A numerical method for simulating three-dimensional convection

We consider the three-dimensional convection of a fluid in a rectangular parallelepiped with isothermic horizontal boundary free from tangent stresses and heated from below. We propose a special spectral difference numerical method for calculating an approximation of the second order in space and the first order in time. Linear analysis of this numerical method showed that the method predicts correctly (with a good quantitative fit in the long wavelength range and a qualitative fit in the short wavelength range) the spectral characteristics of the differential problem for practical values of mesh sizes in time, space, and overcriticity. For testing we simulated two-dimensional cylindrical and turbulent Rayleigh–Bénard convection for the supercriticity equal to 2.2 and 950 and the Prandtl number equal to 10.