M.E. (Water Resources & Hydraulic Engineering) Exam. (6 Sem.), 2022
(1st Semester)
FREE SURFACE FLOW
(Paper - VI)
Time : Three Hours
Full Marks : 100
Answer any five questions.
1. a) The velocity distribution in a semi-circular channel of diameter 2R0 follows the law
u/U0 = (y/R0)1/7, in which y is the distance normal to the surface at which the velocity is
u and U0 is the velocity at the centre of the semi-circle. If R0 = 1.8 m and U0 = 2.6 m/s,
find average velocity, energy and momentum correction factor and define them.
b) “Uniform flow of an ideal fluid on a sloping channel is impossible”, justify this
statement.
c) State the difference between 1-Dimensional Flow, 2-Dimensional flow and 3Dimensional Flow.
13+4+3=20
2. a) Prove that the ratio of the permissible shear stress (on the side of the channel ( ) to the
permissible shear stress on the bottom of the channel ( )when a sand grain to move it
on the bed in a smooth trapezoidal channel represented as
Where K called as tractive force ratio, angle of repose =α
b) What should be the bottom width of a trapezoidal canal (side slopes 2 H: 1 V) having a
uniform slope of 1.2 x10-3 if it is to carry 70 m3/s at a depth of 2.5 m? Find the
maximum shear stresses on the bed and the sides of the channel n =0.013
10+10=20
3. a) Find the normal depth of flow in a triangular section with a bottom angle of 600 for Q =
3 m3/s, n=0.014 and S= 2x10-3. Also find the value of Chezy’s coefficient.
b) A channel has multiple-roughness types in its perimeter. Assuming that the total
discharge in the channel is equal to the sum of discharges in the partial areas, show that
the equivalent roughness is given by
10+10= 20
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AM
4. a) Differentiate between pipe flow and open channel flow
b) A bridge has cylindrical piers 1 m in diameter and spaced 16 m apart. Downstream of the
bridge where the flow disturbance from the piers is no longer present, the flow depth is
2.9 m and the mean velocity is 2.5 m/s.
(i) Calculate the depth of flow upstream of the bridge assuming that the pier coefficient of drag
is 1.3.
(ii) Determine the head loss caused by the pier
5+15= 20
5. a) What are the differences between normal depth, critical depth, transitional depth and
instantaneous depth?
y
y
b) Show that x is positive for M1 and S3 profiles and x is negative for M2 and S2
profiles.
c) Prove that the GVF profile for a horizontal channel can be expressed as


C
C2 y N − M +1
x = − 2 2
y N +1 +
 + Constant
gC1 ( N − M + 1) 
 Q ( N + 1)
3+6+11= 20
6. a) In a rectangular channel three reaches A, B and C in series, with reach A being upstream
of reach B, have the following characteristics.
Reach
Width
Discharge
Slope
n
3
(m)
(m /s)
A
3.5
12.0
0.0004
0.020
B
3.0
12.0
0.0160
0.015
C
2.5
12.0
0.0100
0.018
Sketch the resulting water-surface profiles due to changes in the channel characteristics as
above. The length of the reaches can be assumed to be sufficiently long for the GVF
profiles to develop fully.
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b) A trapezoidal channel has a bed width B = 5 m, bed slop = 1/4000, side slope m = 2H:1V
and n = 0.02. the normal depth of flow is 3 m. if the channel empties into a pool at the
downstream end and the pool elevation is 1.25 m higher than the channel bed elevation at
the downstream end, find the distance between two sections A and B, given yA = 1.80 m,
yB = 2.28 m. Assume α = 1, N = 4, M = 3.75.
9+11 = 20
7. a) Establish that the GVF profile in a frictionless rectangular channel is given by
y
x=
S0
 1  y 3 
1 +  c   + Const.
 2  y  
b) Show that for a wide horizontal channel, by assuming Chezy’s C = constant,
C2 
y4 
x=
 y − 3  + Const.
g 
4 yc 
c) In a very long, wide rectangular channel the discharge intensity is 3 m3/s per meter
width. The bed slope of the channel is 0.004 and Manning’s n = 0.014. At a certain
section in this channel, the depth of flow is observed to be 0.95 m. What type of GVF
profile occurs in the neighbourhood of this section?
7+8+5 = 20
8. a) Establish a relationship between the depths of flow upstream and downstream of
hydraulic jump in terms of upstream Froude number (F1) in case of horizontal
rectangular channel.
b) Establish a relationship to evaluate the loss of energy in hydraulic jump. Determine the
relationship to determine relative loss (EL/E1) and efficiency (E2/E1) of a hydraulic
jump in terms of upstream Froude number (F1).
c) A hydraulic jump occurs in a rectangular channel and the depths of flow before and after
the jump 0.75 m and 2.25 m respectively. Calculate the critical depth of flow.
8+8+4 = 20
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