A$)$ When fluid flows around a circular cylinder whose axis is perpendicular to the free stream, eddies $($vortices$)$ are formed behind the cylinder at a frequency of $\backslash ($ n $\backslash ).$ The eddy shedding frequency depends on a number of factors such as the size of the cylinder, speed of the free stream, density and kinematic viscosity of the fluid. To investigate the variation of eddy frequency, perform dimensional analysis.

a$.$ Using the Buckingham$-$PI theorem, determine the non$-$dimensional parameters relevant to this problem.

b$.$ Explain briefly why Buckingham$-$PI theorem is useful and discuss the relations you obtained.

B$)$ Consider an airplane landing at a speed of $\backslash (50\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ at sea level with standard atmospheric conditions of $1$ atm and $\backslash (15^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ If a $\backslash (1/5\backslash )$ th scale model is tested in the wind tunnel at $10$ atm and $\backslash (15^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ it is found that the loads on the horizontal tail is subject to impulsive fluctuations at a frequency of $20$ cycles per second, due to eddies being shed from the wing$-$fuselage junction.

a$.$ Compare the eddy shedding frequencies for the full$-$scale airplane and scaled model.

b$.$ If the natural frequency of vibration of the horizontal tail of the airplane is $8\mathrm{.}5$ cycles per second, could this represent a dangerous condition?

c$.$ Also, discuss the conditions for the scaled model which is tested in the wind tunnel at $1$ atm and $\backslash (15^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$