Re-entrant hexagons in non-Boussinesq convection

Abstract

While non-Boussinesq hexagonal convection patterns are known to be stable close to threshold (i.e. for Rayleigh numbers $R \,{\approx}\, R_c$), it has often been assumed that they are always unstable to rolls for slightly higher Rayleigh numbers. Using the incompressible Navier–Stokes equations for parameters corresponding to water as the working fluid, we perform full numerical stability analyses of hexagons in the strongly nonlinear regime ($\epsilon\,{\equiv}\, (R-R_c)/R_c\,{=}\,{O}(1)$). We find ‘re-entrant’ behaviour of the hexagons, i.e. as $\epsilon$ is increased they can lose and regain stability. This can occur for values of $\epsilon$ as low as $\epsilon\,{=}\,0.2$. We identify two factors contributing to the re-entrance: (i) far above threshold there exists a hexagon attractor even in Boussinesq convection as has been shown recently and (ii) the non-Boussinesq effects increase with $\epsilon$. Using direct simulations for circular containers we show that the re-entrant hexagons can prevail even for sidewall conditions that favour convection in the form of competing stable rolls. For sufficiently strong non-Boussinesq effects hexagons even become stable over the whole $\epsilon$-range considered, $0 \,{\le}\, \epsilon \,{\le}\, 1.5$.