Horizontal convective rolls in a tilted square duct of conductive and insulating walls

Numerical solutions with a fourth-order scheme show the characteristics of convective rolls that form in the interior of a duct lying horizontally and of square cross-section that can be tilted. The walls are either conductive or insulating and solutions are found which are steady and two-dimensional, without dependence along the duct. The density imposed at the conductive walls is of a uniform and stable stratification. The convection is caused by the existence of insulating walls, which oblige the isopycnals to reach them at right angles. This study shows features of the case reported by Quon [Quon, C., Convection induced by insulated boundaries in a square. Phys. Fluids A, 1983, 26, 632-637] for a duct with two parallel insulating and two parallel conductive walls, and another three possible wall combinations as a function of the tilt angle (α) and Rayleigh (Ra) numbers. All the cases indicate that a number of different rolls are possible, and their characteristics with respect to the tilt angle are described. Integral relationships for density, enstrophy and kinetic energy are used to characterize globally the solutions in the parameters space (α, Ra) with Ra up to 10⁵. These calculations and others with Ra up to 10⁶ and differing spatial resolution are used to obtain error estimates.