A converging-diverging nozzle that has particular application for providing a cooling air flow to ring segments in a gas turbine engine. The engine includes a turbine section that receives a hot working gas. The turbine section includes at least one row of vanes, at least one row of blades and a plurality of ring segments forming at least one ring. The ring segments and the vanes are mounted to a vane carrier, where the vane carrier includes a cooling flow channel for each of the ring segments that receives an air flow to cool the ring segments. A plug is provided in each channel and has an internal bore shaped to define the converging-diverging nozzle through which the air flows so as to create a supersonic flow that reduces the temperature of the air and thus provides more cooling for the same amount of air flow.