Gas turbine engines including embedded electrical machines and associated cooling systems

A gas turbine engine comprising:

• a compressor section and a turbine section arranged in serial flow order, the compressor section and the turbine section together defining a core airflow path;
• a rotary member rotatable with at least a portion of the compressor section and with at least a portion of the turbine section;
• an electrical machine located at least partially inward of the core airflow path in a radial direction, the electrical machine comprising an outer generator stator and an inner generator rotor that is coupled to the rotary member;
• a thermal shield forming an enclosure at least partially around the electrical machine, the thermal shield at least partially defining a cooling airflow path at least partially around the electrical machine;
• and
• a cooling manifold that directs a cooling airflow received from an airflow passageway that extends radially through a strut of a turbine rear frame past the outer generator stator and onto the inner generator rotor disposed radially inward from the outer generator stator.