Numerical study of the influence of forcing terms and fluctuations near onset on the roll pattern in Rayleigh-Bénard convection in a simple fluid

The stochastic Swift-Hohenberg equation is studied as a model of Rayleigh-Bénard convection in a simple fluid. The equation has been solved numerically in two spatial dimensions to obtain the convective heat current and the roll pattern when either a bulk stochastic forcing field or different models of thermal-diffusivity mismatch at the sidewalls of the convective cell are considered. The parameters that enter the equation have been chosen to match the ramping experiments on Rayleigh-Bénard convection by Meyer, Ahlers, and Cannell [Phys. Rev. Lett. 59, 1577 (1987)]. For any combination of forcing mechanisms, we are able to find values of their various amplitudes that lead to excellent fits to the experimentally measured convective current. In the case of a bulk random forcing field, we find an amplitude of ${\mathit{F}}_1$=5×${10}^{\mathrm{-}5}$, compared to ${\mathit{F}}_{\mathrm{th}}$=1.92×${10}^{\mathrm{-}9}$, the value obtained from fluctuation theory. A random, cellular pattern of rolls is observed, in agreement with experiments involving a gel sidewall designed to eliminate the influence of the sidewalls on the onset of convection. A thermal-conductivity mismatch at the sidewall has also been modeled by a variety of forcing fields. In all cases a roll-like pattern that reflects the geometry of the sidewalls is observed. Different combinations of both types of forcing fields have also been studied and found to yield patterns intermediate between cellular and roll-like, while yielding a very reasonable fit to the convective heat current measured experimentally.