Disclosed is a tangential on-board injector (TOBI) system that includes an annulus and a plurality of cooling airflow passages disposed about the annulus. Each cooling airflow passage of the plurality of cooling airflow passages includes an inlet opening having a polygonal inlet cross-section, the inlet opening having an inlet cross-sectional area. Each cooling airflow passage of the plurality of cooling airflow passages further includes an outlet opening having an outlet cross-section and an outlet cross-sectional area. The inlet cross-sectional area is greater in magnitude than the outlet cross-sectional area. Also disclosed are additive manufacturing methods for manufacturing the tangential on-board injector system and gas turbine engines that incorporate the tangential on-board injector system.