The are designed to handle huge amounts of water. Look at [https://www.quora.com/Can-rain-cause-a-jet-engine-to-flame-out](https://www.quora.com/Can-rain-cause-a-jet-engine-to-flame-out) for the test image of a jet engine and water.
A jet engine can perform better with water the just dry air. [https://en.wikipedia.org/wiki/Water_injection_(engine)](https://en.wikipedia.org/wiki/Water_injection_(engine)) has been used historically to get higher energy performance. You can use more fuel without them overheating. The mass of the exhausters also increases and you get better trust. So some jet airplanes from 1950-60 was designed with water tanks and an engine water injection system. Early Boeing 707 and Boeing 747 use that. B-52 bombers still have that, the engine they use are very old just like airplanes.
It is not just jet engines, water injection is possible in internal composition engines in a car too. Some WWII prison engine fighters did have water injection.
It is not always just wate but a 50-50 water and alcohol mixture is common
The most common solution is simply use the spinning low pressure compressor to flung water out to the bypass duct, the core intake is much drier after this.
The engines are tested with huge amount of water ingression under power before they are tested.
The bypass design actually meant that it’s easier to pass the shower test under high power settings than low power or idle. At low power, the flames are less stable and the fan turns slower, making more water flooding into the core. There has been flameout incidents in this situation before low/idle test became a requirement as well.
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