The cooling air flow in a gas turbine is governed by the flow through its internal passages and controlled by restrictors such as circular orifices. If the cooling air flow is incorrectly controlled, the durability and mechanical integrity of the whole turbine may be affected. Consequently, a good understanding of the orifices in the internal passages is important. This study presents experimental results for a range of pressure ratios and length-to-diameter ratios common in gas turbines including even very small pressure ratios. Additionally, the chamfer depth at the inlet was also varied. The results of the chamfer depth variation confirmed its beneficial influence on decreasing pressure losses. Moreover, important effects were noted when varying more than one parameter at a time. Besides earlier mentioned hysteresis at the threshold of choking, new phenomena were observed, e.g., a rise of the discharge coefficient for certain pressure and length-to-diameter ratios. A correlation for the discharge coefficient was attained based on the new experimental data with a generally lower error than previous studies.