Fox J.A. Transient Flow In Pipes, Open Channels And Sewers
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The
programming
of the wave
equations
lch.
12
4
5
6
c
100
TP
(
_SO
),
PO(50
),
PINER
(
50
),
DP
(
50
),
HV, TNKL
(
50
),
DTF
(
50
),
POLY,
DTI
(
50
),
AC
(
50
),
AE
(50
),
TVS(
50
) .
TVF
(
_c0
),
KSR
(
50
),
pArRr
(
50
),
AT
(
50
),
HB
(
5C
),
ICHO
,
TSIM, DT
,
II
,
KK
HRITE(1,100)
FORMAT('
ENTRY OF VALVE')
NET=NETNO(
I
)
AL=PI/4.0rD(I
)'D(
I
)
CALL [4ASP(
I, NET,K,C1
)
CALL
FRINT( I,NET,
HX,
VX,
FX,
1,K,C1
)
IF
( (
T.
LE. TvS
(
I
) )
. AND.
(
KSR
(
I
)
. LT. 0
) )
VALK=VK
(
I
,
21
)
IF(
(T.LE.TVS(
I))
.AND.
iK*sR(I)
.cT.0)
)VALK=VK(
I,1)
IF
(
(
T.cT. T'/F(
I
)
)
. AND.
(KSR (
I
)
.
LT. 0
) )
VALK=VK
(
I, 1
)
IF(
(T-GT.TVF(I
)
).AND.
(I.:SR(I)
.cT.0)
)VALK=VI:(
I,21)
IF(T.
LT. TVS( I
)
.OR.
T.GT.
TVF(I
) )GOTO
Z
DTV=
(TvF(
I
)
-TVS(
I)')
/20
.
o
L=INT
( (
T-TV-s(
I
) ) /DTV )
+1
DELT=(
T-TVS( I
)
-
(L-
1
)
t
DTV
)
IF
(RSR
(
I
)
.
LT .0)L=22-L
fFiKSR(I
)
.
cT.0)VALIi=VK
(
I, Li
+
(VK(I,
L+
j.
)-VK(
I. L)
)
rDELT//DTV
IF(ti-sR(I)
.LT. 0)VALK=VK(I,
L)+
(VK(I,L-1
)-VK(
I,L)
i'DELT,/DTV
IF( IF.OUTE(
I.2)
. EO. 42)GOTO
1
AZ=PI/4.
0tD( I+L
)
'D(
I+1
)
CALL
tlA-sP(
I+1
.
!
,R,CZ)
CALL
FRINT(I+1,
1,
HY,VY.
FY,
-1,K,C2)
IF
(AB-s(OV(I
) )
.LT.
0. 0001
)Ov(I)=0.
0001
AL-VALK,/(2.
0'A1
tA1'c)
xOV(I
)
/ABS(AV(
I
) )
EET=
(
C2/
A2+Ct
/
A1
)
r/G
GAI4=HY-HX-
(
C2'VY+C1
I
VX
) /G+
(
(!
tpy+C1
r
FX
) /G
SURD=BEI
rBET-4
. C
rALrcAM
IF(SURD.
LT. 0.
0
)
SURD=O.
0
AV
(
r
)
= (
-BET+SORT(SURD)
| / Q.0rAL)
n2(I,NET)=GV(I)/A1
V2(I+1,1)=OV(Il/Az
H2
(
I
,NET)
=HX-C1/Gr
(Vz{I,
NET)
-VX+FX)
H2
(
I+1,
Ll
=HY+CZ/G.
(Vz(f
+1,
1
)
-VY+Fy)
IF
(
IROUTE
(T,21
.
EO. 41
)GOTO
3
AL=VALK/
(2.0r41
rA1
tc)
BET=CI,/(GrA1)
GAM=HA+HRES
(
I
,
2)
+Z
(y
,
ll
+Ctr
FXlc-HX-
CLrvX/c
SURD=BETTBET-4
.
0
rAL
rcAlrl
IF
(SURD.
LT. 0.
0
)
SURD=O.
0
AV( I
)=
(
-BET+SQRT(SURD)
)
/
(2.0tAL)
V2(I,NET)=OV(I't/Al
H2
(I
,NET
)
=HX-C
L
lca
(V2 (
I
,
NET
)
-VX+
FX
)
CONTINUE
IF
(H2(I,NET)
-Z(I,2t
.LT.HVlH2(I,NET)
=HV+Z
(I,2')
IF(H2(f+1,1)-Z(I+1,
1
)
.LT.HV)H2(I+1,
1
)=HV+Z(I+1,
1
)
l,tRrTE(1,101)
FORHAT('
EXIT
OF
VALVE')
RETURN
END
3
101
Sec.
12.41
Subroutines
12.4.10
The
midstream
subroutine
MIDSTR
This
subroutine
does
most
of
the
work
of
the
program.
It
handles
the
transmission
of
the
wave
from
one
end
of
the
pipeline
to
the
othet
by
calculating
the
heads
and
velocities
at each
Ax
increment
ihioughout
the length
of
the
pipe network'
SUBROUTINE
MIDSTH
(
I,}:)
coMMoN
/AA/N.
EL
(
50
),
D(
50
),
EK
(
50
),
G,
['l
"
t',lNU'
H1
(
50'2!)'H2(
50',
21
)',
1OP(50),V1(50,27),V2(50,21),2(50,2),DX,TROUTE(50'2)',NETNO(50)'
2
JUNCT(50,
5
),
HRES
(
50,
2
),
Pr,
HA,
HG,
E1
.
82,
RI.ISP,
IP,
vK
{
50,
21
),
ov
(
-c0
)'
3
FK,FE(-q0)
.TK(50)
,T.AP(501
,epi50)
,cP(50)
,PN(-s0),PNS(50)
'
EFF(50)
'
4
Tp
(
50
),
pL)
(50
),
PINER
(50
),
Di
(
50
),
HV,
TNKL
(
50
),
DTF
(
-s0
),
PoLY'
DTI
(
50
)'
5
AC
(
50
),
AE
(50
)
:
Tv-s
(
50
),
TvF
(
50
),
KSR
(
50
),
PATRI
(
-c0
),
AT
(
50
)',
HB
{
50
)'
6
ICHO,TSIM.DT,II,KK
c
l.lRITEi1,100)
1CO
FORMAT('
ENTRY
OF
MIDSTR')
tiET=l'iETNC(
I
)
DO
1.T=1.NET
CALL
IIASP
(
I
,
.T,
}:,
C)
L:ALL
FHINT
(
I
,
.T
,
HX,
VX
,
FX
,
1
,
K
,
C
)
CAIL
FFIINT(
I,.1,HY,VY,
FY,
-1,K.C)
H?
(I..1
)
=
(Hll+HY
i /i+clGr
( (vx-vY)
/2-
{JX,-FYJ
/!)
ZE=Z{I
,
1
)
+
(Z(I
,2J
-Z(I
,
1
}
)
r'
(
J-
1
)
tDXi/EL
(
I
)
IF
(H2(
I..T)
-ZE.
LT'
HV)H2
(
I,
J
)
=HV+ZE
V2(
I..T
)
=VX-G/C'
(HZ{
I
.
J
)
-HX)
-FX
J.
Ct'INTINUE
c
wRrTE(
1.101)
:T1
FOF.HAT{'
E}:IT
OF
MIDSTR')
RETURN
END
l2.4.ll
The
junction
subroutine
JUN
This
subroutine
can
be
written
as an
upstream
or
as
a
downstream
subroutine.
A
case
can be
made
for
having
both
versions
avai
that
there
is only
a
necessity
for
the
dowr
givenhere
is complex
and
can
be
written
n
looget
to
run.
This
procedure
is
more
fr
midstream
subroutine);
so
its
efficiency
is
is,
probably,
the
most
important
boundal
number
of
pipes
joining
at
the
junction
i
choice
and
ian
be
changed
and
set
to
any
value
that
the user
likes.
c
c
c
c
c
c
c
c
r'rt'tttrrttttf*t'trt-:3tl'r'rtt'trlrlttrll''ltlrlrtft**tltl|lt'''l
SUBROUTINE
JUN
CAN
HANDLE
AS
MANY
PIPES
JOINING
AT
THE
JUNCTION
AS
MAY
BE
REOUIRED.
AS
THE
ARRAY
JUNCT
IS
CURRENTLY
SET
TO
BOUNDS
OF
5O'5
ONLY
4
PIPES
CAN
BE
HANDLED.
IF
THE
BOUNDS
ARE
I,J
I
IS
A NUHBER
LARGER
OR
EOUAL
TO
THE
NUMBER
OF
PIPES
IN
THE
NETI']TIRK
AND
J
IS
A
NUMBER
LARGER
OR
EOUAL
TO
THE
LARGEST
NUI''BER
OF
PiPES
'IOINING
AT
ANY
JUNCTION
IN
THE
NETI.JORK
PLUS
ONE.
rtttlfrrrl**t'tttxl'flrltllrr*tlIlIttl:'l:t'rllr:tt*f,'rttlrlratltlt
SUBROUTINE
JUN(I)
DTMENSTON
IS(50),Nr
(50),Ar(50),
JE(50)'
258 The
programming
of the
wave
equations
lch.
12
vY(50,2
) .HY(50,
2),
FU(50.
2
)
coMMoN
/
AAiN.
EL
(
50
),
D
(
50
),
EK
(
_5C
),
c,
[,J,
l,tNU,
H1
(
_c0, 27
l,H2
(
50, 27
),
oP
(
50),
v1
(50,27',),V2 (
50, 2\),2
(
50. 2
),
DX, TROUTE
{50,
2j, NETNO
(50
),
.TUNCT(50,5.),HRE-S(50,2),pI.HA,HG,E1,E2,R[.lSp,rp.VK(50,21),OV(50),
FK,
FE( 5.0
)
.
TIt
(
50
),
T, Ap(
50
),
Bp
(
50
),
Cp
(
5.0
),
pN
(
50
),
pNS
(
50)
.
EFF
(
50
),
TP
(
5O
),
PO
(
50
)
.
PINER
(
50
) .
DF
i
-qO
),
HV,
TNKL
(
50
),
DTF
(
50
),
POLY,
DTI
(
50
),
AC
(
5C'),
AE{50
),
TVS
(
50
) .
TVF
(
50
j,
I.:SR
(
50
},
PAIRI
(
50
} .
AT
(
50
),
HB(50
),
rCHO, TSII,'. DT. ] I
,I.:K
HRITE(1,100)
FORMAT
(
'
ET{TRY OF
"IUN'
)
SG1=0 - 0
SG2=0.0
SG3=0
- 0
SG4=0. 0
JU=,IUNCT(I,1)
.iU=JU+1
DO 1 l{=2,
JU
.TS=JUNCT(I,M)
JA=IABS(JS)
IS(JA)=JS/.IA
AI
(
JA)
=PI
/
4xD
(.TA
)
r
*2
NI
{
JA)
=NETNO
(
JA
)
IF
(
IS(
JA)
.tT.
1
)NI
(JA)
=1
cAtL
HASP(JA,
Nr
(Je1,gg1
(JA,NI
(
JA)
)
)
JE
(
JA)
=2
IF(IS(
JA)
. EO.
1
)
JE(JA)
=1
CALL FRINT(JA,NI
(JA)
,HY(JA,JE(JA)
),VY(JA,
JE(JA)
),FY(JA,JE(JA)
),
1
IS(JA)
,CH1
(JA,NI
(JA)
) )
SG1.=*sG1+fS(JA)
rAI
(
JA)'VY(
JA,
JE(JA)
)
SG2=SG2+G
/Ct
AI
(
JA)'HY
(.TA,
JE
(
JA
) )
SG3=SG3+IS(JA)
rFY(
JA,
JE(JA)
)
tAI
(,IA)
.*iC4=SG4
+
G
//C
I
A I
(
-TA
)
1
CONTINUE
HP=
(-(G1+SG2-SC3
) /Sc4
IF(HP-Z(,TUNCT(
1,2),2)
. LI.
HV)Hp=HV+Z(JUNCT
(T,Z),2)
DO
2 M=2,JU
JS=JUNCT(I,M)
JA=IABS(
J-s
)
H2(JA,NI(JA)
)=HP
2 v2{JA,Nr(JA)
)=vY(JA.-TE{JA}
)-rs(:al'c/c'
(Hp-Hy(JA.,iE(JA)
) )
1
-FY(.TA,.TE(.TA)
)
CONTINUE
llRrTE(1,101)
FORHAT('
EXIT
OF .IUN')
RETLIRN
END
References
[1]
Swaffield,
J.
A.
A study
of
column
separation
following
valve
closure
in
a
pipeline carrying
aviation
kerosine.
Proc.
Inst.
Mech.
Eng.No.
23,
1969'
[2]
Von
Reimann,
E.
Portirh,
Dffirenrtalgleichungen,Braunschweig,
1869'
[fi
AUi"ui,L.
Teoria
generale
del
noto
perturbato
dell'acqua
nei
tubi
in
pressione,
Milan,
1903.
Tranilated
into
English
in 1925
by
E.
E.
Halmosi
The
theory
of
waterhammer,
Ameican
society
of civil
Engineers,1925.
[a]
Schnyder,
O.
Druckstosse
Pumpensteigleitengungen
Schweitz
Bauztg'
94,
Nos
22
and23,1929.
[5]
Bergeton,
L.
Etudes
des
variations
de
r6gime
dans
les
conduites
d'eau.
Rev.
Gtn.
Hydraulique,
Nos
L
and
2,
L935.
[6]
Angus,
R.W.'Waterhammer
in
pipes including_those
supplied
by
centrifugal
purips;
graphical
treatment.
Proi.-Inst.
Mech.
Eng.,
pp. 136, 245,.1973.
[Z]
nngus;
fr.'W.
Waterhammer
in compound
and
branched
pipes
' Trans'
Am'
Soc.
Civ.
Engrs
PP.
104,
340,1939.
[g]
Zienkieweiz,
O.-C.
and
Hawkins,
P.
Transmission
of
water
hammer
pressures
through
surge
tanks.
Proc.
Int.
Mech-
EnT'68,
No'
25,
1954'
[9]
Lax,
F. unJWendroff,
B.
Systems
of
conservation
laws.
Comm-
Pure
AppI'
Matls,
12,217-237
,
L960.
[10]
Streeter,
V.
L.
and
Lai,
C.
Waterhammer
analysis
including
fluid
friction.
Trans.
Am.
Soc.
Civ.
Engrs
128,1963.
[11]
Vardy,
A.
E. On
the
usJ
of
the
method
of
characteristics
for
the
solution
of
'
'
unsteady
flows
in
networks.
BHRA
2nd
Int. Conf.
Pressure
Surges,
PaperHZ,
1976.
[12]
Suter,
p.
Representation
of
pump
characteristics
for
calculation
of
waterham-
mer.
Sulzer
Review,
L966.
[13]
Donsky,
B.
Complete
pump
characteristics
and
the effects
of specific
speeds
on
hydruuii,
ttuotients.
J.
Bastc
Eng.
December,
pp.
685{99
,
196I
-
[1a]
Karplus,
M.
B.
The
velocity
ol sound
in
a
liquid
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gas bubbles'
'
-
Arrnour
Research
Earth
Foundation
report,
June,
1958'
[15]
De
Saint
Venant,
Barrd.
Elementary
demonstration
of
the
propagation
1
1
2
1
4
5
6
100
99
c
io1
260
References
formula for a
wave
or a translatory wave
in a
prismatic
channel; remarks
on the
propagation
of sound and lights
on hydraulic
j
umps
; ,
remarks
on the distinction
between
rivers
and
torrents. Acad.
Sci.
(Paris)
Comptes
Rendus 71, pp.
19G195, 1870.
[16]
Courant,
R., Friedrichs,
K. and
k*y, H.
On the
partial
differential
equations
of
mathematical
physics.
New York
University Institute
of Mathematics
(trans-
lated by P.
Fox),
1956.
[17]
Lax,
P. Weak solutions
of nonlinear
hyperbolic
equations
and their numerical
computations. Comm.
Pure Appl. Maths 7
,
pp.
159-193, L954.
[18]
Strelkov,
T. Numerical
solution of
St
Venant
equations. Jour.,
Hyd.
Div.
Amer. Soc. Civ. Engrs
January,
pp.23T252,1970.
[19]
Preissman, A. and
Cunge,
J.
A.
Calcul des intumescence
sur machine electroni-
que. IX Meeting Assoc.
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Dubrovnik,
196L.
[20J
Edgell, G. Pressure Transien$
in Tunnels:
Extension
of
Theory to lrreversible
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Limited
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UK, L974.
Bibliography
Abbreviations
Adv.
Advances.
e.t.Cft.B.
American
Institute
of
Chemical
Engineers'
An.
Annual.
App.
APPlied.
nlS.C.E.
American
Society
of
Civil
Engineers',
n.S.tU.f
.
American
Society
of Mechanical
Engineers'
B.
Bulletin.
g.H.n.a.
BritishHydromechanicsResearchAssociation'
Br.
British.
Chem.
Chemical'
Commun.
Communications'
Conf.
Conference.
Cong.
Congress.
c.s.'N.I.
coninittee
on
the
Safety
of
Nuclear
Installations.
Div.
Division.
Ed.
Editor.
Ens.
Engineering-
EnErs.
Engineers.
ie.rr.n.
IntErnationalAssociationforHydraulicResearch.
I.C.E.
Institution
of
Civit
Engineers'
i.fvfecn.e.
Institution
of Mechanical
Engineers'
Ind.
Industrial.
Inst.
Institute.
Instn.
Institution.
Int.
International.
J.
Journal.
f
.S.f"f
.B.
Japan
Society
of Mechanical
Engineers'
Lab.
LabortatorY'
Math.
Mathematics'
Nat.
National.
N.E.L.
National
Engineering
Laboratory'
No.
Number.
Pp.
Pages'
Pioc.
Proceedings'
Prog.
Progress.
Pt.
Part.
O.
Quarterly'
Rep.
RePort.
Rev.
Review'
262
Sci.
Soc.
symp.
Tech.
Trans.
Vol.
Bibliography
Science.
Society.
Symposium.
Technology.
Transactions.
Volume.
BIBLIOGRAPHY
OF
BOUNDED
UNSTEADY
FLOW
(Contributed
by Patricia Dawson)
Abbott,
M. B.
Continuous Flows, Discontinuous
Flows
and Numerical Analysis.,I.
Hydraulic.
Research, Vol.
t2(4),
pp.
417467
.
Abbott,
M. B.
Computational Hydraulics:
A
Short Pathology. ,I.
Hydraulic
Research,
Vol. 14(4)
,
7976,
pp.
277-285.
Abbot, M.
8., Verwey, A. Four-Point
Method
of
Characteristics.
Proc.
A.S.C.E.,
I.
Hydraulics Div.,PNLZ, Vol.
96,
Dec.
1970,pp.2549-2564.
Abbott,
M. B.,
McCowaD, A.D.,
Warren,I.
R. Accuracyof
Short-Wave Numerical
Models.
J. Hydraulic.
Eng., A.S.C.E.,
Vol.
110(10),
Oct. 1984,
pp.1287-1301.
Agu,
J., Karterud,
T.
J.,
l.{ielsetr,
T. K .
Testing
oi Transient Fiow and
Coiumn
Separation in
Crude Oil
Pipelines.
B.H.R.A.
Third Int.
Conf. Pressure
Surges,
Paper
C1,
1980,
pp.1,lTl26.
Akagawa,
K.,
Sakaguchi,
T.,
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Index
Abbot,
S
absolute,
116
accelerate,
13,106
acceleration
,
87
,
782,
183,
187
angular,
17,69,106
term,28
actuator,
multi-sPeed,
255
acoustic
velocity,
123
wave
speed,
42,153
adiabatic,
89,240
Adriatic
sea,
198
aerofoil,240
air,
15
AIRV,250,25l
air
vessel,
d,12,14,15,85,
90,112,113,
114,
115'
116,
1 19,
l2l,
r30,
244,
250,
253
aleorithm,
136
Alieni,
7, 36,
41,
42,
50,
5I,
52
Inteilocking
Equations,
16,
45,
51,
52
American
Practice,Z4
amolifier,2l2
amblitude,
lL,
114,
123,
17
4, 205,
212
pressure head,22l
flow,?21
anchor
blocks,
37
angular
velocity,
17
,69,88,
95,
106
Angus,
7,62,63
apex,227
ahay,
81,
87,
96,
98,
I 15,
l3l,
I32,
134,
136,
255,
257
head,
130,
246,247
atmospheric
pressure'
n,
4I,
Il4,l'41
super,114
sub,
114
attenuation,
39,
58,
59,
62,
65,
67,
123,
125,
177
axial
flow
machine,
127
azimuth,
37
back
electromotive
force,
107
base
of tank,
116
barometer
water,246
liquid,249
barrage,
182
bearing,
113
losses,
105
resistance
torque,
103
bed,11,
12,167
dry,11,14
ofthe
channel,
157
shear,226
Bernoulli's
equation,
29,
50,
53,
2Zl
Bergeron,8,62
bi-characteri
stic,
223
,
224
'
233
blade,9,
14,16
angle,95
blower,89
block
data,
246
body
forces,
15,225
booster
pump, 67,94
boundary,
85,
146,
215,
232
node,
78,79
upstream,78,79
Uou'nO"ty
conditions,
49, 51,
5\
5?
: 97 : :69,,7
4'
-
79,8i,
85,
L26,
130,
r57,
159'
168,
196,
215,
2!7,
2lg,
222,
23r,
239,
245,
246
downstream,52
upstream,52
boundary
laYet,224
separation,94
bounds,
115
brace,114
Braun,
T
breadth,
95
British
practice,
24
u"6ur",
!4,
!6,'&,
4!,139,
140,
142,
143,
247,
249
fractional
content'
1/m,
141
buckling,
114
bulb
turbine,
127
Uuif:
rnoOufu:.
,12,42,88,
139'
740,247
,249
burn
out,
90
burst
disc,90
burst
of
a
piPeline,
43,241
bursting,90
C
(computer language),
242