CVE 341 – Water Resources

Chapter 13:Momentum Principles

in Open-Channel

Lecture Notes 4:

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

maF )( 12 VVQF

Momentum Equation in Open Channel Flow

222

2

111

2

hAgA

QhA

gA

Q

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

111

2

hAgA

Qis known as momentum function (M)

h: depth of centroid of the flow area

F relation can be written as

Momentum Equation in Open Channel Flow

3

2

1gA

BQ

Critical flow condition(obtained by dM / dy = 0):

satisfied at the minimum value of the momentum-impulse force

111

2

hAgA

QF

Pressure-Momentum ForceFirst term: dynamic forceSecond term : hydrostatic force

At pt C: momentum flux is miny1 & y2: conjugate depths

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 momentum-impulse force, the critical depth, and the conjugate depth.

SOLUTION

222

2

hAgA

QM

111

2

hAgA

QF

3

2

g

qyc

Momentum

Momentum-impulse force

Critical depthcan also be calculated by

To determine critical depth

& conjugate depth, M-y diagram is constructed.

•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

Classifying Critical Flow

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.

Momentum and conjugate depth relationships for the hydraulic jump.

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.

Conjugate or Sequent Depths

(b) Hydraulic jump occurring on a steep slope.

(a) Hydraulic jump forced upstream.

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 or Sequent Depths

Conjugate Depths in Rectangular or Wide Channels

222

2

111

2

hAgA

QhA

gA

Q

22

21 Fr811

2

yy

21

12 Fr811

2

yy

322

222

1

1Fr81

Fr8Fr

Neglecting friction forces,Momentum equation

Inserting rectangular relations & doing math manipulations:

Four assumptions made!

Conjugate Depths v Alternate Depths

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

The loss of energy:

∆E = E1-E2

Energy Loss in Hydraulic Jump

g2

Vy

g2

VyE

22

2

21

1

The hydraulic jumps involve considerable reduction in the velocity head & increase in the static head

Energy Loss in Rectangular channel

21

312

yy4

yyE

the energy loss per unit weight of water

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)

Weak Jump (1.7<Fr1<2.5)y2/y1=2-3

Oscillating Jump (2.5<Fr1<4.5)

Stable Jump (4.5<Fr1<9)

Strong Jump (Fr1>9) y2/y1=12-20

y2/y1=3-6

y2/y1=6-12

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.