Thermal instability: From cat's eyes to disjoint multiple cells natural convection flow in tall tilted cavities

Abstract

Paper presented to the 10th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Florida, 14-16 July 2014.

Thermal viscous incompressible fluid flows, modeled by the unsteady Boussinesq approximation in primitive variables, are solved numerically, by means of a direct projection method, which involves four steps, at each time level: one for the energy equation (temperature step) and three more for the momentum and continuity equations (motion steps). An operator splitting for a second order time discretization is used, taking care of the nonlinear system of equations. The entire process is semiimplicit, which is a common practice. In the temperature step an elliptic problem is solved. In the first motion step, an intermediate velocity is obtained explicitly, considering only one of the three terms of the approximation of the time derivative, and the linear extrapolation of the nonlinear term from the momentum equation; this velocity does not satisfy the incompressibility constraint. In the second step, a new intermediate velocity is computed using an equation that contains another portion of the approximation of the time derivative and the term associated with the pressure. In this step, an additional assumption is incorporated: the new velocity satisfies the incompressibility constraint, such that its projection onto the free divergence subspace is attained. In the third motion step, the final velocity is calculated through an elliptic problem, which contains the last part of the time derivative and the viscous linear term of the velocity. Additionally, in the second step, an elliptic problem is accomplished for the pressure (Poisson equation), complemented with an appropriated Neumann boundary condition obtained by considering the viscous linear contribution in terms of an irrotational part, which is zero due to the incompressibility constraint, and a solenoidal part approximated by linear extrapolation of known values from the two previous time levels. To solve the elliptic problems that result from this process, efficient solvers exist regardless of the space discretization. Natural convection results describing the flow dynamics in tall tilted cavities are shown, which involve from thermal instability known as "cat's eyes" to disjoint multiple cells. To this end, the Rayleigh number Ra and the aspect ratio of the cavity A (A=ratio of the height to the width of the cavity) are fixed to Ra= 11000 and A=16. The thermal instability we are mentioning is obtained varying the angle φ of the cavity from 0° up to 120°.