Problem $4(25$ points$)$

A long structural component of a bridge has an elliptical cross section shown in the figure. It is known that when a steady wind blows past this type of bluff body, vortices may develop on the downwind side that are shed in a regular fashion at some definite frequency.

Because these vortices can create harmful periodic forces acting on the structure, it is important to determine the shedding frequency. For the specific structure of interest, $D=0\mathrm{.}1m,H=0\mathrm{.}3m,$ and a representative wind velocity is $V=50k\frac{m}{h}r.$ Standard air can be assumed $($density $=1\mathrm{.}23k\frac{g}{m^3},$ viscosity $=1\mathrm{.}79{10}^{-5}Pa.s).$ The shedding frequency is to be determined through the use of a small$-$scale model that is to be tested in a water tunnel. For the model, $D_m=20mm$ and the water temperature is $20C($density ${}_m=998k\frac{g}{m^3},$ viscosity ${}_m={10}^{-3}Pa.s).$

a$)$ Using Buckingham $-$theorem, write a dimensionless functional relationship between the shedding frequency $(5^{-1})$ and the other parameters in the problem: $D,H,V,,$ and $.$

N$.$B$.$ For consistence, chose $D$ among the repeating variables. Note the existence of a dimensionless number called Strouhal number $St=\frac{D}{V}.$ Fot the $-$group corresponding to the dependent variable $,$ show the detail of your calculation. You can find the other $$ groups by inspection and experience.

b$)$ Determine the model dimension $H_m$ and the velocity $V_m$ at which tests should be performed to achieve complete similarity between the model and the prototyp.

c$)$ If the shedding frequency for the model was found to be ${}_m=49\mathrm{.}9s^{-1},$ what is the corresponding frequency for the prototype?