A numerical study was performed to explicate the structure of non-premixed swirl-stabilized CH4/H-2 flames. Swirl-stabilized flames involve a highly complex interaction between turbulence and chemistry that leads to another level of difficulty in modeling. In such cases, the large eddy simulation with a conditional moment closure (CMC) would be a promising tool. However, to achieve results with minimal computational effort without compromising on accuracy, the Reynolds stress model (RSM) with CMC was used in this study. The OpenFOAM-based CMC solver (cmcFoam) earlier proposed by us (Gaikwad and Sreedhara, 2019) was employed to achieve an optimized coupling between RSM and CMC. Three-dimensional (3D) simulations were performed using a transient compressible RSM with the detailed chemical kinetic mechanism, GRI-Mech 3.0, involving 36 species and 219 chemical reactions (excluding NOx chemistry). Results predicted by RSM were compared with the measured data. A good agreement between predicted results and the measured data were achieved by the RSM-CMC method in both conditional and physical spaces. The main features of the swirl-stabilized flame, such as flow recirculation and vortex breakdown, were captured well. The complex structure of the swirl-stabilized flame, which results from the interaction of a swirling flow with fuel jets, has been described in detail with the help of contours of pressure gradient and Reynolds stress. A probable local extinction region was found near the necking zone of the flame, in the outer layer of the toroidal recirculating bubble, where high values of tangential stresses exist. This region is accompanied by a higher hydrodynamic strain rate, a lower hydroxyl mass fraction, and a lower Damkohler number (Da). The predicted features of the swirl-stabilized flame will be helpful in understanding the structure of more complex industrial problems.