Numerical Solutions of the Nonlinear Equations for a Heated Fluid Layer

The equations of the Boussinesq approximation to the Navier—Stokes equations are solved numerically for the problem of a fluid layer heated from below. Solutions are obtained throughout the range of Rayleigh numbers from critical to R = 107. Free boundary solutions are compared with analysis, and rigid boundary solutions are compared with experiment. The dimensionless heat transport varies as R ⅓ for free boundaries and for rigid boundaries a variation of about R 0.296 is observed. Excellent agreement is obtained with both analysis and experiments and by an examination of various modal behaviors a number of the observed properties of the flow can be explained.