E4 Questions on Gradually varied flow: Flow Transitions.

1. E4.1

A wide channel consists of two long reaches. The upstream reach has a bed slope of 1 in 100 and the downstream reach has a bed slope of 1 in 1500. The Chezy C for both lengths can be assumed constant and equal to 45. Determine the profile that occurs when the flow per unit width is $7.5m^{3}/s/m$ and determine whether a hydraulic jump occurs and if one occurs which reach of the channel it occurs.
(Answers: $y_{n1}=1.406m$, $y_{n2}=3.467m$, $y_{c}=1.79m$. Jump in reach 1.)

2. E4.2

A long wide channel consists of an upstream section with a bed slope of 0.025 and a downstream section with a bed slope of 0.0002. The Chezy C is constant for the whole of the channel and equal to 40 in SI units. When the discharge in the channel is $3m^{3}/s/m$. width demonstrates that a hydraulic jump occurs. Determine in which reach of the channel the jump occurs. Complete your answer with a free-hand sketch of the final profile and label each of the gradually varied profiles.
(Answer: $y_{n1}=0.609m$, $y_{n2}=3.041m$, $y_{c}=0.972m$, $y_{n1\;conjugate}=1.459m$, $y_{n1\;conjugate}=0.187m$ )

3. E4.3

A wide channel has three reaches each with a different slope. Each can be considered to be long and at the downstream end of the three reaches there is a free outfall. For a specific case, the Chézy C value for all reaches is 32 and the flow per unit width is $1m^{3}/s/m$. The upstream reach has a slope of 1-in-500, the middle reach has a slope of 1-in-50 and the downstream reach has a slope of 1-in-2,000. Determine which flow profile occurs.
(Answer: $y_{n1}=0.787m$, $y_{n2}=0.366m$, $y_{n3}=1.250m$, $y_{c}=0.467m$, $y_{n2\;conjugate}=0.586m$, $y_{n3\;conjugate}=0.119m$ )

4. E4.4

A wide channel has two reaches with different bed slopes changing from $s_{o}=0.01$ to $S_{o}=0.0005$. Both reaches have the same cross-section shape and bed roughness of $n=0.03$. Sketch the critical flow depth $y_{c}$ line, uniform flow depth $y_{n}$ lines, and water surface profile in the channel and identify the type of water surface profile, if the flow discharge per unit width is $q=0.8m^{3}/s/m$.
(Answer: $y_{n1}=0.425m$, $y_{n2}=1.043m$, $y_{c}=0.403m$, M1 profile )

5. E4.5

A wide channel consists of three long reaches. The upstream reach has a bed slope of 1 in 500, the middle reach a slope of 1 in 50, and the downstream reach slope of 1 in 1,750. The Chezy C for all three lengths can be assumed constant and equal to 27. Determine the profiles that occur if the flow per unit width is $0.75m^{3}/s/m$.
(Answer: $y_{n1}=0.728m$, $y_{n2}=0.338m$, $y_{n3}=1.105m$, $y_{c}=0.386m$, $y_{n2\;conjugate}=0.438m$, $y_{n3\;conjugate}=0.087m$. Jump in reach 2 (middle). )