E2 Questions on Rapidly Varied Flow  Sudden transitions and Hydraulic Jumps

E2.1
For a trapezoidal channel with a $\text{base width}=3.0m$, and side slope 1 vertical 2 horizontal, calculate the critical depth if the discharge is $Q=10m^{3}/s$.
(Answer: 0.86m) 
E2.2
Water is flowing at a normal depth in a $3m$ wide rectangular channel with a bed slope of 1:500. If Manning’s $n=0.025$ and the discharge is $5m^{3}/s$. Calculate the height of a bump that would produce the critical flow without causing backwater upstream (i.e. without raising the upstream water level). (Answer: $y_{n}=1.215m$, Bump height $=0.326m$)

E2.3
Water is flowing at a velocity of $3.4m/s$ and a depth of $3.4m$ in a channel of rectangular section with a width of $3.4m$. Find the changes in depth produced by

(a)
A smooth contraction to a width of 3.0m

(b)
The smallest allowable contraction for the flow to be possible upstream as described.
(Answer: (a) $y_{2}=3.048m$, (b) $y_{2}=2.89m$)

(a)

E2.4
The normal depth of flow in a rectangular channel (with a 5m wide bases and 2m high side walls) is 1m. It is laid to a slope of 1m/km with Manning’s $n=0.02$. Some distance downstream there is a hump of height 0.5m on the stream bed. If critical flow occurs on the bump, determine the depth of flow ($y_{1}$) immediately upstream of the bump and the depth of flow ($y_{2}$) above the bump. If the bump is reduced to 0.1m, what values will $y_{1}$ and $y_{2}$ be?
(Answer: $Q=6.32m^{3}/s$, $y_{1}=1.25m$, $y_{c}=0.54m$, $y_{1}=1.0m$, $y_{2}=0.87m$) 
E2.5
Water is flowing at a rate of $10m^{3}/s$ through a rectangular channel $4m$ wide, at a depth of $0.5m$. A weir downstream causes the water to back up the channel and a hydraulic jump occurs. Find the sequent depth and the loss of energy at the jump.
(Answer: $y_{2}=1.37m$, $\Delta E=0.238m$) 
E2.6
Water flows in a rectangular channel at a depth of 30cm and with a velocity of $16m/s$. If a downstream sill forces a hydraulic jump, what will be the depth and velocity downstream of the jump? What head loss is produced by the jump?
(Answer: $y_{2}=3.81m$, $V_{2}=1.26m/s$, $\Delta E=9.46m$) 
E2.7
Water passes under a sluice gate in a horizontal channel of width 2m. The depths of flow on either side of the sluice gate are $1.8m$ and $0.3m$. A hydraulic jump occurs a short distance downstream. Assuming no energy loss at the gate, calculate:

(a)
The force on the gate

(b)
The depth of flow downstream of the hydraulic jump

(c)
The fraction of the fluid energy that is dissipated in the jump
(Answer: $F=15.7kN$, $y_{3}=1.22m$, Fraction of energy lost (dissipated) $=28\%$)

(a)