Onset of thermal convection and its flow patterns in a rectangular cavity

The onset of thermal convection of a fluid in a rectangular cavity of aspect ratios A(x) and A(y) exposed to a vertically linear temperature field is examined under the assumption that all its walls are rigid, and of perfect thermal conductance. For several A(x) and A(y) smaller than 6, the critical Rayleigh number R-c and the steady flow patterns of most unstable modes are computed by a Galerkin spectral method of high accuracy. Characteristics of flow patterns are examined by using upper-wall flow patterns based on near-wall velocity fields, distributions of vertical velocity, and trajectories of fluid particles. We find that a symmetry mode that is not the most unstable when the difference between A(x) and A(y) is large is the most unstable if both A(x) and A(y) are around 4 or around 5.5. The flow pattern of this mode is consistent with the results in a preceding experimental study. R-c increases rapidly as A(x) or A(y) decreases to zero and decreases slowly as they increase. The validity of the assumption of finite roll is found to be limited, even if the difference between A(x) and A(y) is large. The flow pattern for A(x) = A(y) = A becomes more complicated, and the number of convection cells increases as A increases. The motion of fluid particles in each cell is roughly the circulation along closed curves around an axis that is on the central horizontal plane but is not necessarily parallel to side walls. Moreover, the motion of fluid particles near side walls can be much different from that in the central part of the cavity.