Hydraulics
Prof. B.S. Thandaveswara
36.4 Superelevation
Superelevation is defined as the difference in elevation of water surface between inside
and outside wall of the bend at the same section.
∆y=y0 − y1
(1)
This is similar to the road banking in curves. The centrifugal force acting on the fluid
particles, will throw the particle away from the centre in radial direction, creating
centripetal lift.
Superelevation in other words means the greater depth near the concave bank than
near convex bank of a bend. This phenomenon was first observed by Ripley in 1872,
while he was surveying Red River in Loussiana for the removal of the great raft
obstructing the stream.
36.4.1 Transverse water surface slope in bends
Gockinga was first to derive the following formula for determining the difference in
elevation of the water surface on opposite sides of channel bends.
2
⎛ x⎞
y=0.235V log ⎜ 1- ⎟
⎝ r⎠
in which V is the velocity in ms-1 , x is the distance at which y is to be determined, r is
the radius of the river bend. But above equation is found to fit a particular stream to
which it was designed. Also he found that the transverse slope was twice greater than
the longitudinal slope. He showed that the increased depth in bend is caused by the
helicoidal flow induced by centrifugal force. Fargue while conducting studies on scour in
meandering devised the formula and called " Fargue's law of greatest depth " in 1908.
But unfortunately it was found to be applicable to the stream Garonne at Barsac only.
C1 =0.03H3 -0.23H 2 +0.78H-0.76
in which C1 is the reciprocal of radius of curvature in kilometer and H is the lowest water
depth at the deepest point of the channel in meter.
Mitchell also derived another equation applicable to Delawave river, Philadelphia.
Indian Institute of Technology Madras
Hydraulics
Prof. B.S. Thandaveswara
⎛ 11002 − x 2 ⎞ 289 ⎛ 11002 − x 2 ⎞
y = 33 ⎜
+
⎜⎜
⎟⎟ 2 x
2
⎜ 11002 ⎟⎟
r
1100
⎝
⎠
⎝
⎠
Ripley in 1926 arrived at the formulae based on the field observations, having their own
limitations.
⎛ 4 x2 ⎞
17.52 ⎛ 4 x 2 ⎞
y = D1 ⎜ 1 − 2 ⎟ + D1
⎜ 1 − 2 ⎟⎟ x
⎜
r ⎜⎝
T ⎟⎠
T ⎠
⎝
⎛ 4 x 2 ⎞ 26.28P1 ⎛ 4 x 2 ⎞
y = P1 ⎜1 − 2 ⎟ +
⎜⎜1 − 2 ⎟⎟ x
⎜
r
T ⎟⎠
T ⎠
⎝
⎝
In the above equations, parabolic sections are assumed whose focal distances are
and
T2
D1
T2
respectively and with the origin at a point on the axis at a distance of D1 and P1
P1
from apex respectively.
r
CENTRIFUGAL FORCE
O
ri
W1 = WEIGHT OF FLUID
90
Vmax
dy
dr
y
v
O
r0
MECHANICAL MODEL OF THE HELICOIDAL FLOW AS PROPOSED
BY GRASHOF
The above two equations when combined yield a simplified form in FPS units. Figure
represents the general profile for equation given below.
⎛
⎞ ⎛ 17.52 x ⎞
x2
y = 6.35D ⎜ 0.437 − 2 − 0.433 ⎟ ⎜1 +
⎟
⎜
⎟⎝
r0 ⎠
T
⎝
⎠
Indian Institute of Technology Madras
Hydraulics
Prof. B.S. Thandaveswara
r0
T/2
T/2
0
0
origin
1.445D
Y AXIS
PROFILE FOR EQUATION
Grashof was the first to try an analytical solution for superelevation. He obtained
equation by applying Newton's second law of motion to every streamline and integrating
the equation of motion. Equation gives a logarithmic profile. Referring to figure given
below one may write
2
W1 Vmax
Centrifugal force =
g rc
2
W1 Vmax
2
g rc
Vmax
dy
=
=
W1
grc
dr
Assuming the boundary conditions near inside wall of the bend and integrating above
equation reduces to the form
2
Vmax
r
∆y=2.3
log 0
g
ri
in which ri and r0 are the inner and outer radii of the bend respectively.
Woodward in 1920 assumed the velocity to be zero at banks and to have a maximum
value at the centre of the bend and the velocity distribution varying in between
Indian Institute of Technology Madras
Hydraulics
Prof. B.S. Thandaveswara
according to parabolic curve. Using Newton's second law of motion he obtained the
following equation for superelevation.
∆y =
2
Vmax
⎡
3 ⎛
2 ⎞2 ⎤
⎛ 2r + b ⎞
r
r
r
20
⎛
⎞
⎛
⎞
c
⎢
-16 ⎜ c ⎟ + ⎜ 4 ⎜ c ⎟ -1⎟ ⎥ ln ⎜ c
⎟
⎢3 b
⎝ b ⎠ ⎝⎜ ⎝ b ⎠ ⎠⎟ ⎥ ⎝ 2rc - b ⎠
⎢⎣
⎥⎦
Shukry obtained the following equation for maximum superelevation based on freevortex flow and principle of specific energy.
∆y max =
The Euler equation of motion.
Since
C2
2g
r02 ri2
( r02 - ri2 )
∂
( p + γ z ) + ρa s = 0
∂s
( p + γz ) = γh
∂
∂h
( p + γz ) = − γ
∂s
∂n
2
v
H0 = y +
+z
2g
=h+
v2
2g
(∵ h = y + z )
r0
2
_V__
g
2
r
ri
X
X
n
ri
r0
y
isovels
z
H0
datum
Section xx
dh v 2
=
dn gr
Indian Institute of Technology Madras
Hydraulics
Prof. B.S. Thandaveswara
differentiating above equation
dh v ∂v
+
=0
dn g ∂n
v 2 v ∂v
=0
+
gr g ∂n
v ∂v
+
=0
r ∂n
Thus v decreases and h increases from inner boundary to outer boundary.
∂v ∂r
The equation can be rewritten as
+ = 0.
v
r
Therefore it can be shown as v r = constant which is in the free vortex condition.
Consider a rectangular channel bend,
Discharge per unit width q = vy .
c
y q
q = y or = a constant.
r
r c
r0
ri
2
_V__
g
2
Total Energy Line
Outer
Wall
H0 y
Inner Wall
Subcritical
flow F < 1
Supercritical
flow F > 1
y
0
z
Modification to the bed profile to obtain the horizontal water surface
in a bend
dy y
=
dr r
dy
dz 1
=dr
dr 1-F2
z decreases from inner wall to outer wall for subcritical flow (as shown in the
above figure).
dz
dy
y
=1-F2 = - 1-F2
dr
dr
r
and
dh dz dy
dy
dy
=
+
=1-F2 +
dr dr dr
dr
dr
2
v
dy ⎡
y
-1+F2 + 1⎤ = F2 =
⎦ r
dr ⎣
gr
(
)
(
(
Indian Institute of Technology Madras
)
)
Hydraulics
Prof. B.S. Thandaveswara
Transverse bed profile.
v02
v2
H 0 =z+y +
= y0 +
2g
2g
H 0 =z+
rq C2
+
C 2gr 2
The above equation gives resonably good result as long as the angle of the circular
bend in plan is greater than 90
a correction factor was suggested by Shukry for
circulation constant C, assuming it to vary linearly from 0 to 90 .
⎡θ
vr = CU f =C ⎢ +
⎣ 90
θ ⎞ r VmA ⎤
⎛
⎜1 − ⎟
C ⎥⎦
⎝ 90 ⎠
⎡θ ⎛
v
θ ⎞V ⎤
=w1U X =w1 ⎢ + ⎜1 − ⎟ mA ⎥
r
⎣ 90 ⎝ 90 ⎠ rw1 ⎦
However, by applying Newton's 2nd law of motion based on one dimensional analysis
i.e., all the filamental velocities in the bend are equal to the mean velocity Vmb and that
of all the streamlines having the same radii of curvature rc , an equation can be obtained
for a rectangular channel namely
∆y max =
2
⎛ 2b ⎞
Vmb
⎜ ⎟
2g ⎝ rc ⎠
For the channels other than the rectangular channel, the bed width (b) can be replaced
by the water surface width (T) then
2
⎛ 2T ⎞
Vmb
⎜
⎟
2g ⎝ rc ⎠
The above equation is only a first approximation and gives transverse profile as the
∆y max =
straightline . This assumes that the rise and the drop of the water surface level from the
normal level is equal on either side of center line of bend.
As a better approximation, Ippen and Drinker obtained an equation for superelevation.
The derivation of equation is based on the assumptions of free vortex or irrotational flow
with the uniform specific head over the cross section, and the mean depth in bend being
equal to the mean approaching flow depth.
Indian Institute of Technology Madras
Hydraulics
Prof. B.S. Thandaveswara
⎛
⎞
⎜
⎟
V 2T ⎜ 1 ⎟
∆y=
2g rc ⎜
T2 ⎟
⎜ 1− 2 ⎟
⎜ 4r ⎟
c ⎠
⎝
The bends in nature will not have the symmetry due to entrance conditions, length of
2
curvature and boundary resistance. Hence above equation will not give accurate result.
If the forced vortex condition exists with constant stream cross section and constant
average specific energy, then equation for superelevation assumes the form
⎛
⎞
⎜
⎟
V 2T ⎜ 1 ⎟
∆y=
2g rc ⎜
T2 ⎟
⎜ 1+
⎟
⎜ 12r 2 ⎟
c ⎠
⎝
The above equation is applicable to a smooth rectangular boundary with circular bend
2
with the flowing fluid being ideal.
Better results can be obtained by combining the effects of the free and forced vortex
conditions simultaneously. The minimum angle of bend is to be 90 for applying the
above equation in combination. For the smaller angles the difference in computed
values from the above equation becomes larger than the actual ones.
However, for a rectangular channel, circular bend with , applying the free vortex
formula, velocity at inside wall of the bend becomes as thus a depth of should exist at
the boundary (i.e. at ), which is physically impossible.
Muramoto obtained an equation for superelevation based on equations of motion.
vr = C1
g S0 r 2
C
U=
+ 2
3C1
r
Then
(
)
r
⎡ 2 4 C2 + C2
⎤0
1
2
g
S
r
2S
C
r
0 2 ⎥
∆y = ⎢ 0 2 −
+
2
⎢ 36 C1
3C
2gr
1 ⎥
⎣
⎦ ri
in which C1 and C2 are circulation constants obtained, after integration. The special
feature of above equation lies in including the effect of bed slope on superelevation.
Indian Institute of Technology Madras
Hydraulics
Prof. B.S. Thandaveswara
Thus it can be observed from the above discussion that superelevation in a bend is a
function of shape of the cross section, Reynolds number, approach flow, slope of the
bed, Froude number,
θ rc
,
and boundary resistance. The superelevation is also
180 b
affected by the presence of secondary currents and separation.
However, before applying to field situations, the validity of the transverse flow profile
equation has to be justified. The observations in the field have shown the occurrence of
troughs near the concave profile in the bend. Further, these troughs have been
observed during rising and as well as falling stages of the channels. The channels of
smaller width have exhibited accumulation of debris instead of troughs.
36.4.2 Superelevation and transverse profile
Normalised super elevation ∆y/∆y max was correlated with normalised bend angle θ/θ0
for all the three bends, for two different Reynolds numbers.
From Figure, it may be observed that two peaks of superelevation occur at θ/θ 0 = 0.17
and θ/θ 0 = 0.67 for all the cases except for bend B1 for R e = 42, 280 . The influence of
Reynolds number on the trend of the variation of superelevation at various section of
the bend is insignificant in all the three bends.
Indian Institute of Technology Madras
Hydraulics
Prof. B.S. Thandaveswara
1.0
y
___
ymax
0.34
0.67
1.00
SECTION B1
y
___
ymax
1.0
0.67
0.34
1.00
SECTION B2
y
___
ymax
1.0
0
0.34
0.67
1.00
θ/θ0
SECTION B3
Variation of normalised Super elevation
with normalised angle for
Reynolds number 42280
Indian Institute of Technology Madras
Hydraulics
Prof. B.S. Thandaveswara
1.0
y
___
ymax
0.67
0.34
1.00
SECTION B1
y
___
ymax
y
___
ymax
1.0
0.34
0.67
SECTION B2
1.00
0.67
1.00
1.0
0
0.34
θ/θ0
SECTION B3
Variation of normalised Super elevation
with normalised angle for
Reynolds number 101,700
The maximum value of superelevation was normalised with the approaching velocity
2
head V /2g and correlated with the Froude number. The equations of these lines is in
the form
∆y max
2
=m log F + C
V /2g
in which 'm' is the slope of the line and 'C' a constant.
Bend 1:
Bend 2 :
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∆y max
= − 1.19 log F + 0.20
2
V /2g
∆y max
= 5.33 log F − 0.49
2
V /2g
Hydraulics
Prof. B.S. Thandaveswara
Bend 3:
∆y max
2
= 1.96 log F − 0.06
V /2g
View of SET-UP
Indian Institute of Technology Madras
Hydraulics
Prof. B.S. Thandaveswara
Flow Condtions D/S of B1
The greatest difference in elevation between the longitudinal profiles at outer and inner
walls is maximum at the 30 section of the 180 bend.
Indian Institute of Technology Madras
Hydraulics
Prof. B.S. Thandaveswara
N
2
5
4
100
6
100
9
7
200
10
8
100
11
468
100
468
A
Bend I A
P.G.
314
1
737
P.G. 4
4315
1
12
200
100
782
50
1
Wheel volve
2
Inlet pipe
3
Adjustment valve
4
Entrance chamber
5
Transition
6
Leading Channel
rc = 300
90 0
280
50
7
Rectangular notch
8
Stilling chamber
9
Masonry honey comb
10
3270
50
870
Transition
11 Main channel
12
Tail control
2400
LAYOUT PLAN
1
G.L
3
rc = 140
2
Point gauge 6
230
100
9
Bend II
440
11
56
72
180
Experimental Set - Up
Scale = 1: 100 (all dimensions in cm)
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180 0
Bed slope 1000
1960
Hydraulics
Prof. B.S. Thandaveswara
All dimensions are in cm
0
2m
rc
0
90
0
1 m 75
60
0 15 0 0
30 0
45
0
60
0
75
0
90
1 m Station B
0
0
13
50
45 rc 200
0
30
0
15
13
0
0
rc = 300
70
A
A
G.L
15
100
Section AA
Outer
Inner
Station C
1m
0
0
0
-22.5 -12.5
0
0.4
12.5 22.5
b) points of measurement
180
0
30
150
0
120 90 0
Section and points of measurement
Indian Institute of Technology Madras
60
0
0
Hydraulics
Prof. B.S. Thandaveswara
6.5
5.0
4.0
Bend 2
3.0
2.0
Bend 3
1.0
Bend 1
- 1.05
- 0.60
0
- 0.31 - 0.25
log F
y
______
Variation of _2
V / 2g
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with log F
y
______
_2
V / 2g
Hydraulics
Prof. B.S. Thandaveswara
Outer wall
Inner wall
experimental
24
theoretical
23
0
12.5
25.0
bed in cm
50.0
37.5
Observed
Theoretical : eqn
Experimental : eqn
Comparison of Observed and Theoretical Superelevation
1.2
y/yA 1.0
0.8
Bed slope 1:1000
STN A
B1
STN B
Q = 26.1 lps
B2
STN C
B3
STN D
Re 42280 F = 0.49
1.2
y/yA
1.0
0.8
Bed slope
STN A
B1
STN B
Q = 71.9 lps
B2
STN C
Re 101760 F = 0.55
Longitudinal Water Surface Profile
Indian Institute of Technology Madras
STN D
B3
Scale
x axis 1:100
y axis 5:1
Outer wall
Inner wall
Hydraulics
Prof. B.S. Thandaveswara
CL
1.10
1.11
Inner wall
1.12
1.20
Outer wall
CL
Inner wall
0.95
0.95
1.15
1.12
1.05
1.00
1.18 1.15
0.90
STATION B
CL
CL
Outer wall Inner wall
Outer wall
1.25
1.15
1.20
0.90
1.00
0.90
0.80
0.98
1.00
1.05
1.15
STATION A
Inner wall
Outer wall
1.15
1.10
1.05
0.90
1.05
0.81
0.85
0.98
STATION D
STATION C
ISOVELS in open channel bend [Normalised with Vmax ] Q = 26.1 lps, F = 0.18, Re= 36050
Indian Institute of Technology Madras
Hydraulics
Prof. B.S. Thandaveswara
Outer wall
CL
Outer wall Inner wall
CL
Inner wall
1.30
1.25
1.151.10
0.85
1.45 1.22
1.25 1.00
1.15
0.85
1.0
1.10
0.77
STATION B
STATION A
Outer wall
CL
Outer wall Inner wall
CL
Inner wall
1.30
1.25
1.30
1.25
1.23
1.20
0.92
1.30
1.27
1.00
1.08 1.00
0.95
0.75
1.30
0.95 1.00
0.80
STATION D
STATION C
ISOVELS in open channel bend [Normalised with Vmax ] Q = 71.9 lps, F = 0.44, Re= 95420
Inner wall
CL
Outer wall Inner wall
CL
Outer wall
1.25
1.30
1.15
1.10
1.30
1.25
1.00
0.77
0.93 0.78
STATION B
STATION A
Inner wall
1.15
1.10
Outer wall Inner wall
CL
CL
Outer wall
1.20
1.15
1.05
1.00
0.95
0.80
1.15
1.10
1.00 1.08
0.83
STATION D
STATION C
ISOVELS in open channel bend [Normalised with Vmax ] Q = 83.5 lps, F = 0.41, Re= 103460
Indian Institute of Technology Madras
Hydraulics
Prof. B.S. Thandaveswara
Reference
1. Ippen A.T and Drinker P.A., "Boundary shear stresses in curved trapezoidal
channels" , Proc. ASCE. journal Hydraulic Diivision HY5, Part - I, Volume 88, p 3273, pp
- 143 - 179, September 1962, and discussion by Shukry A, Proc. ASCE. journal
Hydraulic Division, pp 333, May 1963.
2. Henderson F.M. , "Open Channel Hydraulics", Mac Millan Company Limited 1966.
3. Thandaveswara B.S, "Characteristics of flow around a 90° open channel bend", M.Sc
Engineering Thesis, Department of Civil and Hydraulic Engineering, Indian Institute of
Science, Bangalore - 12, May 1969.
Indian Institute of Technology Madras