Open Channel FlowSteady-Uniform Flow: Force BalanceOpen Conduits: Dimensional AnalysisPressure Coefficient for Open Channel Flow?Dimensional AnalysisOpen Channel Flow FormulasManning FormulaWhy Use the Manning FormulaValues of Manning nExample: Manning FormulaDepth as f(Q)SummaryTurbulent Flow Losses in Open ConduitsHydraulic JumpSlide 15ExampleGrand Coulee DamColumbia Basin ProjectPumpsPowerPoint PresentationSlide 21Grand Coulee Feeder CanalColumbia Basin Irrigation ProjectUnsteady Hydraulics!GatesSlide 26Banks LakeOpen Channel FlowOpen Channel FlowJanuary 13, 2019January 13, 2019Hydraulic radiusHydraulic radiusSteady-Uniform Flow: Force BalanceWW sin xabcdShear forceEnergy grade lineHydraulic grade line Shear force =________0sin  xPxAo0sin  xPxAosin PAosin PAohR = PAhR = PAsincossinSsincossinSW cos gV22gV22Wetted perimeter = __Gravitational force = ________oP xPx sinDimensional analysisDimensional analysisRelationship between shear and velocity? ___________________Geometric parameters_________________________________________________________Write the functional relationshipGeometric parameters_________________________________________________________Write the functional relationshipPARhPARhHydraulic radius (Rh)Hydraulic radius (Rh)Channel length (l)Channel length (l )Roughness ()Roughness ()Open Conduits:Dimensional AnalysisOpen Conduits:Dimensional AnalysisRe, , ,ph hlC fR Re� �=� �� �F,M, WRe, , ,ph hlC fR Re� �=� �� �F,M, WglVFglVFReVlrm=Pressure Coefficient for Open Channel Flow?Pressure Coefficient for Open Channel Flow?22CVpp22CVpp22CVghlhl22CVghlhl2f2CfVlgSS2f2CfVlgSSlShl flShl flhplhpPressure CoefficientPressure CoefficientHead loss coefficientHead loss coefficientFriction slope coefficientFriction slope coefficient(Energy Loss Coefficient)(Energy Loss Coefficient)Friction slopeFriction slopeThe friction slope is the slope of the EGL. The friction slope is The friction slope is the slope of the EGL. The friction slope is the same as the bottom slope (the same as the bottom slope (SSoo) for steady, uniform flow.) for steady, uniform flow.Dimensional AnalysisDimensional Analysisf, ,ReSh hlC fR Re� �=� �� �f, ,ReSh hlC fR Re� �=� �� �lRChSflRChSflRVlgSh2f2lRVlgSh2f2hRgSVf2hRgSVf2hRSgVf2hRSgVf2f,ReSh hlC fR Re� �=� �� �f,ReSh hlC fR Re� �=� �� �Head loss  length of channelHead loss  length of channelf,RehShRC fl Rel� �= =� �� �f,RehShRC fl Rel� �= =� �� �2f2fVlgSCS2f2fVlgSCS(like f in Darcy-Weisbach)2f2lhl Vh S lR gl= =2f2lhl Vh S lR gl= =Chezy formulaOpen Channel Flow FormulasOpen Channel Flow FormulasVAQ VAQ 2/13/21ohSARnQ 2/13/21ohSARnQ 1/2o2/3hSR 1nV 1/2o2/3hSR 1nV hSRgV2hSRgV2Dimensions of n?Is n only a function of roughness?hSRCV hSRCV Manning formula (MKS units!)NO!T /L1/3Manning FormulaManning FormulaThe Manning n is a function of the boundary roughness as well as other geometric parameters in some unknown way...___________________________________________________Hydraulic radius for wide channelsHydraulic radius for wide channelsThe Manning n is a function of the boundary roughness as well as other geometric parameters in some unknown way...___________________________________________________Hydraulic radius for wide channelsHydraulic radius for wide channels1/2o2/3hSR 1nV 1/2o2/3hSR 1nV RAPhA bhP b h  2Rbhb hh 2Channel curvature (bends)Cross section geometry12P1 < P2Rh1 > Rh2Why Use the Manning FormulaWhy Use the Manning FormulaTraditionNatural channels are geometrically complex and the errors associated with using an equation that isn’t dimensionally correct are small compared with our inability to characterize stream geometryMeasurement errors for Q and h are largeWe only ever deal with water in channels, so we don’t need to know how other fluids would respondTraditionNatural channels are geometrically complex and the errors associated with using an equation that isn’t dimensionally correct are small compared with our inability to characterize stream geometryMeasurement errors for Q and h are largeWe only ever deal with water in channels, so we don’t need to know how other fluids would respondValues of Manning nValues of Manning nLined Canals nCement plaster 0.011Untreated gunite 0.016Wood, planed 0.012Wood, unplaned 0.013Concrete, trowled 0.012Concrete, wood forms, unfinished 0.015Rubble in cement 0.020Asphalt, smooth 0.013Asphalt, rough 0.016Natural ChannelsGravel beds, straight 0.025Gravel beds plus large boulders 0.040Earth, straight, with some grass 0.026Earth, winding, no vegetation 0.030Earth , winding with vegetation 0.050Lined Canals nCement plaster 0.011Untreated gunite 0.016Wood, planed 0.012Wood, unplaned 0.013Concrete, trowled 0.012Concrete, wood forms, unfinished 0.015Rubble in cement 0.020Asphalt, smooth 0.013Asphalt, rough 0.016Natural ChannelsGravel beds, straight 0.025Gravel beds plus large boulders 0.040Earth, straight, with some grass 0.026Earth, winding, no vegetation 0.030Earth , winding with vegetation 0.050The worst channel has…Roughness at many scales!Example: Manning FormulaExample: Manning FormulaWhat is the flow capacity of a finished concrete channel that drops 1.2 m in 3 km? What is the flow capacity of a finished concrete channel that drops 1.2 m in 3 km? 11223 m3 m1.5 m1.5 msolutionDepth as f(Q)Depth as f(Q)Find the depth in the channel when the flow is 5 m3/sHydraulic radius is function of depthArea is a function of depthCan’t solve explicitlyUse trial and error or solverFind the depth in the channel when the flow is 5 m3/sHydraulic radius is function of depthArea is a function of depthCan’t solve explicitlyUse trial and error or solverQnAR Sh o12 3 1 2/ /SummarySummaryOpen channel flow equations can be obtained in a similar fashion to the Darcy-Weisbach equation (based on dimensional analysis)The dimensionally incorrect Manning equation is the standard in English speaking countriesOpen channel flow equations can be obtained in a similar fashion to the Darcy-Weisbach