This car has was diagnosed with a defective engine coolant pump. Symptoms were overheating and noise from the front timing belt area. Disassembly of the front cover and timing belt, no easy task on at Audi TT, allowed for removal and inspection of the pump. As you can see from the photos, it is a bit of a process to access the pump, so be sure to always replace the timing belt and tensioner at the same time.
The timing belt, scheduled by Audi for replacement at 105K miles, has a history of failing considerably sooner, so consider timing belt service at 70 or 75,000 miles. The mode of failure with this pump appears to be a simple design flaw. The water pump impeller is made from a plastic material, the plastic becomes brittle with age and extended immersion in the hot engine coolant, and simply cracks apart. The defective pump, and pieces of the failed impeller, are on the left side of the photo, the new pump on the right. As this car is an aluminum alloy engine, Its always a good policy to flush the cooling system at least every two years, or 30,000 miles, and to use a quality antifreeze with corrosion inhibiters.