For more clarity, the computation has been donewith the same intake and circulation to

test the effects of value of bon intake discharge without changing the submergence

(Figure 4.11a) and the effects on submergence with same discharge (Figure 4.11b). This

phenomenon is consistent with the physical meaning of the coefficient ' 'band it also

improves the performance of bell-mouth intake.

So far we have been able to verify the model qualitatively based on the fact that no

previous studies are available which describe the variation in water surface profile at the

intake. However, a larger quantity of experimental data has been used by Odgaard (1986)

to verify the equation representing the critical submergence, which is defined as the

submergence when the tip of air-core vortex just reaches the intake (Figure 4.12), in the

absence of surface tension (Eq. 18 in Odgaard, 1986) presented as follows:

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p
J zyx
2221 term)(pressure
To estimate the pressure terms in Equations (3.33), (3.34), the momentum equation in
ζ -direction of Equation (3.25) is considered. If the acceleration and shear stress
components are negligible, the equation can be reduced as:
( )
ρζ
ζζζζζ
ηη
ζ
ηξ
ζ
ξη
ζ
ξξ
p
JJ
GVVVUUVUU
J
zzx
∂
∂++
−=Γ+Γ+Γ+Γ
0
222
00
1 (3.35)
therefore,
( )[ ]ζζηηζηξζξηζξξζζζρζ GVVVUUVUU
p
zzx
+Γ+Γ+Γ+Γ−
++
=
∂
∂
222
1
ζ is a straight axis consequently, 1222 =++ zzx ζζζ ,
( ) ζζηηζηξζξηζξξρζ GVVVUUVUUp +Γ+Γ+Γ+Γ−=∂∂ (3.36)
The expression for pressure distribution is obtained by integrating Equation (3.36) with
respect to ζ . It follows that:
{ }
2
2
0
0
000
hGVVUVVUUUdp
h
ζζ
ηη
ζ
ξη
ζ
ξη
ζ
ξξζρ −Γ+Γ+Γ+Γ=∫ (3.37)
Equations (3.26, 3.33, 3.34) and (3.37) are the depth-averaged equations in a general form
used in calculating water surface profile of flows in the following applications.
29
Notations
t : time
zyx ,, : Cartesian coordinates
ζηξ ,, : generalized curvilinear coordinates
K,,,,,, zyxzyx ηηηξξξ : components of transformation matrix
J : transformation Jacobian
( )wvu ,, : velocity components in Cartesian coordinate
( )WVU ,, : contravariant components of velocity vector
( )sss WVU ,, : contravariant components of velocity vector at water surface
h : water depth in ζ -direction
p : pressure
ρ : water density
g : gravitational acceleration
ζηξ GGG ,, : contravariant components of gravitational vector
Kyyyxxzxyxx τττττ ,,,, : effective shear stress
ηξ ττ bb , : contravariant of shear stress acting on the bottom
i
jkΓ : Christoffel symbol
30
Chapter 4
STEADY ANALYSIS OF WATER
SURFACE PROFILE OF FLOWS
WITH AIR-CORE VORTEX AT
VERTICAL INTAKE
4.1 Introduction
Air-core vortex formation at intakes is a significant hydraulic engineering problem in
many situations such as intakes for irrigation, drainage system, hydropower generation in
which the water is normally drawn from rivers, channels or reservoir,… as well as in
water art-works or sculptures. It occurs typically whenever the submergence is less than a
critical value and causes some detrimental effects as reduction in intake discharge,
resulting vibrations and noises as well as operational difficulties. Figure 4.1 shows an
example of free surface air-core vortex occurred in laboratory.
When the flow in a large body of slowly moving water is diverted and locally accelerated
or drawn off, any associated vortex tube is extended and its rotation is thereby increased.
Higher velocities incur lower pressures and, if a free surface exists, it becomes locally
31
Figure 4.1 An example of free surface air-vortex
Figure 4.2 Various stages of development of air-entraining vortex: S1>S2>S3>S4
(Jain et al., 1978)
Q Q
S1
S2
S4
Q
S3
a. No depression on the surface at large submergence b. Formation of a dimple
d. Air-entraining vortexc. Air core extends deeper as the vortex becomes
stronger
32
depressed. Thus in hydraulic structures, where flow negotiates a vertical shaft, submerged
conduit or gated outlet, the vortex may be sufficiently intense to form a hollow core,
which could transmit air. This may lead to the undesirable effects already mentioned, and
considerable efforts have been made to predict and control such phenomena. Indeed, the
deliberated introduction of a vortex at the vertical intake has been described in Jain et al.
(1978).
Figure 4.2 shows the various stages in the development of an air-entraining vortex when
the water depth is decreased gradually. At the first stage, when the submergence is large,
there is no depression on the surface (Figure 4.2a), but together with decreasing water
depth, a dimple forms as in Figure 4.2b. If the submergence is further decreased the
air-core vortex occurs (Figure 4.2c) and Figure 4.2d shows critical condition under which
a vortex just tends to entrain air (Jain et al. 1978).
Suppose that steady flow occurs in the horizontal yx − plane and that closed path, S ,
is traced out in the fluid (Figure 4.3). The path encloses an area A by linking adjacent
flow particles and subsequently moves with the flow. At a point on the path, one particle
may have velocity components nV and tV normal and tangential to the path. For the
path to remain unbroken and assuming the flow is incompressible, the sum ∑ ∆SVn is
the net inflow into the closed area A and must always be zero. The corresponding sum
along the path ∑ ∆SVt is not necessarily zero and is known as the circulation Γ
(being defined as positive anticlockwise). The Kenvin’s theorem states that the circulation
remains constant with time unless an external shear stress exists along S , as described in
Townson (1991).
33
Figure 4.3 The inflow to and circulation round a closed path in a flow field
(Townson 1991)
Figure 4.4 The concept of simple Rankine vortex including two parts: free vortex in
outer zone and forced vortex in inner zone (Townson 1991)
S
A
nV
S∆ tV
r
∞→r
0→tV
H h
H∆
free vortex
forced vortex
ideal fluid real fluid
34
If we consider an infinitely wide reservoir of constant depth is drawn off at a central point
with rate q per unit depth. At some distance from the center, a circular vortex tube may
be defined with circulation Γ about a vertical axis through the drawn-off point. As the
tube contracts to radius r under the influence of drawn-off, the tangential velocity
rVt π2/Γ= . The radial velocity is rqVr π2/= , and both clearly accelerate, producing a
spiral flow towards the center. Applying the Kelvin’s theorem, the circulation is
conserved therefore the velocity at the center is infinite. If the pressures remain
hydrostatic, the depression of the free surface is also infinite. But the viscosity present in
real fluids prevents this condition arising, and a zone in center of the flow rotates as a
solid mass (Townson 1991). This central part is known as a forced vortex and the
composite system as Rankine’s vortex (Figure 4.4).
Several approaches have been presented in the literature to deal with the problem of
determination and prediction of critical submergence serving in design works. These
approaches basically can be labeled as analytical models (Jain 1984; Odgaard 1986; Hite
and Mih 1994) and physical models (Anwar et al. 1978; Jain et al. 1978; Yildirim and
Kocabas 1995; Yildirim and Kocabas 1997; Yildirim and Kocabas 1998).
Many analytical attempts have been presented in the literature in order to attain a
theoretical view of the far-field velocity; in fact the flow representation has not been
defined so far by any comprehensive analytical analysis. The concept of simple Rankine
vortex normally used in the basic equations (Odgaard 1986; Hite and Mih 1994),
therefore this approach obviously could not be applied for the case of air-entraining
vortex and moreover, the Kelvin’s theorem is invalid for the central region.
Trivellato et al. (1999) set the water surface equal to the stationary headwater while other
experimental works only focused on the critical submergence. Consequently, these
approaches could not be used to predict the water surface profile of flow with air-core
35
vortex.
In this study, the water surface profile of a steady air core vortex flow into a vertical
intake was derived through out a depth-averaged model of open channel flows over the
3-D curvilinear bottom using a generalized and body fitted coordinate system. The
assumption of fully free air-core vortex in the new coordinate allowed us to use the
Kelvin’s theorem of the conservation of circulation for the whole flow field. The vortex
was assumed axisymmetric and steady. The assumption of shallow water and kinetic
boundary condition at water surface were also used. The equation describing water
surface profile was derived and calculated results were compared to the formula
introduced by Orgaard (1986).
The application’s results showed us the ability of the model in analyzing the water surface
profile and can be improved to simulate the flow structure of an air-core vortex.
4.2 Governing equations
Coordinate setting
To consider the vortex occurring at a cavity on the bottom surface (Figure 4.5), the
position of any point say P , is defined by three coordinates ( )ζηξ ,, where ( )ηξ ,
define the position of 0P (projection of P ) on the bottom, and ζ is the distance from
point P to that bottom surface.
Assuming that the shape of bottom surface (i.e at 0=ζ ) has the form of
)( 0
0 ar
bz
−
−
= , (4.1)
where a and b are the coefficients which define the shape of the bottom; 0r is the
distance from any point on the bottom to the z -axis
2
0
2
00 yxr += (4.2)
36
Figure 4.5 Definition of coordinate components
x
r
y
η
a
O
ξ
ζ P P0
r
x
ad 2=
aa−
•
z
l •
37
Based on Figure 4.5, we can derive the relations ηcos00 rx = and ηsin00 ry −= .
Then, the bottom surface is expressed by the following equation:
0)(),,( 00000 =+−=Φ bzarzyx (4.3)
The unit vector )(n normal to the bottom surface is derived as:
k
arz
ar
j
r
zy
arz
i
r
zx
arzgrad
gradn
rrrr
2
0
2
0
0
0
00
2
0
2
00
00
2
0
2
0 )()(
1
)(
1
−+
−
+
−+
+
−+
=
Φ
Φ
=
At ζ the relations between ),,( zyx and ),,( ζηξ can be expressed as:
ζηηζ
24
0
0
0
00
2
0
2
0
0
)(
coscos
)(
1
bar
br
r
zx
arz
xx
+−
−=
−+
+= (4.4a)
ζηηζ
24
0
0
0
00
2
0
2
0
0
)(
sinsin
)(
1
bar
br
r
zy
arz
yy
+−
+−=
−+
+= (4.4b)
ζζ
24
0
2
0
020
2
0
0
0
)(
)(
)()( bar
ar
ar
b
arz
ar
zz
+−
−
+
−
−=
−+
−
+= (4.4c)
Then the covariant base vector components on the bottom surface are denoted as follows:
k
arp
bj
p
i
p
kzjyixe
2
0111
000 )(
sincos
−
−+−=++=
ηη
ξξξξ
rv
(4.5a)
jrirkzjyixe ηηηηη ηη cossin 00000 −−=++=
rr
(4.5b)
j
bar
bi
bar
bkzjyixe
rrr
24
0
24
0
000
)(
sin
)(
cos
+−
+
+−
−=++=
ηη
ζζζζ
k
bar
ar r
24
0
2
0
)(
)(
+−
−
+ (4.5c)
Let’s denote that
2
0
2
0
22
0
2
0
00
2
0
1
)(2)()(
)()(
barlbarlarr
brarbarl
p
+−+−+−
+−+−
= (4.6)
38
in which l is defined in Figure 4.5.
From Figure 4.5, it could be easily seen that
( )
( )arr
barl
r
ar
bl
r
zl
−
+−
=
−
+
=
−
=
00
0
0
0
0
0tanξ (4.7)
then
( )
( ) 2020
00
2
0
00
0
00 )(
)()(tan
arr
brarbarl
arr
barl
rr
−
+−+−
−=⎟⎟⎠
⎞
⎜⎜⎝
⎛
−
+−
∂
∂
=
∂
∂ ξ (4.8)
From (4.6) we can obtain:
( ) ( ) ( )
( )2020
2
0
2
0
22
0
2
02
2
2
tan1
cos
1tan
arr
barlbarlarr
−
+−+−+−
=+==
∂
∂ ξξξ
ξ
(4.9)
but
0
00
0 tan
tan
tantan
r
rr
r
∂
∂
∂
∂
=
∂
∂⇒
∂
∂
⋅
∂
∂
=
∂
∂
ξ
ξ
ξ
ξξ
ξ
ξ
ξ
(4.10)
Thus, substituting (4.8) and (4.9) into (4.10) gives
100
2
0
2
0
2
0
22
0
2
00 1
)()(
)(2)()(
pbrarbarl
barlbarlarrr
−=
+−+−
+−+−+−
−=
∂
∂
ξ (4.11)
From (4.4a):
( )
( )[ ] ζ
ηη 2324
0
3
0
0
cos2
cos
bar
arb
r
x
+−
−
+=
∂
∂ (4.12)
Using the relations in (4.11), (4.12), it can be obtained:
( )
( )[ ] ⎟⎟⎠
⎞
⎜⎜⎝
⎛
−⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−
−
+=
∂
∂
=
1
2324
0
3
0 1cos2cos
pbar
arbxx ζηηξξ
( )
( )[ ] ζ
ηη
⋅
+−
−
−−= 2324
01
3
0
1
cos2cos
barp
arb
p
(4.13)
39
Similarly,
( )
( )[ ] ζ
ηη
ξ ⋅
+−
−
+= 2324
01
3
0
1
sin2sin
barp
arb
p
y (4.14)
( )
( )
( )[ ] ζξ ⋅+−
−
−
−
−= 2324
01
0
2
2
01
2
barp
arb
arp
bz (4.15)
and
( ) ζ
ηηη ⋅
+−
+−=
24
0
0
sinsin
bar
brx (4.16)
( ) ζ
ηηη ⋅
+−
+−=
24
0
0
coscos
bar
bry (4.17)
0=ηz (4.18)
( ) 240
cos
bar
bx
+−
−=
η
ζ (4.19)
( ) 240
sin
bar
by
+−
=
η
ζ (4.20)
( )
( ) 240
2
0
bar
ar
z
+−
−
=ζ (4.21)
and the Jacobian right on the bottom surface is:
2
0
24
00
100 )(
)(1
11
000
000
000
ar
barr
p
zzz
yyy
xxx
JJ
−
+−
===
=
ζηξ
ζηξ
ζηξ
ζ
(4.22)
The gravitational vector can be expressed in the new coordinate system as follows:
ζ
ζ
η
η
ξ
ξ eGeGeGG ++= (4.23)
Substituting the base vector components from (4.5) into (4.23)
40
( ) ( ) ( ) kzjyixGkzjyixGkzjyixGG ζζζζηηηηξξξξ ++++++++=
( ) ( ) ( ) kzGzGzGjyGyGyGixGxGxG ζζηηξξζζηηξξζζηηξξ ++++++++=
but
kgG −=
consequently,
gG
G
G
zzz
yyy
xxx
−
=⋅ 0
0
ξ
η
ξ
ζηξ
ζηξ
ζηξ
(4.24)
hence, the contravariant components of gravitational vector are derived
( ) ( )( ) 240
2
0
1 bar
ar
bgpyxyxgJG
+−
−
=−−= ηζζη
ξ (4.25)
( ) 0=−−= ξζζξη yxyxgJG (4.26)
( ) ( )( ) 240
2
0
bar
ar
gyxyxgJG
+−
−
−=−−= ξηηξ
ξ (4.27)
Using Equations (4.13-4.22) in plugging with (3.15-3.17), it gives:
( )
( ) 240
4
0
1 cos bar
ar
px
+−
−
−= ηξ (4.28)
( )
( ) 240
4
0
1 sin bar
ar
py
+−
−
= ηξ (4.29)
( )
( ) 240
2
0
bar
arb
z
+−
−
−=ξ (4.30)
0
sin
rx
ηη −= (4.31)
0
cos
ry
ηη −= (4.32)
41
0=zη (4.33)
( ) 240
cos
bar
b
x
+−
−=
ηζ (4.34)
( ) 240
sin
bar
b
x
+−
=
ηζ (4.35)
( )
( ) 240
2
0
bar
ar
z
+−
−
=ζ (4.36)
Equation of water surface profile
Dealing with the flows at a vertical intake as in Figure 4.5, the depth-averaged equations
(3.26, 3.33 and 3.34) are applied. With the assumptions of steady vortex ( 0=
∂
∂
t
) and
axisymmetric flow, i.e. the flow is homogenous in η -direction ( 0=
∂
∂
η
), those
equations become:
Continuity equation:
0
0
=
J
M
d
d
ξ or .00 constQJ
M
== (4.37)
Momentum equations:
ξ -direction:
( )
00
0
2
000
2
00
1
J
G
J
hNNMMNM
hJJ
UM
d
d b
ρ
τ
ξ
ξξξ
ηη
ξ
ηξ
ξ
ξη
ξ
ξξ −=Γ+Γ+Γ+Γ+⎟⎟⎠
⎞
⎜⎜⎝
⎛
({ )
⎭⎬
⎫
−Γ+Γ+Γ+Γ
++
−
2
2
0000
0
2
0
2
0
2
0 hGVVUVVUUU
d
d
J
zyx ζζ
ηη
ζ
ξη
ζ
ξη
ζ
ξξξ
ξξξ
(4.38)
η -direction:
( )
00
0
2
000
2
00
1
J
G
J
hNNMMNM
hJJ
UN
d
d b
ρ
τ
ξ
η
ηη
ηη
η
ηξ
η
ξη
η
ξξ −=Γ+Γ+Γ+Γ+⎟⎟⎠
⎞
⎜⎜⎝
⎛
42
({ )
⎭⎬
⎫
−Γ+Γ+Γ+Γ
++
−
2
2
0000
0
000000 hGVVUVVUUU
d
d
J
zzyyxx ζζ
ηη
ζ
ξη
ζ
ξη
ζ
ξξξ
ηξηξηξ
(4.39)
Using the definition in Equation (3.32) in plugging with Equations (4.28 – 4.36), the
Christoffel symbols are hereafter calculated:
( ) ( )[ ]24001
2
0
1
2
1
0
21
bararp
b
dr
dp
p
zyx zyx
+−−
+=
∂
∂
−
∂
∂
−
∂
∂
−=Γ ξ
ξ
ξ
ξ
ξ
ξ
ξξξ
ξ
ξξ (4.40)
000 =Γ=Γ
ξ
ηξ
ξ
ξη (4.41)
( )
( ) 240
4
00
10 bar
arr
p
+−
−
=Γ ξηη (4.42)
000 =Γ=Γ
η
ηη
η
ξξ (4.43)
01
00
1
rp
−=Γ=Γ ηηξ
η
ξη (4.44)
( ) ( ) 2400210
2
bararp
b
+−−
−=Γ ζξξ (4.45)
000 =Γ=Γ
ζ
ηξ
ζ
ξη (4.46)
( ) 240
0
0
bar
br
+−
−=Γ ζηη (4.47)
( )
( ) 240
4
0
2
12
0
2
0
2
0 bar
arp
zyx
+−
−
=++ ξξξ (4.48)
2
0
2
0
2
0
2
0
1
rzyx
=++ ηηη (4.49)
ξ and η introduced in Figure (4.5) are obviously perpendicular with each other, thus it
is given that
0000000 =++ zzyyxx ξηξηξη
43
Therefore, Equation (4.39) becomes:
( )
00
0
2
000
2
00
1
J
G
J
hNNMMNM
hJJ
UN
d
d b
ρ
τ
ξ
η
ηη
ηη
η
ηξ
η
ξη
η
ξξ −=Γ+Γ+Γ+Γ+⎟⎟⎠
⎞
⎜⎜⎝
⎛
(4.50)
By substituting Equations (4.26, 4.43-4.44) and neglecting the effect of friction in
η -direction, Equation (4.50) is reduced as:
0 2
0100
=⎟⎟⎠
⎞
⎜⎜⎝
⎛
−+⎟⎟⎠
⎞
⎜⎜⎝
⎛
rphJ
MN
J
MV
d
d
ξ (4.51)
Applying Equation (4.37) into (4.51) gives
( ) 02
01
00 =⎟⎟⎠
⎞
⎜⎜⎝
⎛
−+
rph
NQV
d
dQ ξ
or
0
0
01
2 2
r
dr
V
dV
rp
V
d
dV
−=⇒=ξ (4.52)
Integrating Equation (4.52) between two arbitrary points A and B with distance Ar0 and
0Br respectively from z -axis, the following relation is obtained:
2
0
2
0
0
02
B
A
A
B
B
A
B
A r
r
V
V
r
dr
V
dV
=⇒−= ∫∫ (4.53)
where AV and BV are the contravariant components of velocity vector at A and B
respectively.
Based on equation (4.53), the depth averaged tangential velocity component at 0r is
defined as:
BBAA
B
A
A
A
B
B
rvrv
r
r
r
v
r
v
VreVv 002
0
2
0
0
0
0 =⇒=⇒== ηr (4.54)
Eq. (4.54) is identical to the free vortex condition which allows us to assume the
conservation of circulation vr02π=Γ for the whole flow regime, thus Vr
2
02π=Γ and
44
from which:
2
02 r
V
π
Γ
= (4.55)
The evaluation of the contravariant component of shear stress acting on the bottom is
approximated using the following relation:
2
0
2
0
0
2
hr
Qf
J
UefUf bb =⇒==
ρ
τ
ρ
τ ξ
ξ
ξ rrV (4.56)
Using the relations described in Equations (4.40-4.49) and (4.45-4.46), Equation (4.38)
becomes
ρ
τ
ξ
ξ
ξξ
ξξ
00
0
0
2
0
2
J
G
J
h
hJ
M
hJ
M b
−=Γ+⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂
( ) ⎥⎦
⎤⎢⎣
⎡
−Γ+Γ
∂
∂
++−
222
1 2
0
2
0
2
2
0
2
0
2
0
0
hGNM
J zyx
ζζ
ηη
ζ
ξξξξξξ (4.57)
Substituting (4.37) into (4.57) gives:
ρ
τ
ξ
ξ
ξξ
ξξ
00
0
0
2
0
0
2
0
1
J
G
J
h
h
JQ
hJ
Q b−=Γ+⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂ ( ) [ ]ζξξξξξξ 02020200
2
0
2
Γ
∂
∂
++− zyxJ
Q
( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
Γ
∂
∂
++
Γ
−
ζ
ηηξξξξπ 040
2
0
2
0
2
0
0
2
2
8 rJ zyx
( ) [ ]2202020
02
1 hG
J zyx
ζ
ξξξξ ∂
∂
+++
Then, using the relations (4.56) and after some manipulation, we obtain the equation for
water surface profile in the form of
),(
),(
021
01
rhfp
rhf
d
dh
=ξ (4.58)
in which
45
-0.1 0.0 0.1 0.2
x (m)
-0.2
0.0
0.2
z
(m
)
Water surface
Critical depth line
Quasi-normal depth line
Bottom
Figure 4.6 An example of computed water surface profile withç quasi-normal
depth line and critical depth line
46
[ ]
[ ]
[ ]
4
2324
0
2
0
3
00
24
0
2
2
2324
0
0
2
02
01
)(
)(2)(3
8
)(
)(
),( h
barr
arrbar
b
bar
arr
gbrhf
⎪⎭
⎪⎬
⎫
+−
−++−Γ
−
⎪⎩
⎪⎨
⎧
+−
−
=
π
[ ]
[ ]
[ ]
2
2524
0
24
0
2
02
02324
00
3
02
0
)(
)()(3
)(
)(2
h
bar
barar
bQ
barr
ar
bQ
⎪⎭
⎪⎬
⎫
+−
−−−
+
⎪⎩
⎪⎨
⎧
+−
−
+
[ ]
2
0
2124
02
0
0
2
03
0
2
2
2
0
2
0
)(
)(
.- 1
4)(
gb
ar
bar
Qfh
r
Q
h
rar
r
−
+−
−⎪⎭
⎪⎬⎫⎪⎩
⎪⎨⎧ Γ−
−
+
π
(4.59)
and
[ ] [ ]
2
0
3
2124
00
2
2
3
2124
0
2
0
2
0
02
)(
1
4)(
)(
),( Qbh
barr
gh
bar
arr
rhf −
+−
Γ
+
+−
−
=
π
(4.60)
Method of Calculation
For convenience of simulating the water surface profile, the transformation of special
coordinates from ξ to 0r is needed. Substitute Equation (4.11) into (4.58) it follows
that
),(
),(
),(
),(1.
02
01
0021
01
01
0
0 rhf
rhf
dr
dh
rhfp
rhf
dr
dh
pd
dr
dr
dh
d
dh
−=⇒=−== ξξ (4.61)
A common method of analysis including singular point often used in hydraulic
engineering is applied. The singular point is the point at which both functions ),( 01 rhf
and ),( 02 rhf in Equation (4.58) are equal to zero. The equation 0),( 01 =rhf
expresses the quasi-normal depth line whereas 0),( 02 =rhf expresses critical depth line
as shown in Figure 4.6. The water surface profile is derived from Equation (4.58) by
using the fourth-order Runge-Kutta scheme with the initial slope near the singular point
defined by the following equation:
47
S
SSSSSS
S
h
f
r
f
h
f
h
f
r
f
h
f
r
f
dr
dh
∂
∂
∂
∂
∂
∂
−⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂
+
∂
∂
±⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂
+
∂
∂
−
=
2
0
12
2
1
0
21
0
2
0 2
4
(4.62)
(The subscript “s” denotes the derivatives at singular point)
4.3 Results and discussions
Posey and Hsu (1950) and Jain et al. (1978) have reported that a large reduction in intake
discharge was due to formation of vortices, especially in case of flow entering the intake
with entrapped air. The model herein derived has been applied with different imposed
circulations (i.e. different strengths of vortex) and the computed results show reduction of
discharge when the circulation increases while keeping the same submergence (water
head) (Figure 4.7). It is observed that the larger the circulation the larger the air core of
the vortex extends into the intake, and in case of air coming into intake the intake
discharge decrease significantly (from sm331031.0 −× , in case TC01 to
sm331006.0 −× , in case TC04 as in Table 4.1). This phenomenon is in agreement with
the previous experiments (Jain et al. 1978; Posey and Hsu 1950).
Intake discharge is inversely proportional to circulation as shown in Figure 4.8 where the
circle marks are the calculated results and solid line is the trend line calculated by the
least squares method.
The results also show that for small decrease in submergence, the air core becomes larger
(Figure 4.9). Similar results were reported by Anwar et al. (1978). In Figure 4.9, it is
noted that when the circulation increases, the submergence also would increase to
maintain the same intake discharge (see Table 4.2 for reference). Based on these results, it
is evident that the model is capable of representing the effects of circulation on the flow
through vertical intake.
48
Table 4.1 Parameters used in the calculations of results in figure 4.7
Case a
(m)
b
(m2)
Q0ç
(10-3m3/s)
Γ
(m2/s)
f
TC 01 0.025 10-4 0.310 0.182 0.015
TC 02 0.025 10-4 0.230 0.200 0.015
TC 03 0.025 10-4 0.135 0.225 0.015
TC 04 0.025 10-4 0.060 0.250 0.015
-0.05 0.00 0.05 0.10
x (m)
-0.05
0.00
0.05
0.10
0.15
z
(m
)
+++++++++
+
+
+
+
+
+
+
+
+
+ + + + + + + + +
+
+
+
+
+
+
+
+
+
+ + + + TC 01
TC 02
TC 03
TC 04
Bottom
Figure 4.7 The effect of circulation on water surface profile andç discharge at the
intake with same water head
49
Table 4.2 Parameters used in the calculations of results in figure 4.9
Case a
(m)
b
(m2)
Q0ç
(10-3m3/s)
Γ
(m2/s)
f
TC 12 0.02 10-4 0.15 0.30 0.015
TC 13 0.02 10-4 0.15 0.25 0.015
TC 14 0.02 10-4 0.15 0.20 0.015
0.3 0.4 0.5 0.6
0
0.001
0.002
0.003
0.004
Γ (m2/s)
Q
(m
3/
s)
Figure 4.8 Variation of intake discharge with circulation
(a=0.025m, b=10-5 m2, water head=0.5m)
-0.1 0.0 0.1 0.2
x (m)
-0.1
0.0
0.1
0.2
z
(m
)
TC 12
TC 13
TC 14
Bottom
Figure 4.9 Different water surface profiles with different values of circulationç
while maintaining the constant intake discharge
50
The effects of the shape of intake have been examined by changing the value of ''b in
equation (4.1). In this case, three simulations were considered with values of
54 10 5 ;10 −− ×=b and 2510 m− while maintaining the radius of intake (i.e value of ''a
in Equation 4.1) at m2102 −× . It can be seen from Figure 4.10 that the water depth
increases when the intake entrance becomes sharper (decreasing value of b as shown in
Table 4.3).
For more clarity, the computation has been done with the same intake and circulation to
test the effects of value of b on intake discharge without changing the submergence
(Figure 4.11a) and the effects on submergence with same discharge (Figure 4.11b). This
phenomenon is consistent with the physical meaning of the coefficient ''b and it also
improves the performance of bell-mouth intake.
So far we have been able to verify the model qualitatively based on the fact that no
previous studies are available which describe the variation in water surface profile at the
intake. However, a larger quantity of experimental data has been used by Odgaard (1986)
to verify the equation representing the critical submergence, which is defined as the
submergence when the tip of air-core vortex just reaches the intake (Figure 4.12), in the
absence of surface tension (Eq. 18 in Odgaard, 1986) presented as follows:
[ ] 5.05.0)(6.5
2.23
FrN
R
d
S e Γ= (4.63)
where circulation function 0)( QSN Γ=Γ , Froude number ( ) 21gdVFr = and Reynold
number dQ υ=Re .
This equation can be applied for the case of vortex with turbulent core, in which υ is
replaced by Γ+ kυ . From calibration the value of k was then estimated to be
5106 −×≈k .
51
Table 4.3 Parameters used in the calculations of results in figure 4.10
Case a
(m)
b
(m2)
Q0ç
(10-3m3/s)
Γ
(m2/s)
f
TC 15 0.025 1.x10-4 0.3 0.225 0.015
TC 16 0.025 5.x10-5 0.3 0.225 0.015
TC 17 0.025 1.x10-5 0.3 0.225 0.015
0.00000 0.00006 0.00012
b (m2)
0.0000
0.0005
0.0010
0.0015
0.0020
Q
(m
3/
s)
0.00000 0.00004 0.00008 0.00012
b (m2)
0
5
10
15
20
25
H
/d
Figure 4.11 The effects of b on discharge (17a) and submergence (17b) at an intake
-0.1 0.0 0.1 0.2
-0.10
0.00
0.10
0.20
z
(m
)
TC 15
TC 16
TC 17
Bottom
Figure 4.10 Changing of water surface profile with different shape of the intake
52
x
z
S
d
Figure 4.12 Definition sketch of critical submergence
100 101 102
Computed Critical Submergence (S/d)
100
101
102
O
dg
aa
rd
's
C
rit
ic
al
S
ub
m
er
ge
nc
e
(S
/d
)
Figure 4.13 Comparison of computed critical submergence by the model
(Equat

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