CIVE3415: Water Engineering III
Steady Open Channel Hydraulics
Dr Andrew Sleigh
School of Civil Engineering
University of Leeds
This version was created on:
- 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.
A rectangular channel is wide, has a 0.01 slope, flow rate of , and . Find its normal depth and critical depth .
(Answer: , )
Water flows in a long rectangular channel at a depth of and discharge of . Determine the minimum channel width if the channel is to be subcritical.
A rectangular channel has a bottom width of 8 m and Manning’s n=0.025
Determine the slope at a normal depth of 2m when the discharge is
Determine the critical slope and the critical depth when the discharge is
Determine the critical slope at the critical depth of 1.5m and compute the corresponding discharge.
(Answer:(a) , (b) , , (c) , )
For a trapezoidal channel with a base width , Manning’s and side slope (i.e. 1 vertical: 2 horizontal), calculate the critical depth, critical velocity and critical slope if its discharge .
(Answer:, , )
A rectangular channel wide carries of water when flowing deep. Work out the flow specific energy. Is the flow sub-critical or super-critical?
(Answer: and flow is subcritical)
Two engineers observed two rivers and recorded the following flow parameters: River 1: flow discharge , flow velocity , water surface width ; River 2: flow discharge , flow velocity , water surface width . Decide the flow regime of two rivers, i.e. sub-critical or super-critical.
(Answer: River 1 is sub-critical and River 2 is super-critical)
A concrete, trapezoidal channel has a bottom slope of and a Manning roughness factor of . The bottom width of the channel is , and the side slopes are 1 in 2. Determine the velocity and discharge when the flow is normal at a depth of . (Answer: , )
A trapezoidal channel has a bottom slope of slope of in and a Manning roughness factor of . The bottom width of the channel is , and the side slopes are 1 in 3. Determine the normal depth in this channel for . (Answer: ).
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 . Use the Chezy formula, assuming that . (Answer: , ).
The cross section of a stream can be approximated by the compound channel shown in figure 1. The bottom slope is . The Manning roughness factor for the main channel and for the overbank areas. Determine the normal depth for a discharge of . Also calculate the energy coefficient and the momentum coefficient for the channel with this flow condition.
(Answer: , , .)
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 in the flow depth for the discharge of . Assuming normal depth still is present in the chanel, determine the minimum allowable channel total width, . (Answer: .)
The flow discharge through a rectangular channel wide is when the slope is 1:100. Is the flow sub-critical or super-critical? Calculate the slope, , that would give critical depth. The Manning roughness coefficient is . (Answer: super-critical, ).
An open channel with is to be designed to carry of water at a slope of 0.0065. Find the most efficient cross section for a rectangular section. (Answer: ).
The cross-section of a river may be idealized as shown in Fig. 2. Determine the discharge carried by the river when its bed slope is and the Manning’s .
For the channel section shown in Fig. 2, estimate the discharge if the channel slope is 0.002.
A rectangular channel has width and normal depth . 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 and are the same in the two cases. Compare the two wetted perimeters.
What are the dimensions of the most efficient rectangular channel section to carry at a slope of 1 in 900. The surface of the channel is of concrete.
What is the most efficient depth for a brick channel of trapezoidal section with sides sloping at to the horizontal to carry . The bed slope is 0.0009.
E2 Questions on Rapidly Varied Flow
- Sudden transitions and Hydraulic Jumps
For a trapezoidal channel with a , Manning’s and side slope 1 vertical 2 horizontal, calculate the critical depth if the discharge is .
Water is flowing at a normal depth in a wide rectangular channel with a bed slope of 1:500. If Manning’s and the discharge is . 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)
Water is flowing at a velocity of and a depth of in a channel of rectangular section with a width of 3.4m. Find the changes in depth produced by
A smooth contraction to a width of 3.0m
The greatest allowable contraction for the flow to be possible upstream as described.
(Answer: 3.0m, 2.89m)
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 . 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 () immediately upstream of the hump and the depth of flow () above the hump. If the hump is reduced to 0.1m, what values will and be?
(Answer: 1.25m, 0.54m, 1.0m, 0.87m)
Water is flowing at a rate of through a rectangular channel wide, at a depth of . 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)
Water flows in a rectangular channel at a depth of 30cm and with a velocity of . 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)
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 and . A hydraulic jump occurs a short distance downstream. Assuming no energy loss at the gate, calculate:
The force on the gate
The depth of flow downstream of the hydraulic jump
The fraction of the fluid energy that is dissipated in the jump
(Answer: 15.7kN, 1.22m, 28%)
E3 Questions on Gradually varied flow
- including integration of the backwater curve.
A Rectangular channel is 3.0m wide, has a 0.01 slope, discharge of and n=0.011. Find and . If the actual depth of flow is 1.7m, what type of profile exists?
(Answer: = 0.4m, = 0.683m, S1 Curve)
A rectangular channel with a bottom width of 4.0m and a bottom slope of 0.0008 has a discharge of . 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, = 0.24m, = 0.43m)
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: , 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: = 2.44m, with 2 steps of integration, = 4469m, however with 10 steps of integration, 5854m )
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 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, =1780m, with 10 steps of integration = 1978m)
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. , 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.
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 and determine whether a hydraulic jump occurs and if one occurs which reach of the channel it occurs.
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 . 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
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 . 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 does occur
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 line, uniform flow depth line and water surface profile in the channel and identify the type of water surface profile, if the flow discharge per unit width is and bed roughness is .
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
E5 Relevant Equations of Steady Open Channel Flow
The Chezy Equation
The Manning’s Equation
Where and these are calculated depending on the shape of the channel cross-section 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 , is much greater than depth , i.e. . This results in:
Also, for wide channels we often specify flow per unit width i.e. in units of .
For a rectangular channel and , so
At critical depth , , so (for a rectangular channel)
Critical slope, is the slope in the uniform flow equation (Manning’s or Chezy) at which depth is equal to critical depth, .
For a wide rectangular channel then as and thus this results in:
Bernoulli and Specific Energy
Where is the coefficient of energy , which can often be taken as 1.
Relationship between critical depth and specific energy
Gradually Varied Flow Equation
Direct step method
Standard step method
Numerical Integration (Euler method)
And these must be addapted for the appropriate channel geometry. Table 1 may help:
|( in radians)|
|Hydraulic mean depth,|