im
Journal of	JET v°iume 8 (2015) p.p. 31-42
Issue 2, October 2015 Typology of article 1.01
Technology	www.fe.um.si/en/jet.html
NUMERICAL AND EXPERIMENTAL INVESTIGATIONS OF TRANSIENT CAVITATING PIPE FLOW
NUMERIČNE IN EKSPERIMENTALNE RAZISKAVE PREHODNEGA KAVITACIJSKEGA TOKA V CEVI
Anton BergantR, Uroš Karadžic1 Keywords: pipe flow, transient cavitation, discrete gas cavity model, unsteady skin friction
Abstract
This paper investigates the effects of transient vaporous cavitation caused by the closure of the downstream end ball valve against the discharge. Numerical results are compared with the results of measurements in the simple reservoir-pipeline-valve apparatus. Pressures measured at the end points and at two equidistant positions along the pipeline are compared with computational results as piezometric heads. Comparisons between the results of two distinct water column separation tests and numerical simulations using an advanced discrete gas cavity model show good agreement. Two distinct column separation runs include active and passive column separation cases.
Povzetek
Prispevek obravnava prehodni parni kavitacijski tok induciran z zapiranjem dolvodnega kroglastega zapirala v sistemu pod pretokom. Računski rezultati so primerjani z rezultati meritev v preprosti preizkusni postaji, ki jo sestavljajo rezervoar, cevovod in zapiralo. Tlaki merjeni na dolvodnem in gorvodnem delu cevi in tlaka merjena na ekvidistantnih dolžinah vzdolž cevi so primerjani z izraču-
R Corresponding author: Anton Bergant, PhD, Litostroj Power d.o.o., Litostrojska 50, SI-1000 Ljubljana, Slovenia, anton.bergant@litostrojpower.eu
1Uroš Karadžic, PhD, Univerzitet Crne Gore, Mašinski fakultet, Džordža Vašingtona bb, ME-81000 Podgorica, Montenegro, uros.karadzic@ac.me
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nomkotpi ezometricne visine. Rezultatimeritevinizracunovdobljenihs f^omc^c^jo naprednfsga dis-k^^^r^^l^a p^stegntavitecy^ke;par^onep sedea ppsebn-primeraferefrganjaka pn'n-^^r^!^r^egnsmkce se jJodro ejemajg.Prviprimer zajemadl<0dnoobiiko pcpf r^^me sz^t^rmeeugi primer pa nredcrav|ja l-asiveo olol^optettgnejn.
1 INTRODUCTION
Ipen-trial nineline -y-tem- dkhrath uper e nruee rapdh nf dkhratipd reaimg-. Inencge nn-teaey flaw- in nine- ane -y-tem cnmnnnent- (palpe, nomp, torniee) are tde -unrce nf many nnwantee luae- in inen-trial in-tallatinp-, inclneina -epere kre--nre knl-atiup- ane kikelipe pinratinn- [1], [2]. Water hammer i- tde krukaaatiup nf nre—nre wape- alnna liqnie-fillee kikeliPe-, ane it i-can-ee by a change in flnw pelncity. Tde cla--ic water hammer effect may be affectee by tran-ient capitatinn, nn-teaey frictinn, flnie--trnctnre interactinn (FSI) ane pi-cnela-tic nedapiuor nf tde nine wall [3]. Tran-ient capitating nine flnw nccnr- a- a re-nlt nf pery lnw nre—nre- enrina water dammer epent-. Tdi- naner eeal- witd tran-ient pannrnn- capitatinn (cnlnmn -enaratinn) tdat nccnr- wden tde nre--nre ernn- tu tde liqnie pannnr nre--nre. Tde amnnnt nf free ane/nr relea-ee ga- in tde liqnie i- a--nmee tn be -mall. Tdi- i- n-nally tde ca-e in mn-t inen-trial nineline -y-tem-. Twn ei-tinct tyne- nf tran-ient pannrnn- capitatinn may nccnr. Tde fir-t tyne i- a lncalizee (ei-crete) pannnr capity witd a large pnie fractinn. A ei-crete pannnr capity may fnrm at a nuupeary (palpe, nnmn, tnrnipe) nr at a digd nuint alupa tde nineline. In aeeitinn, an intermeeiate capity may fnrm a- a re-nlt nf tde interactinn nf twn lnw-nre--nre wape- alnna tde nine. Tde -ecnne tyne nf capitatinn i- ei-trinntee pannrnn- capitatinn tdat may extene uper lung -ectinn-nf tde nine. Tde pnie fractinn fur tdi- ca-e i- -mall (cln-e tn zern). Di-trinntee pannrnn- capitatinn nccnr- wden a rarefactinn wape nrnare--ipely ernn- tde nre—nre in an exteneee reainn nf tde nine tn tde liqnie pannr nre—nre. Tde cnllan-e nf a pannnr capity may inence -dnrt-enratinn nre—nre nnl-e- witd palne- digder tdan tde nre—nre initially aipen by tde Jnnpnw-py eqnatinn. Beraant ane Simn-nn [4] cla—ifiee cnlnmn -enaratinn flnw regime- reaareina tde ndy-ical -tate nf tde liqnie ane tde maximnm nineline nre—nre a-:
(i)	Active column separation flow regime. Tde maximnm nineline nre—nre i- generatee fnllnwina tde cnlnmn -enaratinn at tde palpe ane alnna tde nineline (actipe cnlnmn -enaratinn frnm tde ee-ianer'- ner-nectipe). Tde maximnm nre—nre at tde palpe i- anpernee by tde inten-ity nf tde -dnrt enratinn nre—nre nnl-e.
(ii)	Passive column separation flow regime. Tde maximnm nineline nre—nre i- tde water dammer nre—nre before inten-e capitatinn nccnr-.
Tde palne nf tde frictinn factnr enrina nn-teaey flnw i- eifferent tdan it- palne enrina -teaey flnw. Tde frictinn factnr can be exnre—ee a- a -nm nf twn nart-: 1) -teaey ane 2) nn-teaey [5]. Tde nn-teaey nart mimic- tran-ient-inencee cdanae- in flnw cnneitinn- (pelncity nrnfile, turnulepce inten-ity), ane it i- imnnrtant fur -nme nn-teaey flnw -itnatinn-. Fur nineline- tdat are nut cnmnletely fixee, FSI effect- dape tn be taken intn accnnnt [6]. Vi-cnela-tic nine-wall nedapiuur i-imnnrtant in ca-e- in wdicd tde nine i- maee frnm nla-tic material -ncd a- diad-een-ity nnlyetdylene [7]. Ranie fillina ane emntyina nf tde nineline may be cnn-ieeree tn be a -necific ca-e in wdicd bntd pannrnn- ane aa-enn- capitie- may be nre-ent [8]. Enaineer- -dnnle be able tn nreeict all tde-e epent- in ninina -y-tem- ane take annrunriate mea-nre- tn Seen water dammer luae- witdin tde nre-cribee limit-. Tdere i- a -trnna neee fur well-cnntrnllee
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measurements of the water hammer effects; therefore, a flexible pipeline apparatus for investigating water hammer, transient cavitating flow, unsteady skin friction, fluid-structure interaction, and pipeline filling and emptying has been developed and installed at the University of Montenegro [9]. The small-scale apparatus consists of an upstream end high-pressurized tank, horizontal steel pipeline (total length 55.37 m, inner diameter 18 mm), four valve units positioned along the pipeline including the end points, and a downstream end tank (outflow tank). This paper investigates the effects of vaporous cavitation caused by the closure of the downstream end ball valve against the discharge. Comparisons between the results of two distinct water column separation tests and numerical simulations using an advanced discrete gas cavity model [10] are presented and discussed.
2 THEORETICAL MODELLING
Water hammer in liquid-filled pipelines is fully described by the continuity equation and equation of motion [1], [2],
dH a2 dQ „
-+--— = 0, (2.1)
dt gA dx
dH 1 dQ fQQ | „
-+--* ' = 0 . (2.2)
dx gA dt 2gDA
Note that all symbols are defined in the Nomenclature. Water hammer equations are valid only when the pressure is above the liquid vapour pressure. A quasi-steady approach for estimating skin friction losses (QSF) in the pipeline is satisfactory for slow transients only, [11]. Equations (2.1) and (2.2) are solved by the method of characteristics (MOC) using the staggered numerical grid, [1]. At a boundary (reservoir, valve, turbine), a device-specific equation is used instead of one of the MOC water hammer compatibility equations.
Some numerical models have been developed for simulation of transient vaporous cavitating pipe flow. One of them is a discrete gas cavity model (DGCM) that performs accurately over a broad range of input parameters [4]. The DGCM allows gas cavities to form at all computational sections within the MOC numerical grid. A liquid phase with a constant wave speed is assumed to occupy the computational reach. The DGCM is fully described by the two water hammer compatibility equations as a result of the MOC-transformation of Eqs. 2.1 and 2.2, and two additional equations; the continuity equation for the gas volume and the ideal gas equation with assumption of isothermal behaviour of the free gas, respectively, [1], [4],
dt
V g = a0V
= Qout - Qin ,	(")
(2.4)
( * A Po
v pg
The numerical solution of DGCM equations can be found elsewhere, [1], [4].
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Column separation is a relatively short duration event with a wide range of rapid flow event types. For rapid transients, the unsteady friction model is needed for the proper estimation of skin friction losses during transient events, [11]. The friction factor can be expressed as a sum of the quasi-steady part fq and the unsteady part fu, [5]
f = fq + fu .	(2.5)
The quasi-steady friction factor fq depends on the Reynolds number and relative pipe roughness. A number of unsteady friction models have been proposed in the literature including one-dimensional (1D) and two-dimensional (2D) models. In this paper, an improved convolution based unsteady friction model [12] is used in DGCM, [10]. The convolution-based model (CBM) has been analytically developed by Zielke for transient laminar flow, [13]. This model produces correct results for some flow types using analytical expressions. The unsteady friction term fu is defined as, [12]:
32 vA ^
f = ^3250 § "(()	1261
The quantity yk accounts for weights of past velocity changes. It is expressed as a recursive expression; theoretical derivation for yk is given in, [12].
3 DESCRIPTION OF PIPELINE APPARATUS
A small-scale pipeline apparatus has been designed and constructed at the Faculty of Mechanical Engineering, the University of Montenegro, [9], for investigating water hammer, column separation, fluid-structure interaction, and pipeline filling and emptying. The apparatus is comprised of a horizontal pipeline that connects the upstream end high-pressurized tank and the outflow tank (steel pipe of total length L = 55.37 m; internal diameter D = 18 mm; pipe wall thickness e = 2 mm; maximum allowable pressure in the pipeline pmax, aii = 25 MPa) - see Fig. 1. Four valve units are positioned along the pipeline including the end points. Valve units at the upstream end tank (position 0/3) and at the two equidistant positions along the pipeline (positions 1/3 and 2/3) are comprised of two hand-operated ball valves (valves V//3U and V//3D; i = 0, 1, 2) that are connected to the intermediate pressure transducer block. Recently an additional T-section with two shut-off valves has been added to the upstream end valve unit (position 0/3) to facilitate pipeline filling and emptying tests, [14]. There are four 90° bends along the pipeline with radius R = 3D. The pipeline is anchored against the axial movement at 37 points (as close as possible to the valve units and bends). Loosening of the anchors is planned for fluid-structure interaction tests. The air pressure in the upstream end tank (total volume VHPT = 2 m3; maximum allowable pressure in the tank pHPTmax, aii = 2.2 MPa) can be adjusted up to 800 kPa. The pressure in the tank is kept constant during each experimental run by using a high-precision air pressure regulator in the compressed air supply line, [9]. The upstream end tank is supplied with water from the tap water supply system. The operating air for the electro-pneumatically actuated ball valve (valve V3/3P) can be adjusted to between 200 to 400 kPa, yielding valve opening and closing times from 10 to 20 ms. The V3/3P is operated by a solenoid valve (Burkert 5/2) and a pneumatic actuator (Prisma). In addition, a hand-operated ball valve (valve V3/3H) is positioned next to the electro-pneumatically actuated ball valve.
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The test procedure is as follows. The steady state flow conditions (in advance of a dynamic test) are controlled by a set pressure in the upstream end tank and by a set opening of the downstream end control needle valve (valve V3/3C in Fig. 1). The water level in the upstream end pressurized tank can be adjusted. From initial steady flow conditions (flow, no-flow), a transient event is initiated by some valve manipulations.
^	Pressure transducer 0
jTj1	Electro-pneumatically-operated ball valve
^	Hand-operated ball valve
C
S
Control needle valve
V1/3U: 18.02 m V1/3D: 18.20 m —
Bend (R = 3D) /	Bend (R = 3D )
29.92 m / pM: 36.09 m 36.97 m
V2/3U: 36.00 m V2/3D: 36.20 m
V0/3SV
Upstream end high-pressurized tank
V0/3AV
V0/3U: 1.67 m V0/3D: 1.85 m
1.46 m p0/3: 1.76 m V0/3SH: 1.26 m
x = 0.0 m
6.65 m Bend (R = 3D)
50.33 m p3/3: 53.34 m p333sg: 53.76 m x = L = 55.37 m
V3/3P: 53.47 m-V3/3H: 53.60 m-V3/3C: 53.91 m-V3/3E: 54.00 m-
WW
t —'
Outflow tank
Horizontal steel pipeline
-	internal diameter D = 18 mm
-	full-length L = 55.37 m
Figure 1: Lvysui sf oxvll-onvle pipeline vaavuviuo, [14]
Four dynamic pressure transducers are positioned within the valve units along the pipeline including the end points (see Fig. 1). Pressures p0/3, p1/3, p2/3 and p3/3 are measured with Dytran 2300V4 high frequency piezoelectric absolute pressure transducers (pressure range: from 0 to 6.9 MPa; resonant frequency: 50 kHz; acceleration compensated; discharge time constant: 10 seconds (fixed); uncertainty Ux = ±1 % for Ap duration of 100 ms). The uncertainty in a measured quantity (Ux) is expressed as a sum of bias and precision errors. All four piezoelectric transducers were flush mounted to the inner pipe wall. For evaluation of the initial conditions in the system, two Endress+Hauser PMP131 strain-gauge pressure transducers are positioned at the upstream end pressurized tank and at the control valve V3/3C (pressure range: from 0 to 1 MPa; Ux = ±0.5%). The datum level for all pressures measured in the pipeline and at the tank is at the top of the horizontal steel pipe (elevation 0.0 m in Fig. 1). The downstream end valves V3/3P and V3/3H are equipped with Positek P500.90BL fast-response displacement sensors (measurement range: from 0 to 90°; frequency response: 10 kHz; Ux = ±0.5°). The sensors measure the change of the
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valve angle during valve closing and opening events. Figure 2 shows the layout of the downstream end valve unit with instruments including two pressure transducers and two valve displacement sensors. The initial discharge (Q0) and, consequently, the initial flow velocity (V0) are measured with different methods (Ux = ±1%). For initial flow velocities larger than 0.3 m/s, an electromagnetic flow meter Khrone OPTIFLUX 4000F IFC 300C is used. Smaller steady state velocities are estimated from the Joukowsky pressure head rise or drop resulting from the rapid valve action. The water temperature is continuously monitored with the thermometer (Ux = ±0. 5° C) installed in the outflow tank. The water hammer wave speed was determined as a = 1340 m/s (gx = ±1%). Column separation experiments presented in this paper have shown a good repeatability of the magnitude and timing of the main pressure pulses.
Figure 2: Layout of downstream tnd valvt unit with instruments.
4 COMPARISONS OF COMPUTED AND TEST RESULTS
This section presents numerical and experimental results from two distinct column separation runs including active and passive column separation cases [4]. Numerical results from the discrete gas cavity model with consideration of (1) quasi-steady friction (DGCM+QSF) and (2) unsteady skin friction (DGCM+CBM) [10] are compared with results of measurements performed in the laboratory pipeline apparatus - see Fig. 1. The two runs were carried out for a rapid closure of the hand-operated ball valve positioned at the downstream end of the horizontal pipe (valve V3/3H in Fig. 1). The sampling frequency for each dynamically measured quantity was fs = 2000 Hz. Pressures measured at the end points (positions 0/3 and 3/3) and at the two equidistant positions along the pipeline (positions 1/3 and 2/3) are compared with computational results as
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piezometric heads (heads) with a datum level at the top of the horizontal pipe (elevation 0.0 m in Fig. 1). The number of reaches for all computational runs were N = 108.
4.1 Active column separation case
The active column separation case represents a transient event with a maximum head rise larger than the Joukowsky head rise (AH = (s/n)V0), [4]. The initial flow velocity and the upstream end reservoir head were V0 = 0.44 m/s and Hhpt = 30.5 m, respectively. Numerical and experimental results for this case are depicted in Figs. 3 and 4. The effective valve closure time of 0.025 s was much less than the wave reflection time 2L/v of 0.08 seconds. A rapid valve closure generates a column separation event with limited cavitation. The valve closure first induced Joukowsky head rise at the valve (AHJ = 60 m) and subsequently in time i = 0.09 s column separation at the valve. The negative wave travels along the pipeline and drops the head to the vapour pressure head at all measured positions along the pipeline. The maximum measured head at the valve H3/3xvx = 125 m occurs as a short-duration pressure pulse after the first cavity collapses. The duration of the maximum measured head is very short (0.015 s).
Figure 3: Csxavuioscg sf xevouuea scd DGCM+QSF-nvlnulviea hesdo si the end asicio (H0/3 scd H3/3) scd si the ips equiaioisci asoiiisco vlsca the aiaelice (H1/3 scd H2/3); V0 = 0.44 x/o.
The maximum head obtained by DGCM+QSF (Fig. 3) is slightly higher than the measured one; it is H3/3xvx = 128 m. In contrast, the maximum computed head predicted by DGCM+CBM (Fig. 4) is slightly lower H3/3xvx = 110 m. The difference between the measured and calculated heads is due to the slightly different timing of the cavity collapse. The DGCM+QSF model gives good agreement with measured results for the first two pressure pulses. After that, a phase shift is obvious as well as lesser
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attenuation of pressure traces (Fig. 3). This is not the case for DGCM+CBM results. The results agree well with the measured results during the whole period of observation (Fig. 4).
-Measurement ........DGCM+CBM	-Measurement ........DGCM+CBM
	120.0
a	80.0
s	
m	40.0
	0.0
a)
	120.0
a	80.0
fi *—-	
CI	40.0
	
	0.0
c)
0.0 0.2 0.4 0.6 0.8 Time (s)
0.0 0.2 0.4 0.6 0.8 Time (s)
	120.0
fl	80.0
s	
«1	40.0
	0.0
b)
0.0 0.2 0.4 0.6 0.8 1.0 Time (s)
	120.0
a	80.0
s	
rri	40.0
SÍ	
	0.0
N= 108
d)
0.0 0.2 0.4 0.6 0.8 1.0 Time (s)
Figure 4: Comparisons of measured and DGCM+CBM-calculated heads at the end points (H0/3 and H3/3) and at the two equidistant positions along the pipeline (H1/3 and H2/3); V0 = 0.44 m/s.
4.2 Passive column separation case
The passive column separation case is a transient event with a maximum head rise equal to the Joukowsky head rise (AHJ), [4]. The initial flow velocity and the upstream end reservoir head were V0 = 2.19 m/s and HHPT = 48 m, respectively. Numerical and experimental results for this case are shown in Figs. 5 and 6. The effective valve closure time of 0.020 s was much less than the wave reflection time 2L/a of 0.08 seconds, and it was about 50% of the total closure time. A rapid valve closure generates a column separation event with severe cavitation. The valve closure first induced a Joukowsky head rise at the valve (AHj = 300 m excluding friction effect and AHj = 315 m with friction) and subsequently, in time t = 0.09 s, severe column separation at the valve. The negative wave travels along the pipeline and drops the head to the vapour pressure head at all measured positions along the pipeline. The maximum measured head at the valve H3/3max = 295 m after the first cavity collapsed is less than the Joukowsky head HJ = 340 m. Pressure histories along the pipeline (Figs. 5b to 5c and 6b to 6c, respectively) enable accurate tracing of distributed vaporous cavitation zones and intermediate cavities. For this case the maximum measured head at the valve H3/3max = 340 m occurs as the Joukowsky water hammer head at the valve just before the first liquid column separation.
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Figure S: Comanrisocs of mcnsurcd ncd DGCM+QSF-cnlculnicd hcnds ni ihc ccd aoicis (H0/s ncd Hs/s) ncd ni ihc iwo cquidisinci aosiiiocs nlocg ihc aipclicc (H1/s ncd H2/s); V0 = 2.19 m/s.
Figure S: Comanrisocs of mcnsurcd ncd DGCM+CBM-cnlculnicd hcnds ni ihc ccd aoicis (H0/s ncd Hs/s) ncd ni ihc iwo cquidisinci aosiiiocs nlocg ihc aiaclicc (H1/s ncd H2/s); V0 = 2.19 m/s.
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For the passive column separation case, the maximum measured and calculated pressure heads are in excellent agreement; see Figs. 5a and 6a, respectively. Again the DGCM+QSF model gives good agreement with the measured results for the first two bulk pressure pulses. After that, there are significant differences between the measured and calculated results (Fig. 5). In contrast, the DGCM+CBM results agree well with measured ones during the whole period of observation (Fig. 6).
5 CONCLUSIONS
Numerical results are compared with the results of the measurements given for the closure of the downstream end ball valve in the pipeline apparatus. Pressures measured at the end points (positions 0/3 and 3/3 in Fig. 1) and at the two equidistant positions along the pipeline (positions 1/3 and 2/3) are compared with computational results as piezometric heads (heads). Two distinct column separation runs include active and passive column separation cases. The DGCM model using a quasi-steady friction approach (DGCM+QSF) gives good agreement with the measured results for the first two pressure pulses. After that, there are significant differences between the measured and calculated results. In contrast, the advanced discrete gas cavity model with the consideration of unsteady skin friction (DGCM+CBM) performs well throughout the period of observation. Therefore, the discrete gas cavity model using the convolution-based unsteady friction term is recommended for engineering practice.
References
[1]	E.B. Wylie, V.L. Streeter: Fluid Transients in Systems, Prentice Hall, 1993
[2]	M.H. Chaudhry: Applied Hydraulic Transients, Springer, 2014
[3]	A. Bergant, A.S. Tijsseling, J.P. Vitkovsky, D.I.C. Covas, A.R. Simpson, M.F. Lambert:
Parameters affecting watto-hammto wave ottenuotion, shape and timing. Part 1: Mathematical tools and Paot 2: Case studies, Journal of Hydraulic Research, IAHR, Vol. 46, Iss. 3, Part 1, p.p. 373 - 381 and Part 2, p.p. 382 - 391, 2008
[4]	A. Bergant, A.R. Simpson: Pipeline column separation flow regimes, Journal of Hydraulic Engineering, ASCE, Vol. 125, Iss. 8, p.p. 835 - 848, 1999
[5]	A. E. Vardy, J.M.B. Brown: Evaluation of unsteady wall shear stress by Zielke's method, Journal of Hydraulic Engineering, ASCE, Vol. 136, Iss. 7, p.p. 453 - 456, 2010
[6]	A. S. Tijsseling: Fluid-structure interaction in liquid-filled pipe systems: a review, Journal of Fluids and Structures, Vol. 10, Iss. 2, p.p. 109 - 146, 1996
[7]	D. Covas, I. Stoianov, F.J. Mano, H. Ramos, N. Graham, C. Maksimovic: The dynamic effect of pipe-wall viscoelasticity in hydraulic transients. Part I - experimental analysis and creep characterisation and Part II - model development, calibration and verification, Journal of Hydraulic Research, IAHR, Part - I, Vol. 42, Iss. 5, p.p. 516 - 530 and Part - II, Vol. 43, Iss. 1, p.p. 56 - 70, 2004 and 2005
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[8]
[9]
[10]
[11] [12]
[13]
[14]
A. Malekpour, W.B. Karney: Profile-induced column separation and rejoining during pipeline filling, Journal of Hydraulic Engineering, ASCE, Vol. 140, Iss. 11, p.p. 04014054-1 - 12, 2014
U. Karadžic, V. Bulatovic, A. Bergant: Valve-induced water hammer and column separation in a pipeline apparatus, Strojniški Vestnik - Journal of Mechanical Engineering, Vol.60, Iss. 11, p.p. 742 - 754, 2014
A. Bergant, U. Karadžic, J.P. Vitkovsky, I. Vušanovic, A.R. Simpson: A discrete gas cavity model that considers the frictional effects of unsteady pipe flow, Strojniški vestnik -Journal of Mechanical Engineering, Vol. 51, Iss. 11, p.p. 692 - 710, 2005
A. Bergant, A.R. Simpson, J.P. Vitkovsky: Developments in unsteady pipe flow friction modelling, Journal of Hydraulic Research, IAHR, Vol. 39, Iss. 3, p.p. 249 - 257, 2001
J. P. Vitkovsky, M. Stephens, A. Bergant, M.F. Lambert, A.R. Simpson: Efficient and accurate calculations of Zielke and Vardy-Brown unsteady friction in pipe transients, 9th International Conference on Pressure Surges, BHR Group, Chester, England, 2004
W. Zielke: Frequency-dependent friction in transient pipe flow, Journal of Basic Engineering, ASME, Vol. 90, Iss. 1, p.p. 109 - 115, 1968
U. Karadžic, F. Strunjaš, A. Bergant, R. Mavrič, S. Buckstein: Developments in pipeline filling and emptying experimentation in a laboratory pipeline apparatus, 6th IAHR Meeting of the Working Group Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Ljubljana, Slovenia, 2015
Nomenclature
(Symbols) A
pipe area
water hammer wave speed pipe diameter, diameter friction factor sampling frequency gravitational acceleration piezometric head, head length
number of reaches; number of yk components
pressure
discharge
time
uncertainty in a measured quantity
flow velocity
distance along the pipe
(Symbol meaning)
a
D
f fs
g
H L N P Q
t
Ux V
x
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yk	component of the weighting function in Eq. 2.6
a	void fraction
V	kinematic viscosity AH	head rise
V	volume
(Subscripts)	(Subscripts meaning)
g	gas
HPT	high-pressurized tank, reservoir
in	inflow
J	Joukowsky head
max	maximum
out	outflow
q	quasi-steady part
u	unsteady part
0	initial conditions (Superscripts) (Superscripts meaning)
*	absolute pressure (Abbreviations) (Abbreviations meaning)
CBM	Convolution-Based Model
DGCM	Discrete Gas Cavity Model
FSI	Fluid-Structure Interaction
MOC	Method Of Characteristics
QSF	Quasi-Steady Friction
Acknowledgments
The authors wish to thank Slovenian Research Agency (ARRS) and Ministry of Science, Montenegro (MSM) for their support for this research conducted through the projects BI-ME/14-15-016 (ARRS, MSM) and L2-5491 (ARRS).
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