A river$-$ bridge has an upstream approach section discharge of $1750\frac{m^3}{s},$ width of $95$ m and

flow depth of $3\mathrm{.}05$ m $.$ At the bridge section, one of the abutments protrudes into the river

channel causing contraction with the contracted section discharge being, $2125\frac{m^3}{s}$ and

channel width, $91\mathrm{.}5$ m $.$ The stretch mean flow depth before scour is $2\mathrm{.}95$ m $,$ mean approach

slope; $0\mathrm{.}0005,$ and mean sediment diameter; $3\mathrm{.}4$ mm $.$ A bridge pier of the shape shown is near

the channel centre from the other bank with minor abutment. The approach flow in

longitudinal direction is skewed at $150.$ The plan cross$-$sectional is of length, $L=2\mathrm{.}3m$ and

width, $b=1\mathrm{.}4m.$

$($a$)$ Determine the contraction scour depth due to the protruding abutment.

$($b$)$ Determine a local scour depth due to the bridge pier near the channel centre from the

other bank abutment.