Turbine engines such as the PT6 that have inertial separators have a reputation for being able to deal with intake water very well. How much can they handle? Do the manufacturers torture-test their engines to see how they will operate under wet conditions such as heavy rain?
Of reciprocating, turboprop, or turbojet engines, which type handles large volumes of water most effectively?
Turbofans are the best at taking in a large volume of water. Almost all of it stays in the bypass stream, as it's deflected away from and around the core's inlet. Most turbofans are designed to Part 121 requirements, calling for reliable operation at 20 g/m^3 of water, well beyond typical rain conditions.
As an example, when an A380 had its engine control wires cut, it took 2 hours for the fire crews to drown the engine (page 9 on paper, 19 pdf). Of course, a dip in water would've done that easily, but it's next to impossible to get that much rain.
20 g/m^3 is about the equivalent of delivering about 300 gpm (20L/s) of water into a large jet engine's airflow. That's two fire lines, but not all of the water gets in. This is the amount where the engine is certified to work, drowning it takes more.
A related question goes into more detail: How does a jet engine handle suddenly entering a lot of rain?
Turboprops also deflect the flow of water in flight around the spinner. Some rely on inertial separators, used at takeoff, landing, and in the clouds, for better protection. Turbojets don't have that feature, but haven't experienced a lot of flameouts in practice.
Piston engines also avoid taking in water by having their intake face forward or below rather than above. They're no different from road cars, which also run fine in the rain.
But that applies to ordinary rain that falls on the ground. Conditions in low-altitude clouds, especially on the edge of freezing, can be considerably worse. Reliable operation of piston engines in such is not certain and not recommended.
You may wish to start by looking at a maximum amount of rain possible per given volume of air. Then evaluate air flow into the turbine, as any circular motion will tend to sling droplets to the side.
As turbines do not run anywheres near stoichiometric ratio, water intake generally will not affect the burners enough to either deprive them of oxygen or heat. Water expands to steam at a ratio of 1 to 1600. Using 20 g water per meter$^3$ as a test criteria, evaporating rainwater in the compressor would change the air composition from 20 parts per million water to:
20 ppm × 1600 steam volume/liquid water volume = 32,000 cubic centimeters of water vapor, or around 3%.
Add on another 20 g per meter$^3$ to account for high relative humidity and you have roughly 6 % water vapor to 94 % nitrogen/oxygen atmosphere entering the combustion chamber. Plenty of oxygen to burn fuel.
Since water vapor requires more energy to raise its temperature than dry air, fuel consumption increases slightly. Water vapor being lighter than air reduces mass flow, but water in the compressor intake will increase density by lowering temperature increase during compression.
Even in torrential rains, total moisture content is not enough to significantly affect the operation of the turbine. Hurricane hunters routinely fly through the worst of tropical downpours unscathed.
Interestingly, a reciprocating (piston) engine with a turbocharger will be less affected than an naturally aspirated engine under these conditions, as the turbocharger compression will help prevent icing by heating the intake air.
