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

Abstract

A fully implicit finite difference scheme is developed for the stream function formulation of unsteady thermoconvective flows. An artificial time is added in the equation for the stream function rendering it into an ultra-parabolic type. For each time stage of the real time, a convergence is obtained with respect to artificial time (“internal iterations”). An implicit efficient operator-splitting time stepping is designed and proved to be absolutely stable. Employing a conservative central-difference approximation of the non-linear terms makes the scheme absolutely stable without using upwind differences. As a result, the scheme has no scheme viscosity and has virtually negligible phase error, which makes it a useful tool for investigating the intricate structure of the thermoconvective flow. The scheme is second-order approximation both in time and space. 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.