This article was downloaded by: [University of Sydney]

On: 22 March 2013, At: 00:55

Publisher: Taylor & Francis
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,
37-41 Mortimer Street, London W1T 3JH, UK

Urban Water Journal

Publication details, including instructions for authors and subscription information:
http://www.tandfonline.com/loi/nurw20

On the fundamental equations of water hammer

a
MOHAMED S. GHIDAOUI
a
Department of Civil Engineering Hong Kong University of Science and Technology Clear
Water Bay, Kowloon Hong Kong
Version of record first published: 09 Aug 2010.

To cite this article: MOHAMED S. GHIDAOUI (2004): On the fundamental equations of water hammer, Urban Water Journal,
1:2, 71-83

To link to this article: http://dx.doi.org/10.1080/15730620412331290001

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantial or systematic
reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to
anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contents
will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should
be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims,
proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in
connection with or arising out of the use of this material.
 Urban Water Journal, Vol. 1, No. 2, June 2004, 71 – 83

On the fundamental equations of water hammer

MOHAMED S. GHIDAOUI*

Department of Civil Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon,
Hong Kong

(Revised version received June 2004)

The wide spectrum of practical problems that warrant water-hammer modelling has
increased the importance of careful formulation of the fundamental equations of water
hammer and critical analysis of their assumptions. To this end, this paper reviews the
Downloaded by [University of Sydney] at 00:55 22 March 2013

relation between state equations and wave speeds in single as well as multiphase and
multicomponent transient flows, formulates the various forms of one- and two-
dimensional water-hammer equations and illuminates the assumptions inherent in these
equations. The derivation of the one- and two-dimensional water-hammer equations
proceeds from the three-dimensional Navier – Stokes equations for a compressible fluid.
The governing equations for turbulent water-hammer flows are obtained by applying
ensemble averaging to the simplified form of the Navier – Stokes equations for water-
hammer problems. Unlike time averaging, ensemble averaging is applicable to unsteady
flows where the time scale of the transient is often much smaller than the time scales of
the turbulence. Order of magnitude analysis, physical understanding and recent research
findings are used throughout the paper to evaluate the accuracy of the assumptions made
in the derivation of the one- and two-dimensional water-hammer equations.

Keywords: Unsteady flow; Transient flow; Compressible flow; Water hammer; Funda-
mental equations; Wave speed

protective devices so as to reduce the risk of pipe ruptures,

1. Introduction
cavitation, pitting and corrosion, damage to other hydrau-
The scientific study of transient fluid flow has been lic devices and fire related damages in the case of natural
undertaken since the middle of the nineteenth century. As gas pipelines. In addition, water-hammer models are
is true of every other area of engineering research, a great required for assessing and mitigating transient induced
many advances have been made in the accuracy of analysis hazards to human health associated with the intrusion of
and the range of applications since then. Although only a contaminants through cracks and joints during low
few simple problems were approachable by analytical pressure cycles and the sloughing off of biofilm and re-
methods and the earlier approximate techniques, a much suspension of particulates during transient events. In
broader spectrum of transient problems could be solved addition, transient models are being developed for the
once graphical methods were developed. More recently, the purpose of improving the reliability of inverse transient
application of digital computing techniques has resulted in models. This is very significant since inverse models have
a rapid increase in the range and complexity of problems the potential to utilize field measurements of transient
being studied by water-hammer models. For example, events to accurately and inexpensively calibrate a wide
transient models are becoming widely used in the design range of hydraulic parameters, including pipe friction
and analysis of complex pipeline systems and their factors, system demands and leakage. Traditionally, such

*Corresponding author. Email: ghidaoui@ust.hk

Urban Water Journal

ISSN 1573-062X print/ISSN 1744-9006 online # 2004 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/15730620412331290001
 72 M. S. Ghidaoui

information could only be obtained through costly field

measurements of flow and pressures conducted on a few
individually sampled pipes in the system. Transients, on the
other hand, traverse the entire system, interacting with each
pipe or device in the system. Thus, they embody large
amounts of information about the system’s physical
characteristics. Inverse transient analysis techniques are
now being developed to decode this information for
hydraulic model calibration as well as to identify and Figure 1. Transient flow in an open channel.
locate system leakage, closed or partially closed valves and
damaged pipes (see also Bruno and Ferente 2004, Kapelan
et al. 2004). The potential annual savings in routine
data collection costs for water supply utilities are
enormous.
The existence of a wide range of complex practical
problems that warrant water-hammer modelling along with
the significant advances in fluid mechanics and the
prevalence of computer power and numerical techniques
Downloaded by [University of Sydney] at 00:55 22 March 2013

has, if anything, increased the importance of critical

analysis of both the fundamental equations of water
hammer and the numerical solutions of these equations. Figure 2. Transient flow in a pressurized pipe.
This paper focuses on the derivation of the various one-
and two-dimensional fundamental equations of pipe Following sudden gate closure, the channel stores water
transients. In particular, the relation between state equa- by a change in water depth and an increase in cross-
tions and wave speeds in single as well as multiphase and sectional area by an amount DAc. A bore is generated and
multicomponent transient flows is reviewed and the various moves upstream at a speed of c. The mass stored in the
forms of one- and two-dimensional water-hammer equa- channel after a time Dt is rDAccDt. This mass is equal to the
tions are derived. Special attention is given to the mass inflow at the upstream rQDt. Therefore, mass balance
assumptions and restrictions involved in various governing dictates that:
equations so as to illuminate both their range of applic-
Q U
ability and their limitations. Understanding the governing rQDt ¼ rDAcCDt or c ¼ ¼ ð1Þ
equations that are in use in water-hammer research and DAc DAc
A
practice and their limitations is essential for: (i) interpreting
the results of the numerical models that are based on these For the case of pressurized flow, the pipe stores water by
equations; (ii) judging the reliability of the data obtained a change in pipe area and a change in fluid density. The
from these models; (iii) minimizing misuse of water- valve closure generates a shock wave that moves upstream
hammer models in both research and practice; and (iv) at a speed of a. The additional mass stored in the pipe after
delineating the contribution of physical processes from the a time Dt is (r + Dr)(A + DAp)aDt – rAaDt & (ADr +
contribution of numerical artefacts to the results of water- rDAp)aDt. This mass is equal to the mass inflow at the
hammer models. upstream rQDt. Therefore, mass balance dictates that:

rQDt ¼ ðADr þ rDAp ÞaDt or

2. State equations: bulk modulus and wave speed Q U ð2Þ
a¼ ¼ Dr
An analogy between a transient flow in an open channel ðADr þ rDAp Þ DA A þ r
c

and a transient flow in a pipe is useful for developing

insight into the water-hammer wave speed. Figure 1 shows Clearly, the mass stored due a change in pipe area, DAp,
a transient in an open channel generated by a sudden and and water density, Dr, is much smaller than the mass stored
complete closure at a downstream gate, while figure 2 in a channel due to a change in water depth. That is,
shows a transient in a pipe generated by a sudden and
complete closure of a valve of the downstream end. The DAp Dr DAc
þ
or a  c ð3Þ
initial steady state flow rate (Q) and velocity (U) are A r A
assumed to be the same in both the channel and the pipe.
Therefore, the initial cross sectional area of the flow in the To make an important connection, the related problems
channel A is the same as that of the pipe. in sewer surcharging are commonly modelled using so-
 On the fundamental equations of water hammer 73

called Preissman slot theory. A hypothetical slot in the pipe where Kf = bulk modulus of elasticity of the fluid and
is chosen so that the speed of gravity waves in the slot is Ks = bulk modulus of elasticity of the pipe walls. The
equal to the water-hammer wave speed and the associated expressions in equation (5) represent stress – strain rela-
water level h is equal to the water-hammer head. That is, tions, where dP is the change in stress, and dr/r and dA/A
DA
the size of the slot is chosen so that A p þ Dr DAc
r ¼ A , implying are the bulk strains associated with dP. Both r and A
that DAc is very small. Let w denote the width of the slot, increase with increasing P and vice versa. That is, r and A
DAc = wh or w = DAc/h. Noting that h must be very large if are monotonic functions of P, implying dP/dr = Kf/r 4 0
the water depth in the slot is to be equivalent to the water- and Ar dP/dA=Ks/r4 0. In addition, Kf /r and Ks/r have
pffiffiffiffiffiffiffiffiffiffi
hammer pressure, combined with the fact that DAc is very the dimension of velocity squared. In fact, Kf =r ¼ af is
small, results in a slot of extremely small width containing the wave speed of a flow disturbance in a compressible fluid
very deep water. The combination of large water level and bounded by an infinitely rigid conduit (Chaudhry pffiffiffiffiffiffiffiffiffiffiffi1987,
small width results in numerical instabilities (e.g. Yen 1986, Wylie and Streeter 1993). On the other hand, Ks=r ¼ as
Cardle et al. 1988). Such instabilities can be suppressed by is the wave speed of a flow disturbance in an incompressible
making the slot wider. However, changing the width of the fluid bounded by a flexible conduit (Lighthill 1996). In
slot destroys the equivalence between the water hammer water-hammer applications, the wave speed (a) generally
and the free surface flow equations and predicts incorrect depends on both the fluid compressibility and pipe elasticity
wave speeds and pressure heads (i.e. an incorrect distribu- (i.e. a = a (Kf /r, Ks/r)). The functional dependence of the
tion of mass and momentum). In addition, there is no wave speed on the bulk modulus of the fluid and pipe can
Downloaded by [University of Sydney] at 00:55 22 March 2013

guidance as to what is an acceptable minimum slot, one be obtained as follows. Let dA/A + dr/r define the bulk
that will avoid numerical instability without producing a strain of the fluid-pipe system induced by a change in
large departure from water-hammer theory. pressure dP. Stress – strain relationships dictate that the
change in pressure is equal to the product of the bulk strain
and the effective bulk modulus of the fluid-pipe system (Ke).
2.1 Single phase and single component flow
That is,
Flow disturbances in pipeline systems induce changes in
velocity, pressure, density and pipe area. State equations, dA dr dðrAÞ
dP ¼ Keð þ Þ ¼ Ke ð6Þ
which relate density (r) and pipe area (A) to pressure (P), A r rA
are as follows (Lighthill 1996):
Re-writing equation (6) gives:
r ¼ rðPÞ and A ¼ AðPÞ ð4Þ
drA
1 r
dA
A þ dr
r 1 1 Ks þ Kf
Note the absence of temperature from the above state ¼ ¼ ¼ þ ¼ ¼
Ke0 dP dP Kf Ks Ks Kf
equations. In practice, the heat flux between the water in ð7Þ
1 a2f Kf dA=A
the pipe and the surrounding soil is negligible since the pipe or ¼1þ ¼1þ
has relatively adiabatic walls. In addition, the fact that ra2 a2 Ks r=r
water has a large specific heat value implies that changes in
its temperature, induced by the conversion of mechanical In practice, Kf is a measurable quantity. Values of Kf are
energy into internal energy through the action of viscosity, available in numerous books and monographs of fluid
are negligible. Moreover, the energy loss and change in mechanics and hydraulics. In addition, Ks can be evaluated
both temperature and entropy across a shock wave are of by relating dP/dA/A to known (measurable) pipe properties
the order of the Mach number (e.g. Walker 1975, Courant such as its diameter (D), wall thickness (e) and Young’s
and Friedrichs 1976). In water hammer, the Mach number modulus of elasticity (E) using stress – strain relationships.
is often of order 10 7 3, making the changes in energy, For example, for a thick-walled elastic pipe possessing
entropy and temperature across a water hammer shock expansion joints throughout it length, it can be shown that
wave negligible (i.e. the flow across a water-hammer shock Ks = Ee/D (e.g. see Chaudhry 1987, Wylie and Streeter
is isentropic). Therefore, water-hammer flows are charac- 1993, McInnis et al. 1999; Larock et al. 2000). In fact, it is
terized by time-independent and uniform temperature field, through Ks that the interaction between the fluid and pipe is
thus, the absence of this thermodynamic variable from incorporated.
equation (4).
Writing equation (4) in a differential form gives:
2.2 Multiphase and multicomponent flow
dP Kf A dP Ks Multiphase and multicomponent transient flows in pipes
¼ and ¼ ð5Þ
dr r r dA r occur in numerous engineering applications. For example,
cavitating transient flow is a typical multiphase problem
 74 M. S. Ghidaoui

that is often encountered in water-hammer applications. In Approaches for including viscoelastic and pipe motion
addition, transient flows in pressurized or surcharged effects in water-hammer models are reviewed in Tijsseling
sediment carrying pipes are examples of multicomponent (1995).
water-hammer flows. Moreover, unsteady flows in pressur-
ized or surcharged sewers are typical examples of
3. Fundamental equation of water hammer: Joukowsky
multiphase and multicomponent transient flows in closed
relation
conduits. Again, assuming transient flows are isothermal,
the state equation of the ith phase or component is as Consider a steady flow of a compressible fluid in an elastic
follows: pipe. Flow disturbances caused by accidental or normal
pump shutdowns, pump starts or rapid changes in valve
dP Ki
ri ¼ ri ðPÞ or ¼ ; for all i ð8Þ settings trigger a series of positive and negative pressure
dri ri and velocity waves. The first fundamental relation for
estimating the magnitude of these so-called water-hammer
where ri = density of the ith component or phase. In or surge waves was formulated by Joukowsky in 1898
equation (8), P does not carry an index, implying that the (Rouse and Ince 1957) although it appears that this result
pressure is assumed to be uniform across the different was earlier derived by Von Kries. Therefore, it is both
phases and components within a pipe cross-section. This fitting and instructive that a paper on the fundamental
assumption is often adopted in multiphase and multi- equations of water hammer begins by deriving one of the
Downloaded by [University of Sydney] at 00:55 22 March 2013

component transient flows (e.g. Chaudhry 1987, Wylie and earliest fundamental results.
Streeter 1993, McInnis et al. 1999, Larock et al. 2000). In In his formulation, Joukowsky applied the mass and
practice, this assumption is justified because the magnitude momentum principles to one-dimensional flow in a pipe
of the surface tension is negligible in comparison to the and assumed that the fluid and pipe are homogenous and
water-hammer pressure. that the effects of friction are negligible. Adopting
The change in volume and the bulk strain of the ith Joukowsky’s assumptions and using a control volume that
component or phase are related as follows: moves with the wave front (see figure 3), the equations
d8i =8i ¼ dri =ri . As a result, the change in the volume governing this unsteady flow problem can be formulated
of the mixture d8m ¼ Si 8i dri =ri . Therefore, using steady state theory. In particular, applying the steady
mass balance equation to the control volume of figure 3
 d8
8m
m X 8i dr
ri
i
1 X 8i 1 gives:
¼ or ¼ ð9Þ
dP 8m dP Km 8m Ki
i i
m_ in ¼ m_ out ! rAðU þ aÞ ¼ ðr þ DrÞðA þ DAÞðU þ DU þ aÞ
where rm and Km = density and bulk modulus of the ð10Þ
mixture, respectively. The wave speed of the mixture is
obtained by inserting Km in place of Kf and rm in place of r where m _ in = rate of mass influx in the control volume;
in equation (7). The resulting equation of the mixture wave m_ out = rate of mass outflow from the control volume;
speed is generally in good agreement with available r = density; A = cross-sectional area; a = wave speed;
experimental data (Wylie and Streeter 1993). The mixture U = cross-sectional average velocity; DU, Dr and DA =
wave speed is highly sensitive to the relative values of the wave-induced change in velocity, density and area,
volume fractions. For example, the presence of 1 part of air respectively. The above continuity equation can be re-
to 10 000 parts of water can reduce the value of wave speed written as follows:
by about 50%. Therefore, reliable estimates of the wave
speed in multiphase and multicomponent water hammer
require accurate quantification of the volume fractions in
these flows. In practice, however, the values of the volume
fractions in multiphase and multicomponent flows (e.g.
sewer flows) are often not available. In such cases, extensive
sensitivity analysis of water hammer results to the value of
wave speed is essential.
The wave speed relations given here assume that the
pipe material is elastic and that the pipe stress – strain
relation under an unsteady load is the same as that
obtained under a static load. These assumptions are not
valid in viscoelastic pipes and in pipes with significant Figure 3. Control volume extended to the pipe wall and
motion and inertia (e.g. unsupported, free-hanging pipes). moving with the wave front.
 On the fundamental equations of water hammer 75

DP ¼ raDU ð16Þ
Dr DA DADr DU
þ þ ¼ ð11Þ
r A Ar U þ DU þ a The negative sign in equation (16) arises from the force
balance in the control volume diagram (see figure 3), where
Applying the momentum principle to the control volume in SFx = – ADP. If the disturbance had occurred upstream
figure 3 and neglecting the shear stresses gives: (while keeping the same sign convention of x positive in the
X downstream direction), equation (16) would become
Fx ¼PA  ðP þ DPÞðA þ DAÞ ¼ SFx = + ADP while the sign of the net momentum flux
ð12Þ
m_ fðU þ DU þ aÞ  ðU þ aÞg ¼ mDU_ would be unchanged. Thus, in general, the momentum
equation is
where P = initial (steady state) pressure. Using the
continuity equation, m_ ¼ m_ in ¼ m_ out ¼ rða þ UÞA equa- DP ¼ raDU ð17Þ
tion (12) becomes:
which is the fundamental equation of water hammer or, in
PA  ðP þ DPÞ ðA þ DAÞ ¼ rA ðU þ aÞ DU ð13Þ honour of the person usually credited with first deriving it,
the Joukowsky equation (relation). Note that the Joukows-
Equations (11) and (13) are applicable to one-dimensional ky relation can be re-written as DP/rg = DH = + aDUg,
flow of a compressible fluid in a frictionless pipe. In typical where DH = change in piezometric head and g = gravita-
Downloaded by [University of Sydney] at 00:55 22 March 2013

water hammer applications, the fluid (water) is only slightly tional acceleration.
compressible and the pipe is only slightly deformable. That The Joukowsky relation and its formulation constitute
is, jDr/rj
1 and jDA/Aj
1. Accordingly, DADr/ the ideal starting point for introducing and elucidating
Ar
jDA/Aj
1 and DADr/Ar
jDr/rj
1; thus, equa- water-hammer phenomena to the novice. In addition, this
tion (11) becomes: relation is useful for obtaining a first approximation of the
transient response of a pipe system. Essentially, the
Dr DA DU Joukowsky equation is used to estimate the head rise in a
þ ¼ ð14Þ
r A U þ DU þ a pipe for a given change in velocity. However, this equation
can neither resolve the dynamics of transient waves in pipe
Although Dr/r and DA/A are small, their presence in systems and how these waves interact with the various
equation (14) is essential and provides the essence to the hydraulic and control devices nor estimate the effects of
wave nature of water-hammer flows; in particular, these wall shear on water-hammer waves. To overcome these
terms provide the mechanism of mass storage by fluid limitations, more complete water-hammer equations have
compressibility and pipe elasticity. Therefore, neglecting been formulated. The remainder of this paper is devoted to
them amounts to treating the fluid as incompressible and the derivation and discussion of these more complete water-
the pipe as rigid and results in rigid water hammer hammer equations.
theory. Indeed, although Dr/r and DA/A are small they
must be maintained in the continuity equation for they
4. Navier – Stokes equations for compressible flows
balance the equally small but physically important term
(NSCF)
of DU/(U + DU + a). Together, the three terms in
equation (14) are what gives transient pipe flows their The derivations of one- and two-dimensional flow water-
wave character. hammer equations often proceed by applying the mass
While all the terms in equation (14) must be maintained and momentum principles to a control volume. In this
if the resulting model is to describe water-hammer waves, paper, however, the derivation proceed from the 3-D
the small magnitude of these terms allows for further Navier – Stokes equations. This approach is chosen
simplifications of the governing equation. In particular, because it allows the explicit identification of the
consistent with DU/(U + DU + a) being small is the fact successive approximations needed to arrive at the various
that U
a and DU
a. For example, in water-hammer published forms of the water-hammer equations. Never-
applications, both U and DU are typically of the order of theless, it must be recalled that the 3-D Navier – Stokes
1 m/s while a is of the order of 1000 m/s. Using the equations are themselves derived from the control volume
inequalities U
a, jDUj
a and jDA j
A simplifies approach, where the physical laws are first applied to a
equations (14) and (12) as follows: control volume of finite size, and then the limit of the
resulting difference equations as the size of this fluid
element becomes infinitesimal but without violating the
Dr DA DU continuum hypothesis is performed (e.g. Currie 1993,
þ ¼ ð15Þ
r A a Batchelor 1994, Liggett 1994). The three-dimensional
 76 M. S. Ghidaoui

Navier – Stokes equations for a compressible Newtonian flows (e.g. transient flows involving column separation);
fluid in cylindrical coordinates are (iii) in multicomponent flows (e.g. transient flows in
sewers); and (iv) in instability-induced non-axisymmetric
dr dru 1 drrv 1 drw transient flows. Transient flows with non-axisymmetric
þ þ þ ¼0 ð18Þ
dt dx r dr r dy instabilities are relatively new phenomena and need to be
explained. Recent experimental and theoretical studies
dru dru2 1 drruv 1 drruw show that flow instabilities in transient flows can lead to
þ þ þ ¼
dt dx r dr  r  dy
ð19Þ the breakdown of flow symmetry with respect to the pipe
dp d 1 Dr axis. For example, Das and Arakeri (1998) performed
  ðk þ m=3Þ þ mr2 u þ rfx
dr dx r Dt unsteady pipe flow experiments, where the initial flow was
laminar and the transient event was generated by a piston.
drv druv 1 drrv2 1 drvw w2 dp They found that when the Reynolds number and the
þ þ þ r ¼ 
dt dx  r dr r dy r dr
 ð20Þ transient time scale exceed a threshold value, the flow
d 1 Dr v 2 dw becomes unstable. In addition, they observed that the flow
ðk þ m=3Þ þm r v 2 2
2
þ rfr
dr r Dt r r dy instability results in the formation of non-stationary helical
vortices and that the breakdown of these vortices into
drw druw 1 drrvw 1 drw2 vw dp turbulence is very rapid. The breakdown of the helical
þ þ þ þr ¼ 
dt dx  r dr r dy r rdy
 ð21Þ vortices into turbulence resulted in strong asymmetry in the
Downloaded by [University of Sydney] at 00:55 22 March 2013

d 1 Dr w 2 dv flow with respect to the pipe axis. Das and Arakeri (1998)
ðk þ m=3Þ þ m r2 w  2 þ 2 þ rfy
rdy r Dt r r dy applied linear stability analysis to unsteady plane channel
flow to explain the experimentally observed instability in
where x = longitudinal distance along pipe centreline; unsteady pipe flow, achieving good qualitative agreement
r = radial distance from pipe centreline; y = azimuthal with experimental results. Brunone et al. (2000) carried out
coordinate; t = time; fx = body force along x; fr = body measurements of water hammer velocity profiles in
force along r; fy = body force along y; u = velocity turbulent flows. They also observed strong flow asymmetry
component along x; v = velocity component along r; w = with respect to the pipe axis. In particular, they found that,
velocity component along y; m = dynamic viscosity; a short time after the wave passage, flow reversal no longer
k = second dynamic viscosity (bulk viscosity); r = density; appears simultaneously in both top and bottom sides of the
p = thermodynamic pressure; g = gravitational accelera- pipe. Instead, flow reversal appears to alternate between
tion; DtD
¼ dtd þ u dc
d
þ v drd þ wr dy
d
= material derivative in the bottom and the top of the pipe. This is consistent with
d2 d d2
cylindrical coordinates; and r2 ¼ dn 2 þ r dr ðr drÞ
1 d
dy2
= the asymmetry observed by Das and Arakeri (1998).
Laplace operator in cylindrical coordinates. Ghidaoui and Kolyshkin (2001) studied the stability of
Assuming that water-hammer flows remain axisymmetric base flow velocity profiles in laminar and turbulent water
during a transient event reduces equations (18) to (21) to: hammer flows with respect to axisymmetric and asym-
  metric modes. It is found that the asymmetric mode with
1 dr dr dr du 1 drv azimuthal wavenumber one is the least stable and that the
þu þv þ þ ¼0 ð22Þ
r dt dx dr dx r dr stability results of the laminar and the turbulent velocity
  profiles are consistent with published experimental data
du du du 1 dp k þ m=3 d du 1 drv
þu þv ¼ þ þ þ and successfully explain controversial conclusions in the
dt dx dr r dx r dx dx r dr
 2   literature. Physically, flow instabilities change the structure
d 1d du and strength of the turbulence in a pipe, result in strong
n þ r þ fx ð23Þ
dx2 r dr dr flow asymmetry and induce significant fluctuations in wall
  shear stress. These effects of flow instability are not
dv dv du 1 dp k þ m=3 d du 1 drv
þu þv ¼ þ þ þ represented in water hammer models that assume flow
dt dx dr r dr r dr dx r dr
 2    axisymmetry.
dv 1d dv v
n þ r  þ fr ð24Þ
dx2 r dr dr r2
5. NSCF for unidirectional flow and the 1-d water-hammer
equations
where v = m/r = kinematic viscosity of water. This system
is written in so-called non-conservative form. The classical derivation of the one-dimensional water-
The axisymmetric assumption does, at first, appear to be hammer equations, which proceeds by applying the
evident. However, it is important to realize that this Reynolds transport theorem to a pipe section, is given in
assumption becomes questionable: (i) near flow boundaries detail in Chaudhry (1987) and Wylie and Streeter (1993).
such as valves, pumps and pipe junctions; (ii) in multiphase Here, the derivation begins by applying the Navier – Stokes
 On the fundamental equations of water hammer 77

equations for a compressible fluid to a unidirectional flow to interchange the order differentiation, integration, and
(i.e. v = 0). As a result, equations (22), (23) and (24) assuming that there is no leak leads to
become  
dAU dbAU2 A dP k þ 4m=3 d dUA
þ ¼ þ þ
dr dr du dt dx r dx r dx dx
þu þr ¼0 ð25Þ   ð32Þ
dt dx dx dU
pDn þAfx
dr r¼D=2
   
du du 1 dp k þ 4m=3 d du nd du
þu ¼ þ þ r þ fx Ð
dt dx r dx r dx dx r dr dr where b ¼ A u2 dA=U2 = momentum coefficient. This coef-
ð26Þ ficient appears in the process of averaging the momentum
flux with respect to the pipe area and relating this average
Integrating the momentum equation with respect to the to the average of the velocity distribution. This coefficient is
pipe areas by multiplying equation (25) by 2prdr, integrat- always greater than 1.0 and b = 4/3 for steady laminar
ing from 0 to D/2, and using Leibnitz’s rule (see Chaudhry flow, b = 1 for uniform flow and b&1 for steady turbulent
1987) to interchange the order differentiation, integration pipe flows. In water-hammer flows, b is often much larger
leads to: than one. For example, following a sudden and complete
  valve closure, the velocity profile contains
Ð regions of reverse
drA drAU 1 dr dr flow, where the mean U = 0 but A u2 dA 6¼ 0. Therefore, in
þ  2rA þu ¼ 0: ð27Þ
Downloaded by [University of Sydney] at 00:55 22 March 2013

dt dx R dt dx r¼D=2 this case b tends to infinity. That is, b for unsteady pipe
flows is, in general, very large and must be included in the
The last term in equation (27) represents the mass flux at momentum equation. In practice, however, the term
the pipe wall. Such a flux is non-zero if there is a leak in the involving b is not important because the advective
pipe. If there is no leak, however, this term becomes zero acceleration in water hammer is very small (see the low
because the tangential and normal velocities at the wall are Mach number approximation section given below).
zero. In the absence of a leak, equation (27) is: Denoting g = rg = unit gravity force; tw ¼ mðdU dr Þr¼D=2
wall shear; realizing that fx = – g sina = gravitational force
drA drAU
þ ¼ 0: ð28Þ along x, where a = angle between the pipe and the
dt dx horizontal direction; using the product rule of differentia-
tion; invoking the continuity equation [i.e. equation (30)];
which is the conservative form of the area-averaged mass and dividing through by rA gives the following non-
balance equation for 1-D unsteady and compressible fluids conservative form of the momentum equation:
in a flexible pipe. The non-conservative form of the
continuity equation can be obtained by exploiting the dU dU dðb  1ÞU2 1 dP
þU þ þ þ g sinaþ
product rule of differentiation. The result is as follows: dt dx dx r dx
  ð33Þ
    twpD k þ 4m=3 d dU
¼
1 dr dr 1 dA dA dU rA r dx dx
þU þ þU þ ¼ 0: ð29Þ
r dt dx A dt dx dx

5.1 Low Mach number approximation: classical 1-D water-

Identifying the parenthesized expressions as total deriva-
hammer equations
tives, the non-conservative form of the continuity equation
can be re-written as follows: Let U1 = longitudinal velocity scale; L1 = longitudinal
length scale taken to be equal to the pipe length L;
1 dr 1 dA dU 1 drA dU
þ þ ¼ 0 or þ ¼ 0: ð30Þ r0 = density of the fluid at the undisturbed state; rf U12 /
r dt A dt dx rA dt dx 8 = wall shear scale; and f = Darcy – Weisbach friction
factor; T = zL/a = time scale of the event that generates
Referring to equation (7), d(rA)/rA = dP/ (ra2) and, thus, the transient; and z = is parameter that defines the time
equation (30) becomes scale in terms of the wave travel time L/a. This allows one
  to investigate the relative magnitude of the various terms in
1 dP dU 1 dP dp dU
þ ¼ 0 or þU þ ¼ 0: ð31Þ equations (31) and (33) under different time scales. For
ra2 dt dx ra2 dt dx dx example, the relative magnitude of the various terms in the
unsteady mass and momentum equations for rapid
Integrating the continuity equation with respect to the transient events is obtained when z is of order 1 or smaller.
pipe area by multiplying equation (26) by 2prdr, from the The pressure scale is given by estimating the change of
limits of 0 to D/2, using Leibnitz’s rule (see Chaudhry 1987) momentum of the water column in the pipe:
 78 M. S. Ghidaoui

dU 1 dP tw pD
mU1 =T r0 LAU1 =T r0 LU1 r0 aU1 þ þ g sina þ ¼ 0: ð42Þ
¼ ¼ ¼ ð34Þ dt r dx rA
A A T z
Applying the above scaling to equations (31) and (33) Using the piezometric head definition (i.e., P/gr0 = H 7Z),
gives: equations (41) and (42) become
 
r0 1 dP  dP

dU r0 dH dU
2 
þ zMU 
þ  ¼0 ð35Þ þ ¼0 ð43Þ
r z dt dx dx ra2 dt dx

dU dU dðb  1ÞU2 r0 dP dU r0 dH tw pD


þ zMU  þ zM þ þ þ þ ¼ 0: ð44Þ
dt dx dx r dx dt r dx rA
  ð36Þ
gzL zL f M d2 U
sina þ M tw ¼ z
U1 a D 2 RL dx2 where H = piezometric head and Z = elevation of pipe
centreline from a given datum. Again, noting that in water-
where M = U1/a = Mach number; RL = LU1/(k/r+4v/3) hammer problems, where M
1, r & r0, equations (43)
= a modified Reynolds number based on the pipe length; and (44) become
and the superscript * is used to denote dimensionless
1 dH dU
quantities. When the transient is slow (e.g. z around 3 or 4 þ ð45Þ
Downloaded by [University of Sydney] at 00:55 22 March 2013

or larger), equations (35) and (36) become a2 dt dx

dU dU dH tw pD
¼0 ð37Þ þ þ ¼ 0: ð46Þ
dx dt dx rA

dU r0 dP gzL zL f

þ þ sina þ M tw ¼ 0: ð38Þ which are the classical 1-D water-hammer equations.
dt r dx U1 a D 2 The system described by (45) and (46) is not closed.
This system becomes closed if either the wall shear is
which are the rigid-column water-hammer equations. neglected or an auxiliary relation for it is specified.
When the transient is rapid (i.e. z less than 2) and According to equation (40), the importance of tw
recognizing that water-hammer flows are often character- depends on the magnitude of the dimensionless para-
ized by their very small Mach number (i.e. M
1), meter zLMf=2D. If the time scale of interest is limited to
equations (35) and (36) become a very short time, say one full wave cycle (i.e. z around
4), the effects of friction are negligible if 2Lf M=D
1.
r0 dP dU For example, if M = 1073, D = 0.2 m, and f = 0.02, the
þ  ¼0 ð39Þ
r dt dx wall shear is negligible as long as L
5000 m. Choo-
sing L = 1000 m gives 2Lf M=D ¼ 0:1, which implies
dU r0 dP gzL zL f
þ þ sina þ M tw ¼ 0: ð40Þ that the other terms are about 10 times larger than the
dt r dx U1 a D 2 shear stress. This example highlights that the wall shear
is often negligible for the first wave cycle. In fact, it is
Note that the term involving the momentum correction well known that water-hammer models provide results
coefficient has been dropped from equation (40) on the basis that are in reasonable agreement with experimental data
that M
1. This is the practice in water-hammer modelling. during the first wave cycle irrespective of the wall shear
However, a note of caution is in order. The implicit stress formula being used (e.g. see Bergant and Simpson
assumption here is that b must remain small enough so 1994, Axworthy et al. 2000, Brunone et al. 2000). That is,
that the whole term is negligible. In rapid transients, the b the type of shear stress model used is inconsequential as
coefficient can be very large. Therefore, although this term long as the parameter zLM f=2D is significantly smaller
has been dropped here as is usual in water-hammer than one.
modelling, furtherer investigations are required to assess On the other hand, whenever zLM f=2D is either of
the magnitude of this term in comparison to the temporal order one or larger, a shear stress formula becomes
acceleration, pressure gradient and wall shear. essential. To this end, numerous shear stress models have
Re-writing equations (39) and (40) in dimensional form been formulated and tested in the literature (e.g. Dailey et
gives: al. 1956, Zielke 1968, Safwat and Polder 1973, Trikha
1975, Jelev 1989, Suo and Wylie 1989, Greco 1990, Vardy
1 dP dU
þ ¼0 ð41Þ et al. 1993, Brunone et al. 1995, Vardy and Brown 1995,
ra2 dt dx Shuy 1996, Silva-Araya and Chaudhry 1997, Pezzinga
 On the fundamental equations of water hammer 79

1999, Axworthy et al. 2000, Brunone el al. 2000, Ghidaoui Ghidaoui et al. (2001) found that U2/U1
1. This is not
and Mansour 2002). Current wall shear models can be surprising given that the radial motion is induced by the
broadly divided into three classes. First-class models use radial deformation of the pipe wall, non-prismatic conduits
shear stress equations, derived under steady state equa- and radial turbulence fluctuations. Therefore, equation (48)
tions (e.g. Darcy – Weisbach formula), for water-hammer becomes:
problems. Second-class models add a correction term,
1 dp k þ m=3 d2 u
proportional to the instantaneous acceleration of the flow,  þ þ fr ¼ 0 ð49Þ
to class-one models. Third-class models add a physically r0 dr r dxdr
based convolution integral of the instantaneous accelera-
tion of the flow to class-one models. The various wall Further non-dimensionalization of equation (49) gives:
shear stress formulas are discussed in the review paper by
Ghidaoui et al. (2004). dp M k=m þ 1=3 d2 u L2
 þ þ fr ¼ 0 ð50Þ
dr RL r dx dr aU1

6. NSCF and the 2-D water-hammer equations where RL = Reynolds number based on L. Since
M=RL
1, the radial momentum equation simplifies to
6.1 Plane wave water-hammer equations
the following
The plane wave assumption is central to the formulation
dp
Downloaded by [University of Sydney] at 00:55 22 March 2013

L2 dp
of quasi-two-dimensional water-hammer equations (e.g. ¼ fr or ¼ rfr ¼ gdz=dr ð51Þ
Bratland 1986, Vardy and Hwang 1991, Pezzinga 1999). dr aU1 dr
The analysis of Mitra and Rouleau (1985) indicates that
this assumption is valid everywhere along the pipe except where z = elevation from a datum to a point in the pipe.
in the vicinity of boundary elements such as valves for the The section on the Joukowsky relation showed that
case of laminar water-hammer flows. In most practical jDr/rj = j(r7r0)/rj
1. In addition, Walker (1975),
water-hammer applications, the fluid (water) is only Courant and Friedrichs (1976) and Hinze (1987) showed
slightly compressible (i.e. jDrj/r
1), the lateral pipe that the variation of density in unsteady compressible flows
deformation is negligible (i.e. jDAj/A
1), and the is of the order of the Mach number (i.e.,
conduit is prismatic. As a result, it is expected that radial r=r0 ¼ 1 þ O ðMÞ). Therefore, equation (51) becomes:
inertia and viscous terms are negligible in comparison to
their axial counterparts and, thus, the plane wave d p
ðp þ r0 gzÞ ¼ 0 or þ z ¼ Hðx; tÞ ð52Þ
assumption justified. This section studies the validity of dr g
the plane wave assumption through order of magnitude
analysis. where H(x,t) = piezometric head and g = gr0 = specific
In addition to the previously introduced scales, let weight of the fluid. A water hammer wave, whose piezo-
L2 = radial length scale; and U2 = radial velocity scale. metric head is independent of r [i.e. satisfies equation (52)],
As a result, equations (23) and (24) can be re-written as: is called a plane wave. This plane wave assumption is used
in the water hammer models of Bratland (1986), Vardy and
U1 du U21  du U1 U2 du U1 k þ 4m=3 d2 u Hwang (1991), Pezzinga (1999 and 2000), Ghidaoui and
þ u þ  
T dt L dx L2 dr L21 r dx2 Kolyshkin (2001) and Ghidaoui et al. (2001). The current
 
 ð47Þ
U1 n d  du 1 dp k þ m=3 d 1 drv analysis shows that the plane wave assumption is justified
r ¼  þ þ f x
L22 r dr dr r dx r dx r dr when U2/U1 and M=RL are much smaller than unity, a
condition that is satisfied in the majority of water hammer

U2 U1 dv U21  dv U1 U2  dv U1 d2 n applications.
þ u þ v   2v 2 Using equation (52), system (22), (23) and (24) becomes
U1 T dt L dk L2 dr L1 dk
   

U1 k þ m=3 d 1 dr v v d  dv v
þ r  v ¼ ð48Þ  
L22 r dr r dr r dr dr r2 1 dr dr dr du 1 drv
þu þv þ þ ¼0 ð53Þ
1 dp k þ m=3 d2 u r dt dx dr dx r dr
 þ þ fr
r dk r dkdr  
du du du 1 dp k þ m=3 d du 1 drv
þu þv ¼ þ þ þ
dt dx dr r0 dx r0 dx dx r dr
It is evident that the ratio of the left-hand side of equation  2  
(48) to the left-hand side of equation (47) is of order U2/U1. d u 1d du
n þ r þ fx ð54Þ
Mitra and Rouleau (1985), Vardy and Hwang (1991), dx2 r dr dr
Pezzinga (1999), Ghidaoui and Kolyshkin (2001) and
 80 M. S. Ghidaoui

d p
ðp þ gzÞ ¼ 0 or þ z ¼ Hðx; tÞ ð55Þ Reynolds number and relative roughness. In this section,
dr g the derivation of the two and quasi-two-dimensional
water-hammer equations is performed by progressive
which are the 2-D plane wave water-hammer equations. simplification of the three-dimensional Navier – Stokes
When the pipe is flexible, its area expands and contracts equations.
in response to the transient pressure. The change in pipe The plane wave equations can be further simplified using
area makes the solution of the above system highly order of magnitude analysis. In particular, applying the
complicated. To avoid this problem, one can treat the pipe scaling to equations (56) and (57) gives:
as rigid so long as the continuity equation is altered in    
dp aT  dp

du aT U2  dp

1 dr v
such a way as to account for the mass that would be þ Mu þ þ Mv þ ¼0
dt L dx dx L2 U1 dr r dr
stored by the change in pipe area. This can be easily
performed by replacing dr in the continuity equation by ð59Þ
d(rA)/A (i.e. replacing dr/rdt by d(rA)/rAdt. In terms of
pressure, one can write d(rA)/rAdt = [d(rA)/rAdp]dp/ du U1 T  du U2 T  du aT dp T
þ u þ v ¼ þ fx þ
dt = a 7 2dp/dt. As a result, the equivalent plane wave dt  L dx  L2 dr L dx U1
equations are:   
vT v d du L2 k þ 4m=3 d2 u
   
r  þ 22 þ ð60Þ
1 dp dp dp du 1 drv 2
L2 r dr dr L1 r0 dx2
þu þv þ þ ¼0 ð56Þ 
a2 dt dx dr dx r dr L2 U2 km =3 d 1 dr v
Downloaded by [University of Sydney] at 00:55 22 March 2013

  L U1 r0 dk r dr
du du du 1 dp k þ m=3 d du 1 drv
þu þv ¼ þ þ þ
dt dx dr r0 dx r0 dx dx r dr
 2   Clearly, since M
1, the advective terms in the continuity
du 1d du
n þ r þ fx ð57Þ equations are negligible in comparison to the mass flux
dx2 r dr dr
terms. In addition, the resolution of the radial distribution
d p of the velocity field requires that L2 is much smaller than
ðp þ gzÞ ¼ 0 or þ z ¼ Hðx; tÞ ð58Þ the pipe diameter. In fact, near the pipe wall, L2 must be of
dr g
the order of the viscous sublayer. In general, L2 is a
measure of the radial diffusion of mass and momentum.
6.2 Quasi-2-D water-hammer equations
Obviously L2
L. As a result, equation (60) shows that the
Recently, two- and quasi-two-dimensional models have two viscous terms associated with the compressibility of the
been developed and used to perform numerical experi- fluid are significantly smaller than the viscous term
ments in water-hammer flows (e.g. Mitra and Rouleau associated with the radial variation of the longitudinal
1985, Vardy and Hwang 1991, Eichinger and Lein 1992, velocity. Therefore, equations (59) and (60) become as
Silva-Araya and Chaudhry 1997, and Pezzinga 1999, follows:
2000). For example, Mitra and Rouleau (1985) used a
1 dp du 1 drv
two-dimensional laminar model to investigate the plane þ þ ¼0 ð61Þ
wave assumptions in water-hammer flows. In addition, r0 a2f dt dx r dr
numerical data generated from quasi-two-dimensional  
du du du 1 dp 1d du
water-hammer models, have been used to analyze the þu þv ¼ þn r þ fx ð62Þ
flow behaviour in water hammer flows through the dt dx dr r0 dx r dr dr
computation of instantaneous velocity profiles and shear
stress fields, calibrate and verify one-dimensional water d p
ðp þ gzÞ ¼ 0 or þ z ¼ Hðx; tÞ ð63Þ
hammer models, evaluate the parameters of one-dimen- dr g
sional unsteady friction models, and compare and analyze
various one-dimensional unsteady friction models. In fact, When gravity is the only body force present, fx ¼ gdz=dx
Silva-Araya and Chaudhry (1997) used the data obtained and the above system can be re-written in terms of the
from their turbulence water hammer model to calculate piezometric head as follows:
the required multiplying factor for the Darcy – Weisbach
formula so that the one-dimensional water hammer model g dH du 1 drv
þ þ ¼0 ð64Þ
and the turbulence water hammer model produce similar a2f dt dx r dr
dissipation rates. In addition, Pezzinga (2000) employed
the data obtained from a turbulence water hammer model  
du du du dH 1d du
to derive Moody-like charts for the resistance coefficients þ u þ v ¼ g þn r ð65Þ
dt dx dr dx r dr dr
of one-dimensional unsteady friction as a function of
 On the fundamental equations of water hammer 81

These are the local (i.e. continuum scale) equations of du0 v0 =dr. In addition, using the scaling introduced in
water-hammer flows, where U2/U1, M, M=RL and L2/L are previous sections shows that g=a2 udH=dt, the axial
all significantly smaller than one. advective acceleration and the radial advective acceleration
are all M times smaller than d
u=dt. Therefore, the above
equations become:
6.3 Ensemble averaging turbulent water-hammer equations
g dH du 1 drv
In practice, water hammer problems often belong to the þ þ ¼0 ð68Þ
turbulent regime. In principle, there are two approaches to a2 dt dx r dr
solving turbulent water-hammer flows. The Direct Numer-  
ical Simulation (DNS) approach solves the Navier – Stokes du dH 1d du du0 v0
¼ g þn r  ð69Þ
equations and aims to resolve all the scales of flow motion. dt dx r dr dr dr
The number of grid points required to carry out DNS is of
the order of Reynolds number to the power three. This which are the turbulent water-hammer equations found in
tremendous requirement for computational resources limits Vardy and Hwang (1991). The turbulent equations in Silva-
DNS to simple flows with low Reynolds numbers. The Araya and Chaudhry (1997) and Pezzinga (1999, 2000)
second approach is to formulate the equations that govern neglect the radial mass flux from the continuity equation.
the turbulence field by averaging equations (64) and (65) Ghidaoui et al. (2001) made direct comparison between
and to employ a turbulence model to estimate the models that contain the radial mass flux and those that do
Downloaded by [University of Sydney] at 00:55 22 March 2013

Reynolds’ stresses which emerge from the averaging not and found that the results are virtually indistinguish-
process. The governing equations for turbulent water- able. This suggests that the radial mass flux term is
hammer flows are determined below. inconsequential in water-hammer problems. In fact, all
The velocity and pressure fields in a turbulent flow can be the terms in the continuity equations for the flow field
decomposed into mean and fluctuation components. That between two successive water-hammer fronts are negligibly
is, u ¼ u
þ u0 , v ¼ v
þ v0 and H ¼ H
þ H0 , where the over- small and the turbulent field can be computed from the
bar denotes mean values and the prime denotes turbulent incompressible flow equations, an approach that has been
fluctuations (i.e. deviations from the mean). In water- successfully used by Kita et al. (1980), Eichinger and Lein
hammer problems the flow is highly unsteady and the time (1992) and Silva-Araya and Chaudhry (1997).
scale of the transient is often much smaller than the time The closure of equations (68) and (69) requires a
scale of the turbulence. Therefore, ensemble averaging and turbulence model for the Reynolds stress term u0 v0 . To this
not time averaging must be used in determining the mean end, various turbulence models have appeared in water
flow field. Consider a series of N identical experiments of hammer literature (e.g. Vardy and Hwang 1991, Silva-
the water-hammer flow field, where un, vn and Hn denote the Araya and Chaudhry 1997, Pezzinga 1999, 2000). All
outcome of the nth experiment. The ensemble mean flow existing turbulence water-hammer models assume that eddy
P
field is obtained as follows: u
ðx; r; tÞ ¼ N n¼1 vn =N, viscosity expressions, derived for steady state pipe flows,
PN PN
v
ðx; r; tÞ ¼ n¼1 vn =N and Hðx;
r; tÞ ¼ H =N. In the- remain applicable for water-hammer flows (i.e. all existing
n¼1 n
ory, when N Ð tends to infinity, the
Ð ensemble mean becomes models are based on the quasi-steady assumption).
uÐ
ðx; r; tÞ ¼ uPdu, v
ðx; r; tÞ ¼ vP dv, and Hðx;
r; tÞ= Ghidaoui et al. (2001) found that this approach is
HPdH, where P = probability density function of the applicable when the ratio of the radial diffusion time scale
experiments. Adding u 6 equation (64) to equation (65); to the wave time scale is large.
inserting u ¼ u
¼ u0 , v ¼ v
þ v0 and H ¼ H
þ H0 into both
equation (64) and u 6 (64) + (65); performing the
7. Conclusions
ensemble averaging; and re-arranging gives:
The relation between state equations and wave speeds in
g dH du 1 drv
þ þ ¼0 ð66Þ single as well as multiphase and multicomponent transient
a2 dt dx r dr flows are illustrated and discussed. The various forms of
  one- and two-dimensional water-hammer equations such as
du du2 duv dH 1 d du g dH du2 duv the Joukowsky model, classical 1-D water-hammer equa-
þ þ ¼ g þn r  2u  
dt dx dr dx r dr dr a dt dx dr tions, the 2-D plane wave equations and the quasi-two-D
ð67Þ plane wave equations are derived. The governing equations
of turbulent water-hammer flows are obtained by ensemble
The velocity fluctuations are of the same order of averaging of the quasi-two-D plane wave equations. Order
magnitude as the shear velocity Ut. Therefore, the last of magnitude analysis is used throughout the paper to
two terms in equation (67) are of order Ut2/L and Ut2/L2, evaluate the accuracy of the assumptions in the various
respectively. Therefore, du0w dx is L2/L smaller than forms of water-hammer governing equations.
 82 M. S. Ghidaoui

Kapelan, Z., Savic, D.A. and Walters, G.A., Incorporation of prior

Acknowledgments information in inverse transient analysis for leak detection and rough-
ness calibration. Urban Wat. J., 2004, 1, 129 – 143.
The writer wishes to thank the Research Grants Council of Kita, Y., Adachi, Y. and Hirose, K., Periodically oscillating turbulent flow
Hong Kong, Project No. HKUST6113/03E, for financial in a pipe. Bull. J. S. M. E., 1980, 23, 654 – 664.
support and Mr. Zhao Ming, for his helpful comments. Larock, B.E., Jeppson, R.W. and Watters, G.Z., Hydraulics of Pipeline
Systems, 2000 (CRC Press: New York).
Liggett, J.A., Fluid Mechanics, 1994 (McGraw-Hill: New York).
Lighthill, J., Waves in Fluids, 1996 (Cambridge University Press: Cam-
References bridge).
McInnis, D.A., Karney, B.W. and Axworthy, D.H., TRANSAM Reference
Axworthy, D.H., Ghidaoui, M.S., and McInnis, D.A, Extended thermo-
Manual, 1999 (HydraTek Associates, Ajax, Ontario).
dynamics derivation of energy dissipation in unsteady pipe flow. J.
Mitra, A.K. and Rouleau, W.T., Radial and axial variations in transient
Hydraulic Engng, 2000, 126, 276 – 287.
pressure waves transmitted through liquid transmission lines. J. Fluid
Batchelor, G.K., An Introduction to Fluid Dynamics, 1994 (Cambridge
Engng, 1985, 107, 105 – 111.
University Press: Cambridge).
Pezzinga, G., Quasi-2D Model for unsteady flow in pipe networks. J.
Bergant, A., and Simpson, A.R., Estimating unsteady friction in transient
Hydraulic Engng, 1999, 125, 676 – 685.
cavitating pipe flow, in 2nd International Conference on Water Pipeline
Pezzinga, G., Evaluation of unsteady flow resistances by Quasi-2D or 1D
Systems, Edinburgh, UK, BHRA Group Conference Series No. 110,
models. J. Hydraulic Engng, 2000, 126, 778 – 785.
1994, pp. 3 – 15.
Rouse, H. and Ince, S., History of Hydraulics, 1957 (Dover Publications:
Bratland, O., Frequency-dependent friction and radial kinetic energy
New York).
variation in transient pipe flow, in 5th International Conference on
Safwat, H.H. and Polder, J., Friction-frequency dependence for oscillatory
Downloaded by [University of Sydney] at 00:55 22 March 2013

Pressure Surges, Hanover, Germany, 1986.

flows in circular pipe. J. Hydraulic Div., ASCE, 1973, 99, 1933 – 1945.
Brunone, B. and Ferrante, M., Pressure waves as a tool for leak detection
Shuy, E.B., Wall shear stress in accelerating and decelerating turbulent pipe
in closed conduits. Urban Wat. J., 2004, 1, 145 – 155.
flows. J. Hydraulic Res., 1996, 34(2), 173 – 183.
Brunone, B., Golia, U.M. and Greco, M., Effects of two-dimensionality
Silva-Araya, W.F. and Chaudhry, M.H., Computation of energy dissipa-
of pipe transients modelling. J. Hydraulic Engng, 1995, 121, 906 –
tion in transient flow. J. Hydraulic Engng, 1997, 123, 108 – 115.
912.
Suo, L. and Wylie, E.B., Impulse response method for frequency-
Brunone, B., Karney, B.W., and Mecarelli, M. and Ferrante, M., Velocity
dependent pipeline transients. J. Fluids Engng, 1989, 111, 478 – 483
profiles and unsteady pipe friction in transient flow. J. Wat. Resources
Tijsseling, A.S., Fluid-structure interaction in liquid-filled pipe systems: a
Plann. Mgmt, 2000, VOL, 236 – 244.
review. J. Fluids Structures, 1995, 10, 109 – 146.
Cardle, J.A., Song, C.S.S. and Yuan, M., Measurements of mixed transient
Trikha, A.K., An efficient method for simulating frequency-dependent
flows. J. Hydraulic Engng, 1988, 115, 169 – 182.
friction in transient liquid flow. J. Fluids Engng, 1975, 97, 97 – 105.
Chaudhry, M.H., Applied Hydraulic Transients, 1987 (Van Nostrand
Vardy, A.E., and Brown, J., Transient, turbulent, smooth pipe friction. J.
Reinhold: New York).
Hydraulic Res., 1995, 33, 435 – 456.
Courant, R. and Friedrichs, K.O., Supersonic Flow and Shock Waves, 1948
Vardy, A.E., and Hwang, K.L. (1991). ‘‘A characteristic model of transient
(Springer-Verlag: New York).
friction in pipes.’’ J. Hydraulic Res., 29, 669 – 685.
Currie, I.G., Fundamental Mechanics of Fluids, 1993 (McGraw-Hill: New
Vardy, A.E., Hwang, K.L., and Brown, J.M.B., A weighting function
York).
model of transient turbulent pipe friction. J. Hydraulic Res., 1993, 31,
Dailey, J.W., Hankey, W.L., Olive, R.W., and Jordaan, J.M., Resistance
533 – 548.
coefficient for accelerated and decelerated flows through smooth tubes
Walker, J.S., Perturbation solutions for steady one-dimensional water
and orifices. J. Basic Engng, Series D, 1956, 78, 1071 – 1077.
hammer waves. J. Fluid Engng, 1975, 6, 260 – 262.
Das, D., and Arakeri, J. H., Transition of unsteady velocity profiles with
Wylie, E.B. and Streeter, V.L., Fluid Transient in Systems, 1993 (Prentice-
reverse flow. J. Fluid Mech., 1998, 374, 251 – 283.
Hall: Englewood Cliffs, NJ).
Eichenger, P. and Lein, G., The influence of friction on unsteady pipe flow.
Yen, B.C., Hydraulics of sewers. In Advances in Hydroscience Urban Storm
In Unsteady Flow and Fluid Transients, edited by R. Bettes and J. Watts,
Drainage, Vol. 14, pp. 1 – 115, 1986 (Academic Press: New York).
pp. 41 – 50, 1992 (Balkema: Rotterdam).
Zielke, W., Frequency-dependent friction in transient pipe flow. J. Basic
Ghidaoui, M.S. and Kolyshkin, A.A., Stability analysis of velocity profiles
Engng, Series D, 1968, 90, 109 – 115.
in water hammer. J. Hydraulic Engng, 2001, 127, 499 – 512.
Ghidaoui, M.S. and Mansour, S., Efficient treatment of the Vardy – Brown
unsteady shear in pipe transients. J. Hydraulic Engng, 2002, 128, 102 –
112.
Ghidaoui, M.S., Mansour, S. and Zhao, M., Analysis of quasi-two- Appendix
dimensional water-hammer models. J. Hydraulic Engng, 2002, 128, 917 –
924. Notation
Ghidaoui, M.S., McInnis, D.A, Zhao, M. and Axworthy, H.H., Unsteady
flows in closed conduits: from theory to practice. Appl. Mech. Rev., 2004, The following symbols are used in this paper:
(to be published). A = inside cross-sectional area of conduit;
Greco, M., Some recent findings on column separation during water A = acoustic wave speed of pipe wall material;
hammer. Excerpta, 1990, 5, 261 – 272.
D = inside pipe diameter;
Hinze, J.O., Turbulance, second edn, 1987 (McGraw-Hill: New York).
Jelev, I., The damping of flow and pressure oscillation in water hammer
f = Darcy – Weisbach friction factor;
analysis. J. Hydraulic Res., 1989, 27, 91 – 114. fx = local axial body forces;
 On the fundamental equations of water hammer 83

fr = local radial body forces; U2 = radial velocity scale;

Fx = axial external forces; Ut = shear velocity;
Fext = external forces; v = local radial velocity;
g = gravitational acceleration; v = velocity vector;
H = piezometric head; V = volume;
Kf = bulk modulus of elasticity of the fluid; w = local azimuthal velocity;
Ks = bulk modulus of elasticity of the solid (pipe); x = longitudinal axis;
K = hydraulic conductivity matrix; z = vertical axis;
L = length of pipe; b = momentum correction coefficient;
L1 = longitudinal length scale; y = azimuthal axis;
L2 = radial length scale; z = time scale parameter;
m_ = mass flux; ev = eddy viscosity;
M = Mach number; r = fluid density;
n = normal unit vector; r0 = initial fluid density;
N = number of identical experiments; m = dynamic viscosity of the fluid;
O = order of magnitude function; v = kinematic viscosity of the fluid;
p = local pressure; k = second dynamic viscosity of the fluid;
P = cross-sectional average of pressure; t = shear stress;
Downloaded by [University of Sydney] at 00:55 22 March 2013

P = probability density function of turbulence tw = shear stress at pipe wall;

sampling; . = dot product operator;
r = radial axis; f jg = used to denote that f is evaluated at g;
R = inside pipe radius; * = a superscript * denotes dimensionless quantities;
f = Reynolds number based on L; ’ = a superscript ’ denotes quantities due to turbulent
t = time; fluctuations;
T = time scale; f = ensemble mean of quantity f;
u = local longitudinal velocity; I = a subscript i denotes the ith phase or component;
U = cross-sectional average fluid velocity; m = a subscript m denotes mixture.
U1 = longitudinal velocity scale;