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Download Luận văn Hydraulic modeling of open channel flows over an arbitrary 3-D surface and its applications in amenity hydraulic engineering
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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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)
...
Download miễn phí Luận văn Hydraulic modeling of open channel flows over an arbitrary 3-D surface and its applications in amenity hydraulic engineering
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:
Để tải bản DOC Đầy Đủ xin Trả lời bài viết này, Mods sẽ gửi Link download cho bạn sớm nhất qua hòm tin nhắn.
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Tóm tắt nội dung:
pJ 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)
...