Fluid-structure interaction in non-rigid pipeline systems

ELSEVIER

Nuclear Engineering and Design 172 (1997) 123-135

Nuclear Engineering and Design

Fluid-structure interaction in non-rigid pipeline systems 1 A.G.T.J. Heinsbroek * Delft Hydraulics, Industrial Technology Division, PO Box 177, Delft Hydraulics, 2600 M H DelJt, The Netherlands

Received 18 October 1995; received in revised form 7 September 1996

Abstract Fluid-structure interaction in non-rigid pipeline systems is modelled by water hammer theory for the fluid coupled with beam theory for the pipe. Two different beam theories and two different solution methods in the time domain are studied and compared. In the first method, the fluid equations are solved by the method of characteristics and the pipe equations are solved by the finite element method in combination with a direct time integration scheme. In the second method, all basic equations (fluid and pipe) are solved by the method of characteristics. The solution methods are applied to a straight pipeline system subjected to axial and lateral impact loads and to a one-elbow pipeline system subjected to a rapid valve closure. In comparing the beam theories, the effects of rotatory inertia and shear deformation for practical pipe geometries and loading conditions are investigated. The significance of fluid-structure interaction is demonstrated. The fluid-structure interaction computer code FLUSTRIN, developed by DELFT HYDRAULICS and which solves the acoustic equations using the method of characteristics (fluid) and the finite element method (structure), enables the user to determine dynamic fluid pressures, structural stresses and displacements in a liquid filled pipeline system under transient conditions. To validate FLUSTRIN, experiments are performed in a large scale 3D test facility consisting of a steel pipeline system suspended by wires. Pressure surges are generated by a fast acting shut-off valve. Dynamic pressures, structural displacements and strains (in total 70 signals) are measured under well determined initial and boundary conditions. The experiments are simulated with FLUSTRIN. The agreement between experiments and simulations is shown to be good: frequencies, amplitudes and wave phenomena are well predicted by the numerical simulations. It is demonstrated that the results obtained from a classical uncoupled water hammer computation, would render completely unreliable and useless results. © 1997 Elsevier Science S.A.

1. Introduction T r a n s i e n t s in fluid-filled pipeline systems are g o v e r n e d b y wave p r o p a g a t i o n p h e n o m e n a b o t h in the fluid a n d in the p i p e wall. Pressure waves in the fluid, also k n o w n as w a t e r h a m m e r , interact * Tel.: + 31 15 2858491; fax: + 31 15 2858713. Updated and extended version of the presentation given at the SMiRT Conference, Division J.

with axial, b e n d i n g , shear a n d t o r s i o n a l stress waves in the pipe wall. T h e i n t e r a c t i o n between axial stress waves a n d fluid pressure waves takes place via the r a d i a l e x p a n s i o n a n d c o n t r a c t i o n o f the p i p e wall. A t j u n c t i o n s (elbows, d i a m e t e r changes) interactive c o u p l i n g s b e t w e e n all m e n t i o n e d types o f waves m a y occur. T h e c o m m e r c i a l l y a v a i l a b l e FLUSTRIN ( F l u i d S t r u c t u r e I n t e r a c t i o n ) c o m p u t e r code, written by the a u t h o r , enables the user to n u m e r i c a l l y simu-

0029-5493/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S0029-5493(96)0 1 363-5

124

A.G.T.J. Heinsbroek / Nuclear Engineering and Design 172 (1997) 123-135

late the aforementioned phenomena in serial pipeline systems. As such, the code will be useful tool to process and mechanical engineers in the safe design and operation of pipeline systems in nuclear power plants. The code is based on (extended) water hammer theory describing the behaviour of the fluid in combination with beam theories describing the behaviour of the pipes. Two options for the beam theory have been adopted, the first according to Bernoulli-Euler and the second according to Timoshenko. The latter is considered to yield more accurate results in case of impact loads and the accompanying high frequencies. All relevant interaction mechanisms (Lavooij and Tijsseling, 1989) are implemented in the computer code. The fluid equations are solved using the method of characteristics (MOC) involving an analytical time integration scheme. The structural equations are solved by the finite element method (FEM) with a direct time integration scheme. In each computational time step the fluid and pipe solutions are brought to compatibility by an iteration process which takes care of the fluid-structure interaction. This solution process will be referred to as the M O C - F E M procedure. An alternative is provided by utilising the MOC also for the solution of the structural equations. This, with respect to structural dynamics, somewhat unconventional approach, has the ability to accurately describe the wave phenomena encountered in impact problems. This numerical scheme will be referred to as MOC-MOC procedure. In the present article numerical solutions obtained with the two different beam theories mentioned above and with the two different numerical solution procedures are mutually compared and analysed in order to investigate their performance. The test problem consists of a straight liquid-filled cantilever pipe subjected to axial and lateral impacts. Furthermore, the transient behaviour of a practical pipeline system with one elbow, excited by a closing valve, is numerically simulated with both the M O C - F E M and the MOC-MOC procedures. Fluid pressures and displacements are compared. The agreement of the results obtained is excellent.

To validate the computer code, experiments have been performed by DELFT HYDRAULICS in a large scale 3D test facility, allowing axial, flexural and torsional motion. The experiments have been simulated with the computer code and the results are compared and interpreted.

2. Basic equations

2.1. Assumptions The transient behaviour of fluid-filled pipeline systems is governed by acoustic waves propagating in both fluid and pipe. Pressure waves in the fluid, referred to as water hammer, coexist with axial, lateral and torsional stress waves in the pipe wall. Due to radial expansion and contraction of the pipe wall, fluid and pipe interact at the fronts of axial waves. Interactions between all four types of waves may take place at pipe junctions. The behaviour of the fluid is governed by extended water hammer equations (Wylie, 1978; Lavooij and Tijsseling, 1989), whereas the dynamics of the pipe is modelled by standard beam theory. It is assumed that the pipe is thin-walled and linearly elastic. The radial inertia of the pipe wall is neglected. The basic equations are:

2.2. Fluid equations The fluid behaviour is described by extended equations of momentum and mass conservation: V ,

~H

, fVr]Vrl

1 0H

1 ~V

c~ ~t

g Oz

2v (?o-. - 0 (g)E Ot

(2)

The extended equation of momentum conservation is equal to its classical equivalent, with the exception of the friction term, in which the fluid velocity V is replaced by the relative fluid velocity Vr = V - zi_, (fi== axial velocity of the pipe wall). In the extended equation of mass conservation an extra term is added to account for the Poisson effect: a positive rate of axial stress decreases the storage in the pipe. The pressure wave speed cf is defined as:

A.G.T.J. Heinsbroek / Nuclear Engineering and Design 172 (1997) 123-135 cf = 1/x/pr(1/K f + D/Ee)

(3)

2.3. Structural equations The structural behaviour is described by the equations of motion, applied in axial, lateral and torsional direction. The local coordinate system used is given in Fig. 1.

2.3.2. Lateral motion The equations of motion in the x- and y-direction (see Fig. 1) according to the Bernoulli-Euler beam theory are respectively:

~4Ux

A ~2ux

EIt-~z4 + (ptAt + pf f)-~2 = -- (PtAt +/)fAr) g cos 7

O4Uy

¢92Uz ~ 2u= ptAt-ff~- -- EA t ~z 2 vDAt OP pfAffVrlVrl [ F ptAtg sin y 2e ~z 2D

(4)

The first term at the left hand side represents the axial inertia of the pipe. Note that the fluid mass density is not included. The second term accounts for the axial stiffness of the pipe. At the right hand side the various external loads on the pipe in axial direction are given. The first term represents the force due to the fluid pressure. In this term the Poisson's ratio is present. The second term represents the force due to fluid friction. Again the relative fluid velocity Vr is used. The third term describes the axial component of the gravitational force on the pipe. Note that also here the fluid mass density is not present.

TX

(5)

A (~2uy

EIt--~z4 + (/)tAt -k-/)f f)--~- = 0

2.3.1. Axial motion The equation of motion is given by

--

125

(6)

The first term at the left hand side represents the lateral inertia of the pipe. The fluid inertia is included. The second term describes the binding stiffness to which the fluid does not contribute. At the right hand side, the lateral component of the gravitational force is given, only for the non-horizontal x-direction. The equation o f free lateral motion in the y-direction (see Fig. 1) according to the Timoshenko (Timoshenko and Young, 1955) beam theory is:

4Uy

~ 2Uy

Elt--~z 4 + (/)tAt + p f A f ) - ~

(~2

a 2UyX~

--/)tI-o-~t 2 k2GAt ptI ~i ~021 OZUY\ q k2GA t \(/)tAt -{-/)fAf)~--) = 0

(7)

The first two terms describe the bending stiffness and the transverse inertia. The third and fourth term describe the rotatory inertia and shear deformation. The fifth is a coupling term between the third and fourth. Omitting all terms but the first two leads to the Bernoulli-Euler formulation. Analysis oaf wave dispersion based on Eq. (7) yields the following limits for the wave speeds (Fl(igge, 1942):

Cb = ~

(8)

c s = k~/GAt/(/)fA f +/)tAt)

(9)

2.3.3. Torsional motion The equation of motion is given by:

820z Fig. 1. Local coordinate system.

GJ

{~20z z; = 0

(lO)

126

A.G.T.J. Heinsbroek / Nuclear Engineering and Design 172 (1997) 123-135

3. Solution procedure

TO3

3.1. M O C - M O C

procedure

E

T'/ 2 e 1

1

q

"°i

Fig. 2. Definition of pipe wall stresses and strains. The first and second term represent torsional inertia and stiffness respectively. 2.4. Interaction

The fluid and the structural equations of motion are coupled through various terms. Eq. (2) and Eq. (4) are coupled via the Poisson's ratio v; this is referred to as Poisson coupling. The Eq. (1) and Eq. (4) are coupled via the friction coefficient, which is referred to as friction coupling. The Poisson and friction effects are caused by distributed loads which are modelled in the differential equations. Junction coupling are caused by concentrated loads which are modelled in the boundary conditions. They couple all equations. 2.5. Strains

Internal forces and moments are derived from structural displacements, according to the standard FEM. Normal and shear stresses are derived from the internal forces and moments and the fluid pressure under the assumptions: (1) plane stress conditions, (2) cross sections remain plane and perpendicular to the neutral axes, (3) effective shear area can be applied for shear forces. From the axial stress o-l, hoop stress 0-3 and shear stress 7-13 the strains in the directions 1-3 (Fig. 2) are computed with: el = (or1 -- va3)/E

(11)

e2 = (27-13(1 + v) + (o 1 At-a3)(1 -- v ) ) / 2 E

(12)

E3 =

(0" 3 --

vaO/E

(13)

In this procedure, proposed by (Wiggert et al., 1987), the structural Eq. (4) and Eq. (7) are formulated as first-order partial differential equations. Together with the water hammer Eq. (1) and Eq. (2) this results in a set of eight equations which can be arranged into two uncoupled sets of four equations. One set contains the water hammer and the axial structural equations, the other set consists of the four equations for the lateral motion replacing Eq. (7). B~L__choosing rational numbers for Ct/Cf ( C t = x//'-"E/pt and Cb/Cs it is possible to devise a computational grid in which interpolations with the accompanying numerical dissipation are avoided (Heinsbroek et al., 1991). The axial equations are weakly coupled and lead to nearly exact solutions even with course grids. The lateral equations suffer from the strong coupling between the shear and bending waves and require a fine mesh to obtain convergence. After having done so, however, the results are of high resolution and accuracy. 3.2. M O C - F E M

procedure

In this procedure the water hammer Eq. (1) and Eq. (2) are solved by the MOC. The structural Eq. (4), Eq. (10) and either Eq. (5) and Eq. (6) or Eq. (7) (in x- as well as y-direction) are solved by the F E M with 12 d.f beam elements (torsion included). Both methods are standard in their field of application. The fluid pressures from the water hammer computation act as distributed and concentrated loads in the structural-dynamics computation. In its turn the computed motion of the structure influences the distributed (Poisson effect) and concentrated (junction effect) storage terms in the water hammer computation. This coupling is taken care of by an iteration process during each time step. The stiffness and mass matrices corresponding to the Timoshenko beam theory can be found in (Przemieniecki, 1968). Optionally, a Rayleigh damping matrix may be activated in the solution

A.G.T.J. Heinsbroek / Nuclear Engineering and Design 172 (1997) 123-135 A E

8O

~

4O

----MOC-FEM

127

-MOC

I 0 f... m

-40

CO_

-80

~b

16o

'

~o

Time

'

2~o

(ms)

Fig. 3. Axial motion, pressure head at pipe midspan.

procedure. In the material presented the damping has been set to zero. The direct time-integration scheme is the Newmark fl = 1/4 method.

4. Numerical results

4.1. Axial motion, the influence of the method of solution A water filled steel pipe with length 30 m, internal diameter 0.1 m and wall thickness 3 mm and which is assumed to move only in the axial direction, is fixed at both ends. A pressure head square pulse of amplitude 50 m and duration 5 ms is applied at the end of the pipe. The ratio of wave speeds ct/cf is taken equal to 25/6. Utilising the interpolation-free scheme, the MOC-MOC procedure leads to exact solutions. In this application, two elements and a corresponding time step of 0.5 ms are sufficient for this procedure. The M O C - F E M procedure uses 30 elements and a time step of 0.8 ms for the results presented in Fig. 3, where pressure head histories at pipe midspan calculated with both procedures are compared. The agreement is good, but the MOCF E M procedure requires relatively many elements to describe accurately the higher frequency oscillations which are due to fluid-pipe interaction. Although the interaction is quite unimportant initially, its influence becomes more pronounced further in time.

4.2. Lateral motion, the influence of the method of solution A water filled pipe of the same size and material as in the previous case, but which now is assumed to move only in the lateral direction, has one fixed and one free end (cantilever). A shear force square pulse of amplitude 100 N and duration 5 ms is applied at the free end. Here, the influence of the fluid is in its contribution to the lateral inertia of the system. Calculated responses are shown for 200 ms (the period of the first natural vibration is about 12 s) in Fig. 4(a)-(c). The M O C - F E M and MOC-MOC result are obtained with 60 and 240 elements respectively. The corresponding time steps are 0.4 ms and 0.003 ms. Notice that the M O C - F E M procedure only uses the F E M part of the code, although the MOC part still prescribes the numerical time step. The shear forces in Fig. 4(a) and the bending moments in Fig. 4(b) demonstrate that the M O C - F E M solution agrees with the MOC solution up to fairly high frequencies. The displacements calculated on grids with 30 (MOC-FEM) and 120 (MOC-MOC) elements are shown in Fig. 4(c) for 800 ms. The observed drift is attributed to the numerical integration necessary in the M O C - M O C procedure to compute the displacements from the structural velocities. Fig. 4(a), 4(b) and 4(c) indicate that the M O C - M O C procedure requires about four times the number of elements of the M O C - F E M procedure in order to get results of comparable accuracy.

128

A,G.T.J. Heinsbroek /'Nuclear Engineering and Design 172 (1997) 123 135

~00

,. . . .

Z

,

r

MOCLFE M

l

--MOC

5O L

u L o

0

~

-50

m

-100!

16o

~o

'

15o

Time

'

aoo

(ms)

60

~-

Z

-~

,

~ ....

MOCLFEM

n U'~ .,-t E3

MOC

-30

16o

5o

-

'

~o

Time

4J C (1) E (D U (0

-

0

-60

E

r

30

((D E o

,

O[~

aoo

'

(ms)

MOC-FEM

MOC

-3

-4

-5

2o0

'

4o0

.........

66o

Time

........ ~-oo (ms)

Fig. 4. (a) Lateral motion, shear force at pipe midspan. (b) Lateral motion, bending moment at pipe midspan. (c) Lateral motion, deflection at pipe free end.

4.3. The influence of shear deformation and rotatory inertia The test problem consists of a liquid-filled cantilever pipe with length 20 m, D = 0.797 m, e = 8 m m , p t = 7 9 0 0 k g m 3, E = 210 GPa, v = 0 . 3 , k 2 = 0 . 5 3 , p r = 1220 kg m 3, K = 2 . 1 GPa. At the free end a bending m o m e n t of 200 k N m is applied with a step time function. In Fig. 5 spatial distributions are plotted of bending m o m e n t M~ and shear force Q.v at 1.29 ms after application o f the step m o m e n t load as

computed with the M O C - F E M procedure employing the Timoshenko theory and the BernoulliEuler theory. The number of elements is 2000, but only every 10 TM(201 in total) node is plotted. According to the Eq. (8) and Eq. (9) the shear and bending wave speeds are 1062 m s - ' and 5156 m s - 1 respectively. After 1.29 ms the shear and bending waves have consequently travelled a distance o f 1.37 m and 6.65 m respectively. The Bernoulli-Euler model leads to far too high wave speeds and very inaccurate amplitudes for the higher frequencies and is therefore inadequate

A . G. T,J. Heinsbroek / Nuclear Engineering and Design 172 (1997) 123-135

0.3 0.2 E Z

=#

Timoshenko BePnoulli-Eu]eP

........

(N=2000, At=O .0 lOBms) (N=2000. z~t=O.OlO8ms)

// ../,vll

l l ,~-\ "\ l '~. ~' I , " -

~\ t '

, ~ I ~. ~' 'J '

-0.1(

..

~ / \, /

\ ,1

\

, k ~"

'\/'

/ '\

i

Z ~E

/

20

15

z l

/ '

10

/ J

0.1 0.0

129

(m)

-0.50

ii

-0.25,fl

0.00

i

i

I I j '

'i! ;i

I ~'' ~'I , ,,~, ~ i! s? ~. ' i J ,if , 1 .... ; .........

A

II

!',

, !

'

~' '

',

-0.25

,

II!;

/

~r I

,, II

,

J

r

H

I

q

II

i

Vi

'i

~ . . . .

5

~0

i

', ,

~

I'~'i ~ 1, ~1,','~,

~

~;',,~,

q~,,~;i"

I'J,

iiq

L,, ,

,

J

lWil" l

20

15

Z

[m]

Fig. 5. Bending and shear waves according to the Bernoulli-Euler beam theory versus the Timoshenko beam theory.

for the problem under consideration. This is to be expected since the Bernoulli-Euler equations are parabolic. The results of the hyperbolic Timoshenko model concerning the wave speeds appear to be very good. The bending wave front has covered a distance of approximately 6.85 m and the shear wave front (the large peak) has covered a distance of 1.38 m. The bending wave is accompanied by a weak shear wave travelling ahead of the main shear wave.

4.4. The influence o f the method o f solution In Fig. 6 spatial distributions are plotted of bending moment and shear force at 3.87 ms after appliction of the step moment load as computed with the M O C - M O C and with M O C - F E M procedures utilising the Timoshenko theory. The point in time has been chosen such that the bending wave just reaches the fixed end of the pipe for the first time. The coincidence between the results of the two solution procedures is good except for the sharp

peak at the bending wave front. This peak is computed reasonably well by the M O C - M O C procedure, but is entirely missed by the MOCF E M approach. It is apparently too sharp a signal to be described by the F E M mesh resolution. However the dispersive behaviour and the behaviour of the shear waves are in excellent agreement. The M O C - F E M solution exhibits some high speed vibrations around the M O C - M O C solution. These vibrations were observed to be mesh dependent by the researchers and tend to diminish upon refining the grid. The shape of the bending wave corresponds very well to the analytical solutions provided by (Fliigge and Zajac, 1959). 4.5. Fluid-structure interaction in a practical system with an elbow In Fig. 7 the system analysed is sketched. The system consists of two pipes with lengths of 310 m and 20 m, a diameter of 0.2064 m and a wall thickness of 6.35 mm. The other pipe data are the same as those in the previous section. The fluid data are: p f = 880 kg m 3, K = 1.55 GPa.

130

A.G.T.J. Heinsbroek /Nuclear Engineering and Design 172 (1997) 123 135

0.3 ¸ -

.....

Z ~E

0

MOC-MOC (N=2049, &t=O.OOlgms, 1<2=0.50) MOC-FEM (N=4000, At=O.OO54ms, k2=0.53)

t3. ,~, 1, 0 ,'~ .

2

+

x ~E - 0 . 1(~

5

10

20

15 z

l

(m)

---

-0.50

t3 Z

-0.25

If i

/"

0.00

-0.25

5

10

20

15 z

(m)----~

Fig. 6. Bending and shear waves according to the Timoshenko beam theory, computed with both the M O C - M O C and the M O C - F E M procedures.

A fixed head reservoir is located at the upstream end, a rigidly mounted ball valve at the downstream end. At the reservoir and at the valve the pipeline system is rigidly supported. Along the pipe every 10 m a support is located such that vertical motions are restrained and horizontal motions are not restrained. Hydraulic transients are generated by closing the valve in 0.5 s. In Fig. 8 the time histories of the dynamic

x

310 r~

Fig. 7. The pipeline system analysed.

fluid pressure at the valve and two dynamic displacement components at the elbow are given. The two solution methods MOC-MOC and M O C - F E M have been applied. For the MOCF E M procedure two computations have been carried out: one with and one without fluidstructure interaction (FSI). Due to the substantial motion of the elbow in z-direction (away from the reservoir) the pressure-time behaviour differs significantly from the classical water hammer signal (no FSI). The motion in y-direction (towards the valve) is less prominent and can be regarded a quasi-static indicator of the pressure, following more or less the pressure at the elbow. The signals with FSI as produced by the two solution methods agree excellently; they cannot be distinguished from each other. Without FSI the classical water hammer pressure-time behaviour is calculated. Due to FSI fluid pressures are larger than their classical counter-parts. The displacement amplitudes, however, are smaller as a result of FSI.

A. G.T.J. Heinsbroek / Nuclear Engineering and Design 172 (1997) 123-135

131

adj. spring "G

4

12.

0

Ul (I}

&

-4

-80

Q

l

2

3

4

100 50

E •

Fig. 9. The tested and simulated pipeline system. 0

~ "o - 5 0 I N -CO0 0

1

2

3

4

3 (s)

4

4

E

///~

E

-2,

,

/

-4

/

1

2 Time

- .....

MOC-MOC (N=132. &t=O.O6ms) MOC-FEM (N=33. At=B.4ms) MOC-FEM (N=33, At=B.4ms) NO FSI

Fig. 8. Pressure and displacement histories.

5. Experimental validation

5. I. Experiments To validate the computer code, experiments have been performed by DELFT HYDRAULICS in a large scale 3D test facility, allowing axial, flexural and torsional motion. The experimental set-up is shown in Fig. 9. It consists of a water-filled closed loop with a variable speed pump, an air vessel, 77.5 m of welded pipe, six elbows (square metre bends), a fast acting shut off valve and a control valve. A flexible hose closes the loop between the control valve and the pump. The structural boundary conditions of the system are: • Rigid supports at locations A and H.

• Bend supports at locations B and G, allowing axial motion and torsional rotation. • Suspension wires at about every 6 m along the pipe. • Adjustable spring at location E in the X 1 or the )(3 direction Transients are generated by closing the fast acting shut-off valve starting from a steady state flow condition. Different initial and boundary conditions are obtained by varying the initial flow rate, the closure time, the stiffness and direction of the adjustable spring. In each experiment, the following signals were measured: • The steady state flow rate. • Six dynamic fluid pressures. • Three forces in suspension wires. • The valve disc position. • Two steady state fluid pressures. • Nine structural displacement. • Forty eight pipe wall strains.

751

i

/

50

i 2.5i

°'°o

20

40

6b

Bo

loo

Time (ms) Experiment -FLUSTRIN Computation

Fig. 10. Measured and computed pressure at the shut off valve.

132

A.G.T.J. Heinsbroek /Nuclear Eng&eering and Design 172 (1997) I23-135

Time[msl 40 Pressure -

.

.

.

30

.

.

~ .. . . . . . . .

i

.

;

o-2::::~o" 3

.

.

Stress

.

wave wave

"

._ : : : : : : : : : : : : : : : : : : : : :

10

0

.

.

.

.

..... -

.

"

....

..................................................

F 7.5

E Dist.

[ml

31

Fig. 11. Distance-time diagram of hydraulic and structural wave propagation. The signals were recorded simultaneously with a sample rate of 800 Hz during 5 s. In Fig. 10 the measured and computed pressure-time behaviour at the shut off valve is plotted. The wave speed is 1257 m s-1 according to Eq. (3) with p f = 998.2 kg m 3, K f = 2.29 GPa, D = 108.7 mm, e = 3 . 0 7 mm and E = 2 0 0 G P a substituted. The initial fluid velocity is 0.3 m s - 1 At 20 ms the valve is completely closed. In the computed results the pressure change equals the classical Joukowski pressure change (AP = pfcfAg= 998.2 x 1257 x 0.3 = 3.76 bar). In the measured signal the pressure drops just before valve closure, this is attributed to an axial displacement caused by the closing motion of the valve and an example of structurally induced fluid-structure interaction. In the simulations the rigid and bend supports are assumed to be ideal, although in reality they allow some motion. For this reason FSI effects appear more prominently in the measured signal during the time interval between 20 ms and 40 ms. In Fig. 11 the pressure and stress waves are plotted in a distance-time diagram. At 20 ms the valve is fully closed and the basic water hammer wave starts travelling from the valve through the pipe towards elbow G, where it arrives 3/1257 s later (t = 22.4 ms). Here the first FSI junction coupling takes place: the fluid pressure sets elbow G into motion in the (global) X1 direction, thus causing a pipe stress wave in pipe GH, travelling with a speed of 5000 m s 1. This wave arrives 3/5000 s later at the valve (t = 23 ms). The pois-

son coupling effect of this wave is visible in Fig. 10 in the computed signal as a minor pressure rise. The motion of elbow G is relatively small. The basic water hammer wave arrives 7.5/1257 s after valve closure at elbow F. Via FSI junction coupling the pressure causes two stress waves which propagate towards elbows E and G. The vertical motions are limited because of the rather short length of pipe F G and the bend support at elbow G. Therefore elbow F starts to move mainly in the (global) X~ direction. The tensile pipe stress wave towards elbow E, shown in Fig. 11, arrives at this elbow 23.5/5000 s later (t = 30.7 ms) Elbow E will then be set into motion and via FSI junction coupling water hammer waves will be generated. These pressure waves propagate into the directions towards elbows D and F. The latter pressure wave arrives 31/1257 s later at the valve (t = 55.3 ms). This is indicated in Fig. 11 by an arrow. The frequency with a period of about 10 m s 1 most clearly visible in the time interval between 50 ms and 80 ms, is caused by the stress wave travelling up-and-down through pipe EF. The computer code appears to be able to simulate the pressure and stress wave phenomena in the pipeline system quite well. 5. I. 1. Pressures

The measured and two computed pressure-time histories at the valve are given in Fig. 12. The two computed lines are obtained taking into account FSI (the FLUSTRIN computation) and without FSI

-c

~3 O3 0~3 b-

0@00

0.20 -

0140 0 60 O,BO 1.00 T i m e (S) Experiment FLUSTRIN Computation Classical W a t e r Hammer C o m p u t a t a o n

Fig. 12. Measured and computed pressure at the shut off valve.

A. G.T.J. Heinsbroek / Nuclear Engineering and Design 172 (1997) 123-135 :,

30.0

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,fii~'i !,,

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o:6o

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6.so

o.ao

---Experiment Time ....... F L U S T R I N C o m p u t a t ion -C]assical Water Hammer

Hammer C o m p u t a t i o n

(s)

1.oo ,*

Computation

Fig. 13. Measured and computed displacement J(~ direction of elbow D.

Fig. 14. Measured and computed hoop strain 1.5 m downstram of elbow F.

(the classical water hammer computation). In the measured and FSI computed signals a basic frequency with a period of about 0.2 s is present, whereas in the computation without FSI the basic water hammer period is about 0.25 s (4L/cf=4 x 77.5/1257). This virtual increase in wave speed is also observed by others (Kuiken, 1988) and is attributed to junction coupling. At each elbow the energy in the water hammer wave is partly transformed into stress wave energy and vice versa, thus redistributing and 'smearing out' the waves. In Fig. 13 dynamic displacements (relative to the steady state values) in the global )(3 direction of elbow D are given. The results of the classical computation are obtained by calculating the structural response on the water hammer pressures (without FSI) applied as a time and space dependent load. The measured and FSI signals agree quite well: amplitudes and frequencies differ not too much. The same basic frequency with a period of 0.2 s as in the fluid pressures is recognisable in the displacements. Without FSI, the amplitudes are overpredicted significantly by the computation. The basic frequency with a period of 0.25 s of the classical water hammer wave can be recognised.

Fig. 14. The agreement between the experiment and the FSI computation is excellent. Amplitudes and frequencies coincide very well. The basic frequency with a period of 0.2 s is visible in both signals. The classical solution differs substantially from the other two signals. The basic period of 0.25 s from the classical water hammer theory is visible. The amplitudes exceed the measured and FSI values by 40% (negative) to 80% (positive). The difference in the absolute maxima is 35%, resulting in a far too large safety margin. Measured and computed time-histories of a representative shear or torsional strain are given in Fig. 15. In the measured and the FSI results again close agreement is obtained. Next to the basic water hammer frequency, a structural vibration with a period between 1.2 s and 1.4 s is visible in both signals. ,

150.0 i IOO

'<,

.,

~ ,¢ ,

50.0

~J ,j,,

,s

-100 "00.O0 - .....

5.1.2. Strains The measured and computed time-histories of a representative hoop strain are presented in

,

o

/

~'~ V

0.20

0.40 0.60 0.80 ~.00 Time (s) Experiment FLUSTRIN Computation Classical Water Hammer Computation

Fig. 15. Measured and computed shear or torsional strain at shut off valve.

134

A.G.T.J. Heinsbroek /Nuclear Engineering and Design 172 (1997) 123 135

The non-FSI solution overshoots the other two signals by 45%, resulting in an even further too large safety margin.

6. Conclusions

Two completely different solution procedures (MOC-FEM and MOC-MOC) have been applied to two test problems concerning wave propagation in a liquid-filled pipe. For axial waves the MOC-MOC procedure is preferred, since it leads to almost exact solutions. For lateral waves, the MOC-MOC procedure requires very fine computational grids in comparison with the M O C - F E M procedure. Two items which are of importance in the numerical simulation of FSI phenomena in liquidfilled pipes have been discussed. The first item concerned the choice (Bernoulli-Euler or Timoshenko) of the governing equations for lateral pipe motion, the second item concerned the choice (MOC-MOC or MOC-FEM) of the method of solution. One test problem (a straight pipe under extreme load conditions) was presented in which Bernoulli-Euler theory failed, and in which the two solution methods predicted slightly different pipe responses. A second, more practical, test problem (non-instantaneous valve closure in a pipeline system with one non-restrained elbow) showed excellent agreement between results obtained with: (1) the two different beam models and (2) the two different methods of solution. These observations correspond to the experiences of the author: Bernoulli-Euler theory in combination with a M O C - F E M solution procedure is adequate for the computation of FSI occurrences in practical pipeline systems. Pressure surge experiments have been carried out on a flexible water-filled pipeline system in which fluid pressures, structural displacements and strains have been measured. The measured data have been simulated using the FLUSTRIN computer code in which the significant fluid-structure interaction (FSI) mechanisms are taken into account and also with a more classical approach in which FSI is neglected.

It appears that fluid and structural wave phenomena are numerically simulated well. Junction effects play a predominant role in the transient behaviour of the non-rigid pipeline system. A frequency shift in the basic water hammer signal has been measured and computed. The amplitudes and frequencies of the fluid pressures, the structural displacements and strains as computed taking into account FSI all agree well with the measurements. The computations without FS! yield too low values for the pressures and too high values for the structural quantities. This may lead to uneconomic design, with an unintended large safety margin. Taking into account FSI, it appears to be possible to simulate the transient behaviour of a liquidfilled non-rigid pipeline system significantly more accurate than with a classical water hammer computation followed by a structural dynamics computation. 7. Nomenclature Af

At Cb Cf Cs

C~

D E e

f G g H J

Kf L P t H~ ~F

Uz

V Vr

cross-sectional discharge area cross-sectional pipe wall area bending wave speed pressure wave speed shear wave speed 1-D longitudinal wave speed internal diameter Young's modulus of pipe material pipe wall thickness Darcy-Weisbach friction factor shear modulus gravitational acceleration fluid pressure head moment of inertia polar moment of inertia fluid bulk modulus pipe length fluid pressure time vertical lateral displacement horizontal lateral displacement axial displacement fluid velocity relative fluid velocity

A.G.T.J. Heinsbroek/Nuclear Engineering and Design 172 (1997) 123-135

z el ,2,3

0z v pf Pt 0"1, 3

0"z 2713

distance along pipe axis pipe elevation angle strain torsional rotation Poisson's ratio fluid mass density pipe mass density normal stresses axial stress shear stress loss coefficient

Acknowledgements The work done has been part of the FLUSTRIN by DELFT HYDRAULICS, the Netherlands. The FLUSTRIN project is financially supported by: (France) Bergeron Rateau; Elf Aquitaine: (Germany) Rheinisch-West f/ilischer TUV; (The Netherlands) Dutch State Mines Research; Elf Petroland; Ministry of Social Affairs and Employment, Nuclear Department and Pressure Vessel Division; Nucon Nuclear Technology; Shell Internationale Petroleum Maatschappij BV; National Foundation for the Coordination of Mar-

135

itime Research; (UK) ICI; Nuclear Electric; Powergen.

References F1/igge, W., 1942. Die Ausbreitung von Biegungswellen in St~iben, Zeitschrift f/Jr angewandte Mathematik, 22(6), 312318. Fliigge, W., Zajac, E.E., 1959. Bending impact waves in beams, Ingenieur-Archiv 28, 59-70. Heinsbroek, A.G.T.J., Lavooij, C.S.W., Tijsseling, A.S., 1991. Fluid-Structure Interation in Non-Rigid Piping--A Numerical Investigation, SMiRT 11 Transactions, vol. B, Elsevier, Tokyo, Japan, pp. 309-314. Kuiken, G.D.C., 1988. Amplification of Pressure Fluctuations due to Fluid-Structure Interaction. J. Fluids Structures 2, 425-435. Lavooij, C.S.W., Tijsseling, A.S., 1989. Fluid-structure interaction in compliant piping systems, Proc. of the 6th Int. Conf. on Pressure Surges. BHRA, Cambridge, UK, pp. 85-100. Przemieniecki, J.S., 1968. Theory of Matrix Structural Analysis, McGraw-Hill, New York, pp. 80 and 296. Timoshenko, S., Young, D.H., 1955. Vibration Problems in Engineering, 3rd ed. D. van Nostrand, New York, pp. 331. Wiggert, D.C., Hatfield, F.J., Stuckenbruck, S., 1987. Analysis of liquid and structural transients by the method of characteristics. J. Fluids Eng. ASME, 109 (2) 161-165. Wylie, E.B., Streeter, V.L., 1978. Fluid Transients, McGrawHill, New York, pp. 17-30.