$5.(10$ pts$)$ A tall cylindrical chimney with a circular cross$-$section of diameter $\backslash ($ D $\backslash )$ and height $\backslash ($ H $\backslash )$ is subject to steady winds. It is observed that under these conditions, vortices form and are shed from the downwind side of the chimney at a frequency that can cause structural resonance. The vortex shedding frequency $\backslash ($ f $\backslash )$ depends on the diameter $\backslash ($ D $\backslash ),$ height $\backslash ($ H $\backslash ),$ wind speed $\backslash ($ V $\backslash ),$ as well as the density $\backslash (\backslash$rho $\backslash )$ and viscosity $\backslash (\backslash$mu $\backslash )$ of the air. Use dimensional analysis $($the Buckingham $\backslash (\backslash$Pi $\backslash )$ method$)$ to show that

$\backslash [$

S t$=\backslash$phi$\backslash$left$(\backslash$operatorname$\{$Re$\},\backslash$frac$\{$H$\}\{$D$\}\backslash$right$)$

$\backslash ]$

where $\backslash (\backslash$phi $\backslash )$ denotes a functional dependency. The Strouhal number is

$\backslash [$

S t$=\backslash$frac$\{$f D$\}\{$V$\}$

$\backslash ]$

You may assume that one of the dimensionless groups is the Reynolds number and do not need to derive it$.$ Hint: In selecting the repeating variables, use $\backslash ($ D $\backslash )$ as the reference length.