water Hammer

y andrmsowslencetry innder-pipet day

dis-mericalmass

sionald re-restric-

abil-rrenttaand

tioncited

Mohamed S. Ghidaouiemail: [email protected]

Ming Zhaoemail: [email protected]

Department of Civil Engineering, The Hong KongUniversity of Science and Technology,

Hong Kong, China

Duncan A. McInnisSurface Water Group, Komex International Ltd.,

4500 16th Avenue, Suite 100, N. W. Calgary,Alberta T3B 0M6, Canada

David H. Axworthy163 N. Marengo Avenue, #316, Pasadena,

CA 91101email: [email protected]

A Review of Water HammerTheory and PracticeHydraulic transients in closed conduits have been a subject of both theoretical studintense practical interest for more than one hundred years. While straightforward in teof the one-dimensional nature of pipe networks, the full description of transient fluid flpose interesting problems in fluid dynamics. For example, the response of the turbustructure and strength to transient waves in pipes and the loss of flow axisymmepipes due to hydrodynamic instabilities are currently not understood. Yet, such ustanding is important for modeling energy dissipation and water quality in transientflows. This paper presents an overview of both historic developments and presenresearch and practice in the field of hydraulic transients. In particular, the papercusses mass and momentum equations for one-dimensional Flows, wavespeed, nusolutions for one-dimensional problems, wall shear stress models; two-dimensionaland momentum equations, turbulence models, numerical solutions for two-dimenproblems, boundary conditions, transient analysis software, and future practical ansearch needs in water hammer. The presentation emphasizes the assumptions andtions involved in various governing equations so as to illuminate the range of applicity as well as the limitations of these equations. Understanding the limitations of cumodels is essential for (i) interpreting their results, (ii) judging the reliability of the daobtained from them, (iii) minimizing misuse of water-hammer models in both researchpractice, and (iv) delineating the contribution of physical processes from the contribuof numerical artifacts to the results of waterhammer models. There are 134 refrencesin this review article.@DOI: 10.1115/1.1828050#

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1 IntroductionThus the growth of knowledge of the physical aspect of reacannot be regarded as a cumulative process. The basic Gof this knowledge changes from time to time . . . During thecumulative periods scientists behave as if reality is exactlythey know it except for missing details and improvementsaccuracy. They speak of the laws of nature, for example, whare simply models that explain their experience of reality acertain time. Later generations of scientists typically discothat these conceptions of reality embodied certain implicitsumptions and hypotheses that later on turned out to be inrect. Vanderburg,@1#

Unsteady fluid flows have been studied since man first bwater to his will. The ancient Chinese, the Mayan Indians of Ctral America, the Mesopotamian civilizations bordering the NiTigris, and Euphrates river systems, and many other societhroughout history have developed extensive systems for coning water, primarily for purposes of irrigation, but also for dometic water supplies. The ancients understood and applied fluid flprinciples within the context of ‘‘traditional,’’ culture-based technologies. With the arrival of the scientific age and the mathemcal developments embodied in Newton’sPrincipia, our under-standing of fluid flow took a quantum leap in terms oftheoretical abstraction. That leap has propelled the entire devement of hydraulic engineering right through to the mid-twentiecentury. The advent of high-speed digital computers constituanother discrete transformation in the study and applicationfluids engineering principles. Today, in hydraulics and other areengineers find that their mandate has taken on greater breadtdepth as technology rapidly enters an unprecedented stagknowledge and information accumulation.

As cited in The Structure of Scientific Revolutions, ThomasKuhn @2# calls such periods of radical and rapid change in oview of physical reality a ‘‘revolutionary, noncumulative transtion period’’ and, while he was referring to scientific views

Transmitted by Associate Editor HJS Fernando.

Copyright © 20Applied Mechanics Reviews

itystalt

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reality, his remarks apply equally to our technological abilitydeal with a revised or more complex view of the physical uverse. It is in this condition that the field of closed conduit trasient flow, and even more generally, the hydraulic analysis,sign, and operation of pipeline systems, currently finds itself.

The computer age is still dawning, bringing with it a massidevelopment and application of new knowledge and technoloFormerly accepted design methodologies, criteria, and standare being challenged and, in some instances, outdated and revComputer aided analysis and design is one of the principal menisms bringing about these changes.

Computer analysis, computer modeling, and computer simtion are somewhat interchangeable terms, all describing teniques intended to improve our understanding of physical pnomena and our ability to predict and control these phenomeBy combining physical laws, mathematical abstraction, numerprocedures, logical constructs, and electronic data processthese methods now permit the solution of problems of enormcomplexity and scope.

This paper attempts to provide the reader with a generaltory and introduction to waterhammer phenomena, a general cpendium of key developments and literature references as wean updated view of the current state of the art, both with respectheoretical advances of the last decade and modeling practic

2 Mass and Momentum Equations forOne-Dimensional Water Hammer Flows

Before delving into an account of mathematical developmerelated to waterhammer, it is instructive to briefly note the sociecontext that inspired the initial interest in waterhammer phenoena. In the late nineteenth century, Europe was on the cusp oindustrial revolution with growing urban populations and indutries requiring electrical power for the new machines of prodtion. As the fossil fuel era had not begun in earnest, hydroelecgeneration was still the principal supply of this important enersource. Although hydroelectric generation accounts for a msmaller proportion of energy production today, the problems as

05 by ASME JANUARY 2005, Vol. 58 Õ 49

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ciated with controlling the flow of water through penstocks aturbines remains an important application of transient analyHydrogeneration companies contributed heavily to the devement of fluids and turbomachinery laboratories that studiamong other things, the phenomenon of waterhammer and itstrol. Some of Allievi’s early experiments were undertaken asdirect result of incidents and failures caused by overpressureto rapid valve closure in northern Italian power plants. Frictionleapproaches to transient phenomena were appropriate in thesedevelopments because~i! transients were most influenced by thrapid closure and opening of valves, which generated the majoof the energy loss in these systems, and~ii ! the pipes involvedtended to have large diameters and the flow velocities tended tsmall.

By the early 1900s, fuel oils were overtaking hydrogeneratas the principal energy source to meet society’s burgeoningmand for power. However, the fascination with, and need toderstand, transient phenomena has continued unabated to thiGreater availability of energy led to rapid industrialization aurban development. Hydraulic transients are critical design facin a large number of fluid systems from automotive fuel injectito water supply, transmission, and distribution systems. Todlong pipelines transporting fluids over great distances havecome commonplace, and the almost universal developmensprawling systems of small pipe diameter, high-velocity water dtribution systems has increased the importance of wall frictionenergy losses, leading to the inclusion of friction in the governequations. Mechanically sophisticated fluid control devices,cluding many types of pumps and valves, coupled with increingly sophisticated electronic sensors and controls, providepotential for complex system behavior. In addition, the recknowledge that negative pressure phases of transients can rescontamination of potable water systems, mean that the neeunderstand and deal effectively with transient phenomenamore acute than ever.

2.1 Historical Development: A Brief Summary. The prob-lem of water hammer was first studied by Menabrea@3# ~althoughMichaud is generally accorded that distinction!. Michaud@4# ex-amined the use of air chambers and safety valves for controwater hammer. Near the turn of the nineteenth century, researclike Weston@5#, Carpenter@6# and Frizell@7# attempted to developexpressions relating pressure and velocity changes in a pipe.zell @7# was successful in this endeavor and he also discussedeffects of branch lines, and reflected and successive waveturbine speed regulation. Similar work by his contemporarJoukowsky@8# and Allievi @9,10#, however, attracted greater atention. Joukowsky@8# produced the best known equation in trasient flow theory, so well known that it is often called the ‘‘fundamental equation of water hammer.’’ He also studied wareflections from an open branch, the use of air chambers and stanks, and spring type safety valves.

Joukowsky’s fundamental equation of water hammer is aslows:

DP56raDV or DH56aDV

g(1)

where a5acoustic ~waterhammer! wavespeed,P5rg(H2Z)5piezometric pressure,Z5elevation of the pipe centerline fromgiven datum,H5piezometric head,r5fluid density,V5*AudA5cross-sectional average velocity,u5 local longitudinal velocity,A5cross-sectional area of the pipe, andg5gravitational accelera-tion. The positive sign in Eq.~1! is applicable for a water-hammewave moving downstream while the negative sign is applicafor a water-hammer wave moving upstream. Readers familiar wthe gas dynamics literature will note thatDP56raDV is obtain-able from the momentum jump condition under the special cwhere the flow velocity is negligible in comparison to thwavespeed. The jump conditions are a statement of the cons

50 Õ Vol. 58, JANUARY 2005

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duessearlyerity

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tion laws across a jump~shock! @11#. These conditions are obtained either by directly applying the conservation laws for a ctrol volume across the jump or by using the weak formulationthe conservation laws in differential form at the jump.

Allievi @9,10# developed a general theory of water hammfrom first principles and showed that the convective term inmomentum equation was negligible. He introduced two importdimensionless parameters that are widely used to charactpipelines and valve behavior. Allievi@9,10# also produced chartsfor pressure rise at a valve due to uniform valve closure. Furtrefinements to the governing equations of water hammer appein Jaeger@12,13#, Wood @14#, Rich @15,16#, Parmakian@17#,Streeter and Lai@18#, and Streeter and Wylie@19#. Their combinedefforts have resulted in the following classical mass and momtum equations for one-dimensional~1D! water-hammer flows

a2

g

]V

]x1

]H

]t50 (2)

]V

]t1g

]H

]x1

4

rDtw50 (3)

in which tw5shear stress at the pipe wall,D5pipe diameter,x5the spatial coordinate along the pipeline, andt5temporal coor-dinate. Although Eqs.~2! and ~3! were fully established by the1960s, these equations have since been analyzed, discussederived and elucidated in numerous classical texts~e.g.,@20–23#!.Equations~2! and~3! constitute the fundamental equations for 1water hammer problems and contain all the physics necessamodel wave propagation in complex pipe systems.

2.2 Discussion of the 1D Water Hammer Mass and Mo-mentum Equations. In this section, the fundamental equationfor 1D water hammer are derived. Special attention is given toassumptions and restrictions involved in various governing eqtions so as to illuminate the range of applicability as well aslimitations of these equations.

Rapid flow disturbances, planned or accidental, induce spaand temporal changes in the velocity~flow rate! and pressure~pi-ezometric head! fields in pipe systems. Such transient flows aessentially unidirectional~i.e., axial! since the axial fluxes ofmass, momentum, and energy are far greater than their racounterparts. The research of Mitra and Rouleau@23# for the lami-nar water hammer case and of Vardy and Hwang@25# for turbulentwater-hammer supports the validity of the unidirectional approwhen studying water-hammer problems in pipe systems.

With the unidirectional assumption, the 1D classical water hamer equations governing the axial and temporal variations ofcross-sectional average of the field variables in transient pflows are derived by applying the principles of mass and momtum to a control volume. Note that only the key steps of tderivation are given here. A more detailed derivation can be foin Chaudhry@20#, Wylie et al. @23#, and Ghidaoui@26#.

Using the Reynolds transport theorem, the mass conserva~‘‘continuity equation’’! for a control volume is as follows~e.g.,@20–23#!

]

]t Ecvrd;1E

csr~v"n!dA50 (4)

wherecv5control volume,cs5control surface,n5unit outwardnormal vector to control surface,v5velocity vector.

Referring to Fig. 1, Eq.~4! yields

]

]t Ex

x1dx

rAdx1Ecs

r~v"n!dA50 (5)

The local form of Eq.~5!, obtained by taking the limit as thelength of the control volume shrinks to zero~i.e., dx tends tozero!, is

Transactions of the ASME

de

t

t

gn-

d

q.-

aofeiblesionhis

veof

-r. To

de

rms

s.

]~rA!

]t1

]~rAV!

]x50 (6)

Equation~6! provides the conservative form of the area-averagmass balance equation for 1D unsteady and compressible fluia flexible pipe. The first and second terms on the left-hand sidEq. ~6! represent the local change of mass with time due tocombined effects of fluid compressibility and pipe elasticity athe instantaneous mass flux, respectively. Equation~6! can be re-written as follows:

1

r

Dr

Dt1

1

A

DA

Dt1

]V

]x50 or

1

rA

DrA

Dt1

]V

]x50 (7)

where D/Dt5]/]t1V]/]x5substantial~material! derivative inone spatial dimension. Realizing that the density and pipe avary with pressure and using the chain rule reduces Eq.~7! to thefollowing:

1

r

dr

dP

DP

Dt1

1

A

dA

dP

DP

Dt1

]V

]x50 or

1

ra2

DP

Dt1

]V

]x50

(8)

where a225dr/dP1(r/A)dA/dP. The historical developmenand formulation of the acoustic wave speed in terms of fluid apipe properties and the assumptions involved in the formulaare discussed in Sec. 3.

The momentum equation for a control volume is~e.g., @20–23#!:

( Fext5]

]t Ecvrv;1E

csrv~v"n!dA (9)

Applying Eq. ~9! to the control volume of Fig. 2; consideringravitational, wall shear and pressure gradient forces as exterapplied; and taking the limit asdx tends to zero gives the following local form of the axial momentum equation:

]rAV

]t1

]brAV2

]x52A

]P

]x2pDtw2gA sina (10)

Fig. 1 Control volume diagram used for continuity equationderivation

Fig. 2 Control volume diagram used for momentum equationderivation

Applied Mechanics Reviews

eds inof

thend

rea

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ally

whereg5rg5unit gravity force,a5angle between the pipe anthe horizontal direction,b5*Au2dA/V25momentum correctioncoefficient. Using the product rule of differentiation, invoking E~7!, and dividing through byrA gives the following nonconservative form of the momentum equation:

]V

]t1V

]V

]x1

1

rA

]~b21!rAV2

]x1

1

r

]P

]x1g sina1

twpD

rA

50 (11)

Equations~8! and ~11! govern unidirectional unsteady flow ofcompressible fluid in a flexible tube. Alternative derivationsEqs. ~8! and ~11! could have been performed by applying thunidirectional and axisymmetric assumptions to the compressNavier-Stokes equations and integrating the resulting expreswith respect to pipe cross-sectional area while allowing for tarea to change with pressure.

In practice, the order of magnitude of water hammer waspeed ranges from 100 to 1400 m/s and the flow velocity isorder 1 to 10 m/s. Therefore, the Mach number,M5U1 /a, inwater-hammer applications is often in the range 1022–1023,whereU15 longitudinal velocity scale. The fact thatM!1 in wa-ter hammer was recognized and used by Allievi@9,10# to furthersimplify Eqs. ~8! and ~11!. The small Mach number approximation to Eqs.~8! and~11! can be illustrated by performing an ordeof magnitude analysis of the various terms in these equationsthis end, letr0aU15water hammer pressure scale,r05density ofthe fluid at the undisturbed state, andT5zL/a5time scale, whereL5pipe length, X5aT5zL5 longitudinal length scale,z5apositive real parameter,r f U1

2/85wall shear scale, andf5Darcy-Weisbach friction factorTd5radial diffusion time scale.The parameterz allows one to investigate the relative magnituof the various terms in Eqs.~8! and ~11! under different timescales. For example, if the order of magnitude of the various tein the mass momentum over a full wave cycle~i.e., T54L/a) isdesired,z is set to 4. Applying the above scaling to Eqs.~8! and~11! gives

r0

r

DP*

Dt*1

]V*

]x*50 or

r0

r S ]P*

]t*1MV*

]P*

]x* D1]V*

]x*50 (12)

]V*

]t*1MV*

]V*

]x*1M

1

rA

]~b21!rAV* 2

]x*1

r0

r

]P*

]x*

1gzL

Uasina1

zL

DM

f

2tw* 50 (13)

where the superscript* is used to denote dimensionless quantitieSince M!1 in water hammer applications, Eqs.~12! and ~13!become

r0

r

]P*

]t*1

]V*

]x*50 (14)

]V*

]t*1

r0

r

]P*

]x*1

gzL

Uasina1z

L

DM

f

21zS Td

L/aD tw* 50.

(15)

Rewriting Eqs.~14! and ~15! in dimensional form gives

1

ra2

]P

]t1

]V

]x50 (16)

]V

]t1

1

r

]P

]x1g sina1

twpD

rA50 (17)

Using the Piezometric head definition~i.e., P/gr05H2Z), Eqs.~16! and ~17! become

JANUARY 2005, Vol. 58 Õ 51

r

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ible

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gr0

ra2

]H

]t1

]V

]x50 (18)

]V

]t1g

r0

r

]H

]x1

twpD

rA50 (19)

The change in density in unsteady compressible flows is oforder of the Mach number@11,27,28#. Therefore, in water hammeproblems, whereM!1, r'r0 , Eqs.~18! and ~19! become

g

a2

]H

]t1

]V

]x50 (20)

]V

]t1g

]H

]x1

twpD

rA50 (21)

which are identical to the classical 1D water hammer equatigiven by Eqs.~2! and~3!. Thus, the classical water hammer equtions are valid for unidirectional and axisymmetric flow of a compressible fluid in a flexible pipe~tube!, where the Mach number isvery small.

According to Eq.~15!, the importance of wall shear,tw , de-pends on the magnitude of the dimensionless parameteG5zLM f /2D1zTd/(L/a). Therefore, the wall shear is importanwhen the parameterG is order 1 or larger. This often occurs iapplications where the simulation time far exceeds the first wcycle ~i.e., largez!, the pipe is very long, the friction factor isignificant, or the pipe diameter is very small. In addition, wshear is important when the time scale of radial diffusion is larthan the wave travel time since the transient-induced large ragradient of the velocity does not have sufficient time to relax. Inoted thatTd becomes smaller as the Reynolds number increaThe practical applications in which the wall shear is important athe varioustw models that are in existence in the literature adiscussed in Sec. 4.

If G is significantly smaller than 1, friction becomes negligiband tw can be safely set to zero. For example, for the casL510,000 m,D50.2 m, f 50.01, andM50.001, andTd/(L/a)50.01 theconditionG!1 is valid whenz!4. That is, for the caseconsidered, wall friction is irrelevant as long as the simulattime is significantly smaller than 4L/a. In general, the conditionG!1 is satisfied during the early stages of the transient~i.e., z issmall! provided that the relaxation~diffusion! time scale is smallerthan the wave travel timeL/a. In fact, it is well known thatwaterhammer models provide results that are in reasonable ament with experimental data during the first wave cycle irresptive of the wall shear stress formula being used~e.g., @29–32#!.WhenG!1, the classical waterhammer model, given by Eqs.~20!and ~21!, becomes

g

a2

]H

]t1

]V

]x50 (22)

]V

]t1g

]H

]x50 (23)

which is identical to the model that first appeared in Allievi@9,10#.The Joukowsky relation can be recovered from Eqs.~22! and

~23!. Consider a water hammer moving upstream in a pipelength L. Let x5L2at define the position of a water hammefront at time t and consider the interval@L2at2e,L2at1e#,wheree5distance from the water hammer front. Integrating E~22! and~23! from x5L2at2e to x5L2at1e, invoking Leib-nitz’s rule, and taking the limit ase approaches zero gives

DH52aDV

g(24)

Similarly, the relation for a water hammer wave moving dowstream isDH51aDV/g.


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3 Water Hammer „Acoustic… Wave SpeedThe water hammer wave speed is~e.g.,@8,20,23,33,34#!,

1

a2 5dr

dP1

r

A

dA

dP(25)

The first term on the right-hand side of Eq.~25! represents theeffect of fluid compressibility on the wave speed and the secterm represents the effect of pipe flexibility on the wave speedfact, the wave speed in a compressible fluid within a rigid pipeobtained by settingdA/dP50 in Eq. ~25!, which leads toa2

5dP/dr. On the other hand, the wave speed in an incompressfluid within a flexible pipe is obtained by settingdr/dP50 in~25!, which leads toa25AdP/rdA.

Korteweg @33# related the right-hand side of Eq.~25! to thematerial properties of the fluid and to the material and geometrproperties of the pipe. In particular, Korteweg@33# introduced thefluid properties through the state equationdP/dr5K f /r, whichwas already well established in the literature, whereK f5bulkmodulus of elasticity of the fluid. He used the elastic theorycontinuum mechanics to evaluatedA/dP in terms of the piperadius, thicknesse, and Young’s modulus of elasticityE. In hisderivation, he~i! ignored the axial~longitudinal! stresses in thepipe~i.e., neglected Poisson’s effect! and~ii ! ignored the inertia ofthe pipe. These assumptions are valid for fluid transmission lithat are anchored but with expansion joints throughout. Withsumptions~i! and ~ii !, a quasi-equilibrium relation between thpressure force per unit length of pipeDdP and the circumferential~hoop! stress force per unit pipe length 2edsu is achieved, wheresu5hoop stress. That is,DdP52edsu or dp52edsu /D. Usingthe elastic stress-strain relation,dA5pdjD2/2, where dj5dsu /E5radial ~lateral! strain. As a result,AdP/rdA5eE/Drand

1

a2 5r

K f1

r

Ee

D

or a25

K f

r

11K fD

eE

(26)

The above Korteweg formula for wave speed can be extento problems where the axial stress cannot be neglected. Thaccomplished through the inclusion of Poisson’s effect instress-strain relations. In particular, the total strain becomesdj5dsu /E2npdsx /E, where np5Poisson’s ratio andsx5axialstress. The resulting wave speed formula is~e.g.,@17,23#!

a25

K f

r

11cK fD

eE

(27)

wherec512np/2 for a pipe anchored at its upstream end onc512np

2 for a pipe anchored throughout from axial movemeand c51 for a pipe anchored with expansion joints throughowhich is the case considered by Korteweg~i.e., sx50).

Multiphase and multicomponent water hammer flows are comon in practice. During a water hammer event, the pressurecycle between large positive values and negative values, the mnitudes of which are constrained at vapor pressure. Vapor cavcan form when the pressure drops to vapor pressure. In addigas cavities form when the pressure drops below the saturapressure of dissolved gases. Transient flows in pressurized orcharged pipes carrying sediment are examples of multicompowater hammer flows. Unsteady flows in pressurized or surchasewers are typical examples of multiphase and multicompontransient flows in closed conduits. Clearly, the bulk modulus adensity of the mixture and, thus, the wave speed are influencethe presence of phases and components. The wave speed fortiphase and multicomponent water hammer flows can be obta


Fig. 3 Velocity profiles for steady-state and af-ter wave passages

hr

av

o

sar

a

.

e

e

a

y

t

rob-hatelpn-full

avegedity

flowithard ex-asedmer

allar

ig-at-

ocityearng

-dera-thent-

nd

nds-.

by substituting an effective bulk modulus of elasticityKe and aneffective densityre in place ofK f andr in Eq. ~27!. The effectivequantities,Ke andre , are obtained by the weighted average of tbulk modulus and density of each component, where the pavolumes are the weights~see,@23#!. While the resulting math-ematical expression is simple, the explicit evaluation of the wspeed of the mixture is hampered by the fact that the partialumes are difficult to estimate in practice.

Equation~27! includes Poisson’s effect but neglects the motiand inertia of the pipe. This is acceptable for rigidly anchored psystems such as buried pipes or pipes with high density and sness, to name only a few. Examples include major transmispipelines like water distribution systems, natural gas lines,pressurized and surcharged sewerage force mains. Howevemotion and inertia of pipes can become important when pipesinadequately restrained~e.g., unsupported, free-hanging pipes! orwhen the density and stiffness of the pipe is small. Someamples in which a pipe’s motion and inertia may be significinclude fuel injection systems in aircraft, cooling-water systemunrestrained pipes with numerous elbows, and blood vesselsthese systems, a fully coupled fluid-structure interaction moneeds to be considered. Such models are not discussed inpaper. The reader is instead directed to the recent excellent reof the subject by Tijsseling@35#.

4 Wall Shear Stress ModelsIt was shown earlier in this paper that the wall shear stress t

is important when the parameterG is large. It follows that themodeling of wall friction is essential for practical applications thwarrant transient simulation well beyond the first wave cycle~i.e.,largez!. Examples include~i! the design and analysis of pipelinsystems,~ii ! the design and analysis of transient control devic~iii ! the modeling of transient-induced water quality problem~iv! the design of safe and reliable field data programs for dinostic and parameter identification purposes,~v! the application oftransient models to invert field data for calibration and leakadetection,~vi! the modeling of column separation and vaporocavitation and~vii ! systems in whichL/a!Td. Careful modelingof wall shear is also important for long pipes and for pipes whigh friction.

4.1 Quasi-Steady Wall Shear Models. In conventionaltransient analysis, it is assumed that phenomenological expsions relating wall shear to cross-sectionally averaged velocitsteady-state flows remain valid under unsteady conditions. Thawall shear expressions, such as the Darcy-Weisbach and HaWilliams formulas, are assumed to hold at every instant durintransient. For example, the form of the Darcy-Weisbach equaused in water hammer models is~Streeter and Wylie@36#!


etial

veol-

nipetiff-ionnd, theare

ex-nts,For

delthis

view

rm

at

es,s,g-

geus

ith

res-in

t is,zen-g aion

tw~ t !5tws5r f ~ t !uV~ t !uV~ t !

8(28)

wheretws(t)5quasi-steady wall shear as a function oft.The use of steady-state wall shear relations in unsteady p

lems is satisfactory for very slow transients—so slow, in fact, tthey do not properly belong to the water hammer regime. To hclarify the problems with this approach for fast transients, cosider the case of a transient induced by an instantaneous andclosure of a valve at the downstream end of a pipe. As the wtravels upstream, the flow rate and the cross-sectionally averavelocity behind the wave front are zero. Typical transient velocprofiles are given in Fig. 3. Therefore, using Eq.~28!, the wallshear is zero. This is incorrect. The wave passage creates areversal near the pipe wall. The combination of flow reversal wthe no-slip condition at the pipe wall results in large wall shestresses. Indeed, discrepancies between numerical results anperimental and field data are found whenever a steady-state bshear stress equation is used to model wall shear in water hamproblems~e.g.,@25,30,32,37,38#!.

Let twu(t) be the discrepancy between the instantaneous wshear stresstw(t) and the quasi-steady contribution of wall shestresstws(t). Mathematically

tw~ t !5tws~ t !1twu~ t ! (29)

twu(t) is zero for steady flow, small for slow transients, and snificant for fast transients. The unsteady friction componenttempts to represent the transient-induced changes in the velprofile, which often involve flow reversal and large gradients nthe pipe wall. A summary of the various models for estimatitwu(t) in water hammer problems is given below.

4.2 Empirical-Based Corrections to Quasi-Steady WallShear Models. Daily et al. @39# conducted laboratory experiments and foundtwu(t) to be positive for accelerating flows annegative for decelerating flows. They argued that during acceltion the central portion of the stream moved somewhat so thatvelocity profile steepened, giving higher shear. For constadiameter conduit, the relation given by Daily et al.@39# can berewritten as

Ku5Ks12c2

L

V2

]V

]t(30)

whereKu5unsteady flow coefficient of boundary resistance amomentum flux of absolute local velocity andKs5 f L/D5 steadystate resistance coefficient. Daily et al.@39# noted that the longi-tudinal velocity and turbulence nonuniformities are negligible aKu'K5F/rAV2/25unsteady flow coefficient of boundary resitance, whereF52pDLtw5wall resistance force. Therefore, Eq~30! becomes


Fig. 4 Pressure head traces obtained frommodels and experiment

f

g

e

,i

o

r

b

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lmt

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the

,

for

uch

er

thellerq.

ithl isra-eedsor-nsa-the

ri-nta-t

of

and

tw5r f V2

81

c2rD

4

]V

]t(31)

Denoting c2 by k and r f V2/8 by tws reduces Eq.~31! to thefollowing:

tw5tws1krD

4

]V

]t(32)

The formulations of Daily et al.@39# shows that coefficientc25k is a measure of the deviations, due to unsteadiness, owall shear and momentum flux. Therefore,k generally depends onx and t. This remark is supported by the extended thermodynaics approach used by Axworthy et al.@30#. Figure 4 clearly illus-trates the poor agreement between model and data when usin~32! with a constant value ofk.

The experimental data of Daily et al.@39# show thatk50.01for accelerating flows andk50.62 for decelerating flows. On thother hand, the research of Shuy@40# led to k520.0825 for ac-celerating flows andk520.13 for decelerating flows. In factShuy’s data led him to conclude that unsteady wall frictioncreases in decelerating flows and decreases in accelerating flThis result contradicts the previously accepted hypothenamely, that unsteady wall friction decreases in decelerating fland increases in accelerating flows. Shuy@40# attributed the de-crease in wall shear stress for acceleration to flow relaminartion. Given its controversial conclusion, this paper generateflurry of discussion in the literature with the most notable remabeing those of Vardy and Brown@41#.

Vardy and Brown@41# argued that Shuy’s results should notinterpreted as contradicting previous measurements. Insteadresults indicated that the flow behavior observed in Shuy’s expments may have been different from the flow behavior in previexperiments. Vardy and Brown@41# put forward the time scalehypothesis as a possible explanation for the different flow behior between Shuy’s@40# experiments and previous ones. They aobserved that, while Shuy’s experiments dealt with long tiscales, previous measurements dealt with much shorterscales. Vardy and Brown@41# provided insightful and convincingarguments about the importance of time scale to the flow behain unsteady pipe flows. In fact, the stability analysis of Ghidaand Kolyshkin @42# concurs with the time scale hypothesisVardy and Brown@41#. Moreover, the stability analysis showthat, while other experiments belong to the stable domain,experiments of Shuy belong to the unstable domain.


the

m-

Eq.

n-ows.sis,ws

iza-d aks

e, theeri-us

av-so

eime

vioruif

sthe

Theoretical investigations aimed at identifying the domainapplicability of Eq.~32! have appeared in the literature. For eample, Carstens and Roller@43# showed that Eq.~32! can be de-rived by assuming that the unsteady velocity profiles obeypower law as follows:

u~x,r ,t !

V~x,t !5

~2n11!~n11!

2n2 S 12r

RD 1/n

(33)

where n57 for Reynolds number Re5105 and increases withReynolds number,r 5distance from the axis in a radial directionR5radius of the pipe. An unsteady flow given by Eq.~33! de-scribes flows that exhibit slow acceleration and does not allowflow reversal~i.e., does not contain inflection points!. In fact, Eq.~33! cannot represent typical water hammer velocity profiles sas those found in Vardy and Hwang@25#, Silva-Araya andChaudhry@37#, Pezzinga@38,44#, Eichinger and Lein@45# andGhidaoui et al.@46#. The theoretical work of Carstens and Roll@43# shows only that Eq.~32! applies to very slow transients inwhich the unsteady velocity profile has the same shape assteady velocity profile. Unfortunately, the Carstens and Ro@43# study neither supports nor refutes the possibility of using E~32! in water hammer problems.

The theoretical work of Vardy and Brown@47# shows that Eq.~32! can be derived for the case of an unsteady pipe flow wconstant acceleration. In addition, they show that this modeapproximately valid for problems with time dependent acceletion as long as the time scale of the transient event greatly excthe rising time, which is a measure of time required for the vticity diffusion through the shear layer. Their work also waragainst using Eq.~32! for problems with time dependent accelertion induced by transient events with time scales smaller thanrising time (i.e., L/a!Td).

Axworthy et al.@30# found that Eq.~32! is consistent with thetheory of Extended Irreversible Thermodynamics~EIT! and sat-isfy the second law of thermodynamics. In addition, the EIT devation shows that unsteady friction formulas based on instaneous acceleration such as Eq.~32! are applicable to transienflow problems in which the time scale of interest~e.g., simulationtime! is significantly shorter than the radial diffusion time scalevorticity. Using the vorticity equation, Axworthy et al.@30#showed that for such short time scales, the turbulence strengthstructure is unchanged~i.e., ‘‘frozen’’!, and the energy dissipation


r

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so

enn

fa

,

m

e

h

s

e

-

s a.

asd

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delsver,not

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in-curs

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utionlkeofear

ta-

ts

sly

me

ad-canthecu-

theolu-

behind a wave front is well represented by the degree of shifthe cross-sectional mean value of the velocity~i.e., dV/dt) andthe cross-sectional mean value ofV, itself.

The time scale arguments by Vardy and Brown@47# and Ax-worthy et al.@30# represent two limit cases: very slow transienand very fast transients, respectively. In the former case, theenough mixing such that the acceleration history pattern isstroyed, only the instantaneous acceleration is significant towall shear stress. In the latter case, the pre-existing flow strucis frozen, there is no additional acceleration history develoexcept that of instantaneous acceleration. The Axworthy e@30# argument represents a water hammer flow situation whereacceleration behaves like a pulse, say, the flow drops from a fivalue to zero in a short period.

An important modification of instantaneous acceleration-baunsteady friction models was proposed by Brunone and G@48#, Greco @49#, and Brunone et al.@50,51#. The well knownBrunone et al.@50# model has become the most widely usmodification in water hammer application due to its simplicity aits ability to produce reasonable agreement with experimepressure head traces.

Brunone et al.@50# incorporated the Coriolis correction coeficient and the unsteady wall shear stress in the energy equfor water hammer as follows:

]H

]x1

1

g

]V

]t1

h1f

g

]V

]t1Js50 (34)

whereh5difference from unity of the Coriolis correction coefficient, Js5( f uVuV)/2gD 5steady-state friction term(f/g) (]V/]t)5difference between unsteady friction and its coresponding steady friction. In Eq.~34!, the convective term isdropped as the Mach number of the flow is small in water hamproblems.

A constitutive equation is needed forh1f. Brunone et al.@50# proposed

h1f5kS 12a]V

]xY ]V

]t D (35)

or in terms of wall shear stress

tw5tws1krD

4 S ]V

]t2a

]V

]x D (36)

Equation~36! provides additional dissipation for a reservoir-pipvalve system when the transient is caused by a downstreamden valve closure. The pressure head traces obtained frommodels and experiment are plotted in Fig. 4. It is shown talthough both the Darcy-Weisbach formula and Eq.~32! with con-stantk cannot produce enough energy dissipation in the preshead traces, the model by Brunone et al.@50# is quite successful inproducing the necessary damping features of pressure peaks,fied by other researchers@29,52–55#.

Slight modifications to the model of Brunone et al.@50#, whichrenders this model applicable to both upstream and downstrtransients, were proposed in@44# and in @52#. In particular, Pezz-inga @44# proposed

h1f5kF11signS V]V

]x Da]V

]xY ]V

]t G (37)

and Bergant et al.@53# proposed

h1f5kF11sign~V!aU]V

]xUY ]V

]t G (38)

The dependence ofh1f on x andt as well as the flow acceleration is consistent with the theoretical formulations in@30# and@39#. In addition, the form ofh1f gives significant correction for


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the unsteady friction when the flow is accelerated (V]V/]t.0)and small correction when the flow is decelerated (V]V/]t,0)@50#.

Utilization of the models presented in this section requirereliable estimate of the parameterk. The data of Brunone et al@31#, Daily et al. @39#, and others show thatk is not a universalconstant. An empirical method for estimating this parameter wproposed by Brunone et al.@52# by fitting the decay of measurepressure head history. Moody diagram-like charts fork were de-veloped by Pezzinga@44# using a quasi-two-dimensional turbulence model. Vardy and Brown@47# provided a theoretically-baseexpression for determining the coefficientk. This expression wassuccessfully applied by Bergant et al.@52# and Vitkovsky et al.@55#. Although the charts of Pezzinga@44# and the formula ofVardy and Brown@47# are theory-based, their reliability is limitedby the fact that they rely on steady-state-based turbulence moto adequately represent unsteady turbulence. It should, howebe stressed that modeling turbulent pipe transients is currentlywell understood~see Sec. 9!.

The mechanism that accounts for the dissipation of the psure head is addressed in the discussion by Ghidaoui et al.@46#.They found that the additional dissipation associated with thestantaneous acceleration based unsteady friction model oconly at the boundary due to the wave reflection. It was shownafter nc complete wave cycles, the pressure head is dampedfactor equivalent to@1/(11k)#2nc.

4.3 Physically Based Wall Shear Models. This class of un-steady wall shear stress models is based on the analytical solof the unidirectional flow equations and was pioneered by Zie@56#. Applying the Laplace transform to the axial componentthe Navier-Stokes equations, he derived the following wall shexpression for unsteady laminar flow in a pipe:

tw~ t !54nr

RV~ t !1

2nr

R E0

t ]V

]t8~ t8!W~ t2t8!dt8 (39)

where t85a dummy variable, physically represents the instanneous time in the time history;n5kinematic viscosity of the fluid;W5weighting function

W~ t !5e226.3744~nt/R2!1e270.8493~nt/R2!1e2135.0198~nt/R2!

1e2218.9216~nt/R2!1e2322.5544~nt/R2!

fornt

R2.0.02

W~ t !50.282095S nt

R2D 21/2

21.2500011.057855S nt

R2D 1/2

fornt

R2,0.0210.937500nt

R2 10.396696S nt

R2D 3/2

20.351563S nt

R2D 2

(40)

The first term on the right-hand side of Eq.~39! represents thesteady-state wall shear stresstws and the second term representhe correction part due to the unsteadiness of the flowtwu . Thenumerical integration of the convolution integral in Eq.~39! re-quires a large amount of memory space to store all previoucalculated velocities and large central processing unit~CPU! timeto carry out the numerical integration, especially when the tistep is small and the simulation time large. Trikha@57# used threeexponential terms to approximate the weighting function. Thevantage of using exponential forms is that a recursive formulaeasily be obtained, so that the flow history can be lumped intoquantities at the previous time step. In this way, only the callated quantities at the previous time step needs to be stored incomputer memory, and there is no need to calculate the conv


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tion integral from the beginning at every time step. This reduthe memory storage and the computational time greatly. In Suet al. @58#, for t,0.02, the summation is calculated in a normway; for t.0.02, the recursive formula similar to that of Trikh@57# is used, since each of the five terms included in the weighfunction is exponential. Although Zielke’s formula is derived flaminar flow, Trikha@57# and others@29,52# found that this for-mula leads to acceptable results for low Reynolds number tulent flows. However, Vardy and Brown@47# warned against theapplication of Zielke’s formula outside the laminar flow regimbut did note that the error in applying Zielke’s formula to turblent flows diminishes as the duration of the wave pulse reduc

Vardy et al.@59# extended Zielke’s approach to low Reynoldnumber turbulent water hammer flows in smooth pipes. In a lapaper, Vardy and Brown@60# developed an extension of the modof Vardy et al.@59# that was applicable to high Reynolds numbtransient flows in smooth pipes. In addition, Vardy and Bro@60# showed that this model gives results equivalent to thoseVardy et al.@59# for low Reynolds number flows and to thoseZielke @56# for laminar flows. That is, the Vardy and Brown@60#model promises to provide accurate results for Reynolds numranging from the laminar regime to the highly turbulent regimThis model has the following form:

tw~ t !5r fV~ t !uV~ t !u

81

4nr

D E0

t

W~ t2t8!]V

]t8dt8 (41)

where

W~ t !5a exp~2bt !/Apt; a5D/4An;

b50.54n Rek/D2; k5 log~14.3/Re0.05!

and Re5Reynolds number. Similar to Zielke’s model, the convlution nature of Eq.~41! is computationally undesirable. An accurate, simple, and efficient approximation to the Vardy-Brown usteady friction equation is derived and shown to be eaimplemented within a 1D characteristics solution for unsteapipe flow @32#. For comparison, the exact Vardy-Brown unsteafriction equation is used to model shear stresses in transientbulent pipe flows and the resulting water hammer equationssolved by the method of characteristics. The approximate VaBrown model is more computationally efficient~i.e., requires1

6-ththe execution time and much less memory storage! than the exactVardy-Brown model. Both models are compared with measudata from different research groups and with numerical dataduced by a two-dimensional~2D! turbulence water hammemodel. The results show that the exact Vardy-Brown modelthe approximate Vardy-Brown model are in good agreement wboth laboratory and numerical experiments over a wide rangReynolds numbers and wave frequencies. The proposed appmate model only requires the storage of flow variables fromsingle time step while the exact Vardy-Brown model requiresstorage of flow variables at all previous time steps and themodel requires the storage of flow variables at all radial node

A summary of the assumptions involved in deriving Eqs.~39!and ~41! is in order. The analytical approach of Zielke@56# in-volves the following assumptions:~i! the flow is fully developed,~ii ! the convective terms are negligible,~iii ! the incompressibleversion of the continuity equation is used~i.e., the influence ofmass storage on velocity profile is negligible!, and~iv! the veloc-ity profile remains axisymmetric~i.e., stable! during the transient.In order to extend Zielke’s approach to turbulent flows, Vardy aBrown @60# made two fundamental assumptions in relation toturbulent eddy viscosity in addition to assumptions~i! through~iv!. First, the turbulent kinematic viscosity is assumed to vlinearly within the wall shear layer and becomes infinite~i.e., auniform velocity distribution! in the core region. Second, the tubulent eddy viscosity is assumed to be time invariant~i.e., frozento its steady-state value!. Assumptions~i!, ~ii !, and~iii ! are accu-rate for practical water hammer flows, where the Mach numbe


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often negligibly small and pipe length far exceeds flow develoment length. The validity of assumptions such as that the flremains axisymmetric~stable!, that the eddy viscosity is independent of time, and that its shape is similar to that in steady flowdiscussed later in the paper~see Secs. 6 and 7!.

Understanding the connection between Eq.~32! and the physi-cally based unsteady wall friction models proposed by Zielke@56#and Vardy and Brown@60# further illuminates the limitations ofinstantaneous acceleration, unsteady wall friction models asscribed in the previous section. In particular, it is evident froEqs.~39! and~41! that Eq.~32! is recovered when the acceleratiois constant. In addition, plots ofW in Fig. 5 show that for flowswith large Reynolds number, this function is very small evewhere except whennt/R2 approaches 0, that is, whent8 ap-proachest in Eq. ~41!. The region whereW(t2t8) in Eq. ~41!becomes significant and provides a measure of the time scathe radial diffusion of vorticityTd . If the acceleration variesslowly in the region whereW(t2t8) is significant, it is clear thatEqs. ~39! and ~41! can be accurately approximated by Eq.~32!.This is simply an alternative way to state that Eq.~32! is accept-able when the acceleration is not constant as long as the timeof the flow disturbance far exceeds the time scale of radial dision of vorticity across the shear layer. Moreover, it is obviothat Eq.~32! is a good approximation to Eqs.~39! and~41! whent is small, as the integral interval is so small that the integrandbe considered as a constant. Furthermore, the time interval wW(t2t8) is significant reduces with Reynolds number, whishows that Eq.~32! becomes more accurate for highly turbuleflows.

5 Numerical Solutions for 1D Water Hammer Equa-tions

The equations governing 1D water hammer~i.e., Eqs.~20! and~21!! can seldom be solved analytically. Therefore, numeritechniques are used to approximate the solution. The methocharacteristics~MOC!, which has the desirable attributes of accracy, simplicity, numerical efficiency, and programming simplici~e.g.,@20,23,61#!, is most popular. Other techniques that have abeen applied to Eqs.~20! and ~21! include the wave plan, finitedifference~FD!, and finite volume~FV! methods. Of the 11 com-mercially available water hammer software packages revieweSec. 12, eight use MOC, two are based on implicit FD methoand only one employs the wave-plan method.

5.1 MOC-Based Schemes. A significant development inthe numerical solution of hyperbolic equations was publishedLister @62#. She compared the fixed-grid MOC scheme—acalled the method of fixed time interval—with the MOC grscheme and found that the fixed-grid MOC was much easiecompute, giving the analyst full control over the grid selection aenabling the computation of both the pressure and velocity fie

Fig. 5 Weighting function for different Reynolds numbers


t

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in space at constant time. Fixed-grid MOC has since been uwith great success to calculate transient conditions in pipe sysand networks.

The fixed-grid MOC requires that a common time step (Dt) beused for the solution of the governing equations in all pipes. Hoever, pipes in the system tend to have different lengths and sotimes wave speeds, making it impossible to satisfy the Coucondition ~Courant numberCr5aDt/Dx<1) exactly if a com-mon time stepDt is to be used. This discretization problem canaddressed by interpolation techniques, or artificial adjustmenthe wave speed or a hybrid of both.

To deal with this discretization problem, Lister@62# used linearspace-line interpolation to approximate heads and flows at theof each characteristic line. Trikha@57# suggested using differentime steps for each pipe. This strategy makes it possible tolarge time steps, resulting in shorter execution time and the avance of spatial interpolation error. This increased flexibility comat the cost of having to interpolate at the boundaries, which caa major source of error when complex, rapidly changing conactions are considered.

Wiggert and Sundquist@63# derived a single scheme that combines the classical space-line interpolation with reachout in spinterpolation. Using Fourier analysis, they studied the effectsinterpolation, spacing, and grid size on numerical dispersion,tenuation, and stability. These researchers found that the degrinterpolationj decreases as the ratio of the wavelength of thekthharmonicLk to the reach lengthDx increases. As a result, botnumerical dissipation and dispersion are improved. These consions are not surprising for several reasons. First, every interption technique can be expected to produce better results for wcomponents with larger wavelengths. Second, for a fixed timeDt, larger values ofn imply smaller values ofDx and vice versa,sincenDx represents the total length of the reachout on one sConsequently, this scheme generates more grid points and, tfore, requires longer computational times and computer storFurthermore, an alternative scheme must be used to carry ouboundary computations.

The reachback time-line interpolation scheme, developedGoldberg and Wylie@64#, uses the solution fromm previouslycalculated time levels. The authors observed that reachback tline interpolation is more accurate than space-line interpolafor the same discretization. This is a subjective comparisoncause, as the degree of temporal interpolationj varies from 0 to 1,the degree of spatial interpolationa is only allowed to vary from1/(m11) to 1/m. A fairer comparison would have been to alsdivide the distance step bym so that bothj anda vary equally. Inaddition, Goldberg and Wylie@64# assert that numerical errors areduced by increasingm. This is somewhat misleading becausfor a fixed Dx, increasingm means increasing the numbercomputational steps~i.e., reducing the effective time stepDt)which in turn generates finer interpolation intervals. Moreovercases where the friction term is large and/or when the wave spis not constant, reaching back in time increases the approximaerror of these terms.

Lai @65# combined the implicit, temporal reachback, spatreachback, spatial reachout, and the classical time and spaceinterpolations into one technique called the multimode scheDepending on the choice of grid size (Dt,Dx) and the limit on themaximum allowable reachbacks in timem, this scheme may function as either of the methods or a combination of any two meods. Numerical errors were studied using a mass balanceproach. Stability conditions were derived from Von Neumaanalysis. The multimode scheme gives the user the flexibilityselect the interpolation scheme that provides the best performfor a particular problem.

Yang and Hsu@66,67# published two papers dealing with thnumerical solution of the dispersion equation in 1D and 2D,spectively. The authors propose reaching back in time moreone time step and then using the Holly-Preissmann metho


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interpolate either in space or in time. It is claimed that the reaback Holly-Preissmann scheme is superior to the classical HoPreissmann method. An interesting discussion of this work, plished in Bentley@68#, showed that the solution obtained by thclassical Holly-Preissmann method when the time step eqmDt is identical to that obtained by the reachback in space HoPreissmann~i.e., the foot of the characteristic line is extendeback more than one time step until it intersects the space-l!with m reachbacks and a time step value ofDt. The only differ-ence is that the reachback approach producesm21 extra interme-diate solutions at the cost of more computational time.

Sibetheros et al.@69# showed that the spline technique is wesuited to predicting transient conditions in simple pipelines sject to simple disturbances when the nature of the transient beior of the system is known in advance. The most serious probwith the spline interpolation is the specification of the spliboundary conditions.

The authors point out that the selection procedure was a ‘‘tand error’’ one involving many possibilities. This ‘‘flexibility’’suffers from the curse of ‘‘permutability,’’ i.e., in a complex sytem the number of permutations and combinations of spboundary conditions can become enormous. Moreover, in mmultipipe applications it is not accuracy that directly governsselection of the time step, but the hydraulically shortest pipe insystem. Since the most successful spline boundary conditionsessarily involve several reaches, application of the methodcomes problematic in short pipes. It would appear to require msmaller time steps simply to apply the method at all. Other nessary conditions for the success of spline schemes are:~i! thedependent variable~s! must be sufficiently smooth,~ii ! the compu-tation of the derivatives at internal nodes must be accurate,~iii ! the formulation of the numerical and/or physical derivatiboundary conditions must be simple and accurate. Condition~i!and ~iii ! are a problem in water hammer analysis becauseboundary conditions are frequently nonlinear and complex,the dependent variables may be discontinuous.

Karney and Ghidaoui@70# developed ‘‘hybrid’’ interpolationapproaches that include interpolation along a secondary charaistic line, ‘‘minimum-point’’ interpolation~which reduces the dis-tance from the interpolated point to the primary characterist!,and a method of ‘‘wave path adjustment’’ that distorts the pathpropagation but does not directly change the wave speed.resulting composite algorithm can be implemented as a preprosor step and thus uses memory efficiently, executes quickly,provides a flexible tool for investigating the importance of dcretization errors in pipeline systems. The properties of the arithm are analyzed theoretically and illustrated by example inpaper.

5.2 Other Schemes. The wave plan method@71# is similarto the MOC in the sense that both techniques explicitly incorrate wave paths in the solution procedure. However, the wavemethod requires that flow disturbance functions such as vacurves be approximated by piecewise constant functions. Thaflow disturbances are approximated by a series of instantanchanges in flow conditions. The time interval between any tconsecutive instantaneous changes in flow conditions is fixed.piecewise constant approximation to disturbance functions impthat the accuracy of the scheme is first order in both spacetime. Therefore, fine discretization is required for achieving acrate solutions to water hammer problems.

The wave plan method ‘‘lumps’’ friction at the center of eacpipe. In particular, friction is modeled using a disturbance funtion, where the form of this function is determined using the ‘‘ofice analogy.’’ This disturbance function is friction approximateby piecewise constant functions. The modeling of friction asseries of discrete disturbances in space and time generatesspurious waves. In general, with small values of friction, thewould be observed only as low-amplitude noise on the main tr


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sient signal. It is also unclear as to how additional physics, succonvolution-integral unsteady friction models, can be incorprated with the wave plan methodology.

Wylie and Streeter@72# propose solving the water hammeequations in a network system using the implicit central differemethod in order to permit large time steps. The resulting nonlindifference equations are organized in a sparse matrix form andsolved using the Newton-Raphson procedure. Only pipe juncboundary conditions were considered in the case study. It isognized that the limitation on the maximum time step is set byfrequency of the dependent variables at the boundaries. Two cmercially available water hammer software packages use thepoint implicit scheme~see Sec. 12 Water Hammer Software!. Themajor advantage of implicit methods is that they are stablelarge time steps~i.e., Cr.1 @65,72#!. Computationally, howeverimplicit schemes increase both the execution time and the storequirement and need a dedicated matrix inversion solver sinlarge system of equations has to be solved. Moreover, for mproblems, iterative schemes must also be invoked. From a mematical perspective, implicit methods are not suitable for wpropagation problems because they entirely distort the pathpropagation of information, thereby misrepresenting the maematical model. In addition, a small time step is required forcuracy in water hammer problems in any case@23#. For thesereasons, most of the work done on numerical modeling of hypbolic equations in the last three decades concentrated on deving, testing, and comparing explicit schemes~e.g.,@63,64,73#!.

Chaudhry and Hussaini@74# apply the MacCormack, Lambdaand Gabutti schemes to the water hammer equations. Thesemethods are explicit, second-order~in time and space! finite dif-ference schemes. Two types of boundary conditions are used~i!one characteristic equation and one boundary equation, or~ii ! ex-trapolation procedure boundary condition. The second bouncondition solution method adds one fictitious node upstreamthe upstream boundary and another downstream of the dostream boundary. Using theL1 andL2 norms as indicators of thenumerical errors, it was shown that these second-order findifference schemes produce better results than first-order meof characteristics solutions forCr50.8 and 0.5. Spurious numercal oscillations are observed, however, in the wave profile.

Although FV methods are widely used in the solution of hypbolic systems, for example, in gas dynamics and shallow wwaves~see recent books by Toro@75,76#!, this approach is seldomapplied to water hammer flows. To the authors’ knowledge,first attempt to apply FV-based schemes was by Guinot@77#. Heignored the advective terms, developed a Riemann-type solufor the water hammer problem, and used this solution to devea first-order-based FV Godunov scheme. This first-order schis very similar to the MOC with linear space-line interpolation.the time of writing, a second paper by Hwang and Chung@78# thatalso uses the FV method for water hammer, has appeared. Uin Guinot @77#, the advective terms are not neglected in the woof Hwang and Chung@78#. Instead, they use the conservative forof the compressible flow equations, in which density, andhead, is treated as an unknown. The application of such a schin practice would require a state equation relating density to hso that~i! all existing boundary conditions would have to be rformulated in terms of density and flow rather than head and fland~ii ! the initial steady-state hydraulic grade line would needbe converted to a density curve as a function of longitudinal dtance. At present, no such equation of state exists for water.plication of this method would be further complicated at bounaries where incompressible conditions are generally assumeapply.

5.3 Methods for Evaluating Numerical Schemes. Severalapproaches have been developed to deal with the quantificationumerical dissipation and dispersion. The wide range of methin the literature is indicative of the dissatisfaction and distramong researchers of more conventional, existing techniq


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This section discusses a number of methods employed by tsient modelers to quantify numerical dissipation and dispersio

5.3.1 Von Neumann Method.Traditionally, fluid transient re-searchers have studied the dispersion and dissipation charactics of the fixed-grid method of characteristics using the Von Nmann~or Fourier! method of analysis@63,64#. The Von Neumannanalysis was used by O’Brian et al.@79# to study the stability ofthe numerical solution of partial differential equations. The anasis tracks a single Fourier mode with time and dissipationdetermining how the mode decays with time. Dispersion is evaated by investigating whether or not different Fourier modes trawith different speeds.

There are a number of serious drawbacks to the Von Neummethod of analysis. For example, it lacks essential boundaryformation, it ignores the influence of the wave profile on the nmerical errors, it assumes constant coefficients and that the inconditions are periodic, and it can only be applied to linear nmerical models@69,79–81#. To illustrate, the work by Wiggertand Sundquist@63#, Goldberg and Wylie@64#, and others clearlyshows that the attenuation and dispersion coefficients obtafrom the Fourier analysis depend on the Courant number, the rof the wavelength of thekth harmonicLk to the reach lengthDx,and the number of reachbacks and/or reachouts, but does nopend on the boundary conditions. Yet, the simulation of boundconditions and knowledge of how these boundary conditionstroduce and reflect errors to the internal pipe sections is cruciathe study of numerical solutions of hydraulic problems. In shothe Von Neumann method cannot be used as the only benchmfor selecting the most appropriate numerical scheme for nonlinboundary-value hyperbolic problems.

5.3.2 L1 and L2 Norms Method. Chaudhry and Hussain@74# developedL1 andL2 norms to evaluate the numerical erroassociated with the numerical solution of the water hammer eqtions by the MacCormack, Lambda, and Gabutti schemes. Hever, theL1 andL2 method as they apply it can only be used fproblems that have a known, exact solution. In addition, thesenorms do not measure a physical property such as mass or enthereby making the interpretation of the numerical values of thnorms difficult. Hence, theL1 andL2 norms can be used to compare different schemes, but do not give a clear indication of hwell a particular scheme performs.

5.3.3 Three Parameters Approach.Sibetheros et al.@69#used three dimensionless parameters to study various numeerrors. A discussion followed by Karney and Ghidaoui@82# and aclosure was provided by the authors. Salient points from thecussion and closure are summarized below.

The attenuation parameter is intended to measure the numedissipation by different interpolation schemes by looking atmaximum head value at the valve. This parameter, however,derestimates the numerical attenuation because the computatihead and flow at the downstream end of the pipe uses one cacteristic equation and one boundary equation. The dispersionrameter is intended to measure the numerical dispersion by dient interpolation schemes. This parameter is determinedasserting that the change in the wave shape is governed byconstant diffusion equation with initial conditions describedthe Heaviside function. Although this method allows a rudimetary comparison of simple system responses, general concluscannot be drawn for a hyperbolic equation based on a diffusequation. The longitudinal displacement parameter is intendemeasure the extent by which different numerical schemes acially displace the wave front. However, this parameter only sgests to what degree the interpolation method used is symmcally dispersive and says little about the magnitude of artificdisplacement of the wave by the numerical scheme.

5.3.4 Mass Balance Approach.The mass balance metho@83,84# is a more general technique than the other existing me


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ods since this approach can be applied to a nonlinear tranproblem with realistic boundary conditions. The basic idea ischeck how closely a particular numerical method conserves mNote that the mass balance approach can become ineffectivcases where a numerical scheme conserves mass but not eand momentum.

5.3.5 EHDE Approach. Ghidaoui and Karney@85# devel-oped the concept of an equivalent hyperbolic differential equa~EHDE! to study how discretization errors arise in pipeline appcations for the most common interpolation techniques used towith the discretization problem in fixed-grid MOC. In particular,is shown that space-line interpolation and the Holly-Preissmscheme are equivalent to a wave-diffusion model with an adjuwave speed, but that the latter method has additional sourcesink terms. Further, time-line interpolation is shown to be equilent to a superposition of two waves with different wave speeThe EHDE concept evaluates the consistency of the numescheme, provides a mathematical description of the numericalsipation and dispersion, gives an independent way of determithe Courant condition, allows the comparison of alternativeproaches, finds the wave path, and explains why higher-omethods should usually be avoided. This framework clearly poout that numerical approximation of the water hammer equatifundamentally changes the physical problem and must be vieas a nontrivial transformation of the governing equations. Forample, implicit methods, while noted for their stability charactistics, transform the water hammer problem into a superposiof wave problems, each of which has a wave speed different fthe physical wave speed and at least one of which has an infiwave speed. The infinite numerical wave speed associatedimplicit schemes ensures that the numerical domain of depdence is larger than the physical domain of dependence, andplains why these are highly stable. While good for stability, tlarge discrepancy between the numerical and physical domaindependence hinders the accuracy of these schemes. Anotherlem with implicit schemes is that they are often computationainefficient because they require the inversion of large matrice

5.3.6 Energy Approach.Ghidaoui et al.@86# developed anintegrated energy approach for the fixed-grid MOC to study hthe discretization errors associated with common interpolaschemes in pipeline applications arise and how these errors cacontrolled. Specifically, energy expressions developed inwork demonstrate that both time-line and space-line interpolaattenuate the total energy in the system. Wave speed adjustmon the other hand, preserves the total energy while distortingpartitioning of the energy between kinetic and internal formThese analytic results are confirmed with numerical studiesseveral series pipe systems. Both the numerical experimentsthe analytical energy expression show that the discretization eare small and can be ignored as long as there is continuousin the system. When the work is zero, however, aCr value closeto one is required if numerical dissipation is to be minimized. Tenergy approach is general and can be used to analyze otherhammer numerical schemes.

6 Flow Stability and the Axisymmetric AssumptionExisting transient pipe flow models are derived under

premise that no helical type vortices emerge~i.e., the flow remainsstable and axisymmetric during a transient event!. Recent experi-mental and theoretical works indicate that flow instabilities, inform of helical vortices, can develop in transient flows. Theinstabilities lead to the breakdown of flow symmetry with respto the pipe axis. For example, Das and Arakeri@87# performedunsteady pipe flow experiments where the initial flow was lamiand the transient event was generated by a piston. They foundwhen the Reynolds number and the transient time scale excethreshold value, the flow becomes unstable. In addition, theyserved that the flow instability results in the formation of nons


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tionary helical vortices and that the breakdown of these vortiinto turbulence is very rapid. The breakdown of the helical voces into turbulence resulted in strong asymmetry in the flow wrespect to the pipe axis. Brunone et al.@31,88# carried out mea-surements of water hammer velocity profiles in turbulent flowThey also observed strong flow asymmetry with respect topipe axis. In particular, they found that a short time after the wapassage, flow reversal no longer appears simultaneously inthe top and the bottom sides of the pipe. Instead, flow reveappears to alternate between the bottom and top sides of theThis is consistent with the asymmetry observed by Das and Aeri @87#. The impact of instabilities on wall shear stress in usteady pipe flows was measured by Lodahl et al.@89#. They foundthat inflectional flow instabilities induce fluctuations in the washear stress, where the root mean square of the wall shear sfluctuation in the pipe was found to be as high as 45% ofmaximum wall shear stress.

Das and Arakeri@87# applied linear stability analysis to unsteady plane channel flow to explain the experimentally obserinstability in unsteady pipe flow. The linear stability and the eperimental results are in good qualitative agreement. Ghidaand Kolyshkin @90# investigated the linear stability analysis ounsteady velocity profiles with reverse flow in a pipe subjectthree-dimensional~3D! perturbation. They used the stability results to reinterpret the experimental results of Das and Ara@87# and assess their planar flow and quasi-steady assumptComparison of the neutral stability curves computed with awithout the planar channel assumption shows that this assumpis accurate when the ratio of the boundary layer thickness topipe radius is below 20%. Any point in the neutral stability currepresents the parameters combination such that the perturbaneither grow nor decay. Critical values for any of these paramecan be obtained from the neutral stability curve. For unsteady pflows, the parameters related are Re andt. Therefore, critical Recan be obtained.

The removal of the planar assumption not only improvesaccuracy of stability calculations, but also allows for the flostability of both axisymmetric and nonaxisymmetric modes toinvestigated, and for the experimental results to be reinterpreFor example, both the work of Ghidaoui and Kolyshkin@90# andthe experiments of Das and Arakeri@87# show that the nonaxisymmetric mode is the least stable~i.e., the helical type!.

With the aim of providing a theoretical basis for the emergenof helical instability in transient pipe flows, Ghidaoui and Kolyshkin@42# performed linear stability analysis of base flow velociprofiles for laminar and turbulent water hammer flows. These bflow velocity profiles are determined analytically, where the trasient is generated by an instantaneous reduction in flow rate adownstream end of a simple pipe system. The presence of intion points in the base flow velocity profile and the large velocgradient near the pipe wall are the sources of flow instability. Tmain parameters governing the stability behavior of transiflows are Reynolds number and dimensionless time scale.stability of the base flow velocity profiles with respect to axisymmetric and asymmetric modes is studied and the results are ploin the Reynolds number/time scale parameter space. It is fothat the asymmetric mode with azimuthal wave number one isleast stable. In addition, it is found that the stability results oflaminar and the turbulent velocity profiles are consistent with plished experimental data. The consistency between the stabanalysis and the experiments provide further confirmation~i! thatwater hammer flows can become unstable,~ii ! that the instabilityis asymmetric,~iii ! that instabilities develop in a short~waterhammer! time scale and,~iv! that Reynolds number and the wavtime scale are important in the characterization of the stabilitywater hammer flows. Physically, flow instabilities change tstructure and strength of the turbulence in a pipe, result in str


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flow asymmetry, and induce significant fluctuations in wall shstress. These effects of flow instability are not represented inisting water hammer models.

In an attempt to gain an appreciation of the importance ofcluding the effects of helical vortices in transient modeGhidaoui et al.@46# applied current transient models to flow caswith and without helical vortices. In the case where stabilitysults indicate that there are no helical vortices, Ghidaoui e@46# found that the difference between water hammer modelsthe data of Pezzinga and Scandura@91# increases with time at amild rate. However, for the case where stability results andperiments indicate the presence of helical vortices, it is foundthe difference between water hammer models and the datBrunone et al.@31# exhibits an exponential-like growth. In facthe difference between models and the data of Brunone et al.@31#reaches 100% after only six wave cycles. This marked differebetween models and data suggests that the influence of hevortices on the flow field is significant and cannot be neglecte

7 Quasi-Steady and Frozen Turbulence AssumptionsThe convolution integral analytical models for wall shear

unsteady turbulent flows derived in Vardy et al.@59# and Vardyand Brown@60# assume that eddy viscosity remains ‘‘frozen’’~i.e.,time independent! during the transient. Turbulence closure equtions used by Vardy and Hwang@25#, Silva-Araya and Chaudhry@37#, and Pezzinga@38# assume that the turbulence changes inquasi-steady manner and that the eddy viscosity expressionrived for steady-state pipe flows remain applicable for water hamer flows. An understanding of the response of the turbulefield to water hammer waves is central to judging the accuracusing either the frozen or the quasi-steady turbulence assutions.

There is a time lag between the passage of a wave frontparticular location along the pipe and the resulting change inbulent conditions at this location~e.g.,@46,92,93#!. In particular, atthe instant when a water hammer wave passes a positionx alongthe pipe, the velocity field atx undergoes a uniform shift~i.e., thefluid exhibits a slug flowlike motion!. The uniform shift in veloc-ity field implies that the velocity gradient and turbulent conditioare unaltered at the instant of the wave passage. Howevercombination of the uniform shift in velocity with the no-slip condition generates a vortex sheet at the pipe wall. The subseqdiffusion of this vortex ring from the pipe wall to the pipe corethe mechanism responsible for changing the turbulence conditin the pipe.

A short time after the wave passage, the extent of vorticdiffusion is limited to a narrow wall region and the turbulenfield is essentially frozen. In this case, both the quasi-steadybulence and ‘‘frozen’’ turbulence assumptions are equally apcable. A similar conclusion was reached by Greenblatt and M@92# for a temporally accelerating flow; by Tu and Rampari@94#, Brereton et al.@95#, and Akhavan et al.@96,97# for oscilla-tory flow; He and Jackson@93# for ramp-type transients; anGhidaoui et al.@46# for water hammer flows. As the time after thwave passage increases, the extent of the radial diffusion ofticity becomes more significant and begins to influence the veity gradient and turbulence strength and structure in the buzone. The experiments of He and Jackson@93# show that the axialturbulent fluctuations are the first to respond to the changes inradial gradient of the velocity profile and that there is a time debetween the changes in the axial turbulent fluctuations andredistribution among the radial and azimuthal turbulent comnents. The production of axial turbulent kinetic energy andtime lag between production and redistribution of axial turbulkinetic energy within the buffer zone are not incorporatedsteady-state-based turbulence models. The characteristics oflow in the core region will start to change only when the wavinduced shear pulse emerges from the buffer zone into theregion. On the basis of their unsteady flow experiments, He


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Jackson@93# provided an estimate for the time delay from thmoment the wall vortex ring was generated at the pipe wall tomoment when significant changes in the structure and strengtturbulence appeared near the pipe axis.

Ghidaoui et al.@46# proposed a dimensionless parameterP forassessing the accuracy of quasi-steady turbulence modeling inter hammer problems. This parameter is defined as the ratio otime scale of radial diffusion of vorticity to the pipe core to thtime scale of wave propagation from one end of the pipe toother. This parameter provides a measure for the number of tia wave front travels from one end of the pipe to the other befthe preexisting turbulence conditions start to respond to the tsient event. It follows that the frozen and quasi-steady assutions are~i! acceptable whenP@1, ~ii ! questionable whenP is oforder 1, and~iii ! applicable whenP!1. However, the last casedoes not belong to the water hammer regime. These conclusare supported by the work of Ghidaoui et al.@46#, where theycompared the results of quasi-steady turbulence modelsavailable data and by the work of Ghidaoui and Mansour@32#where they compared the results of frozen eddy viscosity mowith experimental data.

8 Two-Dimensional Mass and Momentum EquationsQuasi-two-dimensional water hammer simulation using tur

lence models can~i! enhance the current state of understandingenergy dissipation in transient pipe flow,~ii ! provide detailed in-formation about transport and turbulent mixing~important forconducting transient-related water quality modeling!, and~iii ! pro-vide data needed to assess the validity of 1D water hammer mels. Examples of turbulence models for water hammer probletheir applicability, and their limitations can be found in Vardy anHwang @25#, Silva-Araya and Chaudhry@37,98#, Pezzinga@38,44#, Eichinger and Lein@45#, Ghidaoui et al.@46#, and Ohmiet al. @99#. The governing equations for quasi-two-dimensionmodeling are discussed in this section. Turbulence modelsnumerical solutions are presented in subsequent sections.

The most widely used quasi-two-dimensional governing eqtions were developed by Vardy and Hwang@25#, Ohmi et al.@99#,Wood and Funk@100#, and Bratland@101#. Although these equa-tions were developed using different approaches and are writtedifferent forms, they can be expressed as the following paircontinuity and momentum equations:

g

a2 S ]H

]t1u

]H

]x D1]u

]x1

1

r

]rv]r

50 (42)

]u

]t1u

]u

]x1v

]u

]r52g

]H

]x1

1

rr

]r t

]r(43)

wherex,t,u,H,r are defined as before,v(x,r ,t)5 local radial ve-locity, andt5shear stress. In this set of equations, compressibis only considered in the continuity equation. Radial momentumneglected by assuming that]H/]r 50, and these equations artherefore, only quasi-two-dimensional. The shear stresst can beexpressed as

t5rn]u

]r2ru8v8 (44)

whereu8 andv85turbulence perturbations corresponding to logitudinal velocity u and radial velocityv, respectively. Turbu-lence models are needed to describe the perturbation2ru8v8 since most practical water hammer flows are turbulen

The governing equations can be further simplified by negleing nonlinear convective terms, as is done in the 1D case sincewave speeda is usually much larger than the flow velocityu or v.Then the equations become the following:

g

a2

]H

]t1

]u

]x1

1

r

]rv]r

50 (45)


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These governing equations are usually solved by numermeans.

For an adequately anchored or restrained pipe, i.e., the piprigid and the radial velocity at the pipe wall is zero. From flosymmetry, the radial velocity at the centerline is also zero. Ingrating Eq.~45! across the pipe section, the radial velocity vaishes, leaving the following:

]H

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a2

gA

]Q

]x50 (47)

]u

]t1g

]H

]x5

1

rr

]r t

]r(48)

Q~x,t !5EAudA (49)

whereQ5discharge. These equations are the same as thosesented by Pezzinga@38#.

In cases where the radial velocity component~mass flux! isnegligible, Eqs.~47!–~49! can be usefully applied. However, thinclusion of radial fluxes in Eqs.~45! and~46! remove the incon-sistency that occurs near boundary elements due to the simneous imposition of the no-slip condition and the plane wavesumption@24#. Since numerical integration of Eq.~49! is neededto relate velocity distribution to discharge, even very small errfrom neglecting radial fluxes can produce spurious oscillationpressure head calculations.

Ghidaoui @26# derived quasi-two-dimensional equations frothe complete 3D continuity equation and Navier-Stokes equatusing an ensemble averaging process in which the assumpinherent in the quasi-two-dimensional equations~such as flow axi-symmetry and the plane wave assumption! are made explicit. Thescaling analysis@26# shows that the viscous terms associated wthe compressibility of the fluid are significantly smaller than tviscous term associated with angular deformation. Therefore,compressibility is neglected in the momentum equations of b1D and 2D models.

In Silva-Araya and Chaudhry@37,98# and Eichinger and Lein@45#, an integration of the momentum equation is also carriedIn each case, the system reduces to a 1D formulation. The qtwo-dimensional momentum Eq.~46! is only used to provide anunsteady friction correction for 1D governing equations. Thecorrections include:~i! an energy dissipation factor, which is thratio of the energy dissipation calculated from the cross-sectiovelocity distribution to that calculated from the Darcy-Weisbaformula @37,98# or ~ii ! direct calculation of wall shear stress, ether by velocity gradient at the pipe wall or through energy dispation @45#.

9 Turbulence ModelsTurbulence models are needed to estimate the turbulent pe

bation term for2ru8v8. In the water hammer literature, thwidely used turbulence models are algebraic mod@25,37,38,98,99# in which eddy viscosity is expressed as somalgebraic function of the mean flow field. Other sophisticamodels~such as thek2e model, which require additional differential equations for eddy viscosity! have also been tried@45#.Similar results for the pressure head traces have been obtain

The algebraic turbulence models used by Vardy and Hw@25# and Pezzinga@38# are discussed further to illustrate somfeatures of algebraic turbulence models. These models wereparatively studied by Ghidaoui et al.@46#. The comparison showsthat very similar dissipation is produced by the two models. Otdifferent variations of algebraic turbulence models are availablRodi @102#.


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9.1 Five-Region Turbulence Model. The model used byVardy and Hwang@25# is a direct extension of the model deveoped by Kita et al.@103# for steady flow

t5r~n1e!]u

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wheree5eddy viscosity,nT5total viscosity, and the other termwere previously defined. The total viscosity distribution is copartmentalized into five regions as follows:

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a

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n5ky* @12~k/4Cm!~y* /R* !#

3k

CB1k2/4CmR*

<y* <1

k2Cm~11A12Cc /Cm!R*

(54)

5. core regionnT

n5CcR*

1

k2Cm~11A12Cc /Cm!R* <y*

<R* (55)

wherey5R2r , y* 5u* y/n, R* 5u* R/n, u* 5Atw /r, and thecoefficients area50.19, CB50.011, k50.41, Cm50.077, andCc5a function of Reynolds number~usually a value of 0.06 isused!. The total viscosity distribution depends on friction velociu* and positiony only. This is true for steady flow since ainformation at interior points will ultimately propagate to the waboundary. Given sufficient time, the velocity profile adjusts afinally depends on wall shear stress only. However, this momay be problematic for unsteady flow since the interior conditiocannot solely be represented by wall shear stress.

9.2 Two-Layer Turbulence Model. In the two-layer turbu-lence model, flow is divided into two layers:~i! a smooth pipe,viscous sublayer is assumed to exist near the wall; and~ii ! outsidethe viscous sublayer, the Prandtl mixing length hypothesis is u

1. viscous sublayer e50 y* <11.63 (56)

2. turbulent region e5 l 2U ]u

]r U y* >11.63 (57)

where

l

R5k

y

Re2(y/R) (58)

k50.37410.0132 lnS 1183100

Re D (59)

and in whichl 5mixing length and Re5Reynolds number for ini-tial flow. The thickness of the viscous sublayer is determinedthe wall shear stress. The eddy viscosity in the turbulent regincludes some information about the velocity profile. The twlayer model appears to be more suitable for unsteady flow si


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lation, but one should note that the expression for mixing lenand the empirical coefficientk are based on steady flow equivlents.

Ghidaoui et al.@46# compared both models~i.e., the five-region and the two-layer model! and obtained very similar resultfor pressure head estimates. The comparative study suggestthe pressure head is not sensitive to eddy viscosity distributiothe pipe core region. As these models are based on steadyprinciples, the application of these models to unsteady flow prlems implicitly includes the quasi-steady assumptions discusseSection 7.

These algebraic turbulence models are widely used, mabecause of their simplicity and robustness. As more powecomputers become available and improvements are made tomerical solution techniques, detailed turbulence structures maobtained using more sophisticated turbulence models, such atwo-equationk2e models, or perhaps even Reynolds stress mels, for which no eddy viscosity hypothesis is needed.

All of the models mentioned above are based on the Reynoaveraged Navier-Stokes~RANS! equation. The averaging procesis clearly a time average and valid for steady flows. For unsteflows, the use of the time average is highly questionable unlessunsteadiness has a much larger time scale than the time scaturbulence. Obviously, this is not the case for fast transients.

As an alternative, large eddy simulation~LES! has been devel-oped recently. In LES, the Navier-Stokes equation is filterlarge-scale motion is resolved while the small-scale motionmodeled. If results from LES were available, then some ofassumptions mentioned previously could, in principle, be mrigorously evaluated. Unfortunately, in carrying out LES, a f3D system of equations must be solved using very fine g@104#. For steady flow simulations, when the turbulence statisreach steady, the ensemble average can be obtained over ainterval from a single run@104#. However, the ensemble averagcannot be obtained from a single run for transient flow. Thequirement of many runs makes the resulting computational pcess prohibitively time consuming. As yet, such analyses havebeen performed in pipe transients.

10 Numerical Solution for 2D ProblemsThe 2D governing equations are a system of hyperbo

parabolic partial differential equations. The numerical solution


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Vardy and Hwang@25# solves the hyperbolic part of governinequations by MOC and the parabolic part using finite differenThis hybrid solution approach has several merits. First, the stion method is consistent with the physics of the flow since it uMOC for the wave part and central differencing for the diffusiopart. Second, the use of MOC allows modelers to take advanof the wealth of strategies, methods, and analysis developeconjunction with 1D MOC water hammer models. For exampschemes for handling complex boundary elements and stratedeveloped for dealing with the 1D MOC discretization proble~e.g., wave speed adjustment and interpolation techniques! can beadapted to quasi-two-dimensional MOC models. Third, althouthe radial mass flux is often small, its inclusion in the continuequation by Vardy and Hwang@25# is more physically correct andaccurate. A major drawback of the numerical model of Vardy aHwang@25#, however, is that it is computationally demanding.fact, the CPU time required by the scheme is of the orderNr

3

whereNr5number of computational reaches in the radial diretion. Vardy and Hwang’s scheme was modified by Zhao aGhidaoui @105# to a much more efficient form. The CPU timrequired is reduced to orderNr , making the scheme more amenable to application to the quasi-two-dimensional modelingpipe networks and for coupling with sophisticated turbulenmodels. Several numerical schemes for quasi-two-dimensiomodeling are summarized in the following material.

10.1 Vardy-Hwang Scheme. The characteristic form ofEqs.~45! and ~46! is as follows@25#:

dH

dt6

a

g

du

dt52

a2

g

1

r

]q

]r6

a

g

1

rr

]~r t!

]r

alongdx

dt56a (60)

whereq5rv.The pipe is divided intoNr cylinders of varying thickness. At

a given timet and locationx along the pipe, two equations applto each cylinder. Since there areNr cylinders in total, the totalnumber of equations is 2Nr . Therefore, the governing equationfor all cylinders can be written in matrix form as follows:Az5b, whereA is a 2Nr32Nr matrix whose form is as follows:

¨

1a

g1eCu2~1! uCq2~1!2eCu3~1!

1 2Fa

g1eCu2~1!G uCq2~1! eCu3~1!

A A A

1 ¯ 2eCu1~ j ! 2uCq1~ j !a

g1eCu2~ j ! uCq2~ j ! 2eCu3~ j !¯

1 ¯ eCu1~ j ! 2uCq1~ j ! 2Fa

g1eCu2~ j !G uCq2~ j ! eCu3~ j !¯

A A A

1 ¯ 2eCu1~Nr ! 2uCq1~Nr !a

g1eCu2~Nr !

1 ¯ eCu1~Nr ! 2uCq1~Nr ! 2Fa

g1eCu2~Nr !G

©

where j 5 index along radial direction;Cu1 ,Cu2 ,Cu35coefficients associated with axial velocityu; Cq1 ,Cq25coefficientsassociated with radial flux q; and e and u are weighting coefficients. The unknown vectorz5Hi

n11 ,ui ,1n11 ,qi ,1

n11 , ¯ ,ui , jn11 ,qi , j

n11 , ¯ ,ui ,Nr21n11 ,qi ,Nr21

n11 ,ui ,Nrn11%T in which i 5 index along axial direction and the superscriptT de-


,is

cceptedt matrixxpressed

ts

notes the transpose operator andb5a known vector that depends on head and velocity at time leveln. Therefore, the solution for headand longitudinal and radial velocities, involves the inversion of a 2Nr32Nr matrix. The sparse nature ofA is the reason the schemeinefficient.

Improving the efficiency of the Vardy-Hwang scheme is essential if quasi-two-dimensional models are to become widely aas tools for analyzing practical pipe systems or for conducting numerical experiments. Algebraic manipulation of the coefficienleads to a highly efficient scheme in which the original system becomes two subsystems with tridiagonal coefficient matrices eas the following:Bu5bu andCv5bv , whereB is a tridiagonalNr3Nr matrix given by

1a

g1eCu2~1! 2eCu3~1!

A

¯ 2eCu1~ j !a

g1eCu2~ j ! 2eCu3~ j ! ¯

A

2eCu1~Nr !a

g1eCu2~Nr !

2The unknown vectoru5$ui ,1

n11 , ¯ ,ui , jn11 , ¯ ,ui ,Nr

n11%T represents longitudinal flow velocity;bu is a known vector whose elemendepend onH, u, andq at time leveln; andC is a tridiagonalNr3Nr matrix given by

S 1 uCq2~1!

0 2@uCq1~2!1uCq2~1!# uCq2~2!

A

¯ uCq1~ j 21! 2@uCq1~ j !1uCq2~ j 21!# uCq2~ j !¯

A

¯ uCq1~Nr21! 2@uCq1~Nr !1uCq1~Nr21!#

D
-
l

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Lastly, v5$Hin11 ,qi ,1

n11 , ¯qi ,Nr21n11 %T is an unknown vector of

head and radial velocities andbv5a known vector whose elements depend onH,u,q at time leveln. Inversion of tridiagonalsystems can be performed efficiently by using the Thomas arithm.

10.2 Pezzinga Scheme.The numerical solution by Pezzinga @38# solves for pressure head using explicit FD from tcontinuity Eq.~47!. Once the pressure head has been obtainedmomentum Eq.~48! is solved by implicit FD for velocity profiles.This velocity distribution is then integrated across the pipe secto calculate the total discharge. The scheme is fast due to depling of the continuity and momentum equations and the adopof the tridiagonal coefficient matrix for the momentum equatioIt has been applied to network simulations.

While the scheme is efficient, the authors have found that this a difficulty in the numerical integration step. Since the integtion can only be approximated, some error is introduced instep that leads to spurious oscillations in the solution for pressTo get rid of these oscillations, a large number of reaches inradial direction may be required or an iterative procedure mneed to be used@37#.

10.3 Other Schemes. In Ohmi et al.@99#, the averaged 1Dequations are solved to produce pressure and mean velocitypressure gradient is then used to calculate a velocity profile uthe quasi-two-dimensional momentum equation, from which wshear stress is determined.

A similar procedure is used in Eichinger and Lein@45#. One-dimensional equations are first solved to obtain the pressuredient. This pressure gradient is used to solve Eq.~46! using afinite difference method. The eddy viscosity is obtained fromk2e model. Once the velocity profile is known, the friction tercan be calculated from the velocity gradient at the wall, whichthen used in the 1D equations. An iterative procedure is emploin this calculation to obtain eddy viscosity. Although there migbe some difference between the discharge calculated from


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equations and that obtained from the velocity profile integrationis neglected, since the calculation of the velocity profile is onused to estimate the friction term. This latter difference has onminor influence on the calculation of the unsteady friction term,argued by Eichinger and Lein@45#.

Silva-Araya and Chaudhry’s@37,98# procedure is similar to theforegoing methods. Once the velocity is obtained, energy disstion and discharge can be calculated. The dissipation is useestimate an energy dissipation ratio, which provides a correcfactor for the friction term in the 1D equations. The adjustedequations are then solved to give a new discharge, which is cpared to that calculated from velocity profile integration. If thdifference is small~say, less than 5%!, the calculation proceeds tothe next time step. Otherwise, the pressure gradient is adjusand the procedure is repeated. A mixing length algebraic turlence model~smooth pipe,@37#, rough pipe@98#! is used in thecalculation of the velocity profile.

11 Boundary ConditionsThe notion of boundary conditions as applied to the analysis

fluid transient problems is analogous to, but slightly differefrom, the conventional use of the terminology in solving differetial equations. Just as a ‘‘boundary value’’ problem in the maematical sense implies conditions that must be satisfied atedges of the physical domain of the problem, boundary conditiin fluid transients implies the need for additional head-discharelations to describe physical system components such as pureservoirs and valves. Thus, one or more simplified auxiliarylations can be specified to solve for piezometric head, flow velity, or other variables associated with the physical devices thselves. Examples of boundary conditions include, but arelimited to, valves, nozzles, pumps, turbines, surge tanks,valves, tanks and reservoirs, heat exchangers, condensersmany other application-specific devices.

This section of the paper discusses a generalized approacincorporating boundary conditions within the method of char


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teristics framework that preserves the complex physical and tological character of the compressible fluid system. The approutilizes unambiguous definitions of the nodes, links, and boundconditions that represent the components of a physical pipetem or network. Attention is restricted to the method of characistics solution because it is the most powerful and physically csistent method for dealing with physically and behavioracomplex devices without imposing unrealistic or difficult modications to the numerical scheme. The discussion begins byviewing the governing equations and the form of the methodcharacteristics solution that has been developed for this purp

11.1 Governing Equations and Their Solution. Twoequations—a relation of mass conservation and a momenequation—are generally used to model transient flow in cloconduits ~e.g., @20–23#!, which can be written from Eqs.~20!,~21!, and~28! as

]V

]t1g

]H

]x1

f VuVu2D

50 (61)

]H

]t1

a2

g

]V

]x50 (62)

To be compatible,x andV must be positive in the same directioEquations~61! and ~62! are valid as long as the flow is 1D, thconduit properties~diameter, wave speed, temperature, etc! areconstant, the ‘‘convective’’ and slope terms are small, andfriction force can be approximated by the Darcy-Weisbach fmula for steady flow. In addition, it is usually assumed thatfriction factor f is either constant or weakly dependent on tReynolds number. Note that, for simplicity, the shear model inmomentum Eq.~61! above is equivalent to Eq.~41! without theconvolution term. Other shear models can be readily adapteduse in the boundary condition framework described herein.

Because the equations governing transient fluid flow candom be solved analytically, numerical solutions are used toproximate the solution. The most widely used procedure isfixed grid method of characteristics, which has the desirabletributes of accuracy, simplicity and numerical efficiency. Tmethod is described in many standard references includChaudhry@20# and Wylie et al.@23#. Again, the procedures described here can be easily adapted for use with any of the inpolation, reach-back, reach-out, and wave speed or pipe leadjustment schemes mentioned previously.

In essence, the method of characteristics combines the momtum and continuity equations to form a compatibility expressin terms of dischargeQ and headH, that is

dH6BdQ6R

DxQuQudx50 (63)

whereB5a/gA and

R5f Dx

2gDA2 (64)

This equation is valid only along the so-calledC1 andC2 char-acteristic curves defined by

dx

dt56a (65)

For this reason, thex-t grid in Fig. 6 is chosen to ensureDx56aDt. Then, if the dependent variables are known atA andB,Eq. ~63! can be integrated along bothAP andBP. Integration ofthe first two terms is straightforward, while the third requires tvariation of Q with x to be known. Although this function isgenerally unknown, the term can usually be approximated@23#. Aconvenient linearization of theA to P integration is given byKarney and McInnis@106# as follows:


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for

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en-on

he

EA

P

QuQudx5@QA1e~QP2QA!#uQAuDx (66)

in which ueu<1.This linearization of the friction term includes the ‘‘classica

QAuQAuDx approximation (e50.0) and the ‘‘modified’’QPuQAuDx linearization (e51.0) as special cases. The approxmation associated withe50.0 has been traditionally employedbut is troublesome for high friction cases; the modified linearition is often more accurate and has improved stability proper@107#, but has not yet been universally adopted. Not only does~66! allow a single program to be used for both approximatiobut intermediate values ofe can be used to optimize accuracy foa givenDt. Preliminary results indicate values near 0.81 are wsuited to most applications. Higher-order approximations ofenergy loss term can also be incorporated, but generally reqiterative solution procedures. The linearized first-order approacresult in explicit formulations and provide acceptable results othe initial wave cycle for systems of low to moderate friction.

If Eq. ~63! is integrated as illustrated above, two equations cbe written for the unknowns atP

HP5CP2BPQP (67)

and

HP5CM1BMQP (68)

in which

CP5HA1QA@B2RuQAu~12e!# (69)

BP5B1eRuQAu (70)

CM5HB2QB@B2RuQBu~12e!# (71)

BM5B1eRuQBu (72)

In more complex systems, a subscript to indicate the pipe numis often added to these equations. At pointsP internal to a pipe-line, HP can be eliminated from Eqs.~67! and ~68! to obtain

QP5CP2CM

BP1BM(73)

At the ends of a conduit, however, the solution of the characistic equations is algebraically complicated by one or mo‘‘boundary conditions.’’

11.2 Boundary Conditions. The subject of what constitutea boundary condition can be treated generally. Karney@108# pre-sents concise terminology for describing pipe networks aboundary conditions. His nomenclature is followed throughothis paper and is briefly reviewed here. Once the time domai

Fig. 6 Method of characteristics grid


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discretized intoDt segments, most conduits in the network adivided into one or morereachesof length Dx. For clarity, theterm ‘‘pipe’’ is henceforth restricted to conduits containing at leone characteristic reach. The end of each reach, where headflow values must be determined, is called asection. At sectionsinternal to a pipe, the discharge can be obtained from Eq.~73!.However, at each end of the pipe, an auxiliary relation betwhead and discharge must be specified. Such a head-discharglation is called aboundary condition.

The term ‘‘node’’ is used herein to indicate a location wheboundary sections meet. Thedegreeof a node is the number opipes~i.e., characteristic sections! connected to it. However, in ageneral network, not only pipes may be connected to a nodevarious other elements as well. For example, a node may reprea suction or discharge flange of a pump, the location of a vadischarging from the network, or a connection for a pressure revalve. All such nonpipe junctions are labeledexternal and thenumber of such connections is called thecomplexityof the node.A node of complexity zero is calledsimple, a node of complexityoneordinary, and a node of complexity greater than onecomplex.In this paper, boundary conditions associated with complex noare referred to asboundary systems.Generally, the difficulty ofsolving a network increases as the complexity of the nodes innetwork increases but, as the following section shows, is indepdent of the degree of any node in the network. The terminolorelated to nodes can be extended in a natural way to networkwell and has been used by Karney@109# to develop a generaapproach for analyzing complex networks.

11.2.1 Simple and Ordinary One-Node Boundary ConditioJunctions of several pipes are usually modeled as frictionlestransient flow applications~e.g.,@109,110#!. Complications arisingby attempting to calculate junction losses at a general nodeconsiderable and are not discussed in this paper. Generally, enlosses at junctions are relatively small and neglecting them dnot appear to significantly impair the accuracy of the methodcharacteristics solution for a simple pipe junction.

The assumption that local losses are negligible is equivalenrepresenting the hydraulic grade line elevation at the node bsingle number, designatedHP .

Consider now Fig. 7, which depicts a junction of any numberpipes at a node. LetN1 be the set of all pipes whose assumed flodirection is toward the node in question andN2 be the set of pipeswhose assumed flow direction is away from the node. Letflow be identified as external and governed by an auxiliary retion. Positive flows are assumed to befrom the junction. Thefollowing derivation is similar to that appearing in Chapter 11Fluid Transientsby Wylie and Streeter@22#, but uses the notation

Fig. 7 Generalized node with one external flow


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proposed by Karney@108#. It differs from the latter only by theinclusion of the variable friction term linearization.

For all pipes belonging to the setN1 , Eq. ~67! holds while Eq.~68! applies for members ofN2 . These equations can be rearanged to obtain

QPi52

HP

BPi

1CPi

BPi

, i PN1 (74)

and

2QPj52

HP

BM j

1CM j

BM j

, j PN2 (75)

in which the second subscript represents the variable at the boary section of a particular pipe in the set.

The continuity equation for the junction requires the sumthe flows entering the node to equal the sum of the flows leavthe node

(i PN1

QPi2 (

j PN2

QPj2Qext50 (76)

Equations~74! and~75! can be substituted directly into Eq.~76! toproduce the following expression forHP :

HP5CC2BCQext (77)

in which

BC5S (i PN1

1

BPi

1 (j PN2

1

BM jD 21

(78)

and

CC5BCS (i PN1

CPi

BPi

1 (j PN2

CM j

BM j

D (79)

Equation ~77! represents a single relationship between junctheadHP and external flowQext in a multipipe frictionless junc-tion. The form of this equation is equivalent to the singleC1

compatibility Eq. ~67! and shows that any one-node boundacondition located in a network can be evaluated in exactlysame manner as if the boundary condition occurred at the dostream end of a single pipe.

Once a functional relationship representing a particulardraulic device is substituted into Eq.~77!, a single equation andunknown results. If this relationship is either linear or quadraan explicit formula for the unknown can be obtained.

For example, the simplest boundary condition occurs whQext is either constant or a known function of time~e.g., constantdisplacement pumps or fixed demands!. In this case, the value oQext can be substituted into Eq.~77! to obtain the junction head. Inparticular, this equation becomesHP5CC whenQext is zero. Thissolution for a simple node is algebraically equivalent to Eq.~73! ifthe node has only two pipes.

Comprehensive treatment of various boundary conditions sas valves, pumps, turbines, accumulators, air valves and mothers can be found in Wylie et al.@23#, Karney@108#, Chaudhry@20#, McInnis @111#, Karney and McInnis@82#, and McInnis et al.@112#. Formulations for many system-specific devices aboundthe literature.

12 Water Hammer SoftwareWith the advent of the Windows operating system, compu

languages such as Visual Basic and Visual C, geographic infortion systems~GIS!, and the World Wide Web, many water hammer models, previously only suited to academics and expert eneering practitioners, are now accessible to even the most noanalyst.


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In this section, we describe several commercially availableter hammer software packages. The information presented heintended to aid readers in locating software appropriate to twater hammer analysis needs.

It is important to note that two of the authors of this paperalso authors of a commercially available water hammer softwpackage. To avoid any conflicts of interest and to be fair towater hammer software developers, a critique of each water hmer software package is not presented here.

Instead, the intention of this section is to summarize the penent features of each computer model. These features includeare not limited to, the available hydraulic devices, selectable suprotection measures, input facilities, and output graphical visization options. Also listed for each software package is themerical method used by the water hammer model to solveunsteady flow problem. The reader is directed to Sec. 5 for baground on these numerical methods.

Each computer model has special features that distinguisfrom the other reviewed models. These differences are most othe result of a desire to serve a specific commercial market.example, some packages are best suited to fire protection sprisystems, fueling systems, or oil pipelines, while others are cletailored to large municipal water distribution systems. Still othmodels specialize in the analysis of hydroelectric systems, sewforce mains, or industrial applications such as cooling water stems. However, despite their obvious market focus, it is ofpossible to analyze just about any piping system with eachthese models.

The software packages described herein are in no particorder and more information on a product~e.g., up-to-date pricingnew features, computer system requirements, etc! can be obtainedupon browsing the appropriate Internet homepage, which is lisat the end of each review. Unless otherwise noted, the softwpackages reviewed below operate within a Windows-based eronment. Please also note that the information summarized beis largely derived from each water hammer modeler’s Interhomepage and is current at the writing of this paper~2003!.

Due to space limitations, all of the water hammer softwapackages now readily available could not be included in this smary. The reader is encouraged to search out alternatives oInternet prior to selecting one of the models described herein

12.1 Pipenet. This fluid flow program predicts pressursurges, calculates hydraulic transient forces, models controltems, and has been commercially available for over 20 years

The interface drag-and-drop facilities are used to build a scmatic of the pipeline or network and the associated boundaryvices. Pipe schedules as well as fitting, lining, pump and vadata are provided on-line for the user’s convenience. The userspecify the units of both the input and output data. Fluid propties such as viscosity and specific gravity can also be input byuser. Boundary devices include pumps, air chambers, reservtanks, caissons, vacuum relief valves, check valves, flow convalves, surge relief valves, and air release valves.

PIPENET performs a surge analysis using the method of chateristics and calculates pressures and flow rates at nodes, pand boundary devices, as well as transient pressure forcepipes and bends. As an option, the program calculates the fotion, growth, and collapse of a vapor cavity if the pressure inpipe system drops to vapor pressure. PIPENETalso has facilities forincorporating control theory~e.g., proportional, integral, derivative loops! in the operation of pumps and valves. Note thaspecial module is available for analyzing sprinkler systems.

Output data can be plotted as time history plots, each with udefined titles. Examples include pressure and flow rate timetory plots at nodes, pipe sections, or boundary devices. In ation, graphs of fluid level in an air chamber versus simulation timay be plotted~Contact: www.sunrise-sys.com!.


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12.2 HAMMER . With HAMMER, a schematic of the pipinglayout for both pipelines and networks can be drawn on-scrand groups of hydraulic elements duplicated to save time duthe input process. As an initial condition, steady-state data caimported from EPANET and WATERCAD. Some of the availableboundary devices include pumps, air vessels, open surge tareservoirs, surge control valves, vacuum relief and air relevalves, and bypass lines with check valves.

Commercially available for over 15 years, this methodcharacteristics-based model can be used to simulate pump pfailure, valve closure, pipe breaks, and pump startup. Time hisanimations and plots of transient pressure, flow rate, and aivapor volume at nodes and along pipes are available for bpipelines and networks. The model also produces profile viewnetwork pipeline paths, showing the initial steady-state pressas well as the maximum and minimum pressure envelopes~Con-tact: www.ehg.dns2go.com!.

12.3 HYTRAN . Drag-and-drop facilities enable on-screeconstruction or deletion of a pipeline or network in either planprofile views. Alternatively, node, pipe, and boundary device dcan be directly imported from EPANET. Some of the selectableboundary devices include pumps, turbines, air chambers, vacrelief valves, check valves, tanks, reservoirs, pressure revalves, pressure regulating valves, and demands. On-line hincluding a database of valve coefficients and pipe material prerties, is available to the user.

A method of characteristics-based solver generates pressureflow rate history traces at nodes and along the pipeline followpump power failure or startup. In addition, the computed transhydraulic grade line at any instant in simulation time can be pted in combination with the pipeline profile. Pressure traces,draulic grade line plots, and pipe flow direction can be animafor real-time viewing. A column separation indicator warns tuser when cavitation is detected~contact: www.hytran.net!.

12.4 HYPRESS. This model has an object-oriented interfathat allows for flexible input of pipe, node, and boundary devdata for pipelines and networks. Some of the boundary devthat can be represented by the model include pumps, turbivalves, reservoirs, surge chambers, and air vessels.

Using a fourth-order implicit finite difference based numericsolver, HYPRESScalculates the maximum, minimum, and instataneous transient hydraulic grade line for a pipeline followipump power failure. The hydraulic grade lines are plotted in cobination with the pipeline elevation profile and the instantanetransient hydraulic grade line, which can be animated in real t~Contact: www.hif.cz!.

12.5 IMPULSE. Liquids such as water, petroleum, chemicproducts, cryogens, and refrigerants can all be modeled usingIMPULSE water hammer model. A piping schematic is createdthe workspace using drag-and-drop facilities and data can be idirectly by the user or obtained from a built-in database containg properties for nine fluids and eight pipe materials. Some ofhydraulic devices that can be incorporated into the pipe netwinclude pumps, reservoirs, liquid accumulators, gas accumulavacuum breaker valves, demands, relief valves, and pressuretrol valves.

IMPULSE will calculate a system steady state and transfer itthe method of characteristics solver. Pipe length adjustmentopposed to wave speed adjustment, is used in combinationthe time step to spatially discretize the piping network. In socases, this means that the modeled pipe length can approxithe true length of the pipe. Transient events~e.g., pump powerfailures, pump starts, valve closures, etc! can be initiated based ontime or a device setpoint. Liquid column separation, vapor catation, and cavity collapse can be modeled. This model will idtify when and where maximum pressures occur and plot flow rpressure, and velocity time histories, which can be formatted


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the user. At each time interval, output, such as maximumminimum pressures, are tabulated for each node, pipe, and boary device~Contact: www.aft.com!.

12.6 WANDA. Both pressurized and nonpressurizbranched and looped pipe systems can be simulated with thister hammer model. Free-surface flow~i.e., a partly filled pipe! ismodeled using a conjugate gradient method with an upwindvection approximation. The effects of draining and filling a pipline can be simulated with this component.

A schematic of the piping system can be created on-screening a palette of boundary devices. A user defined image~e.g., astreet map! can be imported as a background to the schematicproperties of nodes, pipes, and boundary devices are input udialog boxes. Some of the available boundary devices inclpumps, control valves, check valves, taps, air vessels, airvalves, surge towers, pressure relief valves, weirs, and conden

The method of characteristics-based solver can be interruand resumed during a simulation. Cavitation and control the~e.g., proportional integral derivative loops, sensors, etc! modulesare optional. Pressure versus time histories can be plotted atdefined locations within the pipe system. In addition, it is possito view an animation of pressure wave propagation and reflecalong pipeline routes of a network~Contact: www.wldelft.nl/soft/wanda/!.

12.7 FLOWMASTER . This model calculates transient presures and flow rates in piping networks. In addition, calculationheat transfer and simulation of partly empty pipe segments~e.g.,sprinkler systems! is possible.

Pipe networks can be drawn on-screen using a list of pipcomponents and some of the boundary devices that can be rsented include pumps, reservoirs, weirs, orifices, valves, acculators, diaphragms, diffusers, heat-exchangers and pipe fittiUser-defined boundary devices can be programmed in eitherOR-

TRAN or C. Operational issues can be studied using predeficontrollers or user-defined controllers programmed in Visualsic or Java.

The method of characteristics solver generates results thabe viewed graphically or in tabular formats. Note that in additito liquids, gas flow dynamics can be simulated~Contact: www-.flowmaster.com!.

12.8 SURGE2000. With this model, a schematic of the piping layout can be drawn on-screen and over it can be placeimported background image, such as a street or elevation conmap. Boundary devices include pumps, valves, reservoirs, taair vessels, air and vacuum valves, pressure relief valves, santicipating valves, and heat exchangers.

This model uses the wave-plan method as opposed tomethod of characteristics or finite difference methods emploby the other models reviewed in this paper. Pump power failupump startup, and valve operations~e.g., closure! are just some ofthe unsteady fluid flow events that can be simulated wSURGE2000. Output, such as pressures, can be tabulated, plas contours over the system map, and displayed in time hisplots at nodes. In addition, for each pipeline path, the maximminimum, and instantaneous transient hydraulic grade linesbe plotted on an elevation versus distance graphic~Contact: ww-w.kypipe. com!.

12.9 LIQT . First introduced in 1972, LIQT can model fluidtransients in pipelines and networks subject to pump power faiand startup, turbine load loss, and valve closure. Some ofboundary devices that can be selected by the user include puturbines, check valves, air and vacuum valves, surge tanks, spipes, accumulators, and pressure relief valves. LIQT operateswithin a DOS environment window and uses the method of chacteristics to compute pressures and flow rates that can beported to spreadsheets, databases, and graphic software fosentation~Contact: www.advanticastoner.com!.


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12.10 WHAMO . This model uses a four-point implicit finitedifference method to calculate time-varying flow and headpipelines and networks. The user can select boundary devsuch as pumps, turbines, valves, tanks, reservoirs, vented ovented air chambers, pressure control valves, electric governand constant or time varying demands. A schematic of the pipsystem can be drawn on-screen with the help of a paletteboundary device symbols.

Both steady-state and transient conditions are generatedsimulations of pump power failure, valve closure, turbine lorejection, turbine startup, and governor controlled turbine opetion are possible~Contact: www.cecer.army.mil/usmt/whamowhamo.htm!.

12.11 TRANSAM. Using this model, real-time, 3D~i.e., dis-tance, time, and pressure! animations of the transient pressusurface can be viewed along user-defined network and pipepaths. A piping layout map can be created in a designated wspace using point-and-click options and a combination of pdown menus and dialog boxes are available for node, pipe,boundary device data input. An EPANET to TRANSAM conversionutility is supplied. Some of the boundary devices that can be rresented by this model include pumps, turbines, air chambreservoirs, tanks, flow control valves, air and vacuum revalves, check valves, pressure relief valves, surge anticipavalves, pressure reducing/sustaining valves, constant andvarying demands, and bypass lines with check valves.

Pump power failure and startup, variable speed pump and voperations~e.g., full and partial openings or closures!, turbineload rejection, and pipe breaks are just some of the event orinitiated unsteady flow conditions that can be simulated usingmethod of characteristics-based model. Simulation of the formtion, growth, and collapse of vapor cavities is optional. Time htory plots of pressure~and flow rate at nodes! can be produced anodes and along pipes. Real-time animations of the instantantransient, maximum, and minimum hydraulic grade lines canviewed for pipe paths~Contact: www.hydratek.com!.

13 Emerging Applications in Water HammerBy now, the reader is likely aware that the principal use

transient analysis, both historically and present day, is the pretion of peak positive and negative pressures in pipe systems toin the selection of appropriate strength pipe materials and aptenances and to design effective transient pressure controltems.

Two important areas in which transient modeling is now takia key role are parameter estimation for leakage detection andter quality predictions in potable water systems. Brief discussiof these two important areas of application are provided in tsection.

13.1 Parameter Estimation for Leakage Detection and In-verse Models. In many pipeline related industries, such aspotable water supply or in oil or gas transmission, owners knthat information is the key to successful management of thpipeline operation. For example, in the case of a water supphysical system characteristics, customer data, production rmaintenance records, quality assays, and so on, each provideagement, engineering, operations, and maintenance staff withformation they need to keep the system running efficiently asafely, and at a reasonable cost to the consumer. A large bodliterature on the subject of information requirements and dmanagement already exists, and all private and public pipeutilities are aware of the importance of collecting, archiving, aanalyzing data. Perhaps the most costly and time consumingpect of information management, however, is the collectiondata. This section outlines how inverse transient analysis caapplied to gather some types of physical system data. The tnology has the potential to be both cost efficient and accurate


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Using high-frequency pressure transducers, it is now possibsafely measure induced, or naturally occurring, pressure sevents. Coupling water hammer models to inverse models othe possibility of inexpensive data collection with a wide coveraof the system. System demands, leakage, pipe condition~rough-ness!, closed or partially closed valves, even pockets of trapgas or air can~in theory at least! all be detected using recordehigh-frequency pressure data. In addition to pressure measments, transient flow data can also be used in the inverse anaprocedures. However, flow meters capable of accurately resolthe variation in flow rates that occurs during water hammer eveare quite expensive and more troublesome to install. Hifrequency pressure transducers, on the other hand, are relainexpensive and easy to mount at common access points, suvalve and meter chambers, or even at fire hydrants. To datedraulic model parameters such as pipe roughness and wave shave been successfully calibrated using these techniques.

13.1.1 Inverse Analysis of Transient Data.Whether a tran-sient is small or large, accidental or planned, pressure wapropagate from their respective points of origin to other partsthe system. They travel at speeds ranging from about 250 mnearly 1500 m/s, depending on pipe material, soil and anchoconditions. The shock fronts interact with any part of the systthat either dissipates energy or does work in a thermodynasense. Thus, the energy content of the wave is diminishedvirtue of its interaction with the physical system, and its frequencomponents, amplitude, phasing, and attenuation characterbecome modified through successive interaction with the sysIn effect, a pressure signal at a given location constitutes a reof conditions in the system during the course of a given transevent.

Deciphering this record of interaction and extracting its infomation content is precisely what an inverse transient model dThe inverse model evaluates the recorded pressure~or flow! signaland determines which set~or sets! of system parameters, i.e., piproughness, water consumption~leakage!, wave speed, etc, besmatches the measured data. In this way, information~data! ofseveral types can be gathered from those areas of the pipe sythat the transient waves have traversed. For example, prestraces from two pump trip tests can be sufficient to estimate proughness values for every major pipe and consumption valueeach node in a small city. Of course, the accuracy of the estimcan be improved by increasing the number of tests performeby monitoring pressures~flows! at more than one location.

There is extensive literature about inverse analysis in bothentific and engineering journals. The techniques have been apfor many years to structural engineering applications e.g., sysidentification and damage detection@113#. Sykes@114#, Sun andYeh @115#, and Sun@116# have used inverse methods to identiparameters in 2D groundwater flow. Jarny et al.@117# applied theadjoint technique to heat conduction problems. Cacuci et al.@118#and Hall @119# applied the adjoint method to meteorology aclimate modeling. Marchuk@120# applied the adjoint technique tair pollution problems.

Most, though not all, inverse models utilize real measuremein a ‘‘data-fitting’’ exercise that typically provides ‘‘best-fit’’ parameters for the mathematical model postulated to fit the dLeast-squares data-fitting is a simple example of an invemethod that tries to fit the best mathematical model~i.e., linear,exponential, polynomial, etc! to some observed data set. Th‘‘goodness of fit,’’ i.e., how well the particular assumed matematical model represents the data, can be measured statistby an analysis of the errors between the observed data andpredicted by the model. In fact, these errors are explicitly mmized using Lagrangian optimization such that the optimal pareter set is directly solved for.

The same concept can be applied to more complex physsystems using sophisticated models. In an inverse problem, oufrom a ‘‘forward’’ model is used to generate an estimate of one


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more measured data sets using some assumed set of systerameters. System parameters could be pipe wave speed, frifactor, water consumption rates and locations, leakage rateslocations, and so on. A ‘‘merit’’ function is used to compare tgoodness of fit between the observed data and the model ouCommon merit functions are the error sum of squares, sum ofabsolute values of errors, etc. Some sort of search or optimizaprocedure is employed to find the set of parameter valuesminimizes the discrepancy between observed data valuesthose predicted by the forward model. It is the nature of the seatechnique employed in the optimization step that characterizesinverse modeling approach.

13.1.1.1 Adjoint models.Adjoint models use a form of La-grangian optimization coupled with a gradient search to minimthe errors between the observed data and the forward modeldiction. In transient flow applications, the problem statemwould take the following general form~see also Liggett and Che@121#!:

Minimize E5( @~hm2hc!21~qm2qc!2# (80)

subject to the following physical constraints:

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whereE is the error sum of squares and Eqs.~81! and~82! are thecontinuity and momentum equations rewritten in terms of dcharge and assuming steady Darcy-Weisbach friction. The suscript m denotes measured data values and the superscriptc de-notes the values computed by the forward model,h is piezometricpressure head,q is the flow rate,a is the pipeline celerity,f is theDarcy-Weisbach friction factor,t is time,x is a spatial coordinateg is gravitational acceleration, andA andD are the pipeline crosssectional area and diameter, respectively.

Equation ~80! can be combined with Eqs.~81! and ~82! byusing Lagrangian multipliersl1 andl2 as follows:

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The merit ~error! function has now been designatedE* toindicate that it includes the Lagrangian terms for the continuand momentum equations, and has been expressed as an into be consistent with the continuum form of the momentum acontinuity equations. The Dirac delta functions are includedensure that merit function terms are evaluated only at those ltions and times for which observed data exist, i.e.,

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The conventional approach to Lagrangian optimization istake partial derivatives of the merit function with respect to tunknown system parameters (a or f in this simple formulation!and the Lagrangian multipliersl1 andl2, and equate these slopfunctions to zero. This provides four equations from which tfour unknown variablesa, f , l1 , and l2 could be determined.


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However, as these equations are quasi-linear hyperbolic padifferential equations, a more elaborate procedure must be uPartial derivatives of the merit function are taken with respecl1 andl2 and at critical points of the merit function must haveslope of zero. The two derivative functions given in Eq.~85!below are known as the adjoint equations.

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It suffices to say that the adjoint model is solved iterativeValues for physical system parametersa and f are assumed, andthe forward model is run to determine the transient head and flThe adjoint equations are solved using the known heads and flin a backward pass to calculate the Lagrangian parametersl1 andl2 . These values are used in a gradient search step~the conjugategradient technique is often used! to select new estimates of thoptimal parameters. The search procedure terminates whenvalue ofE* cannot be reduced any further.

The advantage of the adjoint method is that it can be extremefficient for a well-conditioned problem. The model can be fmulated to solve for other parameters of interest beside wspeed and friction.

13.1.1.2 Genetic algorithms.Genetic algorithms~GAs! havegained widespread popularity in recent years. There are manysons for this success:~i! GAs can be applied to a wide variety oproblems;~ii ! GAs do not require the development of additioncode needed to solve the adjoint of the forward problem;~iii ! asingle GA can be used with various models that solve the sforward problem;~iv! any model parameters can be specifiedthe unknown system parameters in a GA;~v! GAs are quite suc-cessful in problems containing local extrema; and~vi! GAs canfind not only the global optimum, but can also describe otsuboptimal solutions of interest, particularly for flat merit funtions. Genetic algorithms do not work for every problem, hoever, and one must be aware of their limitations. GAs work bfor problems in which genotypes consist of a small numbergenes that can be expressed in short length strings, i.e., probhaving few decision variables~parameters! that can be identifiedby a small number of binary digits. Problems with large numbof real-valued parameters over an extensive and continuousmain are demanding of computer resources when solved bynetic algorithms. Despite these limitations, the method seemwork well with pipeline problems, albeit solution procedures aslower than those of the adjoint method.

In the simplest sense, genetic algorithms are an efficient fof enumeration. A candidate set of parameters is assumed ordomly generated to form individuals in a population. Subsequiterations use evolutionary~mutation! and reproductive~cross-over! functions to generate further generations of solutions. Tmathematical principle upon which genetic algorithms are bais intended for use with problems in which the decision variabare discrete, and in these situations the method can be extreefficient. Modifications to the method have been developedextend its application to continuous real-valued problems,though the procedures are less efficient in these cases.

Karney and Tang@122# have successfully applied the genealgorithm method to parameter estimation problems in watertribution systems using transient pressure readings. Usingfrom only two pump trip tests~one for model validation and theother for the parameter estimation!, Karney and Tang have successfully estimated pipe roughness factors and wave speedseveral large water distribution systems.

13.1.1.3 Pressure wave method.Brunone@123# and Brunoneand Ferrante@124# conducted numerical and physical experimeto investigate the possibility of using transient data for leaka


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detection. The transient response of a pipeline system to a gflow disturbance with and without leakage points was measurewell as computed. The influence of the size and shape of smleaks, along with discharge conditions and initial flow regime,the transient response of a pipeline system were analyzed.found that the influence of the leak on the shape and amplitudthe pressure signal is quite noticeable, even when the leak floonly a few percent of the total flow in the pipe. ThereforBrunone @123# and Brunone and Ferrante@124# formulated ascheme for leakage detection on the basis of studying the dience in transient response of a pipeline system with and witholeak. It is observed that the measured pressure head traces fopipeline with a leak is different from that for an intact pipe. Whethe transient wave encounters a leak, part of the wave is refleback. The leak location is determined from the time whenreflected wave arrives at the measurement station. The leakduces additional drop in the pressure head traces, the amoudrop depending on the size of the leak. The size of the leausing a formal inverse approach. The agreement between thetual and the computed location and size of leak points is goo

13.1.1.4 Frequency response method.The frequency re-sponse method is used by Mpesha et al.@125,126#. A hydraulicsystem is made up of several components. Each component crepresented by a transfer matrix. Transient flow is caused byperiodic opening and closing of a valve@125# or by the suddenopening or closing of a valve@126#. A frequency response diagram at the valve is developed based on the transform matrix.a system with leaks, this diagram has additional resonant presamplitude peaks that are lower than the resonant pressure atude peaks for the system with no leaks. From the frequencythe peaks, the location of the leak can be detected. Very gagreement have been obtained between the computed and thleak condition.

In Ferrante and Brunone@127#, the governing equations fotransient flow in pipes are solved directly in the frequency domby means of the impulse response method. Therefore, the soluof the response of the system to more attractive transient evenavailable. Harmonic analysis of the transient pressure is useidentify the location and the size of a leak.

13.1.1.5 Mode damping method.Wang et al.@128# investi-gated the damping characteristics of a transient pressure wavwall friction and by system leakage. It is found that wall frictiodamps all modes similarly, but leakage damps different Foumodes differently. In addition, mode damping by leakage is fouto depend on leak location. The marked difference in mode daing between wall friction and system leakage was successfused to identify the location and size of leaks@128#. In particular,Wang et al.@128# were able to accurately identify system leaksinvestigating mode damping characteristics of transient presdata obtained from numerical as well as laboratory studies.damping characteristic technique was successfully appliedsingle and multiple leaks.

13.1.1.6 Wavelet transform method.Frequency analysis canonly deal with a stationary signal~i.e., the signal has to be eitheperiodic or decomposable into a set of periodic signals!. Wavelettransform can be used to detect local singularities in a meassignal. Whenever there is a singularity in a measured signalocal maximum of the transform coefficient for the measured snal appears. The application of the transient wavelet transformleakage detection in a pipeline was pioneered by FerranteBrunone@129#. The wavelet transform of pressure head historyperformed. According to the transform of the signal, the disconuities in pressure head traces are detected. These discontincorrespond to wave reflections at boundary elements and atpoints. Using the time at which a discontinuity is observed,


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13.1.1.7 Identifiability and uniqueness requirements.In or-der for any inverse method to be successfully applied, twomathematical properties of the problem must be satisfied: idefiability and uniqueness. Identifiability refers to the notion thasingle set~or a finite number of distinct sets! of parameter valuesmust reproduce, within an established level of error, the saresponse exhibited by the actual system. Uniqueness meansthe merit function exhibits a single, global minimum. For complproblems, there is no rigorous mathematical procedure thatassure us that identifiability and uniqueness requirements areisfied in general. However, a simple example is described infollowing section that provides insight into the suitability of diferent inverse modeling techniques for pipeline transient prlems.

13.1.1.8 Identifiability. Let Hm5$H0 ,H1 ,H2 , . . . ,Hn% de-note a set of measured values, e.g., piezometric head, at time0 to n corresponding to some sampling rate (tn2t0)/(n21). Lethc5$h0 ,h1 ,h2 , . . . ,hn% similarly denote the set of computeheads at the same time steps but for a particular pair of unknobut desired, parameter valuess1 and s2 . Then, the followingcriterion for identifiability can be stated:

Identifiability criterion: A set of parameter valuess1 ands2 areidentifiable if and only ifHm[hc6e, wheree represents the absolute value of data, measurement, and model error. The ideability criterion can be visualized by plotting the difference btweenHm andhc for each pair of feasible values of parameterss1ands2 in the domain and selecting the zero contour of the diffences. Identifiable parameter pairs for whichHm[hc would ap-pear as intersections~loci of intersecting lines! of all such con-tours.

Uniqueness: The second condition that needs to be met ifjoint methods are to be used with a reasonable expectatiosuccess is uniqueness, i.e., there should ideally be only a scritical point of the merit function within the feasible search dmain.

The adjoint technique can still be useful if the feasible seadomain can be restricted to a smaller region containing the glominimum. To this end, a more robust optimization scheme is oemployed to locate the probable region of the global minimuFollowing this initial screening, the adjoint scheme can thenapplied to refine the solution. This two-phase optimizationproach is only worthwhile if the time required to find a globminimum by other methods is too costly. Compared to the adjotechnique, genetic algorithms are better suited to solving problwith multiple critical points and those that appear to give goresults for inverse modeling in pipeline transient applications.

13.2 Pathogen Intrusion in Water Supply Systems. In thefirst sentence of its proposed Ground Water Rule: Public HeConcerns document, the U.S. EPA Office of Water states t‘‘Assurance that the drinking water is not contaminated by humor animal fecal waste is the key issue for any drinking wasystem.’’ The proposed Ground Water Rule is designed to proagainst pathogenic bacteria and viruses in source water, aggrowth of opportunistic pathogenic bacteria in ground water dtribution systems, and to mitigate against any failure in the enneered systems, such as cross-connections or sewage infiltrinto distribution systems.

There is considerable evidence in the literature that the numof disease outbreaks~including a large number that are [email protected]/orgwdw000/standard/phs.html#! due to fecal con-tamination of distribution systems is already large and mightgrowing. From 1971–1994, 50 of 356 reported waterborneease outbreaks occurred as a result of pathogen entry into dbution systems. More recent statistics from the U.S. CenterDisease Control put the ratio of distribution system intrusions


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other sources of drinking water contamination as high as 1Studies by Payment@130# suggest that one third of the 99 milliogastroenteritis cases in the U.S. each year might involve expoto waterborne pathogens in the distribution system. Consetively estimating that 20% of these cases result from pathointrusion into water pipes, then, in America alone, as many asmillion cases of gastroenteritis annually might be directly cauby contamination of drinking water distribution systems.

Recent research into the problem is now attempting to addfour critical questions that naturally arise in response to thalarming statistics.~i! What is the nature of the pathogen intrusiomechanism~s!? ~ii ! Why don’t routine water sampling and laboratory testing detect intrusion events?~iii ! Is the health of waterconsumers in a particular system at risk?~iv! Can the risk ofdistribution system intrusions be reduced~and by how much! oreliminated altogether? Answers to these four questions depentirely on developing a clear understanding of the complexteractions with hydraulic transients in pipe systems.

13.2.1 Distribution System Intrusion Pathways.There areseveral potential intrusion pathways whereby bacterial, protozoand viral pathogens can enter a water supply, transmissiondistribution pipeline:~i! at the source;~ii ! during loss of pressureand subsequent exposure of the pipe interior to the externalsoil or groundwater~such as may happen during a main brerepair!; ~iii ! via cross-connections on a consumer’s property; a~iv! via cross-connections in the distribution system.

The first two intrusion mechanisms are ‘‘controlled’’ situationinsofar as the quality of finished water is carefully monitored atreated to ensure compliance with drinking water standards, wthe latter two pathways are largely ‘‘uncontrolled.’’ Crosconnections can arise whenever a possible source of contaminwater or other liquid can be introduced into the potable wasystem by virtue of backpressure~an excess of pressure causinflow to occur in a direction opposite to its normal intended flodirection! or siphonage~suction or ‘‘negative’’ pressure inducingflow from a contamination source into the distribution system!.While backpressure and negative pressures are usually eliminthrough proper hydraulic design, there is one major sourcenegative pressures that is not normally accounted for in distrtion system design—hydraulic transients.

Water hammer occurs regularly in some systems and perically in others whenever flow conditions are changed rapidWhether these changes in flow are the result of planned operalike pump starts and stops, or are unplanned events initiatedpower outages, accidental valve closures, or rupturing of a pthe ensuing episodes of negative pressure can introduce connated fluids into the pipeline. Contamination can occur on a ctomer’s property or on the utility side of a service connectioContaminated fluids introduced at a cross-connection wouldlargely transported in the prevailing direction of flow in the pipafter entering the system.

Pressure dependent leakage is commonly known to occur fthe potable system to the surrounding~soil! environment throughpipe joints, cracks, pinholes, and larger orifice-like openinFunk et al.@131# developed analytic hydraulic parameters to asess the potential for transient intrusion in a water distributsystem. Their intrusion model was based on the percentagwater lost through leakage lumped at system nodes and‘‘equivalent orifice’’ needed to pass the discrete leakage flowthe prevailing system pressure.

A paper by McInnis~in progress! extends the work of Funket al. to incorporate alternative intrusion flow models basedlaminar flow, turbulent orifice flow, or a mixture of the two flowtypes. Work done by Germanopoulos and Jowitt@132# on pressuredependent leakage suggests that most distributed leakage isably laminar in nature, occurring through larger numbers of smopenings. The 2D water hammer equations with turbulence mels developed by Vardy and Hwang@25#, Pezzinga@38#, andSilva-Araya and Chaudhry@37,98# will be useful in generating


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models for predicting intrusion volumes, initial distribution ocontaminant concentrations in the pipes, and the ultimate fatcontaminants within the distribution system.

McInnis @133# expands the consideration of transient intrusievents from purely fluid mechanics aspects by developing a rbased framework for comparing the actual human health risksrelative risk reduction achieved by alternative transient-intrusmitigation strategies. McInnis@133# applies transient modelingwith some assumed reference groundwater contamination land computes hypothetical intrusion volumes for a given transevent to predict the transitory impact of the event on system wquality. He has also proposed meaningful risk-based measurprovide quantitative comparisons of the relative reduction inrisk of receptor infection achieved by alternative mitigation stregies.

14 Practical and Research Needs in Water HammerBoth theory and experiments confirm the existence of hel

type vortices in transient pipe flows. The conditions under whhelical vortices emerge in transient flows and the influencethese vortices on the velocity, pressure, and shear stress fieldcurrently not well understood and, thus, are not incorporatedtransient flow models. Future research is required to accompthe following:1! understand the physical mechanisms responsible for the e

gence of helical type vortices in transient pipe flows2! determine the range in the parameter space, defined by

nolds number and dimensionless transient time scale owhich helical vortices develop

3! investigate flow structure together with pressure, velocity, ashear stress fields at subcritical, critical, and supercriticalues of Reynolds number and dimensionless time scale

The accomplishment of the stated objectives would be southrough the use of linear and nonlinear analysis. Understanthe causes, emergent conditions, and behavior of helical vorin transient pipe flows as well as their influence on the velocpressure, and shear stress field are fundamental problems inmechanics and hydraulics. Understanding these phenomena wconstitute an essential step toward incorporating this new pnomena in practical unsteady flow models and reducing sigcant discrepancies in the observed and predicted behavior oergy dissipation beyond the first wave cycle.

Current physically based 1D and 2D water hammer modassume that~i! turbulence in a pipe is either quasi-steady, frozor quasi-laminar; and~ii ! the turbulent relations that have beederived and tested in steady flows remain valid in unsteady pflows. These assumptions have not received much attention inwater hammer literature. Understanding the limitations and acracy of assumptions~i! and ~ii ! is essential for establishing thdomain of applicability of models that utilize these assumptioand for seeking appropriate models to be used in problems wexisting models fail. Preliminary studies by Ghidaoui et al.@46#show that agreement between physically based 1D and 2D whammer models and experiments is highly dependent on thenolds number and on the ratio of the wave to turbulent diffustime scales. However, the lack of in-depth understanding ofchanges in turbulence during transient flow conditions is a signcant obstacle to achieving conclusive results regarding the limtion of existing models and the derivation of more approprimodels. Therefore, a research program whose main objectivedevelop an understanding of the turbulence behavior and endissipation in unsteady pipe flows is needed. This researchgram needs to accomplish the following:

1! improve understanding of and the ability to quantify changin turbulent strength and structure in transient events at difent Reynolds numbers and time scales

2! use the understanding gained in item 1 to determine the ra


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of applicability of existing models and to seek more appropate models for problems where current models are knownfail

The development of inverse water hammer techniques isother important future research area. A number of very promisinverse water hammer techniques have been developed in thedecade. Future work in this area needs to accomplish the folling:1! further investigate the issues of efficiency, reliability, and ide

tifiability of inverse water hammer techniques2! develop more realistic laboratory and field programs in or

to further test existing inverse techniques as well as devenew ones

3! develop systematic approaches~e.g., using stochastic methods!that can incorporate the influence of modeling and measment errors on the reliability of inverse methods

4! develop identifiability-based methods to determine the quanand quality of data necessary to carry out a successful invprogram

The practical significance of the research goals stated aboconsiderable. An improved understanding of transient flow behior gained from such research would permit development of trsient models able to accurately predict flows and pressures bethe first wave cycle. One important consequence of this is thatbehavioral aspects of control devices activated~or reactivated! bylocal flow or pressure is correctly modeled. Most importanthowever, reducing the modeling errors beyond the first wacycle, along with better inverse techniques, will greatly improthe accuracy and reliability of inverse transient models. Thisimportant because inverse models have the potential to utfield measurements of transient events to accurately and inexsively calibrate a wide range of hydraulic parameters, includpipe friction factors, system demands, and leakage. At this tisuch information can only be obtained through costly field msurements of flows and pressures conducted on a few individusampled pipes in the system. Transients, on the other htraverse the entire system, interacting with each pipe or devicthe system. Thus, they contain large amounts of informationgarding the physical characteristics of the system. Inverse tsient analysis techniques are now being developed to decodeinformation for hydraulic model calibration as well as to identiand locate system leakage, closed or partially closed valves,damaged pipes. The potential annual savings in routine datalection costs for water supply utilities world wide is significanEqually important, an improved understanding of the true natof turbulence in transient flows will be a groundbreaking sttoward modeling transient-induced water quality problems. Netive pressure waves can cause intrusion of contaminants frompipe surroundings through cracks, pinholes, joints, and rupturethe pipes. In addition, water hammer events cause biofilm slouing and resuspension of particulates within the pipe, potentileading to unsafe or unpleasant drinking water. Without a beunderstanding of transient flow behavior, the risk and degreecontamination of water supply systems during transient evecannot be quantitatively assessed.

15 SummaryThe scientific study of transient fluid flow has been undertak

since the middle of the nineteenth century. As is true of evother area of engineering research, a great many advancesbeen made in the accuracy of analysis and the range of apptions since then. Although only a few simple problems wereproachable by earlier analytical methods and numerical teniques, a much broader spectrum of transient problems coulsolved once graphical methods were developed. More recethe application of digital computing techniques has resulted irapid increase in the range and complexity of problems be


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studied. This paper provides both a historical perspective andview of water hammer theory and an overview of recent develments in this field of fluid mechanics.

Specifically, advances in the last one or two decades deawith some of the more complex and fundamental fluid mechanissues have been discussed:1! The relation between state equations and wave speeds in s

as well as multiphase and multicomponent transient flowsillustrated and discussed.

2! Various forms of 1D and 2D water hammer equations, suchthe Joukowsky model, classical 1D waterhammer equatiothe 2D plane wave equations, and the quasi-two-dimensioplane wave equations are derived.

3! Governing equations of turbulent water hammer flows aretained by ensemble averaging of the quasi-two-dimensioplane wave equations.

4! Order of magnitude analysis is used throughout the papeevaluate the accuracy of the assumptions in the various foof water hammer governing equations.

Water hammer models are becoming more widely used~i! forthe design, analysis, and safe operation of complex pipelinetems and their protective devices;~ii ! for the assessment and mitgation of transient-induced water quality problems; and~iii ! forthe identification of system leakage, closed or partially closvalves, and hydraulic parameters such as friction factors and wspeeds. In addition, turbulence models have been developedused to perform numerical experiments in turbulent water hamflows for a multitude of research purposes such as the comption of instantaneous velocity profiles and shear stress fieldscalibration and verification of 1D water hammer models, tevaluation of the parameters of 1D unsteady friction models,the comparison of various 1D unsteady friction models. Undstanding the governing equations that are in use in water hamresearch and practice and their limitations is essential for inpreting the results of the numerical models that are based on tequations, for judging the reliability of the data obtained frothese models, and for minimizing misuse of water hammer mels.

AcknowledgmentsThe writers wish to thank the Research Grants Council of Ho

Kong for financial support under Project No. HKUST6179/02E

Nomenclature

A - system matrixA - cross-sectional area of pipea - water hammer wavespeeda - coefficient for five-region turbulence modelB - matrix for subsystem of longitudinal velocity

componentB - coefficient for MOC formulation

BC - lumped quantity for characteristics solution fopipe network

BM - quantity for negative characteristics used for1D MOC solution

BP - quantity for positive characteristics used for1D MOC solution

b - known vector for systembu - known vector for subsystem of longitudinal

velocity componentbv - known vector for subsystem of head and rad

componentC - matrix for subsystem of head and radial velo

ity componentCB - coefficient for five-region turbulence modelCC - lumped quantity for characteristics solution fo

pipe networkCc - coefficient for five-region turbulence model


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r

CM - quantity for negative characteristics used for1D MOC solution

Cm - coefficient for five-region turbulence modelCP - quantity for positive characteristics used for

1D MOC solutionCq1 , Cq2 - coefficients beforeq

Cr - courant numberCu1 , Cu2 , Cu3 - coefficients beforeu

c - parameter associated with pipe anchor condition

c - superscript denoting values predicted by for-ward model

c2 - coefficient used in Daily et al.@39#cs - control surfacecv - control volumeD - diameter of pipeE - Young’s modulus of elasticity of pipe materiaE - errors

E* - merit ~error! functione - thickness of pipe wallF - wall resistance force

Fext - external forcesf - Darcy-Weisbach friction factorg - gravitational acceleration

Hm - set of measured piezometric headH - piezometric head

HA - piezometric head at pointAHB - piezometric head at pointBHP - piezometric head at pointPhc - set of computed piezometric headi - index for pipes

Js - steady friction termj - index for pipes

K - unsteady resistance coefficientKe - effective bulk modulus of elasticityK f - bulk modulus of elasticity of the fluidKs - steady-state resistance coefficientKu - unsteady resistance coefficient and momentu

flux of absolute local velocityk - unsteady friction factorL - pipe lengthl - mixing length

M - Mach numberm - superscript denoting measured data valuesm - time level index

N1 - set of all pipes with flow toward conjuctionnode

N2 - set of all pipes with flow away from conjuc-tion node

Nr - number of computational reaches in radial direction

n - unit outward normal vector to control surfacen - index of measured seriesn - exponential for power law of velocity profile

nc - number of complete water hammer wavecycles

P - parameter for quasi-steady assumptionP - piezometric pressureQ - discharge

QA - discharge at pointAQB - discharge at pointB

Qext - discharge of external flowQP - discharge at pointP

q - radial fluxq - flow rate

R, R* - radius of pipe, dimensionless distance frompipe wall

R - coefficient for MOC formulationRe - Reynolds number


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0,J.

heng.

it

,

d

r - radial coordinateT - time scale

Td - time scale for radial diffusion of vorticityt - time

t8 - time used for convolution integralU1 - longitudinal velocity scale

u - unknown vector for subsystem of longitudinavelocity component

u - local longitudinal velocityu* - frictional velocityu8 - turbulence perturbation corresponding touV - cross-sectional average velocityv - velocity vectorv - unknown vector for subsystem of head and

radial velocity componentv - local radial velocity

v8 - turbulence perturbation corresponding tovW - weighting functionX - longitudinal length scalex - distance along the pipe

y, y* - distance from pipe wall, dimensionless dis-tance from pipe wall

Z - elevation of pipe centerline from a given da-tum

z - unknown vector for systema - angle between pipe and horizontal directiona - coefficient in weighting functionb - momentum correction coefficientb - coefficient in weighting functiong - unit gravity forcee - distance from the water hammer fronte - eddy viscositye - implicit parameter for shear stresse - implicit parameter for frictione - measured and modeled data errorz - a positive real parameterh - difference from unity of Coriolis correctionh - constant for weighting functionu - implicit parameter for radial fluxk - coefficient for weighting functionk - coefficient for five-region turbulence modelk - coefficient for two-layer turbulence model

l1 - Lagrangian multiplierl2 - Lagrangian multipliern - kinematic viscosity

np - Poisson rationT - total viscosity

j - strainr - fluid density

r0 - fluid density at undisturbed statere - effective density

s1 , s2 - unknown but desired parameterssx - axial stresssu - hoop stress

t - shear stresstw - wall shear stress

tws - quasi-steady contribution of wall shear stresstwu - discrepancy between unsteady and quasi-

steady wall shear stressf - coefficient in unsteady friction formula.

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Mohamed Ghidaouireceived his B.A.Sc., M.A.Sc. and Ph.D. all in Civil Engineering frthe University of Toronto, Canada, in 1989, 1991, and 1993, respectively. Since Julyhe has been with the Department of Civil Engineering at the Hong Kong UniversiScience and Technology (HKUST) where he is an Associate Professor. His research ininclude modeling of surface water flows and water hammer: unsteady friction in condturbulence modeling of fast transients, flow stability of time-dependent flows and turbshallow shear flow, numerical modeling of surface and closed conduit flows, and applicof Bolzmann theory in hydraulics. He is a member of the International AssociatioHydraulic Research (IAHR) and the American Society of Civil Engineers (ASCE). Hefounding member of IAHR-Hong Kong and currently serves as its president. He is anciate Editor of the Journal of Hydraulic Research and an advisory board member oJournal of Hydroinformatics. His awards include the Albert Berry Memorial Award, Amcan Water Works Association; runner-up for the Hilgard Award for best paper, JournaHydraulic Engineering; and Teaching Excellence Awards, School of Engineering, HKU

Ming Zhao obtained his Ph.D. in April 2004 from the Department of Civil Engineering,Hong Kong University of Science and Technolgy. He obtained both his B.A.Eng. in hydengineering and B.A.Sc. in eneterprise management in 1999 from Tsinghua Universitresearch interests include numerical simulation of unsteady pipe flows, open channelturbulence modeling in hydraulics, and stability analysis for fluid flows.

David H. Axworthyis a registered professional engineer with a consulting engineeringin Los Angeles, California. He obtained his B.A.Sc. (1991), M.A.Sc. (1993) and Ph.D. (1in civil engineering from the University of Toronto. Axworthy has analyzed pressuresients created by the operation of pump stations and valves and designed surge profor water supply, wastewater, fire protection, deicing, diesel, and jet fuel systems. A mof the ASCE and AWWA, Axworthy is coauthor of a water hammer analysis model (sAM), serves as a reviewer for the ASME Journal of Fluids Engineering, and has publscientific papers in the area of pipe network transients.

Duncan A. McInnis (Ph.D., P.Eng., MHKIE) has degrees in environmental biology and cengineering. He has 20 years of scientific and professional engineering experience inputational hydraulics, simulation, and computer modeling of surface water and pipsystems. McInnis has been a Lecturer of Civil Engineering and Senior Project Managthe Hong Kong University of Science and Technology. He is currently the Manager of WResources with Komex International Ltd., an international environmental consulting fi

76 Õ Vol. 58, JANUARY 2005 Transactions of the ASME

water Hammer

Documents

turbine load

sectionally

shear stress

pump trip

water hammer

coriolis correction

surge anticipating

pump power