  An extended immersed boundary method utilizing a semi-implicit direct forcing
approach for the simulation of confined incompressible viscous thermal flow
problems is presented. The method utilizes a Schur complement approach to
enforce the kinematic constraints of no-slip and the corresponding thermal
boundary conditions for immersed surfaces. The developed methodology can be
straightforwardly adapted to any existing incompressible time marching solver
based on a segregated pressure-velocity coupling. The method accurately meets
the thermal and the no-slip boundary conditions on the surfaces of immersed
bodies for the entire range of Rayleigh numbers $10^3\leqslant
Ra\leqslant10^6$. Strategies for further increasing the computational
efficiency of the developed approach are discussed. The method has been
extensively verified by applying it for the simulation of a number of
representative fully 3D confined natural convection steady and periodic flows.
Complex dynamic phenomena typical of this kind of flow including vortical
structures and convection cells and instability characteristics, were simulated
and visualized and the results were found to compare favorably with results
known from literature.
