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To my knowledge, bypass air produces 80% of total thrust. But I don't understand to how it does that. By accelerating the air, by increasing the speed or increasing the pressure of the air? Is it doing this by Bernoulli's principle or something like that?

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The bypass air is accelerated by the fan at the front of the turbofan engine. This changes its velocity and therefore its momentum, which is the definition of a force (in this case: thrust):

$$ F = \frac{\text{d}}{\text{d}t} p = m \frac{\text{d}}{\text{d}t} v = m \cdot a $$

This thrust contributes to the total thrust of the engine. How much will depend on the bypass ratio of the engine and other parameters, but 80% is plausible for a high bypass turbofan. Bernoulli's principle has nothing to do with this.

You can see the accelerated airflow in the following animation:

Turbofan airflow
(source: Wikimedia)

The air coming from the engine core will move even faster, but there is less of it in a high bypass turbofan resulting in less total thrust coming from the core:

The fan airflow, referred to as the cold air stream, is accelerated by the fan and passes through the engine remaining outside of the engine core. The cold air stream moves much slower than the hot stream gas flow passing through the engine core.

(skybrary.aero)

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    $\begingroup$ "The air coming from the engine core will move even faster, but there is less of it in a high bypass turbofan resulting in less total thrust coming from the core" To expand on that: a jet engine's efficiency is (speaking simply) related to the change in velocity of the airstream. The greater the difference in velocity between intake and exhaust, the lower the efficiency (but the greater the thrust). Therefore, it is more efficient (and therefore cheaper) to give a much smaller velocity increase to the air, and use a much larger volume of air so that it adds up to the same overall thrust. $\endgroup$ – anaximander Dec 16 '19 at 10:54
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A bypass fan provides thrust in the same way a propeller provides thrust: by increasing the energy content of the gas mass passing through the disk.

The added energy is most effectively converted into thrust by allowing it to expand until internal pressure is equal to ambient pressure, so all added energy is converted into kinetic energy. This expansion takes place in the cowling behind the fan.

But the initial energy addition is a mixture of both increased pressure and increased velocity: the gas will require time to accelerate to the outlet velocity.

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It doesn't bypass everything, just the combustion chamber. Notably, the air still goes through a fan.

High-bypass turbofan engines make most of their thrust from the ducted fan. Very much like a turboprop except the blades are smaller and enclosed by the cowling.

@Harper's answer on What is the difference between turbojet and turbofan engines? explains that nicely: a high-bypass turbofan extracts most of the exhaust energy from the jet part and uses it to spin the fan, instead of exhausting really high speed air from the jet part directly.

Low-bypass turbofans have a mix of thrust from fan + jet, while a pure turbojet has zero bypass, just compressor blades and no fan.

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