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°.