Average flow velocity in turbulent tube flow.

$($a$)$ For the turbulent flow in smooth circular tubes, the function ${?}^1$

$\frac{\frac{?}{\text{b}\text{a}\text{r}}(v{)}_z}{\frac{?}{\text{b}\text{a}\text{r}}(v{)}_{z,\text{m}\text{a}\text{x}}}=(1-\frac{r}{R}{)}^{\frac{1}{n}}$

is sometimes useful for curve$-$fitting purposes: near $Re=4{10}^3,n=6$; near $Re=1\mathrm{.}1{10}^5,$

$n=7$; and near $Re=3\mathrm{.}2{10}^6,n=10.$ Show that the ratio of average to maximum velocity is

$\frac{(\frac{\text{:}}{b}ar(v{)}_z\text{:})}{\frac{?}{\text{b}\text{a}\text{r}}(\frac{?}{\text{b}\text{a}\text{r}}(v){)}_{z,\text{m}\text{a}\text{x}}}=\frac{2n^2}{(n+1)(2n+1)}$

and verify the result in Eq$.5\mathrm{.}1-5.$

$($b$)$ Sketch the logarithmic profile in Eq$.5\mathrm{.}3-4$ as a function of $r$ when applied to a circular

tube of radius $R.$ Then show how this function may be integrated over the tube cross section

to get Eq$.5\mathrm{.}5-1.$ List all the assumptions that have been made to get this result.