Code_Aster ®
Version
7.2
Titrate:
FDLV107 - Rigidities added under annular flow

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.107-A Page:
1/6

Organization (S): EDF-R & D/AMA, EDF-DPN/UTO

Handbook 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 inflection knee joint-knee joint and subjected to annular flows the different ones
speeds.

One calculates the rigidity added (function speed) on the first mode of inflection of the cylinder. One checks
decrease of the Eigen frequency of the mode, up to zero value for a critical engine failure speed flow of
fluid.

Handbook of Validation
V8.01 booklet: Fluid HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
FDLV107 - Rigidities added under annular flow

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.107-A Page:
2/6

1
Problem of reference

1.1 Geometry


fluid
U

U
Z
R

E
R I
y
X
With
B
L


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: IH = 1 m;
·
external radius: Re = 1.05 m;
·
length: L = 100 Mr.

1.2
Properties of materials

Structure:

Young modulus: E = 2.1011 Pa;
Poisson's ratio: = 0.3;
density: S = 7800 kg/m3.

Fluid:

density: = 1000 kg/m3.

1.3
Boundary conditions and loadings

Structure:

blocking of the nodes of the interior cylinder;
knee 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.
Handbook of Validation
V8.01 booklet: Fluid HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
FDLV107 - Rigidities added under annular flow

Date:
01/03/04
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NR. GREFFET, G. Key ROUSSEAU
:
V8.01.107-A Page:
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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]:

2
2

R

R
y L
+

E
I
2



1 (R, y)
(
)
=
R +
sin sin
2
2

R - R
R
L
E
I


2
2

R

R
y L
+

E
I
2


2 (R, y)
(
)
=
R +
sin cos

2
2
R - R
R
L

E
I

The rigidity added on the first mode of inflection of the external cylinder considered as a beam
knee joint-knee joint is written [bib1]:

V 2 3R3
R2
K
0
E
I


With = -

2 (
R +
R2 - R
E
2


E
I) L
Re

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 = 2.10­3 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
D 3rd
K = 384EI
= 1.649 10-3 m
5L3
with
I =
8


thus K = 2.533.104
.
NR/m

The equivalent system coupled with the flow is represented by the following diagram:

m
K

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 c0 the critical engine failure speed of flow for which rigidity K is cancelled:

2nd
R - I
R lK
0
V, K + K A (0
V)
2
(2 2)
= 0 with 0
V
=

C
C
C
2


I
R
3 3

+

E
R
E
R



E
R
Handbook of Validation
V8.01 booklet: Fluid HT-66/04/005/A

Code_Aster ®
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7.2
Titrate:
FDLV107 - Rigidities added under annular flow

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.107-A Page:
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then it is shown that:

(V
2
0) = ()
0
1 - X
E


where one posed:

E
(0)
K
=

(
rest

with

fluid

in

system



clean

pulsation
)
M + MR. A

V
X = 0
(

of

reduced

speed
flow)
V0c

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
Mr. A (kg)
3.486E6 3.486E6 3.486E6 3.486E6
3.486E6
KA (NR/m)
­ 876.5 ­ 7888.50
­ 14023.95
­ 16968.98
­ 25330
Mtotal (kg)
3.491E+6 =
=
= =
Ktotal (NR/m)
24453.5 17441.5 11306.05
8361.00 0.
F (V
1.318 1.112 0.896 0.772
0.
0) X 10­2 (Hz)

2.3
Uncertainty on the solution

Semi-analytical solution.

2.4 References
bibliographical

[1]
ROUSSEAU G., LUU H.T. - Masse, damping and stiffness added for a structure
vibrating placed in a potential flow - internal Note EDF/DER, HP-61/95/064/A
(1995).

Handbook of Validation
V8.01 booklet: Fluid HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
FDLV107 - Rigidities added under annular flow

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.107-A Page:
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3 Modeling
With

3.1
Characteristics of modeling

For the geometry on which one evaluates the added coefficients:

Fluid: 1800 thermal elements THER_HEXA8 1560 thermal 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 grid

Grid 1 (hulls cylinders):
1800 meshs HEXA8 1560 meshs QUAD4


Grid 2 (discrete system):
1 mesh SEG2 1 nets POI1

3.3 Functionalities
tested

Commands


CALC_MATR_AJOU OPTION
“RIGI_AJOU”


POTENTIEL



CHAM_NO




4
Results of modeling A

4.1 Values
tested

Identification Reference
Aster
% difference
X 10­2
X 10­2
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


Handbook of Validation
V8.01 booklet: Fluid HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
FDLV107 - Rigidities added under annular flow

Date:
01/03/04
Author (S):
NR. GREFFET, G. Key ROUSSEAU
:
V8.01.107-A Page:
6/6

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 compčte 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.

Handbook of Validation
V8.01 booklet: Fluid HT-66/04/005/A

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