A Nonlinear Spectral Model of Convection in a Fluid Unevenly Heated from Below

Abstract

A two-dimensional form of the Boussinesq equations is integrated numerically for the case of a rectangular channel with a temperature gradient maintained along the bottom. The side walls are insulating, the top wall has a constant temperature, and the velocity obeys free boundary conditions on all four walls. The fields of stream function and temperature departure are represented by truncated double Fourier series, and integration of the initial-value problem for the spectral amplitudes results in steady states which agree qualitatively with those of previous experimental and theoretical investigations. Calculations are presented at two levels of truncation (wave numbers 2 and 3) for a wide range of Prandtl numbers and a moderate range of horizontal Rayleigh numbers and top temperatures. For sufficiently large gravitational stability, a single asymmetric convection cell develops. Its intensity and asymmetry increase markedly with increasing horizontal Rayleigh number, decrease with increasing top temperature, and respond very slightly to changes in Prandtl number. As the top temperature is decreased below the temperature of the warm side of the bottom, however, the possibility is indicated that the single cell may be modified by a Bénard-like multi-cellular structure.