Problem $5$ A student team is to design a human$-$powered submarine for a design competition. The overall length of the prototype submarine is $4\mathrm{.}85$ m $,$ and its student designers hope that it can travel fully submerged through water at $0\mathrm{.}440\frac{m}{s}.$ The water is freshwater $($a lake$)$ at $T=15C.$ For water at $T=15C,=999\mathrm{.}1k\frac{g}{m^3}$ and $=1\mathrm{.}138{10}^{-3}k\frac{g}{m}*s.$

$($a$)$ The design team builds two models. Both models are geometrically similar to the prototype. The first one is one$-$fifth $(\frac{1}{5})$ scale model and the second one is one$-$twenty fourth $(\frac{1}{24})$ model. Both models are tested in their university's wind tunnel. A shield surrounds the drag balance strut so that the aerodynamic drag of the strut itself does not influence the measured drag. The air in the wind tunnel is at $25C$ and at one standard atmosphere pressure. At what air speed do they need to run the wind tunnel in order to achieve similarity for both models? For air at $T=25C,=1\mathrm{.}184k\frac{g}{m^3}$ and $=1\mathrm{.}849{10}^{-5}k\frac{g}{m}*s.(10\%)$

$($b$)$ From the air speed results in $($a$),$ do you notice any anything disturbing or suspicious about your results? Which model will you choose in order to achieve better dynamic similarity? Please state any reason for your choice. $(10\%)$

$($c$)$ The students choose the scaled model in $($b$)$ and measure the aerodynamic drag on the model submarine in the wind tunnel. They are careful to run the wind tunnel at conditions that ensure similarity with the prototype submarine. Their measured drag force is $5\mathrm{.}70$ N in wind tunnel test. Estimate the drag foon the prototype submarine at the conditions given in $($a$).(10\%)$

$2/2$