my car has been giving off a light howling/whirring noise from the engine on cold start ups. it makes noise for like 30 seconds and then calms down when the engine revs drop. it doesnt make the noise later in the day when the engine has already been warmed up.

any ideas?

 

Well I do the easy stuff first. Remove the SC belt and see what the noise does.
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My 2010 XKR makes the same noise. I was watching a couple of YouTube videos on unrelated subjects and those cars make the same noise. Maybe it is normal. Is there a vacuum pump that runs during cold start perhaps?

 

On the 5.0L engines the vacuum pump is gear driven and in operation 100% of the time the engine is running.
I do notice the DI stuff is MUCH noisier when first starting and then quickly quiets down in several minutes. Usually only on cold starts.
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I have a 2013 XF 3.0 AWD SC and when a noise as you describe started with mine it turned out to be the early stages of a supercharger failure.

 

Check EGR

 

I had the same sound, Turned out being the idler pulley below the supercharger pulley. I removed belt. Checked pulleys by hand, most were either too stiff or too loose. ended up replacing all pulleys, rebuilding some, and replacing others. Being ambitious, At the same time, I went ahead and removed the whole supercharger. Replaced the pcv diaphragms , knock sensors, all plastic coolant lines (rear pipe, front pipes, etc), New water pump, new thermostat housing, all new lines and little bits associated with that replacement - these sometimes comes as a kit on amazon/rock auto. I cleaned intake valves, cleaned carbon from oil coolers, replaced all gaskets, changed SC oil, Changed SC coupler (new oem - NOT green solid).

Workshop Manual, YouTube videos, forum writeups absolutely needed.