A cooling assembly for a gas turbine system includes a turbine nozzle having at least one channel comprising a channel inlet configured to receive a cooling flow from a cooling source, wherein the at least one channel directs the cooling flow through the turbine nozzle in a radial direction at a first pressure to a channel outlet. Also included is an exit cavity for fluidly connecting the channel outlet to a region of a turbine component, wherein the region of the turbine component is at a second pressure, wherein the first pressure is greater than the second pressure.