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An extended immersed boundary method utilizing a semi-implicit direct forcingapproach for the simulation of confined incompressible viscous thermal flowproblems is presented. The method utilizes a Schur complement approach toenforce the kinematic constraints of no-slip and the corresponding thermalboundary conditions for immersed surfaces. The developed methodology can bestraightforwardly adapted to any existing incompressible time marching solverbased on a segregated pressure-velocity coupling. The method accurately meetsthe thermal and the no-slip boundary conditions on the surfaces of immersedbodies for the entire range of Rayleigh numbers $10^3\leqslantRa\leqslant10^6$. Strategies for further increasing the computationalefficiency of the developed approach are discussed. The method has beenextensively verified by applying it for the simulation of a number ofrepresentative fully 3D confined natural convection steady and periodic flows.Complex dynamic phenomena typical of this kind of flow including vorticalstructures and convection cells and instability characteristics, were simulatedand visualized and the results were found to compare favorably with resultsknown from literature.

Semi-implicit direct forcing immersed boundary method for incompressible viscous thermal flow problems: a Schur complement approach