B$1.$ As shown in Figure B$1,$ a $5m$ wide spillway with a bed slope of $0\mathrm{.}4$ conveys a $50\frac{m^3}{s}$ discharge from a reservoir. At the end of the spillway, a stilling basin of equal width is designed to dissipate energy carrying by the supercritical flow. A $0\mathrm{.}5m$ high weir is used to control the tail water. The Manning coefficient is $0\mathrm{.}02.$ A hydraulic jump is formed at the stilling basin and the flow becomes subcritical with a depth of $y_3=4m$ after the jump.

$($a$)$ Near the spillway entrance at A$,$ give two types of flow profiles that are likely to occur before the flow establishes a normal depth;

$[4$ marks$]$

$($b$)$ Calculate the normal depth $(y_1)$ of the flow through the spillway;

$[4$ marks$]$

$($c$)$ Calculate the water depth $(y_2)$ before the hydraulic jump and the energy loss dissipated by the jump;

$[5$ marks$]$

$($d$)$ Calculate critical water depth $(($:$y_c\}$ if the flow above the weir crest is critical;

$[2$ marks$]$

$($e$)$ Calculate the force on the weir according to $y_3$ and $y_c,$ assuming the water surface approaching the weir is nearly horizontal and the energy losses can be ignored.

$[10$ marks$]$

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