A turbine assembly in a turbine section of an aircraft engine includes a rotor with blades having blade tips, and a turbine housing radially surrounding the blades. A distance between an inner surface of the housing and the blade tips defines a tip clearance gap. A passive cooling system for optimizing the tip clearance gap includes a cooling airflow passage located radially outward from, and in heat-transfer with, the turbine housing. The cooling airflow passage has an inlet opening located upstream of the rotor and an exit opening located downstream of the rotor. The inlet opening provides air flow into the cooling airflow passage. The exit opening provides air flow communication between the cooling airflow passage and a main gaspath of the turbine section. A flow of cooling air through the cooling airflow passage is induced, to cool the housing.