Mohamed S.Ghidaoui
email:ghidaoui@ust.hk
Ming Zhao
email:cezhm@ust.hk Department of Civil Engineering,The Hong Kong University of Science and Technology,
Hong Kong,China
Duncan A.McInnis Surface Water Group,Komex International Ltd., 450016th Avenue,Suite100,N.W.Calgary,
Alberta T3B0M6,Canada
David H.Axworthy 163N.Marengo Avenue,#316,Pasadena,
CA91101
email:bm300@lafn A Review of Water Hammer Theory and Practice
Hydraulic transients in clod conduits have been a subject of both theoretical study and inten practical interest for more than one hundred years.While straightforward in terms of the one-dimensional nature of pipe networks,the full description of transientfluidflows po interesting problems influid dynamics.For example,the respon of the turbulence structure and strength to transient waves in pipes and the loss offlow axisymmetry in pipes due to hydrodynamic instabilities are currently not understood.Yet,such under-standing is important for modeling energy dissipation and water quality in transient pipe flows.This paper prents an overview of both historic developments and prent day rearch and practice in thefield of hydraulic transients.In particular,the paper dis-cuss mass and momentum equations for one-dimensional Flows,wavespeed,numerical solutions for one-dimensional problems,wall shear stress models;two-dimensional mass and momentum equations,turbulence models,numerical solutions for two-dimensional problems,boundary conditions,transient analysis software,and future practical and re-arch needs in water hammer.The prentation emphasizes the assumptions and restric-tions involved in various governing equations so as to illuminate the range of applicabil-ity as well as the limitations of the equations.Understanding the limitations of current models is esntial for(i)interpreting their results,(ii)judging the reliability of the data obtained from them,(iii)minimizing misu of water-hammer models in both rearch and practice,and(iv)delineating the contribution of
physical process from the contribution of numerical artifacts to the results of waterhammer models.There are134refrences cited in this review article.͓DOI:10.1115/1.1828050
͔
1Introduction
Thus the growth of knowledge of the physical aspect of reality cannot be regarded as a cumulative process.The basic Gestalt of this knowledge changes from time During the cumulative periods scientists behave as if reality is exactly as they know it except for missing details and improvements in accuracy.They speak of the laws of nature,for example,which are simply models that explain their experience of reality at a certain time.Later generations of scientists typically discover that the conceptions of reality embodied certain implicit as-sumptions and hypothes that later on turned out to be incor-rect.Vanderburg,͓1͔
Unsteadyfluidflows have been studied since manfirst bent water to his will.The ancient Chine,the Mayan Indians of Cen-tral America,the Mesopotamian civilizations bordering the Nile, Tigris,and Euphrates river systems,and many other societies throughout history have developed extensive systems for convey-ing water,primarily for purpos of irrigation,but also for domes-tic water supplies.The ancients understood and appliedfluidflow principles within the context of‘‘traditional,’’culture-bad tech-nologies.With the arrival of the scientific age and the mathemati-cal developments embodied in Newton’s Principia,our under-standing offluidflow took a quantum leap in terms of its theoretical abstraction.That leap has propelled the entire develop-ment of hydraulic engi
neering right through to the mid-twentieth century.The advent of high-speed digital computers constituted another discrete transformation in the study and application of fluids engineering principles.Today,in hydraulics and other areas, engineersfind that their mandate has taken on greater breadth and depth as technology rapidly enters an unprecedented stage of knowledge and information accumulation.
As cited in The Structure of Scientific Revolutions,Thomas Kuhn͓2͔calls such periods of radical and rapid change in our view of physical reality a‘‘revolutionary,noncumulative transi-tion period’’and,while he was referring to scientific views of reality,his remarks apply equally to our technological ability to deal with a revid or more complex view of the physical uni-ver.It is in this condition that thefield of clod conduit tran-sientflow,and even more generally,the hydraulic analysis,de-sign,and operation of pipeline systems,currentlyfinds itlf.
The computer age is still dawning,bringing with it a massive development and application of new knowledge and technology. Formerly accepted design methodologies,criteria,and standards are being challenged and,in some instances,outdated and revid. Computer aided analysis and design is one of the principal mecha-nisms bringing about the 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 the phenomena. By combining physical laws,mathematical abstraction,numerical procedures,logical constructs,and electronic data processing, the methods now permit the solution of problems of enormous complexity 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 as an updated view of the current state of the art,both with respect to theoretical advances of the last decade and modeling practice.
2Mass and Momentum Equations for
One-Dimensional Water Hammer Flows
Before delving into an account of mathematical developments related to waterhammer,it is instructive to briefly note the societal context that inspired the initial interest in waterhammer phenom-ena.In the late nineteenth century,Europe was on the cusp of the industrial 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,hydroelectric generation was still the principal supply of this important energy source.Although hydroelectric generation accounts for a much smaller proportion of energy production today,the problems asso-
Transmitted by Associate Editor HJS Fernando.
ciated with controlling theflow of water through penstocks and turbines remains an important application of transient analysis. Hydrogeneration companies contributed heavily to the develop-ment offluids and turbomachinery laboratories that studied, among other things,the phenomenon of waterhammer and its con-trol.Some of Allievi’s early experiments were undertaken as a direct result of incidents and failures caud by overpressure due to rapid valve closure in northern Italian power plants.Frictionless approaches to transient phenomena were appropriate in the early developments becau͑i͒transients were most influenced by the rapid closure and opening of valves,which generated the majority of the energy loss in the systems,and͑ii͒the pipes involved tended to have large diameters and theflow velocities tended to be small.
By the early1900s,fuel oils were overtaking hydrogeneration as the principal energy source to meet society’s burgeoning de-mand for power.However,the fascination with,and need to un-derstand,trans
ient phenomena has continued unabated to this day. Greater availability of energy led to rapid industrialization and urban development.Hydraulic transients are critical design factors in a large number offluid systems from automotive fuel injection to water supply,transmission,and distribution systems.Today, long pipelines transportingfluids over great distances have be-come commonplace,and the almost universal development of sprawling systems of small pipe diameter,high-velocity water dis-tribution systems has incread the importance of wall friction and energy loss,leading to the inclusion of friction in the governing equations.Mechanically sophisticatedfluid control devices,in-cluding many types of pumps and valves,coupled with increas-ingly sophisticated electronic nsors and controls,provide the potential for complex system behavior.In addition,the recent knowledge that negative pressure phas of transients can result in contamination of potable water systems,mean that the need to understand and deal effectively with transient phenomena are more acute than ever.
2.1Historical Development:A Brief Summary.The prob-lem of water hammer wasfirst studied by Menabrea͓3͔͑although Michaud is generally accorded that distinction͒.Michaud͓4͔ex-amined the u of air chambers and safety valves for controlling water hammer.Near the turn of the nineteenth century,rearchers like Weston͓5͔,Carpenter͓6͔and Frizell͓7͔attempted to develop expressions relating p
ressure and velocity changes in a pipe.Fri-zell͓7͔was successful in this endeavor and he also discusd the effects of branch lines,and reflected and successive waves on turbine speed regulation.Similar work by his contemporaries Joukowsky͓8͔and Allievi͓9,10͔,however,attracted greater at-tention.Joukowsky͓8͔produced the best known equation in tran-sientflow theory,so well known that it is often called the‘‘fun-damental equation of water hammer.’’He also studied wave reflections from an open branch,the u of air chambers and surge tanks,and spring type safety valves.
Joukowsky’s fundamental equation of water hammer is as fol-lows:
⌬PϭϮa⌬V or⌬HϭϮa⌬V
g
(1)
where aϭacoustic͑waterhammer͒wavespeed,Pϭg(HϪZ)ϭpiezometric pressure,Zϭelevation of the pipe centerline from a given datum,Hϭpiezometric head,ϭfluid density,Vϭ͐A udA ϭcross-ctional average velocity,uϭlocal longitudinal velocity, Aϭcross-ctional area of the pipe,and gϭgravitational accelera-tion.The positive sign in Eq.͑1͒is applicable for a water-hammer wave moving downstream w
hile the negative sign is applicable for a water-hammer wave moving upstream.Readers familiar with the gas dynamics literature will note that⌬PϭϮa⌬V is obtain-able from the momentum jump condition under the special ca where theflow velocity is negligible in comparison to the wavespeed.The jump conditions are a statement of the conrva-tion laws across a jump͑shock͓͒11͔.The conditions are ob-tained either by directly applying the conrvation laws for a con-trol volume across the jump or by using the weak formulation of the conrvation laws in differential form at the jump.
Allievi͓9,10͔developed a general theory of water hammer fromfirst principles and showed that the convective term in the momentum equation was negligible.He introduced two important dimensionless parameters that are widely ud to characterize pipelines and valve behavior.Allievi͓9,10͔also produced charts for pressure ri at a valve due to uniform valve closure.Further refinements to the governing equations of water hammer appeared in Jaeger͓12,13͔,Wood͓14͔,Rich͓15,16͔,Parmakian͓17͔, Streeter and Lai͓18͔,and Streeter and Wylie͓19͔.Their combined efforts have resulted in the following classical mass and momen-tum equations for one-dimensional͑1D͒water-hammerflows
a2
g
ץV
ץxϩ
ץH
叫卖广告
ץtϭ0(2)
ץV
ץtϩg
ץH
ץxϩ
4
Dwϭ0(3) in whichwϭshear stress at the pipe wall,Dϭpipe diameter,x ϭthe spatial coordinate alon
g the pipeline,and tϭtemporal coor-dinate.Although Eqs.͑2͒and͑3͒were fully established by the 1960s,the equations have since been analyzed,discusd,red-erived and elucidated in numerous classical texts͑e.g.,͓20–23͔͒. Equations͑2͒and͑3͒constitute the fundamental equations for1D water hammer problems and contain all the physics necessary to model wave propagation in complex pipe systems.
2.2Discussion of the1D Water Hammer Mass and Mo-mentum Equations.In this ction,the fundamental equations for1D water hammer are derived.Special attention is given to the assumptions and restrictions involved in various governing equa-tions so as to illuminate the range of applicability as well as the limitations of the equations.
Rapidflow disturbances,planned or accidental,induce spatial and temporal changes in the velocity͑flow rate͒and pressure͑pi-ezometric head͒fields in pipe systems.Such transientflows are esntially unidirectional͑i.e.,axial͒since the axialfluxes of mass,momentum,and energy are far greater than their radial counterparts.The rearch of Mitra and Rouleau͓23͔for the lami-nar water hammer ca and of Vardy and Hwang͓25͔for turbulent water-hammer supports the validity of the unidirectional approach when studying water-hammer problems in pipe systems.
With the unidirectional assumption,the1D classical water ham-mer equations governing the axial and temporal variations of the cross-ctional average of thefield variables in transient pipe flows are derived by applying the principles of mass and momen-tum to a control volume.Note that only the key steps of the derivation are given here.A more detailed derivation can be found in Chaudhry͓20͔,Wylie et al.͓23͔,and Ghidaoui͓26͔.
Using the Reynolds transport theorem,the mass conrvation ͑‘‘continuity equation’’͒for a control volume is as follows͑e.g.,͓20–23͔͒
ץ
ץt͵cvd᭙ϩ
͵cs͑v"n͒dAϭ0(4)
where cvϭcontrol volume,csϭcontrol surface,nϭunit outward normal vector to control surface,vϭvelocity vector.
Referring to Fig.1,Eq.͑4͒yields
ץ
ץt͵x xϩ␦xAdxϩ
͵cs͑v"n͒dAϭ0(5)
The local form of Eq.͑5͒,obtained by taking the limit as the length of the control volume shrinks to zero͑i.e.,␦x tends to zero͒,is
ץ͑A ͒ץt ϩץ͑AV ͒
ץx
ϭ0(6)
Equation ͑6͒provides the conrvative form of the area-averaged
mass balance equation for 1D unsteady and compressible fluids in a flexible pipe.The first and cond terms on the left-hand side of Eq.͑6͒reprent the local change of mass with time due to the combined effects of fluid compressibility and pipe elasticity and the instantaneous mass flux,respectively.Equation ͑6͒can be re-written as follows:
1D Dt ϩ1A DA Dt ϩץV
ץx
ϭ0or
1A D A Dt ϩץV
ץx
ϭ0
(7)
where D /Dt ϭץ/ץt ϩV ץ/ץx ϭsubstantial ͑material ͒derivative in
one spatial dimension.Realizing that the density and pipe area vary with pressure and using the chain rule reduces Eq.͑7͒to the following:
1d dP DP Dt ϩ1A dA dP DP Dt ϩץV
ץx
ϭ0
or
1a 2DP Dt ϩץV
ץx
ϭ0
(8)
where a Ϫ2ϭd /dP ϩ(/A )dA /dP .The historical development and formulation of the acoustic wave speed in terms of fluid and pipe properties and the assumptions involved in the formulation are discusd in Sec.3.
The momentum equation for a control volume is ͑e.g.,͓20–23͔͒:
͚
F ext ϭ
ץץt
͵
cv
v ᭙ϩ
͵
cs
v ͑v "n ͒dA (9)
Applying Eq.͑9͒to the control volume of Fig.2;considering
gravitational,wall shear and pressure gradient forces as externally applied;and taking the limit as ␦x tends to zero gives the follow-ing local form of the axial momentum equation:
ץAV ץt ϩץAV 2ץx ϭϪA ץP
ץx
ϪD w Ϫ␥A sin ␣(10)
where ␥ϭg ϭunit gravity force,␣ϭangle between the pipe and the horizontal direction,ϭ͐A u 2dA /V 2ϭmomentum correction coefficient.Using the product rule of differentiation,invoking Eq.͑7͒,and dividing through by A gives the following nonconr-vative form of the momentum equation:
ץV ץt ϩV ץV ץx ϩ1A ץ͑Ϫ1͒AV 2ץx ϩ1ץP ץx ϩg sin ␣ϩw D A ϭ0
(11)
Equations ͑8͒and ͑11͒govern unidirectional unsteady flow of a compressible fluid in a flexible tube.Alternative derivations of Eqs.͑8͒and ͑11͒could have been performed by applying the unidirectional and axisymmetric assumptions to the compressible Navier-Stokes equations and integrating the resulting expression with respect to pipe cross-ctional area while allowing for this area to change with pressure.
In practice,the order of magnitude of water hammer wave speed ranges from 100to 1400m/s and the flow velocity is of order 1to 10m/s.Therefore,the Mach number,M ϭU 1/a ,in water-hammer applications is often in the range 10Ϫ2–10Ϫ3,where U 1ϭlongitudinal velocity scale.The fact that M Ӷ1in wa-ter hammer was recognized and ud by Allievi ͓9,10͔to further simplify Eqs.͑8͒and ͑11͒.The small Mach number approxima-tion to Eqs.͑8͒and ͑11͒can be illustrated by performing an order of magn
itude analysis of the various terms in the equations.To this end,let 0aU 1ϭwater hammer pressure scale,0ϭdensity of the fluid at the undisturbed state,and T ϭL /a ϭtime scale,where L ϭpipe length,X ϭaT ϭL ϭlongitudinal length scale,ϭa positive real parameter,f U 12/8ϭwall shear scale,and f ϭDarcy-Weisbach friction factor T d ϭradial diffusion time scale.The parameter allows one to investigate the relative magnitude of the various terms in Eqs.͑8͒and ͑11͒under different time scales.For example,if the order of magnitude of the various terms in the mass momentum over a full wave cycle ͑i.e.,T ϭ4L /a )is desired,is t to 4.Applying the above scaling to Eqs.͑8͒and ͑11͒gives
0DP *Dt ϩץV *
ץx *
ϭ0
or
0ͩ
ץP *ץt *ϩM V *ץP *ץx *ͪ
ϩץV *
ץx *
ϭ0
(12)
ץV *ץt *ϩM V *ץV *ץx *ϩM 1A ץ͑Ϫ1͒AV *2ץx *ϩ0ץP *ץx *
ϩ
g L
Ua sin ␣ϩL D M f 2
w *ϭ0(13)
where the superscript *is ud to denote dimensionless quantities.
Since M Ӷ1in water hammer applications,Eqs.͑12͒and ͑13͒become
0ץP *ץt *ϩץV *
ץx *
ϭ0
(14)
ץV *ץt *ϩ0ץP *ץx *ϩg L Ua sin ␣ϩL D M f
2ϩͩT d L /a
ͪ
w *ϭ0.(15)Rewriting Eqs.͑14͒and ͑15͒in dimensional form gives
1a 2ץP ץt ϩץV
ץx
ϭ0
(16)ץV ץt ϩ1ץP ץx ϩg sin ␣ϩw D A
ϭ0(17)
Using the Piezometric head definition ͑i.e.,P /g 0ϭH ϪZ ),Eqs.͑16͒and ͑17͒
become
中卫市安全教育平台
Fig.1Control volume diagram ud for continuity equation
derivation
Fig.2Control volume diagram ud for momentum equation
derivation
g0a2ץH
ץtϩ
ץV
ץxϭ0(18)
ץV
ץtϩg 0
学习感受ץH
ץxϩ
wD
Aϭ0(19)
The change in density in unsteady compressibleflows is of the order of the Mach number͓11,27,28͔.Therefore,in water hammer problems,where MӶ1,Ϸ0,Eqs.͑18͒and͑19͒become
g a2ץH
ץtϩ
ץV
ץxϭ0(20)
ץV
ץtϩg ץH
ץxϩ
wD
Aϭ0(21)
which are identical to the classical1D water hammer equations given by Eqs.͑2͒and͑3͒.Thus,the classical water hammer equa-tions are valid for unidirectional and axisymmetricflow of a com-pressiblefluid in aflexible pipe͑tube͒,where the Mach number is very small.
According to Eq.͑15͒,the importance of wall shear,w,de-pends on the magnitude of the dimensionles
s parameter⌫ϭL M f/2DϩT d/(L/a).Therefore,the wall shear is important when the parameter⌫is order1or larger.This often occurs in applications where the simulation time far exceeds thefirst wave cycle͑i.e.,large͒,the pipe is very long,the friction factor is significant,or the pipe diameter is very small.In addition,wall shear is important when the time scale of radial diffusion is larger than the wave travel time since the transient-induced large radial gradient of the velocity does not have sufficient time to relax.It is noted that T d becomes smaller as the Reynolds number increas. The practical applications in which the wall shear is important and the variousw models that are in existence in the literature are discusd in Sec.4.
If⌫is significantly smaller than1,friction becomes negligible andw can be safely t to zero.For example,for the ca L ϭ10,000m,Dϭ0.2m,fϭ0.01,and Mϭ0.001,and T d/(L/a)ϭ0.01the condition⌫Ӷ1is valid whenӶ4.That is,for the ca considered,wall friction is irrelevant as long as the simulation time is significantly smaller than4L/a.In general,the condition ⌫Ӷ1is satisfied during the early stages of the transient͑i.e.,is small͒provided that the relaxation͑diffusion͒time scale is smaller than the wave travel time L/a.In fact,it is well known that waterhammer models provide results that are in reasonable agree-ment with experimental data during thefirst wave cycle irrespec-tive of the wall shear stress formula being ud͑e.g.,͓29–32͔͒. When⌫Ӷ1,the classical waterhammer model,given by Eqs.͑20͒and͑21͒,becomes
g a2ץH
ץtϩ
ץV
ets是什么意思ץxϭ0(22)
ץV
ץtϩg ץH
巴仙ץxϭ0(23)
which is identical to the model thatfirst appeared in Allievi͓9,10͔.
The Joukowsky relation can be recovered from Eqs.͑22͒and ͑23͒.Consider a water hammer moving upstream in a pipe of length L.Let xϭLϪat define the position of a water hammer front at time t and consider the interval͓LϪatϪ⑀,LϪatϩ⑀͔, where⑀ϭdistance from the water hammer front.Integrating Eqs.͑22͒and͑23͒from xϭLϪatϪ⑀to xϭLϪatϩ⑀,invoking Leib-nitz’s rule,and taking the limit as⑀approaches zero gives
战役的意思是什么
⌬HϭϪa⌬V
g
(24)
Similarly,the relation for a water hammer wave moving down-stream is⌬Hϭϩa⌬V/g.3Water Hammer…Acoustic…Wave Speed
The water hammer wave speed is͑e.g.,͓8,20,23,33,34͔͒,
1
a2
ϭ
d
dP
ϩ
A
dA
dP
(25)
Thefirst term on the right-hand side of Eq.͑25͒reprents the effect offluid compressibility on the wave speed and the cond term reprents the effect of pipeflexibility on the wave speed.In fact,the wave speed in a compressiblefluid within a rigid pipe is obtained by tting dA/dPϭ0in Eq.͑25͒,which leads to a2ϭdP/d.On the other hand,the wave speed in an incompressible fluid within aflexible pipe is obtained by tting d/dPϭ0in ͑25͒,which leads to a2ϭAdP/dA.
Korteweg͓33͔related the right-hand side of Eq.͑25͒to the material properties of thefluid and to the material and geometrical properties of the pipe.In particular,Korteweg͓33͔introduced the fluid properties through the state equation dP/dϭK f/,which was already well established in the literature,where K fϭbulk modulus of elasticity of thefluid.He ud the elastic theory of continuum me
chanics to evaluate dA/dP in terms of the pipe radius,thickness e,and Young’s modulus of elasticity E.In his derivation,he͑i͒ignored the axial͑longitudinal͒stress in the pipe͑i.e.,neglected Poisson’s effect͒and͑ii͒ignored the inertia of the pipe.The assumptions are valid forfluid transmission lines that are anchored but with expansion joints throughout.With as-sumptions͑i͒and͑ii͒,a quasi-equilibrium relation between the pressure force per unit length of pipe DdP and the circumferential ͑hoop͒stress force per unit pipe length2edis achieved,where ϭhoop stress.That is,DdPϭ2edor dpϭ2ed/D.Using the elastic stress-strain relation,dAϭdD2/2,where dϭd/Eϭradial͑lateral͒strain.As a result,AdP/dAϭeE/Dand
1
a2
ϭ
K f
ϩ
E
e
D
or a2ϭ
K f
1ϩ
K f D
eE
(26)
The above Korteweg formula for wave speed can be extended to problems where the axial stress cannot be neglected.This is accomplished through the inclusion of Poisson’s effect in the stress-strain relations.In particular,the total strain becomes dϭd/EϪp dx/E,wherepϭPoisson’s ratio andxϭaxial stress.The resulting wave speed formula is͑e.g.,͓17,23͔͒
a2ϭ
K f
1ϩc
K f D
eE
(27)
where cϭ1Ϫp/2for a pipe anchored at its upstream end only, cϭ1Ϫp2for a pipe anchored througho
ut from axial movement, and cϭ1for a pipe anchored with expansion joints throughout, which is the ca considered by Korteweg͑i.e.,xϭ0).
Multipha and multicomponent water hammerflows are com-mon in practice.During a water hammer event,the pressure can cycle between large positive values and negative values,the mag-nitudes of which are constrained at vapor pressure.Vapor cavities can form when the pressure drops to vapor pressure.In addition, gas cavities form when the pressure drops below the saturation pressure of dissolved gas.Transientflows in pressurized or sur-charged pipes carrying diment are examples of multicomponent water hammerflows.Unsteadyflows in pressurized or surcharged wers are typical examples of multipha and multicomponent transientflows in clod conduits.Clearly,the bulk modulus and density of the mixture and,thus,the wave speed are influenced by the prence of phas and components.The wave speed for mul-tipha and multicomponent water hammerflows can be obtained
by substituting an effective bulk modulus of elasticity K e and an effective densitye in place of K f andin Eq.͑27͒.The effective quantities,K e ande,are obtained by the weighted average of the bulk modulus and density of each component,where the partial volumes are the weights͑e,͓23͔͒.While the resulting math-ematical expression is simple,the explicit evaluation of the wave speed of the mixture
is hampered by the fact that the partial vol-umes are difficult to estimate in practice.
Equation͑27͒includes Poisson’s effect but neglects the motion and inertia of the pipe.This is acceptable for rigidly anchored pipe systems such as buried pipes or pipes with high density and stiff-ness,to name only a few.Examples include major transmission pipelines like water distribution systems,natural gas lines,and pressurized and surcharged werage force mains.However,the motion and inertia of pipes can become important when pipes are inadequately restrained͑e.g.,unsupported,free-hanging pipes͒or when the density and stiffness of the pipe is small.Some ex-amples in which a pipe’s motion and inertia may be significant include fuel injection systems in aircraft,cooling-water systems, unrestrained pipes with numerous elbows,and blood vesls.For the systems,a fully coupledfluid-structure interaction model needs to be considered.Such models are not discusd in this paper.The reader is instead directed to the recent excellent review of the subject by Tijsling͓35͔.
4Wall Shear Stress Models
It was shown earlier in this paper that the wall shear stress term is important when the parameter⌫is large.It follows that the modeling of wall friction is esntial for practical applications t
hat warrant transient simulation well beyond thefirst wave cycle͑i.e., large͒.Examples include͑i͒the design and analysis of pipeline systems,͑ii͒the design and analysis of transient control devices,͑iii͒the modeling of transient-induced water quality problems,͑iv͒the design of safe and reliablefield data programs for diag-
nostic and parameter identification purpos,͑v͒the application of transient models to invertfield data for calibration and leakage detection,͑vi͒the modeling of column paration and vaporous cavitation and͑vii͒systems in which L/aӶT d.Careful modeling of wall shear is also important for long pipes and for pipes with high friction.
4.1Quasi-Steady Wall Shear Models.In conventional transient analysis,it is assumed that phenomenological expres-sions relating wall shear to cross-ctionally averaged velocity in steady-stateflows 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 a transient.For example,the form of the Darcy-Weisbach equation ud in water hammer models is͑Streeter and Wylie͓36͔͒
w͑t͒ϭwsϭ
红烧肥肠
f͑t͉͒V͑t͉͒V͑t͒请假条的正确格式
8
(28)
wherews(t)ϭquasi-steady wall shear as a function of t.
The u of steady-state wall shear relations in unsteady prob-lems is satisfactory for very slow transients—so slow,in fact,that they do not properly belong to the water hammer regime.To help clarify the problems with this approach for fast transients,con-sider the ca of a transient induced by an instantaneous and full closure of a valve at the downstream end of a pipe.As the wave travels upstream,theflow rate and the cross-ctionally averaged velocity behind the wave front are zero.Typical transient velocity profiles are given in Fig.3.Therefore,using Eq.͑28͒,the wall shear is zero.This is incorrect.The wave passage creates aflow reversal near the pipe wall.The combination offlow reversal with the no-slip condition at the pipe wall results in large wall shear stress.Indeed,discrepancies between numerical results and ex-perimental andfield data are found whenever a steady-state bad shear stress equation is ud to model wall shear in water hammer problems͑e.g.,͓25,30,32,37,38͔͒.
Letwu(t)be the discrepancy between the instantaneous wall shear stressw(t)and the quasi-steady contribution of wall shear stressws(t).Mathematically
w͑t͒ϭws͑t͒ϩwu͑t͒(29)wu(t)is zero for steadyflow,small for slow transients,and sig-nificant for fast transients.The unsteady friction component at-tempts to reprent the transient-induced changes in the velocity profile,which often involveflow reversal and large gradients near the pipe wall.A summary of the various models for estimating wu(t)in water hammer problems is given below.
4.2Empirical-Bad Corrections to Quasi-Steady Wall Shear Models.Daily et al.͓39͔conducted laboratory experi-ments and foundwu(t)to be positive for acceleratingflows and negative for deceleratingflows.They argued that during accelera-tion the central portion of the stream moved somewhat so that the velocity profile steepened,giving higher shear.For constant-diameter conduit,the relation given by Daily et al.͓39͔can be rewritten as
K uϭK sϩ2c2
L
V2
ץV
ץt(30) where K uϭunsteadyflow coefficient of boundary resistance and momentumflux of absolute local velocity and K sϭf L/Dϭsteady state resistance coefficient.Daily et al.͓39͔noted that the longi-tudinal velocity and turbulence nonuniformities are negligible and K uϷKϭF/AV2/2ϭunsteadyflow coefficient of boundary resis-tance,where Fϭ2DLwϭwall resistance force.Therefore,Eq.͑30͒
becomes Fig.3Velocity profiles for steady-state and af-ter wave passages
