A Numerical Study of Wavenumber Selection in Finite-Amplitude Rayleigh Convection

Abstract

Numerical integrations are performed for the equations governing two-dimensional convection flows in a fluid layer confined between two horizontal parallel plates and heated uniformly from below with free surface boundary conditions at the bottom and top of the fluid. In comparison with several previous works using a similar approach, a special feature in this work is that a large horizontal domain (10 times the critical wavelength or 28.28 times the height) is covered by the grid net so that the preferred mode of finite-amplitude convection flows is investigated. The Rayleigh number covered here is less than four times the critical Rayleigh number. Either random or sinusoidal perturbations with various wavelengths and with various amplitudes are introduced to initiate the motion. In all cases considered, the system achieves an approximate steady state. It is found that: 1) steady-state solutions are not determined uniquely by only the Rayleigh and Prandtl numbers, but also by the initial conditions; 2) a second stability curve or a nonlinear stability curve exists as the dividing line between those cells which exhibit size-adjustment toward a more preferred mode and those which do not; 3) the preferred modes in steady-state solutions depend not only on the wavelength of the initial sinusoidal perturbations but also on their amplitudes; and 4) the extremum principle, such as the maximum heat transport, may be inapplicable in determining the preferred mode.