© Acta hydrotechnica 18/29 (2000), Ljubljana
ISSN 1581-0267
5
UDK: 532.528:532.542.1 UDC: 532.528:532.542.1
Izvirni znanstveni č lanek Scientific paper
KAVITACIJSKI TOK MED PREHODNIMI REŽIMI V CEVNIH SISTEMIH
TRANSIENT CAVITATING FLOW IN PIPING SYSTEMS
Anton BERGANT
Sprememba pretoč ne hitrosti povzroč i prirastek in padec tlaka v cevnem sistemu. Kontinuiteta
kapljevine je prekinjena, ko se tlak v sistemu zniža na parni tlak kapljevine.  Prispevek zajema fizikalne
osnove kavitacijskega toka pri parnem tlaku kapljevine med prehodnimi režimi (pretrganje
kapljevinskega stebra). Kavitacijski tok je popisan z enač bami faznih stanj, ki jih rešujemo z
analitič nimi in numerič nimi metodami. Metode reševanja enač b so vgrajene v diskretni in kombinirani
kavitacijski model. Numerič ni rezultati so primerjani z rezultati meritev v laboratoriju. Rezultati,
dobljeni s kombiniranim modelom, se v splošnem najbolj ujemajo z meritvami. Kombinirani
kavitacijski model, v primerjavi z diskretnim kavitacijskim modelom, bolj natanč no popisuje fizikalno
sliko kavitacijskega toka.
Ključ ne besede: cevni sistemi, nestacionarni kavitacijski tok, vodni udar, parna kavitacija,
pretrganje vodnega stebra, diskretni kavitacijski model, kombinirani kavitacijski model,
eksperimentalna postaja
A change of flow velocity in a piping system induces either an increase or a decrease in pressure.
Liquid column separation occurs when the pressure drops to the liquid vapour pressure. This paper
deals with physical phenomena of transient vaporous cavitating flow in pipelines. Column
separation in pipelines is described by a set of equations representing a particular physical state of
the fluid. The equations are solved separately by analytical and numerical methods. The solution
methods are integrated into a discrete vapour cavity model and an interface vaporous cavitation
model. Numerical results are compared with the results of laboratory measurements. A comparison
between the results shows that the interface model simulates the physics of the transient cavitating
flow more accurately than the discrete vapour cavity model.
Key words: piping systems, transient cavitating flow, water hammer, vaporous cavitation, water
column separation, discrete cavity model, interface vaporous cavitation model, experimental
apparatus
1. UVOD
Hidravlični cevni sistemi delujejo pri
različnih pretočnih režimih. Sprememba
pretoč ne hitrosti povzroč i prirastek in padec
tlaka v cevnem sistemu. Kontinuiteta tekoč ine
je prekinjena (pretrganje kapljevinskega
stebra), ko se tlak v sistemu zniža na parni tlak
kapljevine (pojav kavitacije). Obstajata dva
tipa pretrganja stebra. Prvi je lokalna parna
kavitacija z velikim kavitacijskim
razmernikom (približno ena). Drugi je
področ je kontinuiranega kavitacijskega toka
vzdolž določ ene dolžine cevovoda. V tem
primeru je kavitacijski razmernik majhen
(približno nič ). Prispevek obravnava fizikalne
zakonitosti in teoretič ne modele kavitacijskega
1. INTRODUCTION
Hydraulic systems operate on a broad range
of operating regimes. A change of flow
velocity in a piping system induces either an
increase or a decrease in pressure. Liquid
column separation occurs when the pressure
drops to the liquid vapour pressure (transient
cavitating flow). Two distinct types of column
separation may occur. The first type is a
localised vapour cavity with a large void
fraction (close to 1). The second type of
column separation is distributed vaporous
cavitation that may extend over long sections
of the pipe. The void fraction for this case is
small (close to 0). This paper deals with
physical phenomena and theoretical models of

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
6
toka med prehodnimi režimi v cevnih sistemih.
Izsledki teoretič nih raziskav so podprti s
preizkusi kavitacije pri parnem tlaku
kapljevine. Glavni poudarek je v predstavitvi
novih teoretič nih modelov, izvedbi originalnih
preizkusov in teoretič no-eksperimentalni
izlušč itvi fizikalnih zakonitosti kavitacijskega
toka v cevovodu.
V teoretič nem delu so podane enač be
kavitacijskega toka in metode reševanja enač b.
Kavitacijski tok med prehodnimi režimi v
cevnih sistemih je popisan z enač bami
vodnega udara, kontinuiranega kavitacijskega
toka, kompresijskega skoka in diskretne
kavitacije. Enač be vodnega udara rešujemo z
metodo karakteristik. Enač be kontinuiranega
kavitacijskega toka rešujemo z generalizirano
analitič no-numerično metodo. Enač be
kompresijskega skoka se rešujejo združeno z
enač bami vodnega udara in kontinuiranega
kavitacijskega toka s pomočjo Newton-
Raphsonove metode. Kontinuitetna enač ba
diskretne kavitacije pa je rešena numerič no, v
kombinaciji s kompatibitetnimi enač bami
vodnega udara ali enač bami kontinuiranega
kavitacijskega toka. Metode reševanja enač b
kavitacijskega toka so kombinirane v diskretni
in kombinirani kavitacijski model. Diskretni
kavitacijski model generira kavitacije pri
robnih pogojih in v numerič nih vozlišč ih
vzdolž cevovoda, ko tlak v kapljevini pade na
parni tlak. Cevni odseki med vozlišč i so
modelirani kot cevni odseki s kapljevinskim
tokom. Temu sledi predstavitev novega
kombiniranega kavitacijskega modela, ki
zajema eksplicitni popis diskretnih kavitacij in
področ ij kontinuiranega kavitacijskega toka.
Model zajema poljubno kombinacijo področ ij
kontinuiranega kavitacijskega toka in
diskretnih kavitacij v cevovodu s poljubnim
nagibom.
Eksperimentalne raziskave kavitacijskega
toka med prehodnimi procesi so nujno
potrebne za boljše razumevanje fizikalnega
pojava in overitev postavljenih teoretič nih
modelov. V prispevku orisana preizkusna
postaja dovoljuje raziskave kavitacijskega toka
pri kontroliranih pretočnih pogojih.
Prilagodljivost postaje, rač unalniško krmiljena
regulacija tlaka v tlačnih rezervoarjih,
zapiralni mehanizem s torzijsko vzmetjo in
transient vaporous cavitating flow in pipelines.
The theoretical findings are verified by column
separation measurements in the laboratory.
The main objective of this paper is to present
novel theoretical models that are verified by
experiments and to improve the understanding
of the column separation phenomena in
pipelines.
The theoretical part of the paper describes
pipeline column separation equations and their
solution methods. Transient cavitating flow in
pipelines is described by water hammer
equations, two-phase flow equations for a
distributed vaporous cavitation region, shock
equations and equations for a discrete vapour
cavity. Water hammer equations are solved by
the method of characteristics. Two-phase flow
equations are solved by a generalized
analytical-numerical method. The shock
equations are coupled with water hammer
compatibility equations and two-phase flow
equations, and then solved by the Newton-
Raphson method. The discrete vapour cavity
continuity equation is solved numerically, and
then coupled with water hammer compatibility
equations or two-phase flow equations. The
solution methods for column separation
equations may be incorporated into discrete
vapour cavity models and an interface
vaporous cavitation model. The discrete cavity
model allows vapour cavities to form at
interior pipe sections and boundaries when
liquid pressure drops to the liquid vapour
pressure. A liquid phase between the
computational sections is assumed. Then a
novel interface vaporous cavitation model is
presented.  The model treats discrete vapour
cavities and distributed cavitation regions
explicitly, and it handles the various
interactions between the particular physical
state of the fluid in a number of pipeline
configurations .
Experimental investigation of transient
cavitating flow in pipelines should be
performed in order to improve an
understanding of the phenomena and to verify
theoretical models. An experimental apparatus
that enables research of column separation
phenomena under controlled flow conditions is
presented. The flexible experimental apparatus
for investigating transient cavitating flow in

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
7
vizualizacija kavitacijskega toka z
rač unalniško krmiljeno hitrosnemalno kamero
predstavljajo novosti na področ ju
eksperimentalnih raziskav prehodnih pojavov s
kavitacijskim tokom v cevih sistemih.
Primerjava rezultatov izrač una in meritev
pokaže, da se rezultati, dobljeni s
kombiniranim kavitacijskim modelom v
splošnem najbolj ujemajo z meritvami.
Kombinirani kavitacijski model v primerjavi z
diskretnim kavitacijskim modelom bolj
natančno reproducira fizikalno sliko
kavitacijskega toka. Odstopanja rezultatov
izrač una med obravnavanimi modeli izhajajo
iz nač ina popisa kontinuiranega kavitacijskega
toka. Iz teoretič no-eksperimentalne analize
kavitacijskega toka v sistemu rezervoar-
cevovod-ventil-rezervoar sledi, da kavitacijski
tok med prehodnimi režimi v cevnih sistemih
zajema področ ja kontinuiranega kavitacijskega
toka in diskretnih kavitacij.
2. ENAČ BE VODNEGA UDARA IN
KA VITACIJSKEGA TOKA
Kavitacijski tok med prehodnimi režimi v
cevnih sistemih opišemo z enač bami vodnega
udara, kontinuiranega kavitacijskega toka,
kompresijskega skoka in diskretne kavitacije.
2.1 ENAČ BE VODNEGA UDARA
Enačbe vodnega udara popisujejo
nestacionarni kapljevinski tok v cevovodu.
Kontinuitetna in gibalna enač ba se glasita
(Wylie & Streeter, 1993):
pipelines incorporates several novel features,
including a computer controlled system for
maintaining a specified pressure in the tanks, a
torsional spring valve actuator and flow
visualization, with the aid of computer
controlled high-speed video. A comparison
between the computational and experimental
results shows that the interface vaporous
cavitation model gives the best fit. The
interface vaporous cavitation model simulates
the physics of the transient cavitating flow
more accurately than the discrete vapour
cavity model. The discrepancies between the
two types of models may be attributed to the
different description of vaporous cavitation
zones. Theoretical and experimental
investigations of column separation events in a
reservoir-pipeline-valve-reservoir system
reveal the occurrence of discrete vapour
cavities and vaporous cavitation zones during
transient regimes in pipelines.
2. PIPELINE COLUMN
SEPARATION EQUATIONS
Transient cavitating flow in pipelines is
described by water hammer equations, two-
phase flow equations for a distributed
vaporous cavitation region, shock equations
and equations for a discrete vapour cavity.
2.1 WATER HAMMER EQUATIONS
Water hammer equations describe the
unsteady liquid flow in pipelines. The
equations are the continuity equation and the
equation of motion (Wylie & Streeter, 1993):
0
g
a
sinθ +
2
=
∂
∂
+ −
∂
∂
∂
∂
x
v
v
x
H
v
t
H
(1)
0
2
| | λ g = +
∂
∂
+
∂
∂
+
∂
∂
D
v v
x
v
v
t
v
x
H
(2)

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
8
2.2 ENAČ BE KONTINUIRANEGA
KAVITACIJSKEGA TOKA
Enač be kontinuiranega kavitacijskega toka
opisujejo enokomponentni dvofazni tok v
cevovodu, ko je tlak tekoč ine enak parnemu
tlaku kapljevine. Pri konstantnem tlaku
tekoč ine udarni valovi ne propagirajo vzdolž
cevovoda. Kontinuitetna in gibalna enač ba
homogene zmesi kapljevine in parnih
mehurč kov se glasita (Bergant & Simpson,
1992; Wylie & Streeter, 1993):
2.2 TWO-PHASE FLOW
EQUATIONS FOR A DISTRIBUTED
VAPOROUS CAVITATION REGION
Two-phase flow equations for a distributed
vaporous cavitation region describe the
unsteady one-component two-phase flow of
liquid-vapour mixture in pipelines at a
pressure set to the liquid vapour pressure.
Pressure waves do not propagate at an
assumed constant vapour pressure. The two-
phase flow equations are the continuity
equation and the equation of motion (Bergant
& Simpson, 1992; Wylie & Streeter, 1993):
0 =
∂
∂
−
∂
∂
+
∂
∂
x
v
x
v
t
m v
m
v
α α
 (3)
0
2
λ + sinθ g + + =
∂
∂
∂
∂
D
v v
x
v
v
t
v
m m
m
m
m
(4)
2.3 ENAČ BE KOMPRESIJSKEGA
SKOKA
Enač be kompresijskega skoka popisujejo
fazno spremembo homogene zmesi kapljevine
in parnih kavitacij (enokomponentni dvofazni
tok, enačbe kontinuiranega kavitacijskega
toka) v kapljevino (enofazni tok, enač be
vodnega udara). Sprememba faze je posledica
potovanja kompresijskega vala enofaznega
toka v področ je dvofaznega toka, kjer je tlak
zmesi enak parnemu tlaku kapljevine. Parne
kavitacije (mehurč ki) kondenzirajo, kapljevina
pa se komprimira. Kontinuitetna in gibalna
enačba kompresijskega skoka se glasita
(Bergant & Simpson, 1992):
2.3 SHOCK EQUATIONS
Shock equations describe a phase change of
a homogeneous mixture of liquid and liquid-
vapour bubbles (one-component two-phase
flow, two-phase flow equations) back to liquid
(one-phase flow, water hammer equations).
The phase change is induced by the
propagation of the compression wave from
liquid into the liquid-vapour mixture at the
pressure set to the liquid vapour pressure.
Liquid-vapour bubbles condense; the liquid is
compressed. Shock equations are the
continuity equation and the equation of motion
(Bergant & Simpson, 1992):
0 ) ( α ) (
2
= − −






+ −
m v sv s s
v v H H
a
g
a (5)
0 ) )( ( ) ( = − − − + −
s m m sv s
a v v v v H H g (6)
2.4 ENAČ BE DISKRETNE
KAVITACIJE
Diskretno kavitacijo obravnavamo kot robni
pogoj (pretrganje kapljevinskega stebra).
Sprememba volumna kavitacije je izražena s
kontinuitetno enačbo diskretne kavitacije
(Streeter, 1969):
2.4 EQUATIONS FOR A DISCRETE
VAPOUR CAVITY
A discrete vapour cavity is treated as a
boundary condition (fluid column separation).
The growth and subsequent decay of the cavity
is defined by the continuity equation (Streeter,
1969):

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
9
dt = V v v A
u
t
t
vc
in
) ( −
∫
 (7)
Pritok kapljevine ali tekoč ine v kavitacijo in
iztok iz nje sta izrač unana z enač bami vodnega
udara ali kontinuiranega kavitacijskega toka.
3. ANALITIČ NE IN NUMERIČ NE
METODE
Enačbe vodnega udara (1) in (2),
kontinuiranega kavitacijskega toka (3) in (4),
kompresijskega skoka (5) in (6), in diskretne
kavitacije (7) lahko rešujemo z eksplicitnimi
ali hibridnimi metodami (Bergant, 1992). V
prispevku obravnavamo eksplicitne metode, ki
so v primerjavi s hibridnimi metodami
numerič no bolj stabilne in fizikalno jasne.
Metode združimo v numerič ni model za
analizo kavitacijskega toka. Uporaba metode
karakteristik za reševanje enač b vodnega udara
(1) in (2) (Wylie & Streeter 1993; Simpson &
Bergant 1994):
a) kompatibilitetna enačba za pozitivno
karakteristiko ∆ x/∆ t=a:
The continuity e quation is coupled to the
water hammer equations and/or two-phase
liquid-vapour mixture flow equations in order
to account for flow velocities.
3. ANALYTICAL AND
NUMERICAL METHODS
A separate or combined approach (Bergant,
1992) to the solution of water hammer (1) and
(2), two-phase flow (3) and (4), and shock
equations (5) and (6), and a discrete vapour
cavity equation (7) may be adopted. The paper
deals with the separate solution of these
equations. This approach gives a more stable
and physically sound solution than the
combined approach. The solution methods are
integrated into a numerical algorithm for
column separation analysis. A method of
characteristics solution for water hammer
equations (1) and (2) (Wylie & Streeter, 1993;
Simpson & Bergant, 1994):
a) compatibility eq. along the positive
characteristic line ∆ x/∆ t=a:
uj P P j
Q B C H − = (8)
b) kompatibilitetna enačba za negativno
karakteristiko ∆ x/∆ t=-a:
b) compatibility e q. along the negative
characteristic line ∆ x/∆ t=-a:
j M M j
Q B C H + = (9)
v kombinaciji z analitič nimi in numerič nimi
metodami kavitacijskega toka omogoč a dokaj
natanč no obravnavanje fizikalnega pojava (Q
= vA). V fizikalni ravnini pot-č as zasledujemo
potovanje ekspanzijskih in kompresijskih
valov, kondenzacijo področ ij kontinuiranega
kavitacijskega toka ter formiranje in zrušitev
diskretnih kavitacij. Vgraditev diskretnih
kavitacij v numerič na vozlišč a v metodi
karakteristik daje diskretni kavitacijski model
(Streeter, 1969; Wylie & Streeter, 1993;
Simpson & Bergant, 1994). Kombinirani
kavitacijski model združuje kompletne enač be
vodnega udara in kavitacijskega toka (Streeter,
1983; Bergant & Simpson, 1992).
coupled with analytical and numerical
methods for cavitating flow regions provides a
reasonable treatment of the physical
phenomena (Q = vA). The propagation of
rarefaction and compression waves,
condensation of vaporous cavitation zones,
and the growth and subsequent decay of
discrete cavities can be visualized in the xt
plane. Incorporating discrete cavities into the
computational sections of the method of
characteristics leads to the discrete cavity
model  (Streeter, 1969; Wylie & Streeter, 1993;
Simpson & Bergant, 1994). Coupling the
complete set of column separation methods
gives the interface vaporous cavitation model
(Streeter, 1983; Bergant & Simpson, 1992).

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
10
3.1 ANALITIČ NO-NUMERIČ NO
REŠEVANJE ENAČ B
KONTINUIRANEGA
KAVITACIJSKEGA TOKA
Enač bi (3) in (4) lahko rešujemo analitič no
ali numerič no (Streeter, 1 993; Simpson, 1 986;
Bergant, 1992). V prispevku je predstavljen
kombinirani analitič no-numerični pristop
reševanja obravnavanih enačb v cevi s
poljubnim nagibom (Bergant, 1992). V prvem
koraku s pomočjo analitične integracije
gibalne enač be (4) izrač unamo pretoč no hitrost
homogene zmesi tekoč ine v
m
, v drugem pa s
pomoč jo numerič ne integracije kontinuitetne
enač be (3) kavitacijski volumski razmernik α
v
.
Integracija hitrosti v
m
 iz (4) je odvisna od
nagiba cevovoda in začetne kavitacijske
hitrosti v
mi
 v č asu  t
i
, ko ekspanzijski val v
poljubni toč ki x vzdolž cevovoda povzroč i
padec tlaka na parni tlak kapljevine. Zač etno
kavitacijsko hitrost v
mi
 izrač unamo iz enač be
(8) ali (9) (H
j
 = H
v
). Rezultati integracije
enač be (4) so (Bergant, 1 992):
a) cevovod z nagibom pri θ θ θ θv
mi
 > 0:
- tok proti smeri osne komponente
gravitacijske sile:
3.1 ANALYTICAL AND
NUMERICAL INTEGRATION FOR
TWO-PHASE LIQUID-VAPOUR
MIXTURE EQUATIONS
Equations (3) and (4) can be solved
analytically or numerically (Streeter, 1993;
Simpson, 1986; Bergant, 1992). A combined
analytical-numerical approach to the solution
of these equations developed for an arbitrary
sloping pipeline is presented (Bergant, 1992).
First the velocity of liquid-vapour mixture v
m
is calculated from (4) by analytical integration;
then the void fraction α
v
 is estimated by the
numerical integration of (3).
Integration of v
m
 from (4) depends on the
pipe slope and the inception velocity of the
liquid-vapour mixture v
mi
 at time  t
i
 at an
arbitrary distance x along the pipeline, at
which the rarefaction wave drops the pressure
to the liquid-vapour pressure. The inception
velocity v
mi
 is calculated from (8) or (9) (H
j
 =
H
v
). The results of integration for equation (4)
are (Bergant, 1992):
a) Sloping pipe with θ θ θ θv
mi
 > 0:
- flow in the reverse direction of the x-
component of the body force (gravity):
()








− −








=
−
i
mt
mt
mi
mt m
t t
D
v
v
v
v v
2
λ ) sign(θ tan tan
1
(10)
kjer je gravitacijska hitrost izražena z enač bo
v
mt 
= (2gD|sinθ|/λ)
1/2
, funkcija predznaka
nagiba cevovoda pa z izrazom sign(θ) = {+1
za θ > 0; -1 za θ < 0}.
- tok v smeri osne komponente gravitacijske
sile:
in which v
mt 
= (2gD|sinθ|/λ)
1/2
 is the terminal
velocity of the liquid-vapour mixture and
sign(θ) = {+1 for θ > 0; -1 for θ < 0} is the
sign function of pipe slope θ.
- flow in the direction of the x-component of
the body force (gravity):
1
1
/ ) ( λ ) sign(θ / ) ( λ ) sign(θ +
−
=
− −
− −
D t t v
D t t v
mt m
r mt
r mt
e
e
v v (11)
kjer je t
r
 č as zaustavitve pretoka: in which the time of the flow reversal t
r
 is:








+ =
−
mt
mi
mt
i r
v
v
v
D
t t
1
tan
λ 2
) sign(θ (12)

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
11
b) cevovod z nagibom pri θ θ θ θv
mi
 < 0: b) Sloping pipe with θ θ θ θv
mi
 < 0:
()
()
D t t v
mi mt mi mt
D t t v
mi mt mt mi
mt m
i mt
i mt
e v v v v
e v v v v
v v
/ ) ( λ ) sign(θ / ) ( λ ) sign(θ − −
− −
+ + −
+ + −
= (13)
c) horizontalni cevovod: c) Horizontal pipe:
) ( )λ sign( 2
2
i mi mi
mi
m
t t v v D
Dv
v
− +
= (14)
kjer je sign(v
mi
) = {+1 za v
mi 
> 0; -1 za v
mi 
< 0}
funkcija predznaka začetne kavitacijske
hitrosti v
mi
.
Numerič na integracija kontinuitetne enač be
(3) v č asu t ob vpeljavi utežnega koeficienta v
č asovni smeri ψ (Wylie, 1984) da:
in which sign(v
mi
) = {+1 for v
mi 
> 0; -1 for v
mi 
<
0} is the sign function of inception velocity
v
mi
.
Numerical integration of continuity
equation (3) at time t, using a weighting factor
ψ in time direction (Wylie, 1984) gives:
() () () () [ ]() () () [ ] { } x t v v v v
t t j m t t j m t j m t j m t t k v t k v
∆ ∆ − − + − + =
∆ − ∆ − + + ∆ − , , 1 , , 1 , ,
ψ 1 ψ α α (15)
kjer je j število gorvodne in j+1 število
dolvodne računske toč ke k-tega cevnega
odseka dolžine ∆ x (∆ x=a∆ t).
3.2 NUMERIČ NA METODA ZA
REŠEVANJE ENAČ B
KOMPRESIJSKEGA SKOKA
Enač bi kompresijskega skoka (5) in (6)
tvorita sistem algebraič nih enač b. Obravnavani
enač bi opisujeta č elo kapljevinskega stebra, ki
propagira v področje kontinuiranega
kavitacijskega toka. Enač bi (5) in (6) rešujemo
združeno z ustrezno kompatibilitetno enač bo
(8) ali (9) (glede na smer potovanja skoka),
kinematič no enač bo gibanja kompresijskega
skoka v č asovnem intervalu ∆ t:
in which j is a number of the upstream node
and j+1 is a number of a downstream node for
the computational reach k of length ∆ x
(∆ x=a∆ t).
3.2 NUMERICAL METHOD FOR
COUPLED SHOCK EQUATIONS
Shock equations (5) and (6) form a system
of algebraic equations. The two equations
describe the movement of the shock wave
front into the vaporous cavitation zone.
Equations (5) in (6) are solved simultaneously
with the appropriate compatibility equation (8)
or (9) (depending on the direction of travel of
the interface), kinematic equation for the
length of the front movement during time step
∆ t:
t v a L L
m s t t t
∆ + + =
∆ −
(16)
in gibalno enač bo za kapljevinski steber
dolžine L
t
:
and equation of motion for the liquid plug of
length L
t
:
() 0 ) (
2
λ ) ( sign
, ,
2
= −
∆ − − −
∆ − ∆ − t t j j
t
t t j j
t
s j s
Q Q
t gA
L
Q Q
gDA
L
H H a (17)

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
12
kjer je sign(a
s
) = {+1 za a
s 
> 0; -1 za a
s 
< 0}
funkcija predznaka hitrosti širjenja
kompresijskega skoka in H
j
 višina na gorvodni
strani kapljevinskega stebra (Q
uj
 = Q
j
)
(Bergant & Simpson, 1999).
V obravnavanem združenem nelinearnem
sistemu enač b (5), (6), (8) ali (9), (1 6) in (1 7)
so neznanke a
s
, H
j
, H
s
, L
t
 in Q
j
. Hitrost v
m
izrač unamo iz ustrezne enač be (1 0) do (1 4) in
α
v
 iz (1 5). Sistem nelinearnih enač b rešujemo
numerično z Newton-Raphsonovo metodo
(Carnahan et al., 1969).
3.3 NUMERIČ NO REŠEVANJE
ENAČ B DISKRETNE KAVITACIJE
Sprememba volumna diskretne kavitacije je
dobljena z numerič no integracijo kontinuitetne
enač be (7). Integracija v deltoidni mreži
metode karakteristik da (Wylie, 1984;
Simpson & Bergant, 1994):
in which sign(a
s
) = {+1 for a
s 
> 0; -1 for a
s 
<
0} is the sign function of shock wave speed a
s
and H
j
 is the head at the upstream side of the
liquid plug (Q
uj
 = Q
j
) (Bergant & Simpson,
1999).
The unknowns in the coupled nonlinear
system of equations (5), (6), (8) or (9), (16)
and (17) are a
s
, H
j
, H
s
, L
t
 in Q
j
. Velocity v
m
 is
computed from one of the equations (10) to
(14) and α
v
 from (15). The nonlinear system of
equations is solved numerically by the
Newton-Raphson method (Carnahan et al.,
1969).
3.3 NUMERICAL SOLUTION OF
DISCRETE VAPOUR CAVITY
EQUATIONS
A change of a discrete vapour cavity
volume is obtained by a numerical integration
of the continuity e quation (7). Integration
within the staggered grid of the method of
characteristics gives (Wylie, 1984; Simpson &
Bergant, 1994):
()() ( ) ( ) ( ) [ ] t Q Q Q Q V V
t uj t j t t uj t t j t t j vc t j vc
∆ − + − − + =
∆ − ∆ − ∆ −
2 ψ ψ 1
, , 2 , 2 , 2 , ,
(18)
Pritok kapljevine oziroma homogene zmesi
kapljevine in parnih mehurč kov v kavitacijo in
iztok iz nje sta izrač unana s kompatibilitetno
enač bo vodnega udara (8) ali (9) (H
j
 = H
v
)
oziroma enač bo za izrač un pretoč ne hitrosti
kontinuiranega kavitacijskega toka (ena od
(10) do (14)).
3.4 DISKRETNI KAVITACIJSKI
MODEL
Diskretni kavitacijski model generira
kavitacije pri robnih pogojih in v numerič nih
vozlišč ih vzdolž cevovoda, ko se tlak v
cevnem sistemu zniža na parni tlak kapljevine.
Cevni odseki med vozlišč i so simulirani kot
odseki s kapljevinskim tokom. Kontinuitetna
enač ba (1 8) za diskretno kavitacijo v
numerič nem vozlišč u (robnem pogoju) je za
obravnavani primer združena s
kompatibilitetnimi enač bami vodnega udara
(8) in (9) (H
j
 = (H
v
)
j
). Kapljevinska faza se
vnovič vzpostavi pri negativnem volumnu
diskretne kavitacije (zrušitev kavitacije). Temu
sledi standardni izrač un vodnega udara s
kompatibilitetnima enač bama (8) in (9), vse
dokler se tlak vonvič ne zniža na parni tlak.
Nerealno visoke amplitude in razlike v
The liquid or liquid-vapour mixture inflow
and outflow are computed either by the water
hammer compatibility equations (8) or (9) (H
j
= H
v
) or by the equation for the liquid-vapour
mixture (one of (10) to (14)).
3.4 DISCRETE CAVITY MODEL
Discrete cavity model allows cavities to
form at boundary and interior pipeline
computational sections when the pressure
drops to the liquid-vapour pressure. A liquid
flow between the computational sections is
assumed. The continuity equation (18) for the
discrete cavity at the computational section is
coupled with the water hammer compatibility
equations (8) and (9) (H
j
 = (H
v
)
j
). The liquid
flow at a section is re-established when the
cavity collapses as a result of negative cavity
volume. The standard water hammer solution
using equations (8) and (9) proceeds until the
pressure drops to the liquid-vapour pressure.
The cause of the unrealistic pressure
amplitudes and phase shifts as computed by

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
13
č asovnem poteku tlakov (višin), dobljenih z
obravnavanim diskretnim kavitacijskim
modelom izvirajo iz aproksimativnega opisa
kontinuiranega kavitacijskega toka (Bergant,
1992). Opis diskretnih kavitacij in
kontinuiranega kavitacijskega toka po tem
modelu je identič en.
Izboljšani diskretni kavitacijski model, z
bolj natanč nim obravnavanjem nastanka in
zrušitve kavitacije, vgrajen v deltoidno mrežo
metode karakteristik poimenujemo
modificirani kavitacijski model (MDKM
(Bergant, 1992)). Obravnavani model daje
zadovoljive rezultate, kadar omejimo število
cevnih odsekov (največ ji volumen diskretne
kavitacije naj ne presega 10 odstotkov
volumna cevnega odseka (Simpson & Bergant,
1994)). Dodatek plina v parno kavitacijo
zaduši nerealne tlač ne pulze (plinski
kavitacijski razmernik α g 
≤ 10
-7
 (Wylie,
1984)). V prispevku je Wylijev model
modificiran z vgraditvijo diskretnih parnih
kavitacij ob robnih pogojih (ventil) in
upoštevanju samo kapljevinskega toka v
vozlišč u ob rezervoarju. Modificirani model
imenujemo splošni diskretni kavitacijski
model (SDKM (Bergant, 1992)).
3.5 KOMBINIRANI KAVITACIJSKI
MODEL
Kombinirani kavitacijski model (KKM
(Bergant, 1992; Bergant & Simpson, 1992))
zajema izrecni popis diskretnih kavitacij in
področ ij kontinuiranega kavitacijskega toka.
Modificirani diskretni kavitacijski model
(MDKM), ki ob uporabi deltoidne mreže
metode karakteristik oblikuje diskretne
kavitacije robnih pogojev (ventil) in v
numerič nih vozlišč ih vzdolž cevovoda, je
podlaga za postavitev kombiniranega
kavitacijskega modela. Analitič no-numerič ni
modul za obravnavanje kontinuiranega
kavitacijskega toka in numerič ni modul za
obravnavanje kompresijskega skoka, sta
temeljna elementa v transformaciji
modificiranega kavitacijskega modela v
kombinirani kavitacijski model.
Rezultati matematično-fizikalne in
eksperimentalne analize indicirajo (Bergant,
1992; Bergant & Simpson, 1999), da
kombinirani kavitacijski model, v primerjavi z
diskretnimi kavitacijskimi modeli, bolj
natančno reproducira fizikalno sliko
kavitacijskega toka. To bomo pokazali tudi v
tem prispevku.
the discrete vapour cavity model has been
attributed to the approximate description of the
distributed vaporous cavitation zones
(Bergant, 1992). The model describes the
discrete cavities and vaporous cavitation zones
in the same manner.
An improved (modified) discrete vapour
cavity model (MDKM) including a more
concise description of the cavity growth and
decay within the staggered grid of the method
of characteristics has been developed (Bergant,
1992). The MDKM gives reasonably accurate
results when the number of computational
reaches is restricted (the maximum cavity size
should be less than 10% of the computational
reach size (Simpson & Bergant, 1994)). The
unrealistic pressure spikes can be suppressed
when a small gas void fraction (α g 
≤ 10
-7
(Wylie, 1984)) is selected. A modified
(generalized) Wylie's model, in which a gas
cavity at the valve is replaced by a vapour
cavity, and liquid flow is assumed at the
reservoir section, has been developed (SDKM
(Bergant, 1992)).
3.5 INTERFACE VAPOROUS
CAVITATION MODEL
The interface vaporous cavitation model
(KKM (Bergant, 1992; Bergant & Simpson,
1992)) explicitly describes discrete cavities
and vaporous cavitation regions. The modified
discrete vapour cavity model (MDKM)
algorithm, which allows cavities to form at
computational sections within the staggered
grid of the method of characteristics, has been
used as a basis for the development of the
interface vaporous cavitation model. The
incorporation of numerical modules for the
description of vaporous cavitation zones and
interfaces (shock waves) are two important
features in modifying the discrete cavity
model.
Results from the numerical and
experimental analysis (Bergant, 1992; Bergant
& Simpson, 1999) showed an improved
performance of the interface vaporous
cavitation model with comparison to the
discrete cavity model. The same conclusions
are expected to be drawn in this paper.

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
14
4. PREIZKUSNA POSTAJA
Preizkusna postaja za raziskave
kavitacijskega toka med prehodnimi režimi v
cevnih sistemih je instalirana v Robinovem
hidravlič nem laboratoriju univerze v Adelaidi,
Avstralija (Bergant, 1992; Bergant &
Simpson, 1995). Merilna postaja je sestavljena
iz cevovoda z nagibom (bakrena cev; premer
D = 0,022 m, dolžina L = 37,2 m, kot nagiba
cevovoda θ = 3,2
o
), priključ enega na tlač ni
rezervoar z leve in tlač ni rezervoar  z desne
strani - glej sliko 1 . Največ ji tlak v cevovodu
je 5 MPa, v tlač nih kotlih pa 0,69 MPa.
Izbrano dolžino cevovoda določ ajo prostorske
dimenzije laboratorija in zahteve po
najmanjših energijskih izgubah v sistemu.
Prehodni pojav je induciran z zapiranjem
kroglastega zasuna. Ventil je lahko instaliran
ob tlač nih rezervoarjih ali v sredini cevovoda.
Lokacija vgradnje ventila in smer pretoka v
cevovodu omogoč ata simuliranje poljubnega
tipa hidravlič nega sistema (pretoč ni sistem
hidroelektrarne, č rpalni sistem). Ventil je zaprt
roč no ali s pomoč jo zapiralnega mehanizma na
torzijsko vzmet (t
c 
< 0,01 s). Smer pretoka
vode v cevovodu je poljubna glede na tlač no
razliko v obeh rezervoarjih. To omogoč a študij
razmer v cevovodu s pozitivnim in negativnim
nagibom. Elektronska regulacija tlaka v kotlih
omogoč a izbiro poljubne pretoč ne hitrosti in
statične višine. Nastavitev parametrov je
omejena z zmogljivostjo rezervoarjev in
kompresorja. Pretoč ni medij v sistemu je
demineralizirana voda.
Merjene stacionarne velič ine so tlak v obeh
rezervoarjih, barometerski tlak in temperatura
zraka. Velič ine, merjene v odvisnosti od č asa,
so tlak na treh ekvidistantnih mestih vzdolž
cevovoda (tlač no zaznavalo Kistler 61 0 B),
odprtje ventila in temperatura vode v sistemu.
Meritev časovno odvisnih veličin je
registrirana s pomoč jo merilnega rač unalnika
Concurrent 6655, z vgrajenim
analogno/digitalnim (A/D) pretvornikom.
Stacionarna pretoč na hitrost v cevovodu je
dobljena z volumetrič no metodo in metodo
vodnega udara. Hitrost zvoka je določ ena s
č asom potovanja primarnega udarnega vala od
ventila do tlač nih zaznaval. Ob ventilu je
vgrajen fleksibilni transparentni blok
(polikarbonat), ki je aktivne dolžine 0,1 2 m.
4. EXPERIMENTAL APPARATUS
An experimental apparatus for investigating
transient cavitating flow in pipelines is
installed in the Robin Hydraulic Laboratory at
the University of Adelaide, Australia (Bergant,
1992; Bergant & Simpson, 1995). The
apparatus is comprised of a sloping pipeline
(copper pipe of diameter D = 0.022 m, length
L = 37.2 m, pipe slope θ = 3.2
o
) connecting
two pressurized tanks (reservoirs) - see Figure
1. The design pressure of the pipeline is 5
MPa, and of the two tanks is 0,69 MPa. Spatial
dimensions of the laboratory and required
minimum energy losses governed the selection
of the pipeline length in the system. The
transient event is initiated by the closure of the
ball valve. The valve can be located at either
reservoir or at the midpoint of the pipeline.
The variable position of the valve and the
arbitrary direction of the initial flow velocity
provide the simulation of various types of pipe
configurations (hydroelectric power plant,
pumping system). The valve is closed
manually by hand, or it is closed by a torsional
spring actuator (t
c 
< 0.01 s). A specified
pressure in each of the tanks governs the
direction of the steady-state flow velocity in a
pipeline; thus, investigation of transient events
in either an upward or a downward sloping
pipeline can be performed. The specified
initial flow velocity and head are adjusted by a
pressure control system. However, the net
water volume in the two tanks and the capacity
of the air compressor limit adjustment of these
parameters. Demineralized water was used as
the fluid.
The measured steady-state quantities are the
pressure in each reservoir, barometric
pressure, and ambient temperature. Time-
dependent quantities are the pressures at three
equidistant points along the pipeline (Kistler
610 B pressure transducers), the valve
opening, and the water temperature in the
pipeline. Data acquisition and processing are
performed with a Concurrent 6655 real-time
UNIX data acquisition computer. The steady-
state velocity in the pipeline is measured by
the volumetric method and the water hammer
method. The water hammer wave speed is
obtained from the measured time for the wave
to travel between the closed valve and the
quarter point nearest the valve. A flexible flow
visualization block (polycarbonate) of 0.12 m

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
15
Prosojna sekcija omogoča opazovanje in
snemanje kavitacijskega toka s hitrosnemalnim
videom Kodak Ektapro 1000 (Bergant &
Simpson, 1996). Merilne napake so obdelane v
poroč ilu (Bergant in Simpson, 1 995).
active length is positioned at the valve. A
high-speed video Kodak Ektapro 1000 was
used to photograph the transient cavitating
flow (Bergant & Simpson, 1996). The
uncertainties in measurements are given in a
report by Bergant and Simpson (1995).
Slika 1. Preizkusna postaja.
Figure 1. Experimental apparatus.
5. PRIMERJA V A REZULTATOV
IZRAČ UNA IN MERITEV
Primerjava rezultatov izrač una z meritvami
tlaka zagotavlja podlago za verifikacijo
postavk v razvitih kavitacijskih modelih
(MDKM, SDKM, KKM). V prispevku
obravnavamo primer zapiranja ventila, ki je
vgrajen na nizvodnem koncu cevovoda s
pozitivno strmino (glej sliko 1 ). Pretoč ni
pogoji so:
− stacionarna pretoč na hitrost v
0 
= 1,50 m/s,
− višina v gorvodnem tlač nem kotlu H
ur 
=
22 m,
− č as zapiranja ventila t
c 
= 0,009 s,
− hitrost širjenja udarnih valov a = 1319 m/s.
V numerič ni analizi smo izbrali naslednje
parametre:
− število cevnih odsekov N = 16,
− utežni koeficient ψ = 1,
− plinski kavitacijski razmernik v SDKM α
g 
=
10
-7
.
5. COMPARISON OF
COMPUTATIONAL AND
EXPERIMENTAL RESULTS
A comparison of computational and
experimental pressures serves as a basis for the
verification of the proposed column separation
models (MDKM, SDKM, KKM). The paper
deals with a rapid closure of the valve
positioned at the downstream end of the
upward sloping pipe (see Figure 1). The flow
conditions are:
− initial steady-state flow velocity v
0 
= 1.50
m/s,
− head in upstream end reservoir H
ur 
= 22 m,
− valve closure time t
c 
= 0.009 s,
− water hammer wave speed a = 1319 m/s.
The selected parameters in numerical
analysis are:
− number of reaches in pipeline N = 16,
− weighting factor ψ = 1,
− gas void fraction in SDKM α
g 
= 10
-7
.
2.03 m
0.0 m
Pipeline
Cevovod
D = 0.022 m
L = 37.2 m
Ventil
Valve
Tlač no zaznavalo
Pressure transducer
Tlač ni rezervoar
Pressurized reservoir
D = 0.485 m
V = 0.378 m
3
Tlač ni rezervoar
Pressurized reservoir
D = 0.566 m
V = 0.509 m
3

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
16
Slika 2. Primerjava izmerjenih in po MDKM izrač unanih višin H
dv
 pri ventilu
in H
cp
 na polovici dolžine cevovoda.
Figure 2. Comparison of measured and MDKM computed heads H
dv 
at the valve
and H
cp
 at the midpoint of the pipeline.
Izmerjeni tlaki so podani kot piezometrič ne
višine (višine) z osnovnico na vrhu cevi
priključ ene na gorvodni rezervoar (kota 0,0 m
na sliki 1 ). Primerjamo č asovni potek (t)
piezometrič nih višin na gorvodni strani hitro
zapornega ventila H
dv 
in polovici dolžine
cevovoda H
cp
. Višini sta izmerjeni s pomoč jo
piezoelektrič nih tlač nih zaznaval Kistler 61 0
B. Višina ob gorvodnem tlač nem rezervoarju
H
ur
 je konstantna. Dodatno izmerjeni višini na
četrtinah dolžine cevovoda s pomoč jo
induktivnih tlač nih zaznaval Druck PDCR 81 0
sta podobni višini na polovici dolžine
cevovoda. Izmerjeni in izrač unani
piezometrič ni višini pri ventilu H
dv
 in na
polovici dolžine cevovoda H
cp
 sta primerjani
na sliki 2 za MDKM, sliki 3 za SDKM in sliki
Measured pressures are presented as
piezometric heads (heads) with a datum level
at the top of the pipe at the upstream end
reservoir (elevation 0.0 m in Figure 1). The
temporal behaviour (t) of piezometric heads at
the valve H
dv 
and at the midpoint of the
pipeline H
cp
 is compared. The two heads are
measured by the Kistler 610 B piezoelectric
pressure transducers. The head adjacent to the
upstream reservoir is the reservoir head H
ur
.
Supplementary measurements of heads at the
two quarter points by the Druck PDCR 810
strain-gauge pressure transducers show similar
behaviour to the results at the midpoint. The
measured and computed piezometric heads at
the valve H
dv
 and at the midpoint H
cp
 are
compared in Figure 2 for MDKM, in Figure 3
for SDKM, and in Figure 4 for KKM. The
-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
-40.0
0.0
40.0
80.0
120.0
160.0
200.0
240.0
280.0
-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
-40.0
0.0
40.0
80.0
120.0
160.0
200.0
240.0
280.0
 Meritev; Measurement   MDKM
H
dv
 (m)
t (s)
b)
a)
 Meritev; Measurement   MDKM
H
cp
 (m)
t (s)

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
17
4 za KKM. Največ ja višina v cevnem sistemu
je višina vodnega udara pri ventilu. Najmanjša
višina vzdolž cevovoda je enaka parni tlač ni
višini kapljevine. Izrač unani č as eksistence
prve diskretne kavitacije pri ventilu (t = 0,321
s za MDKM; t = 0,331 s za SDKM; t = 0,329 s
za KKM) se razlikuje od izmerjenega č asa (t =
0,339 s).  Za drugo, tretje in č etrto pretrganje
vodnega stebra pri ventilu pa se izmerjeni in
izrač unani č asi eksistence kavitacije dobro
ujemajo. Izrač unani in izmerjeni tlač ni pulzi,
ki sekvenčno slede nastanku in zrušitvi
diskretne kavitacije pri ventilu (slike 2a, 3a in
4a) in kondenzaciji kombiniranih področ ij
kontinuiranega kavitacijskega toka in
diskretnih kavitacij vzdolž cevovoda v
kapljevinsko fazo (slike 2b, 3b in 4b), se
ujemajo v zadovoljivi meri.
maximum head in the pipeline is the water
hammer head at the valve. The minimum head
along the pipeline is the liquid-vapour pressure
head. The computed duration of the first cavity
at the valve (t = 0,321 s for MDKM; t = 0,331
s for SDKM; t = 0,329 s for KKM) differs
from the measured time (t = 0,339 s).  There is
a good match between the measured and
computed duration of the cavity for the
second, third and fourth column separation at
the valve. There is a reasonable agreement
between the computed and measured pressures
induced by the collapse of a large discrete
cavity at the valve (Figures 2a, 3a and 4a), and
pressures induced by the condensation of
vaporous cavitation zones and the collapse of
discrete vapour cavities along the pipeline
(Figures 2b, 3b and 4b).
Slika 3. Primerjava izmerjenih in po SDKM izrač unanih višin H
dv
 pri ventilu
in H
cp
 na polovici dolžine cevovoda.
Figure 3. Comparison of measured and SDKM computed heads H
dv 
at the valve
and H
cp
 at the midpoint of the pipeline.
-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
-40.0
0.0
40.0
80.0
120.0
160.0
200.0
240.0
280.0
-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
-40.0
0.0
40.0
80.0
120.0
160.0
200.0
240.0
280.0
 Meritev; Measurement   SDKM
H
dv
 (m)
t (s)
b)
a)
 Meritev; Measurement   SDKM
H
cp
 (m)
t (s)

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
18
Primerjava rezulatov izrač una in meritev
pokaže, da se rezultati, dobljeni s KKM v
splošnem najbolj u jemajo z meritvami.
Dodatek najmanjše količ ine plina (zraka) v
diskretnih kavitacijah zaduši intenziteto
visokofrekvenč nih tlač nih pulzov, ustvarjenih
z MDKM. KKM v primerjavi z diskretnima
kavitacijskima modeloma (MDKM, SDKM)
bolj natanč no podaja fizikalno sliko
kavitacijskega toka.
The KKM model gives the best results. A
small amount of gas which is added to the
discrete vapour cavity attenuates the high-
frequency pressure spikes generated by the
MDKM. The KKM describes transient
cavitating flow more accurately than the two
discrete cavity models (MDKM, SDKM).
Slika 4. Primerjava izmerjenih in po KKM izrač unanih višin H
dv
 pri ventilu
in H
cp
 na polovici dolžine cevovoda.
Figure 4. Comparison of measured and KKM computed heads H
dv 
at the valve
and H
cp
 at the midpoint of the pipeline.
-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
-40.0
0.0
40.0
80.0
120.0
160.0
200.0
240.0
280.0
-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
-40.0
0.0
40.0
80.0
120.0
160.0
200.0
240.0
280.0
 Meritev; Measurement   KKM
H
dv
 (m)
t (s)
b)
a)
 Meritev; Measurement   KKM
H
cp
 (m)
t (s)

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
19
6. ZAKLJUČ KI
Prispevek zajema fizikalne temelje
kavitacijskega toka pri parnem tlaku
kapljevine med prehodnimi režimi v cevih
sistemih (pretrganje kapljevinskega stebra).
Kavitacijski tok je popisan z enač bami faznih
stanj, ki jih rešujemo z analitičnimi in
numeričnimi metodami. Metode reševanja
enač b so vgrajene v diskretni in kombinirani
kavitacijski model. Meritve vodnega udara in
pretrganja vodnega stebra smo izvedli v
preizkusni postaji. Prehodni pojavi nastanejo
zaradi hitrega zapiranja dolvodnega ventila.
Primerjava rezulatov izračuna in meritev
pokaže, da se rezultati, dobljeni s KKM, v
splošnem najbolj ujemajo z meritvami.
Sklepamo, da razlike med rezultati izrač una z
obravnavanimi kavitacijskimi modeli in
meritev izvirajo iz aproksimativnega opisa
področ ij kontinuiranega kavitacijskega toka in
diskretnih kavitacij. Odstopanja v manjši meri
izvirajo tudi iz diskretizacije v numerič nem
modelu (∆ t = L/(aN) = 37,2/(1319×16) =
0,00176 s), aproksimacije nestacionarnega
koeficienta trenja kapljevine s stacionarnim
Darcy-Weisbachovim koeficientom trenja λ
(Bergant & Simpson, 1994) in merilne
negotovosti (Bergant & Simpson, 1995).
6. CONCLUSIONS
The paper deals with the description of the
transient cavitating flow in piping systems
(liquid-column separation). Column separation
is described by a set of equations representing
a particular liquid phase. The analytical and
numerical methods for solving these equations
are integrated into discrete cavity and interface
vaporous cavitation models. Water hammer
and column separation measurements were
performed in a laboratory apparatus. The
transient event was initiated by a rapid closure
of the downstream end valve. The KKM
model results fit the experimental results most
accurately. The basic source of discrepancies
between the computed and measured results
originates from the approximate modelling of
the vaporous cavitation zones and the discrete
cavities along the pipeline.  In addition,
discrepancies may be also attributed to
discretization in the numerical models (∆ t =
L/(aN) = 37.2/(1319×16) = 0.00176 s), the
unsteady friction coefficient being
approximated as a steady-state Darcy-
Weisbach friction coefficient λ (Bergant &
Simpson, 1994), and uncertainties in the
measurement (Bergant & Simpson, 1995).
VIRI - REFERENCES
Bergant, A. (1 992). Kavitacijski tok med prehodnimi režimi v cevnih sistemih (Transient cavitating
flow in piping systems). PhD Thesis, University of Ljubljana, Ljubljana (Slovenia), in Slovene.
Bergant, A., Simpson, A.R. (1992). Interface model for transient cavitating flow in pipelines.
Unsteady flow and fluid transients, R. Bettess and J. Watts, eds., A.A. Balkema, Rotterdam,
333 - 342.
Bergant, A., Simpson, A.R. (1994). Estimating unsteady friction in transient cavitating pipe flow.
Water pipeline systems, D.S. Miller, ed., Mechanical Engineering Publications, London, 3 - 16.
Bergant, A., Simpson, A.R. (1995). Water hammer and column separation measurements in an
experimental apparatus. Report No. R128, Department of Civil and Environmental Engineering,
University of Adelaide.
Bergant, A., Simpson, A.R. (1 996). Vizualizacija kavitacijskega toka med prehodnimi režimi v
cevnih sistemih - Visualisation of transient cavitating flow in pipelines. Strojniški vestnik -
Journal of Mechanical Engineering, 42(1-2), 1 - 16.
Bergant, A. and Simpson A.R. (1999). Pipeline column separation flow regimes. Journal of
Hydraulic Engineering, ASCE, 125(8), 835 - 848.
Carnahan, B., Luther, H.A., and Wilkes, J.O. (1969). Applied numerical methods. John Wiley and
Sons, New York, USA.

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
20
Simpson, A.R. (1986). Large water hammer pressures due to column separation in a sloping pipe,
PhD Thesis, University of Michigan, Ann Arbor, USA.
Simpson, A.R., Bergant, A. (1994). Numerical comparison of pipe column-separation models.
Journal of Hydraulic Engineering, ASCE, 120(3), 361 - 377.
Streeter, V.L. (1969). Water hammer analysis. Journal of the Hydraulic Division, ASCE, 95(6),
1959 - 1972.
Streeter, V.L. (1983). Transient cavitating pipe flow. Journal of Hydraulic Engineering, ASCE,
109(11), 1408 - 1423.
Wylie, E.B. (1984). Simulation of vaporous and gaseous cavitation. Journal of Fluids Engineering,
ASME, 106(3), 307 - 311.
Wylie, E.B., Streeter, V.L. (1993). Fluid transients in systems. Prentice Hall, Inc., Englewood
Cliffs, USA.
OZNAKE - NOTATION
A preč ni prerez, pipe area;
a hitrost širjenja udarnih valov v
kapljevini,
water hammer wave speed;
a
s
hitrost širjenja kompresijskega skoka, shock wave speed;
B
M
, B
P
konstanti v kontinuitetnih enač bah
vodnega udara,
known constants in water hammer
compatibility equations;
C
M
, C
P
konstanti v kontinuitetnih enač bah
vodnega udara,
known constants in water hammer
compatibility equations;
D premer cevovoda, pipe diameter;
g zemeljski pospešek, gravitational acceleration;
H piezometrič na višina (višina):
H=p/(ρg)+z=h+z,
piezometric head (head):
H=p/(ρg)+z=h+z,
H
cp
višina na polovici dolžine cevovoda, head at the midpoint;
H
dv
višina pri ventilu (gorvodna stran), head at the valve (upstream end);
H
s
višina na gorvodni strani kompresijskega
skoka,
head on the water hammer side of the
shock wave front;
H
sv
višina na dolvodni strani kompresijskega
skoka,
head on the distributed vaporous
cavitation side of the shock wave front;
H
ur
višina v gorvodnem tlač nem rezervoarju, head in upstream end reservoir;
H
v
višina pri parnem tlaku kapljevine, liquid-vapour head;
h tlač na višina, pressure head;
j številka vozlišč a, number of computational section;
k številka cevnega odseka, number of computational reach;
L dolžina cevovoda, pipe length;
N število cevnih odsekov, number of reaches in pipeline;
p tlak, pressure;
Q pretok, pretok iz vozlišč a, discharge, discharge at the downstream
side of the computational section;
Q
u
pretok v vozlišč e, discharge at the upstream side of the
computational section;
t č as, time;
t
c
č as zapiranja ventila, valve closure time;
t
i
č as nastopa kavitacije, time of cavitation inception;
t
in
č as nastopa diskretne kavitacije, time of inception of the discrete vapour
cavity;

Bergant, A.: Kavitacijski tok med prehodnimi režimi v cevnih sistemih - Transient Cavitating Flow in Piping Systems
© Acta hydrotechnica 18/29 (2000), 3-21, Ljubljana
21
t
r
č as zaustavitve pretoka, time of flow reversal in liquid-vapour
mixture zone;
V volumen, volume;
V
vc
volumen diskretne kavitacije, discrete vapour cavity volume;
v pretoč na hitrost, hitrost iz vozlišč a, flow velocity, velocity at the
downstream side of the vapour cavity;
v
m
pretoč na hitrost homogene zmesi
kapljevine in parnih kavitacij
(mehurč kov),
liquid-vapour mixture velocity;
v
mi
zač etna kavitacijska hitrost, inception velocity of liquid-vapour
mixture;
v
mt
gravitacijska hitrost, terminal velocity in the distributed
cavitation region;
v
u
hitrost v vozlišč e, velocity at the upstream side of the
vapour cavity;
v
0
stacionarna pretoč na hitrost, initial flow velocity;
x koordinata, distance;
z geodetska višina, pipeline elevation;
α
g
plinski kavitacijski razmernik, gas void fraction;
α
v
parni kavitacijski razmernik, void fraction of vapour;
∆ t č asovni korak, time step;
∆ x dolžina cevnega odseka, reach length;
θ
strmina cevovoda, pipe slope;
λ
Darcy - Weisbachov koeficient trenja, Darcy-Weisbach friction factor;
ρ
gostota tekoč ine, liquid density; and
ψ
utežni koeficient. weighting factor.
Naslov avtorja - Author's Address
doc.dr. Anton Bergant
Litostroj E.I. d.o.o.
Litostrojska 40, SI - 1000 Ljubljana
 Email: anton.bergant@litostroj-ei.si