Code_Aster ®
Version
4.0
Titrate:
FDLV100 Piston coupled to a column of incompressible fluid
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.100-A Page:
1/6
Organization (S): EDF/EP/AMV
Handbook of Validation
V8.01 booklet: Fluid
V8.01.100 document
FDLV100 - Piston coupled to a column of fluid
incompressible
Summary:
This test of the field of the fluids (coupling fluid-structure) validates the calculation of mass added on modal basis and
carry out, within the framework of a modal analysis, the calculation of the Eigen frequency of a system piston-arises
coupled to a column of incompressible fluid. To model the fluid, thermal elements are used
plans; to model the piston, one uses machine elements 2D in plane deformation and an element
discrete to model a spring. Lastly, the fluid interface/structure is modelled by linear elements
thermics modified to introduce a boundary condition of the type “acceleration” into the fluid. The case-test
comprise only one modeling, two-dimensional. The Eigen frequency of the coupled system is found
to 0.01% of the analytical result.
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A

Code_Aster ®
Version
4.0
Titrate:
FDLV100 Piston coupled to a column of incompressible fluid
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.100-A Page:
2/6
1
Problem of reference
1.1 Geometry
y
X
D
Piston
Fluid
With
B
L
L
Comes out from stiffness K
Steel piston connected to the solid mass by a spring and coupled to a column of incompressible fluid:
length:
L = 1.0 m
width:
D = 0.25 m
width AB of the piston:
0.05 m
X-coordinates of the points (in m):
With
B
L
X
­ 0.05
0.
1.
1.2
Material properties
Fluid:
Water: O = 1000.0 Kg.m3
Solid:
Steel: S = 7800.0 Kg.m3; E = 2.E11 Pa; = 0.3
Arises connecting the piston to the solid mass:
Discrete element of type K_T_D_L: K = (1.E5, 1.E5, 1.E5) NR/m
1.3
Boundary conditions and loading
One imposes a pressure (IE by analogy thermal a null temperature [R4.07.03]) in all them
nodes of the end of the fluid column.
One imposes the embedding of the spring on the solid mass and one imposes a displacement of the null piston according to
OY.
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A

Code_Aster ®
Version
4.0
Titrate:
FDLV100 Piston coupled to a column of incompressible fluid
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.100-A Page:
3/6
2
Reference solution
2.1
Method of calculation used for the reference solution
Analytical calculation:
When the structure vibrates in the fluid, it modifies the field of pressure which obeys an equation of
Laplace with boundary conditions of Von Neuman [R4.07.03].
In our case, taking into account symmetries of the problem, the field of pressure depends only on
variable X and checks:
2p
p

=

0


X
N
2




F
S
X
X
= -

X 0
p


=
= 0 in X = L
One notes as well as the field of pressure is a function closely connected of X-coordinate X. Both
boundary conditions on the pressure imply: p
= - xS N (X - L
F
)
The compressive force which is exerted on the structure writes:
F
=
N D
=
L (X N) N
p (0)


S
D
F


As the problem is unidimensional, this force can be expressed in an algebraic way according to
component of acceleration according to OX of the structure:
F = - X L D = - L D X = - m X with m = L D
F
F
has
has
F

It is the linear mass added of the fluid on the structure: it is noticed that it corresponds to the mass
of fluid in the column, i.e. to the mass of fluid moved by the piston.
The equation of the movement of the piston projected on OX is written (free vibration not damped out taking into account
the presence of the fluid):
m X + K X
= F = - m X (m + m) X + K X =
has
has
0
The Eigen frequency of this immersed system is thus written:
1
K
F
= 2 m+ma
The effect of the fluid is thus to lower the Eigen frequency of the system in air.
Practically, in Aster, the matrix of added mass is given on the basis of modal
structure in the vacuum: To calculate the added mass given above, one restricts oneself with the calculation of
clean mode of the system piston-arises which corresponds to a translatory movement normalized with the unit
: one truncates consequently the modal base of the structure to only one mode in air (operator
MODE_ITER_SIMULT option PLUS_PETITE). One determines thanks to this mode the mass added on
the piston.
K
=
5
10 NR/m
m
= 200 kg/m
m
=
has
78 kg/m
The Eigen frequency of the system piston-arises immersed is thus F
= 3.018 Hz
2.2
Results of reference
Analytical
2.3 References
bibliographical
[1]
R.J GIBERT - Vibrations of Structures - Interactions with fluids. Eyrolles (1988).
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A

Code_Aster ®
Version
4.0
Titrate:
FDLV100 Piston coupled to a column of incompressible fluid
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.100-A Page:
4/6
3 Modeling
With
3.1
Characteristics of modeling
Thermal formulation planes for fluid (QUAD4 and SEG2)
Plane deformation formulation and discrete for solid (QUAD4 and SEG2)
4 elements MEDPQU4
Fluid
80 elements THPLQU4
Piston
F
K
G
E
H
D
D
I
C
· · ·
J
With
B
L
Comes out from stiffness K
4 elements of interface THPLSE2
1 element MECA_DIS_T_L
Cutting =
21 meshs QUAD4 according to the x axis
4 meshs QUAD4 according to the y axis
4 meshs SEG2 on the fluid interface/piston
1 mesh SEG2 representing the spring binding the piston to the solid mass
Boundary conditions:
DDL_IMPO: (GROUP_NO: noeupist DY: 0. )
DDL_IMPO: (GROUP_NO: embed DX: 0. DY: 0. DZ: 0.)
Name of the nodes:
GROUP_NO NOEUPIST is consisted of the ten nodes A, B, C, D, E, F,
G, H, I, J
GROUP_NO ENCASTRE is consisted of the node K
3.2
Characteristics of the grid
A number of nodes: 111 nodes
A number of meshs and types: 84 QUAD4, 5 SEG2
3.3 Functionalities
tested
Commands
Keys
AFFE_MODELE
“THERMIQUE”
“PLAN”
[U4.22.01]
CALC_MASS_AJOU
MODE_MECA
[U4.??.??]
NUME_DDL_GENE
NUME_DDL_GENE
“PLEIN”
STOCKAGE
U4.55.07]
MODE_MECA
MODE_ITER_SIMULT
“BANDE”
FREQ
U4.52.01]
concept “matr_asse_gene_r”
COMB_MATR_ASSE
COMB_R
[U4.53.01]
concept “matr_asse_gene_r”
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A

Code_Aster ®
Version
4.0
Titrate:
FDLV100 Piston coupled to a column of incompressible fluid
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.100-A Page:
5/6
4
Results of modeling A
4.1 Values
tested
Identification
Reference
Aster
% difference
(Hz)
(Hz)
Command of the clean mode I: 1
3.018
3.01854
+0.018
4.2 Remarks
Calculations of modes carried out by:
MODE_ITER_SIMULT OPTION: “PLUS_PETITE” NMAX_FREQ: 1.
4.3 Parameters
of execution
Version: 3.05.24
Machine: CRAY C98
System:
UNICOS 8.0
Obstruction memory:
8 megawords
Time CPU To use:
7.09 seconds
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A

Code_Aster ®
Version
4.0
Titrate:
FDLV100 Piston coupled to a column of incompressible fluid
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.100-A Page:
6/6
Intentionally white left page.
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A