Re-entrant hexagons in non-Boussinesq convection

While non-Boussinesq hexagonal convection patterns are known to be stable close to threshold (i.e. for Rayleigh numbers R≈ 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 (∈ ≡ (R - R_c)/R_c= O (1). We find 're-entrant' behaviour of the hexagons, i.e. as ∈ is increased they can lose and regain stability. This can occur for values of ∈ as low as ∈ = 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 ∈. 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 ∈ - range considered, O ≤ ∈ ≤ 1.5.