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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.
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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:
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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)
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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.
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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.
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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.
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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]
Serial no. Structural frequency Fluid frequency
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.
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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.
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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.
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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
Serial no. Structural frequency Fluid frequency
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)
Serial no. Structural frequency Fluid frequency
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.
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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
Serial no. Structural frequency Fluid frequency
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
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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)
Serial no. Structural frequency Fluid frequency
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
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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.
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investigation of this aspect to establish guidelines, which are
better than those that exist today.
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