Presentation on theme: "Chapter 13: Momentum Principles in Open-Channel"— Presentation transcript:
1 Chapter 13: Momentum Principles in Open-Channel CVE 341 – Water ResourcesLecture Notes 4:Chapter 13:Momentum Principlesin Open-Channel
2 Governing Equations in Open Channel Flow 1) Continuity Equation: Q = A1V1 = A2V22) Energy Equation:Energy equation: pipesEnergy equation: open channels
3 Governing Equations in Open Channel Flow 3) Momentum Equation:See also CHAPTER 5 of your text book
4 Momentum Equation in Open Channel Flow F relation can bewritten ash: depth of centroid of the flow areawhere 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 termis known as momentum function (M)
5 Momentum Equation in Open Channel Flow Pressure-Momentum ForceFirst term: dynamic forceSecond term : hydrostatic forceCritical flow condition(obtained by dM / dy = 0):At pt C: momentum flux is miny1 & y2: conjugate depthssatisfied at the minimum value of the momentum-impulse force
6 EXAMPLEA 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.
7 SOLUTION Momentum Momentum-impulse force Critical depth can also be calculated byTo determine critical depth& conjugate depth, M-y diagramis constructed.
8 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 subcriticalWhen y < yc, flow is supercritical– When Fr > 1 flow is supercritical
9 HYDRAULIC JUMPA phenomenon of a sudden water rise is called hydraulic jumpA 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
10 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 watersupply system.See your text book for other applications
11 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 depthy2: actual subcritical depth in the channel* Compare: y’1 > y2 ↔ y’2 > y1For jump: supercritical depth must increasefrom y1 to y’2*Jump will move downstream until y’2 isachieved “running jump”In the opposite case, jump tends to moveupstream.Momentum and conjugate depthrelationships for the hydraulic jump.
12 Conjugate or Sequent Depths Hydraulic jumpforced upstream.(b) Hydraulic jumpoccurring on a steep slope.
13 Conjugate or Sequent Depths Conjugate or Sequent Depths y1’=y2 ideal casey1’>y2 the jump moves downstreamy1’<y2 the jump moves downstream
14 Conjugate or Sequent Depths Different possibilities for tail-water and jump rating curves.
15 Conjugate Depths in Rectangular or Wide ChannelsNeglecting friction forces,Momentum equationInserting rectangular relations & doing math manipulations:Four assumptions made!
16 Conjugate Depths v Alternate Depths The loss of energy:∆E = E1-E2Relation between conjugate and alternative depths.Conjugate depths have the same pressure-momentum forceAlternate depths have the same specific energyTwo conjugate depths can never be alternate depths or vice versa
17 Energy Loss in Hydraulic Jump The hydraulic jumps involve considerable reduction inthe velocity head & increase in the static headthe energy loss per unit weight of waterEnergy Loss in Rectangular channel
18 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 designedaccording to the hydraulic jump.Length of the hydraulicjump (USBR).Lr: length of roller( )Lj
20 Classification of Hydraulic Jumps Undular Jump (1<Fr1<1.7): The water surface exhibits slight undulation. Two conjugate depths are closeWeak Jump (1.7<Fr1<2.5): A number of small eddies and rollers are formedOscillating 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 banksStable 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
21 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.