Code_Aster
®
Version
7.2
Titrate:
FDLV107 - Rigidities added under annular flow
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.107-A
Page:
1/6
Manual of Validation
V8.01 booklet: Fluid HT-66/04/005/A
Organization (S):
EDF-R & D/AMA, EDF-DPN/UTO
Manual of Validation
V8.01 booklet: Fluid
Document: V8.01.107
FDLV107 - Rigidities added under flow
annular
Summary:
This test of the field of the interaction fluid-structure, validates the calculation of rigidity added on a circular cylinder
excited on its first mode of bending ball joint-ball joint and subjected to annular flows the different ones
speeds.
One calculates the rigidity added (function speed) on the first mode of bending of the cylinder. One checks
decrease of the Eigen frequency of the mode, up to zero reference for a critical engine failure speed flow of
fluid.
Code_Aster
®
Version
7.2
Titrate:
FDLV107 - Rigidities added under annular flow
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.107-A
Page:
2/6
Manual of Validation
V8.01 booklet: Fluid HT-66/04/005/A
1
Problem of reference
1.1 Geometry
Z
y
X
fluid
L
R
I
U
R
E
U
B
With
The system represented on the diagram above is composed of two coaxial cylinders and a fluid
in flow at the speed U in annular space enters the two cylinders. Dimensions are:
·
interior radius:
R
I
= 1 m;
·
external radius:
R
E
= 1.05 m;
·
length:
L
= 100 Mr.
1.2
Properties of materials
Structure:
Young modulus: E = 2.10
11
AP;
Poisson's ratio:
= 0.3;
density:
S
= 7800 kg/m
3
.
Fluid:
density:
= 1000 kg/m
3.
1.3
Boundary conditions and loadings
Structure:
blocking of the nodes of the interior cylinder;
ball joint at points A and B of the external cylinder.
Fluid:
one imposes various speeds in input of the fluid field with equal normal heat fluxes
to 4 m/s, 0.5 m/s, 1.5 m/s, 2 m/s, 2.2 m/s and 2.688 m/s (critical engine failure speed).
1.4 Conditions
initial
Without object.
Code_Aster
®
Version
7.2
Titrate:
FDLV107 - Rigidities added under annular flow
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.107-A
Page:
3/6
Manual of Validation
V8.01 booklet: Fluid HT-66/04/005/A
2
Reference solution
2.1
Method of calculation used for the reference solution
The reference solution is an approximate analytical solution. Analytical fluctuating potentials
approached to calculate added rigidity are written [bib1]:
(
)
(
)
(
)
(
)
1
2
2
2
2
2
2
2
2
2
2
2
R
y
R
R
R
R Rr
y
L
R
y
R
R
R
R Rr
y
L
E
E
I
I
L
E
E
I
I
L
,
sin sin
,
sin cos
=
-
+
+
=
-
+
+
The rigidity added on the first mode of bending of the external cylinder considered as a beam
ball joint-ball joint is written [bib1]:
(
)
K
V
R
R
R L R
R
R
With
E
E
I
E
I
E
= -
-
+
2
02 3 3
2
2
2
This rigidity, calculated on a cylindrical geometry, is then assigned to a model with a degree of
freedom are equivalent.
The system with 1 ddl equivalent is a system mass-arises equivalent to which one affects a mass
equalize with the mass of the system increased by the mass added by the fluid and a rigidity equalizes with
rigidity of the system increased by the rigidity added by the flow for various speeds.
The mass of the system in air is of:
M = 10292 kg
for an external cylindrical hull thickness:
C = 2.10
3
m
For equivalent rigidity in air of the system “external hull”, one takes the rigidity of a subjected beam
with a force distributed over all its length:
F
K
= 384EI
5L
3
with
I
=
D
3
E
8
= 1.649 10
- 3
m
thus
K
NR m
= 2.533 10
4
.
/
The equivalent system coupled with the flow is represented by the following diagram:
K
m
with
m
M
M
K
K K
With
With
=
+
= +
The own pulsation of the coupled system evolves/moves according to the square rate of flow. If one
call
V
C
0
the critical engine failure speed flow for which rigidity
K
cancel yourself:
()
(
)
3
3
2
2
2
2
0
0
0
2
0
,
E
E
I
E
I
E
With
R
R
R
R
lK
R
R
V
V
K
K
V
C
C
C
+
-
=
=
+
with
Code_Aster
®
Version
7.2
Titrate:
FDLV107 - Rigidities added under annular flow
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.107-A
Page:
4/6
Manual of Validation
V8.01 booklet: Fluid HT-66/04/005/A
then it is shown that:
()
()
V
X
E
0
2
0
1
=
-
where one posed:
()
(
)
(
)
flow
of
reduced
speed
rest
with
fluid
in
system
clean
pulsation
=
C
V
V
X
M
M
K
With
E
0
0
0
+
=
The pulsation of the fluid at rest is worth:
= 0.085 rad/S.
2.2
Results of reference
One calculates for various rates of flow the Eigen frequency of the system.
(
)
V m S
0
/
0.5.1.5 2. 2.2
2.688
M
With
(kg)
3.486E6 3.486E6 3.486E6 3.486E6
3.486E6
K
With
(NR/m)
876.5 7888.50
14023.95
16968.98
25330
M
total
(kg)
3.491E+6 =
=
= =
K
total
(NR/m)
24453.5 17441.5 11306.05
8361.00 0.
()
F V
0
X 10
2
(Hz)
1.318 1.112 0.896 0.772
0.
2.3
Uncertainty on the solution
Semi-analytical solution.
2.4 References
bibliographical
[1]
ROUSSEAU G., LUU H.T. - Mass, damping and stiffness added for a structure
vibrating placed in a potential flow - internal Note EDF/DER, HP-61/95/064/A
(1995).
Code_Aster
®
Version
7.2
Titrate:
FDLV107 - Rigidities added under annular flow
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.107-A
Page:
5/6
Manual of Validation
V8.01 booklet: Fluid HT-66/04/005/A
3 Modeling
With
3.1
Characteristics of modeling
For the geometry on which one evaluates the added coefficients:
Fluid: 1800 elements
THER_HEXA8
thermal thermics 1560 elements
THER_FACE4
of interface;
Structure: 1200 elements of hull
QUAD4
modeling
“DKT”.
For the system with 1 ddl equivalent: 2 discrete finite elements modeling
“DIS_T'
.
3.2
Characteristics of the mesh
Mesh 1 (hulls cylinders):
1800 meshs HEXA8 1560 meshs QUAD4
Mesh 2 (discrete system):
1 mesh SEG2 1 nets POI1
3.3 Functionalities
tested
Controls
CALC_MATR_AJOU OPTION
“RIGI_AJOU”
POTENTIAL
CHAM_NO
4
Results of modeling A
4.1 Values
tested
Identification Reference
X 10
2
Aster
X 10
2
% difference
frequency to 0.5 m/s
1.318
1.332
+1
frequency to 1.5 m/s
1.112
1.130
+1.6
frequency to 2 m/s
0.896
0.917
+2.3
frequency to 2.2 m/s
0.772
0.795
+2.9
frequency to 2.688 m/s
0.
0.192
Code_Aster
®
Version
7.2
Titrate:
FDLV107 - Rigidities added under annular flow
Date:
01/03/04
Author (S):
NR. GREFFET, G. ROUSSEAU
Key
:
V8.01.107-A
Page:
6/6
Manual of Validation
V8.01 booklet: Fluid HT-66/04/005/A
5
Summary of the results
The variation on the Eigen frequencies increases owing to the fact that when one is close the critical engine failure speed
buckling, the rigidity of the equivalent system must tend towards zero. However, with the errors
of round (since one assigns “to the hand” the values of rigidity added calculated by the operator to one
discrete model) do not make it possible to obtain an own pulsation of the system null at the speed
critical.
Variations on the added values of rigidity also remain because the reference solution is built
on an semi-analytical solution which leaves the approximation according to which the separation of the variables
between the dimension y and the co-ordinates orthoradiales is possible. It will be noticed that the selected potentials
to describe the disturbance generated by the vibration of the structure in the fluid do not check
the equation of Laplace compète but only in one transverse section of the fluid in co-ordinates
orthoradiales. This approximation carried out on the reference solution can explain certain variations
with numerical calculation.