Abstract
A layer of fluid is confined between two horizontal boundaries which are kept at constant temperatures. An infinitely thin layer at the middle of the fluid is heated uniformly at a constant rate so that a constant negative temperature gradient is maintained in the upper half of the fluid and a constant positive temperature gradient in the lower half. When the rate of internal heating exceeds a critical value, convective motion is given rise.
The two-dimensional governing equations with the Boussinesq approximation are transformed to a set of differential equations governing the time variations of the coefficients of Fourier series expanded along the horizontal direction. The system is then truncated by taking into account only a limited number of terms of the Fourier expansions. The equations are then numerically integrated for different values of the Rayleigh number (R) and a fundamental horizontal wavenumber.
The result of these numerical integrations indicates that solutions achieve steady states for R smaller than approximately 38 Rc, where Rc is a critical Rayleigh number, and that convection with periodic time-dependency takes place over a narrow range of R, extending to several tens of. The flow in this range consists of successive generations of a pair of plumes and a single plume. The most striking difference between the steady state motion and the periodic motion as revealed by calculation is that the steady flow consists of one mode whereas a few modes having amplitudes of the same order are contributing to produce periodic timedependency. The non-dimensional upward heat flux is found to increase with R to the point where the periodic motion replaces the steady state solution. A sudden decrease of the upward heat flux is observed after this.
