Lecture10 Momentum Principle
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Transcript of Lecture10 Momentum Principle
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The Momentum Principle
Hydromechanics VVR090
Hydraulic Jump
(photo taken 2008-02-24 in M. Larson s
kitchen sink)
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Specific Momentum
( )' ' ' ' ' '1 3 2 2 2 1 1fF F F P Q u ug
+ =
Horizontal momentum equation is applied to a controlvolume in an open channel (x-direction):
Small s lopes:
( )1 1 2 2 2 1fz A z A P Q u ug
=
2 2
1 1 2 2
1 2
1 2
2
f
f
P Q Qz A z A
gA gA
PM M
QM zA
gA
= + +
=
= +
Continuity equation:
1 2
1 2
,Q Q
u uA A
= =
Rearranging the momentum equation:
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Specific Momentum Curve
Analogous to the speci fic energy curve.
For a given value on M there are in general two
possib le water depths (sequent depths of ahydraulic jump.
The minimum value of M corresponds to the
minimum value of E.
The Hydraulic Jump
A hydraulic jump results when there is a confl ict between
upstream and downstream control.
Example: upstream control induce supercritical flow and
downstream control subcritical flow.
=> flow has to pass from one regime to another
creating a hydraulic jump
Significant turbulence and energy dissipation occurs in a
hydraulic jump.
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Appl ications of the Hydraulic Jump
1. Dissipation of energy in flows over dams, weirs, and otherhydraulic structures
2. Maintenance of high water levels in channels for water
distribution
3. Increase of discharge of a sluice gate by repelling the
downstream tailwater
4. Reduction of uplift pressure under structures by raising
the water depth
5. Mixing of chemicals
6. Aeration of flow
7. Removal of air pockets
8. Flow measurements
Hydraulic Jump Dependence on Froude Number
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Hydraulic Jump in a Sink
Governing Equation for a Hydraulic Jump I
Apply the momentum equation for a fi ni te control
volume encompassing the jump and let Pf = 0:
1 2
2 2
1 1 2 2
1 2
M M
Q Qz A z A
gA gA
=
+ = +
For a rectangular section:
1 1 2 2
1 1 2 2
1 1 2 2
,
/ 2, / 2
Q u A u A
A by A by
z y z y
= =
= =
= =
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Governing Equation for a Hydraulic Jump II
Momentum equation (rectangular channel):
( )2 2
1 22 1
2 2
y y qu u
g
=
Momentum equation for rectangular section:
( )2
2 2
2 1
1 2
1 1 1
2
qy y
g y y
=
Solutions:
( )
( )
221
1
21
2
2
11 8 1
2
11 8 1
2
yFr
y
yFr
y
= +
= +
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Downstream Froude number (Fr2) usually small so:
2
21 8 1 0Fr+
Sometimes useful to do a Taylor series expansion:
2 2 4 6
2 2 2 21 8 1 4 8 32 ...Fr Fr Fr Fr + = + + +
Substituting into the hydraulic jump equation:
2 4 61
2 2 2
2
2 4 16 ...y
Fr Fr Fr y
= + +
2 31 1 11 1 ...
2 8 16x x x x+ = + + +( )
Submerged Jump
If the downstream water level is high enough (> sequent
depth y2) the jump might be submerged. The depth of
submergence (y3) is then a crucial unknown.
Could occur downstream gates etc.
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Submergence of Hydraulic Jump
Govinda (1963) proposed the foll owing formula for
submerged jumps (horizontal, rectangular channels):
Chow (1959):
( )( )
( )
1/ 22
2 2 23 11
1
4 1
2
221
1
21 2
1
11 8 1
2
y FrS Fr
y S
y yS
y
yFr
y
= + +
+
=
= = +
1/ 2
23 4
4
4 1
1 2 1y y
Fry y
= +
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Energy Loss in a Hydraulic Jump I
Dissipation in a horizontal channel:
1 2E E E =
Rectangular section:
( )
( ) ( )
( )( )
3
2 1
1 2
22
2 1 1 1 2
2
1 1
3/ 22 2
1 11
2 22 1 1
4
2 2 / 1 /
2
8 1 4 1
8 2
y yE
y y
y y Fr y yE
E F
Fr Fr E
E Fr Fr
=
+ =+
+ +=
+
Energy Loss in a Hydraulic Jump II
( )
2 2
1 21 2
2 2
1 11 22
2
22 12 1 1 2
2 1
2 2
12
44
u uy y E
g g
u yE y y
g y
y yE y y y yy y
+ = + +
= +
= +
Energy equation across the jump:
( )3
2 1
1 24
y yE
y y
=
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Hydraulic Jump Length I
Jump length hard to derive through theoretical
considerations.
Length of the jump (Lj) is defined as the distance from the
front face of the jump to a poin t on the surface immediately
downstream the roller associated with the jump.
Hydraulic Jump Length II
Empirical equation for jump length g iven by Silvester (1964)
for rectangular channels,
( )1.01
1
1
9.75 1j
LFr
y=
and for triangular channels:
( )0.695
1
1
4.26 1jL
Fry =
Submerged jump:
2
4.9 6.1j
LS
y= +
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Hydraulic Jump in Sloping Channels
A number of di fferent cases have to be considered.