  The simulation of natural convection problem based on the Galerkin
finite-element method, with the penalty finite-element formulation of the
momentum balance equation, is exploited for accurate solutions of equations
describing the problem of H-Form cavity differentially heated side walls. The
cavity is occupied by the air whose Prandtl number is Pr=0.71, the fluid is
assumed to be steady, viscous and incompressible within thermal convection. A
numerical investigation has been made for Rayleigh numbers ranging from 10 to
10^6 for three cases of total internal height aspects of H-Form cavity: 0%,
50%, and 85%. Firstly, the goal is to validate the numerical code used to
resolve the equations governing the problem of this work. For that, we present
a comparison between the profiles at the point (0.5, 0) for the u-component,
and u-component obtained in previous work for simple square cavity. Further, a
comparison of the averaged Nusselt number with previous works for simple square
cavity is realized in order to ensure the numerical accuracy, and the validity
of our considered numerical tool. Secondly, the objective is to investigate on
the hydrodynamic effects of Rayleigh number for different total internal height
aspects of H-Form cavity on the dynamics of natural convection. Shortly after,
the ambition is to assess the heat transfer rate for different Rayleigh number
for three cases of internal height aspects.
