What kind of turbine? Steam or gas?
I have built watercooled exhaust stacks for helicopter-size gas turbines used in speedboats.
With 100 gpm of water forced through the jacket and then injected into the gas stream, you could stick your hand in the exhaust stream briefly and not get burned. Without the water injection, I wouldn't dare.
Absent extra injected water, here's still a bit more (combustion product) water coming out than fuel going in, but there's not enough to cool the gas stream much, and it's not cool enough to become visible steam; the gas is damn not, and the gas mass flow is impressive; maybe ten times as high as a Diesel that would use the same size exhaust pipe. Peak gas velocity in a Diesel stack would not exceed 100 mph. Peak velocity in a gas turbine stack probably wouldn't go below 100 mph. Heat transfer from the plume to anything in it would fall into the realm of 'forced convection', and the boundary layer would be pretty thin.
Gas turbines also become very unhappy with long stacks or tight elbows, and manifest their displeasure in expensive ways, most leading to self destruction.
Exhaust plume temperature profiles are commonly found in helicopter pilots' handbooks. ISTR the kill zone extends beyond the rotors, but you should check.
I have no experience with steam turbines.
Mike Halloran
Pembroke Pines, FL, USA