E2 Questions on Rapidly Varied Flow - Sudden transitions and Hydraulic Jumps

1. 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 $Q=10m^{3}/s$.

2. 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 $=0.326m$)

3. 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

1. (a)

A smooth contraction to a width of 3.0m

2. (b)

The smallest allowable contraction for the flow to be possible upstream as described.

(Answer: (a) $y_{2}=3.048m$, (b) $y_{2}=2.89m$)

4. 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: $Q=6.32m^{3}/s$, $y_{1}=1.25m$, $y_{c}=0.54m$, $y_{1}=1.0m$, $y_{2}=0.87m$)

5. 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: $y_{2}=1.37m$, $\Delta E=0.238m$)

6. 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: $y_{2}=3.81m$, $V_{2}=1.26m/s$, $\Delta E=9.46m$)

7. 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:

1. (a)

The force on the gate

2. (b)

The depth of flow downstream of the hydraulic jump

3. (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) $=28\%$)