrsim}2.3 for \ensuremath{\omega}=15), a pattern consisting of randomly placed cells and short roll segments that was reproducible from one cycle to the next was observed in all three regions. At still higher values of \ensuremath{\delta} (\ensuremath{\delta}>3.0 for \ensuremath{\omega}=13), this pattern was observed to be irreproducible from one cycle to the next. The transition from patterns resembling those predicted by the deterministic theory to irreproducible random patterns as \ensuremath{\delta} is increased is presumed to be due to stochastic perturbations. These perturbations appear to play an important role in those parameter ranges where the amplitude of the pattern decays to a microscopic value during part of the modulation cycle.
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Jan 1, 1999
Yasuhiko Nakanishi + 3
Open Access
