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

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Governing Equations in Open Channel Flow 1) Continuity Equation: Q = A 1 V 1 = A 2 V 2 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 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) h: depth of centroid of the flow area F relation can be written as

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Momentum Equation in Open Channel Flow Critical flow condition (obtained by dM / dy = 0): satisfied at the minimum value of the momentum-impulse force Pressure-Momentum Force First term: dynamic force Second term : hydrostatic force At pt C: momentum flux is min y 1 & y 2 : conjugate depths

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EXAMPLE A 2.0 m wide rectangular channel carries a discharge of 4.0 m 3 /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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When the depth in a channel is y c flow is critical When y > y c, flow is subcritical – When Fr < 1 flow is subcritical When y < y c, flow is supercritical – When Fr > 1 flow is supercritical Classifying Critical Flow

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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 y 1, which is lower than critical depth, to another depth y 2, 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 (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

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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. Momentum and conjugate depth relationships for the hydraulic jump. y 1 : initial supercritical depth y 2 : actual subcritical depth in the channel * Compare: y’ 1 > y 2 ↔ y’ 2 > y 1 For jump: supercritical depth must increase from y 1 to y’ 2 *Jump will move downstream until y’2 is achieved. “running jump” In the opposite case, jump tends to move upstream.

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Conjugate or Sequent Depths (b) Hydraulic jump occurring on a steep slope. (a)Hydraulic jump forced upstream.

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Conjugate or Sequent Depths y 1 ’ =y 2 ideal case y 1 ’ >y 2 the jump moves downstream y 1 ’

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Different possibilities for tail-water and jump rating curves. Conjugate or Sequent Depths

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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 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 = E 1 -E 2

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

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Geometry of Hydraulic Jumps Efficiency of the hydraulic jump: E 1 /E 2 ► 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). L r : length of roller ( )L j

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Classification of Hydraulic Jumps Undular Jump (1

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Classification of Hydraulic Jumps Undular Jump (1

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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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