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**Chapter 13: Momentum Principles in Open-Channel**

CVE 341 – Water Resources Lecture Notes 4: Chapter 13: Momentum Principles in Open-Channel

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**Governing Equations in Open Channel Flow**

1) Continuity Equation: Q = A1V1 = A2V2 2) Energy Equation: Energy equation: pipes Energy equation: open channels

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**Governing Equations in Open Channel Flow**

3) Momentum Equation: See also CHAPTER 5 of your text book

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**Momentum Equation in Open Channel Flow**

F relation can be written as 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 is known as momentum function (M)

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**Momentum Equation in Open Channel Flow**

Pressure-Momentum Force First term: dynamic force Second term : hydrostatic force Critical flow condition (obtained by dM / dy = 0): At pt C: momentum flux is min y1 & y2: conjugate depths satisfied at the minimum value of the momentum-impulse force

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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.

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**SOLUTION Momentum Momentum-impulse force Critical depth**

can also be calculated by To determine critical depth & conjugate depth, M-y diagram is constructed.

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**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

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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

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**Some practical applications of hydraulic jump**

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

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**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.

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**Conjugate or Sequent Depths**

Hydraulic jump forced upstream. (b) Hydraulic jump occurring on a steep slope.

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**Conjugate or Sequent Depths Conjugate or Sequent Depths**

y1’=y2 ideal case y1’>y2 the jump moves downstream y1’<y2 the jump moves downstream

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**Conjugate or Sequent Depths**

Different possibilities for tail-water and jump rating curves.

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**Conjugate Depths in Rectangular**

or Wide Channels Neglecting friction forces, Momentum equation Inserting rectangular relations & doing math manipulations: Four assumptions made!

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**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

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**Energy Loss in Hydraulic Jump**

The hydraulic jumps involve considerable reduction in the velocity head & increase in the static head the energy loss per unit weight of water Energy Loss in Rectangular channel

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**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 ( )Lj

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**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) y2/y1=3-6 Stable Jump (4.5<Fr1<9) Strong Jump (Fr1>9) y2/y1=12-20 y2/y1=6-12

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**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

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**EXAMPLE: A hydraulic jump is formed in a trapezoidal channel of 2**

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.

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