Deriving dimensionless turbine equations with Buckingham Pi theorem

[Valgeirp]

I was wondering if someone is very well into the Buckingham Pi Theory and can assist me in understanding the outcome.
I have two dimensionless equations and I need to get an understanding on how they were made by the Buckingham Pi theory. The situation is considered for the reynolds number 10^5.
The two equations are

and

Where T is N*m force, p water densitykg/m^3, w is (rad/s), d is diamater (m)and u is velocity (m/s). The two variables that we are looking at are the relationship between the rotation (rad/s, w) and Force T (n*m)
Can someone help me understand how these two equations were made I have tried making them with the Buckingham Pi theory. Somehow I never end up with the answer I want.

 

[SteamKing]

I was wondering if someone is very well into the Buckingham Pi Theory and can assist me in understanding the outcome.
I have two dimensionless equations and I need to get an understanding on how they were made by the Buckingham Pi theory. The situation is considered for the reynolds number 10^5.
The two equations are
View attachment 89215 and View attachment 89216
Where T is N*m force, p water densitykg/m^3, w is (rad/s), d is diamater (m)and u is velocity (m/s). The two variables that we are looking at are the relationship between the rotation (rad/s, w) and Force T (n*m)
Can someone help me understand how these two equations were made I have tried making them with the Buckingham Pi theory. Somehow I never end up with the answer I want.

First of all, T is not a force. It is a torque if it has units of N-m.

Second, Newtons (N) are derived units under the SI system.