Some people are saying that after having driven a car with a turbo-charged engine, you need to let the engine run for several seconds (or even minutes) before switching off the engine. But nobody is able to explain why, which makes me wonder if it's even true at all.
Is there any kind of evidence, or technical explanation that it is actually beneficial to respect such a cooling down period?
Claim:
Why does a turbo engine require a “cooling down” period?
It doesn't.
Assuming we're talking about the typical turbocharger fitted to many current models of fairly ordinary car.
Should I leave my engine ticking over before it is turned off ?
Not for normal every day driving, but still worthwhile if the engine has been under load or raced before being turned off. e.g. Towing a caravan or after climbing a long incline.
...
What is a water-cooled turbo ?
The central part of the turbo, housing the bearings, is surrounded by a water jacket through which the engine’s water coolant is passed. This water continues to circulate after the engine is turned off, cooling the turbo, and preventing heat soak.
According to a major manufacturer of turbochargers
Following a hot shutdown of a turbocharger, heat soak begins. This means that the heat in the head, exhaust manifold, and turbine housing finds it way to the turbo’s center housing, raising its temperature. These extreme temperatures in the center housing can result in oil coking.
To minimize the effects of heat soak-back, water-cooled center housings were introduced. These use coolant from the engine to act as a heat sink after engine shutdown, preventing the oil from coking. The water lines utilize a thermal siphon effect to reduce the peak heat soak-back temperature after key-off
No, there is no need to run a turbocharged engine after stopping the vehicle - unless you have an older vehicle with an early type of turbocharger and have been pushing the engine to its limits for an extended time. Check the owner's manual for your car to see if the manufacturer recommends any special procedure, if it doesn't, you can assume none is needed.
TLDR: Yes, some turbochargers need time to decompress otherwise they could be damaged.
Perhaps my answer only applies to old turbocharged vehicles. My experience has only been with an early 90s Nissan Exa and an early 00s Subaru WRX and both of those had a "turbo timer". With the timer the car would continue to run for about 1 minute after switching off the ignition (and removing the key). My understanding was that this was normal for turbo cars.
Over Pressure
Turbos can generate a lot of pressure when it's running. That pressure needs to have somewhere to go. When the engine is running it just blows out the exhaust, but when the engine just stops there is a buildup of pressure (not enough to crack a piston head like I've been told but), enough to blow backwards through the compressor, damaging it.
Here's an excerpt from wikipedia
In this situation, the surge can raise the pressure of the air to a level that can cause damage. This is because if the pressure rises high enough, a compressor stall will occur, where the stored pressurized air decompresses backward across the impeller and out the inlet. The reverse flow back across the turbocharger causes the turbine shaft to reduce in speed more quickly than it would naturally, possibly damaging the turbocharger.
Turbo Lubrication and Excess Wear
The engine feeds lubricant to the turbo. With the engine off the turbocharger isn't being lubricated.
This from turborepair.com:
Once you turn off the engine in your car, the turbocharger will continue to spin for up to a minute or more and during this time oil is not being delivered to the turbochargers bearings, causing wear.
This from myturbodiesel.com
If you were driving hard and hot, a 1 minute idling period or a few minutes of sensible driving before shut down should be enough to let the turbo cool down and receive fresh oil. If the turbo is too hot and does not receive cooler oil upon shutdown, the oil could become burnt and "coking" may occur.
This from allpar.com
Hot shutdowns cause extensive deposits of carbon and shellac on the turbine end. As the deposits break up and flow into the oil they score and wear the bearing bore, bearing and shaft journal.
The twist
Turbos don't run all the time, they only run when they need to. Which means if you've been coasting around a car-park for a minute, the turbo isn't running and there is no pressure buildup. In this case you don't need the turbo timer, you can shut off your engine without waiting.
Sources
You can read about turbos here and here's some rev-heads talking about turbo cooldown and how important it is.
Edit
RedGrittyBrick's answer mentions 2 links:
turbotechnics.com (a manufacturer) say allowing the turbo to decompress is "worthwhile" in some instances.
turbobygarrett.com (also a manufacturer) say hotstopping can "result in oil coking". It also says water cooled systems can reduce this, but not all turbos are water cooled.
Many Garrett turbos are water-cooled for enhanced durability
