Presentation on theme: "Chapter 13: Open Channel Flow"— Presentation transcript:
1 Chapter 13: Open Channel Flow Eric G. PatersonDepartment of Mechanical and Nuclear EngineeringThe Pennsylvania State UniversitySpring 2005
2 Note to InstructorsThese slides were developed1, during the spring semester 2005, as a teaching aid for the undergraduate Fluid Mechanics course (ME33: Fluid Flow) in the Department of Mechanical and Nuclear Engineering at Penn State University. This course had two sections, one taught by myself and one taught by Prof. John Cimbala. While we gave common homework and exams, we independently developed lecture notes. This was also the first semester that Fluid Mechanics: Fundamentals and Applications was used at PSU. My section had 93 students and was held in a classroom with a computer, projector, and blackboard. While slides have been developed for each chapter of Fluid Mechanics: Fundamentals and Applications, I used a combination of blackboard and electronic presentation. In the student evaluations of my course, there were both positive and negative comments on the use of electronic presentation. Therefore, these slides should only be integrated into your lectures with careful consideration of your teaching style and course objectives.Eric PatersonPenn State, University ParkAugust 20051 This Chapter was not covered in our class. These slides have been developed at the request of McGraw-Hill
3 ObjectivesUnderstand how flow in open channels differs from flow in pipesLearn the different flow regimes in open channels and their characteristicsPredict if hydraulic jumps are to occur during flow, and calculate the fraction of energy dissipated during hydraulic jumpsLearn how flow rates in open channels are measured using sluice gates and weirs
4 Classification of Open-Channel Flows Open-channel flows are characterized by the presence of a liquid-gas interface called the free surface.Natural flows: rivers, creeks, floods, etc.Human-made systems: fresh-water aqueducts, irrigation, sewers, drainage ditches, etc.
5 Classification of Open-Channel Flows In an open channel,Velocity is zero on bottom and sides of channel due to no-slip conditionVelocity is maximum at the midplane of the free surfaceIn most cases, velocity also varies in the streamwise directionTherefore, the flow is 3DNevertheless, 1D approximation is made with good success for many practical problems.
6 Classification of Open-Channel Flows Flow in open channels is also classified as being uniform or nonuniform, depending upon the depth y.Uniform flow (UF) encountered in long straight sections where head loss due to friction is balanced by elevation drop.Depth in UF is called normal depth yn
7 Classification of Open-Channel Flows Obstructions cause the flow depth to vary.Rapidly varied flow (RVF) occurs over a short distance near the obstacle.Gradually varied flow (GVF) occurs over larger distances and usually connects UF and RVF.
8 Classification of Open-Channel Flows Like pipe flow, OC flow can be laminar, transitional, or turbulent depending upon the value of the Reynolds numberWhere = density, = dynamic viscosity, = kinematic viscosityV = average velocityRh = Hydraulic Radius = Ac/pAc = cross-section areaP = wetted perimeterNote that Hydraulic Diameter was defined in pipe flows as Dh = 4Ac/p = 4Rh (Dh is not 2Rh, BE Careful!)
9 Classification of Open-Channel Flows The wetted perimeter does not include the free surface.Examples of Rh for common geometries shown in Figure at the left.
10 Froude Number and Wave Speed OC flow is also classified by the Froude numberResembles classification of compressible flow with respect to Mach number
11 Froude Number and Wave Speed Critical depth yc occurs at Fr = 1At low flow velocities (Fr < 1)Disturbance travels upstreamy > ycAt high flow velocities (Fr > 1)Disturbance travels downstreamy < yc
12 Froude Number and Wave Speed Important parameter in study of OC flow is the wave speed c0, which is the speed at which a surface disturbance travels through the liquid.Derivation of c0 for shallow-waterGenerate wave with plungerConsider control volume (CV) which moves with wave at c0
13 Froude Number and Wave Speed Continuity equation (b = width)Momentum equation
14 Froude Number and Wave Speed Combining the momentum and continuity relations and rearranging givesFor shallow water, where y << y,Wave speed c0 is only a function of depth
15 Specific EnergyTotal mechanical energy of the liquid in a channel in terms of headsz is the elevation heady is the gage pressure headV2/2g is the dynamic headTaking the datum z=0 as the bottom of the channel, the specific energy Es is
16 Specific Energy For a channel with constant width b, Plot of Es vs. y for constant V and b
17 Specific Energy This plot is very useful Easy to see breakdown of Es into pressure (y) and dynamic (V2/2g) headEs as y 0Es y for large yEs reaches a minimum called the critical point.There is a minimum Es required to support the given flow rate.Noting that Vc = sqrt(gyc)For a given Es > Es,min, there are two different depths, or alternating depths, which can occur for a fixed value of EsA small change in Es near the critical point causes a large difference between alternate depths and may cause violent fluctuations in flow level. Operation near this point should be avoided.
18 Continuity and Energy Equations 1D steady continuity equation can be expressed as1D steady energy equation between two stationsHead loss hL is expressed as in pipe flow, using the friction factor, and either the hydraulic diameter or radius
19 Continuity and Energy Equations The change in elevation head can be written in terms of the bed slope Introducing the friction slope SfThe energy equation can be written as
20 Uniform Flow in Channels Uniform depth occurs when the flow depth (and thus the average flow velocity) remains constantCommon in long straight runsFlow depth is called normal depth ynAverage flow velocity is called uniform-flow velocity V0
21 Uniform Flow in Channels Uniform depth is maintained as long as the slope, cross-section, and surface roughness of the channel remain unchanged.During uniform flow, the terminal velocity reached, and the head loss equals the elevation dropWe can the solve for velocity (or flow rate)Where C is the Chezy coefficient. f is the friction factor determined from the Moody chart or the Colebrook equation
22 Best Hydraulic Cross Sections Best hydraulic cross section for an open channel is the one with the minimum wetted perimeter for a specified cross section (or maximum hydraulic radius Rh)Also reflects economy of building structure with smallest perimeter
23 Best Hydraulic Cross Sections Example: Rectangular ChannelCross section area, Ac = ybPerimeter, p = b + 2ySolve Ac for b and substituteTaking derivative with respect toTo find minimum, set derivative to zeroBest rectangular channel hasa depth 1/2 of the width
24 Best Hydraulic Cross Sections Same analysis can be performed for a trapezoidal channelSimilarly, taking the derivative of p with respect to q, shows that the optimum angle isFor this angle, the best flow depth is
25 Gradually Varied FlowIn GVF, y and V vary slowly, and the free surface is stableIn contrast to uniform flow, Sf S0. Now, flow depth reflects the dynamic balance between gravity, shear force, and inertial effectsTo derive how how the depth varies with x, consider the total head
26 Gradually Varied Flow Take the derivative of H Slope dH/dx of the energy line is equal to negative of the friction slopeBed slope has been definedInserting both S0 and Sf gives
27 Gradually Varied FlowIntroducing continuity equation, which can be written asDifferentiating with respect to x givesSubstitute dV/dx back into equation from previous slide, and using definition of the Froude number gives a relationship for the rate of change of depth
28 Gradually Varied FlowThis result is important. It permits classification of liquid surface profiles as a function of Fr, S0, Sf, and initial conditions.Bed slope S0 is classified asSteep : yn < ycCritical : yn = ycMild : yn > ycHorizontal : S0 = 0Adverse : S0 < 0Initial depth is given a number1 : y > yn2 : yn < y < yc3 : y < yc
29 Gradually Varied Flow12 distinct configurations for surface profiles in GVF.
30 Gradually Varied FlowTypical OC system involves several sections of different slopes, with transitionsOverall surface profile is made up of individual profiles described on previous slides
31 Rapidly Varied Flow and Hydraulic Jump Flow is called rapidly varied flow (RVF) if the flow depth has a large change over a short distanceSluice gatesWeirsWaterfallsAbrupt changes in cross sectionOften characterized by significant 3D and transient effectsBackflowsSeparations
32 Rapidly Varied Flow and Hydraulic Jump Consider the CV surrounding the hydraulic jumpAssumptionsV is constant at sections (1) and (2), and 1 and 2 1P = gyw is negligible relative to the losses that occur during the hydraulic jumpChannel is wide and horizontalNo external body forces other than gravity
33 Rapidly Varied Flow and Hydraulic Jump Continuity equationX momentum equationSubstituting and simplifyingQuadratic equation for y2/y1
34 Rapidly Varied Flow and Hydraulic Jump Solving the quadratic equation and keeping only the positive root leads to the depth ratioEnergy equation for this section can be written asHead loss associated with hydraulic jump
35 Rapidly Varied Flow and Hydraulic Jump Often, hydraulic jumps are avoided because they dissipate valuable energyHowever, in some cases, the energy must be dissipated so that it doesn’t cause damageA measure of performance of a hydraulic jump is its fraction of energy dissipation, or energy dissipation ratio
36 Rapidly Varied Flow and Hydraulic Jump Experimental studies indicate that hydraulic jumps can be classified into 5 categories, depending upon the upstream Fr
37 Flow Control and Measurement Flow rate in pipes and ducts is controlled by various kinds of valvesIn OC flows, flow rate is controlled by partially blocking the channel.Weir : liquid flows over deviceUnderflow gate : liquid flows under deviceThese devices can be used to control the flow rate, and to measure it.
38 Flow Control and Measurement Underflow Gate Underflow gates are located at the bottom of a wall, dam, or open channelOutflow can be either free or drownedIn free outflow, downstream flow is supercriticalIn the drowned outflow, the liquid jet undergoes a hydraulic jump. Downstream flow is subcritical.Free outflowDrowned outflow
39 Flow Control and Measurement Underflow Gate Schematic of flow depth-specificenergy diagram for flow throughunderflow gatesEs remains constant for idealized gates with negligible frictional effectsEs decreases for real gatesDownstream is supercritical for free outflow (2b)Downstream is subcritical for drowned outflow (2c)
40 Flow Control and Measurement Overflow Gate Specific energy over a bump at station 2 Es,2 can be manipulated to giveThis equation has 2 positive solutions, which depend upon upstream flow.
41 Flow Control and Measurement Broad-Crested Weir Flow over a sufficiently high obstruction in an open channel is always criticalWhen placed intentionally in an open channel to measure the flow rate, they are called weirs
42 Flow Control and Measurement Sharp-Crested V-notch Weirs Vertical plate placed in a channel that forces the liquid to flow through an opening to measure the flow rateUpstream flow is subcritical and becomes critical as it approaches the weirLiquid discharges as a supercritical flow stream that resembles a free jet
43 Flow Control and Measurement Sharp-Crested V-notch Weirs Flow rate equations can be derived using energy equation and definition of flow rate, and experimental for determining discharge coefficientsSharp-crested weirV-notch weirwhere Cwd typically ranges between 0.58 and 0.62