Microsoft PowerPoint - open-channel1 [Compatibility Mode]OPEN CHANNELFLOWFlow in Open Channe/sOPEN-CHANNEL FLOW Open-channel flow is a flow of liquid (basically water) in a conduit with a free surface. That is a surface on which pressure is equal to local atmospheric pressure.Free surfaceClassification of Open-Channel FlowsOpen-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 aquaducts, irrigation, sewers, drainage ditches, etc.Comparison of OCF and Pipe FlowOpen-Channel FlowPipe FlowshapeType32lineWidth8890shapeType32lineWidth6350shapeType32lineDashing6lineWidth30480Energy grade line and hydraulic grade lineEnerlKIO(EGL)yinew ter surtace(WS))uum(c) 2002 vvadswor1h Groijp/ Thomsorj LearnngComparison of OCF and Pipe Flowi) OCF must have a free surfaceA free surface is subject to atmospheric pressureThe driving force is mainly the component of gravity along the flow direction.4) HGL is coincident with the free surface.5) Flow area is determined by the geometry of the channel plus the level of free surface, whch is likely to change along the flow direction and with as wel as time.No free surface in pipe flowNo direct atmospheric pressure, hydraulic pressure only.The driving force is mainly the pressure force along the flow direction.HGL is (usually) above the conduitFlow area is fixed by the pipe dimensions The cross section of a pipe is usually circular..1)2)3)4)5)Comparision of OCF and Pipe FlowThe cross section may be of any from circular to irregular forms of natural streams, which may change along the flow direction and as well as with time.Relative roughness changes with the level of free surfaceThe depth of flow, discharge and the slopes of channel bottom and of the free surface are interdependent.The cross section of a pipe is usually circularThe relative roughness is a fixed quantity.No such dependence.Kinds of Open ChannelCanalFlumeChuteDropCulvertOpen-Flow TunnelKinds of Open Channel CANAL is usually a long and mild-sloped channel built in the ground.Kinds of Open Channel FLUME is a channel usually supported on or above the surface of 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 penChannel CULVERT is a covered channel f lowing partly full, which is installed to drain water through highvvay and railroad embankments.Kinds of Open Channel OPEN-FLOW TUNNEL is a comparatively long covered channel used to carry water through a hill or any obstruction on the ground.Channel Ge ometryA channel built with unvarying cross section and constant bottom slope is called a PRISMATIC CHANNEL.Otherwise, the channel is NONPRISMATIC. THE CHANNEL SECTION is the cross section of a channel taken normal to the direction of the flow.The channel section (B-B)The vertical channel section (A-A) THE VERTCAL CHANNEL SECTON is thevertical section passing through the lowest or bottom point of the channel section.The channel section (B-B)The vertical channel section (A-A)Geometric Elements of Channel Section THE DEPTH OF FLOW, y, is the vertical distance of the lowest point of a channel section from the free surface.Geometric Elements of Channel Section THE DEPTH OF FLOW SECTION, d, is thedepth of flow normal to the direction of flow.0is the channelbottom sloped = ycos0.For mild-slopedchannels y d.Geometric Elements of Channel SectionTHE TOP WIDTH, T,is the width of the channel section at the freesurface.THE WATER AREA, A,is the cross-sectional area of the flow normal tothe direction of flow.THE WETTED PERIMETER, P,is the length of the line of intersection of thechannel wetted surface with a cross-sectionalplane normal to the direction of flow.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 top width.TChannelGeometryAc y(b + y/tan 8)Ar R-(0 - sn 8 cos 8)Rl = =b + 2v/sn 62R0(b) Trapezodal channel0 sn 0 cos 0(a) Crcular channel (8 in rad)y 1000Note That R in Reynold number is Hydraulic RadiusEffect of GravityIn open-channel flow the driving force, that is the force causing the motion is the component of gravity along the channel bottom. Therefore, it is clear that, the effect of gravity is very important in open-channel flow.In an open-channel flow Froude number is defined as:Inertia Force0 VVFr = -:, and Fr2 == or Fr =Gravity ForcegD In an open-channel flow, there are three types of flow depending on the value of Froude number:Fr>1 -- Supercritical FlowFr=1 - Critical FlowFr 1, i.e; V > C V > CC^>+CSince V > C, it CANNOT propagate upstream it can propagate only towards downstream with a pattern as follows:VDisturbance will be feltonly within this region This means the flow at upstream will not be affected.In other words, there is no hydraulic communication between upstream and downstream flow. Now let us consider propagation of a small amplitude wave in asubcritical open channel flow:Fr< ! ie; V < C V < C ^ Since V < C, it CAN propagate both upstream and downstream with a pattern as follows:Vroro+O+Iroro2(D*11 (2) Slope: 50 = tan a = constant^2' x2 " X] = Lcosa = L 1X |x2 xHead loss = elevation losshL = Z\~ z2 = S0Lin ChannelsFlow in pen channels is classif ied as being uniform or nonuniform, depending upon the depth y.bepth in Uniform Flow is called normal depth yUniform depth occurs when the f low depth (and thus the average f low velocity) remains constantCommon in long straight runsAverage f low velocity is called uniform-flow velocity V0Uniform 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 eguals the elevation dropUniform Flow in ChannelsV2V2z1 + yi + gg = z2+V2 ++h2g 2 '2 2g 1velocity headDatumSf Axenergygrade linehydraulic grade lineSfSw SoNon-uniform gradually varied flow. Sf*Sw *SoSteady-Uniform Flow: Force BalanceShear force - T P A xoWetted perimeter - PGravitational force = yA Ax sin0VRelationship between shear and velocity?2Steady-Uniform Flow: Force BalanceRelationship between shear and velocity? Resistance EquationoCOC^II>CM>IIor Wall-Shear StressT0 = Y R SoV=c/rsOManning Equation for Uniform FlowV1R 2/3 Sn1/2o(SI System)Discharge:Q = VAQ =1AR2/3S0/2 nChezy equation (1768) Introduced by the French engineer Antoine Chezy in 1768 while designing a canal for the vvater-supply system of Parisc-60Chezy coeff icientVm .,-iiri Vc ^Fr > 1the limb AC corresponds to supercritical flowand y2, corresponding to supercritical depthIf y > yc,V < Vc ^Fr < 1the limb BC corresponds to subcritical flow and y1 corresponding to subcritical depthFr =VVD^ Fr2Q2TgA3 The depths yi and y2 are called alternatedepths.As Q increase, the curves will move towards rightSpecific Energy For rectangular ChannelyQ = AV = byVQ = q unitdischargebE = y +V22gE = y +Q22gA2E = y +q22gy2Critical Flow:Rectangular channel1 = Q2TcgAQ = qb2 31 =qbgyCb3yc =v g jTc =bAc = ycb2qgy 3c1/3q=VgyOnly for rectangular channels!Given the depth we can find the dischargeVc ='Jy q = VcycCritical Flow Relationships:Rectangular Channelsyc =f ^1/3q2'v g jy3 =' Vc2y2Avgbecauseq=VcycJV.c _= 1JyCFroude number for critical flowyc =V2cyc _ Vc2g2 2gE = y +V22gE= ycvelocity head =0-5 (depth)2yc 3EoCharacteristics of Critical FlowArbitrary Cross SectionRectangular Cross SectionQ2 aC Fr 1 ^ = c9 Tc Fr =1 q2 = gy- y = J qV2 D D - - ^ Ec = yc + c 2g 2 c *c 2V2 y 3 = - ^ Ec = 3 yc 2g 2 c 2*c For a given Q, E = Emin For a given q, E = Emin For a given specific For a given specificenergy,energyEo, Q = QmaxEoj q = qmaxDiscussionA long rectangular channel carries water with a flow depth of y1 on a horizontal channel. If there is a rise on the channel bed: a) What is the relation of total head between section 1 and 2, if head loss assumed to be negligible.h1=h2a) What is the relation of specific energy between section 1 and 2E1 = E2 + Azb) How does the water surface profile react to channel bed elevationctamgeyiSide viewCHANNEL TRANSITION for rectangular channelsChange on the bottom elevation of channelChange on the width of the channelChange on the bottom elevation and width of the channelUpward Step-Constant widthH = H2E1 = E2 + Az1) Subcritical flowVUpward Step-Constant width2) Supercritical Flowq = q2y c =H =E 1 =2 1/3q 2v g yH2E 2 + A zDownward Step-Constant widthQ = VA = V1(y1b) = V2(y2b) ^ q1 = q21) Subcritical flowH,= H2E2 = E, + Az2) Supercritical FlowD"= q 22y cql g JH ,(MII(MLU+LUUpward Step-Constant widthSubcritical flowyity////yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy/////yzyyyyyyyy/"'/ (1)Y2YaAz(2)(3)E, =E> +^zE, =E+^zySpecific Energy:Step UpAdditionalConsiderationq = ^2H, = H2 E1 = E2 + Az E2 = E, - AZEySpecific Energy:Step UpAdditionalConsiderationE1 = E2 + A z E2 = E1 - A ZEECHOKING!SSS51) Upward stepZ-iz)U &r nAz=AzEr=E EEi=AZmax+Ec , AZ >AZmaxEl =Az +Eq = q2B.) Downward Step (Constant Width) 1) Subcritical Flowy c HE 11 / 3)HE22A zChannel Expansion (constant bed elevation)q * q2Channel Contraction (constant bed elevation)q * q2AZei uuq > qAZi UjlUq =qw 3 = Va)=l3(e)(3)(OUOflOBJlUOO (z9NIOHDKoch ParabolaQ = [2gA2(E - y)]1/2For a given E,Q = Q(y) [q=q(y) for rectangular channels]The plot of Q vs. y [q vs y] gives Koch Parabola.E - v (asymptote)lr= IA tenatedepth pontsf/maxci 2002 uVaaswOfth Group,1 Thcmson LearnngVariation of specific energy and (unit) discharge with depth: (a) E versus y for constant q;(b) Q versus y for constant E.Figure E10.7Example 3.3Water is flowing in a rectangular channel. Find the change in depth and in absolute water level produced by a smooth downward step of 0.30 m if the upstream velocity and depth are given as.Vj=3 m/s and y^3 m.Vj=5 m/s and y^0.60 m.Draw the water-surface profiles for both casesV1=3m/s*Y=3 m(1)(2)b>Example 3.3 SolutionEnergy Eqn between (1) and (2) :E+Az=E2 +h/V232E. = y1 + ^ = 3 += 3.46m ^ E2 = 3.46 + 0.30 = 3.76 m11 2g19.622q2E2 = y2 +^= V.y. = 3x3 = 9m2/s2gy2924128y2 +- There are 2 possible Solutions. To determiney23.76 = y2 + 219.62y 2which one occurs we should compute upstream Froude number.V13F1Vy7V9.81 x30.553 < 1 subcriticalflow.Therefore, y2 will correspond tosubcritical flow.Az=0.30 mAz=0.30 my2 must be greater than 3 m. The root of greater than 3 m can found by trial and error as;y2 = 3.40 mAyabs = 0^2 - ( Az +)= 3.40 - ( 3.0 + 0.30 )Ayabs = 0.10 m.^z:i kybEnergy equation between section (1) and (2)=0"V252Az + E1 = E1 + htE1 = y1 + ^ = 0.60 += 1.874m2gE2 = 0.30 +1.874 = 2.174 m = y2 +q2gy;19.62, q = V1y1 = 5x0.6 = 3m2/s320 45872.174 m = y2 +-^ 2.174 m = y2 + Two possible solutions.19.62y 22The upstream Froude number5y22Fr, = V-N/y19.81 x 0.60= 2.06 > 1, supercritical flowAz=0.30my2 must be smaller than 0.60 mcan be solved by trial and error to obtain y2=0.528 m.Or y2 = 0.53 m.&yabs= (Az *0/l)-yz= ( 0.30 + 0.60 ) - 0.53AyQbs = 0D.37 m.Solution of Specific Energy EquationEq2V +2gy2Subcritical RootV* = E -Cv2Supercritical rootE= V +q22gy2^ E = V +CV2V =E - VUCSPECIFIC FORCE CONCEPTMomentum Equation in x-direction:Ft = P1 -P2 -Ff +Fh =pQV2 -pQV1 (1)For uniform vel. distn. and horizontal channel,Ft =pQ(V2 - V1)If the vel. distribution is not uniform,Ft = pQ(P2V2 - p 1V1) where p is the momentum correction factorFt = P, - P2 -F, +Fh = pQV2 - pQV, (1)For uniform vel. distn. and horizontal channel,Ft =pQ(V2 - V1)If the vel. distribution is not uniform,Ft = pQ(p2V2 -p 1V1) where p is the momentum correction factorPressure Force = P = yyA ^ substituti ng into Eq.(1) assuming Ff = Fh = 0YyA - Yy2A2 = PQV2 - PQV1QQ_Ay yA + pQ = y y2A2 +pQQ2Q2F = A1y1 += A2y2 +1y1 gA12*2 gA2dF _ dAAt critical depth = 0 ,= T^ Specific ForcedydyQ2 = A3 g = TSpecific force for any channe/ sectionF = yA +gAy = depth of centroid of a channel section.For a given Q, F = F (y)The plot of F vs y gives specific force curveshapeType32lineWidth21590Characteristics of the Specific Force Curve:Specific force curve has two limps AC and BC Specific force becomes minimum when the flow is critical Lower limp AC corresponds to supercritical flow Upper limp BC corresponds to subcritical flow.For a given F and Q, there are two possible flow regimes represented by the depths y1 and y2 which are called conjugate or sequent depths, corresponding to supercritical and subcritical flows, respectively.Characteristics of the Specific Force Curve:For rectangular channels, the specific force can be written as specific force per unit width asFb1 2y2 +2q2gyHYDRAULIC JUMP Hydraulic jump is a rapidly varied flow in which flow changes abruptly from supercritical state to a subcritical state accompanied by considerable turbulence and energy loss.F1 = FE1 E 2 +Rectangular ChannelsQ2F1 = F2^Yi A1 + ^ = y2A 2hl =1(V1+8F3 -1)(y2 - Yi)34 Yi y 2+ 9A 2Alternate depth ofFor a simple jump, since Fi = F2, yi and y2 are the conjugate (sequent) depths.HYDRAULIC JUMP INTERPRETED BY SPECIFIC-ENERGY AND SPECIFIC-FORCE CURVESPRACTICAL APPLICATIONS OF HYDRAULICJUMPTo dissipate energyTo recover head or raise the water levelTo increase weight on apronTo mix Chemicals used for water purificationTo aerate waterFigure 10.14 - Variation of the momentum function with depth.Rapidly Varied Flow and Hydraulic JumpFlow 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 effectsBackflowsSeparationsHydraulic JumpUsed for energy dissipationOccurs when flow transitions from supercritical to subcritical- base of spillwayWe would like to know depth of water downstream from jump as well as the location of the jumpWhich equation, Energy or Momentum?Rapidly Varied Flowand Hydraulic JumpConsider the CVsurrounding the hydraulicjumpAssumptionsV is constant atsections (1) and (2),and p1 and p2 * 1P = pgytw is negligible relativeto the losses that occurduring the hydraulicjumpChannel is horizontalNo external body forcesother than gravityRapidly Varied Flow and Hydraulic JumpContinuity equationXi pybV = py2bV2 yVx = y2V2 =>= (y/y2)V1P\,avgA\ P2,avg^2 ^(^2) m(Vi)m = rri2 = r = pA\V\ py b\\+ - 2Frl = 0 VQuadratic equation for y2/y1Rapidly Varied Flow and Hydraulic JumpSolving 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 jumpRapidly Varied Flow and Hydraulic Jump Experimental studies indicate that hydraulic jumps can be classified into 5 categories, depending upon the upstream FrTABLE 1 3-4Classification of hydraulic jumps Source: U.S. Bureau of Roclamation (1955).Depth Fraction ofUpstream Ratio EnergySurfaceFrj y^jy Dissipation DescriptionProfile9>1270-85%Oscillating jump. Pulsations caused by entering jets at the bottom generate large waves that can travel for miles and damage earth banks. Should be avoided in the design of stilling basins.Steady jump. Stable, vvell-balanced, and insensitive to downstream conditions. Intense eddy motion and high level of energy dissipation vvithin the jump. Recommended range for design.Strong jump. Rough and intermittent. Very effective energy dissipation, but may be uneconomical compared to other designs.USE OF F-y AND E-y CURVES: EXAMPLES HYDRAULIC JUMPUSE OF F-y AND E-y CURVES: EXAMPLES SLUICE GATEyyFUSE OF F-y AND E-y CURVES: EXAMPLES SLUICE BLOCKS - CREATE HYDRAULIC JUMP1GRADUALLY VARIED FLOWIn a channel in the vicinity ofChannel transitionChannel bed slope change locationEmbankmentFallWeirThe flow will not be uniform. The type flow called asGradually Varied Flow{c] 30*02 WadfiwOrth Group/ Thomson LearningNon-uniform gradually varied flow. Sf*Sw *SoGRADUALLY VARIED FLOW (dy/dx ycHorizontal : S0 = 0Adverse : S0 < 0Initial depth is given anumber1 : y > yn2 : yn < y < yc3 : y < ycGradually Varied Flowin GVF.EXAMPLES OF WATER PROFILESIn a channel in the vicinity ofChannel transitionChannel bed slope change locationEmbankmentFallWeirThe flow will not be uniform. Therefore at any section if the flow depth is greater than the critical depth the state of the flow will be subcritical without considering the slope of the channel.EXAMPLES OF WATER PROFILES Mild Slope (yn>yc)yYnyc" ZONE 1; M1 y > yn > ycZONE 2; M2 yn > y >yc ycZONE 3; M3 yn > yc >y yEMild Slope (yn>yc)M1 y > Yn ^ Sf < S0y > yc ^ Fr < 1dy So - Sfdx 1 - Fr2 > 0(u/s) y ^ yn dydx ^ 0(Sf ^ So)dy(d / s) y ^ ^dx ^ So(Fr ^ 0, Sf ^ 0)E2) Steep Slope yn yn ^ Sf < S0 y < yc ^ Fr > 1dy =dx(u/s) (d/s)So - Sf1 - Fr2< 0y ycy yn(Fr 1)(Sf So)yycynZONE 1; S1 y > yc > ynZONE 2; S2 yc > y > yn^ZONE 3; S3 yc>yn>yThe Occurence of Critical Flow ConditionsIn an open channel for a given discharge at any section thefollowing equation can be written,2H = y + z +q2g y2= E + zdE dz+= 0, E = f[y(x)]dx dxdEdy + dz = 0 dy dx dxdydx+= oOutflow from a reservoir with critical flow at the channel entrance.Figure 10.19 - Representative Controls: (a) sluice gate; (b) change in slope from mild (S01)to steep (S02); (c) entrance to a steep channel; (d) free outfall.Figure 10.20 - Example of profile synthesisFlow Control and MeasurementDischarges*Broad-crestedweirL,y (a) Sluice gate with free outflowFlow ra te in pipes and ducts is controlled byvarious kinds of valvesIn OC flows, flow rate is controlled bypartially blocking the channel.Weir : liquid flows over deviceUnderflow gate : liquid flows under deviceThese devices can be used to control theflow rate, and to measure it.Flow Control and MeasurementUnderflow GateSluici gate>V\ >1rFree outflowJ Vena contractaya t Vy2 -1+Sluice gatenyv,Drowned outflowVcy2 ^2Underflow gates are located at thebottom of a wall, dam, or openchannelOutflow can be either free ordrownedIn free outflow, downstream flow issupercriticalIn the drowned outflow, the liquidjet undergoes a hydraulic jump.Downstream flow is subcritical.Flow Control and MeasurementUnderflow GateSchematic 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)SubcriticalllowEs\ -y +FrctonlessgateDrownedoutflovvSupercriticalflowCritical DepthMinimum energy for qdEdy= 0kinetic = potential!yc=V22 2gFr=1EFr>1 = Supercritical Fr