This paper presents divergent phases of numerical investigations on non-premixed combustion flames. The study has been performed using the computational fluid dynamics software (CFD). In this study, the effect of swirl number and oxidizer composition on the combustion characteristics of air-methane combustion and comparison will be performed with respect to oxy-fuel combustion cases. The oxy-fuel combustion investigations were performed on three difference oxygen fractions, namely, 30, 40, and 50%, to investigate the effect of the oxidizer flexibility on combustion characteristics. Both investigations have been conducted under the non-premixed combustion model, and the flame is anchored over a swirl stabilizer. In this work, validation of a previous experimental work was successfully achieved and a grid independency study was performed. The axial and radial temperature distribution are analyzed and reported. The velocity stream lines and emission concentrations are also reported. Furthermore, the effect of swirl number on NOx emissions has been investigated for air-fuel combustion cases only because nitrogen is not allowed in the oxy-combustion cases. The study revealed that increasing swirl number can reduce the thermal NOx by 95% and this can be attributed to the increase in mixing levels of the combustible mixture.