OPEN-CHANNEL FLOW
OPEN-CHANNEL FLOW Open-channel flow is a flow of liquid (basicallywater) in a conduit with a free surface.That is a surface on which pressure is equal tolocal atmospheric pressure.PatmFree surfacePatm
Classification of Open-Channel FlowsOpen-channel flows are characterized by the presenceof a liquid-gas interface called the free surface. Natural flows: rivers,creeks, floods, etc. Human-made systems:fresh-water aquaducts,irrigation, sewers,drainage ditches, etc.p patm
Open channelsNatural channelsArtificial channelsOpen cross sectionCovered cross section
Total Head at A Cross Section: The total head at a cross section is:Vav2PH z αγ2g whereH total headZ elevation of the channel bottomP/g y the vertical depth of flow (provided thatpressure distribution is hydrostatic)V2/2g velocity headαV2/2gyEGLQzDatumx
Energy Grade Line & Hydraulic Grade Line inOpen Channel FlowSf :the slope of energy grade lineSw :the slope of the water surfaceSo :the slope of the bottomSf :the slope of energy grade lineSw :the slope of the water surface
Comparison of Open Channel Flow and Pipe Pipe centerliney2z1V22/2gChannel bottomDatum linez2z1Datum line1Open-Channel FlowPipe Flowz22
Comparison of Open Channel Flow & Pipe Flow1)OCF must have a free surface2)A free surface is subject toatmospheric pressure3)The driving force is mainly thecomponent of gravity along theflow direction.4)5)1)No free surface in pipe flow2)No direct atmospheric pressure,hydraulic pressure only.3)The driving force is mainly thepressure force along the flowdirection.HGL is coincident with the freesurface.4)HGL is (usually) above the conduitFlow area is determined by thegeometry of the channel plusthe level of free surface,which is likely to change alongthe flow direction and with aswell as time.5)Flow area is fixed by the pipedimensions The cross section of apipe is usually circular.
Comparision of Open Channel Flow & Pipe Flow6)The cross section may be of anyfrom circular to irregular formsof natural streams, which maychange along the flow directionand as well as with time.7)Relative roughness changes withthe level of free surface8)The depth of flow, dischargeand the slopes of channelbottom and of the free surfaceare interdependent.6)The cross section of a pipe isusually circular7)The relative roughness is afixed quantity.8)No such dependence.
Kinds of Open Channel CanalFlumeChuteDropCulvertOpen-Flow Tunnel
Kinds of Open Channel CANAL is usually a long and mild-sloped channelbuilt in the ground.
Kinds of Open Channel FLUME is a channel usually supported on or above the surfaceof the ground to carry water across a depression.
Kinds of Open Channel CHUTE is a channel having steep slopes.
Kinds of Open Channel DROP is similar to a chute, but the change inelevation is affected in a short distance.
Kinds of OpenChannel CULVERT is a coveredchannel flowing partlyfull, which is installedto drain water throughhighway and railroadembankments.
Kinds of Open Channel OPEN-FLOW TUNNEL is acomparatively long coveredchannel used to carrywater through a hill or anyobstruction on the ground.
Channel Geometry A channel built with constant cross section andconstant bottom slope is called a PRISMATICCHANNEL. Otherwise, the channel is NONPRISMATIC.
THE CHANNEL SECTION is the cross section of achannel taken normal to the direction of the flow. THE VERTICAL CHANNEL SECTION is the verticalsection passing through the lowest or bottom pointof the channel section.dThe channel section (B-B)yThe vertical channel section (A-A)
Geometric Elements of Channel Section THE DEPTH OF FLOW, y, is the verticaldistance of the lowest point of a channel sectionfrom the free surface.yθ dhθzDatum
Geometric Elements of Channel Section THE DEPTH OF FLOW SECTION, d, is thedepth of flow normal to the direction of flow.θ is the channelbottom sloped ycosθ.yθ dhθzDatumFor mild-slopedchannels y d.
Geometric Elements of Channel Section THETOP WIDTH, T,is the width of the channel section at thefree surface. THE WATER AREA, A,is the cross-sectional area of the flownormal to the direction of flow. THE WETTED PERIMETER, P,is the length of the line of intersection ofthe channel wetted surface with a crosssectional plane normal to the direction offlow. THE HYDRAULIC RADIUS, R A/P,is the ratio of the water area to its wettedperimeter. THE HYDRAULIC DEPTH, D A/T,is the ratio of the water area to the topwidth.TdPA A(d)
ChannelGeometry The wetted perimeter doesnot include the freesurface.Examples of R for commongeometries shown in Figureat the left.
Geometric elements for different channel cross rabolicBBBBBhflow area1mh1hmbbbh(b mh )hmh 21(θ sin θ )D 28b 2hb 2h 1 m 22h 1 m 21θD2(b mh )hmhb 2h 1 m 22 1 m2Phydraulic radiusbhb 2hRh14bb 2mh2mh(b mh )h1h2orBhydraulic depthhDh*Valid forIf sin θ 1 θ D(sin θ / 2)Dtop width0 ξ 1ξ 1b 2 mhwherethenξ 4h / B[hθAwetted perimeterhD(P (B / 2 ) 1 ξ 2 (1 / ξ ) ln ξ 1 ξ 22 h (D h ) θ sin θ D sin θ / 2 8)]2Bh3B 8 h23 B2B 2h3 B 2 8h 23Ah22h3**
Types of Flow Criterion: Change in flow depth with respect totime and spaceOCFTime is a criterionUnsteady flow( y/ t 0)Steady flow( y/ t 0)Space is a criterionUniform Flow Varied Flow( y/ x 0)( y/ x 0)GVFRVFUniform Flow( y/ x 0)Varied Flow( y/ x 0)GVFRVF
Types of Flow Criterion: Change in discharge with respect totime and spaceOCFTime is a criterionSteady flow( Q/ t 0)Unsteady flow( Q/ t 0)Space is a criterionContinuousFlow( Q/ x 0)SpatiallyVaried Flow( Q/ x 0)ContinuousFlow( Q/ x 0)Spatially-VariedFlow( Q/ x 0)
Classification of Open-Channel Flows Obstructions cause the flow depth to vary.Rapidly varied flow (RVF) occurs over a short distance near theobstacle.Gradually varied flow (GVF) occurs over larger distances and usuallyconnects UF and RVF.
Steady non-uniform flow in a channel.
State of Flow Effect of viscosity:VRRe υNote that R in Reynold number is Hydraulic RadiusLaminar OCF, Re 500OCFTransitional OCF, 500 Re 1000Turbulent OCF, Re 1000
Effect of Gravity In open-channel flow the driving force (that is the forcecausing the motion) is the component of gravity along thechannel bottom. Therefore, it is clear that, the effect ofgravity is very important in open-channel flow. In an open-channel flow Froude number is defined as:Inertia ForceFr ,Gravity Force 2VV2and Fr or Fr gDgDIn an open-channel flow, there are three types of flowdepending on the value of Froude number:Fr 1Fr 1Fr 1Supercritical FlowCritical FlowSubcritical Flow
In wave mechanics, the speed of propagation of a smallamplitude wave is called the celerity, C.If we disturb water, which is not moving, a disturbancewave occur, and it propagates in all directions with acelerity, C, as:CCC gyCCCFor a rectangular channel, the hydraulic depth, D y.VVTherefore, Froude number becomes:Fr gy C
Now let us consider propagation of a small amplitude wavein a supercritical open channel flow:Fr 1, i.e; V C CSince V C, it CANNOT propagate upstream it canpropagate only towards downstream with a pattern asfollows:V CDisturbance will be feltonly within this regionThis means the flow at upstream will not be affected.In other words, there is no hydraulic communicationbetween upstream and downstream flow.
Now let us consider propagation of a small amplitudewave in a subcritical open channel flow:Fr 1, i.e; V C CCSince V C, it CAN propagate both upstream anddownstream with a pattern as follows:V C This means the flow at upstream and downstreamwill both be affected.In other words, there is hydraulic communicationbetween upstream and downstream flow.
Now let us consider propagation of a small amplitudewave in a critical open channel flow:Fr 1, i.e; V CCCSince V C, it can propagate only downstream with apattern as follows:This means the flow at downstream will be affected.
State of Flow Effect of gravity:V gDFr V gDV gDD in Froude Number is Hydraulic DepthVgD
Velocity Profiles In order to understand the velocity distribution, it iscustomary to plot the isovels, which are the equal velocitylines at a cross section.isovel
Velocity is zero on bottom and sides of channel due to no-slip conditionthe maximum velocity is usually below the free surface. It is usually three-dimensional flow. However, 1D flow approximation is usually made with good success formany practical problems.
Velocity DistributionThe velocity distribution in an open-channel flow is quite nonuniformbecause of : Nonuniform shear stress along the wetted perimeter, Presence of free surface on which the shear stress is zero.
Uniform Flow in Channels Flow in open channels is classifiedas being uniform or nonuniform,depending upon the depth y. Depth in Uniform Flow is callednormal depth yn Uniform depth occurs when the flow depth (and thus the averageflow velocity) remains constantCommon in long straight runsAverage flow velocity is calleduniform-flow velocity V0Uniform depth is maintained aslong as the slope, cross-section,and surface roughness of thechannel remain unchanged. During uniform flow, the terminalvelocity reached, and the head lossequals the elevation drop
Uniform Flow in ChannelsV2V2z y1 1 z y 2 2 h12l2g2gvelocity headα1V12hl S f x2gα2V222gS f xenergygrade linehydraulicgrade liney1y2S o x xDatumSf Sw So
Non-uniform gradually varied flow.SfSf Sw Sohl S f x
Chezy equation (1768)Introduced by the French engineerAntoine Chezy in 1768 whiledesigning a canal for the watersupply system of ParisV C RhS fC Chezy coefficientmm60 C 150ssDarcy-Weisbach equation (1840)L V2L V2hf f fD 2g 4Rh 2gLSfL V2 f4R h 2 gV28gRhSf f V RhSf8gfIMPORTANT:In Uniform FlowSf Sowhere60 is for rough and 150 is for smoothalso a function of R (like f in Darcy-Weisbach)
Manning Equation for Uniform Flow1V RnDischarge:2/3S 1/2oQ VA1 2 / 3 1/ 2Q AR Son
Manning Equation (1891)1 2/3 1/2V Rh Sfn(SI System)Notes: The Manning Equation1) is dimensionally nonhomogeneous2) is very sensitive to nIs n only a function of roughness?Dimensions of n?V 1.49 2/3 1/2Rh SfnNO!T /L1/3(English system)
Values of Manning nn 0.031d 1 / 6 d in ftn 0.038 d 1 / 6 d in md median size of bed material
Relation between Resistance Coefficients
Example 1A trapezoidal channel has a base width b 6 m and side slopes 1H:1V.The channel bottom slope is So 0.0002 andthe Manning roughness coefficient is n 0.014.Computea)the depth of uniform flow if Q 12.1 m3/sb)the state of flowyo1yo1b 6m
Solution of Example 1a) Manning’s equation is used for uniform flow;yoQ A 2/ 3RSon1yo1b 6m2A b.yo 2.(yo /2) yo (b yo )P b 2 2 yo 6 2 2 yoSo 0.0002 n 0.014 Q 12.1 m3/sAR 2 / 3Qn 11.98So y o (6 y o ) 11.98 yo (6 yo ) 6 2 2 yo by trial & erroryo 1.5 .976
Solution of Example 1b) The state of flowFr VavegD, D AT, T b 2yoA 1.5 (6 1.5) 11.25 m2T 6 2 x 1.5 9 mD 11.25 / 9 1.25 mQ12.1 1.076 m/s A 11.251.076Fr 0.307 1 Subcritical9.81x1.25V ave
FloodPlain
Compound Channel
Generalized section representationactual cross sectioncompound-composite cross section.
Composite Section A channel section, which is composed, of differentroughness along the wetted perimeter is calledcomposite section. For such sections an equivalentManning roughness can be defined asneqn1,P1ni,PiA 2/3Q RneqSf ni2 Pi Pi Pavlovski' s eq. n F F i i 1
Compound Channel is the channel for which the cross section iscomposed of several distinct subsections312
Discharge computation in Compound Channels To compute the discharge, the channel is divided into 3subsections by using vertical interfaces as shown in the figure:Then the discharge in each subsection is computed separatelyby using the Manning equation.In computation of wetted perimeter, water-to-water contactsurfaces are not included.1mIIn2IIII12m2n12111n3A A Qi i i ni Pi 3Q total Q ii 12/3Soi 1,2,3
Example 2Determine the discharge passing through the crosssection of the compound channel shown below.The Manning roughness coefficients are n1 0.02, n2 0.03 and n3 0.04. The channel bed slope for the wholechannel is So 0.008.II1mIIIn2110mn3I2mn124m5m24m110m11
Solution of Example 2 Divide the channel into 3 subsections by usingvertical interfaces as shown in the figure:1mIIn2IIII2m1n1210mAiQi ni4m Ai Pi35m Q total Q ii 1124mn310m2/3Soi 1,2,311
Example 2 Forthe main channel (subsection I):The main channel is a composite channel too.Therefore, we need to find an equivalent value of n.neq n 2P i i Pi 1/2 1/2neqneq n 25 n 2 5 * 2 n 2 5 * 2 23 1 5 4 5 0.0307412A1 (5 13) * 2 (13 * 1) 31 m2P1 5 2x2 5 13.944 m2/331 31 Q1 0.03074 13.944 1/2 (0.02)2 5 2 5 (0.032 0.042 ) 5 4 5 0.008 154.05 m3 / s
Example 2 For the subsection II:A2 10 * 1 10 m2P2 10 1 11 m10 10 Q2 0.030 11 2/30.008 27.97 m3 / sFor the subsection III:A3 1(10 11) * 1 10.5 m22P3 10 2 11.41 m10.5 10.5 Q3 0.040 11.41 2/30.008 22.21 m3 / sQtotal Q1 Q 2 Q 3 154.05 27.97 22.21 204.23 m3 / s
Energy Concept Component of energy equation1) z is the elevation head2) y is the gage pressure head-potential head3) V2/2g is the dynamic head-kinetic headV2H1 z y1 112gSf xV 122gV 222gEGLHGLy1y2S o x x
Continuity and Energy Equations 1D steady continuity equationcan be expressed asV1A1 V2 A 2 x1D steady energy equationbetween two stationsV2V2z y1 1 z y 2 2 h l122g2g Head loss hLhl S f xV2V2y1 1 y 2 2 (S f So ) x2g2g The change in elevation headcan be written in terms of thebed slope θ(z z2 )So 1 x
Example 3 Water flows under a sluice gate in a horizontalrectangular channel of 2 m wide. If the depths offlow before and after the gate are 4 m, and 0.50 m,compute the discharge in the channel.y1 4 mb 2 my2 0.50 m12x
Solution:The energy equation between sections (1) and (2) is: H1 H2 hf The head loss between sections (1) and (2) can be neglected. Therefore:22V2V1z1 y1 α 1 z2 y2 α22g2gChoose the channel bottom as datum. Thenz1 z2 0, α 1Substituting above and Q V * (b*y) energy equation betweensections (1) and (2) becomes:22QQ y2 y1 2 22 g (b y1 )2 g (b 2 y22 )Q2 1 1 - 2 y1 y22 22 gb y2 y1 Q2 11 3.5 22 2 g * 4 0.504 solving for Q 8.352 m 3 / s
EXAMPLE 4Water flow with a velocity of 3 m/s, and a depth of 3m in a rectangular channel of 2 m wide. Then there is an upward step of30 cm as shown in figure below. Compute the depth of flow over the step.y1 3 mDatumV1 3 m/sy2 ? z 0.30 m(1) (2)Energy Eq. Between Sections (1) & (2):Q2Q2z1 y1 z2 y 2 2 22 gb y12 gb 2 y221821823 0.30 y 2 Ë2 g .2 2 3 22 g.2 2 y22Q V1 (by1 ) V2 (by2 ) 3.2.3 18 m 3 / sy2 4.1284 3.1587y22The last equation contains only one unknown: y2.However, it is a third degree polynomial of y2.
Y3-3.1587y2 4.1284 0 This polynomial has three possible solutions: Y(1) 2.496 2.5 mY(2) 1.66 mY(3) -0.996 -1 m Negative depth is not acceptable But both 2.5 m and 1.66 m depths are quite possible. Which one will occur on the step? Nor Energy equation neither continuity equation will help to decide.Luckily, in 1912, Bakhmeteff introduced the concept ofSPECIFIC ENERGY, which is the key to even the most complex openchannel flow phenomena.
Comparison of Open Channel Flow & Pipe Flow 1) OCF must have a free surface 2) A free surface is subject to atmospheric pressure 3) The driving force is mainly the component of gravity along the flow direction. 4) HGL is coincident with the free
Open Channel Flow I - The Manning Equation and Uniform Flow . Harlan H. Bengtson, PhD, P.E. COURSE CONTENT 1. Introduction . Flow of a liquid may take place either as open channel flow or pressure flow. Pressure flow takes place in a closed conduit such as a pipe, and pressure is the primary driving force for the flow.
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flow in natural channels. Other occurrences of open channel flow are flow in irrigation canals, sewer systems that flow partially full, storm drains, and street gutters. 4.2. Classification of Open Channel Flows A channel in which the cross-sectional shape and size and also the bottom slope are constant is te
principles of open-channel flow (see Chow’s Open Channel Hydraulics, and Henderson’s Open Channel Flow). The basic principles of fluid mechanics (e.g., continuity, momentum, energy) can be applied to open-channel flow with the additional complication that the position of
channel-forming flow events (approximately a 1.5-year flow event). Channel-forming flow or bankfull is defined as the flow that transports the greatest amount of sediment over a long period of time and controls the channel geometry. Bankfull flow DOES NOT mean flow to the top of the channel bank.
past a partially open gate or shortly before a falls), and gradually varied flow (GVF) if the flow depth changes gradually over a long distance along the channel. Discussion The equations of GVF are simplified because of the slow changes in the flow direction. Chapter 13 Open-Channel Flow
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