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Code_Aster
®
Version
3
Titrate:
Fluid coupling - Structure with Free Face
Date:
29/09/95
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.04-A
Page:
1/10
Manual of Reference
R4.02 booklet: Accoustics
HP-61/95/029/A
Organization (S):
EDF/EP/AMV
Manual of Reference
R4.02 booklet: Accoustics
Document: R4.02.04
Fluid coupling - Structure with Free Face
Summary:
One has here the fluid coupling/structure if the fluid has a free face. Elements
of free face were established in Code_Aster to calculate the modes of ballotement of a coupled fluid
with an elastic structure for a three-dimensional problem.
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Code_Aster
®
Version
3
Titrate:
Fluid coupling - Structure with Free Face
Date:
29/09/95
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.04-A
Page:
2/10
Manual of Reference
R4.02 booklet: Accoustics
HP-61/95/029/A
Contents
1 Notations ................................................................................................................................................ 2
2 Introduction ............................................................................................................................................ 2
3 theoretical Formulation of the problem ....................................................................................................... 3
3.1 Recalls on the coupling fluid-structure ......................................................................................... 3
3.1.1 Description of the fluid ............................................................................................................... 3
3.1.2 Description of the structure ...................................................................................................... 4
3.1.3 Description of the interface fluid-structure ............................................................................... 4
3.1.3.1 Formulation of the problem coupled ............................................................................... 4
3.2 Action of gravity on the free face ...................................................................................... 5
3.2.1 Formulation of the problem ....................................................................................................... 5
3.2.2 Discretization by finite elements ............................................................................................ 7
3.2.3 Introduction of an additional variable ............................................................................ 8
4 Establishment in Code_Aster ........................................................................................................... 9
5 Bibliography ........................................................................................................................................ 10
1 Notations
P
:
stationary pressure in the fluid
p
:
fluctuating pressure in the fluid,
X
F
:
displacements in the fluid,
X
S
:
the field of displacements in the structure,
G
:
gravity,
:
potential of displacements of the fluid,
F
S
,
:
density of the fluid, the structure,
T
:
the tensor of the stresses in the fluid,



:
the tensor of the stresses in the structure,



:
the tensor of the deformations in the structure,
C
:
the tensor of elasticity,
C
:
the speed of sound in the fluid,
H
:
the height of the fluid (or average height),
N
:
the normal external of the fluid.
2 Introduction
In order to study the behavior of structures filled of fluid, one can be led to take in
hope the phenomena of shaking i.e. to add to the system coupled fluid-structure, the effect
gravity on the level of the free face of the fluid. The structures concerned are, for example,
tanks of nuclear thermal power stations of the rapid system, swimming pools of fuel storage [bib4].
One thus supplemented the developments already carried out in coupling fluid-structure [bib3] by
the introduction of new surface elements which take into account, in their formulation, the effect
gravity.
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Code_Aster
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Version
3
Titrate:
Fluid coupling - Structure with Free Face
Date:
29/09/95
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.04-A
Page:
3/10
Manual of Reference
R4.02 booklet: Accoustics
HP-61/95/029/A
3
Theoretical formulation of the problem
The problem of interaction heavy structure-fluid amounts solving three problems simultaneously:
·
the structure is subjected to a field of pressure
P
imposed by the fluid on the wall
;
·
the fluid is subjected to a field of displacement
X
S
imposed by the structure on
;
·
gravity acts on the free face where
p
G Z
=
.
It will be considered initially that the fluid is nonheavy before introducing gravity with
paragraph [§3.2].
3.1
Recalls on the coupling fluid-structure
In order to account for the interaction fluid-structure well, we will analyze them separately
equations governing the behavior of the fluid and those which govern that of the structure, without
to consider in this chapter the boundary conditions concerning the free face.
3.1.1 Description of the fluid
It is considered that the studied system is subjected to small disturbances around its state of balance
where the fluid and the structure are at rest: thus,
(
)
P
p
p
X
X X
S
S
=
+
=
=
0
0
0
and
. What allows
to write [bib2]:
= -
= -
F
F
p
C
div ()
div ()
X
X
F
F
of or
2
.
With:
·
p
fluctuating pressure of the fluid,
·
the disturbance of density of the fluid,
F
density of the fluid at rest,
·
()
X
,
F
R T
the field of displacement of a particle of fluid.
The fluid is:
·
perfect (i.e. nonviscous)
·
barotrope
:
p
C
=
2
éq 3.1.1-1
·
and irrotational: there is a potential of displacements
, such as
p
T
F
=
2
2
The behavior of the volume of fluid eulérien is thus described by the following equations:
·
law of behavior:
T
p
ij
ij
= -
·
conservation equation of the momentum in the fluid in the absence of source:
div ()
T
X
2 f2
=
F
T
éq 3.1.1-2
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Code_Aster
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Version
3
Titrate:
Fluid coupling - Structure with Free Face
Date:
29/09/95
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.04-A
Page:
4/10
Manual of Reference
R4.02 booklet: Accoustics
HP-61/95/029/A
·
conservation equation of the mass:
T
T
F
+




=
div
X
S
0
éq 3.1.1-3
By combining the conservation equations of the momentum [éq 3.1.1-2] and of the mass
[éq 3.1.1-3] written in harmonic mode with the pulsation
, one obtains, thanks to [éq 3.1.1-1], the equation
of Helmholtz:
p
C p
+
=
2
2
0
3.1.2 Description of the structure
It is considered that the structure is elastic linear and that one remains in the field of the small ones
disturbances. Taking into account these assumptions, one writes:
·
the law of behavior in linear elasticity:
ij
ijkl kl
C
=
·
the conservation equation of the momentum in the structure in the absence of
voluminal forces others that inertias:
div ()









=
S
T
2 s2
X
·
the equation of compatibility on the tensor of deformation:
kl
K L
L K
U
U
=
+
1
2 (
)
,
,
3.1.3 Description of the interface fluid-structure
With the interface (
) between the fluid and the structure, like the fluid is not viscous, there is continuity of
normal stresses and normal speeds to the wall, and nullity of the tangential stress
(absence of viscous friction). These boundary conditions are written:
ij I
ij I
ij I
N
T N
p
N
T
T
=
= -
=



X .n X .n
F
S
3.1.3.1 Formulation of the coupled problem
Finally, the equation of the problem coupled fluid-structure is written, while taking
p
like variable
describing the field of pressure in the fluid and
X
S
the field of displacements in the structure:
C
X
X
V
p
C p
V
N
C
X
N
p
N
p
N
X N
ijkl
S
S
S
S
F
ij I
ijkl
S
I
ij I
F
F
I
K lj
I
K L
I
,
,
+
=
+
=
=
= -

=




2
2
2
2
0
0
in
in
on
on
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Code_Aster
®
Version
3
Titrate:
Fluid coupling - Structure with Free Face
Date:
29/09/95
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.04-A
Page:
5/10
Manual of Reference
R4.02 booklet: Accoustics
HP-61/95/029/A
Fields of displacements
X
S
for the structure and of pressure
p
for the fluid sought
minimize the functional calculus:
[
]
L (,)
()
()
(grad)
X
X
X
X
X N
S
S
S
S
2
S
p Z
p
D
p
p
C
FD
ij
ij
S
V
F V
S
F
=
-
-
+
-






1
2
1
2
1
2
2
2
2
2
3.2
Action of gravity on the free face
3.2.1 Formulation of the problem
One points out the linearized dynamic equations here describing the small movements of a fluid
perfect. One chooses a description eulérienne fluid:
grad
(
)
P
T
V
F
S
F
=
-
G
X
2
2
in
With balance the particle of fluid was in
M
0
and thus:
grad P
V
F
F
0
=
G
in
.
One considers movements of low amplitude around the state of balance (it is the assumption of
small disturbances): then
(
)
MR. M
X M
F
=
+
0
0
, T
Are
p
the fluctuation in pressure eulérienne and
p
L
the fluctuation in Lagrangian pressure, then:
()
(
) ()
()
()
p
T
P
T
P
p
P
T
P
0
L
0
M
M
M
M
M
0
0
0
,
,
,
=
-
=
-
Taking into account the assumption of small displacements:
p
p gradP (
,)
(
,)
(
,)
L
- =
= -
M
X M
X M
O
F
O
F
O
T
T
G
T
F
éq 3.2.1-1
If one considers the case of a heavy fluid having a free face in contact with a medium to
constant pressure
P
atm
, one can write, by neglecting the effects of surface tension:
()
P
T
P
atm
M,
=
on the free face
SSL
i.e.:
p
L
=
0
. Maybe, with [éq 3.2.1-1],
p
G
F
F
=
(
. )
X Z
Taking into account the assumption of the small movements, the instantaneous slope of the tangent plan is one
infinitely small first order.
X. Z
F
thus merges with the second command near with the front elevation
vertical
H
.
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Code_Aster
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Version
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Titrate:
Fluid coupling - Structure with Free Face
Date:
29/09/95
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.04-A
Page:
6/10
Manual of Reference
R4.02 booklet: Accoustics
HP-61/95/029/A
G
SSL
SSL
M
M
P (M) = P
H
X
F
O
atm
O
O
O
Be reproduced 3.2.1-a: approximation on the free face
Thus, if one adds in the boundary conditions the condition of gravity on the free face, that returns
to consider in
Z H
=
the linearized condition:
p
G Z
F
=
éq
3.2.1-2
The equations of the total problem become:
C
X
X
V
p
C p
V
N
C
X
N
p
N
p
N
X N
SSL
p
G Z
SSL
ijkl
S
S
S
S
F
ij I
ijkl
S
I
ij I
F
F
I
F
K lj
I
K L
I
,
,
+
=
+
=
=
= -

=
=








2
2
2
2
0
0
in
in
on
on
and on
on
To express the functional calculus, one uses the law of behavior on the free face. By considering one
acceptable field of displacement
Z
one obtains [bib2]:
G Z Z ds
p Z ds
SSL
SSL
=
Maybe, finally, the functional calculus of the total system fluid structure subjected to gravity:
[
]
L (,)
()
()
(grad)
X
X
X
X
X .n
S
S
S
s2
S
p Z
p
D
p
p
C
FD
G Z ds
p Z ds
ij
ij
S
V
F V
SSL
SSL
S
F
=
-
-
+
-






+
-
1
2
1
2
1
1
2
2
2
2
2
2
2
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Code_Aster
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Version
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Titrate:
Fluid coupling - Structure with Free Face
Date:
29/09/95
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.04-A
Page:
7/10
Manual of Reference
R4.02 booklet: Accoustics
HP-61/95/029/A
This taking into account of gravity implies two additional terms in the functional calculus
describing the fluid:
·
a term of potential energy related to the free face:
1
2
2
G Z ds
SSL
·
a term due to the work of the hydrodynamic pressure in the displacement of the free face
:
p Z ds
SSL
However it should be noted that it is not the single effect of gravity since in any point of the wall
be exerted a permanent pressure
-
G Z
(or
Z
is the altitude of the point
M
considered: one supposes
that
Z
=
0
on the level of the free face to balance). The point
M
is actuated by a movement
X
S
infinitesimal, the element of surface
D
thus vary and the effort due to the permanent pressure too. This
effort is responsible for an additional term of rigidity being added to the rigidity of structure in
system. It could cause a buckling of the structure by cancelling structural rigidity. This effect is
negligible on the vibratory characteristics ([bib2], [bib1]), one thus does not take it into account.
3.2.2 Discretization by finite elements
To obtain the discretized form of the functional calculus, one replaces each integral by a sum
integrals on each element I of the discretized system, then one uses an approximation by elements
stop unknown functions of displacement and pressure on each element
I
[bib18].
The unknown factors are
(
)
X
S
U v W p Z
,
,
, one has then while posing
Ni
functions of forms (or functions
of interpolation nodal on the element
I
):









=
=


=
=
=



=
=


D
B U
X
NR U
p
NR p
p NR p
X. N NR U
Z NR Z
S
I
I
I
S
(,)
X y Z
I
if
where
, p
, are the unknown factors with the nodes structures and the fluid nodes, and
Z
unknown factors with
free face.
From where the discretized expression of the functional calculus associated with the problem:
L
(
)
=
-
+
-
+
-
-
U (K
M) U p
H
Q
p Z
K Z
p M Z
PC U
T
T
T
Z
T
Z
T
2
2
2
2
2
F
F
F
C
G
with
K
NR B D B NR
it
I
T
I
I
I
=
VI
I
I
FD
stamp stiffness of the structure
M
NR
NR
it
I
=
F
If
I
I
FD
stamp mass of the structure
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Code_Aster
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Version
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Titrate:
Fluid coupling - Structure with Free Face
Date:
29/09/95
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.04-A
Page:
8/10
Manual of Reference
R4.02 booklet: Accoustics
HP-61/95/029/A
and
Q
NR NR
I
T
I
=
VI
I
I
fl
FD
M
NR NR
Z
I
T
I
=






I
I
I
dS
K
NR NR
Z
If
T
If
=
If
I
I
dS
M
NR NR
Z
it
If
=
If
I
I
dS
H
NR NR
I
T
I
=
VI
I
I
fl
FD
where
C
is the speed of sound in the fluid,
F
density of the fluid and where
K
F
corresponds to
potential energy of the fluid,
K
Z
with the potential energy of the free face,
H
with the kinetic energy of
fluid,
M
with the coupling fluid-solid and
M
Z
with the coupling
p Z
-
on the free face.
The approximation by finished parts of the complete problem leads then to the following matric system:
K
C
H
M
K
U
M
C Q
M
U
Z
Z
Z
-
-










-










=
0
0
0
0
0
0
0
0
0
0
2
2
p
Z
C
p
Z
F
F
The first equation corresponds to the movement of the structure subjected to the forces of pressure,
second with that of the movement of the fluid coupled with the structure and the free face, third is
the free equation of face.
However the written problem of the kind has matrices masses and rigidity nonsymmetrical what
prevent the use of the conventional algorithms of resolution of Code_Aster.
3.2.3 Introduction of an additional variable
To make the problem symmetrical and to be able to use the conventional methods of resolution, one
introduced an additional variable: potential of displacements in the fluid
[bib2].
X
F
=
=
- -
grad
i.e.
2
p
This additional unknown factor is related to the unknown factors of the problem, which leads to a matrix of
singular rigidity.
One reformulates the problem coupled heavy structure-fluid:
C
X
X
V
C p
V
p
V
N
C
X
N
N
N
X N
p
G Z
SSL
ijkl
S
S
S
S
F
F
F
F
ij I
ijkl
S
I
F
ij I
F
I
F
K lj
I
K L
I
,
,
+
=
+
=
=
=
= -
=
=








2
2
2
2
2
0
0
in
in
in
on
on
on
background image
Code_Aster
®
Version
3
Titrate:
Fluid coupling - Structure with Free Face
Date:
29/09/95
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.04-A
Page:
9/10
Manual of Reference
R4.02 booklet: Accoustics
HP-61/95/029/A
What leads to the functional calculus of the coupled system:
[
]
L (,)
()
()
(grad)
X
X
X
X
X N
S
S
S
s2
S
p
Z
p
C FD
G Z ds
D
Z ds
p
C
FD
ij
ij
S
V
V
SSL
F
F
SSL
F
V
S
F
F
=
-
+
+
-
+
+
+






1
2
1
2
1
2
2
2
2
2
2
2
2
2
Maybe while discretizing:
L
(
)
=
-
+
+
-
+
+
+




























T
T
T
T
T
T
T
K
M
p Q p
Z K Z
H
C
M Z
p Q
2
2
2
2
1
2
2
1
F
F
Z
F
F
F
Z
C
G
C
What is written, in matric form:
K
Q
K
M
C
Q
C
Q
H
M
M
Z
T
T
Z
Z
T
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
2
2
F
F
F
F
F
F
F
C
G
p
Z
C
C
p
Z














-


















=






4
Establishment in Code_Aster
The library of the finite elements of Code_Aster was enriched by five surface elements
isoparametric having like degrees of freedom the deflection of the free face and the potential of
displacements of the fluid on the free face. They are compatible with the élments 3D which treat it
fluid problem of coupling/structure [bib3]
One names:
MEFP_FACE3
and
MEFP_FACE6
respectively triangles with 3 or 6 nodes,
MEFP_FACE4
,
MEFP_FACE8
and
MEFP_FACE9
respectively quadrangles with 4, 8 or with 9 nodes.
These elements belong to modeling
2d_FLUI_PESA
phenomenon
MECHANICS
.
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Code_Aster
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Version
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Titrate:
Fluid coupling - Structure with Free Face
Date:
29/09/95
Author (S):
G. ROUSSEAU, Fe WAECKEL
Key:
R4.02.04-A
Page:
10/10
Manual of Reference
R4.02 booklet: Accoustics
HP-61/95/029/A
5 Bibliography
[1]
J.TANI, J. TERAKI: Free vibration analysis off FBR vessels partially filled with liquid. SMIRT
1989
[2]
R.J. GIBERT: Vibration of sructures - interaction with the fluids - sources of excitation
random. ECA/EDF/INRIA 1988
[3]
F. WAECKEL: Modal analysis in acoustic vibration in ASTER. Note intern
HP-61/91 160 EDF/DER
[4]
C. LEPOUTERE, F. WAECKEL: Effect of gravity on the free face of a fluid coupled to
a structure, Notes intern HP - 61/93.139