High-Rayleigh-number asymptotic classical scaling in three-dimensional steady natural convection

The high-Rayleigh-number asymptotic behaviour of three-dimensional steady exact coherent states (ECS) in Rayleigh–Bénard convection is studied. The steady square and hexagonal convection cell states, whose horizontal scales are optimised to maximise Nusselt number, persist into the Rayleigh-number regime where a clear asymptotic trend emerges. A detailed asymptotic analysis of the governing equations reinforces that this trend persists in the limit of infinite Rayleigh number, with the corresponding Nusselt number following the classical scaling to leading order. The optimised Nusselt number of the three-dimensional ECS far exceeds that of the two-dimensional roll solutions, which are believed to bound currently available experimental and simulation results, reaching nearly twice the typical experimental values. This is an interesting result from an applied perspective, although our solutions are unstable at high Rayleigh numbers.