$(12$ points, Estimated time $10$min $)$ A venturi meter is used to measure air velocity. A onefifth scale model of the venturi meter is built, and water is used as the test fluid. Assume water is at $15\mathrm{.}6C$ and air is approximately the same temperature.

$($a$)$ If use of a venturi meter implies pipe flow, what common dimensionless group should be used to perform a scaling analysis?

$($b$)$ Use your answer from $($a$)$ to determine the approximate ratio of the model velocity to the air velocity in the full$-$scale meter.