Just to add why your initial calculation came out with 1090mph
The formula you gave is
Downwash Velocity (ft/sec) = √Weight (lbs) ÷ 2 x (air density) x RotorDiskArea (sq. ft.)
Whereas you calculated
√31,085*.002378*8,656.8
If you calculate
√31,085
÷(.002378*8,656.8) you get 38.8 ft/s as The Coyote established earlier. So you're science was right ...
As you also said, in ideal flow the veloctiy will double by the time it gets out of the rotor influence which you need to consider.
I'm sorry I don't have any Mi26 specific data but typical induced velocities for a range of helicopters are between 30 and 65 ft/s.
If you are using this data to carry out any safety related calculations then I would suggest you need much more than the physics used above. These are free stream type approximations. As soon as you get near the ground it gets very different. It's not just a case of the velocity of the air but also the sheer amount of air that is being moved that causes problems ... when close to the ground this all has to go somewhere.