Hexagonal and roll flow patterns in temporally modulated Rayleigh-Bénard convection

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 ε≡ΔT/Δ${\mathit{T}}_{\mathit{c}}$-1 had the form ε(t)=${\mathrm{\epsilon }}_0$+δ sin(ωt). Here the time t and frequency ω are scaled by the vertical thermal diffusion time. Experiments were performed within the ranges 8.0≤ω≤18.0, 0.4≤δ≤3.3, and -0.2≤${\mathrm{\epsilon }}_0$≤0.6. Measurements of the convective threshold shift ${\mathrm{\epsilon }}_{\mathit{c}}$(δ,ω) were in good agreement with theoretical predictions. Comparisons were made with theoretical predictions of a range ${\mathrm{\epsilon }}_{\mathit{A}}$(δ,ω)≤${\mathrm{\epsilon }}_0$${\mathrm{\epsilon }}_{\mathit{R}}$(δ,ω) (${\mathrm{\epsilon }}_{\mathit{A}}$${\mathrm{\epsilon }}_{\mathit{c}}$,${\mathrm{\epsilon }}_{\mathit{R}}$≳${\mathrm{\epsilon }}_{\mathit{c}}$) where only a hexagonal pattern with downflow at the cell centers is predicted to be stable, a range ${\mathrm{\epsilon }}_{\mathit{R}}$≤${\mathrm{\epsilon }}_0$${\mathrm{\epsilon }}_{\mathit{B}}$(δ,ω) where both hexagonal and roll patterns are expected to be stable, and a range ${\mathrm{\epsilon }}_0$≥${\mathrm{\epsilon }}_{\mathit{B}}$ where only a roll pattern should be stable. At low modulation amplitudes (δ≲1.2 for ω=15) only rolls were observed over the range of ${\mathrm{\epsilon }}_0$ studied, although the rolls appeared perturbed for ${\mathrm{\epsilon }}_{\mathit{A}}$≤${\mathrm{\epsilon }}_0$${\mathrm{\epsilon }}_{\mathit{R}}$. At moderately high amplitudes (1.2≲δ≲2.3 for ω=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 ${\mathrm{\epsilon }}_{\mathit{A}}$≲${\mathrm{\epsilon }}_0$≲${\mathrm{\epsilon }}_{\mathit{R}}$. Over the range ${\mathrm{\epsilon }}_0$≳${\mathrm{\epsilon }}_{\mathit{B}}$ roll-like patterns were observed. Over the range ${\mathrm{\epsilon }}_{\mathit{R}}$≲${\mathrm{\epsilon }}_0$≲${\mathrm{\epsilon }}_{\mathit{B}}$, where theory predicts bistability of rolls and hexagons, a coexistence between the two patterns was found. At high values of δ (δ≳2.3 for ω=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 δ (δ>3.0 for ω=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 δ 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.