# Exercise (4): Open Channel Flow - Gradually Varied Flow

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1 Exercise 4: Open Channel Flow - Gradually Varied Flow 1 A wide channel consists of three long reaches and has two gates located midway of the first and last reaches. The bed slopes for the three reaches are S1 = 0.008, S2= , and S3 = The water discharges to the channel from a lake where the water surface is higher than the normal depth at the inlet. If the discharge is m3/sec/m, and n = 0.015, sketch the possible water surface profiles in the channel for the following cases: a The last reach terminates in a sudden fall. b The last reach discharges to a lake of water level higher than the normal depth at the exit. Also trace the variation in water depth through both gates on the S.E.D. 2 Water flows with constant discharge q=1.0 m3/s/m into a wide rectangular channel that consists of two very long reaches where the bed slope changes from So1=6.87x10-3 to So2=3.24x10-4, Calculate: a The depth of both uniform and critical flow in both reaches take n=0.018 b Draw a neat sketch of the water profile at the transition zone and calculate the water depths and the head losses wherever appropriate. 3 A barrage is constructed across a wide river whose discharge is 6 m3/sec/m, and bed slope is 10 cm/km. If the afflux produced at the site of the barrage is 3 m, find the length of the water surface profile produced from the site of the barrage till a point where the water depth is 6.25m. Use the direct step method considering 3 points, Chezy coefficient C = A trapezoidal channel of bed width of 7 m, side slopes 3:2, Manning coefficient n of 0.02 is laid on a slope of and carries a discharge of 30m3/sec. The channel terminates to a free fall. It is required to compute and plot the water surface profile from downstream to a water depth of 0.9 the normal depth. Use the direct step method use 3 steps. 5 A steep long channel takes its water from a lake. Prove that the discharge per unit width in the channel is given by: q 8 27 Two lakes are joined by a wide concrete channel A-B as shown in the figure below, n = The water levels in the lakes are constant. gh 3 2 a Find the flow rate into the channel for each of the following bed slopes: 1

2 i. So = 10 cm/km ii. So = iii. So = 0.01 Compare your results and explain how the flow depth at A affects the flowing discharge. b Sketch the Water surface profile for the above cases and calculate the water depths wherever appropriate. c Does a hydraulic jump occur? If so, How far upstream point B does it occur? Use the direct step method, considering 5 points. 1.5m A 2.0m n = 0.02, Bed Slope = S o B d If a gate is located at point A Cc=0.65 and Cd=0.6 to control the discharge into the channel as in the figure below. Find the flow rate into the channel for the following gate openings So=0.01: i. d = 0.5m ii. d = 0.8m iii. d = 1.2m 1.5m A n = 0.02, Bed Slope = B 2.0m

3 6 A 10 m wide, rectangular concrete-lined canal n = has a bottom slope of 0.01 and a constant level lake at the upstream end. The water level in the lake is 6m above the bottom of the canal at the entrance. If the entrance losses are negligible, determine: a The flow depth 600 m downstream of the channel entrance, use the improved Euler method. b The distance from the lake where the flow depth is 3.0 m. Use the direct step method. 7 The figure below shows a longitudinal section in a channel of wide cross section. Manning s n = 0.02 and the bed slope is q B A S o = 1/1000, n = 0.02 A gate is located at point A. The discharge equation for the gate in case of free outflow is given by: q C d d 2g H C d c and for a submerged outflow is: q Cd d 2g H y Where q is the discharge per unit width m2/sec, Cd is the discharge coefficient and is equal to 0.6, d is the height of the gate opening m, H is the water depth just upstream of the gate m, y is the water depth downstream the gate m, and g is the gravitational acceleration m/sec2. The coefficient of contraction of the gate Cc =

4 a If q = 3 m3/sec/m, and H = 3m, find the gate opening d, and check if a free hydraulic jump will form downstream of the gate. Assume the channel downstream A is long enough to allow for uniform flow to develop. If a jump will form, how far downstream of A will it be? Use an appropriate method to calculate the distance b Point B is located 600m upstream of the gate. Find the water depth at B. Use the improved Euler method considering two steps. If the bed level at A = 20.75m, find the water level at B. c The gate at A is used to maintain a constant water level at B for different values of discharge. What is the required gate opening to keep the water level at B unchanged, for a low flow of 1 m3/sec/m? Use the improved Euler method with two steps. 8 A trapezoidal channel having bottom slope is carrying a flow of 30m3/s. The bottom width is 10.0m and side slope 2H to 1V. A control structure is built at the downstream end which raises depth at the downstream end to 5.0m. Compute and draw the water surface profile. Manning n is

5 Problems to be solved by computer: Water flows in a rectangular channel of 10m bed width, n = 0.018, at a flow rate of 15m 3 /sec. Create an Excel Spread Sheet to plot to a suitable horizontal and vertical scale the bed slope, water surface profile, and the total energy line TEL between points A and B for the following cases water flows at point A with uniform flow depth. In order to perform the required plots, calculate the bed elevation, water surface elevation, and the TEL elevation along the profile. You may assume bed elevation at the start of your profile with any appropriate value. Comment on the relation between the bed slope S o and the slope of the TEL S e for the three cases then calculate and compare the length between A and B. q H = 3m A S o = 15 cm/km B q A S o = 15 cm/km Free Fall B q H = 3m A S o = B

6 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd Model Answer Question 1 q = m3/sec/m, n = 0.015, So1 = 0.008, So2 = , So3 =

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8 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd Question 2 q = 1.0 m3/sec/m, So1 = 6.87 x 10-3, So2 = 3.24 x 10-4, n =0.018 a b then the hydraulic jump will occur on the steep slope 8

9 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd Question 3 q = 6 m3/sec/m, So = 10 cm/km, afflux = 3 m, C = 50. 9

10 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd calculate the length of M1 curve using direct step method from y = 8.24 m to y = 6.25 m 3 points. y m yavg m Fr^2 Se 1-Fr^2/SoSe dy dx dx=

11 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd Question 4 Trapezoidal channel b = 7 m, t = 1.5, n = 0.02, So = 0.001, Q = 30 m3/sec. then yo > yc Mild channel 11

12 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd calculate the length of M2 curve using direct step method from y = 0.9x1.623 = 1.46 m to y = yc = 1.13 m 4 points. y m yavg m A P R B Fr^2 Se 1-Fr^2/So-Se dy dx dx=

13 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd Question 5 neglect the head loss at the inlet relatively short distance But then 13

14 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd a i n = 0.02, So = 10 cm / km assume steep channel yo > yc wrong assumption 14

15 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd assume mild channel from 1, 2 then 15

16 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd a ii n = 0.02, So = assume steep channel yo = yc Critical channel As the water depth at the inlet remains yc, then and the discharge q will remain the same 16 is valid

17 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd a iii n = 0.02, So = 0.01 assume steep channel yo < yc Steep channel 17

18 case i ii iii So type Mild Critical Steep q m3/sec/m "The discharge flowing into the channel from the lake depends on the conditions at the inlet water depth at point A. For a constant specific energy available water head H in the lake, the maximum discharge that can be diverted to the channel occurs when water depth at the inlet is at Ycritical, otherwise, the flowing discharge decreases. In this problem, changing the channel slope from critical to steep slope did not affect the water depth at A and thus the flowing discharge remained te same. On the other hand, when the channel slope changed to be Mild, water depth at A increased to Normal depth and thus the flowing discharge decreased. In addition to the above, when the water depth at A in Ycritical, the flowing discharge in to the channel depends only on the available head in the lake, while in case of mild slope, other channel parameters that affect normal depth n, So also affect the flowing discharge in addition to H." 18

19 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd c For case i, and ii no hydraulic jump will occur For case iii a hydraulics jump will occur 19

20 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd calculate the length of S1 curve using direct step method from y = m to y = 2.0 m 5 points. y m yavg m Fr^2 Se 1-Fr^2/So-Se dy dx dx= m

21 CAIRO UNIVERSITY Irrigation & Hydraulics Department 3 year Civil Eng rd d Cd = 0.6, Cc = 0.65 i d = 0.5 m 21

22 ii d = 0.8 m iii d = 1.2 m q = qmax = m 3 /sec/m 22

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