Code_Aster
®
Version
4.0
Titrate:
FDLV106 Calculation of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-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
V8.01.106 document
FDLV106 - Calculation of added damping
in annular flow
Summary:
This test of the fluid field/structure implements the calculation of mass and damping added on one
cylindrical structure subjected to an annular flow which one supposes potential. One calculates in a first
times mass and damping added by the flow on the structure for various speeds upstream (4 m/s,
4.24 m/s and 6 m/s), this on a model 3D for the fluid and hull for the structure. The structure has one
displacement of rotation around a pivot located at the downstream end of the cylinder compared to the flow.
The determined coefficients, one assigns them to a discrete model are equivalent to 1 ddl mass-arise-shock absorber,
on which one carries out a modal analysis, in order to determine the complex Eigen frequencies of the system
for the various rates of flow:
4 m/s: damping,
4.24 m/s: critical engine failure speed, null damping,
6 m/s: negative damping, flutter.
Code_Aster
®
Version
4.0
Titrate:
FDLV106 Calculation of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A
Page:
2/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
1
Problem of reference
1.1 Geometry
Z
y
X
V
0
input
fluid
L
roll fixed intern
external cylinder
mobile in rotation around the pivot C
X
C
R
I
R
E
N
exit
N
not
swivelling
L
= 50 m
R
I
= 1 m
R
E
= 1.1 m
C
: not pivot of the external structure
1.2
Properties of materials
Fluid: density
G
= 1000 kg/m
3
(water).
Structure:
S
= 7800 kg/m
3
;
E
= 2.10
11
AP;
= 0.3 (steel).
1.3
Boundary conditions and loadings
Fluid:
·
to simulate the permanent flow, one forces on the face of input of the fluid a speed
normal of 4 m/s (by thermal analysis, one imposes a normal heat transfer rate equivalent of
4),
·
to calculate the fluid disturbance brought by the movement of the external cylinder Dirichlet in
a node of the fluid.
Structure:
one imposes on the external cylinder a displacement of the type
&
&
X
L
y Z
I
=
-
2
with the nodes of
mesh of this cylinder.
Code_Aster
®
Version
4.0
Titrate:
FDLV106 Calculation of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-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
For the calculation of the added coefficients:
it is shown [bib1] that the coefficients of mass and added depreciation depend on
permanent potential fluid speeds
as well as two fluctuating potentials
1
2
and
: these
potentials are in the case of written the rotational movement of the external cylinder around the pivot C
[bib1]:
=
=
-
+
+
=
-
=
-
+
V y
R
R
R
R
R
R
y
L
L
y
R V
R
R
R
R
R
E
E
I
I
I
E
E
I
I
0
1
2
2
2
2
2
2
0
2
2
2
2
2
sin
sin
with
X
Z
However the added modal coefficients projected on this mode of rotation are written:
(
)
(
)
M
dS
C
dS
has
I
has
I
=
=
+
1
2
1
X
N
X
N
external cylinder
external cylinder
.
.
.
that is to say:
C
V R
R
R
R
R
R
L
M
R
R
R
R
R
R
L
has
E
E
I
E
I
E
has
E
E
I
E
I
E
= -
-
+
= +
-
+
0
3
2
2
2
2
3
2
2
2
3
3
For the system with 1 degree of freedom are equivalent:
It is about a system mass-arise-shock absorber representing the rotational movement of
roll around the pivot C downstream.
C
J
·
the inertia of the mechanical system subjected to the flow is written:
J
I
M
has
= +
where
I
is the inertia of the external cylinder swivelling compared to axis Cx (cf appears below) in air.
Code_Aster
®
Version
4.0
Titrate:
FDLV106 Calculation of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A
Page:
4/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
It is shown [bib2] that this inertia is worth:
(
)
I
m
R
L
E
=
+
6 3
2
2
2
where
m
is the mass of the cylinder:
m
R E L
S
E
=
2
where
E
is the thickness of the cylinder,
L
its overall length.
S
is the density of the cylinder.
C
Z
y
X
0
X
thus
(
)
J
m
R
L
R
R
R
R
R
R
L
E
E
E
I
E
I
E
=
+
+
-
+
6 3
2
3
2
2
3
2
2
2
3
·
the damping of the mechanical system subjected to the flow is written:
= +
WITH C
has
where
With
is the damping of the mechanical system in air. Usually,
With
is equal to some
% of damping criticizes system:
With
IK
=
2
.
where
I
is the inertia of the cylinder in air calculated above and
K
the rigidity of the spring at the point of
swivelling
C
. Reduced damping is taken
equal to 1%.
Thus, the total damping of the system under flow is written:
=
-
-
+
IK
V
R
R
R
R
R
R
L
E
E
I
E
I
E
0
3
2
2
2
2
·
the rigidity of the mechanical system subjected to flow is written:
K
K
K
has
= +
where
K
is the rigidity of the spring in air.
K
has
is the rigidity added by the flow. One shows [bib1]
that the aforementioned is null on this mode of rotation.
K
has
=
0
Thus the overall rigidity of the system is independent the rate of flow.
K
K
=
·
One calculates then the complex modes of this mechanical system under flow (vibrations
free deadened):
J
C
+
+
=
0
Code_Aster
®
Version
4.0
Titrate:
FDLV106 Calculation of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A
Page:
5/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
The complex Eigen frequencies of this system are written [bib3]:
1
2
2
1
or
R
I
= -
±
-
with
=
=
=
+
2 J
K
J
K
I
M
has
and
: reduced damping of the system
: own pulsation.
·
Numerical applications:
One made three calculations of damping added agent to three rates of flow which
three involve behavior vibratory of the structure:
speed to 4 m/s
speed to 4.24 m/s
speed to 6 m/s
The values of the mechanical system are:
E
m
L
m
R
m
R
m
I
kg m
With
NR m rad S
K
NR m rad
I
=
=
=
=
=
=
=
-
-
-
2 10
50
1
1 1
4 5 10
4 24 10
10
2
2
7
2
8
1
13
1
.
,
.
.
.
.
The added mass and damping brought by the flow are worth:
I
kg m
has
=
1 66 10
10
2
.
(independent of the value rate of flow)
According to the speed of input of the fluid, one a:
V
m S
C
NR m rad S
V
m S
C
NR m rad S
V
m S
C
NR m rad S
has
has
has
0
8
1
0
8
1
0
8
1
4
4 00 10
4 24
4 24 10
6
5 94 10
=
= -
=
= -
=
= -
-
-
-
/
.
.
.
/
.
.
/
.
.
Depreciation of the fluid system/structure is written:
·
with
V
m S
NR m rad S
0
8
1
4
0 24 10
=
=
-
/
:
.
.
The flow does not amplify the vibrations. Structural damping interns is
sufficient important to dissipate the energy brought by the flow to the structure.
The system is still deadened.
·
with
V
m S
0
4 24
0
=
.
/
:
(
)
rate of flow critical
The damping of the system is cancelled.
·
with
V
m S
NR m rad S
0
8
1
6
15 10
=
= -
-
/
:
.
.
(
)
the flow amplifies the vibrations
The damping of the system at this last speed is negative: the system enters then
in instability of flutter.
Code_Aster
®
Version
4.0
Titrate:
FDLV106 Calculation of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A
Page:
6/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
Corresponding reduced depreciation is written:
(
)
V
m S
V
m S
V
m S
0
4
0
5
0
4
4
11 10
4 24
0
1.380 10
6
6.6 10
=
=
=
=
=
=
= -
-
-
-
/
.
.
/
(
)
.
/
.
in theory
with the round-off errors
The own pulsation remains as for it unchanged:
=
12 5
. Hz
.
2.2
Results of reference
Analytical result.
2.3 References
bibliographical
[1]
ROUSSEAU G., LUU H.T.: Mass, damping and stiffness added for a structure
vibrating placed in a potential flow - Bibliography and establishment in
Code_Aster
- HP-61/95/064
[2]
BLEVINS R.D: Formulated for natural frequency and shape mode. ED. Krieger 1984
[3]
SELIGMANN D, MICHEL R: Algorithms of resolution for the quadratic problem
[R5.01.02], Manual of Reference
Aster
.
Code_Aster
®
Version
4.0
Titrate:
FDLV106 Calculation of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A
Page:
7/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
3 Modeling
With
3.1
Characteristics of modeling
For the system 3D on which one calculates the added coefficients:
For the fluid:
480 meshs QUAD4
elements of hulls MEDKQU4
For the solid:
480 meshs QUAD4
elements thermics THER_FACE4
on cylindrical surfaces
360 meshs QUAD4
thermal elements THER_FACE4
on the faces of input and exit of fluid volume
720 meshs HEXA8
thermal elements THER_HEXA8
in fluid annular volume
3.2 Functionalities
tested
Controls
Keys
CALC_MATR_AJOU
OPTION
“MASS_AJOU”
“AMOR_AJOU”
[U4.55.10]
POTENTIAL
MODE_ITER_INV
CALC_FREQ
FREQ
[U4.52.01]
MODE_ITER_SIMULT
APPROACH
“REAL”
[U4.52.02]
Code_Aster
®
Version
4.0
Titrate:
FDLV106 Calculation of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A
Page:
8/8
Manual of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
4
Results of modeling A
4.1 Values
tested
Identification
Reference
Aster
% difference
N°1 mode
with
V
m S
0
4
=
/
frequency
reduced damping
12.5 Hz
1.1 10
4
12.388
1.095 10
4
0.889
0.445
N°1 mode
with
V
m S
0
4 24
=
.
/
frequency
reduced damping
12.5 Hz
1.380 10
5
12.388
1.392 10
5
0.889
+0.895
N°1 mode
with
V
m S
0
6
=
/
frequency
reduced damping
12.5 Hz
6.60 10
4
12.388
6.649 10
4
0.889
0.740
4.2 Parameters
of execution
Version: 3.06.08
Machine: CRAY C98
Overall dimension memory:
80 MW
Time CPU to use:
51.7 seconds
5
Summary of the results
The computational tool of damping under flow (potential assumption) was validated on the mode of
rotation of a cylindrical structure subjected to an annular flow. It is however necessary to note [bib1]
that the very good agreement enters the semi-analytical model suggested for comparison and calculation
numerical is obtained only if the cylinder is sufficiently long.
Indeed, the semi-analytical model is in fact only one approximate solution of the problem arising.