Download CVE 240 – Fluid Mechanics

CVE 341 – Water Resources
Lecture Notes 4:
Chapter 13:
Momentum Principles
in Open-Channel
Governing Equations in Open Channel Flow
1) Continuity Equation:
Q = A1V1 = A2V2
2) Energy Equation:
Energy equation: pipes
Energy equation: open channels
Governing Equations in Open Channel Flow
3) Momentum Equation:
See also CHAPTER 5 of your text book
F  ma
F  Q(V2  V1 )
Momentum Equation in Open Channel Flow
F relation can be
written as


Q2
Q2
 A1 h1 
 A2 h 2
gA1
gA2
h: depth of centroid of the flow area
where A is the cross-sectional area of flow and h is the depth of
centroid of the flow area below the water surface and g is the
acceleration term

Q
 A1 h1
gA1
2
is known as momentum function (M)
Momentum Equation in Open Channel Flow
 
 Q2
F 
 A1 h1 
 gA1

Pressure-Momentum Force
First term: dynamic force
Second term : hydrostatic force
Critical flow condition
(obtained by dM / dy = 0):
Q2B
1
gA 3
At pt C: momentum flux is min
y1 & y2: conjugate depths
satisfied at the minimum value of the
momentum-impulse force
EXAMPLE
A 2.0 m wide rectangular channel carries a discharge of 4.0
m3/s with a depth of flow of 1.0 m. Determine the momentumimpulse force, the critical depth, and the conjugate depth.
SOLUTION

Q
M 
 A2 h 2
gA2
2
 
Q
F 
 A1 h1 
 gA1

Momentum
2
Momentum-impulse force
Critical depth
To determine critical depth
& conjugate depth, M-y diagram
is constructed.
can also be calculated by
2
q
yc  3
g
Classifying Critical Flow
•When the depth in a channel is yc flow is critical
• When y > yc, flow is subcritical
– When Fr < 1 flow is subcritical
• When y < yc, flow is supercritical
– When Fr > 1 flow is supercritical
HYDRAULIC JUMP
A phenomenon of a sudden water rise is called hydraulic jump
A hydraulic jump is formed only if the depth of flow is forced to
change from a depth y1, which is lower than critical depth, to
another depth y2, which is higher than the critical depth.
If the state of flow is changed from supercritical to
subcritical flow
Some practical applications of hydraulic jump
(a) to dissipate the high kinetic energy of water near the toe of
the spillway and to protect the bed and banks of a river near
a hydraulic structure
(b) To increase water level in canals to enhance irrigation
practices and reduce pumping head
(c) Mixing of chemicals and removing of air pockets in water
supply system.
See your text book for other applications
Conjugate or Sequent Depths
Initial and final depths of a hydraulic jump are called conjugate or sequent
depths in the sense that they occur simultaneously.
y1: initial supercritical depth
y2: actual subcritical depth in the channel
* Compare: y’1 > y2 ↔ y’2 > y1
For jump: supercritical depth must increase
from y1 to y’2
*Jump will move downstream until y’2 is
achieved.
“running jump”
• In the opposite case, jump tends to move
upstream.
Momentum and conjugate depth
relationships for the hydraulic jump.
Conjugate or Sequent Depths
(a) Hydraulic jump
forced upstream.
(b) Hydraulic jump
occurring on a steep slope.
Conjugate or Sequent Depths
y1’=y2 ideal case
y1’>y2 the jump moves downstream
y1’<y2 the jump moves downstream
Conjugate or Sequent Depths
Different possibilities for tail-water and jump rating curves.
Conjugate Depths in Rectangular
or Wide Channels


Q2
Q2
 A1 h1 
 A2 h 2
gA1
gA2
Neglecting friction forces,
Momentum equation
Inserting rectangular relations
& doing math manipulations:
Fr12 
2
2
8Fr
 1  8Fr
2
2

1
3

y1
y2 
 1  1  8Fr12
2


y2
y1 
 1  1  8Fr22
2
Four assumptions made!

Conjugate Depths v Alternate Depths
The loss of energy:
∆E = E1-E2
Relation between conjugate and alternative depths.
Conjugate depths have the same pressure-momentum force
Alternate depths have the same specific energy
Two conjugate depths can never be alternate depths or vice versa
Energy Loss in Hydraulic Jump
The hydraulic jumps involve considerable reduction in
the velocity head & increase in the static head

V12  
V22 
   y 2 

E   y1 
2g  
2g 

the energy loss per unit weight of water
Energy Loss in Rectangular channel
y 2  y 1 
3
E 
4 y1 y 2
Geometry of Hydraulic Jumps
Efficiency of the hydraulic jump: E1/E2
► Hydraulic jumps cause intensive scour at their locations
► They should contained in stilling basin.
► Apron length & height of side walls of a stilling basin are designed
according to the hydraulic jump.
Length of the hydraulic
jump (USBR).
Lr: length of roller
(0.4-0.7)Lj
Classification of Hydraulic Jumps
Undular Jump
(1<Fr1<1.7)
Stable Jump
(4.5<Fr1<9)
y2/y1=6-12
Weak Jump
(1.7<Fr1<2.5)
y2/y1=2-3
Strong Jump
(Fr1>9)
y2/y1=12-20
Oscillating Jump
(2.5<Fr1<4.5)
y2/y1=3-6
Classification of Hydraulic Jumps
Undular Jump (1<Fr1<1.7): The water surface exhibits slight
undulation. Two conjugate depths are close
Weak Jump (1.7<Fr1<2.5): A number of small eddies and
rollers are formed
Oscillating Jump (2.5<Fr1<4.5): The incoming jet oscillates
from the bottom to the top. It should be avoided if it is
possible since it may cause erosion to banks
Stable Jump (4.5<Fr1<9): Has many advantages. Well
balanced jump and the jump location is least sensitive to any
variation in y2.
Strong Jump (Fr1>9): Jump is effective and should not be
allowed to exceed 12 as the required stilling basins would be very
massive and expensive
EXAMPLE: A hydraulic jump is formed in a trapezoidal
channel of 2.0-m bed width, 1:1 side slope, and carrying a
discharge of 6.0 m3/s. Construct the momentum diagram and
Find the critical depth.