Code_Aster
®
Version
3
Titrate:
FDLV101 Two cylinders separated by an incompressible fluid
Date:
24/08/99
Author (S):
G. ROUSSEAU
Key:
V8.01.101-A
Page:
1/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
Organization (S):
EDF/EP/AMV
Manual of Validation
V8.01 booklet: Fluid
V8.01.101 document
FDLV101 - Two cylinders separated by a fluid
incompressible
Summary:
This test of the field of the fluids (fluid coupling/structure) validates the calculation of matrix of mass added in
case where one has several structures immersed in the same fluid.
By a modal analysis, one thus determines the coupled modes of the two structures because of the mass of
fluid which separates them. One adopts a modeling planes (thermal for the fluid, and deformation planes for
cylinders).
One finds the modes coupled of the system with less than 0.1% of the analytical result.
Code_Aster
®
Version
3
Titrate:
FDLV101 Two cylinders separated by an incompressible fluid
Date:
24/08/99
Author (S):
G. ROUSSEAU
Key:
V8.01.101-A
Page:
2/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
1
Problem of reference
1.1 Geometry
R
X
R
2
R
1
M
1
()
2
()
R
N
Two cylinders separated by incompressible fluid:
interior radius
R
1
= 1.0 m external radius
R
2
= 1.1 m
1.2
Material properties
Fluid:
Water:
O
=
1000.0 Kg.m
3
Solid:
Steel:
S
=
7800.0 Kg.m
- 3
;
E
= 2.E11 AP;
= 0.3
Arises connecting the piston to the solid mass:
One places a discrete element on mesh POI1 in the center of the cylinder
1
of stiffness
K1
and two
discrete elements on mesh POI1 on the cylinder
2
on the level of the axis OX whose stiffness is worth
K2
.
Discrete elements of the type
K_T_D_L
:
K1
= (1.E7, 1.E7, 1.E7) NR/m
K2
= (5.E6, 5.E6, 5.E6) NR/m
1.3
Boundary conditions and loading
One imposes a pressure (IE by analogy thermal a null temperature [R4.07.03]) in a node
unspecified of the fluid.
One imposes a null displacement of the cylinders according to OY.
1.4 Conditions
initial
Without object for the calculation of added mass and the modal analysis.
Code_Aster
®
Version
3
Titrate:
FDLV101 Two cylinders separated by an incompressible fluid
Date:
24/08/99
Author (S):
G. ROUSSEAU
Key:
V8.01.101-A
Page:
3/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
2
Reference solution
2.1
Method of calculation used for the reference solution
Analytical calculation:
One will suppose that the movements of the cylinders and the fluid are primarily plane. Effects
longitudinal will be neglected in front of the transverse effects. The problem is two-dimensional. Count
held of symmetry, the reference mark used is a cylindrical reference mark
()
R,
bound to the central cylinder (see figure
above). In this frame of reference and with this particular geometry, the normal derivative
N
is equal to the derivative
R
compared to
R
.
In all this part, the variable
p
indicate the hydrodynamic field of pressure in the fluid
created by the natural vibrations of the structures,
X
1
2
or
indicate the clean modes of cylinder 1 or 2
respectively.
Clean modes of the hulls of border
()
1
and
()
2
in the absence of fluid are form (
N
indicate the command of the mode):
1n
2 N
X
() cos
sin
X
() cos
sin
R
R
N
N
N
N
or
and
or
0
0
is the azimuth angle. These modes are uncoupled of course. The component 1ère corresponds to
normal displacement of the interior hull, 2nd with that of the external hull. In volume
fluid, one has thus two problems to solve:
1
1
1
0
0
1
2
N
N
F
N
p
p
N
N
p
=
= -
=
N
cos
sin
N
or
éq 2.1-1
and:
2
2
2
0
0
1
2
N
N
N
F
p
p
p
N
N
=
=
=
N
N
cos
sin
or
éq 2.1-2
The field
p
N
1
corresponds to the field of pressure generated in the fluid if the central hull
1
vibrate
only, the field
p
N
2
is that created by the external hull
2
if it only vibrates. The linearity of
the equation of Laplace makes it possible to solve independently each problem and then of
to superimpose to find the field of pressure total.
The solution of the problem [éq 2.1-1] is, in polar co-ordinates, of the type [bib1]:
1n
p (,)
cos
sin
R
Ar
B R
N
N
N
N
=
+
1
or
One must have
N
0
, bus if not one with the not-conservation of the volume of the fluid.
Code_Aster
®
Version
3
Titrate:
FDLV101 Two cylinders separated by an incompressible fluid
Date:
24/08/99
Author (S):
G. ROUSSEAU
Key:
V8.01.101-A
Page:
4/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
Constant A and B are determined by the boundary conditions:
1
2
1
1
0
R
p
N
N
R
p
N
F
N
N
cos
sin
N
= -
=
or
and
It is found whereas the field of pressure for each of the two problems is written:
() (
) ()
(
)
1n
p (,)
cos
sin
R
R
N
R R
R R
R R
R R
N
N
F
N
N
N
N
=
+
-
1
1
2
1
2
2
2
1
1
or
and:
(
) () ()
(
)
2 N
p
(,)
cos
sin
R
R
N
R R
R R
R R
R R
N
N
F
N
N
N
N
=
+
-
2
2
1
1
2
2
2
1
1
or
Modal coefficients of added mass
m
ijnm
With
are calculated starting from the following formula
[R4.07.03] if
()
I
J
N m
=
=
1 or 2
1 or 2
belongs to
,
,
,
¦
2
.
()
m
p
D
ijnm
With
jn
J
J
=
im
J
X
R
N
()
The indexing is a little more complex here than in the formula presented in [R4.07.03]: indices I and
J refer to the hulls
1
and
2
, and indices
m
and
N
are associated the modes of hull. One
notice that there is coupling of the modes of the various hulls, external and intern.
It is noticed, on the one hand, that the fluid does not couple the modes of indices
N
different bus them
integrals
cos
cos
N
m D
cancel themselves; in addition, the fluid does not couple either the modes
()
cos
sin
cos
sin
N
N
N
N D
and
because
=
0
. The only existing coupling is a coupling between
two hulls for the modes of comparable nature.
With each mode
N
, one associates a matrix of command 4 symmetrical. A submatrix corresponding to
projection on the mode
N
is written:
1
With
M
=
11
12
21
22
N
With
N
With
N
With
N
With
m
m
m
m
The total matrix is written:
1
With
2
With
1
With
With
M
M
M
M
0
0
2
=
with
with
(
)
11
1
1
0
2
N
With
m
LR
R
N
N
D
=
1
p
,
cos
sin
or
That is to say:
(
)
(
)
11
1
2
2
2
1
2
2
1
1
1
N
With
F
N
N
m
N
R L R R
R R
=
+
-
éq 2.1-3
Code_Aster
®
Version
3
Titrate:
FDLV101 Two cylinders separated by an incompressible fluid
Date:
24/08/99
Author (S):
G. ROUSSEAU
Key:
V8.01.101-A
Page:
5/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
one will obtain:
(
)
(
)
22
2
2
2
2
1
2
2
1
1
1
N
With
F
N
N
m
N
R L R R
R R
=
+
-
éq 2.1-4
and:
(
)
(
)
21
12
1
2
2
1
2
2
1
2
1
N
With
N
With
F
N
N
m
m
N
R R L
R R
R R
=
= -
-
éq 2.1-5
L
indicate here the height of the hulls cylinders in the longitudinal direction.
In our case, one considers only the modes of command
N
=
1
hulls: they correspond
respectively with the modes of translation of each hull along an axis passing by the center
central tube: one takes those arbitrarily corresponding to axis OX: coefficients of mass
added linear are written:
(
)
(
)
(
)
(
) ()
(
)
11
1
2
2
2
1
2
2
1
22
2
2
2
2
1
2
2
1
21
12
1
2
2
1
2
2
1
1
1
1
1
2
1
With
F
With
F
With
With
F
m
R
R R
R R
m
R
R R
R R
m
m
R R
R R
R R
=
+
-
=
+
-
=
= -
-
The equation of the generalized movement of the two coupled hulls is written:
m
m
X
X
K
K
X
X
m
m
m
m
X
X
1
2
1
2
1
2
1
2
11
12
12
22
1
2
0
0
0
0
+
= -
The own pulsations of the coupled system are given by the equation of degree 4:
det
m
m
m
m
m
m
K
K
1
11
12
12
2
22
2
1
2
0
0
0
+
+
-
=
Numerical application:
K
1
= 10
7
NR/m
K
2
= 10
7
NR/m
m
11
= 33.060 kg/m
m
22
= 40.004 kg/m
m
12
= 36 200 kg/m
One obtains two Eigen frequencies:
F
1
= 1.696 Hz
F
2
= 4.128 Hz
2.2
Results of reference
Analytical
2.3 References
bibliographical
[1]
R.J GIBERT. Vibrations of the Structures. Interactions with fluids. Eyrolles (1988).
Code_Aster
®
Version
3
Titrate:
FDLV101 Two cylinders separated by an incompressible fluid
Date:
24/08/99
Author (S):
G. ROUSSEAU
Key:
V8.01.101-A
Page:
6/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
3 Modeling
With
3.1
Characteristics of modeling
Thermal formulation planes for fluid (QUAD4 and SEG2)
Plane and discrete deformation formulation for solid (TRIA3, QUAD4 and POI1)
This modeling is designed to determine the modes of command
N
=
1
cylinders. Modes of
hulls of a higher nature cannot be simulated by this type of model, but by one
modeling of the type
COQUE_CYL
[U4.22.01].
Cutting =
64 meshs QUAD4 on the circumference of the cylinders
64 meshs TRIA3 on the interior of the interior cylinder
64 meshs SEG2 on the fluid interface/cylinders
2 meshs QUAD4 following the thickness of the fluid
2 meshs QUAD4 following the thickness of the external cylinder
Boundary conditions:
DDL_IMPO: (Group_no: HANG DY: 0. )
DDL_IMPO: (Group_no: ACCREXT DY:0. )
TEMP_IMPO: (Group_no: TEMPIMPO TEMP: 0.)
Code_Aster
®
Version
3
Titrate:
FDLV101 Two cylinders separated by an incompressible fluid
Date:
24/08/99
Author (S):
G. ROUSSEAU
Key:
V8.01.101-A
Page:
7/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
3.2
Characteristics of the mesh
A number of nodes: 356 QUAD4
A number of meshs and types: 64 TRIA3, 128 SEG2, 3 POI1
3.3 Functionalities
tested
Controls
Keys
AFFE_MODELE
“THERMAL”
“PLANE”
[U4.22.01]
CALC_MASS_AJOU
MODE_MECA
[U4.??.??]
NUME_DDL_GENE
NUME_DDL_GENE
“FULL”
STORAGE
[U4.55.07]
MODE_MECA
MODE_ITER_SIMULT
“TAPE”
FREQ
[U4.52.01]
concept
“matr_asse_gene_R”
COMB_MATR_ASSE
COMB_R
[U4.53.01]
concept
“matr_asse_gene_R”
4
Results of modeling A
4.1 Values
tested
Identification
Reference (Hz)
Aster
(Hz)
% difference
Command of the clean mode I: 1
1.696
1.695808
0.011
Command of the clean mode I: 2
4.128
4.12473
0.079
4.2 Remarks
Calculations of modes carried out by:
MODE_ITER_SIMULT OPTION: “PLUS_PETITE” NMAX_FREQ: 2.
4.3 Parameters
of execution
Version: 3.05.24
Machine: CRAY C98
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
10.28 seconds
Code_Aster
®
Version
3
Titrate:
FDLV101 Two cylinders separated by an incompressible fluid
Date:
24/08/99
Author (S):
G. ROUSSEAU
Key:
V8.01.101-A
Page:
8/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
Intentionally white left page.