Turbomachine having an airflow management assembly

A gas turbine engine defining an axial direction, a radial direction, and a circumferential direction, the gas turbine engine comprising:

• a compressor and a turbine;
• a plurality of rotor blades positioned in the compressor or the turbine and arranged in a plurality of stages along the axial direction;
• an outer drum positioned at least partially radially outward of the plurality of rotor blades and rotatable with the plurality of rotor blades;
• and
• a casing extending over the outer drum and defining a plurality of air cavities with the outer drum,
• wherein the plurality of air cavities are arranged in series airflow communication along the axial direction,
• wherein the compressor and the turbine together define at least in part a core air flowpath,
• wherein one or more of the air cavities of the plurality of air cavities is in airflow communication with the core air flowpath through the outer drum, and
• wherein a pressure in each of the plurality of air cavities is higher than a pressure in the core air flowpath disposed directly on a radial inner side thereof.