The present disclosure is directed to a cooling system for a turbomachine. The cooling system includes a turbomachine component defining a turbomachine component cavity. The cooling system also includes an insert positioned within the turbomachine component cavity for cooling the turbomachine component. The insert includes an insert body and a spring body. The spring body includes a first portion fixedly coupled to the insert body, a second portion in sliding engagement with the turbomachine component, and a third portion in sliding engagement with the insert body.