Code_Aster
®
Version
7.0
Titrate:
ADLS102 - Meridian oscillator fluidelastic
Date
:
06/08/03
Author (S):
G. DEVESA, F. STIFKENS
Key
:
V8.21.102-A
Page:
1/10
Manual of Validation
V8.21 booklet: Accoustics HT-66/03/008/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V8.21 booklet: Accoustics
Document: V8.21.102
ADLS102 - Meridian oscillator fluidelastic
Summary:
The objective is to calculate the displacement of the piston of a “oscillator meridian fluid-rubber band”.
It is about piston-arises coupled with a fluid contained in a channel with rigid and fixed walls; the channel is
crossed by a wave of depressurization.
One considers the plane problem of this meridian model. This two-dimensional problem is brought back to a problem
monodimensional by considering by approximation that rates of transverse flow induced by
movement of the piston are transmitted instantaneously in axial speeds.
Only one modeling is used. The calculation of the modes is in formulation U, p,
.
Elements 2D are thus used; these elements are based on meshs QUAD4 for the fluid and for
piston, on meshs SEG2 for the interface between fluid and piston to take into account the fluid interaction
structure (
PHENOMENE= `MECANIQUE', MODELISATION=' 2D_FLUI_STRU'
).
The boundary conditions of nonreturn of the wave are carried out by modelizing a piston shock absorber with each
end; the excitation is carried out by applying a depression to the piston of input.
The fluid which one considers is water (hot), the model schematizing the interaction fluid-structure in
annular space between tank and envelope of core during a fast depressurization.
An exact analytical solution exists. Its comparison with the results produced by Code_Aster allows
to validate the taking into account of the fluid coupling structure in 2D.
Code_Aster
®
Version
7.0
Titrate:
ADLS102 - Meridian oscillator fluidelastic
Date
:
06/08/03
Author (S):
G. DEVESA, F. STIFKENS
Key
:
V8.21.102-A
Page:
2/10
Manual of Validation
V8.21 booklet: Accoustics HT-66/03/008/A
1
Problem of reference
1.1 Geometry
One describes below the model of meridian oscillator fluidelastic schematizing the interaction
fluid-structure in annular space tank-envelope of core.
The meridian elastic fluid oscillator is a model of annular space tank-envelope of core of
jet engine; it consists of an oscillator (piston side-arises appearing a mobile wall) coupled with
a compressible fluid contained in a channel with rigid and fixed walls.
The channel is crossed by a wave of depressurization.
The figure [Figure 1.1-a] below illustrates the model described.
H
E
Z
D X
Appear 1.1-a: total Diagram of the oscillator meridian fluid-rubber band
The channel is of section rectangular of dimensions
E H
×
the rigid side piston moves according to
Z
perpendicular to a wall.
A wave of depressurization arrives by the left; while moving towards the line (without possibility of
return) this wave aspires the piston which, by its resulting displacement, generates waves
propagating towards the ends of the conduit, supposed infinitely long so that there is no reflection.
One conceives a two-dimensional modeling of this system, represented with the figure [Figure 1.1-b]
below:
O
L
- L
Z
E
dx
K
M
X
(D
)
V
V
2L
Appear theoretical 1.1-b: two-dimensional mechanical Representation
Code_Aster
®
Version
7.0
Titrate:
ADLS102 - Meridian oscillator fluidelastic
Date
:
06/08/03
Author (S):
G. DEVESA, F. STIFKENS
Key
:
V8.21.102-A
Page:
3/10
Manual of Validation
V8.21 booklet: Accoustics HT-66/03/008/A
Its geometrical characteristics are as follows:
length piston of wall
2L = 5,0 m,
height of fluid
E = 0,5 m,
width of fluid
H = 1,0 m,
section of fluid
S =
E H
.
1.2
Properties of materials
The physical characteristics of fluid material (hot water) in the tube are as follows:
density
F
3
3
= 0,75 10 kg/m
,
speed of sound
C
F
=
1,0 10 m/s
3
.
Physical characteristics of materials constituting the piston of wall and the pistons of end
have only one formal role in the calculation of Code_Aster.
These physical material characteristics are as follows:
Young modulus
E = 2,0 10 AP
12
,
Poisson's ratio
= 0,3
,
density
S
3
= 0 kg/m
.
1.3
Characteristics of the springs, masses and shock absorbers
The characteristics of the piston of wall as an oscillator are as follows:
Stiffness
K = 5,0 10 NR/m
10
,
Mass
M = 200,0 10
3
kg
.
Damping
With = 0 NR/m
S
The characteristics of the pistons of end as oscillators are as follows:
Stiffness
K = 0 NR/m
,
Mass
m = 0 kg
,
Damping
= C S has = 37,5 10 NR/m
F
F
4
S
.
1.4
Boundary conditions and loadings
Infinitely rigid piston of wall and with displacement only according to the vertical axis.
Infinite length of fluid thus not of reflection of end of the waves: this C.L is simulated in
model by a piston at each end, of null mass, moving only according to the x axis
and fitted with a shock absorber with adequate damping; these pistons are moreover more infinitely rigid.
Total reflection of the waves on infinitely rigid walls of the tube of fluid: realized simply in
omitting to modelize the wall by elements of structure.
Code_Aster
®
Version
7.0
Titrate:
ADLS102 - Meridian oscillator fluidelastic
Date
:
06/08/03
Author (S):
G. DEVESA, F. STIFKENS
Key
:
V8.21.102-A
Page:
4/10
Manual of Validation
V8.21 booklet: Accoustics HT-66/03/008/A
2
Reference solution
2.1
Method of calculation used for the reference solution
The goal is to determine temporal displacement
)
(T
Z
piston of wall.
One considers the plane problem of this meridian model of which geometrical characteristics,
mechanics and fluids are described on the figure [Figure 1.1-a]; the side piston is length 2L.
The two-dimensional problem is brought back to a monodimensional problem while considering by
approximation that rates of transverse flow
z&
induced by the movement of the piston
transmit instantaneously in axial speeds in the channel.
In the volume of control
D
edx
=
of extent
dx
under the piston of wall one can write:
D V
dxd
E
(
)
&z
=
1
2
In the fluid variation speed and variation of pressure in adiabatic evolution are connected by:
P
C V
=
Pressure at the moment
T
in a point of X-coordinate
X
result from the superposition of the propagation from
all elementary sources distributed on the piston:
The coupling thus consists of this: the movement of the piston of acceleration
&&z ()
T
armature in the channel one
field of pressure
)
,
P (T
X
whose effort resulting on the extent from the piston itself acts as return on
dynamics of the oscillator.
The geometrical, mechanical characteristics and fluids of the model are presented on the figure
[Figure 1.1-b].
It is considered initially that the piston and the fluid are at rest and one carries out to release oscillator with
the moment T = 0 by imposing an initial speed to him.
The expression of the pressure
)
,
P (T
X
in a point of the channel develops:
P (,)
&&z (
)
&&z (
)
X T
X U
C
U X
C
D
C
E
X
L
L
X
T
=
- -
+
- -
+
-
2
0
However one has
Z ()
0
0
=
and
Z ()
T
=
0
for T negative, and
Z (
)
- -
=
L X
C
0
since upstream of the piston (
X
included/understood enters
L
-
and
L
) the quantity enters
bracket is always negative; for the same reason one has
Z (
)
- +
=
L X
C
0
.
Finally it comes:
P (,)
Z () Z (
) Z (
)
X T
T
T
L X
C
T
L X
C
C
E
=
-
- -
-
- +
2
2
2
One integrates this expression on X in order to obtain the resultant of the compressive forces on the piston:
R ()
P (,)
P (,)
T
H
X T dx
H
X T dx
L
L
L
= -
= -
-
+
2
0
:
Indeed
P (,)
X T
is even in
X
; it is thus enough to integrate on half of the piston.
Code_Aster
®
Version
7.0
Titrate:
ADLS102 - Meridian oscillator fluidelastic
Date
:
06/08/03
Author (S):
G. DEVESA, F. STIFKENS
Key
:
V8.21.102-A
Page:
5/10
Manual of Validation
V8.21 booklet: Accoustics HT-66/03/008/A
From where the expression of the resultant of the compressive forces on the piston on the assumption of small
movements:
R ()
Z ()
Z ()
T
L T
C
U of
H C
E
T
T
L
C
= -
-
-
2
2
2
The movement of the oscillator thus obeys the equation:
M
K
HL C
E
HL C
E
C
U of
T
T
L
C
&&z
Z
Z
Z ()
+
+
-
=
-
2
2
0
2
2
2
or:
M
coupl
&&z F
F
int
-
-
=
0
if one poses:
F
Z
Z
int
= -
-
K
HL C
E
2
2
and
-
=
T
T
coupl
C
L
U
C
E
C
HL
2
)
Z (
2
F
2
One now considers the case of the propagation of a wave of decompression to stiff face
of amplitude
P
0 along the conduit. At the moment T = 0, this wave still attack the piston of wall with
rest, creating on this piston a force of excitation such as:
F
excit
Hct P
T
L
C
HL P
T
L
C
=
<
0
0
2
2
2
if
if
The equation of the movement is written then:
M
coupl
excit
&&z F
F
F
int
=
+
+
This equation is solved numerically with the Matlab software for the characteristics presented
meridian oscillator.
2.2 Result
of
reference
Displacement
)
(T
Z
piston of wall.
2.3
Uncertainty of the solution
Analytical solution.
2.4 References
bibliographical
[1]
F. STIFKENS: “Transitory Calculation in Code_Aster with the vibroacoustic elements”.
Note intern R & D HP-51/97/026/A.
[2]
F. TEPHANY, A. HANIFI, C. LEHAUT: “Elements of analysis of the interaction fluid-structure
in annular space tank-envelope of core in the event of APRP " - internal Note SEPTEN
ENTMS/94.057.
Code_Aster
®
Version
7.0
Titrate:
ADLS102 - Meridian oscillator fluidelastic
Date
:
06/08/03
Author (S):
G. DEVESA, F. STIFKENS
Key
:
V8.21.102-A
Page:
6/10
Manual of Validation
V8.21 booklet: Accoustics HT-66/03/008/A
3 Modeling
3.1
Characteristics of modeling
3.1.1 System
vibroacoustic equivalent to be modelized
In order to avoid the waves of return coming from the ends of a modeling inevitably of dimension
finished one fits these ends with systems “piston-shock absorber” as on the figure [Figure 3.1.1-a].
The channel is modelized over an overall length of 28 m sufficient to obtain with certainty, at least
the two first extrema of the curve of displacement of the piston without disturbance of a wave of
reflection at the ends.
fluid
P
stiffness
K
mass
M
density
celerity
C
=
S
C
=
S
C
m
=
0
m =
0
piston
wall
piston
discharger
piston
anechoic
rigid wall
fluid
28 m
5 m
0,5 m
Appear vibroacoustic 3.1.1-a: System are equivalent
3.1.2 Modeling
numerical in finite elements of Code_Aster
One chose to modelize in 2D.
For the fluid: modeling is in formulation p,
.
It is carried out by the assignment on meshs of the type
QUAD4
(quadrilaterals with 4 nodes) of elements
PHENOMENON = ' MECANIQUE', MODELING = ' 2D_FLUIDE'
.
For the structures: modeling is in formulation U.
It is carried out by the assignment on meshs of the type
QUAD4
(quadrilaterals with 4 nodes) of elements
PHENOMENON = ' MECANIQUE', MODELING = ' D_PLAN'
.
For the discrete elements of oscillators: modeling is in formulation U.
It is carried out by the assignment on meshs of the specific type
POI1
elements
PHENOMENON = ' MECANIQUE', MODELING = ' DIS_T'
.
For the interfaces fluid-structure: modeling is in formulation U, p,
.
It is carried out by the assignment on meshs of the type
SEG2
(segments with 2 nodes) of elements
PHENOMENON = ' MECANIQUE', MODELING = ' 2D_FLUIDE_STRU'
.
Code_Aster
®
Version
7.0
Titrate:
ADLS102 - Meridian oscillator fluidelastic
Date
:
06/08/03
Author (S):
G. DEVESA, F. STIFKENS
Key
:
V8.21.102-A
Page:
7/10
Manual of Validation
V8.21 booklet: Accoustics HT-66/03/008/A
3.2
Characteristics of the mesh
560 X 10 elements 2D of fluid
50 elements of structure
specific element stiffness
specific element shock absorber
10 élém. 2D of structure
10 élém. 1D of structure
50 élém. 1D flui-stru
10 élém. 1D flui-stru
10 élém. 2D of structure
10 élém. 1D flui-stru
Appear 3.2-a: two-dimensional Mesh of the model of oscillator fluidelastic
One gathered in the table hereafter, the data characterizing this modeling.
Type of mesh
Numbers
QUAD4
SEG2
POI1
total
A number of elements
2870
80
3
2953
A number of generated nodes
3164
0
0
3164
Table 3.2-1: Characteristics of the two-dimensional mesh of the oscillator fluidelastic
3.3 Calculation
One wishes to validate the elements of interaction fluid-structure in transient state by a loading
of excitation.
One carries out the calculation of the displacement of the piston of wall with the operator
DYNA_LINE_TRAN
.
Code_Aster
®
Version
7.0
Titrate:
ADLS102 - Meridian oscillator fluidelastic
Date
:
06/08/03
Author (S):
G. DEVESA, F. STIFKENS
Key
:
V8.21.102-A
Page:
8/10
Manual of Validation
V8.21 booklet: Accoustics HT-66/03/008/A
3.4 Functionalities
tested
Controls Key word
factor
Key word
MECHANICAL AFFE_MODELE
D_PLAN
2d_DIS_T
2d_FLUIDE
2d_FLUI_STRU
FLUID DEFI_MATERIAU
ELAS
RHO, CELE_R,
E, NAKED,…
AFFE_CARA_ELEM
AFFE_CHAR_MECA DDL_IMPO
LIAISON_SOLIDE
PRES_REP
DX, DY…
CALC_MATR_ELEM
CALC_VECT_ELEM
ASSE_MATRICE
ASSE_VECTEUR
DYNA_LINE_TRAN
Code_Aster
®
Version
7.0
Titrate:
ADLS102 - Meridian oscillator fluidelastic
Date
:
06/08/03
Author (S):
G. DEVESA, F. STIFKENS
Key
:
V8.21.102-A
Page:
9/10
Manual of Validation
V8.21 booklet: Accoustics HT-66/03/008/A
4
Result of modeling
4.1 Values
tested
The results of calculation with Code_Aster are presented graphically on [Figure 4.1-a] in
superposition with the “analytical” reference solution.
The curve of Code_Aster appears very close to the reference during the first 4 oscillations but
the differences, at the same time in amplitude and phase, are increasingly perceptible when T increases.
0
20
40
60
80
100
X 10
- 3
time (S)
- 1.2
- 1.0
- 0.8
- 0.6
- 0.4
- 0.2
0.0
X 10
- 3
displacement (m)
0
20
40
60
80
100
X 10
- 3
time (S)
- 1.2
- 1.0
- 0.8
- 0.6
- 0.4
- 0.2
0.0
X 10
- 3
displacement (m)
agraf 22/12/97 (c) EDF/DER 1992
Subjected meridian oscillator fluidelastic has depressurization
EDF
Electricity
from France
Acoustic department and vibratory Mechanics
Displacements compare piston of
wall, between calculation by Aster and
analytical calculation
Aster
analytical
depl. balance
- 0.00136
- 0
Appear 4.1-a: Comparison between Code_Aster calculation and analytical semi reference
The test relates to the displacement of the piston of wall in two moments given close to both
first extrema.
The table presents comparative of the 2 first extrema curve of displacement of the piston
between analytical points and points calculated by Code_Aster.
The values obtained of the moments of extrema in one and the other case are estimated values
extracted without interpolation from the rough computed values: they do not correspond exactly enters
the analytical curve and the curve of Code_Aster.
One estimates the tolerance of variation relating compared to the analytical value to 1.%.
Reference
analytical
Code_Aster
Relative variation
Inst. (ms) Dépl. (mm)
Inst. (ms)
Dépl. (mm)
%
1
Er
Extremum
20,13
1,3530
20,15
1,3536
0,05
2
me
Extremum
26,05
0,4210
26,05
0,42071
0,07
Test of nonregression of the code:
the tolerance of relative variation compared to the reference is worth 0,1%.
Code_Aster
®
Version
7.0
Titrate:
ADLS102 - Meridian oscillator fluidelastic
Date
:
06/08/03
Author (S):
G. DEVESA, F. STIFKENS
Key
:
V8.21.102-A
Page:
10/10
Manual of Validation
V8.21 booklet: Accoustics HT-66/03/008/A
4.2 Notice
The values of reference finally selected are those obtained by Code_Aster at the time of
restitution of the case-test, which will thus make it possible to check nonthe later regression of the code to the course
of its evolution.
5
Summary of the results
Good precision over the first periods then light error in amplitude and phase due to
the influence of integration in numerical time Newmark
=
=
2
1
,
4
1
.