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I was wondering why some turbofan engine have multi-stage fans.

e.g. F-119 engine:

image1

F100

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Saturn AL-31

enter image description here

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All three engines you show are for fighter jets flying at supersonic speeds. These engines need a low bypass ratio: Why do military turbofan engines use a low bypass ratio?

To achieve more thrust at lower bypass ratios, the bypass air needs to be accelerated more, which requires more than one fan stage because axial compressors can only add a limited pressure difference per stage:

Low bypass engines often have a multistage fan which generates a low volume but relatively high speed air stream whereas high bypass engines usually have a single stage fan which generates a high volume but relatively low speed air stream.

(SKYbrary - Turbofan Engine)

In principle, high bypass ratio engines where more air is accelerated less are more efficient and therefore used on modern airliners, but they are not suited for supersonic flight.

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