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I'm especially interested how long it would take to restart a turbofan engine after a hypothetically erroneous shutdown of an engine immediately after takeoff i.e., at an altitude at which the air is reasonably dense. Is windmilling enough or would you need crossbleed and starters again?

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It can take from a few seconds to eternity. If the core cools down, the engine might lock up.

Windmilling starts are only possible if the aircraft is flying fast and low enough. It will assist the startup process, but needs additional help from the starter or bleed air, typically below 300 knots and above 20,000 ft. Details vary from type to type. If everything works as designed, a windmilling assisted restart is faster than a regular engine start.

The ability to restart in flight must be proven as part of the certification process. However, right after takeoff the speed is too low for a pure windmilling start, and an additional energy source is needed.

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    $\begingroup$ So it seems that for most/all commercial aircraft, starter or bleed air would be necessary, since the question was a hypothetical shut down immediately after take off, and the Concorde was the only (one of the exceptionally few?) commercial aircraft to take off at over 200 knots. $\endgroup$ – FreeMan Mar 23 '15 at 12:27
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    $\begingroup$ @FreeMan: Right, I added a sentence which confirms this. I would expect that the Cocorde needed even more speed for starting the Olympus 593 by windmilling. Same goes for the Tu-144 and its Kusnetzovs, btw. $\endgroup$ – Peter Kämpf Mar 23 '15 at 15:56
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    $\begingroup$ @Freeman: How is bleed air used for relighting? How does bleed air get to the inoperative engine? $\endgroup$ – user7241 Mar 27 '15 at 22:20
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    $\begingroup$ @jjack That's a good question. Why don't you go ahead and ask a new one. $\endgroup$ – FreeMan Mar 28 '15 at 12:19
  • $\begingroup$ @PeterKämpf: Actually, it would probably need less speed for a windmill restart than the engines on most jetliners, since the Olympus 593 is a turbojet (where all the air entering the engine passes through the core - that being the entire engine for a turbojet - and helps to spin the compressor and turbine rotors), whereas modern subsonic jetliners use high-bypass turbofans (where at most 20% of the air coming in passes through the core, and the remaining 80%+ passes uselessly through the bypass duct). $\endgroup$ – Sean Sep 28 at 0:13
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Right after takeoff bleed air would be necessary to restart, preferably from the APU if thrust reduction on the operating engine is unacceptable in the current phase of flight.

I only have data for the Rolls Royce Allison AE3007 A series engines and the windmilling envelope at all altitudes is between 260 KIAS and Vmo/Mmo with the additional limitation that N2 must be above 10%. An aircraft that has just taken off will not be going this fast so an assisted (bleed air) start is the only viable option.

How long the start will take depends on the particular engine, but all turbines (that I'm aware of) have a maximum start time limitation. The engine discussed above on the EMB-145 has a 60 second limitation on starting. Based upon my experience starting the engines with a stiff headwind, I'd assume that an APU start at flying speeds would probably take around 35-40 seconds in the EMB-145. By the book the engine needs another minute before significant thrust is applied.

All of the above information is specific to one airframe/engine combination and will vary for other aircraft and engines.

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    $\begingroup$ I think the way engines are started differs from engine to engine. Some have airturbine starters, others use hydraulic or electric power. $\endgroup$ – user7241 Mar 28 '15 at 16:32

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