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IS 15823 (2009): Hydrometry - Computing stream flow using
and unsteady flow model [WRD 1: Hydrometry]
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Satyanarayan Gangaram Pitroda
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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
