Linear and Nonlinear Pattern Selection in Rayleigh-Benard Stability Problems

A new algorithm is introduced to compute finite-amplitude states using primitive variables for Rayleigh-Benard convection on relatively coarse meshes. The algorithm is based on a finite-difference matrix-splitting approach that separates all physical and dimensional effects into one-dimensional subsets. The nonlinear pattern selection process for steady convection in an air-filled square cavity with insulated side walls is investigated for Rayleigh numbers up to 20,000. The heated lower boundary is augmented with "noisy" boundary conditions to illustrate the transient and bimodal nature of the pattern selection process above the first critical Rayleigh number. Above a second critical Rayleigh number other instability modes may also be excited. The internalization of disturbances that evolve into coherent patterns is investigated and transient solutions from linear perturbation theory are compared with and contrasted to the full numerical simulations. The basins of attraction for transient solutions in a phase plane are shown to be bounded by singular states representing unstable modes. A particular symmetric mode that may be stable to symmetric or random disturbances but unstable to antisymmetric disturbances is investigated.