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Is there any formula for the velocity of the working fluid (air) through /the fan of a turbofan engine? If not, how can I find the velocity?

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here is how to get an estimate.

At cruise conditions, the airflow into the engine is such that (ideally) the circumferential lip of the inlet shroud is a stagnation point in the flow- that is, no part of the inlet flow spills out and around the engine nor do the streamlines of the inlet flow converge outside the inlet before entering it.

This means that the inlet ingests all the air coming at it at cruise. If you know the velocity and density of the entering air and the bypass ratio, you can solve for the mass flow rate of air through the core of the engine.

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