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Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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.
0 1∆y=y y− (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.
2y=0 235V log 1-
rx. ⎛ ⎞
⎜ ⎟⎝ ⎠
in which V is the velocity in -1ms , 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.
3 21C =0.03H -0.23H +0.78H-0.76
in which 1C 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.
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
2 2 2 2
2 21100 289 110033 2
r1100 1100x xy x
⎛ ⎞ ⎛ ⎞− −= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
Ripley in 1926 arrived at the formulae based on the field observations, having their own
limitations.
2 2
1 12 24 17.52 4D 1 D 1
rT Tx xy x
⎛ ⎞ ⎛ ⎞= − + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
2 21
1 2 226.28P4 4P 1 1
rT Tx xy x
⎛ ⎞ ⎛ ⎞= − + −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
In the above equations, parabolic sections are assumed whose focal distances are 2
1
TD
and 2
1
TP
respectively and with the origin at a point on the axis at a distance of 1D and 1P
from apex respectively.
O
90Vmax
v
ydydr
W1
CENTRIFUGAL FORCE
WEIGHT OF FLUID
r
r i
O 0r
=
MECHANICAL MODEL OF THE HELICOIDAL FLOW AS PROPOSEDBY GRASHOF
The above two equations when combined yield a simplified form in FPS units. Figure
represents the general profile for equation given below.
2
20
17 526 35D 0 437 0 433 1rT
x . xy . . .⎛ ⎞⎛ ⎞⎜ ⎟= − − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
r0
T/2T/2
origin
Y AXIS
0 0
1.445D
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
Centrifugal force = 2max1
c
VWg r
2max1
2c max
1 c
VWg r V
W grdydr
= =
Assuming the boundary conditions near inside wall of the bend and integrating above
equation reduces to the form
2max 0
i
V r∆y=2.3 logg r
in which ir and 0r 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
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
according to parabolic curve. Using Newton's second law of motion he obtained the
following equation for superelevation.
23 22 c c c cmax
c
r r r 2r + b20∆y = V -16 + 4 -1 ln3 b b b 2r - b
⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦
Shukry obtained the following equation for maximum superelevation based on free-
vortex flow and principle of specific energy.
( )2
2 2max 0 i2 2
0 i
C∆y = r - r2g r r
The Euler equation of motion.
( ) sρa 0p zs
γ∂+ + =
∂
( )p + γz = γh Since
( ) hp + γz = γ s n∂ ∂
−∂ ∂
2
0H y z2vg
= + +
( )2
h h = y + z2vg
= + ∵
yisovels
r0
ri
n
H0z
2g
V___2
ri
r0
X
X
Section xx
r
datum
2dhdn gr
v=
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
differentiating above equation
2
dh v 0dn g n
v+ 0gr g n
v
v v
∂+ =
∂
∂=
∂
v 0r nv ∂+ =∂
Thus v decreases and h increases from inner boundary to outer boundary.
The equation can be rewritten as r 0vv r∂ ∂
+ = .
Therefore it can be shown as = constantv r which is in the free vortex condition. Consider a rectangular channel bend, Discharge per unit width q yv= .
cq = yr
or y q=r c
a constant.
r
H y
2g
Outer
Total Energy Line
Inner Wall 0Wall
0
ri
z
y0
V___2
Modification to the bed profile to obtain the horizontal water surfacein a bend
Subcritical flow F < 1
Supercriticalflow F > 1
dy y=dr r
2dy dz 1= -dr dr 1-F
z decreases from inner wall to outer wall for subcritical flow (as shown in the above figure).
( ) ( )2 2dz dy y= - 1-F - 1-Fdr dr r
=
and
( )2dh dz dy dy dy= - 1-Fdr dr dr dr dr
+ = +
22 2dy y-1+F 1 = F
dr r grv⎡ ⎤+ =⎣ ⎦
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
Transverse bed profile. 220
0 0
2
0 2
H =z+y y2g 2g
rq CH =z+ +C 2gr
vv+ = +
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 .
mAf
r Vvr = CU =C + 1 90 90 Cθ θ⎡ ⎤⎛ ⎞−⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
mA1 X 1
1
Vv =w U =w + 1r 90 90 rw
θ θ⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
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 mbV and that
of all the streamlines having the same radii of curvature cr , an equation can be obtained
for a rectangular channel namely
2mb
maxc
V 2b∆y =2g r
⎛ ⎞⎜ ⎟⎝ ⎠
For the channels other than the rectangular channel, the bed width (b) can be replaced
by the water surface width (T) then
2mb
maxc
V 2T∆y =2g r
⎛ ⎞⎜ ⎟⎝ ⎠
The above equation is only a first approximation and gives transverse profile as the
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.
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
2
2c
2c
V 2T 1∆y=2g r T1
4r
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎜ ⎟⎝ ⎠
The bends in nature will not have the symmetry due to entrance conditions, length of
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
2
2c
2c
V 2T 1∆y=2g r T1+
12r
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
The above equation is applicable to a smooth rectangular boundary with circular bend
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.
1vr = C
20 2
1
g S r CU = + 3C r
Then
( ) 0
i
r2 22 4 1 20 0 22 2
11r
C + Cg S r 2S C r∆y = + 3C36 C 2gr
⎡ ⎤⎢ ⎥−⎢ ⎥⎣ ⎦
in which 1C and 2C are circulation constants obtained, after integration. The special
feature of above equation lies in including the effect of bed slope on superelevation.
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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, crθ180 b
, and boundary resistance. The superelevation is also
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 max∆y/∆y was correlated with normalised bend angle 0θ/θ
for all the three bends, for two different Reynolds numbers.
From Figure, it may be observed that two peaks of superelevation occur at 0θ/θ 0 17.=
and 0θ/θ 0 67.= for all the cases except for bend B1 for eR = 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.
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
1.0
0.34 0.67 1.00SECTION B1
SECTION B2
SECTION B3
1.0
1.0
0.34 0.67 1.00
0.34 0.67 1.00
Variation of normalised Super elevationwith normalised angle forReynolds number 42280
0
y___ymax
θ/θ0
y___ymax
y___ymax
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
0.34 0.67 1.00
0.34 0.67 1.00
0.34 0.67
1.0
1.0
1.0
1.000
SECTION B1
SECTION B2
SECTION B3
Variation of normalised Super elevationwith normalised angle forReynolds number 101,700
y___ymax
y___ymax
y___ymax
θ/θ0
The maximum value of superelevation was normalised with the approaching velocity
head 2
V /2g and correlated with the Froude number. The equations of these lines is in
the form max2
∆y =m log F + CV /2g
in which 'm' is the slope of the line and 'C' a constant.
max2
∆yBend 1: = 1.19 log F + 0.20V /2g
−
max2
∆yBend 2 : = 5.33 log F 0.49V /2g
−
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
max2
∆yBend 3: = 1.96 log F 0.06V /
2g
−
View of SET-UP
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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.
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
112
1
2
3
4
56
78
910
11
12
Wheel volve
Inlet pipe
Adjustment valveEntrance chamber
Transition
Leading Channel
Rectangular notch
Stilling chamber
Masonry honey comb
Main channel
Tail control
Transition
1 236 9
11
Point gauge
56
18072G.L 230
100
Experimental Set - Up Scale = 1: 100 (all dimensions in cm)
1
2
4
5
6
7
8
9
10
11
100 200100 100100
P.G.
468 468
Bend I314
782
50
1960
Bend II440
2400
870
50
280
3270
rc = 300
900
737
P.G. 44315 50
1800
rc = 140
AA
200 100
LAYOUT PLAN
Bed slope 1000
N
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
Section and points of measurementb) points of measurement
70
5013
G.L15
100Section AA
Inner Outer
-22.5 -12.5 0 12.5 22.50.4
00
15 3045
6075
90
2 m
rc
rc 1 m Station B
A
00
0
0
0
0
200
A
00
30
6090
0
00120150
180
0
0
0
Station C
rc = 30000
1530
4560
7590
0
0
0
00
0
1 m
13
1 m
All dimensions are in cm
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
0
1.0
2.0
3.0
4.0
5.0
6.5
- 0.25- 0.31- 0.60- 1.05log F
Bend 2
Bend 3
Bend 1
Variation of with log F
y______
V_2
2g/
y______
V_2
2g/
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
Comparison of Observed and Theoretical Superelevation
experimental
theoretical
24
23
bed in cm0 12.5 25.0 37.5 50.0
ObservedTheoretical : eqnExperimental : eqn
Inner wall Outer wall
Longitudinal Water Surface Profile
Bed slope 1:1000
Bed slope
1.2
1.0
0.8
1.2
1.0
0.8
STN ASTN B STN C
STN D
STN ASTN B STN C
STN D
B1
Q = 26.1 lps
B2 B3
B1 B2 B3
Re 42280 F = 0.49
Q = 71.9 lps Re 101760 F = 0.55 Scalex axis 1:100y axis 5:1Outer wallInner wall
y/yA
y/yA
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
Inner wall Outer wallCL1.10
1.111.12
1.20 1.15
1.121.051.00 0.90
0.80
0.95 0.95
1.18 1.151.15
1.051.00
0.90
1.20
1.151.10 1.05
0.81
0.90
0.85 0.98
1.05
1.15
1.25
CL
CLCL
Inner wall Outer wall
Inner wall Outer wallInner wall Outer wall
STATION A STATION B
STATION DSTATION C
0.98
1.00
0.90
ISOVELS in open channel bend [Normalised with V ] Q = 26.1 lps, F = 0.18, R = 36050emax
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
Inner wall Outer wallCL
STATION A STATION B
STATION DSTATION C
Inner wall Outer wallCL
Inner wall Outer wallCLInner wall Outer wallCL
0.85
1.45 1.22
1.25 1.000.85
1.301.251.151.10
1.151.30
1.251.101.0
0.77
1.30
1.27
0.92
1.001.001.08
0.950.75
1.301.251.23
1.20
0.95 1.000.80
1.30
ISOVELS in open channel bend [Normalised with V ] Q = 71.9 lps, F = 0.44, R = 95420emax Inner wall Outer wallCL
STATION A STATION B
STATION DSTATION CISOVELS in open channel bend [Normalised with V ] Q = 83.5 lps, F = 0.41, R = 103460emax
1.30
Inner wall Outer wallCL
Inner wall Outer wallCL Inner wall Outer wallCL
1.251.151.10
0.77
1.251.30
1.151.101.00
0.93 0.78
1.20
1.15
1.10
1.081.000.83
1.151.051.00
0.950.80
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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.