FLUID-STRUCTURE INTERACTION AND CAVITATION IN A SINGLE-ELBOW PIPE SYSTEM

Journal of Fluids and Structures (1996) 10, 395 – 420

FLUID-STRUCTURE INTERACTION AND CAVITATION IN A SINGLE-ELBOW PIPE SYSTEM A. S. TIJSSELING, A. E. VARDY

AND

D. FAN

Department of Ciy il Engineering , Uniy ersity of Dundee Dundee DD1 4HN , U .K. (Received 20 October 1995 and in revised form 13 March 1996)

The simultaneous occurrence of fluid-structure interaction (FSI) and vaporous cavitation in the transient vibration of freely suspended horizontal pipe systems is investigated by numerical simulation and physical experiment. Extended waterhammer and beam equations, including the relevant FSI mechanisms, are solved by the method of characteristics. Column separation and cavitation are accounted for by a lumped parameter model. Close agreement is found between numerical results and unique experimental data obtained in a single-elbow pipe system. The study aims at a validated numerical model for combined FSI-cavitation phenomena, which can be incorporated in, and so enhance, conventional waterhammer / pipe-stress computer codes. ÷ 1996 Academic Press Limited

1. INTRODUCTION PRESSURE PULSATIONS AND MECHANICAL vibrations in liquid-transporting pipe systems strongly affect system performance and safety. Bad performance costs money; accidents can cost lives. Severe pressure pulsations (waterhammer ) induced by rapid valve closures can damage piping and machinery; pipe vibrations can lead to excessive noise and fatigue. The incorrect reading of flow meters is another matter of concern. Academic research on the dynamic behaviour of pipe systems is fed by realistic problems from industry. The safe transport of dangerous liquids in chemical plants, the integrity of cooling-water systems in nuclear power stations, the reliability of fuel injection systems in aircraft and intolerable pipe vibrations on offshore platforms are just a few examples. Advanced research in the last two decades has shown that liquid pulsations and pipe vibrations cannot be investigated separately. A simultaneous treatment is necessary because liquid-pipe coupling mechanisms always exist. The significance of this coupling, generally referred to as fluid-structure interaction (FSI), has been demonstrated by physical experiments and numerical simulations by many investigators (Tijsseling 1996). FSI in liquid-filled pipe systems is caused by three interaction mechanisms: friction coupling, Poisson coupling and junction coupling. Friction coupling is the mutual friction between the liquid and the axially vibrating pipe wall. Poisson coupling relates pressures in the liquid to axial stresses in the pipes through the radial vibrations of the pipe walls. While friction and Poisson coupling act along the entire pipe, the more important junction coupling acts wherever there is a branch or a change in area or flow direction. For example, the vibration of a short unrestrained elbow generates pressure 0889 – 9746 / 96 / 040395 1 26 $18.00

÷ 1996 Academic Press Limited

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A. S. TIJSSELING ET AL.

pulsations in the liquid, which in turn excite the elbow and other junctions in the system. Adequate and validated mathematical models describing linear FSI phenomena already exist: time-domain analyses (Wiggert et al. 1985, 1987; Vardy & Fan 1989; Vardy et al. 1996; Lavooij & Tijsseling 1991; Kruisbrink & Heinsbroek 1992); and frequency-domain analyses (Lesmez 1989; Tentarelli 1990; De Jong 1994). The occurrence of cavitation in the liquid has been ignored, however, in most previous work on FSI (Tijsseling 1996). Vaporous cay itation occurs when the pressure pulsations are so large, relative to the steady-state pressures, that vapour pressure results. Then, vapour bubbles will develop in the liquid. In some pipes, the liquid column may break (column separation ). In other pipes, low-pressure (rarefaction) waves may leave bubbly regions behind (distributed cay itation ). Gaseous cay itation refers to gases coming out of solution. This phenomenon is not considered herein. Column separation can be compared with the breaking of a rod under tensile stress; vaporous cavitation is conveniently described as boiling at low temperature. The collapse of column separation often leads to an almost instantaneous pressure rise, which, being an impact load, is followed by transient pipe vibrations where FSI is likely to be of importance. This paper gives results of a project in which the simultaneous occurrence of FSI and vaporous cavitation was investigated (Tijsseling 1993). Earlier results in a single straight pipe have been reported previously (Fan & Tijsseling 1992). Here, results obtained for a single-elbow system are presented. The present research is of importance for an accurate assessment of the dynamic fluid pressures and pipe wall stresses in emergency situations. Its objective is to develop a numerical model for combined FSI and cavitation, suitable for implementation within existing computer codes. The experimental validation of the model is an important prerequisite. Simulations with a validated numerical model will eventually lead to a safer design of pipe systems and may be useful in trouble shooting.

2. MATHEMATICAL MODEL The in-plane vibration of planar, liquid-filled pipe systems is assumed herein to be governed by an eight-equation model that allows for axial (longitudinal) and lateral (flexural) wave propagation along each pipe in the system. Torsional and out-of-plane lateral motion are disregarded. The radial (hoop) motion is assumed to follow the liquid and axial pipe motion quasi-statically. Axial and lateral waves do not influence each other in a straight pipe, but, at an elbow, axial motion induces lateral motion and y ice y ersa . The elbow is represented by boundary conditions coupling the axial and lateral motion of the adjacent pipes, thereby including the important FSI junction coupling . Junction coupling also takes place at closed pipe ends where liquid and axial pipe motion interact. Poisson coupling is included in the axial equations of motion. Friction coupling is disregarded herein, because it is usually relatively unimportant. The initial conditions are taken as zero (i.e. the pipe system is at rest before excitation). When the fluid ceases to be wholly liquid, additional equations are needed to supplement the original eight. Simple two-phase flow equations are given for regions of distributed cavitation. Column separations , which mostly occur at pipe ends and junctions, are treated as boundary conditions.

FSI AND CAVITATION IN PIPE SYSTEMS

2.1. LIQUID

AND

397

PIPE MOTION

The one-dimensional mathematical model for axial and lateral pipe motion, in the absence of cavitation, comprises eight first-order partial differential equations governing the eight unknowns: fluid pressure, P , fluid velocity, V , axial pipe stress, s z , axial pipe velocity, u ~ z , lateral shear force, Qy , lateral pipe velocity, u ~ y , bending moment, Mx , and angular pipe velocity, θ~ x . The model is valid for the acoustic behaviour of straight, slender, thin-walled, prismatic, liquid-filled pipes of circular cross-section. The pipewall material is homogeneous, isotropic, linearly elastic and undergoes small deformations. Neglecting gravity (horizontal systems herein) and friction terms (unimportant herein), the basic equations are Axial Motion ­V 1 ­P 1 50 ­t rf ­z

S

and

­u ~ z 1 ­s z 2 50 ­t rt ­z

D

­V 1 2R ­P 2… ­s z 1 1 2 5 0, ­z K Ee ­t E ­t

(1 , 2)

­u ~ z 1 ­s z … R ­P 2 1 5 0; ­z E ­t Ee ­t

(3 , 4)

and

Lateral Motion ­u ~y 1 ­Qy 1 50 ­t rt At 1 r f Af ­z ­θ~ x 1 ­Mx 1 1 5 Qy ­t r t It ­z r t It

and and

­u ~ y 4 1 3… ­Qy 1 5 2θ~ x , ­z EAt ­t ­θ~ x 1 ­Mx 1 5 0; ­z EIt ­t

(5 , 6) (7 , 8)

where A 5 cross-sectional area, E 5 Young modulus, e 5 wall thickness, I 5 second moment of area, K 5 bulk modulus, R 5 inner pipe radius, t 5 time, z 5 distance along pipe, … 5 Poisson ratio, and r 5 mass density; the subscripts f and t refer to the fluid and the structure (tube), respectively. The shear coefficient is taken as 2(1 1 … ) / (4 1 3… ) in accordance with Cowper (1966). Equations (1, 2) are extended waterhammer equations for the liquid, equations (3, 4) are extended beam equations for the axial motion of the pipe and equations (5 – 8) are Timoshenko beam equations for the lateral motion of the pipe. The pressures, stresses and velocities are mean values for the cross-sections. The assumed radial pipe motion is quasi-static, since inertia forces in the radial direction are neglected in both the liquid and the pipe wall. The hoop stress, s f , the radial stress, s r , and the radial displacement, ur , of the pipe wall are then linearly related to the pressure and the axial stress by Radial Motion

sf 5

R P, e

3 sr 5 2 P 4

and

ur 5

R2 …R P2 (s z 1 s r ). Ee E

(9 , 10 , 11)

By inspection, the radial stress, s r , in thin-walled pipes, is much smaller than the hoop stress, s f . It is therefore neglected herein in comparison with s f and also with s z . 2.2. CAVITATION The cavitation model is based on the strong physical constraint that the absolute fluid pressure, Pabs, is equal to the vapour pressure, Py , Pabs 5 Py

(12)

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A. S. TIJSSELING ET AL.

Vapour pressure (kPa)

100 80 60 40 20 0

0

20

40 60 Temperature (°C)

80

100

Figure 1. Vapour pressure of water as function of temperature.

whenever vaporous cavities (bubbles) exist in the liquid. The vapour pressure is a function of the temperature, as shown for water in Figure 1. Equation (12) is valid if free gas is absent and gas release does not occur. Cavitation is assumed to start at the instant the liquid pressure falls to the vapour pressure and to end when all cavities have vanished. In the theoretical model, the ending is inferred from the void fraction, a , which is the ratio of the vapour volume to the total volume, 9y a5 (13) 9l 1 9y where 9y and 9l are the local vapour and liquid volumes. Column separations are identified by void fractions close to unity, whereas distributed cay itation is identified by void fractions which are small with respect to unity. Column separations occur locally and are therefore treated as boundary conditions in Section 2.6. Distributed cavitation occurs along great lengths of pipe. The liquid equations (1, 2) are not valid in regions of cavitating flow. A simple one-dimensional two-phase flow model describing these regions is used herein as a theoretical frame of reference. The model, developed by Kalkwijk & Kranenburg (1971), Kranenburg (1972, 1974a,b) and extended by Wylie & Streeter (1978), Streeter (1983) and Simpson (1986), is based on equation (12). When vapour cavities exist in the liquid, all the fluid elasticity may be assumed to be due to the vapour, so that the compressibility of the liquid and the elasticity of the pipe wall may be ignored in the fluid equations. Therefore, and because pressure changes are tiny in regions of distributed cavitation, fluid-structure interaction through Poisson coupling is not of importance. With these assumptions, the continuity equation reduces to ­a ­a ­V 1V 5 , (14) ­t ­z ­z where V is the average velocity of the liquid-vapour mixture (Simpson 1986; pp. 50 – 52). The convective term, V (­a / ­z ), may not be neglected here, since the pressure wave speed can be very low in a liquid-vapour mixture and, consequently, the acoustic approximation (V Ô cf or d r f Ô r f) does not hold. However, for transient distributed cavitation, the timescale is still acoustic, so that convective terms can reasonably be

399

FSI AND CAVITATION IN PIPE SYSTEMS

neglected. Equation (14) simply states that the void fraction is a function of the velocity gradient. It is the equivalent of the liquid continuity equation (2), with P constant and the convective term added, allowing for tearing voids in the liquid. The two-phase equivalent of the liquid equation of motion (1) is ­V ­V 1V 50 ­t ­z

(15)

(Simpson 1986; pp. 53 – 54). Whilst equation (12) implies that pressure waves do not propagate through a distributed cavitation region, equations (14) and (15) allow for void-fraction waves (Vreenegoor 1990) propagating at constant speed V . When the convective terms in (14) and (15) are neglected, no waves travel at all. In that case, at a fixed location, the velocity is constant [equation (15)] and the void-fraction, a , grows linearly in time until collapse [equation (14), if ­V / ­z 5 constant ? 0, which depends on the conditions in the waterhammer (with FSI) region preceding the distributed cavitation region]. The three variables P , V and a satisfy equations (12), (14) and (15). Although an analytical / numerical solution can easily be found, the problem with this three-equation model is to keep track of the moving boundaries between the liquid waterhammer regions satisfying equations (1, 2) and the distributed cavitation regions satisfying equations (12), (14) and (15). The difficulties associated with the moving boundaries, or interfaces, concern pressure-dependent wave propagation, shock waves and the occurrence of intermediate column separations. Simpson (Simpson 1986; Bergant & Simpson 1992) finds reliable solutions with an interface model , but a simpler, numerical, approach is developed in Section 3.2.

2.3. IMPACT END The external source of excitation used in the physical experiments herein is the impact of a solid rod onto an unrestrained end of a closed pipe (Figure 2). The impact is assumed to be purely axial, so that it does not generate lateral motion. The boundary conditions during the period that the rod and pipe are in contact, i.e. when Vrod 5 u ~ z, are V 5u ~z

Af P 1 Yrod(u ~ z 2 V0rod) 5 At s z 2 mu¨ z ,

and

Qy 5 0

and

Mx 5 0 ,

(18 , 19)

Control volume Pout

σz (σz)rod

Arod

Af

Figure 2. Definition sketch of axial impact.

At P

σz V0rod

(16 , 17)

At

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A. S. TIJSSELING ET AL.

Control volume

σz Af

P

Pout

Af + At

σz

Figure 3. Definition sketch of the closed end.

where V0rod is the velocity of the rod just before impact and m is the mass of the end plug closing the pipe. The admittance or impedance Yrod of the rod is Yrod 5 Arod4Erodr rod

(20)

and the term Yrod(u ~ z 2 V0rod) in condition (17) is equal to Arod(sz )rod. The rod and pipe separate when the contact force becomes tensile. After separation, the governing boundary conditions are those of a free end.

2.4. FREE END The boundary conditions representing an unrestrained closed end of mass m (Figure 3), without column separation, are V 5u ~z

and

Qy 5 0

Af P 5 At s z Ò mu ¨z,

(21 , 22)

Mx 5 0 .

(23 , 24)

and

The 2 and 1 signs in condition (22) are valid for upstream and downstream ends, respectively. The dimensions and the lateral and rotational inertia of the end piece are neglected.

2.5. ELBOW JUNCTION In the absence of cavitation, the junction of two pipes perpendicular to each other is modelled by the eight boundary conditions hAf (V 2 u ~ z )j1 5 hAf (V 2 u ~ z )j2

and

hP j1 5 hP j2 ,

(25 , 26)

hu ~ z j1 5 hu ~ y j2

and

hAf P 2 At s z j1 5 hQy j2 ,

(27 , 28)

hu ~ y j1 5 h2u ~ z j2

and

h2Qy j1 5 hAf P 2 At s z j2 ,

(29 , 30)

hθ~ x j1 5 hθ~ x j2

and

hMx j1 5 hMx j2 ,

(31 , 32)

401

FSI AND CAVITATION IN PIPE SYSTEMS

Pout Control volume

σz1 P1

σz1

Qy1 y1

Mx1

x1

z1

Pout

Qy1 90°

x2 y2 z2

Qy2

Qy2

σz2

Mx2 P2

σz2

Figure 4. Definition sketch of the elbow junction.

where the indices 1 and 2 refer to either side of the junction (Figure 4). The mass and dimensions of the elbow are neglected, just as the forces due to change in liquid momentum, which is consistent with the acoustic approximation. This most simple model is valid if the length of the elbow is small compared to the lengths of the adjacent pipes. The angle between the pipes remains 908; elbow ovalization and the associated flexibility increase and stress intensification are neglected. 2.6. COLUMN SEPARATION Column separations, which are a new element within FSI analyses, generally occur at pipe boundaries like valves, pumps, elbows, branches and high points. The simple but adequate model of Bergeron (1950; pp. 89 – 95) and Streeter & Wylie (1967; p. 209) is adopted to simulate this phenomenon. 2.6.1. Free End Consider the closed end of a liquid-filled pipe which is unrestrained in the axial direction (Figure 5). When the transient displacements of the closed end and the liquid

P = Pv g

V

.

uz

Fictitious liquid– vapour boundary Figure 5. Definition sketch of column separation at the closed end.

402

A. S. TIJSSELING ET AL.

near to it diverge, the local pressure decreases. At the instant that this pressure reaches the vapour pressure, Py , the displacements may still diverge and a vapour cavity is being formed, the growth of which follows the continuity equation, in terms of velocities, ­9c 5 ÒAf (V 2 u ~ z), (33) ­t where 9c is the volume of the cavity, and V and u ~ z are the velocities of liquid column and pipe end, respectively. The 1 and 2 signs are valid for upstream and downstream ends, respectively. The boundary conditions in the case of column separation at a free end are (12) and (22 – 24). The duration of the column separation is inferred from the cavity volume governed by equation (33).

2.6.2. Elbow Junction When column separation occurs at an elbow, the cavity volume follows from ­9c 5 ÒhAf (V 2 u ~ z )j1 Ú hAf (V 2 u ~ z )j2 , ­t

(34)

where the indices 1 and 2 refer to either side of the elbow (Figure 6). The mass balance (25) in the elbow boundary conditions (25 – 32) is not valid during column separation; it is then replaced by the constant-pressure condition (12).

P = Pv

V1

.

uz 2

V2

Figure 6. Definition sketch of column separation at the elbow.

.

uz 1

403

FSI AND CAVITATION IN PIPE SYSTEMS

3. NUMERICAL METHOD The basic equations (1 – 4) and (5 – 8) form hyperbolic sets, which are solved numerically by the method of characteristics (MOC). Vaporous cavitation is modelled numerically in a simple and practical way. 3.1. METHOD

OF

CHARACTERISTICS

The MOC (Forsythe & Wasow 1960; pp. 38 – 42) transforms the partial differential equations (1 – 4) and (5 – 8) into ordinary differential equations (Tijsseling 1993): Axial Motion

Hc˜c 1 2… Re rr (c /cc˜ /)c˜ 2 1JddPt Ú r c ddVt F

2

F

f

t

F

t

F 2

f F

F

2 2…

H

r f cF / c˜ F ds z rf 1 du ~z Ú 2… r t cF 5 0 , (35) r t (ct / c˜ F )2 2 1 dt r t (ct / c˜ F )2 2 1 dt

J

ds z ct R rf c˜ t / ct du ~z Ò 1 2… 2 r t ct 2 dt ˜c t e rt (˜c t / cF ) 2 1 dt 1…

R 1 dP R c˜ t / ct dV Ú… r f ct 5 0; e (˜c t / cF )2 2 1 dt e (˜c t / cF )2 2 1 dt

(36)

Lateral Motion dQy du ~y Ú (r t At 1 r f Af )cs 5 2(r t At 1 r f Af )c 2s θ~ x , dt dt

(37)

dMx dθ~ x Ú r t It cb 5 Úcb Qy ; dt dt

(38)

which are valid along characteristic lines in the distance-time plane. These are the lines along which disturbances in pressure, axial stress, shear force and bending moment, propagate. The characteristic directions dz / dt (wave propagation speeds) pertaining to the equations (35), (36), (37) and (38) are Ú˜c F , Úc˜ t , Úcs and Úcb , respectively. The wave speeds satisfy (Tijsseling 1993):

H SK1 1 (1 2 … ) 2EeRDJ

c 2F 5 r f

1 ˜c 2F 5 (q 2 2 4q 4 2 4c 2F c 2t ) 2 c 2s 5

21

2

and

EAt (4 1 3… )(r t At 1 r f Af )

and

c 2t 5

E , rt

1 c˜ 2t 5 (q 2 1 4q 4 2 4c 2F c 2t ) , 2 and

c 2b 5

E , rt

(39 , 40)

(41 , 42)

(43 , 44)

with q 2 5 c 2F 1 c 2t 1 2… 2(r f / r t )(R / e )c 2F. The actual axial wave speeds (41, 42) include the effects of FSI Poisson coupling; the classical wave speeds (39, 40) do not. The left-hand sides of equations (35 – 38) can be integrated exactly; the right-hand sides of equations (37, 38) need numerical treatment (trapezoidal rule applied herein). Solutions are calculated by time-marching on a computational grid of the type

404

A. S. TIJSSELING ET AL.

P

∆z / c2

t A3

A4

∆z / c1

∆t A2

A1

∆z

z

Figure 7. Computational grid (example with c2 / c1 5 25 / 6) in distance-time (z -t ) plane. Axial motion: c1 5 c˜ F and c2 5 c˜ t . Lateral motion: c1 5 cs and c2 5 cb . $, grid point; ——, characteristic line.

shown in Figure 7. With known values of P , V , s z , u ~ z or Qy , u ~ y , Mx , θ~ x in the grid points A1, A2, A3 and A4, the unknown values in P are readily derived from the integrated (from A to P) equations (35, 36) or (37, 38), respectively. Interpolations are avoided by a slight adjustment of physical data (Lavooij & Tijsseling 1991; Tijsseling 1993). Section 5.1 gives further details. 3.2. CONCENTRATED CAVITY MODEL The concentrated (or discrete ) cay ity model (Provoost 1976; Kot & Youngdahl 1978; Simpson 1986) is used to simulate y aporous cay itation . Cavities are allowed to form at grid points only. Between grid points pure liquid is assumed to exist (see Figure 8). If at a certain instant and location the pressure is calculated to be lower than the vapour pressure, the calculation at that instant and location is restarted with the pressure held equal to the vapour pressure. Then, condition (12) and the (exactly integrated) equations (35, 36) constitute an undetermined system of five equations for the four unknowns V , P , u ~ z and s z . Instead of introducing the void fraction a as the fifth unknown (as in Section 2.2), to obviate this problem, the liquid velocity V is assumed to be discontinuous at a grid point (z , t ), where V1 and V2 are the values of V at (z 2 , t ) and (z 1 , t ), respectively. A cavity is then formed, the volume 9c of which is governed by ­9c 5 2Af (V1 2 V2) , ­t

(45)

or, when numerically forward integrated,

9c (t ) 5 9c (t 2 Dt ) 1 Af hV2(t ) 2 V1(t )j Dt , Vapour

Liquid

Vapour

Liquid

Grid–point (P =Pv)

(46)

Vapour

Liquid

Grid–point (P =Pv)

Liquid

Grid–point (P =Pv)

Figure 8. Definition sketch of concentrated cavities.

405

FSI AND CAVITATION IN PIPE SYSTEMS

where Dt is the numerical time step. The cavity disappears when its volume is calculated to be negative. Then, to satisfy the overall liquid mass balance, the last positive cavity volume is exactly filled up with liquid, according to V1(t ) 2 V2(t ) 5

9c (t 2 Dt ) . Af Dt

(47)

The model handles distributed cay itation regions and intermediate column separations numerically in the same manner. In distributed cavitation regions small cavity volumes are calculated for a series of grid points, whereas column separations are identified by large cavity volumes calculated for a single grid point. 3.3. COLUMN SEPARATION MODEL Column separation at boundaries is modelled as described in Section 2.6. In principle, the solution procedure conforms to Section 3.2, except that at an upstream closed end: V1 5 u ~ z , and at a downstream closed end: V2 5 u ~ z . The cavity volume at an elbow junction is calculated numerically using

9c (t ) 5 9c (t 2 Dt ) Ò [Af hV (t ) 2 u ~ z (t )jDt ]1 Ú [Af hV (t ) 2 u ~ z (t )jDt ]2 .

(48)

The motion of the liquid-vapour interface(s) (Figures 5 & 6) is disregarded; column separation is modelled as a non-moving boundary condition. 4. PHYSICAL EXPERIMENT The transient response of a water-filled single-elbow pipe system to an impact load has been recorded by pressure transducers, strain gauges and a laser-Doppler vibrometer in a laboratory apparatus. This has provided high-quality data for validation of the theoretical model. 4.1. APPARATUS The experimental apparatus (Figure 9) consists of one long (4?51 m) and one short (1?34 m) stainless steel pipe screwed onto a rigid 908 elbow. The pipes have an inner diameter of 52 mm and a wall thickness of 3?9 mm. The pipe system is closed at both ends and filled with pressurized tap water. It is suspended on three long (about 3?3 m), Remote end cap PT6 1210

SGE Impact end plug

V0rod

574

1114

SGA

1126 SGB

PT1 PT2 PT3 59·7 1107 1124·5 20

1130 SGC

740

1125

125

SGD Elbow

PT4 1130·3

z1 Figure 9. One-elbow pipe system (numerial values in mm).

z2

PT5

406

A. S. TIJSSELING ET AL.

thin, vertical, steel wires, so that it can move freely in a nearly horizontal plane. Transients are generated by striking the closed end of the long pipe axially with a solid steel rod of 5 m length. In the results presented herein, the constant rod velocity just before impact was 0?809 m / s. With this particular velocity, an initial (static) pressure of 2 MPa is sufficiently high to prevent cavitation in the liquid. With lower static pressures, the liquid cavitates during the transient phase. The apparatus is less complex than conventional reservoir-pipe-valve systems, since (i) an initial steady-state pressure gradient is absent, (ii) valve-closure characteristics are not needed and (iii) the influence of pipe supports is negligible. The effects of friction, gravity and gaseous cavitation (Zielke et al. 1989) are unimportant due to the timescale (milliseconds) of the experiment. The experiment isolates y aporous cavitation in combination with fluid-structure interaction phenomena. The rod impact leads to very steep wave fronts (velocity, pressure and strain rises in 0?15 ms) and the vapour-liquid interfaces at column separations are, in contrast to Figure 5, believed to be nearly perpendicular to the pipe axis. It is noted that the observed rise times decrease with increasing impact velocity (Tijsseling & Vardy 1996). 4.2. INSTRUMENTATION The pipes are extensively instrumented. Piezoelectric pressure transducers (Kistler 7031, 701A) are flush mounted near to the impact end, near to the elbow, at the remote end, and at three intermediate points on the long pipe, as indicated in Figure 9. The transducers have a natural frequency of 80 kHz. The amplifiers (Flyde FE428CA, FE128CA) have a cut-off frequency of 50 kHz. Strain gauges (TML-FRA-1-11) are attached to the pipe wall at five locations along the system. At each location four three-way (axial, hoop, shear) strain gauges are equidistantly placed around the pipe circumference (top, bottom and sides). The amplifiers (Flyde FE458AC) have a cut-off frequency of 70 kHz. A laser-Doppler vibrometer (Dantec DISA 55X) is used for non-contact measurements of one-directional pipe-wall velocities at any selected location. In particular, it is used to determine the impact velocity of the rod. The amplifier (DISA 55N21, 55N11) has a cut-off frequency of 26 kHz. The release of the rod is remote-controlled and data acquisition is triggered just before impact by the signal from an accelerometer (PCB 305A05) attached to the rod. Fourteen simultaneously recorded signals are directly transmitted from the 12 bit modules of the 125 kHz data acquisition system (Biodata Microlink) to a personal computer, where they are plotted on the screen. More details on the instrumentation and its performance can be found in Fan (1989; pp. 17 – 26). 4.3. INPUT DATA

TO

SIMULATIONS

The physical dimensions and material properties of the pipes, water and rod are given in Table 1 together with the positions of the pressure transducers (PT) and strain gauges (SG) and the location of the laser-Doppler vibrometer (LDV) measuring point. The method of measurement and the accuracy of the values in Table 1 are given, for long pipe, water and rod, in Fan (1989; p. 28) and Fan & Tijsseling (1992; p. 269). The masses m1 and m2 of the end plug and end cap, respectively, which close the pipe system, are taken into account with regard to their motion in the axial direction: the direction in which the important junction coupling takes place. The dimensions of the end pieces and the masses of the attached instrumentation, given by Fan (1989; pp. 16 and 29), are neglected. The off-the-shelf elbow is assumed to be entirely rigid; that is,

407

FSI AND CAVITATION IN PIPE SYSTEMS

TABLE 1 Input data for numerical simulations Pipes L1 5 4?51 m L2 5 1?34 m R 5 26?01 mm e 5 3?945 mm E 5 168 GPa r t 5 7985 kg / m3 … 5 0?29 m1 5 1?312 kg m2 5 0?3258 kg

Liquid

Rod

K 5 2?14 GPa r f 5 999 kg / m3 Py 5 0?002 MPa P0 5 2?00 MPa 5 1?24 MPa 5 1?08 MPa 5 0?87 MPa 5 0?69 MPa 5 0?30 MPa

Lrod 5 5?006 m Rrod 5 25?37 mm Erod 5 200 GPa r rod 5 7848 kg / m3 V0rod 5 0?809 m / s

Instrumentation z1 z1 z1 z1 z2 z2 z1 z1 z1 z1 z1 z2

(PT1) 5 0?0195 m (PT2) 5 1?1265 m (PT3) 5 2?2510 m (PT4) 5 3?3760 m (PT5) 5 0?13 m (PT6) 5 1?34 m (LDV) 5 0?0465 m (SGA) 5 0?5740 m (SGB) 5 1?6880 m (SGC) 5 2?8140 m (SGD) 5 3?9440 m (SGE) 5 0?74 m

the 908 angle between the two pipes does not change. The mass of the elbow, 0?8807 kg, is not modelled explicitly; it is partly incorporated in the pipe lengths. 4.4. LINING OUT A surveying level and a staff are used to make the pipe system horizontal. A theodolite is used to align the central axes of the long pipe and the rod in a vertical plane. A dial gauge, measuring the vertical difference R 1 e 2 Rrod 5 4?6 mm between the tops of rod and pipe, is used to ensure that the impact of the rod is exactly in the centre of the pipe cross-section. In this way, the direct generation of flexural waves is avoided. Vertical lining out is achieved by changing the lengths of the suspension wires; horizontal alignment is simplified by using small remote-controlled magnets to bring and maintain the pipe system in the right position. 4.5. TESTS PERFORMED The impact velocity of the rod and the water static pressure are the two adjustable parameters determining the amount of cavitation. Tests have been performed with one rod impact velocity (0?809 m / s) and static pressures varying in the range 0 – 2 MPa (gauge). In the cavitation tests, pressures were measured at all six locations along the pipes. The axial pipe wall velocity was measured close to the impact end with the laser beam focused on a small PVC block bolted to the pipe. Axial strains were measured at four positions around the pipe cross-section at each of the locations A, D and E (see Figure 9). The bending moment, Mx, is related to the axial strains by Mx 5

EIt (3) h» (1) z 2 » z j, 2(R 1 e )

(49)

where » (1) and »(3) are axial strains measured at opposite sides of the pipe z z (2) (3) circumference. The cross-sectionally averaged axial strain, » z 5 h» (1) z 1 »z 1 »z 1 (4) » z j / 4, is related to the pressure, P , and the axial stress, s z , by

»z 5

H

J

1 R sz 2 … P . E e

(50)

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A. S. TIJSSELING ET AL.

4.6. DIFFICULTIES The static pressure of the water is highly dependent on the temperature (Berthelottube effect). Due to the gradually changing temperature in the laboratory, it was difficult to obtain successive test results for identical initial static pressures. Three repeat tests were usually performed within 10 min of time, however, in which the three initial pressures were nearly the same. A further complication arises because the elapse time between (de)pressurizing the water with a hand pump (connected via a small valve on the end plug), lining out, and doing the next series of test runs was usually substantial. As a consequence, the measured static pressure—obtained from a manometer on the hand pump—is only an estimate. An accurate value for the initial pressure of the water has to be inferred from the dynamic pressure measured at the remote end (PT6), knowing that the pressure remains at the vapour pressure (evident bottom level in time history), which has at room temperature an almost constant value of 2 kPa (absolute, see Figure 1), as long as a local column separation exists. Several pressure transducers were damaged during the tests. The diaphragm of one of them was holed. Apparently the local, nearby explosion and implosion of small cavitation bubbles is too severe a load for the transducers. 5. RESULTS A representative set of numerical and experimental results is discussed. After the purely axial impact at the end of the longer pipe (Figure 9), lateral motion is generated at the elbow by reflecting / transmitting axial stress and pressure waves. There are three locations where column separation is likely to occur, that is, locations where the liquid column separates from its solid boundary: at the two closed pipe ends and at the elbow. Regions of distributed cavitation are expected any time in both pipes. Therefore, some time after excitation, pressure, axial stress, shear and bending waves, column separations and distributed cavitation regions, exist simultaneously in the system. The mutual interactions between waves and cavities make a detailed interpretation of the results difficult to explain. Results are presented for three different levels of cavitation severity, corresponding to initial absolute pressures P0 5 1?24 (or 1?08) MPa, P0 5 0?87 (or 0?69) MPa and P0 5 0?30 MPa. Short-pipe strains and pressures were recorded only with initial pressures P0 5 1?24 MPa and P0 5 0?87 MPa; long-pipe strains and axial velocities only with P0 5 1?08 MPa and P0 5 0?69 MPa. Non-cavitation (P0 5 2?00 MPa) results are given as a reference. In the calculations, the only measured input data in addition to the geometry and the material properties are the impact velocity of the rod and the initial pressure of the liquid. 5.1. NUMERICAL RESULTS The numerical results were obtained with computational grids of the type shown in Figure 7. Four independent grids, two for each pipe (one for axial motion and one for lateral motion), are used in the calculations. The four grids are coupled at the elbow. The grid properties and mass-density adjustments, giving modified wave-speeds to avoid numerical interpolations, are listed in Table 2. The ratio of the number of elements used in each pipe, N1 / N2 5 138 / 41, corresponds closely to the pipe-length ratio L1 / L2 5 3?366, and the numerical time step was taken 5 times smaller than strictly necessary to keep the adjustments in the mass densities

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FSI AND CAVITATION IN PIPE SYSTEMS

TABLE 2 Properties of the four independent numerical grids N

Axial motion of long pipe Axial motion of short pipe Lateral motion of long pipe Lateral motion of short pipe

Dz (m) Dt (m s)

˜c t* / ˜c F* (-)

cb* / cs* (-)

138

0?033

7?1 / 5

17 / 5

—

41

0?033

7?1 / 5

17 / 5

—

138

0?033

7?1 / 5

—

13 / 5

41

0?033

7?1 / 5

—

13 / 5

r f* Dr f / r f r t* Dr t / r t (kg / m3) (%) 999 8036 998 8035 1006 7929 1006 7929

20?0 10?6 20?1 10?6 10?7 20?7 10?7 20?7

i OUT

40 40 40 40

small (,1%). The small time step, 1?42 m s, could lead, with a simulation time of 20 ms, to huge output files. For practical reasons, computed results were written to an output file every 40 (iOUT) time steps. The geometrical and material properties of the elbow system are listed in Table 1. The adjusted mass-densities, r *, in Table 2 lead to the following propagation speeds for the pressure, the axial stress, the shear and the bending waves, respectively: ˜c F* 5 1354 (1354) m / s, c˜ t* 5 4603 (4618) m / s, cs* 5 1770 (1768) m / s and cb* 5 4603 (4587) m / s, where the asterisks denote modified values and the numbers in parentheses are the original values [equations (41 – 44)]. The shear and bending waves are highly dispersive; the cs* and cb* values represent the speeds of their leading edges. The main timescales for the axial waves are: T1 5 (L1 1 L2) / c˜ F < 4?3 ms, T2 5 L1 / c˜ F < 3?4 ms, T3 5 (L1 1 L2) / c˜ t < 1?3 ms, T4 5 L2 / c˜ F < 1?0 ms, T5 5 L1 / c˜ t < 1?0 ms and T6 5 L2 / c˜ t < 0?3 ms. 5.2. REPRODUCIBILITY The reproducibility of the experiments is examined in Figure 10. Dynamic pressures measured at the remote end in two successive experimental runs, the second run about 5 min after the first, are compared. The initial pressures, displayed in the bottom right corner of each frame, are those inferred (see Section 4.6) from the first run. The non-cavitation experiment in the upper graph shows excellent reproducibility. In the cavitation experiments for P0 5 1?24 MPa and P0 5 0?87 MPa the reproducibility is slightly less good, probably because of a slightly increased static pressure in the second run (temperature effect). In the P0 5 0?30 MPa experiment in the bottom graph, the initial static pressure in the two runs is practically the same. The times of occurrence and the magnitudes of the first and highest pressure peaks can be closely reproduced. The subsequent pressure peaks are less reproducible, possibly due to the large amount and the stochastic nature of cavitation. The level of reproducibility is considered to be sufficient for the purposes of the present work. Note that the vapour pressure of water, Py 5 0?002 MPa, is negligible compared to the dynamic pressures. 5.3. PRESSURES Measured and calculated pressures close to the elbow (PT5) and at the remote end (PT6) are shown in the Figures 11 and 12, respectively. The non-cavitation results in

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4

(a)

3 2 1 0

P0 = 2·00 MPa

4 (b) 3

Absolute pressure (MPa)

2 1 0

P0 = 1·24 MPa

4 (c)

3 2 1 0

P0 = 0·87 MPa

4 (d)

3 2 1 0

P0 = 0·30 MPa 0

5

10 Time (ms)

15

20

Figure 10. Reproducibility of physical experiment for the pressure at the remote end (PT6 in Figure 9): ——, measurement 1; ---, measurement 2; (a) P0 5 2?00 MPa; (b) P0 5 1?24 MPa; (c) P0 5 0?87 MPa; (d) P0 5 0?30 MPa.

the upper graphs are considered first. The pressure history close to the elbow (Figure 11) is the more interesting one, since the interactions between axial and lateral waves take place here. It is not possible to provide a definitive wave path diagram in the distance-time plane because of the dispersive character, and hence non-constant propagation speeds, of the lateral waves. Therefore, only a few early-time effects are explained. The impact of the rod generates a pressure wave in the fluid and an axial stress wave in the pipe wall. The arrival of the compressive axial stress wave at the elbow, about 1 ms after impact, causes a pressure drop of 0?7 MPa (Figure 11, top frame). The pressure drop leads to an unbalanced force in the short pipe, which makes the remote

411

FSI AND CAVITATION IN PIPE SYSTEMS

4

(a)

3 2 1 0

P0 = 2·00 MPa

4 (b) 3

Absolute pressure (MPa)

2 1 0

P0 = 1·24 MPa

4 (c)

3 2 1 0

P0 = 0·87 MPa

4 (d)

3 2 1 0

P0 = 0·30 MPa 0

5

10 Time (ms)

15

20

Figure 11. Measured (——) and calculated (---) pressures close to the elbow, at location PT5 (see Figure 9): (a) P0 5 2?00 MPa; (b) P0 5 1?24 MPa; (c) P0 5 0?87 MPa; (d) P0 5 0?30 MPa.

end move away from the liquid. This happens about 1?3 ms after impact, as can be inferred from the first pressure drop of 0?3 MPa in Figure 12. The low-pressure wave generated at the elbow arrives and reflects at the remote end 2?0 ms after impact, giving a total pressure drop of about 1?7(50?3 1 2 3 0?7) MPa relative to the initial pressure (Figure 12, top frame). It is this pressure drop that causes the first column separation when the initial pressure is 1?24 MPa and 0?87 MPa. In the P0 5 0?30 MPa case, however, the first column separation occurs at the elbow. After 3?4 ms, the initial compressive pressure wave generated by the rod impact reaches the elbow. The pressure rise at the elbow causes a large unbalanced force in the short pipe which, 3?7 ms after impact, makes the end cap move in the direction of the fluid, thereby increasing the pressure (Figure 12, top frame). The original pressure wave, transmitted

412

A. S. TIJSSELING ET AL.

4

(a)

3 2 1 0

P0 = 2·00 MPa

4 (b) 3

Absolute pressure (MPa)

2 1 0

P0 = 1·24 MPa

4 (c)

3 2 1 0

P0 = 0·87 MPa

4 (d)

3 2 1 0

P0 = 0·30 MPa 0

5

10 Time (ms)

15

20

Figure 12. Measured (——) and calculated (---) pressures at the remote end, at location PT6 (see Figure 9): (a) P0 5 2?00 MPa; (b) P0 5 1?24 MPa; (c) P0 5 0?87 MPa; (d) P0 5 0?30 MPa.

past the elbow, arrives at the remote end after 4?4 ms, thereby raising the pressure above the initial value of 2 MPa (Figure 12, top frame). The agreement between experiment and numerical simulation is, in view of the complexity of the phenomena, good in the non-cavitation case. The first 12 ms in most of the cavitation experiments are also predicted rather accurately. Column separations, not regions of distributed cavitation, dominate the first 10 ms of the event (Fan & Tijsseling 1992; Tijsseling & Fan 1992). Later on, the agreement between experiment and prediction becomes poorer. In particular, the P0 5 0.30 MPa simulation shows cavitation (P 5 0 MPa) where the experiment does not. It is possible that the model of column separation at the elbow is too simple. Another plausible reason is that

FSI AND CAVITATION IN PIPE SYSTEMS

413

distributed cavitation might occur due to lateral pipe accelerations (e.g., along one side of the shorter pipe when the longer one suddenly excites it). Such cavitation cannot be simulated with the present one-dimensional model. Tensile Stresses Tensile stresses—i.e. short-duration negative absolute pressures—can be seen clearly in the pressure traces before the liquid separates from the solid end pieces (Figure 12, lower three frames, t 5 2?0 ms). The liquid sustains this tensile stress for about 0?15 ms before it begins to cavitate and the pressure rises to the vapour pressure. Tensile stresses, with values up to 0?7 MPa, have been observed in the present apparatus (Fan & Tijsseling 1992) preceding the first occurrence of cavitation and, in particular, of a column separation. Thereafter, the susceptibility to cavitation (Oldenziel 1982) is sufficiently high to prevent significant tensile stresses. 5.4. VELOCITIES Axial pipe wall velocities, measured and calculated near to the impact end, are shown in Figure 13. If the dynamic pressure at the impact end is above the vapour pressure, the liquid and the closed end are in contact, so that the local liquid velocity is equal to the velocity of the pipe end. Hence, the axial pipe wall velocities measured near to the closed end give an indirect indication of the convective velocities in the liquid. The overall agreement between measurement and calculation is good in view of the pressure results, even for the cavitation experiments. 5.5. AXIAL STRAINS

AND

BENDING MOMENTS

Axial strains, measured and calculated at position A (on top of circumference) on the long pipe (Figure 9), are shown in Figure 14. The first 2L1 / c˜ t < 2 ms, when the rod and pipe are in contact, a huge strain-wave travels up (compression) and down (decompression) the long pipe. Later on, after rod-pipe separation, a freely vibrating pipe system remains. The frequency of the axial strain oscillations is predicted accurately for all levels of cavitation severity. The calculated amplitudes are too large, especially in the non-cavitation results (Figure 14, top frame), which is attributed to the lateral wave solutions, since this was not the case for the purely axial vibration of the single pipe (Tijsseling 1993). The use of finer computational grids might give some improvement, but the end plug (of 60 mm length) must then be modelled as a continuous solid member and column separations as moving boundary conditions. This has not been attempted. Measured bending moments, according to equation (49), are compared with predictions in the Figures 15 and 16. The bending moments at position A on the long pipe (Figure 15) exhibit the dispersive character of the lateral waves; small oscillations of relatively high frequency are recorded before the arrival of larger oscillations of low frequency. Although there is good qualitative agreement between predictions and observations, the calculated maximum moments are too high (Figure 15). Again, this is attributed to too coarse a computational grid for the lateral wave solutions. The bending moments in the short pipe (Figure 16) show a considerably better agreement between experiment and calculation than those in the long pipe. An explanation could be that the initial bending of the short pipe is structure-induced (stronger effect), whereas the bending of the long pipe is mainly fluid-induced (weaker effect). The

414

A. S. TIJSSELING ET AL.

1·5

(a)

1·0

0·5 P0 = 2·00 MPa 0·0 1·5 (b)

Axial pipe wall velocity (m/s)

1·0

0·5 P0 = 1·08 MPa

0·0 1·5

(c) 1·0

0·5 P0 = 0·69 MPa 0·0 1·5 (d) 1·0

0·5 0·0 0

P0 = 0·30 MPa 5

10 Time (ms)

15

20

Figure 13. Measured (——) and calculated (---) axial pipe-wall velocities close to the impact end (see Figure 9): (a) P0 5 2?00 MPa; (b) P0 5 1?08 MPa; (c) P0 5 0?69 MPa; (d) P0 5 0?30 MPa.

axial impact of the rod onto the long pipe is a lateral impact for the short pipe. The extreme bending moments in the short pipe are therefore larger, by about a factor 2, than those in the long pipe. The main period of the calculated bending moments is somewhat too short in both pipes. 6. CONCLUSIONS Established numerical models for fluid-structure interaction (FSI) (Wiggert et al. 1987) and vaporous cavitation (Provoost 1976) have, for the first time, been combined in a

415

FSI AND CAVITATION IN PIPE SYSTEMS

75

(a)

0

–75 P0 = 2·00 MPa –150 75 (b)

Axial strain (+ 10–6)

0

–75 P0 = 1·08 MPa

–150 75

(c) 0

–75 P0 = 0·69 MPa –150 75 (d) 0

–75

–150 0

P0 = 0·30 MPa 5

10 Time (ms)

15

20

Figure 14. Measured (——) and calculated (---) axial strains at 0?57 m from the impact end, at location SGA (see Figure 9): (a) P0 5 2?00 MPa; (b) P0 5 1?08 MPa; (c) P0 5 0?69 MPa; (d) P0 5 0?30 MPa.

study of the transient behaviour of liquid-filled pipe systems. The resulting FSIcavitation model is validated against experimental data obtained in a freely suspended single-elbow pipe system. The physical experiment is, as far as the authors know, the only well-documented FSI experiment with cavitating liquid in axially / laterally vibrating pipes. The overall agreement found between experimental and numerical results is good in view of the complexity of the phenomena and the simplicity of, in particular, the cavitation model. The magnitude and timing of the extreme pressures, the extreme

416

A. S. TIJSSELING ET AL.

400

(a)

200 0 –200 P0 = 2·00 MPa

–400 400

(b)

200

Bending moment (Nm)

0 –200 P0 = 1·08 MPa

–400 400

(c) 200 0 –200 P0 = 0·69 MPa

–400 400

(d) 200 0 –200 –400

P0 = 0·30 MPa 0

5

10 Time (ms)

15

20

Figure 15. Measured (——) and calculated (---) bending moments at 0?57 m from the impact end, at location SGA (see Figure 9): (a) P0 5 2?00 MPa; (b) P0 5 1?08 MPa; (c) P0 5 0?69 MPa; (d) P0 5 0?30 MPa.

axial strains and the extreme bending moments are predicted accurately. Discrepancies between measurements and calculations are attributed to (i) the use of too simple a model for column separation at the elbow (Figure 6), (ii) ignoring distributed cavitation induced by lateral pipe motion, (iii) numerical error resulting from the inability to use a sufficiently fine grid, and (iv) experimental uncertainty. Tensile stresses in the water have been observed in all cavitation tests. The relative simplicity of the cavitation model makes an implementation within any numerical FSI model possible if column separations are modelled in the FSI-style described in Section 2.6 of this paper.

417

FSI AND CAVITATION IN PIPE SYSTEMS

400

(a)

200 0 –200 P0 = 2·00 MPa

–400 400

(b)

200

Bending moment (Nm)

0 –200 P0 = 1·24 MPa

–400 400

(c) 200 0 –200 P0 = 0·87 MPa

–400 400

(d) 200 0 –200 –400

P0 = 0·30 MPa 0

5

10 Time (ms)

15

20

Figure 16. Measured (——) and calculated (---) bending moments close to the middle of the short pipe, at location SGE (see Figure 9): (a) P0 5 2?00 MPa; (b) P0 5 1?24 MPa; (c) P0 5 0?87 MPa; (d) P0 5 0?30 MPa.

ACKNOWLEDGEMENTS This paper is based on the first author’s Ph.D. Thesis (Tijsseling 1993) presented to the Faculty of Civil Engineering of Delft University of Technology in The Netherlands. The Ph.D. project, supervised by Prof. J.A. Battjes and Dr H.L. Fontijn of Delft University, was financially supported by the Industrial Technology Division of Delft Hydraulics, Delft, The Netherlands. The experimental work was undertaken at the University of Dundee with financial support from the SERC (grant GR / D / 99942); Mr E.W. Kuperus is thanked for his technical assistance.

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REFERENCES BERGANT, A. & SIMPSON, A. R. 1992 Interface model for transient cavitating flow in pipelines. In Proceedings of the International Conference on Unsteady Flow and Fluid Transients , HR Wallingford, IAHR, Durham, U.K., September – October 1992, pp. 333 – 342. ´ lectricite´ . BERGERON, L. 1950 Du Coup de Be´ lier en Hydraulique —Au Coup de Foudre en E (Waterhammer in hydraulics and wave surges in electricity.) Paris: Dunod (in French). (English translation by ASME committee; New York: John Wiley & Sons, 1961) COWPER, G. R. 1966 The shear coefficient in Timoshenko’s beam theory. Journal of Applied Mechanics 33, 335 – 340. FAN, D. 1989 Fluid-structure interactions in internal flows. Ph.D. Thesis, The University of Dundee, Department of Civil Engineering, Dundee, U.K. FAN, D. & TIJSSELING, A. 1992 Fluid-structure interaction with cavitation in transient pipe flows. ASME Journal of Fluids Engineering 114, 268 – 274. FORSYTHE, G. E. & WASOW, W. R. 1960 Finite -Difference Methods for Partial Differential Equations. New York: John Wiley & Sons. JONG, C. A. F. DE 1994 Analysis of pulsations and vibrations in fluid-filled pipe systems. Ph.D. Thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, ISBN 90-3860074-7. KALKWIJK, J. P. TH. & KRANENBURG, C. 1971 Cavitation in horizontal pipelines due to water hammer. ASCE Journal of the Hydraulics Diy ision 97, 1585 – 1605. KOT, C. A. & YOUNGDAHL, C. K. 1978 Transient cavitation effects in fluid piping systems. Nuclear Engineering and Design 45, 93 – 100. KRANENBURG, C. 1972 The effect of free gas on cavitation in pipelines induced by water hammer. In Proceedings of the 1st International Conference on Pressure Surges , BHRA, Canterbury, U.K., September 1972, Paper C4, pp. 41 – 52. KRANENBURG, C. 1974a Transient cavitation in pipelines. Ph.D. Thesis, Delft University of Technology, Faculty of Civil Engineering; Communications on Hydraulics , Report No. 73 – 2, Delft, The Netherlands. KRANENBURG, C. 1974b Gas release during transient cavitation in pipes. ASCE Journal of the Hydraulics Diy ision 100, 1383 – 1398. KRUISBRINK, A. C. H. & HEINSBROEK, A. G. T. J. 1992 Fluid-structure interaction in non-rigid pipeline systems—large scale validation tests. In Proceedings of the International Conference on Pipeline Systems , BHR Group, Manchester, U.K., March 1992, pp. 151 – 164, ISBN 0-7923-1668-1. LAVOOIJ, C. S. W. & TIJSSELING, A. S. 1991 Fluid-structure interaction in liquid-filled piping systems. Journal of Fluids and Structures 5, 573 – 595. LESMEZ, M. W. 1989 Modal analysis of vibrations in liquid-filled piping systems. Ph.D. Thesis, Department of Civil and Environmental Engineering, Michigan State University, East Lansing, U.S.A. OLDENZIEL, D. M. 1982 A new instrument in cavitation research: the cavitation susceptibility meter. ASME Journal of Fluids Engineering 104, 136 – 142. PROVOOST, G. A. 1976 Investigation into cavitation in a prototype pipeline caused by water hammer. In Proceedings of the 2nd International Conference on Pressure Surges , BHRA, London, U.K., September 1976, Paper D2, pp. 13 – 29. SIMPSON, A. R. 1986 Large water hammer pressures due to column separation in sloping pipes. Ph.D. Thesis, Department of Civil Engineering, The University of Michigan, Ann Arbor, U.S.A. STREETER, V. L. & WYLIE, E. B. 1967 Hydraulic Transients. New York: McGraw-Hill. STREETER, V. L. 1983 Transient cavitating pipe flow. ASCE Journal of Hydraulic Engineering 109, 1408 – 1423. TENTARELLI, S. C. 1990 Propagation of noise and vibration in complex hydraulic tubing systems. Ph.D. Thesis, Department of Mechanical Engineering, Lehigh University, Bethlehem, U.S.A. TIJSSELING, A. S. & FAN, D. 1992 Fluid-structure interaction and column separation in a closed pipe. In Proceedings of the Second National Mechanics Congress , Kerkrade, The Netherlands, November 1992, pp. 205 – 212; Dordrecht, The Netherlands: Kluwer Academic Publishers, ISBN 0-7923-2442-0. TIJSSELING, A. S. 1993 Fluid-structure interaction in case of waterhammer with cavitation. Ph.D. Thesis, Delft University of Technology, Faculty of Civil Engineering; Communications on

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Hydraulic and Geotechnical Engineering , Report No. 93 – 6, ISSN 0169-6548, Delft, The Netherlands. TIJSSELING, A. S. 1996 Fluid-structure interaction in liquid-filled pipe systems: a review. Journal of Fluids and Structures 10, 109 – 146. TIJSSELING, A. S. & VARDY, A. E. 1996 Axial modelling and testing of a pipe rack. In Proceedings of the 7th International Conference on Pressure Surges and Fluid Transients in Pipelines and Open Channels , BHR Group, Harrogate, U.K., April 1996, pp. 363 – 383. VARDY, A. E. & FAN, D. 1989 Flexural waves in a closed tube. In Proceedings of the 6th International Conference on Pressure Surges , BHRA, Cambridge, U.K., October 1989, pp. 43 – 57. VARDY, A. E., FAN, D. & TIJSSELING A.S. 1996 Fluid / structure interaction in a T-piece pipe. Submitted for publication in Journal of Fluids and Structures . VREENEGOOR, A. J. N. 1990 Macroscopic theory of two-phase bubbly flow. Ph.D. Thesis, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands. WIGGERT, D. C., OTWELL, R. S. & HATfiELD, F. J. 1985 The effect of elbow restraint on pressure transients. ASME Journal of Fluids Engineering , 107, 402 – 406. WIGGERT, D. C., HATfiELD, F. J. & STUCKENBRUCK, S. 1987 Analysis of liquid and structural transients by the method of characteristics. ASME Journal of Fluids Engineering 109, 161 – 165. WYLIE, E. B. & STREETER, V.L. 1978 Column separation in horizontal pipelines. In Proceedings of the IAHR / ASME / ASCE Joint Symposium on Design and Operation of Fluid Machinery , Colorado State University, Fort Collins, U.S.A., June 1978, Vol. 1, pp. 3 – 13. ZIELKE, W., PERKO, H.-D. & KELLER, A. 1989 Gas release in transient pipe flow. In Proceedings of the 6th International Conference on Pressure Surges , BHRA, Cambridge, U.K., October 1989, pp. 3 – 13.

APPENDIX: NOMENCLATURE A c ˜c E e g I K L M m N P Q R T t u u ~ u ¨ V 9 x Y y z a Dt

cross-sectional area wave propagation speed (classical) wave propagation speed (with FSI Poisson coupling) Young modulus of elasticity pipe wall thickness gravitational acceleration second moment of cross-sectional area liquid bulk modulus length bending moment mass number of pipe-dividing elements pressure (cross-sectional average) lateral shear force (inner pipe) radius period of time; timescale time pipe displacement pipe velocity pipe acceleration fluid velocity (cross-sectional average); rod velocity volume lateral coordinate (out-of-plane, vertical) admittance, impedance lateral coordinate (in-plane, horizontal) axial coordinate (distance along pipe) void fraction time step (numerical grid length on t -axis)

420 Dz » θ~ … r s

A. S. TIJSSELING ET AL.

element length (numerical grid length on z -axis) strain rotational velocity of pipe Poisson ratio mass density normal stress

Subscripts abs absolute b bending c cavity F fluid f fluid l liquid out outer, external r radial direction rod rod s shear t tube; pipe y vapour x lateral direction (out-of-plane, vertical) y lateral direction (in-plane, horizontal) z axial direction f hoop 0 initial value 1 one side of junction or cavity; long pipe 2 other side of junction or cavity; short pipe Superscripts * modified value (1) position on pipe circumference