CIVE3415: Water Engineering III
Steady Open Channel Hydraulics
EXAMPLE SHEETS
Dr Andrew Sleigh
School of Civil Engineering
University of Leeds
October 2022
This version was created on:
Contents
 E1 Questions on Uniform Flow and Critical Flow.  Calculation of normal and critical depth.
 E2 Questions on Rapidly Varied Flow  Sudden transitions and Hydraulic Jumps
 E3 Questions on Gradually varied flow  including integration of the backwater curve.
 E4 Questions on Gradually varied flow: Flow Transitions.
 E5 Relevant Equations of Steady Open Channel Flow
E1 Questions on Uniform Flow and Critical Flow.
 Calculation of normal and critical depth.

E1.1
A rectangular channel is $3.0m$ wide, has a 0.01 slope, flow rate of $5.3{m}^{3}/s$, and $n=0.011$. Find its normal depth ${y}_{n}$ and critical depth ${y}_{c}$.
(Answer: ${y}_{n}=0.4m$, ${y}_{c}=0.683m$) 
E1.2
Water flows in a long rectangular channel at a depth of $1.22m$ and discharge of $Q=5.66{m}^{3}/s$. Determine the minimum channel width if the channel is to be subcritical.
(Answer: $B=1.34m$) 
E1.3
A rectangular channel has a bottom width of 8 m and Manning’s n=0.025

(a)
Determine the slope at a normal depth of 2m when the discharge is $12{m}^{3}/s$

(b)
Determine the critical slope and the critical depth when the discharge is $12{m}^{3}/s$

(c)
Determine the critical slope at the critical depth of 1.5m and compute the corresponding discharge.
(Answer:(a) ${S}_{o}=0.0002$, (b) ${S}_{c}=0.009$, ${Q}_{c}=0.61m$, (c) ${y}_{c}=0.008$, $Q=46.1{m}^{3}/s$)

(a)

E1.4
For a trapezoidal channel with a base width $b=3.0m$, Manning’s $n=0.025$ and side slope $x=2$ (i.e. 1 vertical: 2 horizontal), calculate the critical depth, critical velocity and critical slope if its discharge $Q=10{m}^{3}/s$.
(Answer:${y}_{c}=0.86m$, ${V}_{c}=2.46m/s$, ${S}_{oc}=0.0076$) 
E1.5
A rectangular channel $9m$ wide carries $7.6{m}^{3}/s$ of water when flowing $1.0m$ deep. Work out the flow specific energy. Is the flow subcritical or supercritical?
(Answer: ${E}_{s}=1.04m$ and flow is subcritical) 
E1.6
Two engineers observed two rivers and recorded the following flow parameters: River 1: flow discharge $Q=130{m}^{3}/s$, flow velocity $V=1.6m/s$, water surface width $B=80m$; River 2: flow discharge $Q=1530{m}^{3}/s$, flow velocity $V=5.6m/s$, water surface width $B=90m$. Decide the flow regime of two rivers, i.e. subcritical or supercritical.
(Answer: River 1 is subcritical and River 2 is supercritical) 
E1.7
A concrete, trapezoidal channel has a bottom slope of ${S}_{o}=0.0009$ and a Manning roughness factor of $n=0.013$. The bottom width of the channel is $b=2.5m$, and the side slopes are 1 in 2. Determine the velocity and discharge when the flow is normal at a depth of $1.8m$. (Answer: $v=2.37m/s$, $Q=26{m}^{3}/s$)

E1.8
A trapezoidal channel has a bottom slope of slope of ${S}_{o}=1$ in $40$ and a Manning roughness factor of $n=0.016$. The bottom width of the channel is $b=6.0m$, and the side slopes are 1 in 3. Determine the normal depth in this channel for $Q=42.3{m}^{3}/s$. (Answer: ${y}_{n}=0.75m$).

E1.9
A trapezoidal channel has side slopes of 1:3/4 and the slope of the bed is 1 in 2000. Determine the optimum dimensions of the channel if it is to carry water at $0.5{m}^{3}/s$. Use the Chezy formula, assuming that $C=80{m}^{1/2}/s$. (Answer: ${y}_{n}=0.552m$, $b=0.552m$).

E1.10
The cross section of a stream can be approximated by the compound channel shown in figure 1. The bottom slope is ${S}_{o}=0.0009$. The Manning roughness factor $n=0.025$ for the main channel and $n=0.035$ for the overbank areas. Determine the normal depth for a discharge of $197{m}^{3}/s$. Also calculate the energy coefficient $\alpha $ and the momentum coefficient $\beta $ for the channel with this flow condition.
(Answer: ${y}_{n}=5.5m$, $\alpha =1.23$, $\beta =1.09$.)

E1.11
The total width of the channel considered in Question E1.10 is to be decreased by reducing the overbank portions symetrically; however, this reduction must not cause an increase of more than $0.15m$ in the flow depth for the discharge of $197{m}^{3}/s$. Assuming normal depth still is present in the chanel, determine the minimum allowable channel total width, $B$. (Answer: $B=16.6m$.)

E1.12
The flow discharge through a rectangular channel $4.6m$ wide is $11.3{m}^{3}/s$ when the slope is 1:100. Is the flow subcritical or supercritical? Calculate the slope, ${S}_{c}$, that would give critical depth. The Manning roughness coefficient is $n=0.012$. (Answer: supercritical, ${S}_{c}=0.0023$).

E1.13
An open channel with $n=0.011$ is to be designed to carry $1.0{m}^{3}/s$ of water at a slope of 0.0065. Find the most efficient cross section for a rectangular section. (Answer: $b=2y=0.946m$).

E1.14
The crosssection of a river may be idealized as shown in Fig. 2. Determine the discharge carried by the river when its bed slope is ${S}_{o}=2\times {10}^{4}$ and the Manning’s $n=0.02$.

E1.15
For the channel section shown in Fig. 2, estimate the discharge if the channel slope is 0.002.

E1.16
A rectangular channel has width $B=3m$ and normal depth $y=1m$. What is the diameter of a semicircular channel that will have the same discharge as in the rectangular channel, when flowing just full in uniform flow. Assume that $n$ and ${S}_{o}$ are the same in the two cases. Compare the two wetted perimeters.

E1.17
What are the dimensions of the most efficient rectangular channel section to carry $5{m}^{3}/s$ at a slope of 1 in 900. The surface of the channel is of concrete.

E1.18
What is the most efficient depth for a brick channel of trapezoidal section with sides sloping at ${45}^{\circ}$ to the horizontal to carry $3{m}^{3}/s$. The bed slope is 0.0009.
E2 Questions on Rapidly Varied Flow
 Sudden transitions and Hydraulic Jumps

E2.1
For a trapezoidal channel with a $\text{base width}=3.0m$, Manning’s $n=0.025$ and side slope 1 vertical 2 horizontal, calculate the critical depth if the discharge is $Q=10{m}^{3}/s$.
(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 $5{m}^{3}/s$. Calculate the height of a hump which would produce the critical flow without causing backwater upstream (i.e., raising the upstream water level).
(Answer: 0.31 m) 
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 greatest allowable contraction for the flow to be possible upstream as described.
(Answer: 3.0m, 2.89m)

(a)

E2.4
The normal depth of flow in a rectangular channel (2m deep and 5m wide) is 1m. It is laid to a slope of 1m/km with a 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 hump, determine the depth of flow (${y}_{1}$) immediately upstream of the hump and the depth of flow (${y}_{2}$) above the hump. If the hump is reduced to 0.1m, what values will ${y}_{1}$ and ${y}_{2}$ be?
(Answer: 1.25m, 0.54m, 1.0m, 0.87m) 
E2.5
Water is flowing at a rate of $10{m}^{3}/s$ 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: 1.37m, 0.234 m) 
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: 3.81m, 1.26m/s, 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: 15.7kN, 1.22m, 28%)

(a)
E3 Questions on Gradually varied flow
 including integration of the backwater curve.

E3.1
A Rectangular channel is 3.0m wide, has a 0.01 slope, discharge of $5.3{m}^{3}/s$ and n=0.011. Find ${y}_{n}$ and ${y}_{c}$. If the actual depth of flow is 1.7m, what type of profile exists?
(Answer: ${y}_{n}$ = 0.4m, ${y}_{c}$ = 0.683m, S1 Curve) 
E3.2
A rectangular channel with a bottom width of 4.0m and a bottom slope of 0.0008 has a discharge of $1.50{m}^{3}/s$. In a gradually varied flow in this channel, the depth at a certain location is found to be 0.30m assuming n=0.016, determine the type of GVF profile, the critical depth and normal depth?
(Answer: M2, ${y}_{c}$ = 0.24m, ${y}_{n}$ = 0.43m) 
E3.3
The figure below show a backwater curve in a long rectangular channel. Determine using the direct step method the profile for the following high flow conditions: $Q=10{m}^{3}/s$, b=3m, n=0.022 and a bed slope of 0.001. Take the depth just upstream of the dam as the control point equal to 5m. At what distance is the water level not affected by the dam?
(Answer: ${y}_{n}$ = 2.44m, with 2 steps of integration, $x$ = 4469m, however with 10 steps of integration, $x=$ 5854m ) 
E3.4
A trapezoidal, concrete lined, channel has a constant bed slope of 0.0015, a bed width of 3m and side slopes 1:1. A control gate increased the depth immediately upstream to 4m. When the discharge is 19 ${m}^{3}/s$ compute the water surface profile upstream and identify the distance when the water depth is 1.8m. (n=0.017)
(Answer: with 2 steps of integration, $x$=1780m, with 10 steps of integration $x$ = 1978m) 
E3.5
Using the figure below, determine the profile for the flood conditions using firstly the direct step method and secondly the standard step method using a standard step length of 100m. $Q=600{m}^{3}/s$, n=0.04, the bed slope of the rectangular channel is 0.002 and has a width of 50m. The sill height of the weir is 2.5m and the water depth over the weir is 4m. Compare both methods.
E4 Questions on Gradually varied flow: Flow Transitions.

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 $7.5{m}^{3}/s/m$ and determine whether a hydraulic jump occurs and if one occurs which reach of the channel it occurs.

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 $3{m}^{3}/s/m$. 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 of the final profile and label each of the gradually varied profiles

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. Assume that the middle reach has a steep slope and that the other two have mild slopes, for a specific case, the Chézy C value for all reaches is 32 and the flow per unit width is $1{m}^{3}/s/m$. The upstream reach has a slope of 1in500, the middle reach has a slope of 1in50 and the downstream reach has a slope of 1in2,000. Determine which flow profile does occur

E4.4
A ’wide and shallow’ channel has two sections with different bed slopes but with the same cross section shape and bed roughness as depicted in the figure below. Sketch the critical flow depth ${y}_{c}$ line, uniform flow depth ${y}_{n}$ line and water surface profile in the channel and identify the type of water surface profile, if the flow discharge per unit width is $q=0.8{m}^{3}/(s.m)$ and bed roughness is $n=0.03$.

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$0.75{m}^{3}/s/m$
E5 Relevant Equations of Steady Open Channel Flow
The Chezy Equation
$Q$  $=AC\sqrt{R{S}_{o}}$  
$=AC{R}^{1/2}{S}_{o}^{1/2}$  (Eq1) 
The Manning’s Equation
$Q$  $={\displaystyle \frac{1}{n}}A{R}^{1/6}\sqrt{R{S}_{o}}$  
$={\displaystyle \frac{1}{n}}A{R}^{2/3}{S}_{o}^{1/2}$  (Eq2) 
Where $R=A/P$ and these are calculated depending on the shape of the channel crosssection e.g. Rectangular, Trapezoidal, etc. Table 1, below, may help.
Note that when a channel is said to be wide then it is assumed to be approximately rectangular and that the width $B$, is much greater than depth $y$, i.e. $B\gg y$. This results in:
$R$  $={\displaystyle \frac{A}{P}}={\displaystyle \frac{By}{B+2y}}\approx {\displaystyle \frac{By}{B}}$  
$R$  $\approx y$ 
Also, for wide channels we often specify flow per unit width i.e. $q$ in units of ${m}^{3}/s/m$.
Froude Number $F\mathbf{}r$
$Fr$  $={\displaystyle \frac{V}{\sqrt{g{D}_{m}}}}={\displaystyle \frac{V}{\sqrt{g\frac{A}{B}}}}$  (Eq3)  
$={\displaystyle \frac{Q}{A\sqrt{g\frac{A}{B}}}}$  (Eq4)  
$={\left({\displaystyle \frac{{Q}^{2}B}{g{A}^{3}}}\right)}^{1/2}$  (Eq5) 
and
For a rectangular channel $A=By$ and $q=Q/B$, so
$Fr$  $={\displaystyle \frac{V}{\sqrt{gy}}}$  (Eq7)  
$={\displaystyle \frac{Q}{By\sqrt{gy}}}$  (Eq8) 
and
At critical depth ${y}_{c}$, $Fr=1$, so (for a rectangular channel)
$1$  $={\displaystyle \frac{Q}{By\sqrt{g{y}_{c}}}}$  (Eq10)  
$1$  $={\displaystyle \frac{{Q}^{2}}{{B}^{2}g{y}_{c}^{3}}}$  (Eq11)  
${y}_{c}$  $=\sqrt[3]{{\displaystyle \frac{{Q}^{2}}{{B}^{2}g}}}$  (Eq12)  
${y}_{c}$  $=\sqrt[3]{{\displaystyle \frac{{q}^{2}}{g}}}$  (Eq13) 
Critical Slope, ${S}_{c}$
Critical slope, ${S}_{c}$ is the slope ${S}_{o}$ in the uniform flow equation (Manning’s or Chezy) at which depth is equal to critical depth, ${y}_{c}$.
Rearranging the Manning’s equation Eq2 and combining with the Froude number equated to 1, equation Eq5 (and using the subscript c to indicate parameters evaluated at critical depth) results in:
For a wide rectangular channel then as $P\approx B$ and thus $R=y$ this results in:
Bernoulli and Specific Energy
Where $\alpha $ is the coefficient of energy $\alpha =\frac{{\scriptscriptstyle \int \rho {u}^{3}\mathit{d}A}}{\rho {V}^{3}A}$, which can often be taken as 1.
Relationship between critical depth and specific energy
Specific Force
Gradually Varied Flow Equation
Direct step method
Standard step method
Numerical Integration (Euler method)
or
${S}_{f}$ is calculated from the uniform flow equations (Chezy or Manning’s) where it is equated to teh slope of the channel ${S}_{o}$. From Eq1 and Eq5 we get:
And these must be addapted for the appropriate channel geometry. Table 1 may help:
Rectangle  Trapezoid  Circle  
($\varphi $ in radians)  
Area, $A$  $by$  $(b+sy)y$  $\frac{1}{8}(\varphi \mathrm{sin}\varphi ){D}^{2}$ 
Wetted perimeter, $P$  $b+2y$  $b+2y\sqrt{1+{s}^{2}}$  $\frac{1}{2}\varphi D$ 
Top width, $B$  $b$  $b+2sy$  $(\mathrm{sin}(\varphi /2))D$ 
Hydraulic Radius, $R=\frac{A}{P}$  $\frac{by}{b+2y}$  $\frac{(b+sy)y}{b+2y\sqrt{1+{x}^{2}}}$  $\frac{1}{4}\left(1\frac{\mathrm{sin}\varphi}{\varphi}\right)D$ 
Hydraulic mean depth, ${D}_{m}=\frac{A}{B}$  $y$  $\frac{(b+sy)y}{b+2sy}$  $\frac{1}{8}\left(\frac{\varphi \mathrm{sin}\varphi}{\mathrm{sin}(\varphi /2)}\right)D$ 