A turbine section includes a stator assembly having an inner diameter end wall, an outer diameter end wall, and a stator vane; a turbine rotor assembly including a rotor blade extending into the mainstream gas flow path; a housing including an annular shroud that circumscribes the rotor blade and at least partially defines the mainstream hot gas flow path; a first baffle arranged to define a first cavity with the outer diameter end wall of the stator assembly; a second baffle; and a third baffle arranged to define a second cavity with the second baffle and a third cavity with the shroud. The first cavity is fluidly coupled to the second cavity and the second cavity is fluidly coupled to the third cavity such that cooling air flows from the first cavity to the second cavity and from the second cavity to the third cavity.