a new finite element formulation based on the velocity

8/3/2019 A New Finite Element Formulation Based on the Velocity

1/14

A new finite element formulation based on the velocityof flow for water hammer problems

Jayaraj Kochupillai1, N. Ganesan, Chandramouli Padmanabhan*

Machine Dynamics Laboratory, Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600 036, India

Received 26 July 2003; revised 21 June 2004; accepted 21 June 2004

Abstract

The primary objective of this paper is to develop a simulation model for the fluidstructure interactions (FSI) that occur in pipeline systemsmainly due to transient events such as rapid valve closing. The mathematical formulation is based on waterhammer equations, traditionally

used in the literature, coupled with a standard beam formulation for the structure. A new finite element formulation, based on flow velocity,

has been developed to deal with the valve closure transient excitation problems. It is shown that depending on the relative time-scales

associated with the structure, fluid and excitation forces, there are situations where the structural vibration response increases with FSIs. This

is in contrast to what is normally accepted in the literature, i.e. FSI reduces the structural displacements.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Fluidstructure interaction; Finite element method; Waterhammer

1. Introduction

Even though many researchers have used hybrid modelsfor waterhammer problems, with the method of charac-

teristics (MOC) modeling the waterhammer equations and

the finite element method (FEM) modeling the structure,

few have used the wave equation resulting from the

elimination of one of the variables from the waterhammer

equation in FEM. The wave equation can be formed with

flow velocity as the fluid variable, which is appropriate for

the valve closure excitation. This equation is elliptical in

nature and hence can be readily modeled using FEM. In this

investigation, this feature is exploited to develop a coupled

FEM formulation of both the structure and the fluid. Effects

such as junction coupling and Poisson coupling are includedwhile friction coupling has been neglected due to the short

time-scales associated with the excitation. Model reduction,

based on the structural and fluid vibration modes, has been

used to reduce the size of the problem and care has been

exercised to include axial mode shapes since the interaction

occurs through the axial equations of the beam.Tijsseling [1] presented a very detailed review of

transient phenomena in liquid-filled pipe systems. He

dealt with waterhammer, cavitation, structural dynamics

and fluidstructure interaction (FSI). The main focus was on

the history of FSI research in the time-domain. One-

dimensional FSI models were classified based on the

equations used. The two-equation (one-mode) model refers

to classical waterhammer theory, where the liquid pressure

and velocity are the only unknowns, the four-equation (two-

mode) model allows for the axial motion of straight pipes;

axial stress and axial pipe-wall velocity are additional

variables. The six-equation model is necessary if radialinertia forces are to be taken into account; hoop stress and

radial pipe-wall velocity are the additional unknowns. The

state-of-the-art fourteen equation model describes axial

motion (liquid and pipes), in and out-of-plane flexure, and

torsional motion of three-dimensional pipe systems.

Wiggert et al. [2] used the MOC to study transients in

pipeline systems. They identified seven wave components,

coupled axial compression of liquid and pipe material,

coupled transverse shear and bending of the pipe elements

0308-0161/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijpvp.2004.06.009

International Journal of Pressure Vessels and Piping 82 (2005) 114

www.elsevier.com/locate/ijpvp

* Corresponding author. Tel.: C91-44-2257-8192; fax: C91-44-

2257-0509

E-mail address: [email protected] (C. Padmanabhan).1 Currently with Government College of Engineering, Thiruvanantha-

puram, Kerala, India.
http://www.elsevier.com/locate/ijpvphttp://www.elsevier.com/locate/ijpvp


2/14

in two principal directions and torsion of the pipe wall. The

fourteen characteristic hyperbolic partial differential

equations were converted to ordinary differential equations

by the MOC transformation. The formulation was applied to

two systems of three mutually perpendicular pipes.

Heinsbroek [3] reported an application of FSI in the

nuclear industry. His analysis was based on a combinationof MOC and FEM. His conclusion was that while the MOC

technique was superior for axial dynamics, FEM was more

robust for transverse/lateral dynamics. The investigation

also highlighted the fact that FSIs do take place and a model

based only on the fluid gives erroneous results. This is

corroborated by data from experiments. Lee and Kim [4]

used a finite element formulation for the fully coupled

dynamic equations of motion and applied it to several

pipeline systems. Wang and Tan [5] combined MOC and

FEM to study the vibration and pressure fluctuation in a

flexible hydraulic power system on an aircraft. Casadei et al.

[6] presented a method for the numerical simulation of FSI

in fast transient dynamic applications. They had used both

finite element and finite volume discretization of the fluid

domain and the peculiarities of each with respect to the

interaction process were highlighted.

An earlier study carried out by Kellner et al. [7] showed

that FSI reduced displacements and the corresponding loads

on the snubber below the elbow by a factor of almost four.

In this investigation junction coupling was considered

whereas Poisson coupling was neglected. Lavooij and

Tijsseling [8] suggested a provisional guideline to judge

when the FSI is important. This guideline is based on the

characteristic time-scales of the system under consideration.

One of the objectives of this study is to re-examine thoseproposed guidelines using the new finite element formu-

lation based on flow velocity.

2. Finite element formulation

2.1. Waterhammer problem

For studying the FSIs in pipelines, the model proposed by

Wiggert et al. [2] has been used. The first four equations are

related to the structure while Eqs. (5) and (6) are the

waterhammer equations. This model accounts for the

Poisson coupling, which appears in the axial structuralequation (Eq. (1)) and the influence of the structural response

on the pressure (Eq. (6)). The set of pipe dynamic equations

suggested by Wiggert et al. (1987) is shown below:

EApu00KmuC2nAp0Z 0 (1)

EIpw0000Cm wZ0 (2)

EIpv0000CmvZ0 (3)

GJt00KrpJtZ 0 (4)

rw _VCp0Z0 (5)

_pCrwa2V0K2rwa

2n _u

0Z0 (6)

where

a2Z

Kf=rw

1CKfD=Et; EZ

E

1Kn2 ;

Kfis the fluid bulk modulus, mp, E, G, n,Ip,Ap,D,rw, t, u, v, w,

p and V, are the mass per unit length of pipe, Youngs

modulus of elasticity, Poissons ratio, the second moment of

area, the cross-sectional area, the inner diameter, density of

fluid, the thickness of the pipeline, displacement of pipe in

x-direction, displacement of pipe in y-direction, displace-

ment in z-direction, pressure and velocity of flow, respec-

tively. If the derivative of Eq. (5) with respect to the axial

direction and Eq. (6) with respect to time respectively is

taken, one of the variables can be eliminated. Two wave

equations can then be obtained, either in terms of pressure or

in terms of velocity. The wave equations obtained are

elliptical in nature and suitable for solution by the FEM.

Since the boundary condition for the valve closure event is in

terms of flow velocity it is easier to use the wave equation in

terms of velocity and is given by:

v2V

vx2K

1

a2

v2V

vt2K2n

v3u

vx2vtZ0 (7)

The 3D beam element with six degrees-of-freedom per node

is used to model the pipe Eqs. (1)(4) resulting in the

equation below

MfugC KfugK S2fpg

Z

fftg (8)where [M]and[K] are the mass matrix and stiffness matrix of

the pipe and the interaction of pressure with the structure due

to the Poisson coupling S2Z2n

l0 Ns

TN0p dxHere Ns

represents the shape function matrix for the axial displace-

ment of the structure and [Np] the shape function matrix for

fluid pressure. The matrix [N0p] represents the gradient of the

shape function in the x-direction. The junction coupling is

modeled as a force term {f(t)} at the nodes on the junctions

given by the area of cross-section multiplied by the pressure

at the respective node. The finite element form of the wave

Eq. (7) is formulated using the Galerkin technique and is

Gf VgC HfVgK Sf _ugZ f0g (9)

where

GZ1

a2

l

0N

TvNv dv; HZ

l

0N0

Tv N

0v dv;

SZ 2n

l

0N0

Tv

N0s dv

The relation between pressure and velocity given by Eq. (6)

is used to obtain the pressure from velocity. Eq. (6) is

converted to the finite element form using the Galerkin

procedure. This leads to the following equation:

J. Kochupillai et al. / International Journal of Pressure Vessels and Piping 82 (2005) 1142


3/14

Af _pgC BfVgK S1f _ugZ f0g (10)

with

AZ

l

0Np

TNp dx; BZrwa2

l

0Np

TN0v dx;

S1Z2nrwa2 l

0NpT

N0 s dx:

In the above equation [Nv] represents the shape function

matrix for flow velocity. The Runge-Kutta fourth order

integration scheme is used to evaluate the transient response

for the valve closure event. The fully coupled equation in

state space form is given by:

Since the size of the problem is large due to the finite

element discretization, overflow errors tend to occur if one

uses the above form. In order to overcome this difficulty, the

modal reduction technique is used to reduce the size of both

structural and fluid matrices. The first few mode shapes of

the structure [4s] as well as the fluid [4f] are used for

transforming the respective variables by substituting:

fugZ 4Sfxg (12)

fVgZ 4ffVmg (13)

If the frequencies of the fluid are much higher than those of

the fundamental frequency of the structure, one has still to

include a few mode shapes of the structure having

frequencies in the range of fluid frequencies, as it can

resonate. After substitution and multiplying throughout by

[4s]T and [4f]

T respectively, one gets:

4sTM4sf x gC 4s

TK4sfxgK 4sTS2fpg

Z 4sTfftg 14

4fTG4ff V mgC 4f

TH4ffVmg

K 4fTS4Sf _xgZ f0g 15

Af _pgC fBg4ffVgK S14Sf _xg (16)

The respective reduced matrices are used in the fourth order

Runge-Kutta integration scheme. The values of the

structural variables as well as that of the fluid flow velocity

available in the modal coordinates are transformed to the

nodal coordinates by multiplying with [4s] a n d [4f],

respectively. The pressure values can be used directly as

no transformation is carried out.

2.2. Pressure transient problems

In problems where pressure alone is prescribed at certain

nodes, it is preferable to use the wave equation in terms of

pressure, which is obtained by eliminating the velocity of

flow from Eqs. (5) and (6) as shown below:

v2p

vx2K

1

a2

v2p

vt2K2rwn

v3u

vt2 vxZ0 (17)

The structural equation remains the same as given by Eq.

(8), while the fluid finite element equation in terms of

pressure as the variable is given by:

GfpgC HfpgKrwST2 fugZ f0g (18)

where [G] and [H] are the same as in Eq. (9) but ST2 is

multiplied by rw. In the finite element model if a

nodal variable is specified, that multiplied by the corre-

sponding columns is brought to the right side of the

equation. In this case also to alleviate the large dimension-

ality problem the modal reduction technique as explainedearlier can be made use of. In this case the coupled equation

can be integrated using the well-known Newmark-Beta

method. The coupled equation is shown below:

M

rwST2 G

" #u

p

( )C

K AfS2

H

" #u

p

( )Z

fpbt

fpt

( )

19

where

S2Z2n

l

0

NTs N

0f dxZ 2

n

l

0N

Tf

N0 s dx T

;

{fpb(t)} is the junction coupling and {fp(t)} is the pressure

excitation.

3. Validation studies

3.1. Benchmark 1

Heinsbroek [3] used the water hammer theory for the

fluid coupled with beam theory for the pipe to model FSI

problems in non-rigid pipelines systems. He compared two

_u

u

_V

V

_p

8>>>>>>>>>>>>>:

9>>>>>>>=>>>>>>>;Z

0 I 0 0 0

KMK1K 0 0 0 MK1S2

0 0 0 I 0

0 GK1S KGK1H 0 0

0 AK1S1 KAK1B 0 0

2666666664

3777777775

u

_u

V

_V

p

8>>>>>>>>>>>>>:

9>>>>>>>=>>>>>>>;C

0

MK1fftg

0

0

0

8>>>>>>>>>>>>>:

9>>>>>>>=>>>>>>>;

(11)

J. Kochupillai et al. / International Journal of Pressure Vessels and Piping 82 (2005) 114 3


4/14

different beam theories and two different solution methods

in the time domain. First he used a hybrid method, i.e. the

fluid equations are solved by the MOC and the pipe

equations are solved by the FEM in combination with a

direct time integration scheme. In the second method, he

used only the MOC for the pipe as well as for the fluid

equations. The system analyzed consists of two pipes with

lengths of 310 and 20 m. The diameter of the pipe is0.2064 m and its wall thickness is 6.35 mm. The material

properties are rsZ7900 kg/m3, EZ210 GPa, nZ0.3,

k2Z0.53, rfZ880 kg/m3, KZ1.55 GPa.

The structural boundary conditions for the pipeline

system are no displacements at the ball valve as well as at

the upstream reservoir end. Further the vertical motion at

every 10 m along the pipe is arrested by supports such that

only horizontal motion is allowed. Hydraulic transients are

generated by closing the valve in 0.5 s. It is assumed that the

flow velocity decreases linearly.

Fig. 1 shows the pressure history at the valve due to valve

closure; a comparison of the results from the presentformulation with those of Heinsbroek [3] shows good

agreement. Some higher frequency ripples are seen in the

results of the present formulation. It can be noted that while

the magnitudes agree very well, there seems to be a phase

difference of 1808 in the pressure response predicted, as can

be seen from Fig. 1. Fig. 2 shows the pressure time histories

at the valve with and without FSI. The effect of the vibration

of the structure on the fluid is to increase the peak values

of pressure, when interaction is included in the model.

The displacement history of the pipe at the bend is shown

in Fig. 3 and the maximum magnitude matches well with

the result of Heinsbroek[3]. Once again higher frequencies

are present in the results of the present formulation, which is

due to the possible smaller time steps taken during

simulation. Fig. 4 shows the displacement histories of the

z-direction at the bend, of which the z displacement becomes

unstable without FSI but with FSI it is much smaller

and stable. This is similar to the example shown in Kellner

et al. [7]. The natural frequencies of the structure and that ofthe fluid are found out separately and are given in Table 1.

These were obtained using LAPACK [9] eigenvalue solver

routines. The added mass effect of the fluid is included while

evaluating the structural frequencies. The fluid frequencies

are evaluated from the finite element form of Eq. (9) without

Fig. 1. The Heinsbroek [3] pipeline system and pressure comparison at valve.

Fig. 2. Pressure at the valve with and without FSI.



5/14

including the FSI term, i.e. the last term. From the table it

can be seen that the structural frequencies are much below

the fluid frequencies. Hence, very good FSI can be expected,

as seen from the pressure and displacement plots in Figs. 2

and 4, respectively. At the same time, in the y-direction,

there is not much interaction as seen in Fig. 5.

3.2. Benchmarks 2 and 3

Wiggert et al. [2] analyzed the liquid and structural

transients in piping by the MOC. The pipe and fluid

dynamic equations presented in Ref. [2] are made use of in

the present study also. The formulation was demonstrated

for two cases of a system with three pipes directed

orthogonally and connected in series as shown in Fig. 6.

For the first case (benchmark 2), the piping is made of

Fig. 4. Displacement in the z-direction, at the bend, with and without FSI.

Table 1

Structural and fluid frequencies, in Hz, for Heinsbroek [3] geometry

Serial no. Structural frequency Fluid frequency

1 178.1 55.3

2 178.1 166.1

3 191.5 277.3

4 202.7 389.2

5 308.5 502.3

6 308.5 616.7

7 377.0 732.8

8 409.2 850.9

9 436.7 971.3

10 437.1 1094.3

11 553.4 1220.3

12 619.8 1349.5

13 620.9 1482.3

14 716.9 1618.7

15 736.0 1759.1

Fig. 3. Displacement at the bend in z-direction.

Fig. 5. Displacement in the y-direction, at the bend, with and without FSI.



6/14

copper with mitred bends and an inside diameter of 26 mm

with a wall thickness of 1.27 mm; each reach is 2 m long.

The conveyed liquid is water and damping is neglected for

both structure and liquid. The boundary conditions are

obtained by completely restraining the motion of points A,

B and D. The system is excited by closing the valve in

2.2 ms linearly from a velocity of flow of 1 m/s. It isassumed that the static pressure is of sufficient magnitude

that dynamic pressure will not reach vapour pressure. The

pressure history result of Wiggert et al. [2] is compared with

the present formulation in Fig. 7; it is clear that those results

agree very well with that of the present formulation.

A comparison of pressure histories with and without FSI

can be seen in Fig. 8. In this case also the peak values of

Fig. 7. Pressure history comparison between (a) Wiggert et al. [2] and (b) present formulation.

Fig. 6. Layout of piping used for benchmarks 2 and 3.



7/14

pressure with FSI are higher than that without FSI resulting

from the flow of energy from the structure to the fluid.

Consequently, the structural displacement reduces. In the

present finite element formulation, it is observed that higher

Fig. 8. Pressure at the valve, for Benchmark 2 case, with and without FSI.

Fig. 9. Velocity of the pipe at the bend C in x and z-directions, (a) Ref. [2]

and (b) present formulation.

Table 2

Frequencies of structure and fluid, in Hz, for benchmark 2 geometry [2]


1 6.3 6.9

2 11.9 20.8

3 12.6 34.8

4 19.1 49.0

5 25.8 63.3

6 26.4 78.2

7 32.7 93.1

8 39.5 109.0

9 46.0 124.5

10 45.7 141.6

11 51.8 157.8

12 55.7 175.8

13 69.6 192.3

14 61.8 211.0

15 70.4 228.2

Fig. 10. FFT of the structural response without the effect of structure on

fluid.

Fig. 11. FFT of fluid response without effect of structure on fluid.



8/14

frequency content is always present for all benchmark cases.

This may be due to the fact that the time step used in the

present case is very small (10 ms for benchmark 2). Otherinvestigators have not reported the time step used, but it is

believed that they have used larger time steps and hence are

unable to capture the high frequency dynamics. In this case,

the peak values reached a pressure of 3.2 MPa from 2 MPa

as in the case without FSI. The structural velocity of the

bend C in the x and z-directions is compared in Fig. 9. It is

found that the peak values match very well, but there is a

qualitative difference in the shape due to the presence of

higher frequencies in the present formulation.The natural frequencies of the structure and fluid are

found separately without coupling the structure and fluid for

this case. In the structure the added mass effect is included.

The first fifteen of them are given in Table 2. In this case, the

fluid frequency is lower than that of the structure as the pipe

is short and the bends A, B and D are fully constrained. Even

though the fundamental structural frequency is higher than

the lowest fluid frequency, second frequency of the fluid

onwards, there are a number of frequencies in the fluid and

the structure in the same range, so one must expect good FSI

in this case and that is seen in Fig. 8. Some of the peak

values reduce to 50% of the value without FSI.In order to verify the major frequency components of

excitation, a Fast Fourier Transform (FFT) of the pressure

Fig. 13. Pressure history comparison for benchmark 3 between (a) Wiggert et al. [2] and (b) present formulation.

Fig. 12. FFT of the structural response in z-direction at the bend with full

FSI.



9/14

pulse as well as the structural response without the structural

effects on the fluid is carried out and plotted in Figs. 10

and 11. It is seen from these figures, that most of the

frequencies of the structure and the fluid are present in

the FFT of the structural response. The FFT of the structural

response including the effect of vibration of the structure on

the fluid is shown in Fig. 12. Modal damping is added for

both structure and fluid with a damping factor of 0.0016. In

this case, some of the frequencies are suppressed and some

are slightly deviated from the original values. The dominant

frequency of excitation of the system is 277 Hz.Wiggert et al. [2] presented a second case (benchmark 3)

with mutually perpendicular sections as in the previous case

but with lengths 28, 7.35 and 12.3 m. The diameter,

thickness and the material properties are same as in the

previous case. The pressure history at the valve D of this

case, when the valve is closed linearly in 2.2 ms having an

initial flow velocity of 1 m/s is shown in Fig. 13(a). The

results of the present formulation using finite elements and

MOC are given in Fig. 13(b). The magnitude as well as the

shape of the curve matches well with the results of Ref. [2].

Fig. 14 shows the comparison of the pressure response

with and without FSI. The peak magnitude of the pressure

response is higher when full FSI is considered. This is

shown up to 80 ms. Fig. 15 shows a comparison of the

velocity of the pipe at bend C in the x-direction. There is

Fig. 14. Benchmark 3 pressure variation with and without FSI.

Fig. 15. Structural velocity at C in x-direction, (a) from Ref. [2] and (b) present formulation.



10/14

a phase shift of 1808 in the present formulation results and

higher frequencies show up due to smaller time steps used

for integration. Nevertheless, the magnitudes match very

well. The x-direction velocity with and without FSI is

almost the same. The fundamental natural frequencies given

in Table 3, for the structure and the fluid show that they are

very close. In spite of this feature, the interaction is small.

This is due to the fact that the excitation time-scale is also

important for FSI. The valve closing time in the case of

benchmark 3 changed from 2.2 ms to 0.15 s and the result is

shown in Fig. 16. It is clear that now the displacement time

histories are not the same, although there is no significantchange in the amplitude of the response. This would indicate

that the valve closing time, i.e. the pressure rise time is

important in FSI.

Table 3

Structural and fluid frequencies, in Hz, for Wiggert et al. [2] benchmark 3

case


1 0.03 1.0

2 0.04 3.0

3 0.08 5.0

4 0.10 7.0

5 0.17 9.1

6 0.20 11.1

7 0.30 13.1

8 0.33 15.1

9 0.45 17.1

10 0.50 19.1

11 0.64 21.1

12 0.69 23.2

13 0.86 25.2

14 0.92 27.2

15 1.12 29.3

Fig. 16. Z-direction velocities when the valve is closed in 0.15 s.

Table 4

Structural and fluid frequencies, in Hz, for modified Heinsbroek [3]

geometry (with each segment 165 m long)


1 0.01 0.9

2 0.03 2.6

3 0.06 4.4

4 0.09 6.1

5 0.14 7.9

6 0.20 9.6

7 0.26 11.4

8 0.33 13.1

9 0.34 14.9

10 0.42 16.6

11 0.51 18.4

12 0.62 20.1

13 0.73 21.9

14 0.85 23.7

15 0.92 25.4

Fig. 17. Pressure at the bend with and without FSI for modified Heinsbroek

[3] geometry with each section being 165 m.

Fig. 18. Velocity of the structure without FSI at the bend in x andz-direction

for modified Heinsbroek[3] geometry with each section being 165 m.



11/14

3.3. Parameter study

In order to understand the role of structural and fluid time

scales as well as the excitation time scales, in the presence

or absence of FSI, a parametric study has been carried out by

varying section lengths while keeping the total length

constant. This is done so that the fluid time-scales are

constant while the structural time-scales are varied due to

the change in the geometric configuration. The Heinsbroek

[3] piping system is considered where the total length is330 m, with two sections of 310 and 20 m, respectively (see

Fig. 1). Now, this is divided into two sections of equal

length keeping all other properties the same. The first fifteen

frequencies of the structure as well as the fluid are given in

Table 4 where the fluid frequencies are same as in the

original Heinsbroek[3] case (see Table 1). The fundamental

structural frequency is increased in this case as the

maximum length of a section is reduced.

Fig. 19. Velocity of the structure with and without FSI at the bend in the (a) z-direction and (b) x-direction.

Fig. 20. Addition of another length of piping with a bend to the original

Heinsbroek[3] geometry.

Table 5

Structural and fluid frequencies, in Hz, corresponding to Fig. 20


1 178.1 23.7

2 178.1 71.13 191.5 118.6

4 202.7 166.1

5 308.5 213.6

6 308.5 261.3

7 376.9 309.2

8 409.3 357.2

9 436.7 405.3

10 437.4 453.7

11 553.0 502.3

12 619.9 551.1

13 622.7 600.3

14 637.8 649.7

15 716.9 699.6



12/14

The pressure response with and without FSI and the

velocity of the structure in the x and z-directions at the bend

is given in Figs. 1719, respectively. From the figures it can

be seen that the pressure peak values are altered by

the structural vibration. In this case, one can observe that

the effect of FSI is to increase the structural response in

addition to the pressure response. This is most likely due to

the matching of the fluid frequency with the axial vibration

of the structure and the excitation time-scale being smaller

than the structural time-scale. In Fig. 20, an additional

section of 50 m is added to the Heinsbroek[3] configurationwith a bend. The structural and fluid frequencies are shown

in Table 5 of which the lowest structural frequency is

0.001 Hz, which implies that the structure is very flexible.

There is a transfer of energy from the fluid to the structure

and the structural response increases in this case while the

pressure response comes down. These structural response

results are shown in Fig. 21.

As a last case, the fluid frequency of Wiggert et al. [2],

benchmark 2, is altered by extending the last pipe section to

10 m and constraining all degrees-of-freedom of the new

portion of the pipe. This is shown in Fig. 22. The lowest

fluid frequency is reduced to 23.7 Hz from 55.3 Hz as seen

from Table 6. The pressure variations in Fig. 23 as well as

structural displacements in Fig. 24 show little change due to

fluid frequency reduction.

Fig. 21. Displacement with and without FSI at the bend C.

Fig. 22. Modification of benchmark 2 case of Wiggert et al. [2].

Table 6

Structural and fluid frequencies, in Hz, for modified benchmark 2 geometry

(see Fig. 22)


1 0.009 1.0

2 0.03 3.03 0.05 5.0

4 0.09 7.0

5 0.13 9.1

6 0.19 11.1

7 0.25 13.1

8 0.33 15.1

9 0.41 17.1

10 0.50 19.1

11 0.60 21.2

12 0.71 23.2

13 0.84 25.2

14 0.97 27.5

15 1.11 29.1



13/14

4. Conclusions

For modeling waterhammer problems most researchershave adopted the MOC, by converting the first-order

hyperbolic partial differential waterhammer equations to

total differential equations. Few of them have used the

wave equation, which is elliptical in nature and more

suitable for the FEM. The waterhammer phenomenon,

which occurs due to sudden valve closure, has been

modeled using a new velocity based finite element

formulation. The above formulation can be coupled with

the beam finite element formulation for the structure.

Poisson coupling and Junction coupling are also included

in the formulation. The comparison of the results of

the present formulation with three benchmark problems

published in the literature validates the present

formulation.A formulation using pressure as the primary variable is

also developed so that if the excitation is in terms of

pressure, this formulation can be made use of. Pressure

histories, velocity histories and the displacement histories

are compared with and without FSI for a variety of piping

geometries to understand when FSI effects are important. It

has been found that there are situations where changing the

time-scales associated with the structure, increases the

structural response. This behaviour is contrary to what is

generally believed, i.e. FSI will cause structural displace-

ments to reduce. However, there is a need for an in-depth

Fig. 23. Pressure response at the valve and at the bend C.

Fig. 24. Displacement at the bend C with and without FSI, in the z-direction.



14/14

investigation of this aspect to establish guidelines, which are

better than those that exist today.

References

[1] Tijsseling AS. Fluidstructure interaction in liquid-filled pipe systems:

a review. J Fluids Struct 1996;10:10946.[2] Wiggert DC, Hatfield FJ, Struckenbruck S. Analysis of liquid and

structural transients in piping by the method of characteristics. ASME

J Fluids Eng 1987;109:1615.

[3] Heinsbroek AGTJ. Fluidstructure interactions in non-rigid pipeline

systems. Nucl Eng Des 1997;172:12335.

[4] Lee U, Kim J. Dynamics of branched pipeline systems conveying

internal unsteady flow. ASME J Vibration Acoustics 1999;121:11421.

[5] Wang ZM, Tan S-K. Vibration and pressure fluctuation in a

flexible hydraulic power system on an aircraft. Comput Fluids 1998;

27:19.

[6] Casadei F, Halleux JP, Sala A, Chille F. Transient fluidstructure

interaction algorithms for large industrial applications. Comput

Methods Appl Mech Eng 2001;190:3081110.

[7] Kellner AP, Grenenboom HL, de Jong JJ. Mathematical models for

steam generator accident simulation Proceedings of the IAEA,IWGFR/50, Specialist Meeting, The Hague, Netherlands 1983 pp.

115121.

[8] Lavooij CSW, Tijsseling AS. Fluidstructure interaction in liquid-filled

piping systems. J Fluids Struct 1991;5:57395.

[9] Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J, Du

Croz J, Greenbaum A, Hammarling S, Mckenney A, Sorensen D.

LAPACK users guide. Philadelphia: SIAM; 1999.


a new finite element formulation based on the velocity

Documents