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I understand the difference between a compressor stall and surge.

Stall: One or two blade stages exceed the critical AoA, the airflow is turbulent, and the smooth airflow through the compressor is disrupted

Surge: Occurs after a compressor stall where most or all the blade stages are affected from it, there’s a complete disruption of airflow through the compressor and restricts airflow to the engine.

A stall produces a big bang, flames at the back and sometimes in front of the engine due to flow inversion, and severe vibrations and engine damage in a severe stall. In the cockpit we can see fluctuations in RPM and raise in EGT.

But what noises/indications a pilot from the cockpit would hear/see in a compressor surge? Are they similar?

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Stall and surge are two different, although related, physical phenomena.

Stall: One or more blade stages exceed the critical AoA, the airflow is turbulent, and the smooth airflow through the compressor is disrupted

Correct.

After a stall initiates, the airflow is disrupted. In an axial compressor, a stalled stage causes increased AoA on neighbouring blades, which in turn may enter a stall. The original blades may, at this point, exit the stall due to the decreased airflow, hence we have a rotating stall, or the stall can spread to the whole inlet section.

If the stall affects a large part of the compressor stage, the airflow from inlet to outlet cannot be guaranteed, so the (higher) pressure downstream of the stage causes a backflow, and

big bang, flames at the back and sometimes in front of the engine due to flow inversion, and severe vibrations and engine damage

This inversion is called surge.

In the end, it is surge that you experience in the way you describe, and it is a consequence of a severely stalled compressor. For brevity sake, people say "the compressor stalled" instead of "stalled and now entered surge".

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