Heat transport bounds for a truncated model of Rayleigh–Bénard convection via polynomial optimization

Abstract

Upper bounds on time-averaged heat transport are obtained for an eight-mode Galerkin truncation of Rayleigh’s 1916 model of natural thermal convection. Bounds for the ODE model—an extension of Lorenz’s three-ODE system—are derived by constructing auxiliary functions that satisfy sufficient conditions wherein certain polynomial expressions must be nonnegative. Such conditions are enforced by requiring the polynomial expressions to admit sum-of-squares representations, allowing the resulting bounds to be minimized using semidefinite programming. Sharp or nearly sharp bounds on mean heat transport are computed numerically for numerous values of the model parameters: the Rayleigh and Prandtl numbers and the domain aspect ratio. In all cases where the Rayleigh number is small enough for the ODE model to be quantitatively close to the PDE model, mean heat transport is maximized by steady states. In some cases at larger Rayleigh number, time-periodic states maximize heat transport in the truncated model. Analytical parameter-dependent bounds are derived using quadratic auxiliary functions, and they are sharp for sufficiently small Rayleigh numbers.