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CAIRO UNIVERSITY HYDRAULICS

Faculty of Engineering 3rd

year Civil Eng.

Irrigation & Hydraulics Department 2010 2011

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Exercise (1): Open Channel Flow Uniform Flow

1) Derive expressions for the flow areaA, wetted perimeterP, hydraulic radiusR,top water surface width B, and hydraulic depth D for the following channelcross sections:

a. Rectangular (bottom width =b)b. Trapezoidal (bottom width =b, side slope =t :1)c. Triangular (side slope =t :1)d. Partially full circular (diameter =D)

2) At a point in a rectangular channel 15.2 m wide, 2.1 m deep, and 15 cm/km bedslope, the discharge is 20 m3/sec and C= 62 (metric) in Chezy's formula.

a. Show whether the flow is uniform or non-uniform.b. Show whether the flow is shooting or streaming.

3) A rectangular canal has a bed slope of 8 cm/km, and a bed width of 100 m. If ata depth of 6 m the canal carries a discharge of 860 m3/sec at uniform flow. Find

the roughness n, Chezy's coefficient C, and the coefficient of friction f. Also

find the average shear stress on the bed.

4) Figure (1) shows the section of a channel with flood way. Manning coefficient istaken 0.025 for the main channel and 0.035 for the flood way. Knowing that the

bed slope is 10 cm/km, calculate the discharge.

5) The following table shows values of average Manning coefficient (n) and thecorresponding water depth (y) for a laboratory flume laid at a slope of 1:100.

The flume has a rectangular cross section of widthb= 30 cm, the sides are made

of glass while the bed is made of stainless steel.

y (m) 0.1 0.2 0.3 0.4 0.5 0.6

n (metric) 0.02 0.016 0.013 0.011 0.01 0.01

a. Draw the relation between y and n then explain why the average roughnesscoefficient decreases while increasing the water depth. Do you expect the

same behavior in natural open channels? Why?

b. Find the water flow depth that corresponds to a discharge of 100 lit/sec if theslope is kept the same.

6.0 2.0 3.0

0.75

3.0

2.0

Figure (1)


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6) Design a channel to irrigate 54000 feddans at a water duty of 60 m3/fed/day. Thesoil allows side slope of 1:1,n= 0.025 (metric), and the bed slope is 12 cm/km.

Find the channel dimensions for each of the following cases:

a. The maximum allowable velocity being 0.7 m/s.b. The maximum allowable boundary shear stress being 0.27 kg/m2.c. y = 0.1(b)(0.5S0+ 4) where S0is in cm/Km.d. The channel is of best hydraulic cross section.

7) Two channels designed to be most economic sections, the first one is trapezoidalwith side slope of 1:1, and the second one is of semi-circular section. Both have

the same area, discharge, and Manning coefficient (n= 0.02). If the trapezoidal

channel is 3m in bed width and is laid at a slope of 15 cm/km, what will be the

diameter and the slope of the semi-circular channel?

8)

A trapezoidal channel of side slopes 1:1, bed width b = 5.0m and roughnesscoefficient n = 0.02 was used to convey water at a depth of 1.5m until it's

decided to decrease the side slopes of the channel to be 2:1 due to slope stability

problems. What is the new bed width if the discharge, water depth and the bed

slope are to be kept the same?

9) A wide rectangular navigation channel consists of three reaches with an averageManning coefficient of 0.025. The bed slopes for the three reaches are 8, 7, and

10 cm/km. The monthly flows are as follows:

Month 1 2 3 4 5 6 7 8 9 10 11 12

Q (m3/sec) 16 11 14 13 15 18 17 19 10 10 13 11

It is required to design the channel such that the water depths through the

channel are always greater than 1.5 m and less than 3 m. Find the suitable range

of the bed width.

10)In a river of bed width 600 m, and bed slope 10 cm/km, it is found that the bedmaterial just begins to move when the discharge is 150 million m3/day.

Supposing the mean velocity to vary with the depth and the slope according to

the relation:

V = 120 S(2/3)

y (in m/s)

Find the slope at which the same tractive stress on the bed would be produced

with a discharge of 600 million m3/day.

11)A canal of bed slope 8 cm/km and side slope of 1:1 has a bed width of 10 m anddepth of 2.5 m. The area irrigated by this canal is drained by a drain of 1:1 side

slope and 10 cm/km longitudinal slope. Half of the water for irrigation is

escaped to the drain, the Manning coefficient of the drain is 4/3 that of the canal.

Design the drain if the relation between depth and width is :345.1 by =


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12)Design a canal for non-silting, non-scouring condition given the following data:Discharge = 50 m3/sec, bed slope = 9 cm/km, n = 0.025, the constant in

Kennedy's formula = 0.28 and side slope =1:1, where Kennedy's formula may be

written as:

V = C y(2/3)

13)An irrigation canal 16 m bed width, 1:1 side slope, and of 10 cm/km bed sloperuns at a normal depth of 4 m. The canal is to be widened to serve 25% more

area. Find the necessary bed width; all other data are the same. Use Manning's

formula.

14)A circular conduit 2 m in diameter carries different discharges under uniformflow conditions with a free surface and 10 cm/km bed slope. If n= 0.01, Draw

the curves showing the variations of mean velocity Vmand discharge Qwith the

water depth. Find the depths of maximum Qand maximum V.

15)A circular sewer 1.2 m in diameter, laid at a slope of 1:100 is to run at maximumvelocity. What would be the discharge passing? Take C= 60 (metric).

16)A transmission cast iron pipeline (n= 0.01) is proposed to convey a maximumdischarge Qmaxof 1.45 m

3

/sec of treated water from the waste water treatmentplant (WWTP) of a city to a main drain: The transmission pipeline will be laid at

a. slope of 1/1000.

a. Select the minimum pipeline diameterD such that the flow depth does notexceed half the pipe diameter. The available pipe sizes are:

1.0, 1.25, 1.5, 1.75, 2.0,... m.

b. Calculate the mean flow velocity.c. Calculate the flow depth and mean flow velocity for a low flow of Qmax/4.


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Problems to be solved by computer:

a) Create an Excel spread sheet that can calculate the flow area A, wetted perimeterP, hydraulic radiusR, top water surface widthB, and hydraulic depthDfor the

following channel cross sections:

a. Rectangular (bottom width =b)b. Trapezoidal (bottom width =b, side slope =t:1)c. Triangular (side slope =t:1)d. Partially full circular (diameter =D)Make your spread sheet dynamic as much as you can. You can follow the

instructions in the tutorial files to get familiar with the Microsoft Excel

interface.

b) For the channel section with flood way shown in figure 1 (problem 4), calculateusing Microsoft Excel the discharge flowing into the channel for water depths

0.1, 0.2, 0.3 8.0m. Plot the variation of Q with water depth (in the specified

range). Make your spread sheet as dynamic as possible so it can be used for

different cross section geometries and manning coefficients. (Hint: The formulae

used in calculating the discharge will vary when water depth is more than 3m.

You can follow the example in the tutorial files for a simpler problem).

c) A circular conduit 2m in diameter carries different discharges under uniformflow conditions with a free surface, lies at a slope of 10 cm/km with manning

coefficient of 0.01 (metric). Using Microsoft Excel,

a. Calculate the velocity and discharge under uniform flow conditions for thefollowing depths: 0.05, 0.10, 0.15, ,1.95 m

b. Plot two curves showing the variations of mean velocity Vmand dischargeQ with the water depth.

c. Find the depths of maximum Q and maximum V.d) Make an Excel spread sheet to calculate the depth of uniform flow for the

following channel cross sections (use Manning equation):

a. Rectangularb. Triangularc. Trapezoidald. Semi-CircularThe spread sheet should be able to calculate the normal depth for the above

cross section shapes for different values of discharge and channel geometry

(side slope (t), bed width (b), bed slope (So), diameter (d), and manning

coefficient (n).


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Model Answer


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When the water depth increases, the

wetted perimeter (contact length

causing friction to the flow) of the sides

made of glass increases. On the other

hand, the contact length with the bed

(stainless steel) remains the same. This

implies that the overall average

manning coefficient of the cross section

will approach that of glass (will

decrease) with increasing the water

de th.

We dont expect to have the same behavior in natural open channels. In natural open channels,

roughness increases with increasing the water depth due to the growth of weeds on the berms.


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