Rumbling - What could be the reason?

I would start with changing the rear differential oil and making sure it has the correct friction modifier for limited slip differentials.

 

Agreed
Symptoms coming on after a turn(particularly if after a long stretch of straight road)is indicative of the clutches in the differential chattering.

 

Do not overlook the bearing inside the torque tube where the splined end meets the diff. If you have never serviced it, it is most definately beginning to seize up, and cause you to lack power to the wheels and is virtually tearing the heart out of your automatic tranny pump, if you have an auto tranny. Jq.

 

Are you sure the hand brake isn't stuck half way on?
Go for a short ride and then compare the rim temperature to the rest.

 

I have experienced the same noise with my 365GT. It was a rumble accompanied by a shutter at the right rear. It only occurred after make a full stop and accelerating into a right turn but not a left turn. It was not noticeable until the car came up to full temp. As suggested here, I replaced the oil in the differential with limited slip 80-90W gear oil. It appeared that the differential had been over filled with a higher viscosity oil. The sound is almost completely gone, just a subtle occasional shutter when tuning right. Is there anything else I should check?

 

From your post it appears the condition has improved. The clutch packs in this type of limited slip consists of alternating plates of steel and steel plates covered with a friction type material. These packs usually have between 9 to 15 plates and have strong springs pushing them together. It can take a while for the new lube to fully work its way between all of those parts. Some shop manuals recommend finding a parking lot and doing a figure 8 pattern (multiple times) to help to work the new lube in place. You can also add some friction modifier to the lube if the problem continues. The downside of adding too much friction modifier is it will reduce the effectiveness of the limited slip.

 

Thanks for your advise. Since my posting the noise and shutter has returned, although still not as bad as before. Is the differential on its way out? Is there another component I should check?

 

I had rumbling in the right rear that turned out to be a bad wheel bearing.

 

Also check for a grumbling mother in law in the right rear passenger seat!