fluid-structure interaction in a t-piece pipeatijssel/pdf_files/vardy-fan-tijsseling_1996.pdf ·...

Journal of Fluids and Structures (1996) 10 , 763 – 786

FLUID-STRUCTURE INTERACTION IN A T-PIECE PIPE

A . E . V ARDY , D . F AN AND A . S . T IJSSELING

Ci y il Engineering Department , The Uni y ersity , Dundee , U .K .

(Received 17 April 1995 and in revised form 11 March 1996)

Experimental measurements in a simple laboratory apparatus are presented and shown to be suitable as benchmark data for the validation of computer software . The apparatus consists of suspended horizontal pipes which are struck externally by a long horizontal rod . Attention is focused on interactions between stress waves in the pipes and pressure waves in the contained liquid . Cavitation , external restraints and pre-existing pressure gradients are all absent . It is demonstrated that coupling at boundaries and , to a lesser extent , coupling at wavefronts propagating along a pipe can have a major influence on stress and pressure histories . It is also shown that coupling changes the fundamental frequencies of vibration in comparison with those deduced by considering the liquid and solid components alone .

÷ 1996 Academic Press Limited

1 . INTRODUCTION

I N STEADY PIPE FLOWS , THE BEST KNOWN interaction between the fluid and the pipe is the force on a bend due to pressure forces and the change of direction of the fluid momentum . In large diameter pipelines , substantial supports are often provided locally to withstand such forces . In smaller pipes , especially suspended ones , it is more common to choose suf ficiently thick-walled pipes to enable the forces to be carried by the pipe material .

Forces on bends and other boundaries can be even greater in unsteady flows than in steady flows . This is particularly so in the case of water-hammer because the velocity can change extremely rapidly . The most common source of such flows is a rapidly closing valve , but other possibilities include pipeline ruptures and pump trips . They also include structure-induced excitation due to , say , vibrating machinery and earthquakes .

Whatever the origin of the disturbances , whether fluid-induced or structure-induced , the integrity of the structure is potentially at risk , either through over-stressing or through fatigue . The principal objectives of analyses are therefore to estimate the likely stresses in the pipework and the resonant frequencies of the system .

The most obvious interactions between the fluid and the pipe occur at bends and at other boundaries such as valves and junctions . For example , the pressure rise upstream of a closing valve exerts a force on the valve that has to be resisted structurally . Less obvious interactions occur during the propagation of stress and pressure waves along pipes . For example , changes in the fluid pressure cause changes in the pipe diameter , thus increasing the ef fective compressibility of the medium (in comparison with the same fluid in a rigid container) and thereby reducing the wave speeds .

Hoop stresses / strains associated with the waves give rise to axial stresses / strains in the pipe wall because of Poisson’s ratio ef fects . Thus axially propagating pressure waves

0889 – 9746 / 96 / 070763 1 24 $25 . 00 ÷ 1996 Academic Press Limited

A . E . VARDY ET AL . 764

induce both circumferential and axial stresses / strains in the pipe wall . Likewise , axial stress (or strain) waves give rise to circumferential stresses / strains and hence to pressure changes , although these are less important than the pressure-induced interactions . Both ef fects are known as Poisson coupling .

1 . 1 . A NALYTICAL D EVELOPMENTS

Several authors have presented theoretical analyses of fluid – structure interactions (FSI) in piping systems . Wilkinson & Curtis (1980) gave a comprehensive account of axial wave interactions and Wilkinson (1978) also included flexural and torsion waves in a study of vibrating , liquid-filled pipework . Wiggert et al . (1987) presented numerical analyses of these phenomena based on the Method of Characteristics . Lavooij & Tijsseling (1991) used the Method of Characteristics in the fluid , but solved the pipe wall equations by the finite element method . Fan & Tijsseling (1992) extended the scope of these analyses to include vaporous cavitation and FSI simultaneously .

The analytical basis of the theoretical results presented herein is outlined in Appendix A . The method follows closely the work of Wiggert et al . (1987) .

1 . 2 . E XPERIMENTAL D ATA

The various methods of analysis need to be validated by comparison with experimental data . Wilkinson & Curtis (1980) constructed an ingenious apparatus in which air was evacuated above an air / water interface , causing the interface to rise and to impact on an air extract vent . This provided a fluid-induced impact without the need for an externally triggered valve closure . Wood & Chao (1971) used two pipes in series , with 30 , 60 , 90 , 120 and 150 8 changes in flow direction at their intersection , connecting a reservoir to a rapid acting valve . They also used a T-piece configuration . Wiggert et al . (1985) used several pipes in series with elbows and a range of supports . Once again , water hammer was induced by rapid closure of a valve at the downstream end of the pipe .

All these experiments give data which can be correlated quite well with theoretical analyses . However they all have significant disadvantages for validation purposes . In particular , there are pre-existing flows—and hence friction pressure gradients—before the initiation of water hammer . The gradients influence the flow during the water-hammer phase . Also , all the experiments involve boundary conditions that are dif ficult to model precisely . In the case of a rapidly acting valve , for example , either the behaviour of the valve must be modelled or the adjacent fluid pressure and the pipe wall stresses and motion must be measured . Usually , neither of these options is achievable with high precision .

Uncertainties such as these are unavoidable in experiments resembling realistic operating conditions , even simple ones , but they are undesirable in data to be used for benchmark purposes in the validation of mathematical models . The primary purpose of this paper is to present experimental measurements that are largely uninfluenced by such uncertainties . A second purpose is to use the data to assess the importance of fluid – structure interactions in suspended pipework .

FSI IN A T-PIECE PIPE 765

(a) (b)

hA B C D

1 2 3 4 5

Figure 1 . Single pipe axial impact experiments : (a) vertical pipe ; (b) horizontal pipe .

2 . SINGLE PIPE EXPERIMENTS

2 . 1 . V ERTICAL P IPE

The authors (Vardy & Fan 1986) described an experiment in which a vertical steel pipe , closed at both ends , was dropped onto a nominally rigid platform [Figure 1(a)] . At the instant before impact , the pipe and its contained liquid were moving at a speed V 0 5 4 (2 gh ) , where g and h denote the gravitational acceleration and the initial height of the bottom of the pipe above the platform . On impact , the bottom of the pipe and the liquid were suddenly brought to rest , causing a compressive axial stress wave to travel up the pipe wall and a compression pressure wave to travel up the liquid column .

The primary ef fect of the stress wave , labelled S1 in the wave diagram in Figure 2 , was to stop the downward motion of the wall . Because of Poisson ratio ef fects , however , it also caused a circumferential expansion and hence a small decrease in the liquid pressure . This is analogous to the precursor waves sometimes reported in water-hammer measurements . In this particular instance , the pressure change

Dis

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(Impact) Time

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Figure 2 . Single pipe wave diagram : S1 , S2 , S3 , S4 , stress waves in the pipe wall ; L1 , L2 , L3 , L4 , pressure waves in the liquid .


associated with the precursor wave was negative and so it caused a (small) increase in the downward water velocity .

The primary ef fect of the first liquid pressure wave L1 was to stop the downward motion of the water . Since the velocity of the latter exceeded V 0 (for the reason just explained) , the magnitude of L1 slightly exceeded r f c f V 0 , where c f denotes the speed of the wavefront . The combined ef fect of the two waves S1 and L1 was a pressure rise slightly smaller than r f c f V 0 .

2 . 1 . 1 . Reflection at top of pipe

The reflection of the stress wave S1 from the (closed) top of the pipe comprised two new wavefronts , S2 and L2 , in the wall and the liquid respectively . The (tensile) wavefront S2 caused the wall to move upwards —at a speed less than V 0 , but not greatly so . Since the liquid remained in contact with the upper end of the pipe (in the absence of cavitation) , the overall result in the water was a velocity change from V 0 downwards to almost V 0 upwards . That is , the wavefront L2 was strongly decompressive .

This result is of particular interest because it shows the magnitude of L2 to be almost double that of L1 (Vardy & Fan 1986) . In fact , L2 was the largest single pressure change at any stage during the experiment . This demonstrates once and for all that the common practice of estimating the ef fects of fluid – structure interaction by analyses that do not include coupling explicitly has the potential to be seriously misleading . If , for example , the above experiments were modelled initially as water hammer in a rigid pipe and the resulting pressures were then used as input data for a subsequent structural analysis , the first liquid event at the top of the pipe would be the arrival of the pressure wave L1 . That is , the biggest individual pressure change in the whole experiment (i . e . L2) would be overlooked .

2 . 2 . H ORIZONTAL P IPE

Notwithstanding its elegance and simplicity , the vertical pipe experiment had two important drawbacks . First , the platform on which the pipe landed was not rigid and its behaviour could not be simulated easily . Second , accelerations due to gravity continued to influence the behaviour of the pipe and the liquid even after impact . It is possible to make reasonably accurate allowances for these complications in theoretical comparisons , but they are nevertheless undesirable in experiments intended for benchmark purposes .

The results reported in this paper have been obtained in a similar , but more useful apparatus in which pipes are suspended on long wires in a horizontal plane . The pipes are initially stationary and waves are generated by impact of a long steel rod [Figure 1(b)] . Cavitation is avoided by pressurizing the pipe , typically to about 20 atmospheres . The new apparatus is more adaptable than its predecessor , enabling alternative pipe configurations and impact directions to be chosen freely . Preliminary results with the simplest configuration—a single pipe—were reported by Vardy & Fan (1989) .

The single pipe results presented herein serve two purposes . First , they are given in suf ficient detail for use as benchmark data . Second , they provide valuable guidance for the interpretation of results presented in Sections 3 and 4 for fluid – structure interactions in a more complex configuration (T-piece) .

The first five boxes in Figure 3(a) show pressures at equally spaced positions along the pipe [see Figure 1(b)] , namely very close to the ends and at the quarter and half

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points . The last box in Figure 3(a) shows the axial velocity of the pipe wall at the impact end of the pipe (measured with a laser Doppler vibrometer) . In each case , the continuous lines are experimental measurements and the broken lines are theoretical predictions . The latter are provided for guidance only ; they are not the focus of this paper .

Figure 3(b) shows axial strains on the top of the pipe at four intermediate locations shown in Figure 1(b) , namely midway between the pressure transducers . The intervals between the main events are shorter than in Figure 3(a) because structural waves travel more quickly than fluid waves . The last two boxes are comparisons of strains measured on opposite faces of the pipe at Position B . The close similarity between all four strain histories demonstrates that flexural ef fects are very small , thus confirming that the alignment of the impact rod was accurate .

Figure 3(c) reproduces two of the above results at a larger scale , and selected data for these cases are tabulated in Appendix C . In the case of axial strain , the experimental data are an average of the four strains—at the top , bottom and sides of the pipe at Position B . The agreement between the measured and theoretical curves is suf ficiently close to give confidence in quite fine details in the experimental data .

The agreement also gives confidence in the particular mathematical model used herein . The numerical implementation utilizes the Method of Characteristics in both the liquid and the pipe wall (Fan 1989) . Other suitable analyses have been given by Wiggert et al . (1987) , Heinsbroek et al . (1991 , 1993) and Tijsseling (1993) .

The wavefronts S1 , L1 , S2 , L2 , S3 in Figure 2 are identified in Figure 3(c) . They illustrate the great importance of boundary coupling because wavefront L2 would not exist if coupling were absent . They also illustrate Poisson coupling . That is , the influence of the stress waves can be seen in the corresponding pressure measurements . Likewise , the influence of the pressure waves can be seen in the measured strains .

2 . 2 . 1 . Stress wa y es in the impact rod

The rod is uniform , elastic and longer than the liquid-filled pipe . Stress waves within it can be simulated accurately by an analysis similar to that used to model the more complex conditions in the pipe .

In the case of axial impact , the analysis is especially simple because the wavefront S2 arrives at the pipe / rod interface before the corresponding reflection from the remote end of the rod . Separation then ensues and the rod plays no further part in the experiment (in the time scales of interest herein) . Throughout the period of contact , the force exerted by the rod is constant , being equal to r rod A rod c rod ( u ~ 0 ,rod 2 V ) , where u ~ 0 ,rod denotes the velocity of the rod just before impact and V is the velocity of the pipe , liquid and rod at the interface during impact .

In other experiments not reported herein , the rod strikes the pipe laterally and the subsequent behaviour is more complex . Several impacts and separations occur during the (longer) periods of interest .

2 . 3 . P HYSICAL D ATA

The overall external length of the single pipe is 4502 mm ( Ú 2 mm) and its internal and external diameters are 52 ? 0 mm ( Ú 0 ? 5 mm) and 59 ? 9 mm ( Ú 0 ? 5 mm) . There is negligible circumferential variation in the diameters or the wall thickness . The


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Time (ms) Figure 3(c) . Single pipe pressure and axial strain : S1 , S2 , S3 , stress waves in the pipe wall ; L1 , L2 , pressure

waves in the liquid ; —— , experiment ; - - - - , theory .

density , Young’s modulus and Poisson’s ratio of the steel are 7985 kg / m 3 ( Ú 0 ? 1%) , 168 GPa ( Ú 10% , see Section 4) and 0 ? 29 ( Ú 10%) respectively , these values having been determined by weighing a 200 mm long machined section cut from the pipe and by loading it axially in an Instron testing machine . The density of the water is 999 kg / m 3 and its bulk modulus is assumed herein to be 2 ? 14 GPa .

The ends of the pipe are sealed by steel caps . They are screwed tightly onto the pipe to prevent relative movement . Account is taken of their mass in the theoretical results presented herein , but the ef fect is small . More detail is given in Appendix B .

The overall length of the impact rod is 5006 mm ( Ú 2 mm) and its diameter is 50 ? 7 mm ( Ú 0 ? 5 mm) . Its density and Young’s modulus are 7850 kg / m 3 ( Ú 0 ? 1%) and 200 GPa ( Ú 3%) .

2 . 4 . I NSTRUMENTATION

The apparatus is extensively instrumented with (i) Kistler 7031 acceleration-compensated pressure transducers and Flyde FE428CA

charge amplifiers , having a maximum frequency response of 50 kHz , (ii) TMA FRA-1-11 strain gauges with Flyde FE491CCS & FE458AC charge

amplifiers having a maximum frequency response of 70 kHz ,


(iii) Bru ̈ el & Kjaer 8309 and PCB 305A05 accelerometers with Flyde FE428CA charge amplifiers , and (iv) a Dantec Laser-Doppler Vibrometer with a maximum frequency response of 26 kHz .

The Biodata Microlink data acquisition system is composed of : (i) 16 channels , 12 bit words , 125 000 samples per sec (ii) 6 channels , 12 bit words , 1 000 000 samples per sec .

2 . 4 . 1 . Calibration

Each pressure transducer is supplied with a calibration factor that is found to be extremely reliable . Extensive comparisons have been made (a) in a purpose-designed calibration rig , (b) in shock tube experiments and (c) with transducers supplied by other manufacturers .

The accelerometers are supplied with individual calibration factors . Their validity is assessed by comparison with one another and with measurements made with the laser Doppler vibrometer .

The strain gauges are supplied with manufacturer’s gauge factors that are assessed in situ from steady-state pressurization of the pipe .

In addition to these methods of assessing the reliability of the instrumentation , there is considerable redundancy in the range of measurements taken in the experiments . Even if none of the above calibrations had been possible , the inter-relationships between the various measurements would have been suf ficient to enable calibration factors to be deduced . For example , if the initial impact velocity is assumed , the pressures and strains can be predicted with high accuracy , thereby enabling the calibration factors of the pressure transducers and strain gauges to be deduced (or confirmed) .

3 . T-JUNCTION EXPERIMENTS

Figure 4 depicts three pipes , PQ , QR and QS , connected to a symmetrical 90 8 T- junction at Q . The junction is suf ficiently stif f to ensure negligible relative rotation between the three pipes . The physical properties of the pipes are given in Section 2 . 3 and Figure 4 .

Impact

P

1 2 3 4

4597

S8

A B C DE

F

R 1339·5

444·5

1106

7 Q

5

Figure 4 . T-piece apparatus—plan ; dimensions in mm .


Pressures were measured at the locations 1-8 in Figure 4 and four axial strains—top , bottom and both sides—were measured at each of six intermediate locations , A – F . In addition , a laser-Doppler vibrometer was used to measure the velocity at six locations .

There were more measuring locations than data acquisition ports so it was necessary to undertake many repeat experiments . These were labour intensive , but they did not have a serious impact on the accuracy of the results . The procedure was as follows :

(i) set up the apparatus with all transducers and strain gauges ; (ii) connect the data acquisition equipment to the first set of instruments ;

(iii) conduct the experiment three times and check that the results are self consistent ; (iv) connect the data acquisition equipment to the next set of instruments , including at least one used in the previous set of measurements ; (v) go to item (iii) .

The repeatability of the experiments has been found to be extremely good . In many cases , it was so good that negligible dif ference can be detected between equivalent graphs from dif ferent experiments . This helps to give a high degree of confidence in the data .

Everything was so good that something had to go wrong—but it waited until some months after the experiments had been completed . At that stage , a major hard disk failure led to the discovery that not all the data had been backed up . This is unforgivably stupid , but we have done it anyway . As a consequence , the results presented herein deal only with (a) pressure at one location , (b) velocity at four locations and (c) strain at six locations .

3 . 1 . W AVES IN PQ

Figure 5(a) shows axial strains and velocities in the limb PQ . The strains are averages of those at the top , bottom and sides of the pipe . One strain and one velocity are reproduced in greater detail in Figure 5(b) and in Appendix C .

Immediately after the rod strikes the pipe axially at P , the conditions are identical to those in the single pipe . This situation persists until the arrival of the reflection S2 from the junction (shortly before 2 ms at position A , for instance) .

When S1 reaches the junction (at approximately 1 ms) , the member R-Q-S resists the axial extension of PQ , but cannot prevent it . In this member , movement in the direction PQ represents lateral motion and so flexural waves propagate from the junction Q towards the ends R and S . The dispersive nature of these waves causes the velocity of the junction Q to vary , resulting in the continuous propagation of axial stress waves from Q towards P . This is seen most easily by comparing the axial strains at the point D [Figure 5(a)] with those in the single pipe case [Figure 3(b)] during the period 1 – 2 ? 5 ms .

The reflection of S1 at the junction causes a pressure drop . The associated pressure wave L2 in the limb PQ is smaller than in the single pipe case , partly because R-Q-S resists the velocity change , but mostly because the pressure drop induces flow in all three limbs .

The subsequent behaviour in PQ is influenced by (i) S3 , namely the reflection of S2 at the impact end , (ii) the pressure wavefront L1 and its reflection from the junction and (iii) other waves propagating into PQ from the limbs QR and QS (see below) . The pressure , velocity and strain histories dif fer increasingly from the single pipe case as time increases . Nevertheless , symmetry ensures that there is negligible rotation at Q so the only significant stresses in PQ are axial .


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strains on opposite sides of the pipe at E (north & south) ; —— , experiment ; - - - - , theory .

3 . 2 . W AVES IN QR AND QS

Figure 5(c) shows strain measurements at position E , lateral velocities at positions 5 and 6 and the pressure at position 7 . The axial strains are averages of the four strains at the top , bottom and sides of the pipe . The ‘bending’ strains are dif ferences between axial strains on opposite surfaces of the pipe . The pressure at position 7 and the axial strain at position F are shown in greater detail in Figure 5(d) . Because of symmetry , the conditions at the positions F are almost identical to those at E .

Structurally , the initial influence of PQ on R-Q-S can be interpreted as a suddenly imposed , lateral load at Q . This induces flexural waves (shear and bending) in QR and QS , but negligible axial waves , large deflection ef fects being absent . The flexural waves propagate outwards along the limbs QR and QS in a dispersive manner .

Flexural waves do not interact significantly with pressure waves except through coupling at boundaries , notably at the junction Q in this instance . When a flexural wave arrives at the junction from RQ or SQ , it induces lateral motion in R-Q-S and hence axial motion in QP . The latter induces axial strain in PQ and also a pressure change (because velocity changes are necessary to satisfy continuity at the end of PQ) . Such pressure changes propagate along all three pipes , not only along PQ .


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Figure 5(d) . T-piece pressure and bending strain (member R-Q-S) : —— , experiment ; - - - - , theory .

The most important axial stress waves in R-Q-S (i . e . other than precursor waves) result from pressure wave reflections at the ends R and S . These stress waves subsequently reflect at R and S , but only very small reflections (in R-Q-S) occur at the junction , these being due to small pressure dif ferences associated with Poisson coupling . The absence of stronger reflections at this location is a consequence of the symmetry of the configuration ; equal and opposite waves arrive at Q simultaneously , thereby causing no lateral disturbance to PQ .

The only available pressure measurement is at the position 7 on limb QS of the apparatus . The first event at this location is a reduction in pressure originating when the wavefront S1 reflected at the junction Q . This pressure wave is denoted L2* [see Figure 5(d)] because it is ef fectively the same pressure disturbance as L2 (in the pipe PQ) . The second significant event is L2** , namely the reflection of L2* from the closed ends R and S . The next , denoted L1* , is the transmitted part of the pressure wavefront L1 from the original impact . The gradual decrease in pressure between L2* and L2** is a consequence of the gradual nature of the reflection of S1 at the junction (described in the discussion of waves in PQ) .


3 . 3 . E XPERIMENTAL A CCURACY

A necessary (but not suf ficient) measure of the reliability of experiments is repeatability . This is exceptionally good in this apparatus ; it is usually dif ficult to distinguish between measurements from nominally identical runs .

Additional assessments are possible in the case of the T-junction configuration because the apparatus and the loading condition are symmetrical . First , a comparison of strain histories at the points E and F (not shown) has demonstrated that the behaviour of the limbs QR and QS is closely symmetrical . Second , the out-of-plane (vertical) bending strain at the point E (magnified 10 times in Figure 5(c) , upper right hand box) is only about 1% of the in-plane (horizontal) bending strain .

4 . RESONANT FREQUENCIES

Fourier Transform routines enable frequency spectra to be deduced relatively easily from the pressure and velocity time histories . Figure 6 shows frequency spectra of pressure and pipe wall velocity at opposite ends of the single - pipe experiment . Both measured (continuous lines) and theoretical (broken lines) spectra are shown .

The results presented have been obtained from a dif ferent test from those considered in Section 2 . A very low impact velocity has been used because a long sample period is needed , and yet the Laser Doppler Vibrometer goes out of focus when large displacements occur . Nevertheless , the agreement between the predicted and measured values is good .

The discrepancy in the velocity measurements / predictions at very low frequencies may be disregarded . It is a consequence of the pipe swinging like a pendulum on its supporting wires . No account is taken of this ef fect in the theoretical simulation .

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The agreement between the remainder of the results is very good—as it must be when the time-histories are so similar . The frequency domain presentation is helpful , though , because it highlights a small discrepancy in the predicted wavespeeds that is less obvious in the time domain . The cause of the dif ferences has not been identified with certainty . One possibility is error in the measured Young’s modulus of the steel . Another is over-simplification of the end caps in the theoretical model . Although account is taken of their masses , no account is taken of their lengths .

4 . 1 . C OUPLED AND U NCOUPLED F REQUENCY S PECTRA

The influence of coupling ef fects can be explored theoretically . Figure 7 gives comparisons between theoretical predictions for three T-piece cases , namely : (i) the base case—with coupling between the structural and liquid waves ; (ii) the same apparatus , but filled with air—only structural waves are significant ; (iii) the same configuration , but with rigid , immovable pipework—liquid waves only .

Each box in Figure 7 shows the frequency spectrum for the coupled case (broken lines) together with the corresponding spectrum for one of the uncoupled cases (continuous lines) . Five of the ‘uncoupled’ spectra correspond to the air-filled apparatus . One (the pressure) corresponds to the rigid pipework .

The principal natural frequencies for these cases are listed in Table 1 . The first natural frequency of the coupled system (27 ? 0 Hz) is close to that of the structure-only system (33 ? 5 Hz) , but is modified by the presence of the water . In part , the modification can be attributed to the mass of the water , which is about 15% of the total mass in this apparatus . However , there is no universal way of allowing for this in the absence of FSI analyses . In flexural modes , the velocities of the pipe and the liquid are ef fectively identical so it is simply a matter of adding the two masses . In axial modes , however , the velocities of the two components are dif ferent and there is no simple relationship between them . The proportion of the mass to be included will depend upon the system geometry (closed or open system) . Also , the allowance to be made for the compressibility (elasticity) of the water is uncertain . In flexural modes , it has negligible influence ; in axial modes , it is fundamental to the propagation of pressure waves .

The second natural frequency of the coupled system (112 Hz) is nearly the same as the first natural frequency in an equivalent water-filled system with rigid , immovable pipes (111 Hz) . The third and fourth natural frequencies of the coupled system (159 and 226 Hz) dif fer by about 12 and 9% respectively from the second natural frequencies of the structure-only and water-only cases (181 and 246 Hz) .

It is concluded that account must be taken of the influence of the water when attempting to predict the natural frequencies of the system—even when the pipes are as thick-walled as those in the present apparatus ( R / e < 6 ? 6) .

5 . CONCLUSIONS

The principal conclusions may be listed as follows . 1 . A simple apparatus has been shown to produce high quality data on fluid –

structure interactions in liquid-filled pipes . There are no complications from cavitation , structural supports or pre-existing pressure gradients due to friction . The data are highly reproducible and include suf ficient redundancy for their self-consistency to be confirmed .


0·2

0·1

0·0

0·2

0·1

0·0

0·0

0·2

0·4

0·0

0·2

0·4

0·0

0·2

0·4

Fre

quen

cy (

kHz)

0·2

0·1

0·0

0·2

0·1

0·0

0·4

0·2

0·0

15 10 5 0

v2 v7sA

p7

v3v6

Velocity (m/s) Velocity (m/s)

Velocity (m/s)

Velocity (m/s) Pressure (MPa)

Microstrain

Fig

ure

7 . T

-pie

ce c

oupl

ed a

nd u

ncou

pled

fre

quen

cy s

pect

ra : v

2 , v3

, axi

al v

eloc

itie

s at

2 a

nd 3

; v6 ,

v7 , l

ater

al v

eloc

itie

s at

6 a

nd 7

; sA

, axi

al s

trai

n at

A ; p

7 , p

ress

ure

at 7 .

In

all

case

s , t

he c

oupl

ed s

pect

ra a

re f

or a

wat

er-fi

lled

pipe

. In

case

p7 ,

the

unc

oupl

ed s

pect

ra a

re a

lso

for

a w

ater

-fille

d pi

pe . A

ll ot

her

unco

uple

d sp

ectr

a ar

e fo

r an

air-

fille

d pi

pe . —

— , U

ncou

pled

; - - -

- , co

uple

d .


T ABLE 1 Natural frequencies (Hz) : coupled and uncoupled cases

Coupled (FSI) Air-filled pipework Rigid pipe system Mode (Structure 1 water) (structure-only) (water-only)

1 27 . 0 33 . 5 111 2 112 181 246 3 159 416 386 4 226 527 506

2 . The transients are generated structurally by striking the apparatus with a long rod . This leads to clean , steep-fronted wavefronts with very little distortion . Many reflections can be identified unambiguously .

3 . Some of the pressure waves are larger than those normally expected in pipework where the excitation is fluid-induced . These cases demonstrate that the influence of interactions must be taken into account in a reliable analysis / design .

4 . Coupling influences the fundamental frequencies of vibration of liquid-filled pipework .

ACKNOWLEDGEMENTS

The authors are indebted to Mr Ernie Kuperus , who undertook many of these experiments , always with great diligence and cheerfulness . Thanks are also due to reviewers of the original manuscript for suggestions and illuminating comments . The EPSRC has provided continuing financial support through grants GR / D / 99942 and GR / J / 54857 .

R EFERENCES

F A N , D . 1989 Fluid-structure interactions in internal flows . Ph . D . thesis , Department of Civil Engineering , University of Dundee , U . K .

F A N , D . & T I J S S E L I N G , A . S . 1992 Fluid – structure interaction with cavitation in transient pipe flows . ASME Journal of Fluids Engineering 114 , 268 – 274 .

H E I N S B R O E K , A . G . T . J . 1993 Fluid – structure interaction in non-rigid pipeline systems— comparative analyses . In Proceedings ASME / TWI 12 th international conference on Of fshore Mechanics and Arctic Engineering , Glasgow U . K ., Paper OMAE-93-1018 , pp . 405 – 410 .

H E I N S B R O E K , A . G . T . J ., L A V O O I J , C . S . W . & T I J S S E L I N G , A . S . 1991 Fluid – structure interaction in non-rigid piping—a numerical investigation . In SMiRT 11 Transactions , Volume B , Tokyo , Japan , pp . 309 – 314 .

L A V O O I J , C . S . W . & T I J S S E L I N G , A . S . 1991 Fluid – structure interaction in liquid-filled piping systems . Journal of Fluids and Structures , 5 , 573 – 595 .

T I J S S E L I N G , A . S . 1993 Fluid – structure interaction in case of waterhammer with cavitation . Ph . D . thesis , Civil Engineering Department , Delft University of Technology , The Netherlands .

V A R D Y , A . E . & F A N , D . 1986 Water hammer in a closed tube . In Proceedings 5 th International Conference on Pressure Surges , BHRA , Hannover , Germany , pp . 123 – 137 .

V A R D Y , A . E . & F A N , D . 1989 Flexural waves in a closed tube . In Proceedings 6 th International Conference on Pressure Surges , BHRA , Cambridge , U . K ., pp . 43 – 57 .

W I G G E R T , D . C ., O T W E L L , R . S . & H A T F I E L D , F . J . 1985 The ef fect of elbow restraint on pressure transients . ASME Journal of Fluids Engineering 107 , 402 – 406 .

W I G G E R T , D . C ., H A T F I E L D , F . J . & S T U C K E N B R U C K , S . 1987 Analysis of liquid and structural transients by the method of characteristics . ASME Journal of Fluids Engineering 109 , 161 – 165 .


W I L K I N S O N , D . H . & C U R T I S , E . M . 1980 Water hammer in a thin-walled pipe . In Proceedings 3 rd International Conference on Pressure Surges , BHRA , Canterbury , U . K ., pp . 221 – 240 .

W I L K I N S O N , D . H . 1978 Acoustic and mechanical vibrations in liquid-filled pipework systems . In Proceedings BNES International Conference on Vibration in Nuclear Plant , Keswick , U . K ., Paper 8 . 5 , pp . 862 – 878 .

W O O D , D . J . & C H A O , S . P . 1971 Ef fect of pipeline junctions on water hammer surges . ASCE Journal of Transportation Engineering 97 , 441 – 456 .

APPENDIX A : THEORETICAL BASIS

The principal purpose of this paper is to present experimental data . Theoretical comparisons are provided primarily to assist in the interpretation and validation of the measurements . The basis of the analytical method is outlined briefly in the following paragraphs . Fuller details are given by Fan (1989) and Tijsseling (1993) .

The one-dimensional model of wave propagation is based on four equations describing axial motion and four equations describing lateral motion , as follows .

Axial motion , fluid : extended water - hammer equations

Û V Û t

1 1 r f

Û P Û z

5 0 , (A . 1)

Û V Û z

1 S 1 K

1 2 R Ee

D Û P Û t

2 2 … E

Û s z

Û t 5 0 ; (A . 2)

Axial motion , structure

Û u ~ z

Û t 2

1 r t

Û s z

Û z 5 0 , (A . 3)

Û u ~ z

Û z 2

1 E

Û s z

Û t 1

… R Ee

Û P Û t

5 0 ; (A . 4)

Lateral motion , structure : Timoshenko beam equations

Û u ~ y

Û t 1

1 r t A t 1 r f A f

Û Q y

Û z 5 0 ,

Û u ~ y

Û z 1

4 1 3 … EA t

Û Q y

Û t 5 2 θ ~ x , (A5 , 6)

Û θ ~ x

Û t 1

1 r t I t

Û M x

Û z 5

1 r t I t

Q y , Û θ ~ x

Û z 1

1 EI t

Û M x

Û t 5 0 . (A7 , 8)

The assumed radial pipe motion is quasi-static . The hoop stress , s f , and radial displacement , u r , are assumed to be linearly related to the pressure and axial stress by

s f 5 R e

P , (A . 9)

u r 5 R 2

Ee P 2

… R E

s z . (A . 10)

BOUNDARY CONDITIONS There are three types of boundary condition , namely (i) a closed end , free to move in any direction ; (ii) a closed end in contact with the impact rod ; (iii) a T-piece junction .

The equations describing these conditions ensure conservation of mass , linear momentum and angular momentum . Consider , for example , the end P of the limb PQ during axial impact by the rod . The equations are

V 5 u ~ z 5 u ~ rod , (A . 11)

A f P 1 r rod A rod c rod ( u ~ rod 2 u ~ 0 , rod ) 5 A t s z 2 mu ̈ z , (A . 12)


where m denotes the mass of the end cap and u ~ rod and u ~ 0 ,rod denote the current velocity of the rod and its velocity just before impact . For axial impact , the shear force , Q y , and bending moment , M x , are both zero . The acceleration term in equation (A . 12) is represented in the numerical analysis by finite dif ference relationships .

Equations (A . 11) and (A . 12) are used from the initial impact until reflections from the remote end of PQ (and / or the rod) cause the predicted contact force between the rod and the pipe to become tensile . At that instant , the rod and the pipe are assumed to separate . Thereafter , the second term on the left-hand side of equation (A . 12) is discarded and equation (A . 11) is replaced by V 5 u ~ z .

COUPLING Poisson coupling exists between the axial equations for the liquid (A . 1 , A . 2) and the pipe (A . 3 , A . 4) . Thus , for example , pressure changes in the liquid cause radial displacements in the pipe , leading to hoop and axial stresses and strains . In thin-walled pipes , this can be an important ef fect . In thicker-walled pipes such as those considered herein , it is less important , but it is nevertheless easily seen in the experimental and theoretical results .

The most important coupling for present purposes occurs at boundaries . At an end cap , for example , the liquid and pipe velocities are equal . At junctions and bends , changes in pressure and momentum in the liquid give rise to forces on the structure .

The axial and lateral equations of wave propagation (A . 1 – A . 4 and A . 5 – A . 8) are not coupled because the pipes are initially straight and undergo only small deflections .

APPENDIX B : APPARATUS DETAILS

The pipe fittings are illustrated in Figure 8 . All connections are designed to prevent relative movement as well as to prevent leakage . All fittings are mild steel .

End cap End cap

Locking ring

Impact end plug

Rod

Figure 8 . Pipe fittings .


END CAP : IMPACT END The end cap at the impact end is a plug that is screwed tightly into the inside of the pipe , ensuring axial compression between the rebate on the plug and the end of the pipe . This is an ideal condition for the initial impact . Sealing is provided by an O-ring in a slightly over-sized groove . The mass of the 60 mm long end cap , 1 ? 29 kg , is taken into account in the numerical simulations . Its influence on any particular event is small , but there is a cumulative ef fect in simulations of long duration .

END CAPS : REMOTE ENDS The end caps at the remote ends are screwed to the outside of the pipes . They are designed to deflect negligibly under pressure and they are much smaller than the impact cap . The mass of each cap , 0 ? 29 kg , is allowed for in the simulations , but the influence is very small .

T-PIECE FITTING The T-piece is a screw fitting of a type readily available commercially . The casting is stif fened suf ficiently for distortion to be neglected . Continuity with the axial pipe is enhanced by a locking ring . The mass of the fitting slightly exceeds the value implied by the (physically impossible) one-dimensional representation of three pipes meeting at a point . No allowance has been made for this mass in the numerical simulations .

APPENDIX C : NUMERICAL DATA FOR FIGURES 3(c) , 5(b) and 5(d)

The numerical values used to produce the experimental curves in Figures 3(c) , 5(b) and 5(d) are summarized below . The measurements were taken at intervals of 8 m s , but only one-tenth of these are listed . These should be suf ficient for most purposes .

The data are presented to more significant figures than is justified by their accuracy . This is partly for convenience of presentation and partly to enable noise levels to be assessed .

Figure 3(c) Figure 5(b) Figure 5(d) ————————————————– ————————————– ————————————

Time (ms)

Pressure at 3

(MPa)

Axial strain at B

( m -strain)

Velocity at 3

(m / s)

Axial strain at B

( m -strain)

Pressure at 7

(MPa)

Axial strain at F

( m -strain)

0 ? 000 0 ? 000 0 ? 205 2 0 ? 001 0 ? 363 0 ? 000 0 ? 345 0 ? 080 0 ? 002 0 ? 066 2 0 ? 001 2 0 ? 190 2 0 ? 003 0 ? 000 0 ? 160 0 ? 002 2 0 ? 590 2 0 ? 002 0 ? 155 0 ? 003 0 ? 691 0 ? 240 2 0 ? 002 2 0 ? 624 0 ? 001 0 ? 173 2 0 ? 002 2 2 ? 418 0 ? 320 0 ? 004 2 0 ? 003 2 0 ? 001 0 ? 035 2 0 ? 002 1 ? 036 0 ? 400 0 ? 002 2 0 ? 452 0 ? 000 2 10 ? 967 0 ? 003 2 0 ? 345 0 ? 480 0 ? 002 2 0 ? 037 2 0 ? 001 2 94 ? 642 2 0 ? 003 0 ? 345 0 ? 560 2 0 ? 042 0 ? 101 2 0 ? 298 2 128 ? 025 2 0 ? 003 2 0 ? 691 0 ? 640 2 0 ? 105 0 ? 861 2 0 ? 574 2 125 ? 348 0 ? 000 2 1 ? 727 0 ? 720 2 0 ? 118 2 0 ? 452 2 0 ? 601 2 124 ? 934 2 0 ? 002 0 ? 691 0 ? 800 2 0 ? 111 2 15 ? 028 2 0 ? 591 2 124 ? 053 0 ? 005 1 ? 036 0 ? 880 2 0 ? 127 2 49 ? 914 2 0 ? 595 2 123 ? 639 2 0 ? 002 2 1 ? 727 0 ? 960 2 0 ? 123 2 80 ? 137 2 0 ? 593 2 124 ? 105 2 0 ? 002 1 ? 727 1 ? 040 2 0 ? 111 2 101 ? 760 2 0 ? 590 2 124 ? 865 2 0 ? 003 2 0 ? 691 1 ? 120 2 0 ? 123 2 113 ? 124 2 0 ? 587 2 125 ? 866 2 0 ? 107 2 1 ? 727 1 ? 200 2 0 ? 127 2 113 ? 642 2 0 ? 603 2 126 ? 246 2 0 ? 289 2 22 ? 106 1 ? 280 2 0 ? 123 2 110 ? 671 2 0 ? 600 2 125 ? 676 2 0 ? 388 8 ? 290 1 ? 360 2 0 ? 120 2 110 ? 948 2 0 ? 598 2 132 ? 015 2 0 ? 450 156 ? 469 1 ? 440 2 0 ? 120 2 112 ? 882 2 0 ? 598 2 133 ? 241 2 0 ? 438 296 ? 359 1 ? 520 2 0 ? 089 2 113 ? 434 2 0 ? 564 2 133 ? 586 2 0 ? 457 335 ? 736 1 ? 600 2 0 ? 042 2 113 ? 124 2 0 ? 603 2 136 ? 850 2 0 ? 485 330 ? 555 1 ? 680 0 ? 036 2 112 ? 122 2 0 ? 715 2 137 ? 559 2 0 ? 487 320 ? 193 1 ? 760 0 ? 568 2 112 ? 916 2 0 ? 847 2 122 ? 965 2 0 ? 494 292 ? 905


Numerical data for Figures 3(c) , 5(b) and 5(d) (continued)


Time (ms)

Pressure at 3

(MPa)

Axial strain at B

( m -strain)

Velocity at 3

(m / s)

Axial strain at B

( m -strain)

Pressure at 7

(MPa)

Axial strain at F

( m -strain)

1 ? 840 0 ? 836 2 112 ? 191 2 0 ? 894 2 97 ? 543 2 0 ? 493 257 ? 328 1 ? 920 0 ? 758 2 112 ? 709 2 0 ? 913 2 77 ? 060 2 0 ? 505 233 ? 495 2 ? 000 0 ? 762 2 112 ? 674 2 0 ? 956 2 72 ? 087 2 0 ? 499 207 ? 590 2 ? 080 0 ? 771 2 113 ? 331 2 0 ? 960 2 66 ? 612 2 0 ? 501 181 ? 684 2 ? 160 0 ? 764 2 113 ? 676 2 0 ? 982 2 59 ? 220 2 0 ? 504 174 ? 085 2 ? 240 0 ? 766 2 113 ? 814 2 0 ? 994 2 58 ? 167 2 0 ? 505 118 ? 475 2 ? 320 0 ? 769 2 113 ? 814 2 0 ? 996 2 55 ? 852 2 0 ? 485 50 ? 775 2 ? 400 0 ? 782 2 114 ? 954 2 1 ? 002 2 31 ? 916 2 0 ? 543 56 ? 647 2 ? 480 0 ? 787 2 114 ? 574 2 0 ? 977 3 ? 005 2 0 ? 538 127 ? 110 2 ? 560 0 ? 816 2 115 ? 023 2 0 ? 805 17 ? 564 2 0 ? 515 149 ? 216 2 ? 640 0 ? 773 2 117 ? 407 2 0 ? 709 16 ? 839 2 0 ? 552 97 ? 750 2 ? 720 0 ? 145 2 119 ? 894 2 0 ? 723 26 ? 458 2 0 ? 596 2 ? 763 2 ? 800 2 0 ? 532 2 121 ? 793 2 0 ? 710 31 ? 570 2 0 ? 590 2 115 ? 021 2 ? 880 2 0 ? 573 2 121 ? 724 2 0 ? 672 28 ? 548 2 0 ? 578 2 176 ? 503 2 ? 960 2 0 ? 530 2 121 ? 172 2 0 ? 643 27 ? 080 2 0 ? 643 2 176 ? 849 3 ? 040 2 0 ? 584 2 120 ? 377 2 0 ? 730 20 ? 845 2 0 ? 745 2 158 ? 197 3 ? 120 2 0 ? 579 2 114 ? 160 2 0 ? 799 19 ? 636 2 0 ? 744 2 147 ? 834 3 ? 200 2 0 ? 568 2 97 ? 546 2 0 ? 781 34 ? 990 2 0 ? 782 2 135 ? 400 3 ? 280 2 0 ? 613 2 76 ? 269 2 0 ? 699 42 ? 261 2 0 ? 800 2 173 ? 394 3 ? 360 2 0 ? 584 2 57 ? 478 2 0 ? 628 29 ? 705 2 0 ? 794 2 245 ? 930 3 ? 440 2 0 ? 590 2 40 ? 657 2 0 ? 668 7 ? 547 2 0 ? 603 2 282 ? 889 3 ? 520 2 0 ? 622 2 32 ? 782 2 0 ? 746 0 ? 501 2 0 ? 226 2 263 ? 891 3 ? 600 2 0 ? 568 2 33 ? 231 2 0 ? 778 13 ? 385 2 0 ? 281 2 241 ? 785 3 ? 680 2 0 ? 448 2 34 ? 302 2 0 ? 764 21 ? 640 2 0 ? 380 2 192 ? 392 3 ? 760 2 0 ? 512 2 34 ? 405 2 0 ? 771 19 ? 619 2 0 ? 430 2 117 ? 093 3 ? 840 2 0 ? 439 2 34 ? 854 2 0 ? 742 22 ? 002 2 0 ? 422 2 83 ? 243 3 ? 920 2 0 ? 336 2 33 ? 473 2 0 ? 717 18 ? 376 2 0 ? 405 2 78 ? 753 4 ? 000 2 0 ? 319 2 32 ? 713 2 0 ? 648 2 0 ? 173 2 0 ? 476 2 52 ? 157 4 ? 080 2 0 ? 307 2 33 ? 576 2 0 ? 598 2 18 ? 169 2 0 ? 419 2 23 ? 142 4 ? 160 2 0 ? 316 2 33 ? 714 2 0 ? 615 2 21 ? 847 2 0 ? 334 2 15 ? 543 4 ? 240 2 0 ? 305 2 33 ? 507 2 0 ? 576 2 14 ? 127 2 0 ? 339 2 27 ? 287 4 ? 320 2 0 ? 292 2 33 ? 404 2 0 ? 466 2 17 ? 288 2 0 ? 322 2 80 ? 480 4 ? 400 2 0 ? 314 2 31 ? 884 2 0 ? 397 2 33 ? 884 2 0 ? 278 2 189 ? 283 4 ? 480 2 0 ? 296 2 31 ? 815 2 0 ? 384 2 46 ? 112 2 0 ? 312 2 273 ? 217 4 ? 560 2 0 ? 261 2 30 ? 364 2 0 ? 410 2 51 ? 949 2 0 ? 279 2 297 ? 741 4 ? 640 2 0 ? 096 2 26 ? 461 2 0 ? 476 2 56 ? 491 2 0 ? 207 2 285 ? 306 4 ? 720 0 ? 218 2 19 ? 553 2 0 ? 514 2 51 ? 828 2 0 ? 212 2 278 ? 398 4 ? 800 0 ? 430 2 7 ? 498 2 0 ? 517 2 43 ? 038 2 0 ? 276 2 266 ? 309 4 ? 880 0 ? 321 3 ? 797 2 0 ? 511 2 34 ? 593 2 0 ? 330 2 272 ? 181 4 ? 960 0 ? 243 12 ? 674 2 0 ? 521 2 27 ? 598 2 0 ? 304 2 289 ? 797 5 ? 040 0 ? 472 26 ? 076 2 0 ? 479 2 18 ? 013 2 0 ? 292 2 347 ? 825 5 ? 120 0 ? 755 21 ? 378 2 0 ? 486 2 5 ? 440 2 0 ? 326 2 439 ? 358 5 ? 200 0 ? 889 17 ? 959 2 0 ? 564 0 ? 328 2 0 ? 289 2 528 ? 128 5 ? 280 0 ? 802 14 ? 953 2 0 ? 654 12 ? 797 2 0 ? 083 2 567 ? 504 5 ? 360 0 ? 824 14 ? 953 2 0 ? 641 42 ? 951 0 ? 148 2 592 ? 373 5 ? 440 0 ? 820 15 ? 748 2 0 ? 569 66 ? 405 0 ? 096 2 631 ? 059 5 ? 520 0 ? 769 15 ? 368 2 0 ? 513 60 ? 343 0 ? 064 2 680 ? 107 5 ? 600 0 ? 684 14 ? 850 2 0 ? 457 44 ? 109 0 ? 201 2 716 ? 720 5 ? 680 0 ? 497 13 ? 261 2 0 ? 450 36 ? 199 0 ? 264 2 739 ? 862 5 ? 760 0 ? 334 12 ? 847 2 0 ? 487 27 ? 857 0 ? 337 2 749 ? 879 5 ? 840 0 ? 345 13 ? 572 2 0 ? 533 23 ? 298 0 ? 410 2 761 ? 623 5 ? 920 0 ? 328 13 ? 745 2 0 ? 600 27 ? 304 0 ? 626 2 774 ? 403


Numerical data for Figures 3(c) , 5(b) and 5(d) (continued)


Time (ms)

Pressure at 3

(MPa)

Axial strain at B

( m -strain)

Velocity at 3

(m / s)

Axial strain at B

( m -strain)

Pressure at 7

(MPa)

Axial strain at F

( m -strain)

6 ? 000 0 ? 002 11 ? 879 2 0 ? 678 41 ? 328 0 ? 742 2 786 ? 147 6 ? 080 2 0 ? 377 11 ? 983 2 0 ? 700 50 ? 499 0 ? 678 2 758 ? 860 6 ? 160 2 0 ? 307 15 ? 679 2 0 ? 757 38 ? 306 0 ? 741 2 714 ? 993 6 ? 240 2 0 ? 261 21 ? 136 2 0 ? 834 28 ? 721 0 ? 786 2 681 ? 834 6 ? 320 2 0 ? 312 25 ? 178 2 0 ? 950 34 ? 593 0 ? 786 2 642 ? 112 6 ? 400 2 0 ? 310 27 ? 423 2 0 ? 959 45 ? 507 0 ? 756 2 573 ? 376 6 ? 480 2 0 ? 307 26 ? 939 2 0 ? 921 55 ? 472 0 ? 647 2 491 ? 169 6 ? 560 2 0 ? 310 27 ? 112 2 0 ? 851 62 ? 415 0 ? 541 2 428 ? 651 6 ? 640 2 0 ? 203 26 ? 179 2 0 ? 759 58 ? 857 0 ? 499 2 407 ? 926 6 ? 720 2 0 ? 018 25 ? 765 2 0 ? 703 42 ? 295 0 ? 480 2 407 ? 235 6 ? 800 0 ? 103 24 ? 798 2 0 ? 689 25 ? 146 0 ? 469 2 406 ? 544 6 ? 880 0 ? 087 23 ? 658 2 0 ? 719 9 ? 741 0 ? 483 2 401 ? 363 6 ? 960 0 ? 045 22 ? 518 2 0 ? 768 2 11 ? 606 0 ? 567 2 381 ? 330 7 ? 040 2 0 ? 272 17 ? 751 2 0 ? 798 2 28 ? 496 0 ? 678 2 337 ? 117 7 ? 120 2 0 ? 680 9 ? 703 2 0 ? 830 2 41 ? 432 0 ? 650 2 289 ? 797 7 ? 200 2 0 ? 633 1 ? 552 2 0 ? 813 2 44 ? 868 0 ? 568 2 255 ? 601 7 ? 280 2 0 ? 546 2 5 ? 184 2 0 ? 785 2 54 ? 229 0 ? 471 2 244 ? 548 7 ? 360 2 0 ? 577 2 13 ? 370 2 0 ? 836 2 71 ? 154 0 ? 512 2 261 ? 473 7 ? 440 2 0 ? 561 2 18 ? 275 2 0 ? 922 2 75 ? 333 0 ? 669 2 306 ? 031 7 ? 520 2 0 ? 573 2 14 ? 372 2 0 ? 964 2 58 ? 547 0 ? 680 2 331 ? 246 7 ? 600 2 0 ? 639 2 15 ? 408 2 0 ? 960 2 40 ? 447 0 ? 658 2 329 ? 864 7 ? 680 2 0 ? 762 2 12 ? 645 2 0 ? 927 2 26 ? 786 0 ? 582 2 306 ? 722 7 ? 760 2 0 ? 842 2 11 ? 194 2 0 ? 875 2 17 ? 340 0 ? 439 2 283 ? 579 7 ? 840 2 0 ? 840 2 9 ? 363 2 0 ? 871 2 18 ? 842 0 ? 326 2 261 ? 473 7 ? 920 2 0 ? 644 2 3 ? 629 2 0 ? 832 2 20 ? 897 0 ? 218 2 260 ? 783 8 ? 000 2 0 ? 151 6 ? 975 2 0 ? 822 2 21 ? 899 0 ? 199 2 273 ? 217 8 ? 080 0 ? 236 2 0 ? 383 2 0 ? 808 2 13 ? 609 0 ? 201 2 274 ? 599 8 ? 160 0 ? 312 2 ? 657 2 0 ? 794 2 0 ? 604 0 ? 143 2 258 ? 019 8 ? 240 0 ? 301 2 11 ? 367 2 0 ? 765 4 ? 214 0 ? 143 2 248 ? 348 8 ? 320 0 ? 499 2 29 ? 673 2 0 ? 747 1 ? 122 0 ? 176 2 262 ? 164 8 ? 400 0 ? 735 2 29 ? 259 2 0 ? 778 2 2 ? 383 0 ? 206 2 301 ? 195 8 ? 480 0 ? 820 2 22 ? 730 2 0 ? 832 2 10 ? 759 0 ? 273 2 347 ? 825 8 ? 560 0 ? 764 2 20 ? 969 2 0 ? 895 2 13 ? 419 0 ? 358 2 367 ? 513 8 ? 640 0 ? 853 2 15 ? 270 2 0 ? 955 2 1 ? 451 0 ? 377 2 354 ? 733 8 ? 720 0 ? 960 2 8 ? 223 2 0 ? 981 10 ? 120 0 ? 367 2 310 ? 176 8 ? 800 0 ? 992 2 5 ? 253 2 0 ? 969 12 ? 677 0 ? 388 2 238 ? 676 8 ? 880 0 ? 978 2 11 ? 436 2 0 ? 937 17 ? 409 0 ? 391 2 166 ? 486 8 ? 960 0 ? 844 2 14 ? 890 2 0 ? 912 25 ? 733 0 ? 304 2 137 ? 817 9 ? 040 0 ? 617 2 11 ? 228 2 0 ? 876 32 ? 572 0 ? 261 2 150 ? 943 9 ? 120 0 ? 459 2 7 ? 947 2 0 ? 799 27 ? 218 0 ? 312 2 169 ? 940 9 ? 200 0 ? 310 2 10 ? 054 2 0 ? 757 12 ? 400 0 ? 439 2 166 ? 832 9 ? 280 0 ? 178 2 7 ? 843 2 0 ? 770 7 ? 599 0 ? 458 2 134 ? 018 9 ? 360 2 0 ? 029 2 4 ? 907 2 0 ? 764 6 ? 925 0 ? 416 2 90 ? 842 9 ? 440 2 0 ? 067 2 2 ? 731 2 0 ? 727 0 ? 242 0 ? 402 2 56 ? 647 9 ? 520 0 ? 033 2 2 ? 904 2 0 ? 706 2 7 ? 668 0 ? 341 2 60 ? 446 9 ? 600 2 0 ? 040 2 4 ? 942 2 0 ? 692 2 13 ? 678 0 ? 166 2 90 ? 151 9 ? 680 2 0 ? 127 2 6 ? 185 2 0 ? 650 2 22 ? 590 0 ? 022 2 107 ? 076 9 ? 760 2 0 ? 194 2 8 ? 051 2 0 ? 608 2 34 ? 144 2 0 ? 009 2 111 ? 912 9 ? 840 2 0 ? 238 2 12 ? 610 2 0 ? 597 2 42 ? 071 2 0 ? 046 2 118 ? 820 9 ? 920 2 0 ? 207 2 18 ? 206 2 0 ? 606 2 44 ? 368 2 0 ? 091 2 149 ? 907

10 ? 000 2 0 ? 076 2 17 ? 757 2 0 ? 616 2 40 ? 499 2 0 ? 193 2 189 ? 283


APPENDIX D : NOMENCLATURE

A cross-sectional area c wave speed E Young’s modulus e pipe wall thickness g gravitational acceleration h height K fluid bulk modulus m mass of an end cap M bending moment P pressure Q shear force R radius t time u , u ~ , u ̈ displacement , velocity , acceleration (pipe and rod) V fluid velocity x , y , z Cartesian coordinates » strain … Poisson’s ratio θ angular velocity r density s stress

Suf fices

f liquid (fluid) r radial direction rod rod t pipe (tube) x , y , z cartesian coordinate directions 0 initial conditions f circumferential

fluid-structure interaction in a t-piece pipeatijssel/pdf_files/vardy-fan-tijsseling_1996.pdf ·...

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