Code_Aster ®
Version
4.0
Titrate:
FDLV106 Calcul of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A Page:
1/8
Organization (S): EDF/EP/AMV
Handbook of Validation
V8.01 booklet: Fluid
V8.01.106 document
FDLV106 - Calcul 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-damping device,
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, undulation.
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
Code_Aster ®
Version
4.0
Titrate:
FDLV106 Calcul of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A Page:
2/8
1
Problem of reference
1.1 Geometry
L
V0
Z
input
N
IH
N left
y
R
X
E
C
X
not
swivelling
roll fixed intern
fluid
external cylinder
mobile in rotation around the pivot C
L = 50 m
IH = 1 m
Re = 1.1 m
C: not pivot of the external structure
1.2
Properties of materials
Fluid: density G = 1000 kg/m3 (water).
Structure: S = 7800 kg/m3; E = 2.1011 Pa; = 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 flow equivalent of
4),
· to calculate the fluid disturbance brought by the movement of the external cylinder Dirichlet in
a node of the fluid.
Structure:
&
L
&
one imposes on the external cylinder a displacement of the type X =
- y Z
I
with the nodes of
2
grid of this cylinder.
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
Code_Aster ®
Version
4.0
Titrate:
FDLV106 Calcul of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A Page:
3/8
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 and 2: these
potentials are in the case of written the rotational movement of the external cylinder around the pivot C
[bib1]:
= V y
0
R2
R2
L
L
1 =
E
R + I y +
sin
with X
y Z
2
2
I
R
2
2
E - IH
R
=
-
R2 V
R2
0
2 =
E
R + I sin
R2
2
E - IH
R
However the added modal coefficients projected on this mode of rotation are written:
M =
X. N dS
has
1
I
external cylinder
C =
2 +
.
has
(
1) (Xi. N) dS
external cylinder
that is to say:
V Re
2
0
3
R
C = -
R
I
+
L2
has
2
E
Re2 - R
R
I
E
R3
R2 L
3
M
E
= +
R
I
+
has
2
2
E
R - R
R 3
E
I
E
For the system with 1 degree of freedom are equivalent:
It is about a system mass-arise-damping device representing the rotational movement of
roll around the pivot C downstream.
J
C
· the inertia of the mechanical system subjected to the flow is written: J = I + My
where I is the inertia of the external cylinder swivelling compared to axis Cx (cf appears below) in air.
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
Code_Aster ®
Version
4.0
Titrate:
FDLV106 Calcul of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A Page:
4/8
It is shown [bib2] that this inertia is worth:
m
I =
(3R2 +2 L2
E
)
6
where m is the mass of the cylinder:
m = 2 R E L
S
E
where E is the thickness of the cylinder, L its overall length.
S is the density of the cylinder.
Z
C
0
y
X
X
m
R3
R2 L3
2
2
E
I
thus J =
(3R +2 L +
+
E
)
R
2
2
E
6
R - R
R 3
E
I
E
· the damping of the mechanical system subjected to the flow is written: = A + Ca
where A is the damping of the mechanical system in air. Usually, A is equal to some
% of damping criticizes system: TO = 2 IK.
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. One takes reduced damping equal to 1%.
Thus, the total damping of the system under flow is written:
R3
2
E
R
= IK - V
R
I
2
0
2
2
E +
L
Re - IH
Re
· the rigidity of the mechanical system subjected to flow is written: K = K + Ka
where K is the rigidity of the spring in air. Ka is the rigidity added by the flow. One shows [bib1]
that this one 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
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
Code_Aster ®
Version
4.0
Titrate:
FDLV106 Calcul of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A Page:
5/8
The complex Eigen frequencies of this system are written [bib3]:
R
2
= - ± -
1
2
I
1
or
K
K
with =
and
=
=
2
J
J
I + My
: reduced damping of the system
: own pulsation.
· Numerical applications:
One made three calculations of damping added correspondent 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 = 2 1
. 0 2 m
L = 50 m
R = 1 m
R2 = 11
, m
I
I = 4 5
. 107 kg m2
8
-
To = 4 2
. 4 10 N.m rad S
1
13
-
K = 10
N.m rad 1
The added mass and damping brought by the flow are worth:
I = 16
. 6 1010 kg m2
has
(independent of the value rate of flow)
According to the speed of input of the fluid, one a:
-
V = 4 m/s
C
8
= - 4 00
.
10 N.m rad S
1
0
has
-
V = 4 2
. 4 m/s
C
8
= - 4 24
.
10 NR. m rad S
1
0
has
-
V = 6 m/s
C
8
= - 594
.
10 N.m rad S
1
0
has
Depreciation of the fluid system/structure is written:
·
8
1
with V = 4 m S
=
-
/
:
0 24
.
10 N.m rad S
0
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 criticizes)
The damping of the system is cancelled.
·
8
1
with V = 6 m S
= -
-
/
:
15
. 10 N.m rad S
0
(the flow amplifies the vibrations)
The damping of the system at this last speed is negative: the system enters then
in instability of undulation.
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
Code_Aster ®
Version
4.0
Titrate:
FDLV106 Calcul of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A Page:
6/8
Corresponding reduced depreciation is written:
-
V = 4 m/s
4
=
0
11
. 10
= 0 (in theory)
V = 4 2
. 4 m/s
0
5
=
-
1 380
.
10 (with the round-off errors)
-
V = 6 m/s
4
= -
0
6.6
. 10
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 - Bibliographie 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], Manuel de Référence Aster.
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
Code_Aster ®
Version
4.0
Titrate:
FDLV106 Calcul of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A Page:
7/8
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 output of fluid volume
720 meshs HEXA8
thermal elements THER_HEXA8
in fluid annular volume
3.2 Functionalities
tested
Commands
Keys
CALC_MATR_AJOU
OPTION
“MASS_AJOU”
[U4.55.10]
“AMOR_AJOU”
POTENTIEL
MODE_ITER_INV
CALC_FREQ
FREQ
[U4.52.01]
MODE_ITER_SIMULT
APPROCHE
“REEL”
[U4.52.02]
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A
Code_Aster ®
Version
4.0
Titrate:
FDLV106 Calcul of damping added in annular flow
Date:
12/01/98
Author (S):
G. ROUSSEAU
Key:
V8.01.106-A Page:
8/8
4
Results of modeling A
4.1 Values
tested
Identification
Reference
Aster
% difference
N°1 mode
with V = m/s
12.5 Hz
12.388
0.889
0
4
frequency
0.445
reduced damping
1.1 104
1.095 104
N°1 mode
with V =.
m/s
12.5 Hz
12.388
0.889
0
4 24
frequency
+0.895
reduced damping
1.380 105
1.392 105
N°1 mode
with V = m/s
12.5 Hz
12.388
0.889
0
6
frequency
0.740
reduced damping
6.60 104
6.649 104
4.2 Parameters
of execution
Version: 3.06.08
Machine: CRAY C98
Obstruction 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.
Handbook of Validation
V8.01 booklet: Fluid
HP-51/96/031 - Ind A