On the onset of natural convection in differentially heated shallow fluid layers with internal heat generation

This paper reports the results of an analytical and numerical investigation to determine the effect of internal heat generation on the onset of convection, in a differentially heated shallow fluid layer. The case with the bottom plate at a temperature higher than the top plate mimics the classical Rayleigh Benard convection. However, internal heat generation adds a new dimension to the problem. Linear stability analysis is first carried out for the case of an infinitely wide cavity. The effect of aspect ratio on the onset of convection is studied by solving the full Navier-Stokes equations and the equation of energy and observing the temperature contours. A bisection algorithm is used for an accurate prediction of the onset. The numerical results are used to plot the stability curves for eight different aspect ratios. A general correlation is developed to determine the onset of convection in a differentially heated cavity for various aspect ratios. For an aspect ratio of 10, it is seen that the cavity approaches the limit of an infinite cavity. Analytical results obtained by using linear stability analysis agree very well with the “full” CFD simulations, for the above aspect ratio.