water hammer with column separation

WATER HAMMER WITH COLUMN SEPARATION:

A REVIEW OF RESEARCH IN THE TWENTIETH CENTURY

ANTON BERGANT

Litostroj E.I. d.o.o., Litostrojska 40,

1000 Ljubljana, Slovenia; [email protected]

ANGUS R. SIMPSON

School of Civil and Environmental Engineering, University of Adelaide,

Adelaide, South Australia, 5005; [email protected]

ARRIS S. TIJSSELING

Department of Mathematics and Computer Science, Eindhoven University of Technology,

P.O. Box 513, 5600 MB Eindhoven, The Netherlands; [email protected]

ABSTRACT

Column separation refers to the breaking of liquid columns in fully filled pipelines. This may occur in a water

hammer event when the pressure drops to the vapor pressure at specific locations such as closed ends, high

points or knees (changes in pipe slope). A vapor cavity, driven by the inertia of the parting liquid columns, will

start to grow. The cavity acts as a vacuum, a low-pressure point, retarding the liquid columns, which finally

starts to diminish in size when the liquid columns change flow direction. The collision of two liquid columns,

or of one liquid column with a closed end, moving towards the shrinking cavity, may cause a large and nearly

instantaneous rise in pressure. The large pressure rise travels through the entire pipeline and forms a severe load

for hydraulic machinery, individual pipes and supporting structures. The situation is even worse: in one water-

hammer event many repetitions of cavity formation and collapse may occur.

This report reviews water-hammer-induced column-separation from the discovery of the phenomenon in the

late 19th century, the recognition of its danger in the 1930s, the development of numerical methods in the 1960s

and 1970s, to the standard models used in commercial software packages in the late 20th century. A

comprehensive survey of laboratory tests and field measurements is given. The review focuses on transient

vaporous cavitation. Gaseous cavitation and steam-condensation are beyond the scope of the report. There are

more than 300 references cited in this review report.

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TABLE OF CONTENTS

1 INTRODUCTION ________________________________________________________________ 4 1.1 Fluid Transients_________________________________________________________ 4 1.2 Scope of this Report ______________________________________________________ 4 1.3 Previous Reviews ________________________________________________________ 4 1.4 Cavitation _____________________________________________________________ 5 1.5 Column Sparation _______________________________________________________ 5 1.6 Mathematical Models_____________________________________________________ 5 1.7 Report Outline__________________________________________________________ 6

2 WATER HAMMER, COLUMN SEPARATION AND CAVITATION_____________________ 6 2.1 Water Hammer in a Historical Context _______________________________________ 7 2.2 First Observations of Sub-Atmospheric Pressures during Water Hammer Events _______ 7 2.3 Vaporous Cavitation _____________________________________________________ 9

2.3.1 Cavitation inception and tensile stress _______________________________________ 9 2.3.2 Local vapor cavities _____________________________________________________ 9 2.3.3 Intermediate vapor cavities ______________________________________________ 10 2.3.4 Distributed vaporous cavitation or two-phase (bubble) flow______________________ 11

2.4 Gaseous Cavitation _____________________________________________________ 13 2.5 Severe Pressure Peaks Following Cavity Collapse ______________________________ 14 2.6 Severity of Cavitation / Cavitation Intensity / Measures of Cavitation _______________ 19

3 MATHEMATICAL MODELS AND NUMERICAL METHODS ________________________ 21 3.1 Assumptions___________________________________________________________ 22 3.2 Water Hammer Equations ________________________________________________ 23 3.3 Discrete Single Cavity Models______________________________________________ 24 3.4 Discrete Multiple Cavity Models____________________________________________ 27

3.4.1 The discrete vapor cavity model (DVCM) _______________________________________ 28 3.4.2 The discrete gas cavity model (DGCM) _________________________________________ 33

3.5 Shallow Water Flow or Separated Flow Models ________________________________ 36 3.6 Two Phase or Distributed Vaporous Cavitation Models __________________________ 38

3.6.1 Two-phase flow equations for vaporous cavitation region __________________________ 40 3.6.2 Shock equations for condensation of vaporous cavitation region_____________________ 42

3.7 Combined Models / Interface Models ________________________________________ 44 3.8 Other Models __________________________________________________________ 47

3.8.1 Jordan algebraic model _________________________________________________ 46 3.8.2 FEM _______________________________________________________________ 47 3.8.3 Other _______________________________________________________________ 47

3.9 A Comparison of Models _________________________________________________ 48

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3.10 Modeling of Friction_____________________________________________________ 49 3.11 State of the Art - The Recommended Models __________________________________ 50

3.11.1 Discrete cavity models: limitations ____________________________________________ 51 4 EXPERIMENTS IN THE LABORATORY AND MEASUREMENTS IN THE FIELD ______ 53

4.1 Photographs of Cavity Formation___________________________________________ 53 4.2 Laboratory Experiments with Liquid Column Separation ________________________ 54 4.3 Gas Release ___________________________________________________________ 61 4.4 Fluid Structure Interaction (FSI) ___________________________________________ 61 4.5 Problems with Pressure Transducers ________________________________________ 62 4.6 Field Measurements _____________________________________________________ 62

5 CONCLUSIONS ________________________________________________________________ 64 6 ACKNOWLEDGEMENT ________________________________________________________ 66 7 REFERENCES _________________________________________________________________ 66 8 NOMENCLATURE _____________________________________________________________ 87

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1 INTRODUCTION

1.1 Fluid Transients

Pipes installed in water supply systems, irrigation networks, hydropower stations, nuclear power stations and

industrial plants are required to convey liquid reliably, safely and economically. Modern hydraulic systems

operate over a broad range of operating regimes. Any change of flow velocity in the system induces a change

in pressure. The sudden shut-down of a pump or closure of a valve causes fluid transients which may involve

large pressure variations, local cavity formation, distributed cavitation, hydraulic and structural vibrations

and excessive mass oscillations. In particular, the occurrence of liquid column separation may have a

significant impact on subsequent transients in the system. Large pressures with steep wave fronts may occur

when vapor cavities collapse and the practical implications are therefore significant. As an outcome, fluid

transients may lead to severe accidents (Jaeger 1948; Bonin 1960; Parmakian 1985; Kottmann 1989; De

Almeida 1991).

1.2 Scope of this Report

Cavitation is a broad field of research. This review is confined to the macroscopic aspects of transient vaporous

cavitation and to the important case of column separation. The basis for this report is formed by the Ph.D.

theses of the three authors (Simpson 1986; Bergant 1992; Tijsseling 1993). The literature reviews of these three

theses have been combined and updated with additional contributions and with publications from the 1990s and

early 2000s. The aim was to give a historical account, to summarize the state of the art and to have a list of

references as complete as practically possible.

1.3 Previous Reviews

Previous literature reviews on column separation have been published. A selected bibliography on column

separation was presented by Jaeger et al. (1965). Martin (1973) provided a valuable summary of the

historical development of many aspects of water hammer analysis and included an extensive bibliography.

Wylie and Streeter in their three books (1967, 1978a and 1993) give account of previous work, including

their own, on column separation. De Almeida (1987) reviewed the period 1978 to 1987. During the period

between 1971 and 1991 an international Working Group of the International Association for Hydraulic

Research (IAHR) carried out a major research effort with respect to column separation in industrial

hydraulic systems. The scope of the work included topics on bubble nucleation and dynamics, two-phase

flow, mathematical modeling of the basic physical phenomena, numerical simulation of transients in

industrial hydraulic systems, instrumentation, and laboratory and prototype measurements. The Group

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focused its efforts on the investigation of transients in long pipelines (aqueducts) and power plant cooling

water systems. One of the main aims of the Group was the development of computer codes and the

validation of these codes against experimental results. The outcomes of the work by the Group, presented at

nine international round-table meetings, were finally summarized in an extensive synthesis report (Fanelli

2000). Material from this publication has been included in this review.

1.4 Cavitation

Microscopic aspects of cavitation, cavitation erosion, cavitation noise and advanced cavitation theory are not

considered in this review. These aspects are covered elsewhere (Knapp et al. 1970; Hammitt 1980; Trevena

1987; Young 1989; Brennen 1995; Li 2000). Valuable information may also be obtained from Arndt's (1981)

review. The presence of air in water may also have a significant impact on transients. Air presence is considered

briefly in this review, but the focus is on systems with very little air as these situations will, in most

circumstances, result in the highest pressures.

1.5 Column Separation

Column separation can be compared with the breaking of a solid rod; Galilei already described this analogy

(Rouse and Ince 1957, 1963, p. 57). The phenomenon can nicely be demonstrated in a simple toy apparatus

(Wylie 1999). The general policy in hydraulic design involving liquid transients is to avoid liquid column

separation, as described in the water hammer design criteria of Parmakian (1955). Lupton (1953) noted that

cumulative high surge pressures are dangerous and recommended “that in order to ensure immunity from

surge danger, vacuous separation should be prevented throughout the system concerned”. Martin (1973)

pointed out that both in Europe and in the USA the design for column separation or the avoidance of

cavitation was still subject of many debates. Even today there is reluctance by many designers to accept the

occurrence of any column separation. Surge tanks, air chambers, or large motor rotating moments of inertia

are suggested to avoid liquid column separation. Several authors concluded that if it cannot be avoided,

then steps should be taken to minimize the impact when the liquid columns rejoin. For example, condenser

cooling water systems are low-head systems where macro-cavitation cannot be avoided because of the

topographic and operational characteristics of the plant.

1.6 Mathematical Models

More than a century ago Joukowsky (1900) mathematically described many of the physical aspects of wave

propagation in liquid systems. He also observed and explained column separation. Water hammer

numerical models give physically accurate results, especially as to the first pressure rise, when the pressure

is above the liquid vapor pressure (Jaeger 1977; Chaudhry 1979; Fox 1989; Wylie and Streeter 1993). The

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decay of the pressure trace is usually quicker in reality due to effects of free air, unsteady friction and

structural vibration. Cavitation occurs when the pressure drops to the liquid vapor pressure and the one-

phase flow is transformed to two-phase flow. The classical water hammer equations are not valid in regions

of cavitation. The first objective of modeling column separation is to predict the pressures that occur when

large vapor cavities collapse. The second objective is to predict the timing of the events. A third objective

might be to predict the structural response of pipes and supports.

1.7 Report Outline

This report is divided into five parts. Following this introduction, water hammer and the different types of

cavitation (including column separation) that may occur in pipelines are discussed in detail. The third

section considers the popular mathematical models and their numerical implementation. Large pressure

peaks or short-duration pulses due to vapor cavity collapse are also considered in this section. Experimental

studies of column separation and field measurements are summarized in Section 4. Finally, conclusions are

drawn in Section 5.

2 WATER HAMMER, COLUMN SEPARATION AND CAVITATION

Column separation can have a devastating effect on a pipeline system. A famous example is the accident at

Oigawa hydropower station in 1950 in Japan (Bonin 1960). Three workers died. The plant was designed in

the early 20th century. A fast valve-closure due to the draining of an oil control system during maintenance

caused an extremely high-pressure water hammer wave that split the penstock open. The resultant release

of water generated a low-pressure wave resulting in substantial column separation that caused crushing

(pipe collapse) of a significant portion of the upstream pipeline due to the external atmospheric pressure. As

an outcome, designing pipelines to withstand full atmospheric pressure on the outside when vapor pressure

occurs on the inside has been standard practice for many years. Jaeger (1948) reviewed a number of the

most serious accidents due to water hammer in pressure conduits. Many of the failures described were

related to vibration, resonance and auto-oscillation. Two of the failures involved liquid column separation.

In one case a governor caused a valve to open suddenly and thus produced a negative wave that resulted in

a local liquid column separation at a break in the pipe profile. When the liquid columns rejoined, quite

strong pressure rises caused cracking of a concrete section of the penstock. Kottmann (1989) described two

accidents related to column separation in which two workers died. List et al. (1999) reported severe damage

to the lining of a 7 km long, 0.6 m diameter, pump discharge pipeline, finally resulting in leaks. The cause

was vapor-cavity collapse.

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2.1 Water Hammer in a Historical Context

During the second half of the 19th century and the first quarter of the 20th century, the majority of the

publications on water hammer came from Europe. The conception of the theory of surges can, amongst

others, be traced to Ménabréa (1858, 1862), Michaud (1878), Von Kries (1883), Frizell (1898), Joukowsky

(1900) and Allievi (1902, 1913). Joukowsky performed classic experiments in Moscow in 1897/1898 and

proposed the law for instantaneous water hammer in a simple pipe system. This law states that the

(piezometric) head rise ∆H resulting from a fast (Tc < 2L/a) closure of a valve, is given by:

gaV

H 0=∆ (1)

in which, a = pressure wave speed, V0 = initial flow velocity, g = gravitational acceleration, L = pipe length

and Tc = valve closure time. The period of pipe, 2L/a, is defined as the return time for a water hammer

wave to travel from a valve at one end of the pipeline to a reservoir at the other end, and back to the valve.

The theoretical analyses performed independently by Joukowsky (1900) and Allievi (1902, 1913) formed

the basis for classical water-hammer theory. Joukowsky’s work was translated by Simin in 1904. Allievi's

work was not known generally outside Europe until Halmos made an English translation in 1925. Gibson

(1908) presented one of the first important water hammer contributions in English. He considered the

pressures in penstocks resulting from the gradual closure of turbine gates.

2.2 First Observations of Sub-Atmospheric Pressures during Water Hammer Events

Joukowsky (1900, pp. 31-32) was the first to observe (see Fig. 1) and understand column separation. He

explained the events in his experimental main-pipeline-gate system literally as follows (translation from

German by the third author of this review): "Starting at the moment of closure of the gate the water in the pipe

is continuously being stopped, whereby it is being compressed, the pipe expands and the pressure increases

with ∆p. When this state travels with the celerity a up to the main, the latter transmits back along the pipe the

main pressure (a little raised due to water hammer in the main itself) and a velocity of the water, which is

directed in the direction of the main. This phase first passes the cabins II and III (measuring points), as a result

of which the pressure in the indicators (gauges) in these cabins falls to the pressure of the main. When however

the mentioned phase reaches the gate, this instantaneously causes, since the velocity of the water is directed

away from the gate, a decrease of the pressure at the gate. If thereby the velocity V is so large, that according

to the theory the reduced pressure would be negative, a break of the water mass will occur. The water column

will be separated from the gate, ahead of which a small rarefied void develops. Similar separations can also

form in other parts of a liquid column, the parts towards which the reduced pressure propagated." and "The

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condition, that the water column is separated from the gate, prolongs the duration of the reduced pressure and

makes the second impact stronger than the first, because it takes place with the velocity, at which the liquid

column speeds into the rarefied void." See also Simin (1904, pp. 381-382).

Figure 1 Pressure record exhibiting column separation. Horizontal axis: time (each dot

indicates half a second). Vertical axis: pressure (Joukowsky-pressure of 15.3 bar). The upper

horizontal line is the static pressure and the lower horizontal line is the atmospheric pressure.

(Joukowsky 1900, Fig. 17; also shown by Simin 1904, Fig. 13, and by Moshnin and

Timofeeva 1965, Fig. 1)

Gibson (1908) performed water hammer experiments with closure and opening of a downstream valve in a

laboratory pipeline apparatus. He indicated that gas release in a part of the pipe section was initiated by a

low-pressure (negative) wave. Strowger and Kerr (1926) indicated that a full column separation in the draft

tube of a reaction water turbine could cause severe hydrodynamic loads following the turbine load

rejection. Thorley (1976) attributed the first work on vapor cavities to Hogg and Traill (1926) and Langevin

(1928). Mostowsky (1929) presented the first theoretical analysis of column separation in an explanation of his

laboratory measurements.

The first major United States involvement in the study of water hammer came in the form of a symposium

on water hammer in 1933 in Chicago (Proceedings 1933). The ASME Committee on Water Hammer

presented an extensive bibliography. The committee reviewed and summarized existing theory on water

hammer and evaluated various methods employed, which included mainly arithmetic methods of solution.

The symposium focused on water hammer in simple conduits, complex conduits, compound conduits,

pump discharge conduits and surge tanks. A set of standardized nomenclature was developed at this

symposium. Liquid column separation did not receive any mention in the committee report. A paper

presented at the symposium by Billings et al. (1933) dealt with “parting of the water column” related to

high-head penstock design. The authors noted that dangerous instantaneous pressure rises often originate in

the upper portion of the penstock, when the liquid column parted and rejoined abruptly.

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2.3 Vaporous Cavitation

Considering liquid transients in a pipeline system – there are two different types of flow regimes. The first

is referred to as the water hammer regime (or “no cavitation” case) where the pressure is above the vapor

pressure of the liquid. The second is the cavitation regime where the pressure is at the vapor pressure of the

liquid. Two types of cavitation in pipelines are now distinguished. The magnitude of the void fraction is the

basis for identifying the two types (Kranenburg 1974a). It is defined as the ratio of the volume of the vapor

to the total volume of the liquid/vapor mixture and has also been referred to as the vapor fraction (Streeter

1983). The symbol α is used for void fraction following the nomenclature introduced by Wallis (1969). The

void fraction depends on the magnitude of the velocity gradient in the cavitating flow. The two types of

cavitation in pipelines are: (i) discrete vapor cavity or local liquid column separation (large α) and (ii)

distributed vaporous cavitation or bubbly flow (small α).

2.3.1 Cavitation inception and tensile stress

The liquid's vapor pressure is adopted as the cavitation inception pressure in numerical models for transient

cavitation (Wylie and Streeter 1993). However, there are a number of reported experiments with cavitation

inception pressures (negative peaks) much lower than the liquid vapor pressure (Lee et al. 1985; Takenaka

1987; Fan and Tijsseling 1992; Simpson and Bergant 1996). Washio et al. (1994) even observed traveling

tensile waves (negative absolute pressure waves) in a dead end branch of a main oil pipe. The magnitude of the

negative absolute pressure is the tensile strength of the liquid. It is governed by the flow conditions and the

cavitational properties of liquid and pipe walls. Intense pre-pressurization of the liquid helps.

Plesset (1969), Overton et al. (1984) and Trevena (1984, 1987) gave an in-depth treatment of the subject of

tensile stresses in liquids. Recent results have been presented by Williams et al. (1999), Williams and Williams

(2000) and Brown and Williams (2000). Tensile stress is a meta-stable condition for the liquid, which in a

transient event should be described by non-equilibrium thermodynamics.

Shinada and Kojima (1987) measured small negative absolute pressures (not negative peaks) in transient

cavitation tests. They attributed this to the effect of surface tension, which can be important when the cavitation

zone consists of many miniscule bubbles.

2.3.2 Local vapor cavities

A large vapor cavity or liquid column separation is considered to have a local character (Kranenburg

1974a). The local void fraction is comparable with unity and the local velocity gradient is large. Examples

include the formation of a cavity adjacent to a closed valve, below a reaction turbine runner (at the draft

9

tube inlet), at a high point in a pipeline, or at an intermediate location along the pipeline as the result of two

crossing low-pressure waves (intermediate vapor cavity – see Section 2.3.3).

After the descriptions of column separation by Joukowsky (1900) and Mostowsky (1929), it was LeConte

(1937) who presented some of the first experimental results for a local liquid column separation at a rapidly

closing valve at the downstream end of a pipeline. LeConte (1937) also proposed an analytical development

based on rigid liquid-column analysis. A restitution coefficient was introduced to provide a match between

the analytical and experimental results. This coefficient was proposed to change for each pipe, and even

between waves in a single pipe, and therefore it was concluded to be difficult to estimate.

2.3.3 Intermediate vapor cavities

Lupton (1953) introduced the possibility of the formation of an internal gap or intermediate vapor cavity,

not located adjacent to a hydraulic device (valve, turbine, etc.) or at a high point. Moores (1953) initiated

consideration of the topic in a discussion of Lupton's paper, when he posed a question as to what happened

when surge or water hammer waves meet. Lupton's reply stated that: “if the meeting of the waves were

negative, and their sum exceeded the absolute head H existing initially, a gap would be formed”. Lupton

(1953) presented an example that exhibited the sequence of events leading to formation of an intermediate

vapor cavity after instantaneous stoppage of a pumping system. The example illustrated that an

intermediate cavity did not necessarily have to be located at a high point or change of slope of the pipeline.

In a Master's thesis at The University of Melbourne, O'Neill (1959) investigated the occurrence of

“intermediate” vapor cavities using the graphical method. He noted that most previous studies overlooked

the formation of these intermediate cavities, which impose an internal boundary condition in the pipe.

O'Neill (1959) presented a method to account for the formation of intermediate cavities. Experimental

results for a simple reservoir-pipeline-valve system were also presented. In addition, visualization studies of

growth and decay of intermediate cavities that used a high-speed movie camera were presented. Some of

the analytical examples exhibited short-duration pressure pulses - due to cavity collapse - that exceed the

initial pressure rise, Eq. (1), due to the valve closure. The experimental pressure trace records did not

exhibit this short-duration pressure pulse behavior.

Sharp (1960, 1965a, 1965b) continued O'Neill's (1959) work, and considered the growth and collapse of

small vapor cavities produced by a rarefaction wave. An ideal spherical cavity was analyzed using force

plus momentum principles (Sharp 1965a). Experimental results, which included high-speed photography of

single intermediate cavities, were presented. Sharp proposed that another type of cavity also existed

“during the first and succeeding rupture phases for an entirely different reason”. A succession of a small

number of cavities at regular intervals was asserted to form moving away from the valve (Sharp 1965a,

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1977). Reversal of the transient flow in the pipeline caused these cavities to collapse and a series of regular

pulses resulted when the liquid column rejoined. A modification of the graphical method of solution was

proposed, which assumed secondary type cavities at equal intervals along the pipeline. This phenomenon

appears to be similar to the occurrence of distributed vaporous cavitation in pipelines.

Jordan (1961) investigated column separation in a pumping system after pump failure. He developed an

analytical method for computation of exact locations of intermediate cavities along the pipeline. He

asserted that the location of the intermediate cavity found by the standard Schnyder (1932) - Bergeron

(1935) graphical method is approximate.

Simpson (1986), Simpson and Wylie (1989), and Bergant and Simpson (1999a) showed clear experimental

evidence of intermediate vapor cavity formation. Experimental evidence of short-duration pressure pulses

was also presented.

2.3.4 Distributed vaporous cavitation or two-phase (bubble) flow

Distributed vaporous cavitation is a region of two-phase flow consisting of both vapor and liquid. In

contrast to local or intermediate vapor cavities, distributed vaporous cavitation occurs over an extended

length of the pipe with the pressure at the vapor pressure of the liquid. The occurrence of distributed

vaporous cavitation is illustrated in an example of an upward sloping pipe by Simpson (1986, p. 38) and

Wylie and Streeter (1993, pp. 197-198). Once a low-pressure wave has passed along the pipe, the velocity

varies from one end of the pipe such that the fluid is tearing apart and the void fraction is growing. The

void fraction in distributed cavitation zones remains small (close to zero).

The difference between a local liquid column separation and distributed cavitation in a pipeline was first

described by Knapp (1937b) in a paper at the second water hammer symposium (Proceedings 1937) which

expanded an earlier reference (Knapp 1937a). He presented an example in which a pressure drop was

produced by a rupture of the pipe just below the shutoff valve. The negative wave traveled up the pipeline

unchanged, provided the wave front did not intersect the “zero pressure line”. The liquid column, between

this point of intersection and the reservoir, cavitated partially, but no liquid column separation occurred.

Reflection of the water hammer wave from the reservoir brought the liquid column back to its original state

without cavitation and produced a corresponding pressure rise. He concluded that “further investigations

were necessary to clear up completely such water hammer conditions with cavitation”. Knapp (1939)

further developed the concept of vaporous cavitation in a discussion of LeConte's (1937) paper. Knapp

(1939) concluded that cases involving vaporous cavitation could not be solved by the graphical method.

Knapp (1939) attributed many penstock failures in the upper portion of the penstock to the pressure rise that

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resulted from liquid column separation at a “knee” in the profile. Previously cited causes for these failures

had been described as of “obscure origin”.

De Haller and Bédué (1951) presented an analytical treatment of liquid column separation in a discharge

pipe resulting from a low-pressure wave propagating upwards along a pipe. They suggested that cavities

could occur along long sections of pipeline rather than forming immediately over the entire pipe cross

section. Lupton (1953) presented a summary of the graphical method, with one section devoted to

“separation of water columns”. The treatise considered water hammer problems in pumping systems.

Lupton (1953) provided a description of the sequence of events associated with distributed vaporous

cavitation. He distinguished between a localized liquid column separation and a region of distributed

cavitation. The transmission of a negative wave along an upward sloping frictionless pipe was considered.

The wave dropped the pressure to the vapor pressure of the liquid somewhere along the pipe. Lupton stated

that further along the pipeline

“the drop brought about in the velocity must diminish as the gradient is further ascended. In

other words, the residual velocity will increase as the wave reaches higher portions of the

main. The liquid column within such rising reaches will therefore become separated from the

more slowly moving portion and will degenerate into a series of thin slugs of liquid

interspersed with vacuous spaces (or more accurately, spaces at absolute pressure equal to the

vapor pressure of the liquid plus the partial pressure of any released dissolved gases)”.

Jordan (1965) investigated localized column separation and distributed vaporous cavitation in pumping

systems with horizontal, upward and downward sloping pipe sections. He developed an analytical method

for treatment of the distributed vaporous cavitation zones. Jordan (1965) studied the effect of hydraulic

grade line (HGL) and pipe slope on the formation of distributed vaporous cavitation zones. There was a

reasonable agreement between the computed results and results from measurements performed in a

laboratory apparatus at Turboinstitute, Ljubljana, Slovenia. The apparatus comprised a fast-closing

upstream valve and an upward sloping pipeline.

A vaporous cavitation region may result from the passage of a negative wave through part of a pipeline, in

which the static pressure decreases in the direction of wave propagation owing to friction or pipeline slope

(Kranenburg 1974a). In contrast, if the static pressure increases in the direction of wave propagation a

vaporous cavitation zone cannot occur (Zielke and Perko 1985; Simpson 1986; Simpson and Wylie 1989).

See Figure 2. This type of cavitation may extend over long parts of the pipeline. The void fraction is usually

much less than unity for small velocity gradients. A decrease in the pressure-change results in a smaller

velocity-change as the water hammer wave propagates along the pipe. Thus fluid particles, as the wave

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passes, end up having a faster velocity than the "earlier" particles. This causes the liquid to pull apart when

the negative wave drops the pressure to vapor pressure.

Figure 2(a) (left) Fast closing valve generating cavitation in (a) downward sloping pipe, (b)

horizontal pipe, and (c) upward sloping pipe. (Zielke and Perko 1985, Fig. 3)

Figure 2(b) (right) Cavity formation at a knee (change in pipe slope). (Zielke and Perko

1985, Fig. 4)

Comprehensive experimental evidence of the occurrence of distributed vaporous cavitation was given by

Simpson (1986), Simpson and Wylie (1989), Bergant (1992) and Bergant and Simpson (1999a). Simpson

and Wylie (1989) gave an illustrative example showing the formation of a distributed vaporous cavitation

zone.

2.4 Gaseous Cavitation

Many papers, starting in the 1970s and early 1980s, have addressed the effects of dissolved gas and gas

release on transients in pipelines (Swaffield (1969-1970, 1972a, 1972b); Enever (1972); Kranenburg

(1974b); Wiggert and Sundquist (1979); Wylie (1980); Baasiri and Tullis (1983); Hadj-Taieb and Lili

13

(1998); Kessal and Amaouche (2001)). One of the main features of liquids is their capability of absorbing a

certain amount of gas with which they come into contact through a free surface. In contrast to vapor

release, which takes only a few microseconds, the time for gas release is in the order of several seconds.

Gas absorption is slower than gas release (order of minutes) (Zielke et al. 1989). Gas release occurs in

several types of hydraulic systems (cooling water systems, long pipelines with high points, oil pipelines,

etc.). Dissolved gas is an important consideration in sewage water lines and aviation fuel lines. Gases come

out of solution when the pressure drops in the pipeline. If a cavity forms, it may be assumed that released

gas stays in the cavity and does not immediately redissolve following a rise in pressure. Kobori et al. (1955)

and Pearsall (1965) showed that the presence of entrained air or free gas reduces the wave propagation

velocity and accordingly the transient pressure variations. A significant limitation in the numerical models

proposed in each of the above studies was the need to make rather arbitrary assumptions regarding the rate

of release of gas. Dijkman and Vreugdenhil (1969) investigated theoretically the effect of dissolved gas on

wave dispersion and pressure rise following column separation.

2.5 Severe Pressure Peaks Following Cavity Collapse

Angus (1935) presented one of the first papers dealing with large short-duration pressure pulses due to the

collapse of a cavity in a pipeline. In one example, the collision of a liquid column with a closed valve

resulted in a short-duration pressure pulse of large magnitude. The duration of the pressure pulse was about

one tenth of a 2L/a period (Angus 1937).

Heath (1962), in a Master's thesis at Georgia Institute of Technology, investigated a local liquid column

separation. Both incompressible (rigid-column) and weakly-compressible (elastic) analytical analyses were

presented. The graphical method was used for the weakly-compressible analysis and exhibited short-

duration pressure pulses due to cavity collapse. Short-duration pressure pulses due to cavity collapse were

not apparent in the experimental results presented by Heath (1962).

A short-duration (t < 2L/a) pressure pulse (Simpson 1986) due to cavity collapse is defined as a delayed

step in the pressure-time trace superimposed on a pressure wave as shown in Figure 3. Both Simpson

(1986) and Bergant (1992) described in detail the sequence of events leading to a short-duration pressure

pulse. A rapid closure of the valve in a reservoir-horizontal pipe-valve system as shown in Figure 3(a) can

be used to illustrate this phenomenon. The small effect of friction is neglected in this example.

Instantaneous valve closure stops the liquid flow. The hydraulic grade line (HGL) along the pipe is initially

constant at H0. The instantaneous head rise at the valve is predicted by the Joukowsky Eq. (1) as

VgaH ∆−=∆ (2)

14

A

H0

Water HammerModel

Discrete Cavity Model

Hv / H0

H / H0

x / L

t / (L / a)

t / (L / a)

AB B

A

B

V0

0.0.

1.

2.

4.

6.

8.

8.6.4.2.0.0.

-1.

1.

2.

3.

a)

c)

b)

Figure 3 A short-duration pressure pulse. (a) Reservoir-pipe-valve system. (b) Wave paths in

distance-time plane. (c) Piezometric head history at valve.

The pressure wave travels towards the reservoir, where it reflects negatively, and returns to the closed valve

arriving at time 2L/a after the valve closure. Figure 3(b) shows the pattern of propagation and reflection of

pressure waves in the system. In addition, Figure 3(c) shows pressure traces next to the valve that are

predicted by a pure water hammer model (thin solid line) and by a single cavity model (thick solid line).

The liquid flows in the reverse direction (towards the reservoir) after time 2L/a. Complete stoppage of the

flow at the valve now requires a head drop of (a/g)V0. This pressure-head drop for the case shown in Figure

3(a) (thin solid line) would result in a head less than the vapor pressure-head (horizontal dashed line).

When the pressure drops to the liquid vapor pressure (thick solid line in Fig. 3(c)), the velocity in the

reverse direction is not reduced to zero when the water hammer wave reflects from the valve, but to

(Mostowsky 1929)

apHHgVVV vb

vc)/( *

000

γ−+−=∆− (3)

where H0 = static HGL at the valve, Hb = barometric head,, pv* = absolute vapor pressure at temperature T,

and γ = specific weight of liquid. Now, at the closed valve, the flow is still towards the reservoir. The liquid

detaches from the valve and a cavity at the valve begins to grow. The cavity acts as a fixed-pressure

boundary condition and water hammer waves with travel time L/a propagate to and from the reservoir. The

15

reverse liquid velocity decreases by ∆ (given by Eq. (3)) each time a water hammer wave reflects off

the vapor cavity. Once the velocity next to the valve becomes positive, the cavity begins to shrink until it

finally collapses (point A in Fig. 3). The head directly caused by the cavity collapse is less than the water-

hammer head generated at the beginning of the transient event, but the maximum pressure-head (greater

than Joukowsky) occurs, in this example, at a time of about 6L/a in the form of a narrow short-duration

pulse (point B in Fig. 3). This pulse is the superposition of the cavity-collapse head and the reservoir wave

head doubled by its reflection from the closed valve. The resulting maximum head (thick solid line) is

greater than the maximum head predicted by the Joukowsky equation (thin solid line). If the cavity collapse

would occur exactly at the arrival of a pressure wave front, then a short-duration pulse would not occur.

vcV

Wylie and Streeter (1993, pp. 192-196) also give a detailed description of the formation of a short-duration

pressure pulse. They showed that the duration of existence of the first vapor cavity approximately is

aL

HH

aL

HgaV

aLV

HgaT

ininincs

222 00 ∆

∆=∆

=∆

= (4)

where ∆Hin = H0 + Hb - pv*/γ is the head drop at which vaporization starts (cavitation inception head drop),

thereby confirming the formula found by Mostowsky (1929) shown in Figure 6(a) herein.

Walsh (1964) in a Master's Thesis at Syracuse University and Li and Walsh (1964) considered the

prediction of the maximum pressure generated by the collapse of the first cavity. Photographs showed a

cavity behind a closing valve. Minute bubbles probably indicated the presence of distributed vaporous

cavitation. One-dimensional equations of motion for a weakly-compressible liquid were applied assuming a

frictionless system. Laplace transforms were used to solve the system for velocity and head (Walsh 1964).

A “linear throttling” case was considered in detail. The pressure generated by the collapse of the cavity was

divided into two components: the first component was caused by the velocity of the liquid column at the

valve just before the cavity collapsed. The second component was caused by the maximum pressure in the

pipe at the instant just before the collapse of the cavity. The maximum head due to collapse of a cavity was

given by

max 2 RVfaH V Hg= + (5)

where Vf = velocity of the liquid column at the valve just before cavity collapse and HRV = difference in

elevation between the downstream reservoir and the vapor pressure-head elevation at the valve. Similar

equations were presented by Moshnin (1961) and Moshnin and Timofeeva (1965), Simpson and Wylie

(1985) and Wylie and Streeter (1993, p. 194: Hmax = ∆H + 2∆Hin) for the case of instantaneous closure of a

16

downstream valve, where Hmax can be twice the Joukowsky value. The situation can even be worse when

intermediate cavities form. Kottmann (1989) used rigid-column theory to show that the collapse of a

midpoint cavity caused pressure rises of three times Joukowsky.

For “non-linear throttling,” Li and Walsh (1964) stated that the maximum pressure in the pipe before

collapse of the cavity was HRV, if the valve discharge monotonically decreased during the period from the

time of valve closure to the beginning of liquid column separation. Eq. (5) also held for these conditions. Li

and Walsh (1964) carried out experimental tests in a 2-inch plastic pipe to substantiate the analytical

results. Walsh (1964) presented results for a number of experimental runs. Small steps in the pressure

diagram or short-duration pressure pulses due to cavity collapse may be noted in a couple of the

experimental results. The time of existence of the first cavity was quite long with respect to 2L/a. The

velocity Vf was computed from experimental results using Eq. (5).

Yamaguchi and Ichikawa (1976) and Yamaguchi at al. (1977) presented oscilloscope traces of experimental

results exhibiting what appear to be the first results in the literature that clearly show short-duration

pressure pulses. An upstream and a downstream closing valve were considered, with oil column separation

in laminar flow being the focus of these studies. Photographs of cavity formation and collapse at a valve

were also shown.

Gottlieb et al. (1981) investigated maximum pressures following the collapse of cavities in a pipeline. They

presented experimental results and numerical calculations. The numerical model was a simple discrete

cavity model for each calculation point based on the method of Streeter and Wylie (1967). Four different

configurations of a steel and a plastic pipeline were considered. Extremely high-pressure peaks were

recorded immediately upon collapse of the vapor cavity. The pressure then dropped to about 40% of the

pressure peak level and maintained this level for 2L/a seconds. The authors concluded the presence of

peaks resembled the pressure peaks associated with the implosion of gas bubbles in pumps. See also

Section 4.5 on the specific properties of pressure transducers.

Martin (1983) measured transient cavitation in a simple reservoir-pipe-valve system. The water contained a

minimal amount of dissolved gas. Limited cavitation (where the duration of the existence of the cavity at

the valve is relatively short) was emphasized in contrast to previous studies (on severe cavitation) reported

in the literature. The experimental results showed that the maximum pressure may exceed the Joukowsky

pressure rise (Eq. (1)) in the form of a short-duration pressure pulse. See Figure 4 (top). Unfortunately, the

reservoir pressure was rising during the experiment because the tank was too small. The measured

reservoir-pressure shown in Figure 4 (bottom) was thought to be typical for all experimental runs.

17

Figure 4 Experimental result showing short-duration pressure pulses (Martin 1983, 1989).

Absolute hydraulic grade line (HGL) variation with time at valve (top) and in reservoir

(bottom).

Graze and Horlacher (1983) investigated the possible occurrence of severe pressure peaks following cavity

collapse. Previously reported severe pressure peaks at the beginning of the cavity-collapse pressure-rise

were the focus of their study. Numerical results based on a discrete vapor cavity model with no air or gas

evolution matched the experimental results reasonably well. The experimental runs performed in various

pipe configurations had a time of existence of the first cavity of approximately 16 to 20 L/a. Short-duration

pressure pulses were not evident in the results, probably due to the length of time of existence of the first

cavity at the valve. The authors used both “piezo-resistive” and “inductive” types of transducers and found

overshoot occurring with the inductive (strain-gage type) transducer, as exhibited by the measurement of

pressures less than vapor pressure-head. Different inductive transducers produced different pressure peaks.

In contrast, the piezo-resistive transducers did not produce these pressure peaks. The authors stated that

“the pressures caused by the collapse of the vaporous cavity in the case of water column separation are of

the same, or less, magnitude as those for the water hammer due to rapid valve closure”.

18

De Almeida (1983), in referring to experimental measurements by Van de Sande and Belde (1981) and

Gottlieb et al. (1981), each of whom presented pressure peak values higher than those calculated by the

Joukowsky formula, commented: “though old, this apparently very simple problem has not been resolved

yet”. De Almeida (1983) cited five possible reasons for obtaining large pressures due to cavity collapse

including: non-uniform velocity distribution, unsteady friction effects, “column elastic effect”, local and

point effects. The “column elastic effect” was the result of the time of existence of the cavity not being an

integer value of 2L/a. He presented an expression for estimating the upper bound of the overpressure in a

frictionless system.

Kojima et al. (1984) presented a mathematical model for predicting both the pressure rise associated with

cavity collapse and the duration of the column separation. Photographs of cavity formation and collapse on

the downstream side of a valve were presented. Their experimental results exhibited short-duration pressure

pulses. The influence of unsteady laminar friction was studied. The authors concluded that the effects of

energy dissipation due to non-adiabatic behavior of gas bubbles in a liquid column and the surface tension

in the gas bubbles forming the separated cavity may be neglected. A “gas non-bubbly flow” model using

unsteady pipe friction accurately predicted the experimental results.

Simpson (1986) showed a range of short-duration pressure pulses measured in a reservoir - upward sloping

pipeline - valve system. See the Figures 13, 18 and 19 herein. Due to the upward slope of the pipe the vapor

cavity was confined to be adjacent to the valve with no distributed cavitation along the pipe (at least until

the collapse of the first vapor cavity).

2.6 Severity of Cavitation / Cavitation Intensity / Measures of Cavitation

The location and intensity of column separation is influenced by several system parameters including the

cause of the transient regime (rapid valve-closure, pump failure, turbine load-rejection), layout of the piping

system (pipeline dimensions, longitudinal profile and position of the valves) and hydraulic characteristics

(steady flow-velocity, static pressure-head, skin friction, cavitational properties of the liquid and pipe walls)

(Bergant 1992; Bergant and Simpson 1999a). The combination of several influential parameters creates

difficulties in defining design criteria on cavitation severity. For a simple reservoir-pipeline-valve system

the pressure rise following cavity collapse at the valve may or may not exceed the Joukowsky pressure rise

Eq. (1) and cavities may form at the valve and/or along the pipe (Martin 1983; Simpson 1986; Carmona et

al. 1987, 1988; Brunone and Greco 1990; Anderson et al. 1991; Simpson and Wylie 1991; Bergant 1992;

Bergant and Simpson 1999a).

Martin (1983) inferred the severity of cavitation from the duration, Tcs , of the first, mostly largest, column

separation at the valve. He introduced the cavitation severity index S = Tcs a /(2L). He classified the intensity

19

of column separation as limited, moderate or severe cavitation with respect to the “number of bubbles”

which may form in the pipeline. Paredes et al. (1987) defined the parameter indicating the 'severity of water

column separation' as the ratio of the instantaneous pressure just before the first negative wave reaches the

cavity boundary minus the instantaneous pressure that would be reached immediately afterwards, assuming that

no column separation occurs, to the absolute instantaneous pressure just before the first negative wave reaches

the downstream boundary. Carmona et al. (1987, 1988) and Anderson et al. (1991) proposed a somewhat

different index and Fan and Tijsseling (1992) based a measure on the structural time scale of a vibrating closed

pipe excited by external impact.

Bergant and Simpson (1999a) performed a parametric numerical analysis on a reservoir-pipe-valve system

to compute the critical flow conditions that classify different column-separation regimes according to the

maximum pressure. A broad range of values was taken for the initial flow velocity (V0), the static upstream

tank head (HT) and the pipe slope θ. The time scales L/a and Tc were constant. The numerical results

(DGCM) shown in Figure 5 clearly show the influence of the initial velocity, static upstream tank head and

(small) pipe slope on the maximum head at the valve. The maximum dimensionless head is presented as the

ratio of maximum head rise (Hmax−H0)v at the valve (H0 = steady-state head at the valve) to the Joukowsky

head rise aV0/g. Water hammer without column separation occurs for low initial flow velocities. The

magnitude of the short-duration pressure pulse (see Section 2.5) is related to the magnitude of the reflection

at the reservoir of the low-pressure wave and the intensity of cavitation along the pipeline. The maximum

head for higher-velocity column-separation cases gradually decreases to the Joukowsky head. The

maximum cavity volume occurs at the valve, whereas smaller cavities are formed along the pipe. The

stronger attenuation of short-duration pressure pulses in a downward sloping pipe may be attributed to the

more intense cavitation along the pipe.

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.600.5

1.0

1.5

2.0

2.5

3.0

θ = -3.2o

HT,1=5. m HT,1=10. m HT,1=15. m HT,1=20. m HT,1=25. m

(

Hm

ax-H

0) v / (a

V 0/g)

(-)

V0 (m/s)

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.600.5

1.0

1.5

2.0

2.5

3.0 HT,2=7. m HT,2=12. m HT,2=17. m HT,2=22. m HT,2=27. m

θ = +3.2o

(b)(a)

(H

max-H

0) v / (a

V 0/g)

(-)

V0 (m/s)

Figure 5 Computed maximum head at valve as function of initial velocity (adapted from

Bergant and Simpson 1999a). (a) downward sloping pipe, (b) upward sloping pipe.

20

Bergant and Simpson (1999a) classified transient regimes according to the physical state of the liquid and

the maximum pipeline pressure as:

(1) Water hammer flow regime. No column separation occurs during transients. Joukowsky pressure.

(2) Active column separation flow regime. The maximum pipeline pressure is generated following the first

column separation at the valve and along the pipeline (active column separation from the designer’s point

of view). The maximum pressure at the valve is governed by the intensity of the short-duration pressure

pulse. Joukowsky pressure exceeded.

(3) Passive column separation flow regime. The maximum pipeline pressure is the water hammer pressure

before intense cavitation occurs. Joukowsky pressure.

This classification is based on maximum pressures. The steepness of wave fronts is another important

parameter in the assessment of dynamic loads on pipe systems and their supporting structures. Column-

separation collapse probably is the most important source of very steep wave fronts.

For water hammer and column separation events, Fanelli (2000) has classified the following regimes:

(1) no cavitation,

(2) cavitation, but cavity does not reclose,

(3) cavitation with cavity reclosure, but no high pressures,

(4) cavitation with cavity reclosure, but with excessive pressures.

3 MATHEMATICAL MODELS AND NUMERICAL METHODS

Until the 1960s, a comprehensive investigation of liquid column separation and cavitation in pipelines was

not possible due to the unavailability of computers. In a report on the status of liquid transients in Western

Europe, Martin (1973) stated that “only in the last decade there has been concerted activity on transient

problems for which vapor or gas was present in the liquid”. Until around 1960, most studies used graphical

and arithmetic procedures originally set forth by Gibson (1919-1920), Schnyder (1932) (see Hager (2001)),

Angus (1935), Stepanoff (1949), Bergeron (1950) and Parmakian (1955). The first computer-oriented

procedures for the treatment and analysis of water hammer include work by Thibessard (1961), Lai (1961),

Streeter and Lai (1962), Streeter (1963, 1964, 1965-1966), Van De Riet (1964), Vreugdenhil (1964) and

Contractor (1965). In a preface to the proceedings of the Third Symposium on Water Hammer - Pumped

Storage Projects (Proceedings 1965), Streeter (1965-1966) observed that the development of digital

computer methods of water hammer analysis had occurred during the previous 5 years.

21

Some of the first studies of liquid column separation using a numerical computer simulation model include

the work by Thibessard (1961) at Liège in Belgium, Streeter and Wylie (1967), Baltzer (1967a, 1967b) at

the University of Michigan, Siemons (1967) at Delft in The Netherlands, and Weyler (1969) and Weyler et

al. (1971) at the University of Michigan.

Column separation (formation and collapse of large discrete cavities, evolution and condensation of

vaporous cavitation regions) may be described by a set of one-dimensional equations each representing a

particular physical state of the liquid. These include the states of liquid, a mixture of liquid and vapor

bubbles, and discrete vapor cavities (Bergant and Simpson 1997, 1999a). A negligible amount of free or

released gas in the liquid is assumed (Blake 1986; Hansson et al. 1982; Wylie 1984). The set of equations

include

(1) Water hammer equations describing the liquid state,

(2) Equations for a discrete vapor cavity or an intermediate cavity that separates liquid and vaporous

cavitation regions along the pipeline or at boundaries (e.g. valve, knee),

(3) Two-phase flow equations for a homogeneous mixture of liquid and liquid-vapor,

(4) Shock equations for the condensation of liquid-vapor mixtures back to the liquid phase.

Each of these various types of flow regime is described in the subsequent sections. The derived equations

may be applied to all types of pipe configurations and to various cases of phase interactions.

3.1 Assumptions

There are a number of assumptions made in relation to the development of mathematical models (Bergant

and Simpson 1997, Bergant and Tijsseling 2001, Bergant et al. 2003). The assumption of one-dimensional

flow is made including average cross-sectional values of flow velocity, pressure, density and void fraction.

Only the assumptions related to column separation are detailed here:

(1) All liquids have entrained free gases or dissolved gases that evolve when the pressure drops below

saturation pressure. This review mainly considers flow situations where the presence of substantial

quantities of air is not considered. Air release is considered in detail by Wylie and Streeter (1993, pp.

178-184). Vaporous cavitation is inertia-driven cavitation and sub-cooled (the equations are valid up to

a temperature of about 330 K, above which thermodynamic effects become important (Hatwin et al.

1970)).

(2) The volume of the vapor cavity must be significantly less than the volume of a reach in the numerical

model. The pressure in the cavity is equal to the vapor pressure. Water hammer waves are reflected off the

22

cavity, which is assumed to occupy the total cross-sectional area of the pipe. The vapor cavity does not

move. Mass and momentum of the vapor in the cavity are negligible. Isothermal conditions in the cavity

prevail. The vapor condenses completely prior to the instant of rejoining of liquid columns or cavity

collapse against the boundary. The formation of cavities has no effect on friction losses in the pipeline.

(3) The void fraction of the vapor in a mixture of liquid and vapor bubbles (distributed vaporous cavitation

zone) is much smaller than unity, so the mass and momentum of the bubbles can be neglected. The

surface tension effect that results in a pressure difference across the vapor bubbles is ignored

(Sundquist 1977). The liquid and the vapor-bubble velocities in the mixture are the same during

vaporous cavitation (Prosperetti and Van Wijngaarden 1976). The vapor bubble is not influenced by

the expansion and compression of the neighboring liquid-vapor bubbles (Brennen 1973). The influence

of gravity on the bubbles is neglected.

(4) There is an infinitesimal width of discontinuity between the interface (shock wave front) of the one-

phase fluid (liquid) and the one-component two-phase fluid (homogeneous mixture of liquid and vapor

bubbles). Increase in temperature across a shock wave front is small and therefore isothermal

conditions across the interface prevail (Campbell and Pitcher 1958).

3.2 Water Hammer Equations

The water hammer equations are applied to calculate unsteady pipe liquid flow when the pressure is (always

and everywhere) greater than the vapor pressure. They comprise the continuity equation and the equation of

motion (Wylie and Streeter 1993):

2

+ sinθ∂ ∂ ∂− +∂ ∂ ∂H H VaV Vt x g x

0= (6)

g| |D

∂∂

+ ∂∂

+ ∂∂

+H =x

Vt

V Vx

f V V2

0 (7)

where H = piezometric head (hydraulic grade line HGL), t = time, V = flow velocity, x = distance along the

pipeline, θ = pipe slope, a = pressure wave speed, g = gravitational acceleration, f = Darcy-Weisbach

friction factor (its history is nicely described by Brown 2002) and D = pipe diameter. For most engineering

applications, the convective terms V(∂H/∂x), V(∂V/∂x) and Vsinθ are very small compared to the other terms

and therefore neglected. Research by Streeter and Wylie (1967) led the world to the direct use of the method of

characteristics as a numerical method on a digital computer to provide solutions to Eqs. (6) and (7). The method

23

of characteristics has been the standard solution method for solving water hammer in pipeline systems for the

last 40 years.

3.3 Discrete Single Cavity Models

Single vapor cavity numerical models deal with local column separations as described in Section 2.3.2. A

single cavity is used either at a boundary, at a high point in the pipeline, or at a change in pipe slope. Most

graphical solutions of water hammer problems employed this modeling approach. Rigid column theory has

also been used to compute the behavior of systems with single cavities.

Figure 6(a) Theoretical analysis of column separation: pressure as function of time (Mostowsky

1929, Fig. 7).

Figure 6(b) (left) Theoretical analysis of column separation: velocity as function of time

(Mostowsky 1929, Fig. 8).

Figure 6(c) (right) Laboratory measurement of column separation: pressure as function of time

(Mostowsky 1929, Fig. 16).

24

Mostowsky (1929) analyzed column separation at a downstream valve. Figure 6(a) shows the pressure-time

diagram for a frictionless pipe where the duration Tcs of the column separation is an integer multiple of the

wave travel time 2L/a. This figure is a confirmation of Eq. (4). Figure 6(b) is the corresponding velocity-

time diagram that determines the time of separation. Mostowsky performed measurements in a 29.5 m long,

2 inch diameter pipe. The measured pressure history in Figure 6(c) shows that, unlike Figure 6(a), the

second pressure rise is lower than the first one, and the measured number of 11-times 2L/a is not the 12-

times predicted by Eq. (4). Mostowsky, attributing these discrepancies to friction, performed a rigid column

analysis with a quadratic friction law. He developed closed-form solutions pertaining to rigid-column

theory, for example a formula relating the effective friction coefficient to P1 and P2 (these are indicated in

Fig. 6(c)), and an improved formula for Tcs giving a value of 10.6-times 2L/a in agreement with the

experiment.

Angus (1935) presented one of the first mathematical models of a single vapor cavity at a boundary. A

graphical method was presented (Fig. 7). For a pump failure on a discharge line, the possibility was

investigated of a cavity forming on the pipe side of a check valve near the pump. After the cavity formed

and the liquid column returned to the closed valve, the pressure rise was found to be about “four times” the

normal value. This was some of the first literature dealing with large short-duration pressure pulses due to

the collapse of a cavity in a pipeline. Angus (1937) prepared a bulletin on water hammer that included an

example of a power failure for a pump with a small inertia. The example assumed a pump discharge valve

closed instantly when the pump failed. A vapor cavity formed on the pipe side of the valve and existed for

about 2L/a seconds. The eventual reversal of the liquid column resulted in a short-duration pressure pulse

of large magnitude after the cavity collapsed. The duration of the pressure pulse was about one tenth of a

2L/a period.

Figure 7 Graphical method by Angus (1935, Fig. 6).

25

In a discussion of LeConte's paper (1937), Bergeron (1939) presented an example of the formation of a

cavity at a valve. The graphical method was used which included "lumped" friction losses. Bergeron (1950)

published a book that provided an extensive treatment of water hammer in hydraulics and surges in

electricity generation with the English translation being published in 1961. Treatment of a local liquid

column separation was similar to that of Angus (1935), however, examples were presented in which the

cavity existed for longer than 2L/a seconds. Improvements were made in the schematization for local liquid

column separations. Bergeron indicated that the underpressure in a cavity was not the “barometric

vacuum”, but rather the vapor pressure at the temperature of the liquid. He neglected the influence of

gravity on the shape of the liquid-vapor interface. The interface was assumed to be perpendicular to the

pipe and the pressure at the local liquid column separation was equated to the vapor pressure of the liquid.

In one example, a cavity − formed at the valve after 2L/a seconds − collapsed at an exact multiple of 2L/a

seconds; in other words, just as the initial wave due to the instantaneous valve closure arrived back at the

valve. In another example conditions were selected such that the cavity collapsed at a time that was not a

multiple of 2L/a seconds after the valve closed. The collapse of the cavity generated a second water-

hammer wave in the system with a different timing than the initial water-hammer wave created by the

closure of the valve. When the initial valve-closure wave returned to the valve for the first time after

collapse of the cavity, a distinct step in the pressure-time trace was generated. The maximum pressure of

the step did not exceed the magnitude of the initial Joukowsky pressure rise due to the valve closure. In this

application of the graphical method, friction was lumped at the upstream end by using an orifice. A number

of cycles of successive formation and collapse of a local liquid column separation was considered. Friction

loss resulted in damping and the mean value of head rise decreased with every cycle, as did the maximum

volume of the cavity.

Kephart and Davis (1961) used rigid-column theory to determine the magnitude of the pressure rise when

liquid columns rejoined in pump discharge lines equipped with check valves at the pump outlet. This

method served as a preliminary design technique. Escande (1962) included cavities and lumped non-linear

friction in his calculations.

Streeter and Wylie (1967) presented a computer model describing vaporous cavitation using only a single

vapor cavity in a pipeline. Rupture of a pipeline just below an upstream reservoir was considered. A single

cavity was assumed to form at the point in the pipeline that first dropped to the vapor pressure of the liquid.

Weyler (1969) continued studies of liquid column separation at The University of Michigan. His numerical

model used a single vapor cavity at the valve. Cavities were not permitted to form at the other

computational sections. A semi-empirical “bubble shear stress” was proposed to predict the increased

momentum loss observed under liquid column separation conditions. The “bubble shear stress” was

26

attributed to the non-adiabatic expansion and collapse of gas bubbles present throughout the low-pressure

flow. Microscopic bubbles of permanent gas, usually air, served as nuclei for the explosive vaporous

growth. More air was picked up by diffusion of dissolved air in the liquid during the initial phase of the

transient. Air remained undissolved accounting for the decrease in wave speed. A single spherical bubble

was examined and compressive dissipation was concluded to be large compared with the viscous

dissipation for a single spherical bubble. De-aerated liquid was found to undergo much more violent

opening and closing of cavities, a behavior characteristic of distributed vaporous cavitation. Safwat (1972b)

also considered the wave attenuation problem and introduced an equivalent shear stress concept.

Kranenburg (1974a) disagreed with Weyler and Safwat and contended that the thermodynamic behavior

was essentially isothermal (because of the small size of the bubbles) and that no dissipation would occur

due to this bubble shear stress mechanism.

The growth and subsequent decay of the discrete vapor cavity is calculated by the continuity equation:

(8) tVVA u

t

tvc

in

d)( −=∀ ∫

where ∀vc = discrete vapor cavity volume, tin = time of inception of the discrete vapor cavity, A = cross-

sectional pipe area, V = outflow velocity at the downstream side of the vapor cavity, and Vu = inflow velocity at

the upstream side of the vapor cavity.

3.4 Discrete Multiple Cavity Models

This type of model includes the concentrated or discrete vapor cavity models (DVCM). Liquid column

separations and regions of vaporous cavitation are both modeled using discrete cavities at all computational

sections. Liquid is assumed to be in between all computational sections and the method of characteristics is

applied throughout the pipeline, even in vaporous cavitation regions. The discrete free gas cavity model

(DGCM) (Wylie 1984, Zielke and Perko 1985) is similar to the discrete vapor cavity model, with a quantity

of free air assumed to be concentrated at each computational section. The only difference with DVCM is in

the ∀ - p* curve as shown in Figure 8.

Bergeron (1950) considered the formation of a cavity at a location away from a pump. Potential cavity

locations were identified from the profile of the pipeline. The cavity at the valve was treated correctly.

However, O'Neill (1959) and Sharp (1965a, 1965b) pointed out that Bergeron's work did not account for an

intermediate vapor cavity that formed away from the control valve, following the first collapse of the cavity

at the valve. The meeting of two low-pressure water-hammer waves produced a potential pressure

considerably less than vapor pressure and resulted in the formation of an intermediate cavity.

27

Figure 8 p*-∀ curves for DVCM (thick solid line, ∀0 = 0 mm3) and DGCM (thin solid lines,

∀0 = 0.01 mm3, ∀0 = 0.1 mm3, ∀0 = 1 mm3, ∀0 = 5 mm3, ∀0 = 10 mm3) with p0* = 1 bar and

pv* = 0.2 bar. The value of pv* (water at 60 oC) is chosen large to show its influence.

3.4.1 The discrete vapor cavity model (DVCM)

The discrete vapor cavity model (DVCM) continues to be the most commonly used model for column

separation and distributed cavitation at the present time. One significant advantage of the DVCM is that it

is easily implemented and that it reproduces many features of the physical events of column separation in

pipelines. Bergeron (1950, pp. 89-95) and Streeter and Wylie (1967, p. 209), described in general terms a

procedure for calculating liquid column separation and rejoining (single cavity). The first development of

the discrete vapor cavity model (multiple cavities) appears to be by Thibessard (1961) and independently

by Streeter (1969) and Tanahashi and Kasahara (1969).

Figure 9 Definition sketch for discrete cavity model (Tijsseling et al. 1996, Fig. 8)

28

Wylie and Streeter (1978a, 1993) have described the DVCM in detail and they provided a FORTRAN

computer code for its implementation. Cavities are allowed to form at any of the computational sections if

the pressure is computed to become below the vapor pressure of the liquid. The DVCM does not

specifically differentiate between localized vapor cavities and distributed vaporous cavitation (Simpson and

Wylie 1989, Bergant and Simpson 1999a). Vapor cavities are thus confined to computational sections, and

a constant pressure wave speed is assumed for the liquid between computational sections. See Figure 9.

Upon formation of a cavity, a computational section is treated as a fixed internal boundary condition. The

pressure is set equal to the vapor pressure of the liquid until the cavity collapses. Both upstream and

downstream discharges for the computational section are computed, using the C+ and C– compatibility

relationships for each of the positive and negative characteristics within the method of characteristics

(MOC) solution. These are (see Fig. 10 for the MOC x-t staggered grid):

PupcpcP QBCH −= (9)

PmcmcP QBCH += (10)

where Cpc and Bpc are constants for the positive or plus characteristic (pc), and Cmc and Bmc are constants for

the negative or minus characteristic (mc) based on HP and QP at the previous time step at adjacent

computational sections A and B in Fig. 10.

t ∆x

0. L

C-

x

1 2 j N N+1

P t+∆t

t

∆t

B A

C+

Figure 10 The method of characteristics staggered grid in reservoir-pipe-valve system.

29

The vapor cavity volume change at a computational section in terms of the difference in discharge at

the downstream (Q

vc∆∀

P) and upstream (QPu) side of the section is

( )∫∆+

−=∀∆tt

tPuPvc tQQ d (11)

The volume of the cavity is cumulated at each computational section where vapor pressure exists. If the

cavity volume becomes zero or negative, the computation returns to the standard method of characteristics

(water hammer solution). Provoost (1976) used a closure condition that exactly satisfies the mass balance.

Wylie and Streeter (1993, p. 69) showed a comparison of the results of the DVCM and the experimental

results by Li and Walsh (1964) of an isolated cavity formation at the downstream side of a valve. The

magnitudes of the pressures were reasonably predicted, whereas the timing of existence of column

separation was not well predicted.

Although the discrete vapor cavity model was correctly formulated by Thibessard (1961), Streeter (1969)

and Tanahashi and Kasahara (1969), these investigators did not apply the model to regions of distributed

cavitation. This was done by De Vries (1972, 1973, 1974) when he simulated the experiments at Delft

Hydraulics Laboratory performed by Vreugdenhil et al. (1972) and Kloosterman et al. (1973). De Vries

(1972) was the first to report on numerical oscillations induced by the simulated annihilation of a region of

distributed cavitation. See Figure 11. To suppress these oscillations he added small amounts of free gas to

the discrete cavities. In fact he then used the discrete (concentrated, lumped) free gas cavity model

developed by Brown (1968). This model, described in Section 3.4.2 and used by Enever (1972), Tullis et al.

(1976), Ewing (1980) and Suda (1990), allowed for the presence of free gas in the liquid. Ewing (1980)

also discussed various damping mechanisms in liquid-gas mixtures. Evans and Sage (1983) had confidence

in the DVCM and used it for the water-hammer analysis of a practical situation. Capozza (1986) applied the

method to condenser cooling circuits. Wylie (1984) and Zielke and Perko (1985) gave thorough treatises of

both the vapor and the free gas discrete cavity models.

Bergant and Simpson (1999b) incorporated cavitation inception with “negative” pressure into the discrete vapor

cavity (DVCM) numerical model. Numerical and measured results of experimental runs with “negative”

pressure spikes were compared. The local "negative" pressure spike at cavitation inception did not significantly

affect the column separation phenomena.

30

Figure 11 Numerical oscillations in DVCM results by De Vries (1972, Fig. 42). Pressure-

head histories at different positions after pump shut-down.

Bergant (1992) classified methods to attenuate unrealistic pressure spikes due to multi-cavity collapse as

follows:

(1) Interpolation within the MOC grid (Numerical damping). Kot and Youngdahl (1978a) used spatial

interpolation for a complete (with convective terms) set of water hammer compatibility equations.

A similar approach has been adopted by Anderson and Arfaie (1991). The authors introduced a

variable length of the liquid column due to formation, growth and collapse of vapor cavities along

the pipe. Miwa et al. (1990) artificially corrected the numerical time step to ∆t = 0.95 ∆x / a and

then used spatial interpolations. Spatial interpolation causes numerical damping (Goldberg and

Wylie 1983), which may suppress physical pressure pulses as observed in measurements.

(2) Additional damping mechanisms (Physical damping). Weyler et al. (1971) attributed additional

(apparent) 'bubble shear stress' to the non-adiabatic expansion and collapse of gas bubbles present

throughout the low-pressure cavitating flow. Similar models were developed by Safwat (1975) and

De Bernardinis et al. (1975). Alexandrou and Wylie (1986) proved analytically that during column

separation the 'thermodynamic' losses are much smaller than the losses due to skin friction at the

pipe wall. Kojima et al. (1984) implemented Zielke's (1968) unsteady friction model into a column

separation model. The authors investigated an oil-hydraulic system with transient laminar flow.

They compared computational results with results of measurements and found that the

consideration of unsteady friction in DVCM improved the numerical results. Similar approaches

using unsteady friction model have been used by Shuy and Apelt (1983), Brunone and Greco

(1990), Bergant and Simpson (1994a), Bughazem and Anderson (1996, 2000) and Bergant and

Tijsseling (2001). Implementation of unsteady friction models into DVCM is discussed in Section

3.10.

(3) Accounting for the discrete vapor cavity volume (Numerical damping). Safwat and Van Den

Polder (1973) allowed discrete cavities to form only at predetermined locations like boundaries

31

and thus avoided problems with multi-cavity collapse. When the pressure at the internal

computational section dropped below the vapor pressure, the discharge at this section was taken

the average of the two discharges calculated from Eqs. (9) and (10), with HP set to Hv. Carmona et

al. (1988) adopted a similar approach. A disadvantage of this method is that the user must

preselect the potential locations where localized vapor cavities may form. Also, the averaging of

discharges has no physical meaning. Golia and Greco (1990) allowed cavities to form at a selected

number of internal pipeline computational sections. At other sections the pressure was allowed to

drop below the liquid vapor pressure, which is physically not realistic. Several authors tried to

control the numerical spikes with an adequate integration of Eq. (11). A generalized numerical

integration scheme for the continuity equation describing the vapor cavity volume is (adopted

from Wylie (1984) for the MOC with rectangular grid):

tP P P P{ ( ) (1 ) ( )}t t t t t t t t

v v u uQ Q Q Qψ ψ−∆ −∆ −∆∀ = ∀ + − + − − ∆t (12)

where ∀vt is the vapor cavity volume at time t, ψ is a weighting factor, and QPu

t and QP t are the

upstream and downstream discharges at the computational section at time t. The standard form of

(12) is the trapezoidal rule which uses a ψ of 0.5 (Tanahashi and Kasahara 1969, 1970; Streeter

1972; Wylie and Streeter 1978a; Safwat et al. 1986; Liou 1999). Streeter (1969), Provoost (1976)

and Bergant and Simpson (1999a) used a ψ value of 1.0. Bergant (1992) and Simpson and Bergant

(1994a) performed numerical computations taking ψ values in the range from 0.5 to 1.0 and found

that when using a ψ close to 1.0, which was also recommended by Liou (2000), that the pressure

spikes were suppressed significantly.

(4) Allowing free gas in a cavity. (Physical damping). As stated in one of the previous paragraphs it

was De Vries (1973) who first employed a discrete gas cavity model (DGCM) to suppress

spurious pressure spikes and this model has since been used by several researchers. The DGCM

and its properties are discussed in the next section.

(5) Filters. (Numerical damping). Kranenburg (1974a) used a numerical filter to suppress spurious

oscillations.

Bergant (1992), Bergant and Simpson (1994b), and Simpson and Bergant (1994a) compared a

number of pipe column separation models. They found that within the MOC the staggered grid is

preferred above the rectangular grid, which actually comprises two independent staggered grids

(chess-board instability). The model with staggered grid and a ψ value of 1.0 (DVCM) gave

reasonably accurate results when the number of reaches was restricted (the ratio of maximum

cavity size to reach volume should be below 10%).

32

3.4.2 The discrete gas cavity model (DGCM)

Brown (1968) presented the first attempt at describing liquid column separation with the effects of

entrained air. Entrained air was assumed to be evenly distributed in concentrated pockets at equal distances

along the pipeline. The presence of air decreased the overall pressure wave celerity. A pressure-volume

relation was assumed to describe the behavior of the concentrated air pockets, whose locations were

assumed to be permanent with no change due to the prevailing direction of flow. The presence of entrained

air was neglected above a certain head, where the solution reverted back to normal water hammer

computations.

The DGCM is a numerical model that has been used for modeling column separation over the last 30 years

but not as widely as DVCM. The numerical model utilizes free gas volumes to simulate distributed free gas

(Wylie 1992). After Brown (1968), this model was further developed by De Vries (1973), Provoost (1976),

Provoost and Wylie (1981) and Wylie (1984). The discrete free gas cavity model is a modification of the

discrete vapor cavity model. Figure 8 shows that the DVCM is a limit-case of the DGCM.

Provoost (1976) and Wylie (1984) presented a detailed description of a DGCM, in which cavities were

concentrated at “grid points” (or computational sections). Pure liquid was assumed to remain in each

computational reach. A quantity of free gas was introduced at each computational section. Wylie (1984)

and Wylie and Streeter (1993) described the discrete free gas model for simulating vaporous and gaseous

cavitation. Gas volumes at each computational section expanded and contracted as the pressure varied and

were assumed to behave as an isothermal perfect gas. This model exhibited dispersion of the wave front

during rarefaction waves and steepening of the wave front for compressive waves. Distributed cavitation in

pipelines may be successfully simulated by using very small quantities of free gas.

According to the ideal gas law, the void fraction α varies inversely with the absolute partial pressure of the

gas as *gp

*g g

g m

M R Tp

α =∀

(13)

where Mg = mass of free gas, Rg = specific gas constant, T = absolute temperature, and∀ = unit mixture

volume. The amount of gas concentrated at each computational section is determined by coalescing to the

section the distributed gas from the adjacent half reaches. The volume of fluid mixture ∀ in each adjacent

reach is taken to be constant. Eq. (13) can be written for a reference void fraction α

m

m

0 at a given reference

33

pressure *0gp . As the absolute pressure of the gas changes throughout the transient, the volume of

isothermal gas from Eq. (13) and from the reference condition is

*gp

Cz

∀ =− +

p

g

dtd∀

t∀ = (1ψ ψ−

*

0 0* /

g mg m

g P b v

pp H H pα

αγ

∀∀ = =

− (14) *

where C = *0g α0 m∀ / γ, HP = piezometric head at the computational section, z = elevation of the

centerline of the pipe, Hb = barometric pressure-head, pv* = absolute vapor pressure at temperature T, and γ

= specific weight of liquid. Note that p* = pg* + pv

*. A value of the reference void fraction α0 of 10-7 or

smaller is taken for simulation of vaporous cavitation. For this order of void fraction, there is only a small

change in pressure wave speed due to the presence of the free gas, even at low pressures.

Continuity of the discrete gas cavity volume (also the vapor volume) is given by

uPP QQ −= (15)

where QP and QPu are the flow rates at the downstream and upstream side of the computational section,

respectively. Provoost and Wylie (1981) presented various schemes for integrating the gas volume equation

(15). The trapezoidal rule yielded high frequency pressure oscillations, whereas a forward integration

procedure damped out the oscillations. Wylie (1984) integrated Eq. (15) and introduced a time-direction

weighting-factor ψ to control numerical oscillations. The solution written for the staggered grid of the

MOC was

2 t 2 2

P P P P{ ( ) ) ( )} 2t t t t t t tg g u uQ Q Q Q− ∆ − ∆ − ∆∀ + − + − ∆ (16) t

in which ∀gt and ∀g

t−2∆t are the gas volumes at the current time and at 2∆t earlier. The range of values of ψ

was from zero to unity, however, selecting a value greater than 0.5 was recommended to avoid severe

oscillations. The forward integration scheme (this is the Euler-backward method for ODE Eq. (15)) used a

value of ψ of 1.0 resulting in minimal numerical oscillations (Bergant 1992; Wylie and Streeter 1993;

Simpson and Bergant 1994a). Liou (1999, 2000) recommended a value of ψ close to or equal to unity in

combination with a sufficient number of computational reaches. In a rigorous treatise of the numerical

method, applying Von Neumann analysis to a linearized set of equations, Liou nicely showed that the

numerical wave speed converges to the theoretical (physical) one (see Figure 12). He explained why the

DCGM exhibits non-linear variable wave-speed features.

34

Figure 12 Wave speed ratios (Liou 2000, Fig. 5). Dots: numerical (DGCM) wave speed /

gas-free wave speed. Lines: theoretical mixture wave speed / gas-free wave speed.

Wylie and Streeter (1993, pp. 195-196) compared the performance of the DGCM against experimental

results (Fig. 13) where only one vapor cavity formed adjacent to the valve with no distributed cavitation.

The results clearly show the occurrence of short-duration pressure pulses of different relative magnitudes

and widths.

Figure 13 DGCM results versus experimental results (Simpson 1986, Fig. 7.8). Hydraulic

grade line (HGL) variation with time.

Wylie (1992) and Wylie and Streeter (1993, pp. 202-205) compared results from the DGCM with analytical

and experimental data in a low void-fraction system during rapid transient events. Favorable comparisons

were obtained although highly non-linear behavior was observed. The wave speed variation with pressure

in a system with free gas was demonstrated. Wylie and Streeter (1993, pp. 188-192) tested the DGCM for a

bubbly flow case in which air was dispersed in the continuous liquid phase. The experimental results were

35

taken from Akagawa and Fujii (1987). Wylie and Streeter (1993) considered three different void fraction

distributions in applying the DGCM. The comparison of numerical and experimental results for two of the

three cases was quite favorable despite the results being extremely sensitive to the amount of free gas in the

system. In the third case, with a uniform void fraction in one section and a single phase upstream, the

DGCM produced oscillations that were not present in the experimental results.

Bergant and Simpson (1999a) validated the DGCM (including DVCM and GIVCM (generalized interface

vaporous cavitation model) results) against experimental results for the rapid closure of a downstream

valve. First the results for two different initial flow velocities (V0 = 0.30 and 1.40 m/s) in an upward sloping

pipe were compared. The low velocity case represented a column separation event with a maximum

pressure larger than the Joukowsky value (due to a short-duration pressure pulse); the high velocity column

separation case generated a maximum pressure lower than the valve-closure pressure. Next the results for

two different static heads (H0 = 12.0 and 22.0 m) in the upstream reservoir and an initial flow velocity V0 =

0.30 m/s were considered. Valve closure for the lower static head generated column separation with a wide

short-duration pressure pulse. The influence of pipe slope was investigated by comparing experimental runs

in a downward and an upward sloping pipe for an identical initial flow velocity V0 = 0.71 m/s and identical

static pressure-head at the valve of 20.0 m. The authors found a slightly different timing of the cavity

collapse at the valve and a slightly changed intensity of cavitation along the pipeline for the two runs. The

differences were caused by the difference in steady head envelope (hydraulic grade line) and direction of

action of gravity. In addition, the authors presented a global comparison of DGCM and experimental results

for a number of flow regimes in downward and upward sloping pipes (30 cases). A comparison was made

for the maximum head at the valve and the duration of maximum cavity volume at the valve. The

agreement between the computed and measured results was acceptable. Barbero and Ciaponi (1992)

performed a similar global comparison analysis between DGCM simulations and measurements.

3.5 Shallow Water Flow or Separated Flow Models

This type of numerical model describes liquid column separation or cavitation regions with shallow water

(open channel) theory. The water hammer (liquid) regions are calculated by the method of characteristics.

They are separated from the shallow water regions by moving boundaries. Shallow water flow modeling of

regions of vapor pressure provided the first real attempt at a more realistic description of transient

cavitation. Vapor bubbles were assumed to form, rise quickly and agglomerate to form a single long thin

cavity when the pressure reached the vapor pressure (Provoost 1976).

Li (1962, 1963) presented a study of the mechanics of pipe flow following a local liquid column separation

at an upstream closing valve. A numerical example of the motion and spreading of the vapor-liquid

interface was considered when an upstream valve closed instantaneously on an upward sloping pipe of

36

3.9 m length. A topological study of the phase plane was used. Spreading of the interface was described by

shallow water theory using two quasi-linear partial differential equations. The study revealed that the

spreading of the surface may be neglected in computing the water hammer pressure resulting from the

cavity collapse. A small variation in vapor pressure occurred during the existence of the cavity (Li 1962,

1963), which nevertheless may be regarded as constant (Li and Walsh 1964). Rigid liquid column theory

was used, which required the assumption that the time of existence of the cavity was long compared with

2L/a.

In a Ph.D. thesis at The University of Michigan, Baltzer (1967a) developed a shallow water flow numerical

model for column separation at a valve while the water hammer equations were applied in the remaining

part of the pipe. The shape, movement, and collapse of a vapor cavity formed at the upstream side of a

valve were considered. Once the vapor cavity formed, it usually expanded and propagated in the direction

of the flow as a bubble. A detailed description was presented of the sequence of events during the genesis,

growth, and collapse of a cavity at a valve. This is illustrated in Figure 14.

Figure 14 Simulated transient pressures at gate valve and concurrent, free-surface profiles at

vapor cavity (Baltzer, 1967, Fig. 4).

Siemons (1966, 1967) at Delft Hydraulics Laboratory also developed a shallow water flow model of liquid

column separation to describe cavitating flow. This model was referred to as a “separated flow” model by

Vreugdenhil et al. (1972). The thickness of the cavity was assumed to be small in comparison to the

diameter of the pipe. A Lax finite-difference scheme was used to solve the equations. Siemons (1967)

stressed that the rise in pressure at the collapse would not be great. Dijkman (1968) and Dijkman and

37

Vreugdenhil (1969) extended Siemons' (1967) “separated flow” model, by considering the case of gas

release into a single cavity at a high point in a pipeline. The pressure rise, after compression of the cavity

incorporating dissolved gases, was concluded to be less serious than for the vapor-only case. Gas flow

equations were solved in conjunction with the shallow water flow equations. The method of characteristics

was applied to solve the fourth-order hyperbolic system. The authors concluded that it was uncertain how

the collapse pressures may be computed and suggested an approach of attempting to prevent the occurrence

of cavitation.

Kalkwijk and Kranenburg (1971) noted that Siemons' results did not maintain a mass balance at the

boundary of the cavity, and therefore they questioned the validity of the conclusion concerning the

generation of high pressures. The transition from the water hammer region to the cavitating region was one

of the major problems with the shallow water flow approach. For this reason, Kranenburg (1974a)

concluded that the description of cavitating flow and liquid column separation by shallow water flow theory

did not seem attractive. Another problem concerned the appearance of gravity waves. Furthermore, the model

was concluded to be physically incorrect for vertical pipes.

Marsden and Fox (1976) presented a numerical model in which the low-pressure flow region was described

by the partial differential equations of open channel flow. The authors used a similar approach to that taken

by Baltzer (1967a, 1967b). No evidence of prediction of high pressures was found in their study and the

authors concluded that uneconomic overdesign of pipelines resulted if the cavity collapse was computed by

normal techniques.

Fox and McGarry (1983) presented a variant of the discrete cavity model with a cavity assumed to occupy

the upper portion of the pipe and to be spread over a ∆x length. Thermodynamic effects were included, but

the authors concluded that their influence was insignificant if the vapor pressure of the liquid was small.

3.6 Two Phase or Distributed Vaporous Cavitation Models

This type of numerical model distinguishes between water hammer regions (with pure liquid) and

distributed vaporous cavitation regions (with a homogeneous liquid-vapor mixture). Velocity and vapor

void fraction are computed in the mixture region. The void fraction is defined as

m

vv ∀

∀=α (17)

where = volume of vapor and = volume of mixture of liquid and vapor. v∀ m∀

38

In the early 1970s research related to pipeline cavitation was carried out at Delft Hydraulics Laboratory and

at Delft University of Technology in The Netherlands. Kalkwijk and Kranenburg (1971, 1973) presented a

theory to describe the occurrence of distributed vaporous cavitation in a horizontal pipeline. They referred

to this as the “bubble model”. Their approach ignored free gas content in the liquid and the diffusion of gas

towards cavities. An upstream end pump failure was considered. Wylie and Streeter (1978b) presented a

similar analytical development of a model for vaporous cavitation in a horizontal pipeline. An example was

considered that involved the rupture of a pipeline at the upstream reservoir.

Kalkwijk and Kranenburg (1971) presented two approaches to the theory. The first approach was based on

the dynamic behavior of nuclei or gas bubbles. However, the method failed at the point where the radii of

the bubbles exceeded a critical value. At this size the bubbles became unstable and the characteristics

became imaginary. The incorporation of added mass might solve this problem (Geurst 1985). The second

approach distinguished between the regions with and without cavitation. Different systems of equations

held for the water hammer and the vaporous cavitation region (Kranenburg 1972). The wave celerity in the

cavitation region, with respect to the fluid particles, is reduced to zero. Kalkwijk and Kranenburg (1971)

assumed that the liquid in a cavitation region had a pressure equal to the vapor pressure. Analytical

expressions were developed for the velocity and void fraction for the vaporous zone in a horizontal pipe.

When a cavitation region stopped growing, a shock formed at the transition from the water hammer to the

vaporous region, which penetrated into the cavitation region. This interface was described using the laws of

conservation of mass and momentum, which resulted in “shock equations” analogous to the equations for a

moving hydraulic jump. A shock-fitting technique was used (Kranenburg 1972). Use of analytical methods

for the treatment of the cavitation region and the explicit shock calculation was implemented to avoid

numerical distortions (Vreugdenhil et al. 1972). A number of systematic computations for a simple

horizontal pump discharge line showed that the maximum pressure after cavitation did not exceed the

steady state operating pressure for the pump.

In a Ph.D. thesis at Delft University of Technology, Kranenburg (1974a) presented an extensive work on

the effect of free gas on cavitation in pipelines. A detailed description was given of the phenomenon of

transient cavitation in pipelines. A simplified one-dimensional mathematical model, referred to as the

“simplified bubble flow” model, was presented (Kranenburg 1972, 1974b). Continuity and momentum

equations in conservation form were presented for the vaporous regions. The slope of the pipe and the

influence of gas release were both considered. Kranenburg (1974a) found that there was considerable

difficulty in using the method of characteristics due to the pressure dependence of the wave celerity

because of the presence of free gas. He asserted that discontinuities or shocks between the water hammer

and vaporous region should be fitted in the continuous solution only for simple cases. To simplify the

modeling approach, the bubble flow or vaporous cavitation regime was assumed for the whole pipe, even

for the water hammer regions. A modified surface tension term was used to achieve this simplification. As

39

a result, this model did not show explicit transitions between the water hammer and vaporous cavitation

regions.

Kranenburg (1972) used a Lax-Wendroff two-step scheme despite the occurrence of shock waves. A

numerical viscosity was used to suppress the non-linear instability resulting in the spreading of the

developing shock wave over a number of “mesh points” as described by Richtmeyer and Morton (1967). In

addition, a smoothing operator was introduced to reduce oscillations and instability of computations caused

by the pressure dropping to vapor pressure. Liquid column separation was explicitly taken into account at

mesh points where it may be expected to occur.

Kalkwijk and Kranenburg (1971) also presented experimental results. Dispersion of the negative wave was

observed for pressures below atmospheric pressure. This was attributed to the growth of nuclei. Some gas

bubbles were observed to remain after the passage of the shock wave that collapsed the vaporous region,

and this suggested that gas content played a certain role. Computations did not exhibit the dispersive effect

observed for the experimental results. In conclusion, the authors stated “this method gives a reasonable

description of the overall behavior of the process”.

Fanelli (2000) also addressed methods for solving ordinary and partial differential equations including

shock-fitting and shock-capturing procedures.

Wylie and Streeter (1978b, 1993) developed a case-specific model involving vaporous cavitation and

referred to it as an “analytic model”, while Streeter (1983) referred to a more general model as a

“consolidation model”. A distributed vaporous cavitation region (zone) is described by the two-phase flow

equations for a homogeneous mixture of liquid and liquid-vapor bubbles (liquid-vapor mixture). A

homogeneous mixture of liquid and liquid-vapor bubbles in pressurized pipe flow is assumed to occur

when a negative pressure wave traveling into a region of decreasing pressure along the pipe drops the

pressure to the liquid-vapor pressure over an extended length of the pipe. Pressure waves do not propagate

through an established distributed vaporous cavitation zone.

3.6.1 Two-phase flow equations for vaporous cavitation region

The two equations describing a vaporous cavitation region are the following continuity equation and

equation of motion (Bergant and Simpson 1992, Wylie and Streeter 1993)

∂α∂

+∂α∂

−∂∂

=vm

v m

tV

xVx

0 (18)

40

+ + sin +2m mm m

mf V VV V

V gt x D

θ∂ ∂=

∂ ∂0 (19)

where αv = void fraction of vapor and Vm = liquid-vapor mixture velocity. As the pressure is assumed

constant (vapor pressure), only gravitational and friction forces act. The two equations are valid for small

void fractions (αv << 1) and up to a temperature of about 330 K (Hatwin et al. 1970). The Darcy-Weisbach

friction factor f for liquid flow is assumed in Eq. (19). The friction loss effect due to vaporous bubbles in

the liquid-vapor mixture can be ignored for small void fractions (Griffith 1987).

Eqs. (18) and (19) can be solved analytically (Streeter 1983). Bergant (1992) and Bergant and Simpson

(1999a) presented the solution of the two equations for an upward and downward sloping pipe, and for a

horizontal pipe. Introducing the total derivative, the velocity Vm of the liquid-vapor mixture is first

calculated from Eq. (19) by analytical integration, then the void fraction αv is estimated by numerical

integration of Eq. (18).

The solution of Eq. (19) for Vm depends on the pipe slope (upward, downward or horizontal) and the

inception (initial) velocity of the liquid-vapor mixture Vmi at time of cavitation inception tin at distance x

along the pipeline at which the pressure drops to the liquid vapor pressure. The inception velocity Vmi is

calculated from Eq. (9) or Eq. (10) within the MOC numerical grid with the pressure set to the vapor

pressure. Noting that Vm is a uniform velocity - independent of x - of one individual distributed cavitation

zone, the different results of integration for Vm are as follows (Bergant 1992, Bergant and Simpson 1999a):

(1) Sloping pipe with θ Vmi > 0:

There are two situations for this case: (i) the pipe slope angle θ is positive and Vmi is positive or (ii) θ is

negative and Vmi is negative.

(1.1) Before flow reversal:

( )

−−

= −

inmt

mt

mimtm tt

DfV

VVVV

2)sign(tantan 1 θ (20)

in which Vmt = (2gD|sinθ|/f)1/2 = terminal velocity of liquid-vapor mixture in the sloping pipe and sign(θ) =

{+1 for θ > 0 or −1 for θ < 0}.

(1.2) After flow reversal:

41

sign( ) ( ) /

sign( ) ( ) /1 tanh{ sign( ) ( )}

21

mt r

mt r

f V t t Dmt

m mt mt rf V t t Df VeV V V t t

De

θ

θ θ− −

− −−= = −+

− (21)

in which the time of flow reversal tr is:

+= −

mt

mi

mtinr V

VfV

Dtt 1tan2)sign(θ (22)

(2) Sloping pipe with θ Vmi < 0:

There are two situations for this case: (i) θ is positive and Vmi is negative or (ii) θ is negative and Vmi is

positive.

( )( ) DttVf

mimtmimt

DttVfmimtmtmi

mtminmt

inmt

eVVVVeVVVVVV /)()sign(

/)()sign(

−−

−−

++−++−= θ

θ (23)

(3) Horizontal pipe:

)()sign(22

inmimi

mim ttfVVD

DVV−+

= (24)

With Vm given by one of the Eqs (20), (21), (23) or (24), numerical integration of Eq. (18) at time t for the

void fraction αv over a time step ∆t assuming a weighting factor ψ in time direction gives:

( ) ( ) ( ) ( ) ( ) ( ) ( ){ }1 11t t t t t t t t tv v m m m mk k j j j j

tV V V Vx

α α ψ ψ−∆ −∆ −∆

+ +

∆ = + − + − − ∆ (25)

in which j is the number of the upstream node and j+1 the number of the downstream node for the

computational reach k of length ∆x (∆x=a∆t).

3.6.2 Shock equations for condensation of vaporous cavitation region

A distributed vaporous cavitation region expands in size as a result of a low-pressure wave propagating into

a water-hammer region. Eventually, the distributed vaporous cavitation region stops expanding and the

boundary separating the water hammer and vaporous cavitation regions commences to move back into the

cavitation region. The progression of the liquid into the liquid-vapor mixture or the collapse of a discrete

42

vapor cavity separating the mixture zone(s) (intermediate cavity) condenses the liquid-vapor mixture back

to pure liquid. The liquid is then compressed to a pressure that is unconditionally greater than the liquid

vapor pressure. The movement of the interface (shock wave front) separating the one-phase fluid (liquid)

and the two-phase fluid (liquid-vapor mixture) is described by shock equations. Isothermal conditions

across the interface of infinitesimal width are assumed (Campbell and Pitcher 1958). The shock equations

developed for a generalized movement of the interface (movement in either direction along the pipe) are

derived from the continuity equation and the equation of motion (Bergant 1992, Bergant and Simpson

1992) are

a ga

H H V Vs s sv v m2 0( ) ( )− +

− − =α (26)

(27) g H H V V V V as sv m m s( ) ( )( )− + − − − = 0

where as = shock wave speed, Hs = piezometric head on the water-hammer side of the shock wave front,

and Hsv = piezometric head on the distributed vaporous cavitation side of the shock wave front. The shock

wave equations are coupled with water hammer and two-phase flow equations.

Eqs. (26) and (27) form a system of algebraic equations describing the movement of the shock wave front

into the liquid-vapor mixture. The shock equations are coupled with Eqs. (9) or (10) depending on the

direction of travel of the interface, the kinematic equation for the length of the front movement and the

equation of motion for the liquid plug condensed over a part of the reach. Let L be the distance to the shock

interface measured from the nearest computational section (through the liquid plug). The kinematic

equation for the position of the shock interface as it moves from Lt−∆t to Lt during time step ∆t is:

t t t

s mL L a V t−∆= + + ∆ (28)

The equation of motion written for the liquid plug of length Lt is:

( ) 2 (2

t tt t t t ts

j s j j j js

a f L LH H Q Q Q Qa gDA g A t

−∆ −∆− − − − =∆

) 0 (29)

in which Hj = piezometric head at the upstream side of the liquid plug at computational section j. Note that

Quj = Qj at the upstream end of the liquid plug.

The unknowns in the above system are Hj, Hs at the water-hammer side of the interface, Qj, as and Lt. Vm is

calculated directly from one of the Eqs. (20) through (24) and αv from Eq. (25). This system of non-linear

43

equations is solved by the Newton-Raphson method (Carnahan et al. 1969). Development of shock

equations for the collapse of an intermediate cavity located between two distributed vaporous cavitation

regions and for the end boundary conditions (e.g. reservoir, valve) is described in the literature (e.g.

Simpson 1986; Bergant 1992; Bergant and Simpson 1992).

3.7 Combined Models / Interface Models

This type of model allows for distributed vaporous cavitation in the same way as the previous type,

however, the formation of local column separations at any point in the pipeline is taken into account

(Kranenburg 1974b, Streeter 1983). Flow regions with different characteristics (that is – water hammer,

distributed vaporous cavitation, end cavities and intermediate cavities) are modeled separately, while the

region interfaces are tracked (Simpson 1986, Bergant 1992).

Kranenburg (1974b) presented consideration of a liquid column separation at a valve, in conjunction with

the description of vaporous cavitation regions using his “bubble flow” model. The numerical model was

applied to the inclined pipe experiments of Baltzer (1967a, 1967b). Gas release in the vaporous region was

concluded to cause damping of the pressure peaks caused by the collapses of liquid column separation. To

support this contention a hydrodynamic energy balance was computed for a liquid column separation in a

reservoir/horizontal-pipeline/valve system. Contributions to the energy balance included elastic energy of

the liquid and pipe wall, elastic energy of the free gas, work done at the reservoir, dissipation caused by

shock waves, and dissipation caused by pipe friction. The elastic energy of the free gas was small compared

with the dissipation terms. This explained the relatively small influence of gas release at the liquid column

separation void. There was only slight damping due to wall friction, noting that the contribution of

numerical damping was smaller. The conclusion was drawn that the marked energy loss may be attributed

to the dissipation at the shock wave fronts due to heat-transfer and viscosity effects.

In summary, Kranenburg (1974b) concluded that the inclusion of gas release had no effect where only

cavitating flow occurred, whereas the influence was considerable where liquid column separation occurred

in combination with cavitating flow. Gas release in the cavitating flow region adjacent to a liquid column

separation diminished the duration of subsequent liquid column separations and thus the maximum

pressures upon collapse.

Streeter (1983) was the first to develop a combined analysis for modeling local liquid column separations at

high points and a number of distributed vaporous cavitation regions, while retaining the shock-fitting

approach to explicitly compute the locations of transitions between water hammer and vaporous cavitation

regions. Gas release was not considered, thereby removing the problem associated with the variable wave

speed due to the presence of free gas. This model was referred to as an “analytical approach” or the

44

“consolidation model”. The model was applicable to pipes at any angle with the horizontal. Many separate

distributed vaporous cavitation zones could be modeled, as well as the collapse and reforming of vaporous

regions. All water-hammer regions were described by the method of characteristics. The equations

developed for the vaporous region were for various combinations of slope and initial (inception) velocity. A

computational section became vaporous once a pressure less than the vapor pressure had been computed

from the method of characteristics. A time-line interpolation scheme was used to find the first occurrence of

vapor at a computational section.

The development by Streeter (1983) considered the formation of discrete vapor cavities (column

separations) in the pipeline. It was asserted that a vapor cavity may only form if the angle with the

horizontal between two adjacent pipe sections decreased in the downstream direction, such as at a high

point in the pipeline. If vapor pressure occurred at such a section, a local liquid column separation was

assumed to form. For all other pipe slope conditions a vaporous cavitation region was assumed and both the

velocity and void fraction in the vaporous region were computed.

The model assumed that a vapor cavity could not form at computational sections for which the pipe slope

was the same in the upstream and downstream reaches. The same assumption was applied for the case

where the angle with the horizontal between two adjacent sections increased in the upstream direction. In

making these assumptions, this approach did not account for the possibility of intermediate cavities due to

the interaction of the two low-pressure water hammer waves in the pipeline.

Wylie and Streeter (1993, pp. 196-207) give a detailed presentation of the so-called interface model for

modeling distributed cavitation. The discrete vapor cavity may be located between the following regions

(Streeter 1983, Simpson 1986, Bergant 1992, Bergant and Simpson 1992):

(1) Two liquid zones,

(2) Two distributed vaporous cavitation zones,

(3) Liquid and distributed vaporous cavitation zone,

(4) End boundary and liquid zone,

(5) End boundary and distributed vaporous cavitation zone.

Bergant (1992) and Bergant and Simpson (1992) used a standard DVCM algorithm that allows cavities to

form at computational sections as a basis for the development of an interface model and referred to it as a

generalized interface vaporous cavitation model (GIVCM). Distributed vaporous cavitation zones, shock

waves and various types of discrete cavities were important features in modifying the standard DVCM. The

model handles a number of pipeline configurations (sloping and horizontal) and various interactions

between water hammer, distributed vaporous cavitation, intermediate column separation (along the

45

pipeline), and column separation at boundaries (valve, high point). For example, Figure 15 shows a typical

sequence of transient events in a horizontal pipe including growth and collapse of a discrete cavity,

propagation of a vaporous cavitation zone along the pipe and two shock-wave fronts. In essence, the

GIVCM algorithm maintains the same basic structure as the DVCM and is therefore simpler than previous

interface models. A loop for the shock treatment at appropriate computational sections was added to the

basic DVCM loop and a module for combined discrete vapor cavity and distributed vaporous cavitation

computation was incorporated. Flags to control the correct physical behavior of various phase interactions

and to identify possible new interactions support the algorithm.

j

t+∆t

t

t+2∆t

t+3∆t

t+4∆t

Discrete vapor cavity

Propagation of distributed vaporous cavitation zone

Propagation of shock wave front

Cavity collapse

j-1 j+1 j+2 j+3 j+4

Figure 15 Cavitation and shock formation in a horizontal pipe (adapted from Bergant and Simpson 1992).

Although interface models give reliable results (Bergant 1992, Bergant and Simpson 1999a), they are quite

complicated for general use. More accurate treatment of distributed vaporous cavitation zones, shock waves

and various types of discrete cavities contribute to improved accuracy of the pipe column separation model.

The drawback of this type of model in comparison with discrete cavity models is the complex structure of

the algorithm and the longer computation times.

46

3.8 Other Models

3.8.1 Jordan algebraic model

Jordan (1965, 1975) developed an analytical method for the treatment of distributed vaporous cavitation

zones. He asserted that pressure waves cannot propagate through an established mixture of liquid and vapor

(vaporous cavitation zone) and that the Schnyder-Bergeron graphical method could not be used in this

region. Jordan developed dynamic equations for the distributed vaporous cavitation zone and equations for

the movement of the upstream and downstream liquid columns into the vaporous cavitation region. He

calculated the time when the two columns met (condensation of the vaporous cavitation zone) and

consequently the re-establishment of the liquid phase. Tarasevich (1975, 1997) developed a similar

analytical method. The method was applied to a horizontal pipeline with a valve at the upstream end.

3.8.2 FEM

Howlett (1971) modeled the liquid contained within a pipe system by means of solid beams without bending

stiffness in a finite-element (FEM) solution procedure. Watt et al. (1980), Bach and Spangenberg (1990) and

Shu (2003a) applied the FEM to the classical water hammer equations. Giesecke (1981) mentioned the discrete

cavity model but did not show any results. Axisa and Gibert (1982) and Schwirian (1982, 1984) employed the

DVCM within the context of the FEM; gaps were allowed to form between the axial beam elements simulating

the liquid. See Figure 16. They compared numerical results obtained with and without cavitation.

Figure 16 DVCM in FEM context (Schwirian 1984, Fig. 4). Dynamic element with gap

which allows the formation of a cavity.

3.8.3 Other

Mansour (1996) used the DVCM in combination with a special finite difference scheme for the

waterhammer equations. Equations for a simple condenser cooling water system were developed by

Fanelli (2000) including unsteady pipe flow equations with variable wave speed and friction losses due to

the presence of gas bubbles. Equations for water column separation were presented. In addition, a number

47

of boundary conditions were fully described, including pumps, valves, condensers, siphons, surge tanks and

non-return valves.

3.9 A Comparison of Models

De Vries et al. (1971), Kalkwijk et al. (1972) and Vreugdenhil et al. (1972) compared Siemons' “separated

flow” model with Kalkwijk and Kranenburg's (1971, 1973) “bubble flow” model. Experimental results were

presented for a horizontal 1450 meter test circuit. The construction of some large water supply pipelines in

The Netherlands prompted these studies. Allowing the occurrence of pipeline cavitation was considered as

an alternative to expensive water-hammer control devices such as surge tanks, air vessels and flywheels.

The authors concluded that the results obtained from both computer programs exhibited adequate

agreement with the experimental results for the horizontal test circuit.

Provoost (1976) also compared the results of the “separated flow” (open channel flow) model and

Kranenburg's (1974a) “simplified bubble flow” model for a horizontal pipeline and a pipeline with high

points. Provoost (1976) concluded that the “separated flow” model did not reproduce the field

measurements for the pipeline system with two high points. The “simplified bubble model” was not suited

to describe the local liquid column separations at the high points. This model assumed that the equations for

the cavitation region were applied to the entire pipeline. Explicit transitions were not shown between water-

hammer regions and vaporous cavitation zones. A filtering procedure was required to suppress a slowly

developing instability in the cavitation regions. As a result, the discrete free gas model was developed by

De Vries (1973) and Provoost (1976) in order to deal with local liquid column separations at high points.

In his Ph.D. thesis, Simpson (1986) compared predictions of both his interface model and the discrete cavity

model with experimental data obtained in a 36 m long upward sloping pipe of 20 mm diameter. Bergant (1992)

and Bergant and Simpson (1999a) compared numerical results from discrete vapor (DVCM), discrete gas

(DGCM) and generalized interface vaporous cavitation models (GIVCM) with results of measurements

performed in a 37.2 m long sloping pipeline of 22 mm diameter. The principle source of discrepancies between

the computed and measured column separation results was found to originate from the method of physical

description of vaporous cavitation zones and resulting phenomena along the pipeline.

Dudlik et al. (2000) compared the DVCM with a three-phase model that allowed for the calculation of sudden

changes of gas content in the liquid. Shu (2003b) compared the DVCM with a two-phase model and with

experimental data from Sanada et al. (1990).

48

3.10 Modeling of Friction

The friction term in unsteady pipe flow differs from the friction term in steady pipe flow. Experimental

validation of quasi-steady friction models for rapid transients without column separation has shown significant

discrepancies in attenuation and timing of pressure histories. Computational results have been compared with

the results of measurements by Holmboe and Rouleau (1967), Vardy (1980), Brunone and Greco (1990), Golia

(1990), Bergant and Simpson (1994a), Brunone et al. (1995), Bughazem and Anderson (1996), Bergant et al.

(2001) and Shu (2003b). The quasi-steady friction model may be used for transients where the wall shear stress

is in phase with the flow velocity (cross-sectional average).

Traditionally, the quasi-steady friction term has been incorporated in most column separation models. This term

may contribute to inaccuracies of numerical model results. The influence of unsteady friction effects during

column separation has not been studied in any depth. Researchers have attempted to incorporate a number of

unsteady friction models for liquid flow into standard discrete cavity models that are used in most engineering

transient simulation software packages (Safwat et al. 1986; Dudlik 1999). The numerically stable water-

hammer compatibility equations, written in a finite-difference form for a computational section with index i,

are:

− along the C+ characteristic line (∆x/∆t = a):

0)(2

}){( 1211 =∆+−+− ∆−−

∆−−

∆−−

ttj

tju

ttj

tju

ttj

tj QQ

gDAxfQQ

gAaHH (30)

− along the C− characteristic line (∆x/∆t = −a):

0)(2

})({ 1211 =∆−−−− ∆−+

∆−+

∆−+

ttju

tj

ttju

tj

ttj

tj QQ

gDAxfQQ

gAaHH (31)

The unsteady friction factor f used in Eqs (30) and (31) can be expressed as the sum of a quasi-steady part fq and

an unsteady part fu, i.e. f = fq + fu (Zielke 1968; Vardy 1980; Vardy and Brown 2000; Bergant et al. 2001).

Numerous unsteady friction models have been proposed to date including one-(1D) and two-(2D) dimensional

models (Stecki and Davis 1986; Brereton and Mankbadi 1995; Gündogdu and Çarpinlioglu 1999). The 1D

models approximate the actual cross-sectional velocity profile and the corresponding viscous losses in different

ways.

Shuy and Apelt (1983) performed numerical analysis with five different friction models (steady, quasi-steady

and three unsteady friction models - those of Carstens and Roller (1959), Trikha (1975) and Hino et al. (1977) -

49

that had been incorporated into a standard DVCM. The authors studied 'slow' transients in two long pipelines

(2.3 km and 9 km). They found small differences in the results of the five models for the case of pure water

hammer. Large discrepancies between the five different results occurred for the column-separation case.

Column separation leads to very fast variations in pressure and flow velocity when voids collapse. Brunone and

Greco (1990) and Golia and Greco (1990) used a DVCM in combination with Golia's (1990) unsteady friction

model. Numerical results were compared with results of measurements of 'rapid' water hammer and of column

separation. Significant discrepancies between experiment and theory were found for all runs when using a

quasi-steady friction term. Golia's model results showed an improved agreement between the computed and

measured results. The agreement was better for the water hammer case than for the column separation case.

Brunone et al. (1991) applied their unsteady friction model in the DVCM. The results for the column separation

case showed similar behavior as the results with Golia's model. Bergant and Simpson (1994a) investigated the

performance of the quasi-steady and the three distinct types of unsteady friction models - those of Zielke

(1968), Hino et al. (1977) and Brunone et al. (1991). The friction models had been incorporated into a standard

DVCM using the staggered grid of the method of characteristics. Results of calculations were compared with

experimental results for a fast valve-closure in a reservoir-pipeline-valve system. The Zielke and the Brunone

(et al.) unsteady friction models gave the best fit for the case of no column separation (water hammer). The

effectiveness of the numerical models differed for the case of column separation because the DVCM is

inconsistent due to multi-cavity collapse (Simpson and Bergant 1994a). Bughazem and Anderson (2000)

extended the earlier study by Anderson and Arfaie (1991) to the modified DVCM that simulates apparently

variable wave speed. Their study showed that the effectiveness of the Brunone unsteady friction model did not

appear to be influenced by the choice of the column separation model.

3.11 State of the Art - The Recommended Models

The discrete vapor cavity model (DVCM) gives acceptable results when clearly defined isolated cavity

positions occur rather than distributed cavitation. The DVCM and its variations like DGCM involve a

relatively simple numerical algorithm in comparison to the interface models. The latest developments that

were aimed at models physically better than DVCM and DGCM are due to Simpson (1986) and Bergant

(1992). Simpson's work (Simpson 1986; Simpson and Wylie 1987, 1989, 1991; Bergant and Simpson 1992) is

based on that of Streeter (1983). However, Simpson’s (1986) interface model allowed column separations to

form at any point in a pipe system whenever two low-pressure waves meet. Furthermore, the GIVCM (Bergant

1992; Bergant and Simpson 1999a) enables direct tracking of actual column separation phenomena (e.g.

discrete cavities, vaporous cavitation zones) and consequently it gives better insight into the transient event.

The discrepancies between measured data and DVCM, DGCM and GIVCM predictions found by temporal

and global comparisons (Bergant and Simpson 1999a) may be attributed to approximate modeling of

column separation along the pipeline (distributed vaporous cavitation region, actual number and position of

50

intermediate cavities) resulting in slightly different timing of cavity collapse and different superposition of

waves. In addition, discrepancies may also originate from discretization in the numerical models, the

unsteady friction term being approximated as a quasi-steady friction term, and uncertainties and stochastic

behavior in the measurement (Simpson and Bergant 1996). At the present stage, the GIVCM is used as a

research tool whereas the DVCM and DGCM models are used in most commercial software packages for

water-hammer analysis.

3.11.1 Discrete cavity models: limitations

When the absolute pressure reaches the vapor pressure, cavities or bubbles will develop in the liquid. In the

DVCM these cavities are concentrated, or lumped, at the grid points. Between the grid points pure liquid is

assumed for which the basic water hammer equations remain valid. This means that the pressure wave speed a

is maintained (and convective terms neglected) between grid points in distributed cavitation regions. However,

in bubble flow the pressure wave speed is very low and pressure-dependent. These matters are implicit in the

model (Liou 1999, 2000). Pressure waves actually do not propagate through an established distributed

cavitation region, since this is at an assumed constant vapor pressure. The annihilation of a distributed

cavitation region by a pressure wave causes a delay in propagation, which must be regarded as a reduction of

the wave speed.

In the DVCM the cavities, concentrated at grid points, do not move. This is consistent with the acoustic

approximation: since the overall time scale is acoustic (water hammer), the displacements of vapor bubbles are

small. Vreugdenhil (1964) took into account, within the DVCM, the motion of liquid-vapor boundaries.

The maximum length, /vc v AL = ∀ , of a cavity must be small compared to the spatial grid size. Simpson and

Bergant (1994a) recommended

0.1vcL < x∆

(32)

where ∀v = the volume of the vapor cavity, A = area of the pipe and ∆x = reach length used in the numerical

simulation.

For distributed cavitation regions, condition (32) is mostly fulfilled. If not, the DVCM model is not valid and

the application of models for two-phase plug flow, slug flow or open-channel flow should be considered. For

column separations, condition (32) may sometimes be violated, which is acceptable in the opinion of the

authors, since a column separation is a local phenomenon, and only a few grid points are concerned. However,

51

care should be taken when /vc xL ∆ becomes larger than 1. In that case liquid-vapor boundaries moving from

grid point to grid point should be explicitly modeled. The mass of vapor is neglected, just as the influence of

radial pipe displacements on the cavity volume.

From a physical "microscopic" point of view the "macroscopic" DVCM is not correct; a two-phase flow

(Wallis 1969) approach would be better. However, during cavitation,

(33) *v

* pp =

is physically a strong condition, if there is no free gas or gas release involved. Furthermore, the continuity

equation, that is the mass balance, is satisfied throughout.

As stated, the discrete cavity model is a relatively simple model, which is able to cover the essential phenomena

in transient cavitation. It fits in with the standard MOC approach, so that it can be used in general water-

hammer computer-codes. Its main deficiency is in the numerical oscillations and unrealistic spikes appearing in

the calculated pressure histories, when regions of distributed cavitation occur (De Vries 1972, Bergant and

Simpson 1999a). One way of partly suppressing the oscillations and spikes (see Section 3.4.1) is by assuming

small amounts of initial free gas in the grid points (De Vries 1973; Provoost 1976; Wylie 1984; Zielke and

Perko 1985; Simpson 1986; Barbero and Ciaponi 1991; Bergant 1992; Simpson and Bergant 1994a). Condition

(Eq. 33) is then replaced by

(34) constant)( * =∀− gv* pp

where the free gas is assumed to behave isothermally. The recommended free gas void fractions to be used are

of the order of 10−7. The numerical integration of equation (Eq. 15) may also affect the amount of oscillations

and spikes (Provoost and Wylie 1981; Simpson and Wylie 1985; Simpson and Bergant 1994a; Liou 1999,

2000). The application of a numerical filter may be considered (Vliegenthart 1970; Kranenburg 1974a). Bergant

(1992), and Simpson and Bergant (1994a), who performed a systematic study of the numerical oscillations and

non-physical spikes, came to the conclusion that the discrete cavity model does not converge and, consequently,

the number of grid points should be limited. Liou (1999, 2000), however, showed that the DGCM converges

with respect to the wave speed (Fig. 12). This is why the DGCM can capture the steepening of a positive

pressure wave and the spreading of a negative pressure wave.

The opinion of the authors is the following. Numerical oscillations and spikes are due to multi-cavity collapse

during the annihilation of a cavitation region. In fact, discrete cavity models assume small column separations

at every grid point in such a region. Upon collapse of a single column separation a pressure rise or spike occurs,

which, once generated, does not disappear. The spike travels in a cavitation-free region to and from the

52

boundaries and when two of these spikes meet a higher spike is the result. However, the one-dimensional long-

wavelength water-hammer theory is not valid for these short-duration pressure spikes. (Note that the duration of

the spikes decreases when the computational grid is refined.) Physical dispersion should be introduced in the

model, so that spikes travel with attenuation and more realistic results are obtained. On the other hand, high-

frequency oscillations are to a certain extent observed in cavitation measurements: they are unstable and

physically not repeatable (Fan and Tijsseling 1992; Simpson and Bergant 1996). This instability, reflected in

the unstable numerical oscillations of the discrete cavity models, is part of the cavitation phenomenon. It is

acceptable to smooth the grid-dependent highest frequencies by means of a numerical filter.

The overall conclusion is that discrete cavity models are adequate, but improvements concerning the numerical

oscillations are welcome, because engineers tend to take the highest pressure, which might be an unrealistic

peak, as a measure for design and operation. Unsteady friction models help to predict more accurately the time

intervals between successive column separations.

4 EXPERIMENTS IN THE LABORATORY AND MEASUREMENTS IN THE FIELD

In all experiments described in this section: unless otherwise stated, water hammer is initiated by the rapid

closure of a valve.

4.1 Photographs of Cavity Formation

Escande and Nougaro (1953) reported an early flow visualization study. They conducted column separation

experiments in a laboratory apparatus comprised of a 25 m long horizontal pipeline of 200 mm internal

diameter. A transparent section was positioned next to the valve for flow visualization using a high-speed

camera with 1700 images per second. A large vapor cavity was observed following the closure of the valve.

Bunt (1953), Smirnov (1954) and Blind (1956) made similar observations; they presented photographs

exhibiting the shape of the cavity at the valve during its growth and collapse. The vapor-liquid interface

sloped gently over quite a long distance.

Duc (1959) investigated liquid column separation, for three different piping configurations at a high point,

due to the shut down of pumps in a 1 km long discharge line at a field installation. A clear piece of

Plexiglas pipe at the high point allowed a sequence of photographs to be taken, which exhibited the changes

in cavity during a local liquid column separation. These tests contradicted the generally accepted

supposition of a complete dynamic separation of the liquid column at an elevated point in the line. Liquid

remained in the pipe at the separation point until return-flow filled the void.

53

O'Neill (1959) presented some photographs from visual studies using a high-speed movie camera. He

concluded that intermediate cavities did not form over the entire pipe cross section, as ideally assumed in

the analytical analysis. Most cavities appeared in the upper portion of the cross section only. He concluded

that the existence of any shape of intermediate cavity acted to form a control point in the pipeline, but the

shape might have had an effect on the subsequent pressure surge generated by the cavity collapse.

Li and Walsh (1964), Baltzer (1967b) and Safwat (1972a) presented photographs of a discrete cavity at the

downstream side of a closing valve upstream in a simple pipeline system. The appearance of tiny bubbles

was observed across the whole cross section, which extended along a large portion of the pipe. Pressure

peaks due to cavity collapse were also presented. Tanahashi and Kasahara (1970) studied the formation and

collapse of a discrete cavity at the high point of a pumping system. Nonoshita et al. (1991, 1992, 1999)

presented photographs of liquid column separation in a laboratory draft tube of a water turbine following

wicket gates closure. They studied the effect of the draft tube inlet swirl on column separation events. The

swirl flow generated gas release and subsequent attenuation of maximum pressure following cavity

collapse. Dudlik et al. (1997, 1999) and Dudlik (1999) presented photographs of column separation in a

230 m long test rig with complex pipe geometry.

Swaffield (1969-1970) and Kojima et al. (1984) employed photography to visualize column separation in

liquids other than water (kerosene, mineral oil). Differences in the column separation mechanism for

different liquids were not observed.

The development of new flow visualization technologies in the 1990s included high-speed video and

electrical capacitance tomography techniques. Bergant (1992) and Bergant and Simpson (1996) presented

photographs of column separation in their sloping pipeline experimental apparatus using high-speed video-

equipment. The system comprised a polycarbonate flow-visualization pipe section, high-intensity

illuminator and high-speed video Kodak Ektapro 1000. A discrete vapor cavity and a vaporous cavitation

zone were observed following rapid valve closure. Adam et al. (1998) used an electrical capacitance

tomography technique to observe cross-sectional images from changes in fluid capacitance in a 31.5 m long

pipeline of 41.6 mm diameter. A similar approach was used by Dudlik et al. (1997, 1999) and Dudlik

(1999) and referred to as 'wire-mesh' visualization technique.

4.2 Laboratory Experiments with Liquid Column Separation

At the end of 19th century Joukowsky (1900) performed water hammer and column separation experiments

in Aleksejew's water supply system in Moscow, Russia. He connected three pipelines ((L, D) = ((320, 51);

(320, 102); (325, 152)) (m, mm)), each with a downstream valve, to the main supply pipeline of diameter

610 mm. The pressure records were taken at the valve and along the pipeline. Joukowsky was the first to

54

qualitatively explain water column separation phenomena in pipelines (see Section 2.2). In his paper, he

noted that it was Carpenter (1894) who was the first one to record sub-atmospheric pressures.

Langevin (1928) was probably the first person using piezo-electric pressure transducers for the

measurement of column separation. His record shown in Fig. 17 is typical for all later measurements.

Figure 17 Typical pressure history for repeated column separation and collapse (Bergeron 1950, Fig. 56).

Binnie and Thackrah (1951) performed an investigation that used an experimental apparatus consisting of a

pipeline with an automatic air-inlet valve. When the air-valve was removed, a series of violent impacts took

place following a local liquid column separation and the collapse of a vapor cavity. Maximum recorded

pressures were actually somewhat greater than the theoretical estimates. Pressure traces indicated the

existence of a short-duration pressure pulse due to cavity collapse, which exceeded the “main shock

pressure”. The authors attributed the higher pressures to the existence of additional pressure waves caused

by reflections from bends, sockets and other discontinuities in the pipeline. Their analysis was based on

rigid liquid column theory and on elastic column theory. Lupton (1953), in discussing Binnie and

Thackrah's (1951) paper, did not agree with the use of air valves because of the unpredictably high

pressures that may result from the presence of air in a pipeline.

Bunt (1953) presented findings of a laboratory investigation that considered the possibility of occurrence of

liquid column separation during a water hammer event. In addition to local liquid column separations at the

valve and at high points, the possibility of the formation of distributed vaporous cavitation region was

recognized.

Carstens and Hagler (1964, 1966) considered water hammer in a pipeline transporting phosphate-ore slurry.

Results were presented from both a model study and an analytical analysis that described the liquid column

separation phenomenon. Jordan (1965) investigated column separation and distributed vaporous cavitation

in a laboratory apparatus comprising a 202 m long upward sloping pipeline of 52 mm diameter. The first

25.3 m of the pipe had a slope of 28% and the rest a slope of 1.9%. Baltzer (1967a, 1967b) tested in a

coiled copper-pipe laboratory apparatus and he found that his experimentally measured pressure rises were

appreciably smaller than the simulated ones. In addition, the recurrence intervals between successive

55

experimental pressure rises were of appreciably shorter duration than the predicted intervals. Baltzer

concluded that both of these differences pointed to “higher-than-anticipated” energy dissipation and

believed it to be the result of highly turbulent, meta-stable, two-phase flow that occurred ahead of the main

column separation. Tanahashi and Kasahara (1970) presented a comparison of experimental and analytical

results of water hammer with a local liquid column separation. Liquid column separation occurred at the

valve and at the pipe mid-point. Photographs were also presented of the growth of a cavity at a high point.

Safwat (1972a) performed measurements in a 46 m long horizontal Plexiglas pipeline of 90 mm diameter.

A quantitative analysis of pressure records and photographs of column separation was presented. In

addition, Safwat (1972b, 1972c) and Safwat and de Kluyver (1972) presented results of a comprehensive

experimental study of pressure surges in a laboratory condenser apparatus. Vreugdenhil et al. (1972)

conducted tests in a 1450 m long horizontal pipe of internal diameter of 100 mm. Cavitation was generated

by the sudden decrease of pressure at the upstream end of the pipeline. Piga and Sambiago (1974) tested a

laboratory cooling water system with three possible configurations (length 25 to 30 m, diameter 100 mm).

Thorley and Chohan’s (1976) experimental test facility comprised a horizontal pipe set-up (length 16.3 m,

diameter 38 mm) for the study of compressive and rarefaction waves. Provoost (1976) added two high

points to the original apparatus of Vreugdenhil et al. (1972) and performed the tests.

Krivchenko et al. (1975) presented results of column separation measurements in a draft tube of the Kaplan

turbine test rig at MISI, Moscow, Russia. A large cavity formed after rapid closure of the wicket gates

(closing time of 0.085 s). Column separation first occurred in the space between the guide vanes and the

runner, the cavity then grew into the draft tube inlet cone. A large pressure pulse occurred after cavity

collapse. The authors provided measurements of the pressures under the turbine head cover and the draft

tube inlet, as well as measurements of the axial hydraulic force acting on the turbine runner. Nonoshita et

al. (1991, 1992, 1999) performed similar experimental tests.

Katz and Chai (1978) carried out experiments in 0.3 m long tubes of 5 or 6 mm diameter with 2 ms valve

closures. They showed one diagram of column separation duration as a function of initial velocity ranging

from 1 to 20 m/s. This is the only paper describing a beneficial use of column separation, namely as

feedback mechanism in liquid oscillators that can be used for industrial cleaning. Their theory, based on

rigid column theory, included the effect of gas release.

Kot and Youngdahl (1978a, 1978b) gave a clear explanation of a discrete cavity model and used experiments in

a 9.15 m long closed tube for validation. Aga et al. (1980) applied the DVCM to oil flow in a 250 m long, 90

mm diameter, test rig. Van De Sande and Belde (1981) tested a U-tube laboratory apparatus (length of

approximately 9 m and diameter of 45 mm). They measured pressure peaks higher than the Joukowsky pressure

and contributed it to the delay in rheological response of both vapor and liquid. They used discrete cavities in

their simulations. Gottlieb et al. (1981) constructed four different configurations of a steel and a plastic pipeline.

56

Extremely high pressure peaks were recorded immediately upon collapse of the vapor cavities. The pressure

dropped to about 40% of the pressure peak level and maintained this level for 2L/a seconds. The authors

concluded the presence of peaks resembled the pressure peaks associated with the implosion of gas bubbles in

pumps. Graze and Horlacher (1983) built two horizontal test rigs (120 m and 86 m, and diameters of 82 mm

and 200 mm, respectively). The authors stressed the importance of using adequate pressure transducers and

recording equipment to avoid unrealistic pressure spikes, see Section 4.5 herein. Fox and McGarry (1983)

developed a test rig for the study of pressure transients in pipelines carrying volatile liquids (18 m length, 55

mm diameter). Martin (1983) tested in a coiled copper tube (102 m length, 13.4 mm diameter). He presented

experimental results with limited, moderate and severe cavitation, the severity based on the ratio between the

Joukowsky pressure and the tank pressure, see also Section 2.6.

Borga and De Almeida (1985) tested in a horizontal pipeline of 105 m length. They concluded that the type of

pressure transducer should be a sophisticated low-inertia instrument. In addition, they studied the influence of

an in-line non-return valve on reducing the pressure pulses.

Simpson (1986) studied short-duration pressure pulses resulting from cavity collapse. An experimental pipeline

apparatus was designed and constructed in the G.G. Brown Hydraulics Laboratory at the University of

Michigan (USA), in which measurements were done for eight levels of cavitation severity. Experimental results

obtained for three levels of cavitation severity were compared with predictions by the DVCM in Fig. 18. The

test rig, shown in Fig. 19, consisted of a 36 m long copper pipeline with an inner diameter of 19 mm and a wall

thickness of 1.6 mm. The pipeline connected two reservoirs and was upward sloping. The difference in

elevation between the downstream end and the upstream end is 1 m, so that the elevation angle θ = – 0.028

radians. The pipeline was rigidly supported every 2.5 m by brackets affixed to a wall. The elbow in the pipeline

is also rigidly fixed, by two brackets, so that fluid-structure interaction effects were insignificant. Ordinary tap

water was used in the experiments.

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Figure 18 DVCM results against experimental results of Simpson (1986). Gauge pressure-heads

at valve for four different steady-state situations (Tijsseling 1993, Fig. 6.1).

58

Figure 19 Schematic representation of experimental apparatus (Simpson 1986, Fig. 5.1).

Paredes et al. (1987) constructed a 1460 m long and 104 mm diameter horizontal iron pipe apparatus. Carmona

et al. (1987) conducted extensive measurements in a laboratory set-up resembling that of Provoost (1976) and

they showed the corresponding numerical results. Golia and Greco (1990) found excellent agreement between

computations and experimental data provided by Martin (1983). Barbero and Ciaponi (1991) reported on 23

experiments performed in a nearly 500 m long, 110 mm diameter, test circuit. In their calculations they

examined the influence of initial free gas and gas release. Anderson et al. (1991) and Anderson and Arfaie

(1991) discussed several aspects of the DVCM and showed results of laboratory measurements for three levels

of cavitation severity.

The laboratory apparatus of Shinada and Kojima (1987, 1989, 1995) was a small-scale physical model of a

hydraulic press. The oil-filled test pipe was 5 m long with a diameter of 19 mm. The test results were compared

with the results of a single-vapor-cavity model that included laminar unsteady friction and a dynamic equation

of motion for the valve.

59

Tijsseling and Fan (1991a, 1991b, 1992) carried out measurements in a 4.5 m long stainless steel pipe with two

closed ends. The pipe (outer diameter 59.9 mm, wall thickness 3.9 mm) was suspended by thin steel wires and

filled with highly pressurized water. Transients were generated through the impact of a solid steel rod at one

end of the pipe. The apparatus is very suitable to perform cavitation tests. By taking the initial pressure of the

water low enough, the rod impact will cause transient cavitation. Two complications encountered in the

conventional reservoir-pipe-valve system are absent: the initial steady state pressure gradient and the nonlinear

valve closure. The water is stored under pressure in a closed container so that the amount of free gas is

negligible. Due to the time scale of the experiment (in milliseconds): there is no release of dissolved gas and

friction effects are unimportant. The way of generating transients leads to very steep wave fronts (pressure

rises in microseconds) and the vapor-liquid interfaces at column separations are believed to be nearly

perpendicular to the pipe axis. The experiment isolates vaporous cavitation (in combination with fluid-

structure interaction). The severity of cavitation was regulated by the initial pressure of the water and the

initial velocity of the rod. The experimental results were used to validate the DVCM.

From 1989 to 1992, Bergant and Simpson performed a comprehensive experimental test programme for the

investigation of column separation events at the University of Adelaide in Australia (Bergant 1992; Bergant

and Simpson 1995; Simpson and Bergant 1996). The adjustable experimental apparatus comprised a

straight 37.2 m long sloping copper pipeline of 22 mm internal diameter and 1.6 mm wall thickness

connecting two computer-controlled pressurized tanks. The pipe slope was constant at about 5.4%. The

excitation valve could be located at either end of the pipeline adjacent to either pressurized tank or at the

midpoint of the pipeline. The valve is closed by a torsional spring actuator (valve closure times from 5 to

10 ms) and equipped with an optical valve position recording system. A high-pressure flow visualization

section enabled a high-speed video to film the growth of transient cavitation. Five pressure transducers

were located along the pipeline. The influence of the following quantities on the magnitude, shape and

timing of column separation induced pressure pulses was investigated in the experimental programme:

(1) Initial flow velocity in pipeline,

(2) Static pressure-head in each tank,

(3) Pipe slope (upwards, downwards),

(4) Position of the fast-closing valve in the pipeline system (downstream end, midpoint, upstream

end),

(5) Valve closure time.

Repeat tests and uncertainty analyses were performed (Bergant and Simpson 1995; Simpson and Bergant

1996). The results and documentation of all 116 measurements are available on CD-ROM and can be

obtained through the School of Civil and Environmental Engineering at the University of Adelaide.

60

It is difficult to directly measure flow rates in transient flows. Washio et al. (1996) developed a procedure to

accurately deduce flow rates in transient laminar flow from two closely located pressure transducers. The

procedure is based on frequency-dependent friction theory (Zielke 1968) and relies on a specially constructed

differential amplifier. The method was shown to work very well for oil column-separation in the test rig of

Washio et al. (1994). The typical changes of flow rates during column separation and rejoining were

successfully measured for the first time.

Mitosek (1997, 1998, 2000) took cavitation measurements in plastic pipes. He recorded extremely high

pressure peaks that might be attributed to the strain-gauge type pressure transducers used (see Section 4.5).

Greenshields and Leevers (2000) studied the brittle fracture behavior of plastic pipes. Their laboratory

apparatus consisted of a closed vertically falling pipe where column separation occurred at the bottom end.

Surge and cavity collapse were able to fracture pipes with a defect.

The IAHR working group (Fanelli 2000) is holding experimental records of the laboratory tests by Paredes

et al. (1987) performed in a 1460 m long, 104 mm diameter horizontal iron pipeline and totaling 13

experiments with different initial flow velocities (0.35 to 0.82 m/s) and static heads (16 to 80 m).

Lai et al. (2000) carried out tests with initial voids of different air content in a pipeline with one vertical

branch. Deaerated water was the test liquid. The test results were used to validate their DGCM code, which

had a polytropic gas law, but no gas release mechanism.

Tabei et al. (2003) added small amounts of the noble gases xenon and argon to initially degassed water to

provoke light emission at bubble collapse. In this way, they were able to accurately determine the speed of

a shock wave entering a region of bubbly flow. Their theoretical study showed that local temperatures up to

7000 K may occur when small bubbles collapse.

4.3 Gas Release

Keller and Zielke (1976) measured free gas variations subsequent to a rapid drop in pressure in a 32 m long

plastic pipe with a diameter of 125 mm that was connected to a cavitation tunnel. Wiggert and Sundquist

(1979) conducted experiments using a 129 m and a 295 m long coiled copper-tube apparatus with a diameter

of 25 mm. They investigated gas release during transients at different initial gas concentrations. The effects

of gas release, cavitation nuclei and turbulence were studied. Martin (1981) used a Plexiglas pipe apparatus

(length 32 m, diameter 26 mm) where the water was saturated with injected air.

Kazama (1983) performed experiments with water-methanol mixtures. A test section of 2 m length

contained an initial cavity of vapor and foam at its vertical closed end. Sudden valve opening created a

61

pressure wave collapsing the cavity. The pressure at one location and the volume of remaining air were

measured.

Shinada (1994) studied experimentally and theoretically column separation with gas release. In a 2.5 m

long, 19 mm diameter pipe he tested with saturated and deaerated oil. He measured air content, surface

tension and diffusion rate. On the basis of experimental results, the proposed bubble-diffusion model

allowed for gas release only at the column separation. Gas release has a significant effect on column

separation in saturated oil.

4.4 Fluid Structure Interaction (FSI)

The repeated collapse of column separations, and the almost instantaneous pressure rises associated with

them (see Figure 17), form a severe load for pipelines and their supporting structures. Structural vibration is

likely to occur. Fluid-induced structural motion, structure-induced fluid motion and the underlying coupling

mechanisms are commonly referred to as FSI (fluid-structure interaction). Most of the researchers

mentioned in this review paper prevented unwanted FSI effects by rigidly anchoring their pipes. Fan and

Tijsseling (1992), however, focused on the simultaneous occurrence of cavitation and FSI. They performed

experiments in a closed pipe, the vibrating ends of which interacted with transient column separations.

They observed distributed vaporous cavitation caused by a stress wave in the pipe wall. Tijsseling et al.

(1996) investigated experimentally and theoretically column separation in a freely moving one-elbow pipe

system. More information on combined cavitation/FSI models and on FSI in general can be found in review

papers by Tijsseling (1996) and Wiggert and Tijsseling (2001).

4.5 Problems with Pressure Transducers

Care must be taken with the selection (and way of mounting) of pressure transducers to be used in

cavitation measurements. Because the collapses of column separations are impact loads for the pipe wall,

they must be compensated for accelerations. The pressure transducer's natural frequency must not be too

low, thereby noting that exploding and imploding cavitation bubbles may lead to pressure signals with a

frequency spectrum up to 1 MHz (Oldenziel and Teijema 1976, p. 14). Graze and Horlacher (1983) and

Simpson and Bergant (1991, 1994b) reported unrealistic pressure spikes and oscillations for inductive and

strain-gauge type pressure transducers and attributed these to the fact that the natural frequency of the

transducer was too low. Sayir and Hausler (1991), who performed cavitation experiments in a 20 m long,

110 mm diameter, closed tube of transparent PVC, had also problems with a too low natural frequency of

piezoelectric transducers. Le et al. (1989, p. 3) developed their own special transducers with a natural

frequency of 1.7 MHz to overcome this problem. Arndt et al. (1995) described how to make your own

pressure transducers. Mitosek (1997, 1998, 2000) found unrealistically high-pressure peaks with strain-

gauge type pressure transducers. Greenshields and Leevers (2000) suggested that the cause may be the

62

frequencies in the measured signal in relation to the natural frequency of the piezoelectric pressure

transducer.

Pressure transducers may be easily damaged during cavitation tests, because nearby explosions and implosions

of small bubbles are a too severe load for them (Chen and Israelachvili 1991; Broos 1993, p. 13).

4.6 Field Measurements

Apelt (1956) measured pressures in the field for water hammer in pump discharge lines that undergo liquid

column separation. His investigations verified that elastic column theory may be applied with confidence,

but that it could not account for the phenomenon of liquid column separation. Richards (1956) also

presented some field test data for pumping plants. He contended that it was virtually impossible to

analytically analyze such systems. O'Brien (1956) took issue with this point and cited some extensive

calculations that involved six or seven separate liquid column separations. Whiteman and Pearsall (1965)

conducted field tests at a pumping station on reflux-valve characteristics and pressure rises after pump

shutdown. Heavy flywheels were used to lengthen the run-down time and thus prevented liquid column

separation.

Duc (1959) measured liquid column separation in a field installation. Duc (1965) presented further field test

results at the Third Symposium on Water Hammer (Proceedings 1965). Design of protective measures was

emphasized to prevent any pressure dropping to vapor pressure. In addition to recording pressure peaks due

to rejoining of liquid columns, the phenomena were observed visually. Results exhibited some narrow high-

pressure peaks of short duration. These may have resulted from the type of pressure recording device being

used (see discussion in Section 4.5).

Brown (1968) presented field measurements of transients in two pump discharge lines with distinct “knees”

in their profiles. A mathematical model was also presented in which a local liquid column separation was

assumed to occur at the “knee”. The pressures measured in the field were greater than those predicted by

the graphical method in the design stage. This was attributed to the presence of air and gases entrained in

solution in the liquid. Brown noted that the presence of air may result in large pressure surges and higher

reverse speeds of pumps due to the prolonging of liquid column separation.

De Vries (1975a, 1975b) carried out measurements in a thermal power plant cooling water system (pipeline

at the upstream end of the condenser: length of 76 m and diameter of 1.8 m; pipeline at the downstream end

of the condenser: length of 182 m and diameter of 1.8 m). Provoost (1976) conducted an investigation of a

27.9 km long pipeline with a diameter of 1.8 m. The pipeline was carefully de-aerated before each test. Six

cases of pump shutdown were presented. Sharp (1977) presented field measurements in a 12.3 km long, 250

63

mm diameter, pipeline. In his calculation he allowed cavities to form at not more than six locations.

Nevertheless he obtained reasonable results. Siccardi (1979) presented a comprehensive summary of

measurements and computations of water hammer and column separation in a cooling water system at La

Casella thermal power plant (pipeline at the upstream end of the condenser: length of 440 m and diameter

of 2 m; pipeline at the downstream end of the condenser: length of 234 m and diameter of 1.6 mm). The

IAHR working group (Fanelli 2000) is holding the experimental records in the La Casella industrial plant

through Daco and Meregalli (1981) who provided 25 experimental runs in which pump start-up and pump

power-failure tests at different operating conditions were recorded.

De Almeida and Hipolito (1981) performed measurements at a 70 MW thermal power-plant cooling-water

system featuring a non-return butterfly valve. Water column separation occurred upstream from the condenser

in a series of pump start-up and trip-off tests. Jolas (1981) investigated cavitation effects during pump start-up

in a 900 MW Saint-Laurent nuclear (PWR) power plant cooling water system. Enever (1983) performed water

hammer and column separation measurements in cooling water systems at the Fawley and the Grain thermal

power station. Yow et al. (1985) measured the pressures at two locations after pump restart with a vapor gap in

the water cooling system of a nuclear power station. Wang and Locher (1991) found surprisingly good

agreement between simulations and field data obtained in a 47 km long cross-country pipeline with a diameter

of 840 mm.

5 CONCLUSIONS

During the 20th century there has been considerable research into column separation during water hammer

or transient events. This report attempts to span all of the significant research that has been carried out

during this period. The occurrence of low pressures and associated column separation during water hammer

events has been a concern for much of the twentieth century in the design of pipe and water distribution

systems. The closure of a valve or shutdown of a pump may cause low pressures during transient events.

The collapse of vapor cavities and rejoinder of water columns can generate extremely large pressure that

may cause significant damage or ultimately failure of the pipe system.

As early as 1900, Joukowsky had identified the physical occurrence of column separation. The 1930s

produced the first mathematical models of vapor cavity formation and collapse based on the graphical

method. The identification of the various physical attributes of column separation occurred in the mid-20th

century (distributed or vaporous cavitation in the 1930s; intermediate vapor cavities in the 1950s). These

both led to a better physical understanding of the process of column separation and ultimately laid out the

groundwork for the development of computer based numerical models. The late 1960s saw the

development of the first computer models of column separation within the framework of the method of

64

characteristics solution of the water hammer equations. A variety of alternative numerical models were

developed from the late 1960s to the early 1990s. The most significant models that have been developed

include: the discrete vapor cavity model (DVCM), the discrete gas cavity model (DGCM) and the

generalized interface vapor cavity model (GIVCM). The first two are the easiest to implement. The DVCM

is the most popular model used in currently available commercial computer codes for water hammer

analysis.

The GIVCM handles a number of pipeline configurations (sloping and horizontal pipe) and various

interactions between water hammer regions, distributed vaporous cavitation zones, intermediate cavities

(along the pipeline) and cavities at boundaries (valve, high point). More accurate treatment of distributed

vaporous cavitation zones, shock waves and various types of discrete cavities, contribute to improved

performance of the pipe column separation model. Although the interface model gives reliable results, it is

quite complicated for general use. The drawback of this model in comparison to the discrete cavity models is

the complex structure of the algorithm and longer computational time. At the present stage, the GIVCM is

useful as a research tool but not in commercial codes.

Numerous experimental studies have been carried out over the last 40 years. From all the validation tests

presented in the research literature it may be concluded that, despite its simplicity, the discrete vapor cavity

model (DVCM) reproduces the essential features of transient cavitation. The versatility of the model has been

demonstrated by the variety of pipe systems used in the tests. The major deficiency of the model is the

appearance of non-physical oscillations in the results. The DGCM is recommended in developing and

revising industrial engineering water hammer computer codes.

Bergant and Simpson (1999a) compared numerical results from discrete vapor (DVCM), discrete gas (DGCM),

and generalized interface vaporous cavitation models (GIVCM) with results from measurements. The

discrepancies between the measured, DVCM, DGCM and GIVCM results found by temporal and global

comparisons may be attributed to approximate modeling of column separation along the pipeline

(distributed vaporous cavitation regions, actual number and position of intermediate cavities) resulting in

slightly different timing of cavity collapse and superposition of pressure waves. In addition, discrepancies

may also originate from discretization in the numerical models, the unsteady friction term being

approximated as a quasi-steady friction term and uncertainties in the measurements.

In the last 10 years, the amount of research effort into column separation has slowed. There is still room for

further research in a number of areas. Laboratory testing of the impact of various cases of distributed

cavitation (or vaporous cavitation) in conjunction with the testing of the performance of the three most

commonly used models (DVCM, DGCM and GIVCM) needs to be undertaken. Over the last 10 years a

number of advances in unsteady friction have occurred. The impact of these new models on the behavior of

65

column separation modeling also needs to be investigated. Finally, further work could also be carried out

on the GIVCM to explore whether simplifications could be made to the complexity of the approach so that

it becomes viable to introduce this approach into commercial water hammer codes.

6 ACKNOWLEDGEMENT

The Surge-Net project (www.surge-net.info) is supported by funding under the European Commission’s

Fifth Framework ‘Growth’ Programme via Thematic Network “Surge-Net” contract reference: G1RT-CT-

2002-05069. The authors of this paper are solely responsible for the content and it does not represent the

opinion of the Commission. The Commission is not responsible for any use that might be made of data

herein.

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Abbreviations in References

ASCE American Society of Civil Engineers

ASME American Society of Mechanical Engineers

AWWA American Water Works Association

BHRA British Hydromechanics Research Association

BHR Group British Hydromechanics Research Group

BNES British Nuclear Engineering Society

FED Fluids Engineering Division

IAHR International Association of Hydraulic Research

JSME Japan Society of Mechanical Engineers

MISI Moscow Institute of Civil Engineering

PVP Pressure Vessels and Piping

3R Rohre Rohrleitungsbau Rohrleitungstransport

SMiRT Structural Mechanics in Reactor Technology

86

8 NOMENCLATURE

Abbreviations

DGCM discrete gas cavity model

DVCM discrete vapor cavity model

FEM finite element method

FSI fluid-structure interaction

GIVCM generalized interface vaporous cavitation model

MOC method of characteristics

ODE ordinary differential equation(s)

Scalars

A cross-sectional pipe area, m2

a pressure wave speed, m/s

B known constant in water hammer compatibility equations

C known constant in water hammer compatibility equations

D inner diameter of pipe, m

f Darcy-Weisbach friction factor

g gravitational acceleration, m/s2

H (piezometric) head, m

HGL hydraulic grade line, m

L pipe length, m

M mass, kg

N number of computational reaches

p fluid pressure, Pa

Q discharge, m3/s

R inner radius of pipe, m

Rg specific gas constant, m2/s2/K

S cavitation severity index

T absolute temperature, K

cT valve closure time, s

csT duration of the first vapor cavity, s

t time, s

V flow velocity, m/s

∀ volume, m3

x distance along the pipeline, m

z elevation of centerline of pipe, m

87

88

α void fraction

γ specific weight, kg/m2/s2

∆ numerical step size, change in magnitude

θ pipe slope, rad

ν Poisson ratio

ρ mass density, kg/m3

ψ weighting factor

Subscripts

b barometric

cs column separation

f final

g gas

in inception, initial

j computational section index

k reach index

m mixture

max maximum

mc negative (minus) characteristic

mi mixture-inception

mt mixture-terminal

P point P in x-t plane

pc positive (plus) characteristic

q quasi-steady

r reversal

RV reservoir-valve (see Eq. 5)

s shock

sv shock-vapor

T tank

u upstream, unsteady

v vapor

vc vapor cavity

0 initial (steady) state at t = 0

Superscripts

* absolute pressure

water hammer with column separation

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