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FUNDAMENTALS of FLUID MECHANICS Chapter 10 Flow In FUNDAMENTALSFUNDAMENTALS OFOF FLUIDFLUID MECHANICSMECHANICS ChapterChapter 1010 FlowFlow inin OpenOpen ChannelsChannels JyhJyh--CherngCherng ShiehShieh Department of Bio-Industrial Mechatronics Engineering National Taiwan University 1 MAINMAIN TOPICSTOPICS GeneralGeneral CharacteristicsCharacteristics ofof OpenOpen--ChannelChannel FlowFlow SurfaceSurface WavesWaves EnergyEnergy ConsiderationsConsiderations UniformUniform DepthDepth ChannelChannel FlowFlow GraduallyGradually VariesVaries FlowFlow RapidlyRapidly VariesVaries FlowFlow 2 IntroductionIntroduction Open channel flow involves the flows of a liquid in a channel or conduit that is not completely filled. There exists a free surface between the flowing fluid (usually water) and fluid above it (usually the atmosphere). The main deriving force is the fluid weight-gravity forces the fluid to flow downhill. Under steady, fully developed flow conditions, the component if the weight force in the direction of flow is balanced by the equal and opposite shear force between the fluid and the channel surface. 3 OpenOpen ChannelChannel FlowFlow vs.vs. PipePipe FlowFlow There can be no pressure force driving the fluid through the channel or conduit. For steady, fully developed channel flow, the pressure distribution within the fluid is merely hydrostatic. 4 ExamplesExamples ofof OpenOpen ChannelChannel FlowFlow The natural drainage of water through the numerous creek and river systems. The flow of rainwater in the gutters of our houses. The flow in canals, drainage ditches, sewers, and gutters along roads. The flow of small rivulets, and sheets of water across fields or parking lots. The flow in the chutes of water rides. 5 VariablesVariables inin OpenOpen--ChannelChannel FlowFlow Cross-sectional shape. Bends. Bottom slope variation. Character of its bounding surface. Most open-channel flow results are based on correlation obtained from model and full-scale experiments. Additional information can be gained from various analytical and numerical efforts. 6 GeneralGeneral CharacteristicsCharacteristics ofof OpenOpen--ChannelChannel FlowFlow 7 ClassificationClassification ofof OpenOpen--ChannelChannel FlowFlow For open-channel flow, the existence of a free surface allows additional types of flow. The extra freedom that allows the fluid to select its free-surface location and configuration allows important phenomena in open- channel flow that cannot occur in pipe flow. The fluid depth, y, varies with time, t, and distance along the channel, x, are used to classify open-channel flow: 8 ClassificationClassification -- TypeType II Uniform flow (UF): The depth of flow does not vary along the channel (dy/dx=0). Nonuniform flows: ►Rapidly varying flows (RVF): The flow depth changes considerably over a relatively short distance dy/dx~1. ►Gradually varying flows (GVF): The flow depth changes slowly with distance dy/dx <<1. 9 ClassificationClassification -- TypeType IIII R = ρVR / μ V is the average velocity of the fluid. e h Rh is the hydraulic radius of the channel. Laminar flow: Re < 500. Transitional flow: Turbulent flow: Re > 12,500. Most open-channel flows involve water (which has a fairly small viscosity) and have relatively large characteristic lengths, it is uncommon to have laminar open-channel flows. 10 ClassificationClassification -- TypeType IIIIII Fr = V / gl Critical Flow: Froude number Fr =1. Subcritical Flow: Froude number Fr <1. Supercritical Flow: Froude number Fr >1. 11 SurfaceSurface WaveWave 12 SurfaceSurface WaveWave The distinguishing feature of flows involve a free surface (as in open-channel flows) is the opportunity for the free surface to distort into various shapes. The surface of a lake or the ocean is usually distorted into ever- changing patterns associated with surface waves. 13 KindsKinds ofof SurfaceSurface WaveWave Some of the surface waves are very high, some barely ripple the surface; some waves are very long, some are short; some are breaking wave that form white caps, others are quite smooth. Dsmall amplitude DFinite-sized solitary DContinuous sinusoidal shape 14 smallsmall amplitudeamplitude WaveWave SpeedSpeed 1/51/5 Consider a single elementary wave of small height, by δy, is produced on the surface of a channel by suddenly moving the initially stationary end wall with speed δV. 15 smallsmall amplitudeamplitude WaveWave SpeedSpeed 2/52/5 The water in the channel was stationary at the initial time, t=0. A stationary observer will observe a single wave move down the channel with a wave speed c, with no fluid motion ahead of the wave and a fluid velocity of δV behind the wave. The motion is unsteady. For a observer moving along the channel with speed c, the flow will appear steady. v v To this observer, the fluid velocity will be V = -c i on the observer’s v v right and V = (-c + δ V) i to the left of the observer. MomentumMomentum EquationEquation ++ ContinuityContinuity EquationEquation 16 smallsmall amplitudeamplitude WaveWave SpeedSpeed 3/53/5 With the assumption of uniform one-dimensional flow, the continuity equation becomes − cyb = (−c + δV)(y + δy)b (y + δy)δV δV ⇒ c = = y (1) δy δy δy << y Similarly, the momentum equation 1 1 F = γy A = γy2b F = γy A = γ(y + δy)2 b 2 c2 2 2 1 c1 1 2 1 1 F − F = γy2b − γ(y + δy)2 b = ρbcy[](c − δV) − c 2 1 2 2 17 smallsmall amplitudeamplitude WaveWave SpeedSpeed 4/54/5 (δy)2 << yδy δV g ⇒ = (2) δy c (1)+(2) c = gy (3) EnergyEnergy EquationEquation ++ ContinuityContinuity EquationEquation 18 smallsmall amplitudeamplitude WaveWave SpeedSpeed 5/55/5 The single wave on the surface is seen by an observer moving with the wave speed, c. Since the pressure is constant at any point on the free surface, the Bernoulli equation for this frictionless flow is V2 VδV + y = cons tan t → + δy = 0 2g g The continuity equation Vy = cons tan t → yδV + Vδy = 0 Combining these two equations and using the fact V=c c = gy 19 finitefinite--sizedsized solitarysolitary WaveWave SpeedSpeed More advanced analysis and experiments show that the wave speed for finite-sized solitary wave 1/ 2 1/ 2 ⎛ δy ⎞ ⎛ δy ⎞ c = gy⎜1+ ⎟ c = gy⎜1+ ⎟ > gy ⎝ y ⎠ ⎝ y ⎠ The larger the amplitude, the faster the wave travel. 20 ContinuousContinuous sinusoidalsinusoidal shapeshape WaveWave SpeedSpeed 1/21/2 A more general description of wave motion can be obtained by considering continuous (not solitary) wave of sinusoidal shape. DBy combining waves of various wavelengths, λ, and amplitudes, δy. DThe wave speed varies with both the wavelength and fluid depth as 1/ 2 ⎡gλ ⎛ 2πy ⎞⎤ c =& ⎢ tanh⎜ ⎟⎥ (4) ⎣2π ⎝ λ ⎠⎦ 21 ContinuousContinuous sinusoidalsinusoidal shapeshape WaveWave SpeedSpeed 2/22/2 1/ 2 ⎡gλ ⎛ 2πy ⎞⎤ c =& ⎢ tanh⎜ ⎟⎥ ⎣2π ⎝ λ ⎠⎦ y ⎛ 2πy ⎞ → ∞ ⇒ tanh⎜ ⎟ = 1 λ ⎝ λ ⎠ gλ ⇒ c = Deep layer 2π y ⎛ 2πy ⎞ 2πy → 0 ⇒ tanh⎜ ⎟ = λ ⎝ λ ⎠ λ ⇒ c = gy Shallow layer Wave speed as a function of wavelength.22 FroudeFroude NumberNumber EffectsEffects 1/31/3 Consider an elementary wave travelling on the surface of a fluid. If the fluid layer is stationary, the wave moves to the right with speed c relative to the fluid and stationary observer. When the fluid is flowing to the left with velocity V. DIf V<c, the wave will travel to the right with a speed of c-V. DIf V=c, the wave will remain stationary. DIf V>c, the wave will be washed to the left with a speed of V-c. Froude Number Fr = V / g = V / c Is the ratio of the fluid l velocity to the wave speed. 23 FroudeFroude NumberNumber EffectsEffects 2/32/3 When a wave is produced on the surface of a moving stream, as happens when a rock is thrown into a river. DIf V=0, the wave speeds equally in all directions. DIf V<c, the wave can move upstream. Upstream locations are said to be in hydraulic communication with the downstream locations. Such flow conditions, V<c, or Fr<1, are termed subcritical. DIf V>c, no upstream communication with downstream locations. Any disturbance on the surface downstream from the observer will be washed farther downstream. Such conditions, V>c, or Fr>1, are termed supercritical. 24 FroudeFroude NumberNumber EffectsEffects 3/33/3 DIf V=c or Fr=1, the upstream propagating wave remains stationary and the flow is termed critical. 25 EnergyEnergy ConsiderationsConsiderations 26 EnergyEnergy ConsiderationsConsiderations 1/31/3 z1 − z 2 The slope of the channel bottom or bottom S0 = l slope is constant over the segment Very small for most open-channel flows. x and y are taken as the distance along the channel bottom and the depth normal to the bottom. 27 EnergyEnergy ConsiderationsConsiderations 2/32/3 With the assumption of a uniform velocity profile across any section of the channel, the one-dimensional energy equation become 2 2 p1 V1 p2 V2 + + z = + + z + h (5) γ 2g 1 γ 2g 2 L hL is the head loss due to viscous effects between sections (1) and (2). 2 2 V1 V2 (5) y1 + + Sol = y2 + + h L (6) 2g 2g z1 − z2 = Sol p1 / γ = y1 p2 / γ = y2 28 EnergyEnergy ConsiderationsConsiderations 3/33/3 2 2 V2 − V1 (6) y1 − y2 = + (Sf − So )l (7) 2g Sf = hL / l V2 − V2 (7) y − y = 2 1 1 2 2g For a horizontal channel bottom (S0=0) and negligible head loss (Sf=0) 29 SpecificSpecific EnergyEnergy 1/41/4 Define specific energy, E V2 E = y + (8) 2g (9) E1 = E2 + (Sf − So )l (9) E1 + z1 = E2 + z2 The sum of the specific energy and the elevation of the channel bottom remains constant. Head losses are negligible, Sf=0 This a statement of the Bernoulli −Sol = z2 − z1 equation. 30 SpecificSpecific EnergyEnergy 2/42/4 If the cross-sectional shape is a rectangular of width b q2 E = y + (10) 2gy2 Where q is the flowrate per unit width, q=Q/b=Vyb/b=Vy For a given channel b= constant q = constant E = E (y) Specific energy diagram 31 SpecificSpecific EnergyEnergy 3/43/4 q2 E = y + (10) 2gy2 For a given q and E, equation (10) is a cubic equation with three solutions , ysup, ysub, and yneg.
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