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I know momentum of the airflow inside jet engine generates thrust, but what about the pressure difference in front of and at the back of the engine? The air sucking intake at the front has lower than atmospheric pressure and the exhaust has higher than atmospheric pressure, does this pressure differential contribute to the thrust generated by the engine in adition to the momentum thrust?

The high pressure zone push the engine from the back side and low pressure zone pulls it from the front. For example, if the exhaust has 1 square meter surface area, and the exhaust pressure is 2 atmosphere, then there is pressure equal to 10000 kilograms pushing it from the back since 1 atmosphere is ten thousands kilograms per square meter.

Which of these two is true?

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    $\begingroup$ Have you looked at any textbooks? Such as Engineering Thermodynamics Work and Heat Transfer by Rogers & Mayhew. $\endgroup$ – Solar Mike Nov 23 '19 at 6:00
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The easiest way to view the problem is to draw a control volume around the engine that is big enough so that the air pressure is constant on the boundary surface of the control volume and the upstream air velocity is just the free-stream velocity everywhere on the upstream boundary surface of the control volume. The downstream velocity is faster, and the difference in the momentum across the boundary can be equated to net thrust.

When you start asking about the forces and pressure distributions over the engine components, you need a more detailed near field flow model. The simplest that helps with your question is the actuator disk model. A result is that half of the flow acceleration happens in front of the disk, and half happens downstream of the disk as the jet continues to contract. So if you have cowls and ducts around a jet engine, they are in a region of accelerating flow. Upstream, the pressure is lower than freestream pressure, and carefully designed intake ducts contribute to thrust. Or perhaps a better way to say it is that tranferring all the engine thrust via high speed shaft bearings is doing things the hard way. Better is to design the rotating components of jet so that they are largely thrust neutral, and let the fixed stators and cowls and nozzles produce most of the reaction thrust.

So there is an inlet duct effect, but it is not in addition to the thrust computed for the momentum change, it is just a normal part of it.

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