The small size of orifice flow meters in the pipeline and their acceptable accuracy are two reasons for their wide use in measuring flow rates in pipelines. An orifice flow meter has several issues, including an inability to recover pressure loss, the inability to take very accurate flow rate measurements, and vortex shedding behind the orifice, resulting from the shear forces that the walls apply to the flow. An open-source computational fluid dynamics (CFD) code, Open FOAM, has been used to measure flow through square-fringed orifices within a given pipe’s diameter and thickness. The ratio of the diameter of an orifice to the diameter of a pipe ranges from 0.25 to 0.75, whereas the ratio of the thickness of an orifice to the diameter of a pipe ranges from 0.125 to 2. Different discharge coefficients were determined for different orifice characteristics at different Reynolds numbers. The two flow regimes examined through simulations are the turbulent flow regime, at 2500 ≤ Re ≤ 40,000, and the laminar flow regime, at 0 ≤ Re ≤ 400. The stream-wise velocity contours, Q-yardstick curves, and streamlines are introduced to evaluate the impact of various thicknesses, Re values, and β values on flow characteristics. Validation was conducted by comparing the results of the current study with the results of previous studies. A satisfactory agreement was found between the discharge coefficient data in this study and the experimental and numerical data in previous studies.