Stability of a fluid in a rectangular region heated from below

Finite‐difference methods were used to compute hydrodynamic instability due to natural convection in an enclosed horizontal rectangular region heated from below. Critical Rayleigh numbers were determined for a series of Prandtl numbers and length‐to‐height ratios. For Prandtl numbers greater than unity excellent agreement was obtained between these calculations and the values predicted by Kurzweg on the basis of a linearized theory. However, for Prandtl numbers less than unity the critical Rayleigh numbers exhibited a dependence on NPr, which was not predicted by the linearized theory. For Rayleigh numbers greater than the critical, complete temperature and velocity fields were determined. The calculations assumed that the fluid motion is two dimensional. Experiments have indicated that the flow may be two or three dimensional depending on minor perturbations in the boundary conditions. Although a number of metastable two‐dimensional circulations are possible for symmetrical initial conditions, the calculation always converged to a single, unique solution for any asymmetric initial condition.