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 R _c and the steady velocity and temperature fields of most unstable modes are computed by a Galerkin spectral method of high accuracy for aspect ratios A _x and A _y either or both of which are small. We find that if A _x is decreased to 0 with A _y being kept constant, R _c 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^-\tfrac{1{2}}. Moreover, as A _x is decreased, we observe the changes of the symmetry of most unstable mode that occur more frequently for smaller A _x . However, if {A}_x={A}_y=A is decreased to 0, although we again observe the increase in R _c 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 R _c and velocity fields in the limit of {A}_x\to 0 or A\to 0 are obtained by an asymptotic analysis in which the dependences of R _c and the magnitude and length scale of velocity fields of most unstable modes on A _x and A _y in the numerical computations are used. For example, R _c is approximated by {π }⁴{A}_x^-4 and 25{π }⁴{A}^-4 in the limits of {A}_x\to 0 and A\to 0, respectively. Moreover, analytical expressions of some components of velocity fields in these limits are derived. Finally, we find that for small A _x or A the agreement between the numerical and analytical results on R _c and velocity field is quite good except for the velocity field in thin wall layers near the top and bottom walls.