Question $1$: Open Channel Flows $-$ Total and Specific Energy

For a rectangular channel having a very gentle slope, state what is the difference between Total Energy and Specific Energy.

$[1]$

For a rectangular channel of width $b,$ conveying a total volumetric discharge $Q,$ give an expression for the Specific Energy $(E),$ in terms of the flow depth $d,$ and the specific discharge $q.$

$[4]$

Give an expression for the minimum value of specific energy $E_{\text{m}\text{i}\text{n}}$ and hence derive an expression for the Critical Depth $d_c.$ Use this result with the equation from $($ b $)$ to show that the Froude Number at this condition is equal to $1\mathrm{.}0.$

$[5]$

A sluice gate is lowered into a rectangular channel. The velocity of flow upstream is $2\mathrm{.}0\frac{m}{s}$ and the Specific Energy upstream is $2\mathrm{.}2$ m $.$

a$.$ Calculate the depth upstream $($ y $1)$

$[4]$

b$.$ Calculate the Specific Discharge $($discharge per unit width$)$

$[2]$

c$.$ Calculate the Critical Depth for this flow

$[2]$

d$.$ Is the flow upstream Sub$-$Critical? Or Super$-$Critical?

$[4]$

e$.$ If the gate opening $(yg)$ is set at $1\mathrm{.}4$ m $,$ will the water depths upstream be different from those downstream?

$[3]$