nele

nfs

e

io

t

intcertain time. Later generations of scientists typically discoverthat these conceptions of reality embodied certain implicit as-sumptions and hypotheses that later on turned out to be incor-rect. Vanderburg, @1#Unsteady fluid flows have been studied since man first bent

water to his will. The ancient Chinese, the Mayan Indians of Cen-tral America, the Mesopotamian civilizations bordering the Nile,Tigris, and Euphrates river systems, and many other societiesthroughout history have developed extensive systems for convey-ing water, primarily for purposes of irrigation, but also for domes-tic water supplies. The ancients understood and applied fluid flowprinciples within the context of traditional, culture-based tech-nologies. With the arrival of the scientific age and the mathemati-cal developments embodied in Newtons Principia, our under-standing of fluid flow took a quantum leap in terms of itstheoretical abstraction. That leap has propelled the entire develop-ment of hydraulic engineering right through to the mid-twentiethcentury. The advent of high-speed digital computers constitutedanother discrete transformation in the study and application offluids engineering principles. Today, in hydraulics and other areas,engineers find that their mandate has taken on greater breadth anddepth as technology rapidly enters an unprecedented stage ofknowledge and information accumulation.

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

Computer aided analysis and design is one of the principal mecha-nisms bringing about these changes.

Computer analysis, computer modeling, and computer simula-tion are somewhat interchangeable terms, all describing tech-niques intended to improve our understanding of physical phe-nomena and our ability to predict and control these phenomena.By combining physical laws, mathematical abstraction, numericalprocedures, logical constructs, and electronic data processing,these methods now permit the solution of problems of enormouscomplexity and scope.

This paper attempts to provide the reader with a general his-tory and introduction to waterhammer phenomena, a general com-pendium of key developments and literature references as well asan updated view of the current state of the art, both with respect totheoretical advances of the last decade and modeling practice.

2 Mass and Momentum Equations forOne-Dimensional Water Hammer Flows

Before delving into an account of mathematical developmentsrelated to waterhammer, it is instructive to briefly note the societalcontext that inspired the initial interest in waterhammer phenom-ena. In the late nineteenth century, Europe was on the cusp of theindustrial revolution with growing urban populations and indus-tries requiring electrical power for the new machines of produc-tion. As the fossil fuel era had not begun in earnest, hydroelectricgeneration was still the principal supply of this important energysource. Although hydroelectric generation accounts for a muchsmaller proportion of energy production today, the problems asso-Transmitted by Associate Editor HJS Fernando.

Copyright 2005 by ASMEApplied Mechanics Reviews JANUARY 2005, Vol. 58 49Mohamed S. Ghidaouiemail: ghidaoui@ust.hk

Ming Zhaoemail: cezhm@ust.hk

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: bm300@lafn.org

A ReviewTheory aHydraulic transients iintense practical interof the one-dimensionapose interesting problstructure and strengthpipes due to hydrodystanding is importantflows. This paper preresearch and practicecusses mass and momsolutions for one-dimeand momentum equatproblems, boundary csearch needs in watertions involved in varioity as well as the limimodels is essential forobtained from them, (ipractice, and (iv) deliof numerical artifactsin this review article.

1 IntroductionThus the growth of knowledge of the physical aspect of realitycannot be regarded as a cumulative process. The basic Gestaltof this knowledge changes from time to time . . . During thecumulative periods scientists behave as if reality is exactly asthey know it except for missing details and improvements inaccuracy. They speak of the laws of nature, for example, whichare simply models that explain their experience of reality at aof Water Hammernd Practiceclosed conduits have been a subject of both theoretical study and

st for more than one hundred years. While straightforward in termsnature of pipe networks, the full description of transient fluid flowsms in fluid dynamics. For example, the response of the turbulenceto transient waves in pipes and the loss of flow axisymmetry inamic instabilities are currently not understood. Yet, such under-or modeling energy dissipation and water quality in transient pipeents an overview of both historic developments and present dayin the field of hydraulic transients. In particular, the paper dis-ntum equations for one-dimensional Flows, wavespeed, numericalnsional problems, wall shear stress models; two-dimensional massons, turbulence models, numerical solutions for two-dimensionalnditions, transient analysis software, and future practical and re-

hammer. The presentation emphasizes the assumptions and restric-us governing equations so as to illuminate the range of applicabil-ations of these equations. Understanding the limitations of current(i) interpreting their results, (ii) judging the reliability of the datai) minimizing misuse of water-hammer models in both research andeating the contribution of physical processes from the contributiono the results of waterhammer models. There are 134 refrences cited@DOI: 10.1115/1.1828050#

reality, his remarks apply equally to our technological ability todeal with a revised or more complex view of the physical uni-verse. It is in this condition that the field of closed conduit tran-sient flow, and even more generally, the hydraulic analysis, de-sign, and operation of pipeline systems, currently finds itself.

The computer age is still dawning, bringing with it a massivedevelopment and application of new knowledge and technology.Formerly accepted design methodologies, criteria, and standardsare being challenged and, in some instances, outdated and revised.

ciated with controlling the flow of water through penstocks andturbines remains an important application of transient analysis.

tion laws across a jump ~shock! @11#. These conditions are ob-tained either by directly applying the conservation laws for a con-Hydrogeneration companies contributed heavily to the develop-ment of fluids and turbomachinery laboratories that studied,among other things, the phenomenon of waterhammer and its con-trol. Some of Allievis early experiments were undertaken as adirect result of incidents and failures caused by overpressure dueto rapid valve closure in northern Italian power plants. Frictionlessapproaches to transient phenomena were appropriate in these earlydevelopments because ~i! transients were most influenced by therapid closure and opening of valves, which generated the majorityof the energy loss in these systems, and ~ii! the pipes involvedtended to have large diameters and the flow velocities tended to besmall.

By the early 1900s, fuel oils were overtaking hydrogenerationas the principal energy source to meet societys burgeoning de-mand for power. However, the fascination with, and need to un-derstand, transient phenomena has continued unabated to this day.Greater availability of energy led to rapid industrialization andurban development. Hydraulic transients are critical design factorsin a large number of fluid systems from automotive fuel injectionto water supply, transmission, and distribution systems. Today,long pipelines transporting fluids over great distances have be-come commonplace, and the almost universal development ofsprawling systems of small pipe diameter, high-velocity water dis-tribution systems has increased the importance of wall friction andenergy losses, leading to the inclusion of friction in the governingequations. Mechanically sophisticated fluid control devices, in-cluding many types of pumps and valves, coupled with increas-ingly sophisticated electronic sensors and controls, provide thepotential for complex system behavior. In addition, the recentknowledge that negative pressure phases of transients can result incontamination of potable water systems, mean that the need tounderstand and deal effectively with transient phenomena aremore 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 controllingwater hammer. Near the turn of the nineteenth century, researcherslike Weston @5#, Carpenter @6# and Frizell @7# attempted to developexpressions relating pressure and velocity changes in a pipe. Fri-zell @7# was successful in this endeavor and he also discussed theeffects of branch lines, and reflected and successive waves onturbine speed regulation. Similar work by his contemporariesJoukowsky @8# and Allievi @9,10#, however, attracted greater at-tention. Joukowsky @8# produced the best known equation in tran-sient flow theory, so well known that it is often called the fun-damental equation of water hammer. He also studied wavereflections from an open branch, the use of air chambers and surgetanks, and spring type safety valves.

Joukowskys fundamental equation of water hammer is as fol-lows:

DP56raDV or DH56aDV

g (1)

where a5acoustic ~waterhammer! wavespeed, P5rg(H2Z)5piezometric pressure, Z5elevation of the pipe centerline from agiven datum, H5piezometric head, r5fluid density, V5*AudA5cross-sectional average velocity, u5local longitudinal velocity,A5cross-sectional area of the pipe, and g5gravitational accelera-tion. The positive sign in Eq. ~1! is applicable for a water-hammerwave moving downstream while the negative sign is applicablefor a water-hammer wave moving upstream. Readers familiar withthe gas dynamics literature will note that DP56raDV is obtain-able from the momentum jump condition under the special casewhere the flow velocity is negligible in comparison to thewavespeed. The jump conditions are a statement of the conserva-

50 Vol. 58, JANUARY 2005trol volume across the jump or by using the weak formulation ofthe conservation laws in differential form at the jump.

Allievi @9,10# developed a general theory of water hammerfrom first principles and showed that the convective term in themomentum equation was negligible. He introduced two importantdimensionless parameters that are widely used to characterizepipelines and valve behavior. Allievi @9,10# also produced chartsfor pressure rise at a valve due to uniform valve closure. Furtherrefinements to the governing equations of water hammer appearedin 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 momen-tum equations for one-dimensional ~1D! water-hammer flows

a2

g]V]x

1]H]t

50 (2)

]V]t

1g]H]x

14

rD tw50 (3)

in which tw5shear stress at the pipe wall, D5pipe diameter, x5the spatial coordinate along the pipeline, and t5temporal coor-dinate. Although Eqs. ~2! and ~3! were fully established by the1960s, these equations have since been analyzed, discussed, red-erived and elucidated in numerous classical texts ~e.g., @2023#!.Equations ~2! and ~3! constitute the fundamental equations for 1Dwater hammer problems and contain all the physics necessary tomodel wave propagation in complex pipe systems.

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

Rapid flow disturbances, planned or accidental, induce spatialand temporal changes in the velocity ~flow rate! and pressure ~pi-ezometric head! fields in pipe systems. Such transient flows areessentially unidirectional ~i.e., axial! since the axial fluxes ofmass, momentum, and energy are far greater than their radialcounterparts. 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 approachwhen studying water-hammer problems in pipe systems.

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

Using the Reynolds transport theorem, the mass conservation~continuity equation! for a control volume is as follows ~e.g.,@2023#!

]

]t Ecvrd;1Ecsr~v"n!dA50 (4)where cv5control volume, cs5control surface, n5unit outwardnormal vector to control surface, v5velocity vector.

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

]

]t Exx1dx

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

where g5rg5unit gravity force, a5angle between the pipe andthe horizontal direction, b5*Au2dA/V25momentum correction]~rA !]t

1]~rAV !

]x50 (6)

Equation ~6! provides the conservative form of the area-averagedmass balance equation for 1D unsteady and compressible fluids ina flexible pipe. The first and second terms on the left-hand side ofEq. ~6! represent the local change of mass with time due to thecombined effects of fluid compressibility and pipe elasticity andthe instantaneous mass flux, respectively. Equation ~6! can be re-written as follows:

1r

DrDt 1

1A

DADt 1

]V]x

50 or1

rADrADt 1

]V]x

50 (7)

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

1r

drdP

DPDt 1

1A

dAdP

DPDt 1

]V]x

50 or1

ra2DPDt 1

]V]x

50

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

The momentum equation for a control volume is ~e.g., @2023#!:

( Fext5]

]t Ecvrv;1Ecsrv~v"n!dA (9)Applying Eq. ~9! to the control volume of Fig. 2; considering

gravitational, wall shear and pressure gradient forces as externallyapplied; and taking the limit as dx tends to zero gives the follow-ing local form of the axial momentum equation:

]rAV]t

1]brAV2

]x52A

]P]x

2pDtw2gA sin a (10)

Fig. 1 Control volume diagram used for continuity equationderivation

Fig. 2 Control volume diagram used for momentum equationderivation

Applied Mechanics Reviewscoefficient. Using the product rule of differentiation, invoking Eq.~7!, and dividing through by rA gives the following nonconser-vative form of the momentum equation:

]V]t

1V]V]x

11

rA]~b21 !rAV2

]x1

1r

]P]x

1g sin a1twpD

rA

50 (11)Equations ~8! and ~11! govern unidirectional unsteady flow of acompressible fluid in a flexible tube. Alternative derivations ofEqs. ~8! and ~11! could have been performed by applying theunidirectional and axisymmetric assumptions to the compressibleNavier-Stokes equations and integrating the resulting expressionwith respect to pipe cross-sectional area while allowing for thisarea to change with pressure.

In practice, the order of magnitude of water hammer wavespeed ranges from 100 to 1400 m/s and the flow velocity is oforder 1 to 10 m/s. Therefore, the Mach number, M5U1 /a , inwater-hammer applications is often in the range 1022 1023,where U15longitudinal velocity scale. The fact that M!1 in wa-ter hammer was recognized and used by Allievi @9,10# to furthersimplify Eqs. ~8! and ~11!. The small Mach number approxima-tion to Eqs. ~8! and ~11! can be illustrated by performing an orderof magnitude analysis of the various terms in these equations. Tothis end, let r0aU15water hammer pressure scale, r05density ofthe fluid at the undisturbed state, and T5zL/a5time scale, whereL5pipe length, X5aT5zL5longitudinal length scale, z5apositive real parameter, r f U12/85wall shear scale, and f5Darcy-Weisbach friction factor Td5radial diffusion time scale.The parameter z allows one to investigate the relative magnitudeof the various terms in Eqs. ~8! and ~11! under different timescales. For example, if the order of magnitude of the various termsin 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

r0r

DP*Dt* 1

]V*]x*

50 or

r0r S ]P*]t* 1MV* ]P*]x* D1 ]V*]x* 50 (12)

]V*]t*

1MV*]V*]x*

1M1

rA]~b21 !rAV*2

]x*1

r0r

]P*]x*

1gzLUa sin a1

zLD M

f2 tw*50 (13)

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

r0r

]P*]t*

1]V*]x*

50 (14)

]V*]t*

1r0r

]P*]x*

1gzLUa sin a1z

LD M

f2 1zS TdL/a D tw*50.

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

1ra2

]P]t

1]V]x

50 (16)

]V]t

11r

]P]x

1g sin a1twpD

rA 50 (17)

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

JANUARY 2005, Vol. 58 51

gr02

]H1

]V50 (18)

3 Water Hammer Acoustic Wave Speed

ra ]t ]x

]V]t

1gr0r

]H]x

1twpD

rA 50 (19)

The change in density in unsteady compressible flows is of theorder of the Mach number @11,27,28#. Therefore, in water hammerproblems, where M!1, rr0 , Eqs. ~18! and ~19! become

ga2

]H]t

1]V]x

50 (20)

]V]t

1g]H]x

1twpD

rA 50 (21)

which are identical to the classical 1D water hammer equationsgiven by Eqs. ~2! and ~3!. Thus, the classical water hammer equa-tions are valid for unidirectional and axisymmetric flow of a com-pressible 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 parameter G5zLMf /2D1zTd/(L/a). Therefore, the wall shear is importantwhen the parameter G is order 1 or larger. This often occurs inapplications where the simulation time far exceeds the first wavecycle ~i.e., large z!, the pipe is very long, the friction factor issignificant, or the pipe diameter is very small. In addition, wallshear is important when the time scale of radial diffusion is largerthan the wave travel time since the transient-induced large radialgradient of the velocity does not have sufficient time to relax. It isnoted that Td becomes smaller as the Reynolds number increases.The practical applications in which the wall shear is important andthe various tw models that are in existence in the literature arediscussed in Sec. 4.

If G is significantly smaller than 1, friction becomes negligibleand tw can be safely set to zero. For example, for the case L510,000 m, D50.2 m, f 50.01, and M50.001, and Td/(L/a)50.01 the condition G!1 is valid when z!4. That is, for the caseconsidered, wall friction is irrelevant as long as the simulationtime 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 time L/a . In fact, it is well known thatwaterhammer models provide results that are in reasonable agree-ment with experimental data during the first wave cycle irrespec-tive of the wall shear stress formula being used ~e.g., @2932#!.When G!1, the classical waterhammer model, given by Eqs. ~20!and ~21!, becomes

ga2

]H]t

1]V]x

50 (22)

]V]t

1g]H]x

50 (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 pipe oflength L . Let x5L2at define the position of a water hammerfront at time t and consider the interval @L2at2e ,L2at1e# ,where e5distance from the water hammer front. Integrating Eqs.~22! and ~23! from x5L2at2e to x5L2at1e , invoking Leib-nitzs rule, and taking the limit as e approaches zero gives

DH52aDV

g (24)

Similarly, the relation for a water hammer wave moving down-stream is DH51aDV/g.

52 Vol. 58, JANUARY 2005The water hammer wave speed is ~e.g., @8,20,23,33,34#!,

1a2

5drdP 1

r

AdAdP (25)

The first term on the right-hand side of Eq. ~25! represents theeffect of fluid compressibility on the wave speed and the secondterm represents the effect of pipe flexibility on the wave speed. Infact, the wave speed in a compressible fluid within a rigid pipe isobtained by setting dA/dP50 in Eq. ~25!, which leads to a25dP/dr . On the other hand, the wave speed in an incompressiblefluid within a flexible pipe is obtained by setting dr/dP50 in~25!, which leads to a25AdP/rdA .

Korteweg @33# related the right-hand side of Eq. ~25! to thematerial properties of the fluid and to the material and geometricalproperties of the pipe. In particular, Korteweg @33# introduced thefluid properties through the state equation dP/dr5K f /r , whichwas already well established in the literature, where K f5bulkmodulus of elasticity of the fluid. He used the elastic theory ofcontinuum mechanics to evaluate dA/dP in terms of the piperadius, thickness e , and Youngs modulus of elasticity E . In hisderivation, he ~i! ignored the axial ~longitudinal! stresses in thepipe ~i.e., neglected Poissons effect! and ~ii! ignored the inertia ofthe pipe. These assumptions are valid for fluid transmission linesthat are anchored but with expansion joints throughout. With as-sumptions ~i! and ~ii!, a quasi-equilibrium relation between thepressure force per unit length of pipe DdP 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

1a2

5r

K f1

r

Ee

D

or a25

K fr

11K fDeE

(26)

The above Korteweg formula for wave speed can be extendedto problems where the axial stress cannot be neglected. This isaccomplished through the inclusion of Poissons effect in thestress-strain relations. In particular, the total strain becomes dj5dsu /E2npdsx /E , where np5Poissons ratio and sx5axialstress. The resulting wave speed formula is ~e.g., @17,23#!

a25

K fr

11cK fDeE

(27)

where c512np/2 for a pipe anchored at its upstream end only,c512np

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

Multiphase and multicomponent water hammer flows are com-mon in practice. During a water hammer event, the pressure cancycle between large positive values and negative values, the mag-nitudes of which are constrained at vapor pressure. Vapor cavitiescan form when the pressure drops to vapor pressure. In addition,gas cavities form when the pressure drops below the saturationpressure of dissolved gases. Transient flows in pressurized or sur-charged pipes carrying sediment are examples of multicomponentwater hammer flows. Unsteady flows in pressurized or surchargedsewers are typical examples of multiphase and multicomponenttransient flows in closed conduits. Clearly, the bulk modulus anddensity of the mixture and, thus, the wave speed are influenced bythe presence of phases and components. The wave speed for mul-tiphase and multicomponent water hammer flows can be obtained

Transactions of the ASME

~iii! the modeling of transient-induced water quality problems,~iv! the design of safe and reliable field data programs for diag-

ments and found twu(t) to be positive for accelerating flows andnegative for decelerating flows. They argued that during accelera-nostic and parameter identification purposes, ~v! the application oftransient models to invert field data for calibration and leakagedetection, ~vi! the modeling of column separation and vaporouscavitation and ~vii! systems in which L/a!Td. Careful modelingof wall shear is also important for long pipes and for pipes withhigh friction.

4.1 Quasi-Steady Wall Shear Models. In conventionaltransient analysis, it is assumed that phenomenological expres-sions relating wall shear to cross-sectionally averaged velocity insteady-state flows remain valid under unsteady conditions. That is,wall shear expressions, such as the Darcy-Weisbach and Hazen-Williams formulas, are assumed to hold at every instant during atransient. For example, the form of the Darcy-Weisbach equationused in water hammer models is ~Streeter and Wylie @36#!

tion the central portion of the stream moved somewhat so that thevelocity profile steepened, giving higher shear. For constant-diameter conduit, the relation given by Daily et al. @39# can berewritten as

Ku5Ks12c2LV2

]V]t

(30)

where Ku5unsteady flow coefficient of boundary resistance andmomentum flux of absolute local velocity and Ks5 f L/D5 steadystate resistance coefficient. Daily et al. @39# noted that the longi-tudinal velocity and turbulence nonuniformities are negligible andKuK5F/rAV2/25unsteady flow coefficient of boundary resis-tance, where F52pDLtw5wall resistance force. Therefore, Eq.~30! becomes

Applied Mechanics Reviews JANUARY 2005, Vol. 58 53by substituting an effective bulk modulus of elasticity Ke and aneffective density re in place of K f and r in Eq. ~27!. The effectivequantities, Ke and re , are obtained by the weighted average of thebulk modulus and density of each component, where the partialvolumes are the weights ~see, @23#!. While the resulting math-ematical expression is simple, the explicit evaluation of the wavespeed of the mixture is hampered by the fact that the partial vol-umes are difficult to estimate in practice.

Equation ~27! includes Poissons effect but neglects the motionand inertia of the pipe. This is acceptable for rigidly anchored pipesystems such as buried pipes or pipes with high density and stiff-ness, to name only a few. Examples include major transmissionpipelines like water distribution systems, natural gas lines, andpressurized and surcharged sewerage force mains. However, themotion and inertia of pipes can become important when pipes areinadequately restrained ~e.g., unsupported, free-hanging pipes! orwhen the density and stiffness of the pipe is small. Some ex-amples in which a pipes motion and inertia may be significantinclude fuel injection systems in aircraft, cooling-water systems,unrestrained pipes with numerous elbows, and blood vessels. Forthese systems, a fully coupled fluid-structure interaction modelneeds to be considered. Such models are not discussed in thispaper. The reader is instead directed to the recent excellent reviewof the subject by Tijsseling @35#.

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

is important when the parameter G is large. It follows that themodeling of wall friction is essential for practical applications thatwarrant transient simulation well beyond the first wave cycle ~i.e.,large z!. Examples include ~i! the design and analysis of pipelinesystems, ~ii! the design and analysis of transient control devices,tw~ t !5tws5r f ~ t !uV~ t !uV~ t !

8 (28)

where tws(t)5quasi-steady wall shear as a function of t .The use of steady-state wall shear relations in unsteady prob-

lems is satisfactory for very slow transientsso slow, in fact, thatthey do not properly belong to the water hammer regime. To helpclarify the problems with this approach for fast transients, con-sider the case of a transient induced by an instantaneous and fullclosure of a valve at the downstream end of a pipe. As the wavetravels upstream, the flow rate and the cross-sectionally averagedvelocity behind the wave front are zero. Typical transient velocityprofiles are given in Fig. 3. Therefore, using Eq. ~28!, the wallshear is zero. This is incorrect. The wave passage creates a flowreversal near the pipe wall. The combination of flow reversal withthe no-slip condition at the pipe wall results in large wall shearstresses. Indeed, discrepancies between numerical results and ex-perimental and field data are found whenever a steady-state basedshear stress equation is used to model wall shear in water hammerproblems ~e.g., @25,30,32,37,38#!.

Let twu(t) be the discrepancy between the instantaneous wallshear stress tw(t) and the quasi-steady contribution of wall shearstress tws(t). Mathematically

tw~ t !5tws~ t !1twu~ t ! (29)twu(t) is zero for steady flow, small for slow transients, and sig-nificant for fast transients. The unsteady friction component at-tempts to represent the transient-induced changes in the velocityprofile, which often involve flow reversal and large gradients nearthe pipe wall. A summary of the various models for estimatingtwu(t) in water hammer problems is given below.

4.2 Empirical-Based Corrections to Quasi-Steady WallShear Models. Daily et al. @39# conducted laboratory experi-

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

namely, that unsteady wall friction decreases in decelerating flowsand increases in accelerating flows. Shuy @40# attributed the de-

@43# study neither supports nor refutes the possibility of using Eq.~32! in water hammer problems.crease in wall shear stress for acceleration to flow relaminariza-tion. Given its controversial conclusion, this paper generated aflurry of discussion in the literature with the most notable remarksbeing those of Vardy and Brown @41#.

Vardy and Brown @41# argued that Shuys results should not beinterpreted as contradicting previous measurements. Instead, theresults indicated that the flow behavior observed in Shuys experi-ments may have been different from the flow behavior in previousexperiments. Vardy and Brown @41# put forward the time scalehypothesis as a possible explanation for the different flow behav-ior between Shuys @40# experiments and previous ones. They alsoobserved that, while Shuys experiments dealt with long timescales, previous measurements dealt with much shorter timescales. Vardy and Brown @41# provided insightful and convincingarguments about the importance of time scale to the flow behaviorin unsteady pipe flows. In fact, the stability analysis of Ghidaouiand Kolyshkin @42# concurs with the time scale hypothesis ofVardy and Brown @41#. Moreover, the stability analysis showsthat, while other experiments belong to the stable domain, theexperiments of Shuy belong to the unstable domain.

The theoretical work of Vardy and Brown @47# shows that Eq.~32! can be derived for the case of an unsteady pipe flow withconstant acceleration. In addition, they show that this model isapproximately valid for problems with time dependent accelera-tion as long as the time scale of the transient event greatly exceedsthe rising time, which is a measure of time required for the vor-ticity diffusion through the shear layer. Their work also warnsagainst using Eq. ~32! for problems with time dependent accelera-tion induced by transient events with time scales smaller than therising 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 deri-vation shows that unsteady friction formulas based on instanta-neous acceleration such as Eq. ~32! are applicable to transientflow problems in which the time scale of interest ~e.g., simulationtime! is significantly shorter than the radial diffusion time scale ofvorticity. Using the vorticity equation, Axworthy et al. @30#showed that for such short time scales, the turbulence strength andstructure is unchanged ~i.e., frozen!, and the energy dissipation

54 Vol. 58, JANUARY 2005 Transactions of the ASMEtw5r f V2

8 1c2rD

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, of thewall shear and momentum flux. Therefore, k generally depends onx and t . This remark is supported by the extended thermodynam-ics approach used by Axworthy et al. @30#. Figure 4 clearly illus-trates the poor agreement between model and data when using Eq.~32! with a constant value of k .

The experimental data of Daily et al. @39# show that k50.01for accelerating flows and k50.62 for decelerating flows. On theother hand, the research of Shuy @40# led to k520.0825 for ac-celerating flows and k520.13 for decelerating flows. In fact,Shuys data led him to conclude that unsteady wall friction in-creases in decelerating flows and decreases in accelerating flows.This result contradicts the previously accepted hypothesis,Theoretical investigations aimed at identifying the domain ofapplicability of Eq. ~32! have appeared in the literature. For ex-ample, Carstens and Roller @43# showed that Eq. ~32! can be de-rived by assuming that the unsteady velocity profiles obey thepower law as follows:

u~x ,r ,t !

V~x ,t ! 5~2n11 !~n11 !

2n2 S 12 rR D1/n

(33)

where n57 for Reynolds number Re5105 and increases withReynolds number, r5distance from the axis in a radial direction,R5radius of the pipe. An unsteady flow given by Eq. ~33! de-scribes flows that exhibit slow acceleration and does not allow forflow reversal ~i.e., does not contain inflection points!. In fact, Eq.~33! cannot represent typical water hammer velocity profiles suchas 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 Roller@43# shows only that Eq. ~32! applies to very slow transients inwhich the unsteady velocity profile has the same shape as thesteady velocity profile. Unfortunately, the Carstens and Roller

Fig. 4 Pressure head traces obtained frommodels and experiment

behind a wave front is well represented by the degree of shift inthe cross-sectional mean value of the velocity ~i.e., dV/dt) and

the unsteady friction when the flow is accelerated (V]V/]t.0)and small correction when the flow is decelerated (V]V/]t,0)the cross-sectional mean value of V , itself.The time scale arguments by Vardy and Brown @47# and Ax-

worthy et al. @30# represent two limit cases: very slow transientsand very fast transients, respectively. In the former case, there isenough mixing such that the acceleration history pattern is de-stroyed, only the instantaneous acceleration is significant to thewall shear stress. In the latter case, the pre-existing flow structureis frozen, there is no additional acceleration history developedexcept that of instantaneous acceleration. The Axworthy et al.@30# argument represents a water hammer flow situation where theacceleration behaves like a pulse, say, the flow drops from a finitevalue to zero in a short period.

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

Brunone et al. @50# incorporated the Coriolis correction coef-ficient and the unsteady wall shear stress in the energy equationfor water hammer as follows:

]H]x

11g

]V]t

1h1f

g]V]t

1Js50 (34)

where h5difference from unity of the Coriolis correction coeffi-cient, Js5( f uVuV)/2gD 5steady-state friction term,(f/g) (]V/]t)5difference between unsteady friction and its cor-responding steady friction. In Eq. ~34!, the convective term isdropped as the Mach number of the flow is small in water hammerproblems.

A constitutive equation is needed for h1f . Brunone et al.@50# proposed

h1f5kS 12a ]V]x Y ]V]t D (35)or in terms of wall shear stress

tw5tws1krD

4 S ]V]t 2a ]V]x D (36)Equation ~36! provides additional dissipation for a reservoir-pipe-valve system when the transient is caused by a downstream sud-den valve closure. The pressure head traces obtained from themodels and experiment are plotted in Fig. 4. It is shown thatalthough both the Darcy-Weisbach formula and Eq. ~32! with con-stant k cannot produce enough energy dissipation in the pressurehead traces, the model by Brunone et al. @50# is quite successful inproducing the necessary damping features of pressure peaks, veri-fied by other researchers @29,5255#.

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

h1f5kF11signS V ]V]x D a ]V]x Y ]V]t G (37)and Bergant et al. @53# proposed

h1f5kF11sign~V !aU]V]x UY ]V]t G (38)The dependence of h1f on x and t as well as the flow accel-

eration is consistent with the theoretical formulations in @30# and@39#. In addition, the form of h1f gives significant correction for

Applied Mechanics Reviews@50#.Utilization of the models presented in this section requires a

reliable estimate of the parameter k . The data of Brunone et al.@31#, Daily et al. @39#, and others show that k is not a universalconstant. An empirical method for estimating this parameter wasproposed by Brunone et al. @52# by fitting the decay of measuredpressure head history. Moody diagram-like charts for k were de-veloped by Pezzinga @44# using a quasi-two-dimensional turbu-lence model. Vardy and Brown @47# provided a theoretically-basedexpression for determining the coefficient k . 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 modelsto adequately represent unsteady turbulence. It should, however,be stressed that modeling turbulent pipe transients is currently notwell understood ~see Sec. 9!.

The mechanism that accounts for the dissipation of the pres-sure head is addressed in the discussion by Ghidaoui et al. @46#.They found that the additional dissipation associated with the in-stantaneous acceleration based unsteady friction model occursonly at the boundary due to the wave reflection. It was shown thatafter nc complete wave cycles, the pressure head is damped by afactor 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 solutionof the unidirectional flow equations and was pioneered by Zielke@56#. Applying the Laplace transform to the axial component ofthe Navier-Stokes equations, he derived the following wall shearexpression for unsteady laminar flow in a pipe:

tw~ t !54nr

R V~ t !12nr

R E0t ]V]t8

~ t8!W~ t2t8!dt8 (39)

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

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

2!1e2135.0198~nt/R2!

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

2!

fornt

R2.0.02

W~ t !50.282095S ntR2D21/2

21.2500011.057855S ntR2D1/2

fornt

R2,0.0210.937500nt

R2 10.396696S ntR2D3/2

20.351563S ntR2D2

(40)

The first term on the right-hand side of Eq. ~39! represents thesteady-state wall shear stress tws and the second term representsthe correction part due to the unsteadiness of the flow twu . Thenumerical integration of the convolution integral in Eq. ~39! re-quires a large amount of memory space to store all previouslycalculated velocities and large central processing unit ~CPU! timeto carry out the numerical integration, especially when the timestep is small and the simulation time large. Trikha @57# used threeexponential terms to approximate the weighting function. The ad-vantage of using exponential forms is that a recursive formula caneasily be obtained, so that the flow history can be lumped into thequantities at the previous time step. In this way, only the calcu-lated quantities at the previous time step needs to be stored in thecomputer memory, and there is no need to calculate the convolu-

JANUARY 2005, Vol. 58 55

tion integral from the beginning at every time step. This reducesthe memory storage and the computational time greatly. In Suzuki

et al. @58#, for t,0.02, the summation is calculated in a normalway; for t.0.02, the recursive formula similar to that of Trikha@57# is used, since each of the five terms included in the weightingfunction is exponential. Although Zielkes formula is derived forlaminar flow, Trikha @57# and others @29,52# found that this for-mula leads to acceptable results for low Reynolds number turbu-lent flows. However, Vardy and Brown @47# warned against theapplication of Zielkes formula outside the laminar flow regime,but did note that the error in applying Zielkes formula to turbu-lent flows diminishes as the duration of the wave pulse reduces.

Vardy et al. @59# extended Zielkes approach to low Reynoldsnumber turbulent water hammer flows in smooth pipes. In a laterpaper, Vardy and Brown @60# developed an extension of the modelof Vardy et al. @59# that was applicable to high Reynolds numbertransient flows in smooth pipes. In addition, Vardy and Brown@60# showed that this model gives results equivalent to those ofVardy et al. @59# for low Reynolds number flows and to those ofZielke @56# for laminar flows. That is, the Vardy and Brown @60#model promises to provide accurate results for Reynolds numbersranging from the laminar regime to the highly turbulent regime.This model has the following form:

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

8 14nrD E0

t

W~ t2t8!]V]t8

dt8 (41)

where

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

b50.54n Rek/D2; k5log~14.3/Re0.05!and Re5Reynolds number. Similar to Zielkes model, the convo-lution nature of Eq. ~41! is computationally undesirable. An accu-rate, simple, and efficient approximation to the Vardy-Brown un-steady friction equation is derived and shown to be easilyimplemented within a 1D characteristics solution for unsteadypipe flow @32#. For comparison, the exact Vardy-Brown unsteadyfriction equation is used to model shear stresses in transient tur-bulent pipe flows and the resulting water hammer equations aresolved by the method of characteristics. The approximate Vardy-Brown model is more computationally efficient ~i.e., requires 16-ththe execution time and much less memory storage! than the exactVardy-Brown model. Both models are compared with measureddata from different research groups and with numerical data pro-duced by a two-dimensional ~2D! turbulence water hammermodel. The results show that the exact Vardy-Brown model andthe approximate Vardy-Brown model are in good agreement withboth laboratory and numerical experiments over a wide range ofReynolds numbers and wave frequencies. The proposed approxi-mate model only requires the storage of flow variables from asingle time step while the exact Vardy-Brown model requires thestorage of flow variables at all previous time steps and the 2Dmodel requires the storage of flow variables at all radial nodes.

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 Zielkes approach to turbulent flows, Vardy andBrown @60# made two fundamental assumptions in relation to theturbulent eddy viscosity in addition to assumptions ~i! through~iv!. First, the turbulent kinematic viscosity is assumed to varylinearly within the wall shear layer and becomes infinite ~i.e., auniform velocity distribution! in the core region. Second, the tur-bulent 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 number is

56 Vol. 58, JANUARY 2005often negligibly small and pipe length far exceeds flow develop-ment length. The validity of assumptions such as that the flowremains axisymmetric ~stable!, that the eddy viscosity is indepen-dent of time, and that its shape is similar to that in steady flow, isdiscussed 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 as de-scribed in the previous section. In particular, it is evident fromEqs. ~39! and ~41! that Eq. ~32! is recovered when the accelerationis constant. In addition, plots of W in Fig. 5 show that for flowswith large Reynolds number, this function is very small every-where except when nt/R2 approaches 0, that is, when t8 ap-proaches t in Eq. ~41!. The region where W(t2t8) in Eq. ~41!becomes significant and provides a measure of the time scale ofthe radial diffusion of vorticity Td . If the acceleration variesslowly in the region where W(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 time scaleof the flow disturbance far exceeds the time scale of radial diffu-sion of vorticity across the shear layer. Moreover, it is obviousthat Eq. ~32! is a good approximation to Eqs. ~39! and ~41! whent is small, as the integral interval is so small that the integrand canbe considered as a constant. Furthermore, the time interval whereW(t2t8) is significant reduces with Reynolds number, whichshows that Eq. ~32! becomes more accurate for highly turbulentflows.

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, numericaltechniques are used to approximate the solution. The method ofcharacteristics ~MOC!, which has the desirable attributes of accu-racy, simplicity, numerical efficiency, and programming simplicity~e.g., @20,23,61#!, is most popular. Other techniques that have alsobeen 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 reviewed inSec. 12, eight use MOC, two are based on implicit FD methods,and only one employs the wave-plan method.

5.1 MOC-Based Schemes. A significant development inthe numerical solution of hyperbolic equations was published byLister @62#. She compared the fixed-grid MOC schemealsocalled the method of fixed time intervalwith the MOC gridscheme and found that the fixed-grid MOC was much easier tocompute, giving the analyst full control over the grid selection andenabling the computation of both the pressure and velocity fields

Fig. 5 Weighting function for different Reynolds numbers

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

interpolate either in space or in time. It is claimed that the reach-back Holly-Preissmann scheme is superior to the classical Holly-and networks.The fixed-grid MOC requires that a common time step (Dt) be

used for the solution of the governing equations in all pipes. How-ever, pipes in the system tend to have different lengths and some-times wave speeds, making it impossible to satisfy the Courantcondition ~Courant number Cr5aDt/Dx

sient signal. It is also unclear as to how additional physics, such asconvolution-integral unsteady friction models, can be incorpo-

This section discusses a number of methods employed by tran-sient modelers to quantify numerical dissipation and dispersion.rated with the wave plan methodology.Wylie and Streeter @72# propose solving the water hammer

equations in a network system using the implicit central differencemethod in order to permit large time steps. The resulting nonlineardifference equations are organized in a sparse matrix form and aresolved using the Newton-Raphson procedure. Only pipe junctionboundary conditions were considered in the case study. It is rec-ognized that the limitation on the maximum time step is set by thefrequency of the dependent variables at the boundaries. Two com-mercially available water hammer software packages use the fourpoint implicit scheme ~see Sec. 12 Water Hammer Software!. Themajor advantage of implicit methods is that they are stable forlarge time steps ~i.e., Cr.1 @65,72#!. Computationally, however,implicit schemes increase both the execution time and the storagerequirement and need a dedicated matrix inversion solver since alarge system of equations has to be solved. Moreover, for mostproblems, iterative schemes must also be invoked. From a math-ematical perspective, implicit methods are not suitable for wavepropagation problems because they entirely distort the path ofpropagation of information, thereby misrepresenting the math-ematical model. In addition, a small time step is required for ac-curacy in water hammer problems in any case @23#. For thesereasons, most of the work done on numerical modeling of hyper-bolic equations in the last three decades concentrated on develop-ing, testing, and comparing explicit schemes ~e.g., @63,64,73#!.

Chaudhry and Hussaini @74# apply the MacCormack, Lambda,and Gabutti schemes to the water hammer equations. These threemethods 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 boundarycondition solution method adds one fictitious node upstream ofthe upstream boundary and another downstream of the down-stream boundary. Using the L1 and L2 norms as indicators of thenumerical errors, it was shown that these second-order finite-difference schemes produce better results than first-order methodof characteristics solutions for Cr50.8 and 0.5. Spurious numeri-cal oscillations are observed, however, in the wave profile.

Although FV methods are widely used in the solution of hyper-bolic systems, for example, in gas dynamics and shallow waterwaves ~see recent books by Toro @75,76#!, this approach is seldomapplied to water hammer flows. To the authors knowledge, thefirst attempt to apply FV-based schemes was by Guinot @77#. Heignored the advective terms, developed a Riemann-type solutionfor the water hammer problem, and used this solution to developa first-order-based FV Godunov scheme. This first-order schemeis very similar to the MOC with linear space-line interpolation. Atthe time of writing, a second paper by Hwang and Chung @78# thatalso uses the FV method for water hammer, has appeared. Unlikein Guinot @77#, the advective terms are not neglected in the workof Hwang and Chung @78#. Instead, they use the conservative formof the compressible flow equations, in which density, and nothead, is treated as an unknown. The application of such a schemein practice would require a state equation relating density to headso that ~i! all existing boundary conditions would have to be re-formulated in terms of density and flow rather than head and flow,and ~ii! the initial steady-state hydraulic grade line would need tobe converted to a density curve as a function of longitudinal dis-tance. At present, no such equation of state exists for water. Ap-plication of this method would be further complicated at bound-aries where incompressible conditions are generally assumed toapply.

5.3 Methods for Evaluating Numerical Schemes. Severalapproaches have been developed to deal with the quantification ofnumerical dissipation and dispersion. The wide range of methodsin the literature is indicative of the dissatisfaction and distrustamong researchers of more conventional, existing techniques.

58 Vol. 58, JANUARY 20055.3.1 Von Neumann Method. Traditionally, fluid transient re-searchers have studied the dispersion and dissipation characteris-tics of the fixed-grid method of characteristics using the Von Neu-mann ~or Fourier! method of analysis @63,64#. The Von Neumannanalysis was used by OBrian et al. @79# to study the stability ofthe numerical solution of partial differential equations. The analy-sis tracks a single Fourier mode with time and dissipation bydetermining how the mode decays with time. Dispersion is evalu-ated by investigating whether or not different Fourier modes travelwith different speeds.

There are a number of serious drawbacks to the Von Neumannmethod of analysis. For example, it lacks essential boundary in-formation, it ignores the influence of the wave profile on the nu-merical errors, it assumes constant coefficients and that the initialconditions are periodic, and it can only be applied to linear nu-merical models @69,7981#. To illustrate, the work by Wiggertand Sundquist @63#, Goldberg and Wylie @64#, and others clearlyshows that the attenuation and dispersion coefficients obtainedfrom the Fourier analysis depend on the Courant number, the ratioof the wavelength of the kth harmonic Lk to the reach length Dx ,and the number of reachbacks and/or reachouts, but does not de-pend on the boundary conditions. Yet, the simulation of boundaryconditions and knowledge of how these boundary conditions in-troduce and reflect errors to the internal pipe sections is crucial tothe study of numerical solutions of hydraulic problems. In short,the Von Neumann method cannot be used as the only benchmarkfor selecting the most appropriate numerical scheme for nonlinearboundary-value hyperbolic problems.

5.3.2 L1 and L2 Norms Method. Chaudhry and Hussaini@74# developed L1 and L2 norms to evaluate the numerical errorsassociated with the numerical solution of the water hammer equa-tions by the MacCormack, Lambda, and Gabutti schemes. How-ever, the L1 and L2 method as they apply it can only be used forproblems that have a known, exact solution. In addition, these twonorms do not measure a physical property such as mass or energy,thereby making the interpretation of the numerical values of thesenorms difficult. Hence, the L1 and L2 norms can be used to com-pare different schemes, but do not give a clear indication of howwell a particular scheme performs.

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

The attenuation parameter is intended to measure the numericaldissipation by different interpolation schemes by looking at themaximum head value at the valve. This parameter, however, un-derestimates the numerical attenuation because the computation ofhead and flow at the downstream end of the pipe uses one char-acteristic equation and one boundary equation. The dispersion pa-rameter is intended to measure the numerical dispersion by differ-ent interpolation schemes. This parameter is determined byasserting that the change in the wave shape is governed by theconstant diffusion equation with initial conditions described bythe Heaviside function. Although this method allows a rudimen-tary comparison of simple system responses, general conclusionscannot be drawn for a hyperbolic equation based on a diffusionequation. The longitudinal displacement parameter is intended tomeasure the extent by which different numerical schemes artifi-cially displace the wave front. However, this parameter only sug-gests to what degree the interpolation method used is symmetri-cally dispersive and says little about the magnitude of artificialdisplacement of the wave by the numerical scheme.

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

Transactions of the ASME

ods since this approach can be applied to a nonlinear transientproblem with realistic boundary conditions. The basic idea is to

tionary helical vortices and that the breakdown of these vorticesinto turbulence is very rapid. The breakdown of the helical vorti-check how closely a particular numerical method conserves mass.Note that the mass balance approach can become ineffective incases where a numerical scheme conserves mass but not energyand momentum.

5.3.5 EHDE Approach. Ghidaoui and Karney @85# devel-oped the concept of an equivalent hyperbolic differential equation~EHDE! to study how discretization errors arise in pipeline appli-cations for the most common interpolation techniques used to dealwith the discretization problem in fixed-grid MOC. In particular, itis shown that space-line interpolation and the Holly-Preissmannscheme are equivalent to a wave-diffusion model with an adjustedwave speed, but that the latter method has additional source andsink terms. Further, time-line interpolation is shown to be equiva-lent to a superposition of two waves with different wave speeds.The EHDE concept evaluates the consistency of the numericalscheme, provides a mathematical description of the numerical dis-sipation and dispersion, gives an independent way of determiningthe Courant condition, allows the comparison of alternative ap-proaches, finds the wave path, and explains why higher-ordermethods should usually be avoided. This framework clearly pointsout that numerical approximation of the water hammer equationsfundamentally changes the physical problem and must be viewedas a nontrivial transformation of the governing equations. For ex-ample, implicit methods, while noted for their stability character-istics, transform the water hammer problem into a superpositionof wave problems, each of which has a wave speed different fromthe physical wave speed and at least one of which has an infinitewave speed. The infinite numerical wave speed associated withimplicit schemes ensures that the numerical domain of depen-dence is larger than the physical domain of dependence, and ex-plains why these are highly stable. While good for stability, thelarge discrepancy between the numerical and physical domains ofdependence hinders the accuracy of these schemes. Another prob-lem with implicit schemes is that they are often computationallyinefficient because they require the inversion of large matrices.

5.3.6 Energy Approach. Ghidaoui et al. @86# developed anintegrated energy approach for the fixed-grid MOC to study howthe discretization errors associated with common interpolationschemes in pipeline applications arise and how these errors can becontrolled. Specifically, energy expressions developed in thiswork demonstrate that both time-line and space-line interpolationattenuate the total energy in the system. Wave speed adjustment,on the other hand, preserves the total energy while distorting thepartitioning of the energy between kinetic and internal forms.These analytic results are confirmed with numerical studies ofseveral series pipe systems. Both the numerical experiments andthe analytical energy expression show that the discretization errorsare small and can be ignored as long as there is continuous workin the system. When the work is zero, however, a Cr value closeto one is required if numerical dissipation is to be minimized. Theenergy approach is general and can be used to analyze other waterhammer numerical schemes.

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

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, in theform of helical vortices, can develop in transient flows. Theseinstabilities lead to the breakdown of flow symmetry with respectto the pipe axis. For example, Das and Arakeri @87# performedunsteady pipe flow experiments where the initial flow was laminarand the transient event was generated by a piston. They found thatwhen the Reynolds number and the transient time scale exceed athreshold value, the flow becomes unstable. In addition, they ob-served that the flow instability results in the formation of nonsta-

Applied Mechanics Reviewsces into turbulence resulted in strong asymmetry in the flow withrespect to the pipe axis. Brunone et al. @31,88# carried out mea-surements of water hammer velocity profiles in turbulent flows.They also observed strong flow asymmetry with respect to thepipe axis. In particular, they found that a short time after the wavepassage, flow reversal no longer appears simultaneously in boththe top and the bottom sides of the pipe. Instead, flow reversalappears to alternate between the bottom and top sides of the pipe.This is consistent with the asymmetry observed by Das and Arak-eri @87#. The impact of instabilities on wall shear stress in un-steady pipe flows was measured by Lodahl et al. @89#. They foundthat inflectional flow instabilities induce fluctuations in the wallshear stress, where the root mean square of the wall shear stressfluctuation in the pipe was found to be as high as 45% of themaximum wall shear stress.

Das and Arakeri @87# applied linear stability analysis to un-steady plane channel flow to explain the experimentally observedinstability in unsteady pipe flow. The linear stability and the ex-perimental results are in good qualitative agreement. Ghidaouiand Kolyshkin @90# investigated the linear stability analysis ofunsteady velocity profiles with reverse flow in a pipe subject tothree-dimensional ~3D! perturbation. They used the stability re-sults to reinterpret the experimental results of Das and Arakeri@87# and assess their planar flow and quasi-steady assumptions.Comparison of the neutral stability curves computed with andwithout the planar channel assumption shows that this assumptionis accurate when the ratio of the boundary layer thickness to thepipe radius is below 20%. Any point in the neutral stability curverepresents the parameters combination such that the perturbationsneither grow nor decay. Critical values for any of these parameterscan be obtained from the neutral stability curve. For unsteady pipeflows, the parameters related are Re and t . Therefore, critical Recan be obtained.

The removal of the planar assumption not only improves theaccuracy of stability calculations, but also allows for the flowstability of both axisymmetric and nonaxisymmetric modes to beinvestigated, and for the experimental results to be reinterpreted.For example, both the work of Ghidaoui and Kolyshkin @90# andthe experiments of Das and Arakeri @87# show that the nonaxisym-metric mode is the least stable ~i.e., the helical type!.

With the aim of providing a theoretical basis for the emergenceof helical instability in transient pipe flows, Ghidaoui and Koly-shkin @42# performed linear stability analysis of base flow velocityprofiles for laminar and turbulent water hammer flows. These baseflow velocity profiles are determined analytically, where the tran-sient is generated by an instantaneous reduction in flow rate at thedownstream end of a simple pipe system. The presence of inflec-tion points in the base flow velocity profile and the large velocitygradient near the pipe wall are the sources of flow instability. Themain parameters governing the stability behavior of transientflows are Reynolds number and dimensionless time scale. Thestability of the base flow velocity profiles with respect to axisym-metric and asymmetric modes is studied and the results are plottedin the Reynolds number/time scale parameter space. It is foundthat the asymmetric mode with azimuthal wave number one is theleast stable. In addition, it is found that the stability results of thelaminar and the turbulent velocity profiles are consistent with pub-lished experimental data. The consistency between the stabilityanalysis 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 wavetime scale are important in the characterization of the stability ofwater hammer flows. Physically, flow instabilities change thestructure and strength of the turbulence in a pipe, result in strong

JANUARY 2005, Vol. 58 59

flow asymmetry, and induce significant fluctuations in wall shearstress. These effects of flow instability are not represented in ex-

Jackson @93# provided an estimate for the time delay from themoment the wall vortex ring was generated at the pipe wall to theisting water hammer models.In an attempt to gain an appreciation of the importance of in-

cluding the effects of helical vortices in transient models,Ghidaoui et al. @46# applied current transient models to flow caseswith and without helical vortices. In the case where stability re-sults indicate that there are no helical vortices, Ghidaoui et al.@46# found that the difference between water hammer models andthe data of Pezzinga and Scandura @91# increases with time at amild rate. However, for the case where stability results and ex-periments indicate the presence of helical vortices, it is found thatthe difference between water hammer models and the data ofBrunone et al. @31# exhibits an exponential-like growth. In fact,the difference between models and the data of Brunone et al. @31#reaches 100% after only six wave cycles. This marked differencebetween models and data suggests that the influence of helicalvortices on the flow field is significant and cannot be neglected.

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

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 equa-tions used by Vardy and Hwang @25#, Silva-Araya and Chaudhry@37#, and Pezzinga @38# assume that the turbulence changes in aquasi-steady manner and that the eddy viscosity expressions de-rived for steady-state pipe flows remain applicable for water ham-mer flows. An understanding of the response of the turbulencefield to water hammer waves is central to judging the accuracy ofusing either the frozen or the quasi-steady turbulence assump-tions.

There is a time lag between the passage of a wave front at aparticular location along the pipe and the resulting change in tur-bulent conditions at this location ~e.g., @46,92,93#!. In particular, atthe instant when a water hammer wave passes a position x alongthe pipe, the velocity field at x 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 conditionsare unaltered at the instant of the wave passage. However, thecombination of the uniform shift in velocity with the no-slip con-dition generates a vortex sheet at the pipe wall. The subsequentdiffusion of this vortex ring from the pipe wall to the pipe core isthe mechanism responsible for changing the turbulence conditionsin the pipe.

A short time after the wave passage, the extent of vorticitydiffusion is limited to a narrow wall region and the turbulencefield is essentially frozen. In this case, both the quasi-steady tur-bulence and frozen turbulence assumptions are equally appli-cable. A similar conclusion was reached by Greenblatt and Moss@92# for a temporally accelerating flow; by Tu and Ramparian@94#, Brereton et al. @95#, and Akhavan et al. @96,97# for oscilla-tory flow; He and Jackson @93# for ramp-type transients; andGhidaoui et al. @46# for water hammer flows. As the time after thewave passage increases, the extent of the radial diffusion of vor-ticity becomes more significant and begins to influence the veloc-ity gradient and turbulence strength and structure in the bufferzone. The experiments of He and Jackson @93# show that the axialturbulent fluctuations are the first to respond to the changes in theradial gradient of the velocity profile and that there is a time delaybetween the changes in the axial turbulent fluctuations and itsredistribution among the radial and azimuthal turbulent compo-nents. The production of axial turbulent kinetic energy and thetime lag between production and redistribution of axial turbulentkinetic energy within the buffer zone are not incorporated insteady-state-based turbulence models. The characteristics of theflow in the core region will start to change only when the wave-induced shear pulse emerges from the buffer zone into the coreregion. On the basis of their unsteady flow experiments, He and

60 Vol. 58, JANUARY 2005moment when significant changes in the structure and strength ofturbulence appeared near the pipe axis.

Ghidaoui et al. @46# proposed a dimensionless parameter P forassessing the accuracy of quasi-steady turbulence modeling in wa-ter hammer problems. This parameter is defined as the ratio of thetime scale of radial diffusion of vorticity to the pipe core to thetime scale of wave propagation from one end of the pipe to theother. This parameter provides a measure for the number of timesa wave front travels from one end of the pipe to the other beforethe preexisting turbulence conditions start to respond to the tran-sient event. It follows that the frozen and quasi-steady assump-tions are ~i! acceptable when P@1, ~ii! questionable when P is oforder 1, and ~iii! applicable when P!1. However, the last casedoes not belong to the water hammer regime. These conclusionsare supported by the work of Ghidaoui et al. @46#, where theycompared the results of quasi-steady turbulence models withavailable data and by the work of Ghidaoui and Mansour @32#where they compared the results of frozen eddy viscosity modelswith experimental data.

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

lence models can ~i! enhance the current state of understanding ofenergy 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 mod-els. Examples of turbulence models for water hammer problems,their applicability, and their limitations can be found in Vardy andHwang @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-dimensionalmodeling are discussed in this section. Turbulence models andnumerical solutions are presented in subsequent sections.

The most widely used quasi-two-dimensional governing equa-tions 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 written indifferent forms, they can be expressed as the following pair ofcontinuity and momentum equations:

ga2

S ]H]t 1u ]H]x D1 ]u]x 1 1r ]rv]r 50 (42)]u

]t1u

]u

]x1v

]u

]r52g

]H]x

11rr

]rt

]r(43)

where x ,t ,u ,H ,r are defined as before, v(x ,r ,t)5local radial ve-locity, and t5shear stress. In this set of equations, compressibilityis only considered in the continuity equation. Radial momentum isneglected by assuming that ]H/]r50, and these equations are,therefore, only quasi-two-dimensional. The shear stress t can beexpressed as

t5rn]u

]r2ru8v8 (44)

where u8 and v85turbulence perturbations corresponding to lon-gitudinal velocity u and radial velocity v , respectively. Turbu-lence models are needed to describe the perturbation term2ru8v8 since most practical water hammer flows are turbulent.

The governing equations can be further simplified by neglect-ing nonlinear convective terms, as is done in the 1D case since thewave speed a is usually much larger than the flow velocity u or v .Then the equations become the following:

ga2

]H]t

1]u

]x1

1r

]rv]r

50 (45)

Transactions of the ASME

]u1g

]H5

1 ]rt (46)9.1 Five-Region Turbulence Model. The model used by

Vardy and Hwang @25# is a direct extension of the model devel-

]t ]x rr ]r

These governing equations are usually solved by numericalmeans.

For an adequately anchored or restrained pipe, i.e., the pipe isrigid and the radial velocity at the pipe wall is zero. From flowsymmetry, the radial velocity at the centerline is also zero. Inte-grating Eq. ~45! across the pipe section, the radial velocity van-ishes, leaving the following:

]H]t

1a2

gA]Q]x

50 (47)

]u

]t1g

]H]x

51rr

]rt

]r(48)

Q~x ,t !5EAudA (49)

where Q5discharge. These equations are the same as those pre-sented by Pezzinga @38#.

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

Ghidaoui @26# derived quasi-two-dimensional equations fromthe complete 3D continuity equation and Navier-Stokes equationsusing an ensemble averaging process in which the assumptionsinherent 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 withthe compressibility of the fluid are significantly smaller than theviscous term associated with angular deformation. Therefore, thecompressibility is neglected in the momentum equations of both1D and 2D models.

In Silva-Araya and Chaudhry @37,98# and Eichinger and Lein@45#, an integration of the momentum equation is also carried out.In each case, the system reduces to a 1D formulation. The quasi-two-dimensional momentum Eq. ~46! is only used to provide anunsteady friction correction for 1D governing equations. Thesecorrections include: ~i! an energy dissipation factor, which is theratio of the energy dissipation calculated from the cross-sectionalvelocity distribution to that calculated from the Darcy-Weisbachformula @37,98# or ~ii! direct calculation of wall shear stress, ei-ther by velocity gradient at the pipe wall or through energy dissi-pation @45#.

9 Turbulence ModelsTurbulence models are needed to estimate the turbulent pertur-

bation term for 2ru8v8. In the water hammer literature, thewidely used turbulence models are algebraic models@25,37,38,98,99# in which eddy viscosity is expressed as somealgebraic function of the mean flow field. Other sophisticatedmodels ~such as the k2e model, which require additional differ-ential equations for eddy viscosity! have also been tried @45#.Similar results for the pressure head traces have been obtained.

The algebraic turbulence models used by Vardy and Hwang@25# and Pezzinga @38# are discussed further to illustrate somefeatures of algebraic turbulence models. These models were com-paratively studied by Ghidaoui et al. @46#. The comparison showsthat very similar dissipation is produced by the two models. Otherdifferent variations of algebraic turbulence models are available inRodi @102#.

Applied Mechanics Reviewsoped by Kita et al. @103# for steady flow

t5r~n1e!]u

]r5rnT

]u

]r(50)

where e5eddy viscosity, nT5total viscosity, and the other termswere previously defined. The total viscosity distribution is com-partmentalized into five regions as follows:

1. viscous layernTn

51 0

lation, but one should note that the expression for mixing lengthand the empirical coefficient k are based on steady flow equiva-

Vardy and Hwang @25# solves the hyperbolic part of governingequations by MOC and the parabolic part using finite difference.

irigorously evaluated. Unfortunately, in carrying out LES, a full3D system of equations must be solved using very fine grids@104#. For steady flow simulations, when the turbulence statisticsreach steady, the ensemble average can be obtained over a timeinterval from a single run @104#. However, the ensemble averagecannot be obtained from a single run for transient flow. The re-quirement of many runs makes the resulting computational pro-cess prohibitively time consuming. As yet, such analyses have notbeen performed in pipe transients.

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

parabolic partial differential equations. The numerical solution of

1

a

g 1eCu2~1 !uCq2~1 !2eCu3~1

1 2Fag 1eCu2~1 !G uCq2~1 ! eCu3~1 !A A

1 fl 2eCu1~ j ! 2uCq1~ j !

1 fl eCu1~ j ! 2uCq1~ j !A A

1 fl

1 fl

where j5index along radial direction; Cu1 ,Cu2 ,Cu35coefficassociated with radial flux q; and e and u5Hi

n11,ui ,1

n11,qi ,1

n11, fl ,ui , jn11 ,qi , jn11 , fl ,ui ,Nr21n11 ,qi ,Nr21n11 ,ui ,Nrn11%T

62 Vol. 58, JANUARY 2005dt 6g dt 52 g r ]r 6g rr ]r

alongdxdt 56a (60)

where q5rv .The pipe is divided into Nr cylinders of varying thickness. At

a given time t and location x along the pipe, two equations applyto each cylinder. Since there are Nr cylinders in total, the totalnumber of equations is 2Nr . Therefore, the governing equationsfor all cylinders can be written in matrix form as follows: Az5b, where A is a 2Nr32Nr matrix whose form is as follows:

!

A

a

g 1eCu2~ j ! uCq2~ j ! 2eCu3~ j !fl

2Fag 1eCu2~ j !G uCq2~ j ! eCu3~ j !flA

2eCu1~Nr ! 2uCq1~Nr !a

g 1eCu2~Nr !

eCu1~Nr ! 2uCq1~Nr ! 2Fag 1eCu2~Nr !G

'ents associated with axial velocity u; Cq1 ,Cq25coefficientsare weighting coefficients. The unknown vector zin which i5index along axial direction and the superscript T de-

Transactions of the ASMElents.Ghidaoui et al. @46# compared both models ~i.e., the five-

region and the two-layer model! and obtained very similar resultsfor pressure head estimates. The comparative study suggests thatthe pressure head is not sensitive to eddy viscosity distribution inthe pipe core region. As these models are based on steady flowprinciples, the application of these models to unsteady flow prob-lems implicitly includes the quasi-steady assumptions discussed inSection 7.

These algebraic turbulence models are widely used, mainlybecause of their simplicity and robustness. As more powerfulcomputers become available and improvements are made to nu-merical solution techniques, detailed turbulence structures may beobtained using more sophisticated turbulence models, such as thetwo-equation k2e models, or perhaps even Reynolds stress mod-els, for which no eddy viscosity hypothesis is needed.

All of the models mentioned above are based on the Reynolds-averaged Navier-Stokes ~RANS! equation. The averaging processis clearly a time average and valid for steady flows. For unsteadyflows, the use of the time average is highly questionable unless theunsteadiness has a much larger time scale than the time scale ofturbulence. 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 filtered,large-scale motion is resolved while the small-scale motion ismodeled. If results from LES were available, then some of theassumptions mentioned previously could, in principle, be more

This hybrid solution approach has several merits. First, the solu-tion method is consistent with the physics of the flow since it usesMOC for the wave part and central differencing for the diffusionpart. Second, the use of MOC allows modelers to take advantageof the wealth of strategies, methods, and analysis developed inconjunction with 1D MOC water hammer models. For example,schemes for handling complex boundary elements and strategiesdeveloped for dealing with the 1D MOC discretization problem~e.g., wave speed adjustment and interpolation techniques! can beadapted to quasi-two-dimensional MOC models. Third, althoughthe radial mass flux is often small, its inclusion in the continuityequation by Vardy and Hwang @25# is more physically correct andaccurate. A major drawback of the numerical model of Vardy andHwang @25#, however, is that it is computationally demanding. Infact, the CPU time required by the scheme is of the order Nr

3

where Nr5number of computational reaches in the radial direc-tion. Vardy and Hwangs scheme was modified by Zhao andGhidaoui @105# to a much more efficient form. The CPU timerequired is reduced to order Nr , making the scheme more ame-nable to application to the quasi-two-dimensional modeling ofpipe networks and for coupling with sophisticated turbulencemodels. Several numerical schemes for quasi-two-dimensionalmodeling are summarized in the following material.

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

dH a du a2 1 ]q a 1 ]~rt!

notes the transpose operator and b5a known vector that depends on head and velocity at time level n . Therefore, the solution for head,and longitudinal and radial velocities, involves the inversion of a 2Nr32Nr matrix. The sparse nature of A is the reason the scheme is

aemr

l

q

A

N10.2 Pezzinga Scheme. The numerical solution by Pezz-inga @38# solves for pressure head using explicit FD from thecontinuity Eq. ~47!. Once the pressure head has been obtained, themomentum Eq. ~48! is solved by implicit FD for velocity profiles.This velocity distribution is then integrated across the pipe sectionto calculate the total discharge. The scheme is fast due to decou-pling of the continuity and momentum equations and the adoptionof the tridiagonal coefficient matrix for the momentum equation.It has been applied to network simulations.

While the scheme is efficient, the authors have found that thereis a difficulty in the numerical integration step. Since the integra-tion can only be approximated, some error is introduced in thisstep that leads to spurious oscillations in the solution for pressure.To get rid of these oscillations, a large number of reaches in theradial direction may be required or an iterative procedure mayneed to be used @37#.

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

A similar procedure is used in Eichinger and Lein @45#. One-dimensional equations are first solved to obtain the pressure gra-dient. This pressure gradient is used to solve Eq. ~46! using afinite difference method. The eddy viscosity is obtained from ak2e model. Once the velocity profile is known, the friction termcan be calculated from the velocity gradient at the wall, which isthen used in the 1D equations. An iterative procedure is employedin this calculation to obtain eddy viscosity. Although there mightbe some difference between the discharge calculated from 1D

foregoing methods. Once the velocity is obtained, energy dissipa-tion and discharge can be calculated. The dissipation is used toestimate an energy dissipation ratio, which provides a correctionfactor for the friction term in the 1D equations. The adjusted 1Dequations are then solved to give a new discharge, which is com-pared to that calculated from velocity profile integration. If thedifference is small ~say, less than 5%!, the calculation proceeds tothe next time step. Otherwise, the pressure gradient is adjusted,and the procedure is repeated. A mixing length algebraic turbu-lence 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 of

fluid transient problems is analogous to, but slightly differentfrom, the conventional use of the terminology in solving differen-tial equations. Just as a boundary value problem in the math-ematical sense implies conditions that must be satisfied at theedges of the physical domain of the problem, boundary conditionsin fluid transients implies the need for additional head-dischargerelations to describe physical system components such as pumps,reservoirs and valves. Thus, one or more simplified auxiliary re-lations can be specified to solve for piezometric head, flow veloc-ity, or other variables associated with the physical devices them-selves. Examples of boundary conditions include, but are notlimited to, valves, nozzles, pumps, turbines, surge tanks, airvalves, tanks and reservoirs, heat exchangers, condensers andmany other application-specific devices.

This section of the paper discusses a generalized approach toincorporating boundary conditions within the method of charac-

Applied Mechanics Reviews JANUARY 2005, Vol. 58 63inefficient.Improving the efficiency of the Vardy-Hwang scheme is essenti

as tools for analyzing practical pipe systems or for conducting numleads to a highly efficient scheme in which the original system becoas the following: Bu5bu and Cv5bv , where B is a tridiagonal N

1a

g 1eCu2~1 !2eCu3~1 !

fl 2eCu1~ j ! ag

2

The unknown vector u5$ui ,1n11

, fl ,ui , jn11 , fl ,ui ,Nrn11%T representsdepend on H , u , and q at time level n; and C is a tridiagonal Nr3

S 1 uCq2~1 !0 2@uCq1~2 !1uCq2~1 !# uCAfl uCq1~ j21 ! 2@uCq1~ j !1fl uCq1~

Lastly, v5$Hin11

,qi ,1n11

, flqi ,Nr21n11 %T is an unknown vector ofhead and radial velocities and bv5a known vector whose ele-ments depend on H ,u ,q at time level n . Inversion of tridiagonalsystems can be performed efficiently by using the Thomas algo-rithm.l if quasi-two-dimensional models are to become widely acceptedrical experiments. Algebraic manipulation of the coefficient matrixes two subsystems with tridiagonal coefficient matrices expressed

3Nr matrix given by

A

1eCu2~ j ! 2eCu3~ j ! flA

eCu1~Nr !a

g 1eCu2~Nr !

2ongitudinal flow velocity; bu is a known vector whose elementsNr matrix given by

2~2 !

uCq2~ j21 !# uCq2~ j !fl