Onset of thermal convection in a rectangular parallelepiped cavity of small aspect ratios

Onset of thermal convection of a fluid in a rectangular parallelepiped cavity of small aspect ratios is examined both numerically and analytically under the assumption that all walls are rigid and of perfect thermal conductance exposed to a vertically linear temperature field. Critical Rayleigh number Rc and the steady velocity and temperature fields of most unstable modes are computed by a Galerkin spectral method of high accuracy for aspect ratios Ax and Ay either or both of which are small. We find that if Ax is decreased to 0 with Ay being kept constant, Rc increases proportionally to A x − 4 , the convection rolls of most unstable mode whose axes are parallel to the shorter side walls become narrower, and their number increases proportionally to A x − 1 2 . Moreover, as Ax is decreased, we observe the changes of the symmetry of most unstable mode that occur more frequently for smaller Ax. However, if A x = A y = A is decreased to 0, although we again observe the increase in Rc proportional to A − 4 , we obtain only one narrow convection roll as the velocity field of most unstable mode for all A. The expressions of Rc and velocity fields in the limit of A x → 0 or A → 0 are obtained by an asymptotic analysis in which the dependences of Rc and the magnitude and length scale of velocity fields of most unstable modes on Ax and Ay in the numerical computations are used. For example, Rc is approximated by π 4 A x − 4 and 25 π 4 A − 4 in the limits of A x → 0 and A → 0 , respectively. Moreover, analytical expressions of some components of velocity fields in these limits are derived. Finally, we find that for small Ax or A the agreement between the numerical and analytical results on Rc and velocity field is quite good except for the velocity field in thin wall layers near the top and bottom walls.