Non-linear waves of the steady natural convection in a vertical fluid layer: A numerical approach

By means of the scheme developed, the convective flow in a vertical slot with differentially heated walls and vertical temperature gradient is studied for very large Rayleigh numbers. The model involves Boussinesq approximation and consists of a coupled system of a fourth-order in space equation for the stream function and a convection-diffusion equation for the temperature. The numerical results show that with increasing stratification parameter, the mode of the instability changes from traveling-wave to stationary-wave, which is consistent with the predictions of the linear theory of hydrodynamic instability. The role of the dimensionless wavelength is investigated and the issue of most dangerous wave is addressed numerically.