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IS 15823: Hydrometry - Computing stream flow using and unsteady flow model PDF

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इंटरनेट मानक Disclosure to Promote the Right To Information Whereas the Parliament of India has set out to provide a practical regime of right to information for citizens to secure access to information under the control of public authorities, in order to promote transparency and accountability in the working of every public authority, and whereas the attached publication of the Bureau of Indian Standards is of particular interest to the public, particularly disadvantaged communities and those engaged in the pursuit of education and knowledge, the attached public safety standard is made available to promote the timely dissemination of this information in an accurate manner to the public. “जान1 का अ+धकार, जी1 का अ+धकार” “प0रा1 को छोड न’ 5 तरफ” Mazdoor Kisan Shakti Sangathan Jawaharlal Nehru “The Right to Information, The Right to Live” “Step Out From the Old to the New” IS 15823 (2009): Hydrometry - Computing stream flow using and unsteady flow model [WRD 1: Hydrometry] “!ान $ एक न’ भारत का +नम-ण” Satyanarayan Gangaram Pitroda ““IInnvveenntt aa NNeeww IInnddiiaa UUssiinngg KKnnoowwlleeddggee”” “!ान एक ऐसा खजाना > जो कभी च0राया नहB जा सकता हहहहै””ै” Bhartṛhari—Nītiśatakam “Knowledge is such a treasure which cannot be stolen” IS 15823 :2009 Indian Standard HYDROMETRY- COMPUTING STREAM FLOW USINGAN UNSTEADY FLOWMODEL ICS17.120.20 © BIS 2009 BUREAU OF INDIAN STANDARDS MANAK BHAVAN, 9 B AHADUR SHAH ZAFAR MARG NEWDELHI 110002 June 2009 Price Group 5 HydrometrySe.ctionalCommittee,WRD 1 FOREWORD This Indian Standardwasadoptedbythe BureauofIndian Standards,afterthedraftfinalized bythe Hydrometry Sectional Committee had been approved bythe Water Resources Division Council. Unsteady flow modelsare appropriate for computingstream flowrecordsat locations where: a) a single-valued stage-discharge relation does not exist, b) backwateraffects the discharge underselected or all conditions, c) flows are affected bytides, or d) it is not possible to gauge the flow using velocity-area methods. Unsteadyflow modelsare alsoappropriate forevaluatingthe effectsofchangesinamanagedflow regimeon downstream conditions priorto the implementation ofany changes. In the formulation ofthis standard considerable assistance has been derived from ISO/TR 11627 : 1998 'Measurementofliquid flow inopen channels- Computingstream flow usingan unsteady flow model'. Forthe purposeofdecidingwhetheraparticularrequirementofthis standard iscompliedwith,the final value, observed or calculated, expressing the result ofa test or analysis shall be rounded offin accordance with IS2: 1960 'Rules for roundingoffnumerical values(revised)'.The numberofsignificantplacesretained in the rounded offvalue should be the same as that ofthe specified value in this standard. IS 15823: 2009 Indian Standard HYDROMETRY- COMPUTING STREAM FLOW USING AN UNSTEADY FLOW MODEL 1 SCOPE ISNo. Title This standard lays down a method for computing 1191 :2003 Hydrometric determinations continuousrecordsofstreamflow inanopen channel Vocabularyandsymbols(second through the numerical solution of the one revision) dimensional unsteady flow equations. Such an 1192:1981 Velocity area methods for approach istypically identified as an unsteady flow measurement of flow in open model and generally involves the use ofcomputers channels (first revision) for solution ofthe flow equations. 2912: 1999/ Liquid flowmeasurementinopen The method described inthis standard is applicable ISO 1070:1992 channels - Slope - Area to steady and unsteady flows and to tidal flows in method (first revision) which there are no significant longitudinal and 3910:1992 Requirements for rotating verticaldensitygradients. The method isconsidered elementcurrentmeters(cup type) equivalent to, or better than, the commonly used forwaterflow measurement(first stage-fall-dischargetechnique [see 7.2 and AnnexC revision) of IS 15119 (Part I») because the method uses information on the physical characteristics of the 3912: 1993 Sounding rods - Functional channel, including the cross-sectional geometry, requirements (first revision) channel rugosity and channel slope, and the method 9163 Dilution methods of isbasedonamathematicaldescriptionofthe physics (Part1): 1979 measurement of steady flow: offluid flow. Part I Constant rate injection method This standard also describes the theoretical basis and fundampntal assumptions ofthe technique, and 15118:20021 Measurement of liquid flow in provides a summary ofselected numerical methods IS04373: 1995 open channel - Water level used to solve the unsteady flow equations. The measuring devices detailsonthe applicationofanunsteadyflow model. 15119 Measurement of liquid flow in including data requirements. procedures for model open channels: calibration, testing, and applications, and (Part I):20021 Establishment and operation of identification of uncertainties associated with the ISO1100-1:1996 a gauging station method have also been included. (Part2):20021 Determination of the stage This standard, howeverdoes not provide sufficient ISO1100-2:1998 discharge relation information for the development of a computer 15122:20021 Measurement of liquid flow in programmefor solvingthe unsteadyflow equations, ISO2425:1999 open channels under tidal but rather it is based on the assumption that an conditions adequately documented computer programme is available. 3 DEFINITIONS 2 REFERENCES For the purpose of this standard. the definitions given inIS 1191 and the followingshall apply. The following standards contain provisions which Co0>i through reference in this text. constitute provisions 3.1 Boundary Condition- Aboundarycondition z ofthis standard. At the time of publication. the is a condition that a dependent variable of a iii .ii.i editions indicated were valid. All standards are differentialequationmust satisfyalongtheboundary -.t subject to revision and parties to agreements based ofthe model domain. Boundary conditions for the 1 on this standard are encouraged to investiga te the dependent variables should be specified at the possibility of applying the most recent editions of physical extremities ofthe modeled region for the the standards indicated below: durationofmodelapplication. IS 15823 :2009 3.2 CourantCondition- The usual condition for the physical system into corresponding thenumericalstabilityofthe explicitformulationofa characteristic equations. The characteristic numerical scheme which requires that the ratio of equations are ordinary differential equations and the propagation speed ofa physical disturbance to generally are more amenable to numerical solution that ofa numerical signal should not exceed unity. than are the original partial differential equations. 3.3 Explicit Finite - Difference Numerical 3.9 Momentum Coefficient- The momentum Scheme- Explicit numericalschemesconverteither coefficient,alsoknownasthe Boussinesqcoefficient, the characteristic equations or the governing quantifies the deviation ofthe velocity at any point equations to a system of Iinear algebraic equations inacross-sectionfrom uniform velocitydistribution from which the unknowns may be solved directly inthe samecross-section.A valueofunity indicates (explicitly) without iterative computations. that a uniform velocity distribution ispresent inthe Dependent variables onthe advanced time level are cross-section. The momentum coefficient generally determined one point at a time from known values variesbetweenabout 1.0Iand 1.12forfairlystraight, and conditionsat the presentor previoustime levels. prismaticchannels;coefficientsaretypicallysmaller Explicit schemes are only conditionally stable, for large, deep channelsthan for small channels. meaning that errors may grow as the solution 4 UNITSOFMEASUREMENT progresses.and the errors are a function ofthe time and distance finite-difference step sizes. Explicit The units ofmeasurement used in this standard are schemes are generally stable when the courant SI units conditions ismet,which results in limitationson the distancestepand maximum time which can be used. 5 PRINCIPLESOFUNSTEADYFLOWMODELS 3.4 Gradually- Varied,UnsteadyFlow-Generally 5.1 GoverningEquations non-uniform tlow in which there are no abrupt The foundations for the fundamental derivation of changes indepth along the longitudinal axis ofthe the governing one-dimensional unsteady flow channel. and in which depth (and velocity and equationswerelaid bythe 19thcenturyhydraulicians discharge)change with time. Coriolis, Boussinesq, and Saint Venant. The" 3.5 Hydrograph - A relation in graphical, governingequationsaretheone-dimensional,cross equational, or tabular form between time and flow sectionallyaveragedexpressions for the following: variables such as discharge depth. velocity. and a) Conservation of mass (or equation of stage. Stageanddischargehydrographsaretypically continuity) used foropen channel flows. d4. CQ 3.6 Implicit Finite- Difference Numerical -a+-D=: q ...(1) Scheme-Implicitnumericalschemes convert either the characteristic equations or the governing b) Conservation oflinearmomentum equations to a system on non-linear algebraic co c( Q2 ) ,",:,: equations from which the unknowns must be solved iteratively. All of the unknowns within the model --C:=1-+-C;:f- 13A- +g.A-c:r-+gA (SI' - S,,)='flt'...(-Y.)" domain are determined simultaneously. rather than point-by-point as with explicit methods" Implicit where methods are generally stable. and are more computationallyefficientthan explicitschemes. but A cross-sectional area ofthe channel. implicit schemes require more complex computer and varieswith .r.t. and r; algorithmsthan do explicitschemes. time: Q discharge. and varies with x and t: 3.7 Initial Conditions - A description of the dynamic conditions (typically. discharge and depth II' longitudinal component of the of flow for unsteady flow models) in the model lateral intlow velocity, and varies domain atsomespecifiedtime,usuallvthebeainninc with x and t: times~ ofthe simulation period. For all subsequent x longitudinal position along the the governing equations and the boundarv channelaxis; conditions describe the state of the system. • z depth offlow,and varies withxandt; g acceleration ofgravity; 3.8 Method ofCharacteristics - The method of characteristicsisamathematical approachforsolving fJ momentum coefficient, and varies boundary-value problems by transforming th e with .r,z,and t: original partial differential equations representing q lateral inflow per unit length of 2 IS 15823:2009 channel,and varies withx and t: 2. An unsteady flow model which is based on bed slope, and varies with.r:and Equations I and ::! should generally be applied to friction slope, and varies with x, ( those conditions in which none of the major and z. The momentum coefficient assumptionsareseverely violated.Theassumptions may be computed as: are as follows: , U- dA a) Flow is approximately one-dimensional. p= J- , ...(3) meaningthatthepredominantspatialvariation V-A indynamic conditions(discharge,velocityand where stage) is inthe longitudinal direction. /I ~ velocityillsomeelementalarea b) Fluiddensity ishomogeneousthroughoutthe d.i,and modeled reach. V mean velocity in the same c) Vertical accelerations are negligible (the cross-section having a total hydrostatic pressure distribution is areaA. applicable). Thefrictionslope,Sfaccountsfortheresistancedue d) Velocity is uniformly distributed in a given to external boundary stresses. The friction slope is cross-section. Inclusion of the momentum generallywrittenas: coefficient in Equation 2 allows this assumptiontobeviolated somewhat,butthere Sf ==Q---/-=Q--/n : ...(4) should be no flow separation. and streamline AR4/1 should not be highlycurvilinear. . e) Neither aggradation nor degradation of the where flowchanneloccurs. R = hydraulic radius. and t) Turbulence and energy dissipation can be n = manningcoefficient. described by resistance laws formulated for Both Rand n can vary as a function ofx, z, and t. steady,uniform flow(requiredforEquation4). Equation 4 is based on the assumption that the g) There are noabruptchangesinchannel shape manning equation for steady uniform flow provides or alignment. The velocity is zero at the areasonable approximation for Sfin unsteady, non channel boundary. uniformflow. h) There is no superelevation ofthe water level Equation 2 can be modified to include a term at any cross-section. accounting for the momentum impartedtothe water J) Surface tension and the density ofair at the byatemporallyandspatiallyvaryingwind.Equations freesurfaceare negligible. 1and2also canbewritten with the following: 5.3Simplified Models a) Depth and velocity, b) Stage and velocity,or Anumber oftechniques have been used to simplify Equations I and 2to provide approximate unsteady c) Stage and discharge as the dependent flow models. These simplified models generally variables. provide results with less computational effort and Equations J and 2 apply to the unsteady, spatially fewer data than is required for solution ofthe full varied, turbulent free-surface flow of an equations. However, the models have limited incompressible viscous fluid inan open channel of applicability, and it is more appropriate to use a arbitrarycross-sectionand alignment.Theequations generalunsteady flowmodel based on Equations I aresolvedsimultaneouslyfortheunknownsz(depth and2toobtainreliablerecords ofdischargeundera offlow)and Q(discharge) as a function oftime (I) wide range of conditions. A brief summary of and longitudinalposition (x). simplifiedmodelsfollows. 5.2 AssumptionsUponWhich GoverningEquations 5.3.1 Empirical Models areBased Empiricalmodelsare based on observationsofpast Equations J and 2 are derived from first principles floodevents. Thesemodelsare limitedtoapplications and may be obtained directly from the three in which sufficient observations of inflows and dimensional equation of mass continuity and the outflows ofariver sectionare available tocalibrate Navier-Stokes equations which are general, three essential empirical relations or routingcoefficients. dimensional statements of the conservation of These models are typically applied to slowly momentum for any flu id flow. A number of fluctuatingriverswith negligible lateral inflowsand assumptions are required to derive Equations I and backwatereffects. 3 IS 15823 : 2009 functionofdepth ofllow.Moreover.kinematicwaves 5.3.2 Hydrologic Models travel without attenuationofthe peak flow. but the Hydrologic models are based on the continuity shape ofthe flood wave is modified as the wave is equation written as: translated downstream. The kinematic wave model I- O =dS/Jt ...(5) allows only the downstream propagation of flow disturbances so that backwater and tidal effects where cannot be modeled. Numerous analytical solutions I inflow tothe modelledriversection. exist forapplicationsofthekinematicwave modelto o outflow from the section.and specific flowgeometries, andthese modelsaremost dS change in storage within the section widely used in the routing of overland flow of during the time interval dt. precipitation runoff. The storage is generally assumed to be related to 5.3.5 Diffusion Analogy Model theinlloworoutflow bysomeempirically-determined storage constant. Hydrologic models are limited to Thediffusionanalogy model isobtainedbyassuming applications. in which the stage-discharge relation thai the channel is prismatic. that the local and is single-valued. and are not applicable to flows convective acceleration terms in the momentum having backwatereffects.significant lateral inflows, equationare negligible. and that there is no lateral or looped stage-discharge relations. Difficulties in inflow,Thecontinuityandmomentumequationsmay solvingEquation5areoftenencounteredwhen flows then becombined to formasingleparabolic partial are changing rapidlywithtime. differential equation, which is inthe form ofthe so called convective-diffusion equationwiththesingle 5.3.3 l.inearized Models unknown of discharge. The local and convective accelerationterms,thefirsttwoterms in Equation2, Linearized modelsare derived fromEquations Iand are oftensmall insteepstreams. 2 by ignoring or linearizing nonlinear terms in the equations. The linearized Equations can then be Thediffusionanalogymodel can beused to compute analyticallyintegratedwith lesscomputationaleffort flows affected by backwater conditions. However, than is required for numerical integration of the diffusion model islimitedtoapplicationsinwhich Equations I and 2. The most common simplifying flows change relatively slowly, and in which the assumptions for these models are: channel has a rather uniform geometry throughout a) Acceleration term (second term) in the the modeled reach. momentum Equation (equation 2) is 5.4 Numerical Techniques for Solution of negligible: GoverningEquations b) Cross-sectional area A and channel bottom No known analytical solutions existfor Equations I slope ('\')are constant;the frictionslope(Sf) and 2. Consequently,numerical techniquesareused is linearized with respect to discharge and toconvert Equations Iand2 intoalgebraicEquations depth.. thatmay be solved for : and Qat finite. incremental c) There is no lateral inflow; and valuesofxand t.Thissolutiondependsonthe proper d) Routed flood wave has a simple shape description ofthe cross-sectional area as a function described byananalytical expression. of x and t, and on the availability of accurate boundary condition data. These assumptions severely limit the applicability oflinearized models. A variety of numerical techniques have been proposed and used to solve the unsteady flow 5.3.4 Kinematic Wavt' Model equations.Although finite elementmethods may be The kinematic wave model is derived by assuming used to solve the Equations. finite-difference thatallterms inthemomentumequationare negligible techniques generally are more appropriate for the relative to the friction slope (Sf) and the bed slope solution ofthe one-dimensional partial-differential (So),and that there is no lateralinflow. so that Equations.describing unsteady open-channel flow. The three broad categories of numerical techniques ...(6) are as given below: AsaconsequenceofEquation6. the dischargefor a a) Method ofcharacteristics, kinematicflow isequaltothe normaldischarge.This b) Explicitfinite-difference methods,and means that the momentum of the unsteady fl ow is c) Implicitfinite-differencemethods. described by an expression such as the Manning or Chezy equations. in which flow is a single-valued Numerous variations of each of these general 4 IS 15823 :2009 categories of techniques exist. The methods are oflarge rivers. which makes the models somewhat briefly reviewed to provide some perspective on computationallyinefficient.Explicitfinite-difference advantages and disadvantages ofeach method. schemesalsorequire thatthecomputationaldistance stepsbeequal throughout the model domain.which 5.4.1 Method ofCharacteristics may be a disadvantage forsome systems. The method ofcharacteristics is a mathematical 5.4.3 Implicit Finite-difference Methods approach for solving boundary-value problems by transforming the original partial differential Implicit numerical schemes convert either the equations representing the physical system into characteristic equations or the governing equations corresponding characteristic equations. In this to a system on nonlinear algebraic equations from context. the characteristic is the speed of a wave which the unknowns must be solved iteratively. relative to a stationary observer. Characteristic Consequently.a systemof2 N algebraic equations equations are ordinary differential equations and is generated for a model having N cross-sections generally are more amenable to numerical solution alongthex-axis.Alloftheunknownwithinthemodel than the original partial differential equations. The domain arc determined simultaneously. rather than characteristic equations are solved using either point-by-point aswithexplicit methods. explicit orimplicitfinite-differencemethods. Weighting factors are typically required in the The method of characteristics can be used \\ith a application of implicit schemes. These factors curvilineargridorarectangulargrid inthex-tdomain. determine the time between adjacent time levels at Thecurvilineargridgenerallyisnotusedforsolution which(a) the spatial derivatives. and (b) functional of the unsteady flow equations in natural open quantities are evaluated: functional quantities are channels. The nature ofcharacteristics is such that such featuresascross-sectional area. top width.and the wave trains in the x-t domain usually are not hydraulic radius. all of which are functions of the orthogonal, so solutions of the characteristic computed depthofflow.Somejudgement isrequired equations typically do not coincide with a point on in selecting these weighting factors. and the therectangulargrid representing the natural system. weighting factors often are adjusted as part ofthe Consequently. an interpolation scheme is required model calibration process. The accuracy of the totransferresults from thecharacteristic networkto numerical scheme generallydecreasesas the factor the rectangular grid representing the flow system. approaches one. where the terms in the governing The accuracy of the interpolation scheme plays a equationsareexpressedentirelyintermsofthefuture major role in determining the performance of the method ofcharacteristics in solving the governing time step. equations. Fewernumerical stabilityproblemsareencountered 5.4.2 Explicit Finite-difference Methods with implicit schemes than with explicit schemes. Numerical instabilities can occur when modeling Explicit numerical schemes convert either the rapidly varying flows ifthe time step islargeand if characteristic equations or the governingequations the spatialderivativesare not sufficiently weighted toasystem oflinearalgebraicequationsfromwhich toward the future time step. Non-Iinearities caused the unknowns may be solved directly (explicitly) by irregular cross-sections having widths that vary without iterative computations.Dependentvariables rapidlyalongthechannelorwithdepthalsocancause ontheadvanced time levelaredeterminedonepoint numerical instabilitiesinimplicit models. at a time from known values and conditions at the present orprevioustime levels.Explicitschemesare 6 DATA REQUIREMENTS only conditionally stable. meaning that errors may Data are required to construct, calibrate. test. and growasthesolutionprogresses.and theerrorsarea apply unsteady flow models. Referenced Indian function ofthe time and distance finite-difference Standards for the measurement of velocity and stepsizes.Explicitschemesaregenerallystablewhen discharge.forcollectionofwater-levelanddischarge the courant condition is mer. which results in limitations on the distance step and maximumtime records should be followed. which can be used. Ingeneral,dataare required atmodel boundaries for Inorder tomeetnumerical stabilityrequirements,the the entire period for which flow isto be computed computational time step must decrease as the using the unsteady flow model. Short-term records hydraulic depth increases. Consequen tly, and discrete measurements are needed at locations computational time steps may be required to be on withinthemodeldomain fortheperiod whichisused theorderofafewminutes forunsteadyflow modeis for model calibrationand testing. 5

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