Problem $5$: $($a$)$ Find the governing non$-$dimensional Pi products for the case of the wind turbine

shown below. Note that $,,F_T$ are the shaft torque, angular

velocity, and axial thrust force, respectively.

u

$($b$)$ If we are to test a model turbine at $\frac{1}{100}$ scale of the full$-$

size turbine using water as the test fluid, how much faster or

slower would the free$-$stream velocity $($U$)$ of the model need to

be compared to the full$-$scale? For this velocity, how much

faster would the model need to rotate compared with the full$-$

scale? If the full$-$scale rotates at $17$ rpm $,$ do you think this is a

feasible rotational speed for the model? You may use ${}_{\text{w}\text{a}\text{t}\text{e}\text{r}}=$

$0\mathrm{.}001Pa**s$ and ${}_{\text{w}\text{a}\text{t}\text{e}\text{r}}=1000k\frac{g}{m^3}$ for the model and air

at standard conditions for the full$-$scale turbine.