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Code_Aster
®
Version
5.0
Titrate:
Coupling fluid-structure for the tubular structures and the hulls
Date:
23/09/02
Author (S):
T. KESTENS, Mr. LAINET
Key
:
R4.07.04-B
Page
:
1/34
Manual of Reference
R4.07 booklet: Coupling fluid-structure
HI-86/02/008/A
Organization (S):
EDF/MFTT, CS IF















Manual of Reference
R4.07 booklet: Coupling fluid-structure
Document: R4.07.04



Coupling fluid-structure for the structures
tubular and coaxial hulls




Summary:

This document describes the various models of coupling fluid-structure available starting from the operator
CALC_FLUI_STRU
. These models make it possible to simulate the forces of coupling fluid-rubber band in
following configurations:
·
beams of tubes under transverse flow (primarily, tubes of Steam generator),
·
passage stem of control/plate of housing (exclusively for the control rods),
·
coaxial cylindrical hulls under annular flow (for example, space ferments/envelope of
core),
·
beams of tubes under axial flow (for example, fuel assemblies).

For each configuration, the model of forces fluid-rubber bands is initially presented. The resolution of
modal problem is then described. The methods of resolution employed integrate specificities of
various models of forces fluid-rubber bands.
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Coupling fluid-structure for the tubular structures and the hulls
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1 Presentation
general
1.1 Recalls
The dynamic fluid forces being exerted on a structure moving can be classified in
two categories:
·
forces independent of the movement of the structure, at least in the range of small
displacements; they are mainly random forces generated by turbulence or
diphasic nature of the flow,
·
fluid forces dependant on the movement of the structure, known as “forces
fluid-rubber bands ", persons in charge for the coupling fluid-structure.
In this document, one is interested in the four models of forces fluid-rubber bands integrated in
the operator
CALC_FLUI_STRU
. The data-processing aspects related to the integration of these models made
the object of notes of specifications [bib1], [bib2].
1.2 Modeling
The dependence of the forces fluid-rubber bands with respect to the movement of the structure is translated, in
range of the low amplitudes, by a matrix of transfer enters the force fluid-rubber band and it
vector displacement. The projection of the equation of the movement of the system coupled fluid-structure
on the basis of modal structure alone is written, in the field of Laplace:
[]
[] []
()
[
]
{
}
() ()
T
ij
II
II
II
S
U
S
S
Q
Q
,
B
K
C
M
=
-
+
+
2
éq
1.2-1
where
[] [] []
II
II
II
K
C
,
M
and
the diagonal matrices of mass indicate respectively,
of structural damping and stiffness in air;
()
Q
indicate the vector of the displacements generalized in air;
()
T
Q
indicate the vector of the generalized random excitations (forces independent of
movement);
and
()
[
]
S
U
ij
,
B
represent the matrix of transfer of the forces fluid-rubber bands, projected on the basis
modal of the structure alone. This matrix depends in particular on
U
, characteristic speed
flow, as well as frequency of the movement via the variable of
Laplace
S
.
A priori,
()
[
]
S
U
ij
,
B
is an unspecified matrix whose extradiagonaux terms, if they are not null,
introduce a coupling between modes. In addition, terms of
()
[
]
S
U
ij
,
B
evolve/move in manner not
linear with the complex frequency
S
.
With each model of force fluid-rubber band is associated a specific matrix of transfer.
In all the cases, the formulation of the modal problem under flow can be characterized
by the relation [éq 1.2-1].
For the various types of configurations being able to be simulated using the operator
CALC_FLUI_STRU
, the representations of the matrices of transfer of the forces fluid-rubber bands are
clarified in the continuation of this document.
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Coupling fluid-structure for the tubular structures and the hulls
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2
Excitation fluid-rubber band acting on the beams of
tubes under transverse flow (primarily for
tubes of Steam Generator)
The integration of this model of excitation fluid-rubber band in Code_Aster was approached in the note
specifications [bib1]. The note of principle of software FLUSTRU [bib3] constitutes documentation
theoretical of reference. The main main ones of modeling are pointed out hereafter.
2.1
Description of the studied configuration
One considers a beam of tubes excited by a transverse external flow. Physically, them
transverse external flows tend to destabilize the mechanical system when
rate of the flow increases.
An industrial case to treat in practice is that of the vibrations of the tubes of steam generators. On
this component, the transverse flows are observed in the input area of the beam of
tubes (monophasic flow liquidates), and in the bent part of the tube out of U (flow
diphasic) [Figure 2.1-a].
Excited area
by
flow
monophasic
Excited area
by
flow
diphasic
Primary fluid exit
Primary fluid input
Food water
Water return
Tubular plate
Plate spacer
Beam of tubes
Exit vapor
Separators
Appear 2.1-a: Diagram of steam generator
From the point of view of the coupling fluid-rubber band, the study of the dynamic behavior of different
tubes of a beam subjected to a transverse flow is brought back to the study of an equivalent tube;
the definition of the equivalent tube depends on the environment of the tube to treat.
When the tube considered has vibratory characteristics appreciably different from those
of its neighbors, this tube can be compared to only one tube, vibrating in the middle of a beam of tubes
rigid.
In the contrary case, the problem is more complex because one must consider a mechanical system
with coupling between tubes of the beam and thus comprising a great number of degrees of freedom.
To treat this kind of configuration, a model was developed at Department TTA, “the model
total " [bib7]; this model allows the definition of a system equivalent to a degree of freedom, which
represent the complete coupled system.
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Titrate:
Coupling fluid-structure for the tubular structures and the hulls
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Author (S):
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Key
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The approach adopted to lead calculations can be summarized in the following way [Figure 2.1-b]:
·
Taking into account the telegraphic nature of the structures studied, the calculation of the coupling
fluid-rubber band in the beam of tubes is carried out by describing the tube by its X-coordinate
curvilinear.
·
In calculation, the fluid environment of the tube is characterized, at the same time by the properties
physiques of the fluid circulating inside the tube (fluid primary education), and by those of the fluid
circulating outside the tube (fluid exiting secondary). These physical properties, such
that the density, can vary along the tube, according to the curvilinear X-coordinate.
·
The rate of flow taken into account for the calculation of coupling fluid-rubber band is
component, normal with the tube in the plan of the tube, the speed of the secondary fluid. This
speed can vary along the tube.
·
In order to be able to take into account the various possible types of excitation, several areas
of excitation can be defined along the structure. In the case of the generator of
vapor, for example, one may find it beneficial to distinguish, on the one hand the areas where the excitation is exerted
by a fluid in a monophasic state, which is located in foot of tube, and in addition, the area where
the excitation is diphasic atstrong rate of vacuum, localized in the bent part of the tube.
·
The calculation of coupling is carried out starting from the mechanical characteristics of the structure in
“fluid at rest”. The forces fluid-rubber bands of coupling are estimated from
adimensional correlations which are obtained on analytical experiments in
similarity. On each area of excitation, one can thus apply the adequate correlations;
the areas of excitation must be disjoined.
0
X
has X E D E the fib Re N E U tre D U you B E
(B C M is S.E has. C U rv ilig N E)
S U p p O rts
Z O N E 2
Z O N E 3
Z O N E 1
Appear 2.1-b: Representation of the configuration to be studied
For this configuration of coupling fluid-rubber band, the following notations will be used:

L
Overall length of the tube
L
K
Length of the area
K
D
E
Diameter external of the tube
I
D
Internal diameter of the tube
I
Modal deformation of mode I
()
X
E
Density of the external fluid to the curvilinear X-coordinate
X
()
X
I
Density of the fluid interns with the curvilinear X-coordinate
X
T
Density of the tube (structure alone)
()
X
eq
Density equivalent to the curvilinear X-coordinate
X
U
Speed of the external fluid specified by the user in the operator
DEFI_FLUI_STRU
()
X
V
Speed of the external fluid to the curvilinear X-coordinate
X
()
X
V
K
Speed of the external fluid to the curvilinear X-coordinate
X
(area of excitation
K
) challenge I
L
D it D
U
T of
yarn D
it
ifié E
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Code_Aster
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Titrate:
Coupling fluid-structure for the tubular structures and the hulls
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Key
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HI-86/02/008/A
K
) defined by the product of
U
and of a profile speed specified by
the user in the operator
DEFI_FLUI_STRU
K
U
Mean velocity of the external fluid calculated from
()
V X
K
for
area of excitation
U
Average speeds
U
K
on all the areas of excitation

2.2
Stages of calculation
·
The first stage of calculation consists in calculating the structural features in “fluid with
rest ". One proceeds by considering an equivalent mass of the tube; this equivalent mass
gather, on the one hand the mass of the tube alone, and on the other hand the masses added by the fluids
intern and external.
An equivalent density is thus defined along the tube according to the X-coordinate
curvilinear
X
by the expression:
(
)
(
)
[
]
D
(X).
D
D
.
D
(X).
D
D
(X)
eq
E
I
E
T
I
I
I
E
eq
2
2
2
2
2
2
1
+
-
+
-
=
éq
2.2-1
with
D
.
.C
D
E
m
eq
2
2
2
=
éq 2.2-2
In the equation [éq 2.2-1], the term
D
(X).
eq
E
2
represent the mass added by the external fluid.
This term depends, via the parameter
m
C
,
arrangement of the beam of tubes
(not square or triangular), and of the containment of the beam (not reduced). For calculations of
coupling fluid-rubber band of the beams of tubes subjected to a transverse flow, one uses
usually, to estimate the coefficient
m
C
, of the given analytical expressions from
experimental results.
The whole of the data necessary to the estimate of the coefficient
m
C
is collected by the operator
DEFI_FLUI_STRU
.
·
Knowing the equivalent density of the tube, the elementary matrices of mass and of
stiffness out of water at rest are then calculated by means of the profile of density
equivalent, by the operator
CALC_MATR_ELEM
; the options are used
MASS_FLUI_STRU
and
RIGI_FLUI_STRU
. The operator
MODE_ITER_SIMULT
allows, after assembly of the matrices
elementary, to directly calculate the modes out of water at rest of the studied structure.
·
The forces fluid-rubber bands of coupling are calculated by the operator
CALC_FLUI_STRU
to leave
adimensional correlations established on analytical models in similarity. These
forces of coupling,
()
[
]
S
U
ij
,
B
, dependant on the movement of the structure are then taken
in account in the general equation of the movement [éq 1.2-1] to calculate the characteristics of
system coupled flow-structure for a given speed of flow.
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2.3
Form of the matrix of transfer of the forces fluid-rubber bands
In the case of beams of tubes excited by a transverse flow, the forces fluid-rubber bands
of coupling are forces distributed along the structure. They are characterized by
linear adimensional coefficients of added damping and stiffness, named
respectively
Cd
and
Ck
. The expression of the coefficients of the fluid matrix of transfer of the forces
rubber bands projected on the basis of modal structure in “fluid at rest” is then the following one:
()
() ()
(
) ()
() () () ()
ij
I
R
E
L
L
I
R
E
E
ij
dx
X
X, S
Ck
X
V
X
S
dx
X
S
X
Cd
D
X
V
X
U, S










+




=
2
2
2
2
1
2
1
,
B
éq
2.3-1
Dependence of the coefficients
Cd
and
Ck
with respect to the movement the structure and speed of
the flow of the fluid is represented by their evolution according to the complex reduced frequency
S
R
,
defined by:
U
sD
S
R
=
éq 2.3-2
The expression [éq 2.3-1] shows that one retains a diagonal matrix of transfer. That implies that:
·
the various clean modes of the structure are rather distant from/to each other so that
one can suppose that there is not coupling between modes.
·
the modal deformations of the structure in “fluid at rest” are not disturbed by the setting
in flow of the fluid.
These two assumptions could be checked in experiments on the beams of tubes subjected to one
transverse flow.
In practice, taking into account the various areas of excitation taken into account along the structure,
the diagonal coefficients of the matrix of efforts fluid-rubber bands projected on modal basis are written
:
()
() ()
()
() ()
()


















+










=
K
I
K
E
K
K
E
L
I
K
E
K
E
K
E
L
II
dx
X
U
U
U
sd
Ck
X
V
X
S
dx
X
U
U
U
sd
Cd
D
X
V
X
U, S
K
K
2
2
2
2
1
2
1
B
éq
2.3-3
where
K
Cd
and
K
Ck
indicate respectively the adimensional coefficients of coupling,
of damping and stiffness, retained for the area of excitation
K
. Fluid speed
U
U
U
K
intervening in the reduced frequency complex in argument of the coefficients of coupling corresponds
at the mean velocity on the area of excitation
K
, after renormalization of the profile
()
X
V
K
, so that
its average on all the areas of excitation is worth
U
.
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2.4
Resolution of the modal problem under flow
In the configuration “Beam of tubes subjected to a transverse flow”, the problem is solved
on the modal basis characterizing the structure in “fluid at rest”.
Generally, the characteristics of the system coupled flow-structure are obtained
by seeking the solutions of the equation:
[]
[] []
()
[
]
{
}
() ()
0
2
=
-
+
+
Q
,
B
K
C
M
S
U
S
S
II
II
II
II
éq
2.4-1
where
[] [] []
II
II
II
K
C
,
M
and
the diagonal matrices of mass indicate respectively,
of damping and stiffness structural features in “fluid at rest”;
()
Q
indicate the vector of the displacements generalized in “fluid at rest”.
As the matrix of efforts fluid-rubber bands retained is diagonal, and that modal deformations
are supposed not to be modified under flow, the problem of coupling fluid-rubber band
bring back to the resolution of
NR
scalar problems,
NR
indicating the number of modes taken into account
in the modal base.
For each mode
I
and each rate of flow
U
, the problem to be solved is written:
() ()
()
() ()
()
0
2
1
2
1
2
2
2
2
=


















-
+


















-
+
K
I
K
E
K
K
E
L
II
K
I
K
E
K
E
K
E
L
II
II
dx
X
U
U
U
sd
Ck
X
V
X
S
dx
X
U
U
U
sd
Cd
D
X
V
X
S
K
K
K
C
M
éq
2.4-2
It will be noted that the equation [éq 2.4-2] is non-linear in
S
; its solutions are obtained using one
iterative method of Broyden type.
For each mode
I
, a solution is obtained
S
I
equation [éq 2.4-2]. One deduces then from
S
I
,
for this mode, the pulsation
I
and damping
I
system coupled flow-structure, in
using the relation:
2
1
I
I
I
I
I
J
S
-
+
-
=
with
1
2
-
=
J
éq
2.4-3
The coupled system dynamically becomes unstable when one of the damping coefficients
I
becomes negative or cancels themselves.
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3 Excitation fluid-rubber band acting on the stem of
order on the level of the plate of housing
(exclusively for the control rods)
The forces fluid-rubber bands acting on this type of configuration were identified on the model
GRAPPE2 of department TTA. The theoretical aspects of the identification of these sources are
developed in reference [bib4]. The integration of model GRAPPE2 in Code_Aster is approached
in the note of specifications [bib2].
3.1
Description of the studied configuration
Model GRAPPE2 represents the stem of control, the higher part of the guide of bunch, and
the thermal cuff of a jet engine of the type 900 or 1300 MWe [Figure 3.1-a].
Cuff
thermics
Plate
housing
Tube envelope
Core
power station
Appear 3.1-a: Schematic diagram of the model BUNCH 2
This model primarily consists of a hollow cylindrical tube low thickness, fixed on
a full cylindrical central core. The hollow tube is entirely immersed in water with
ambient temperature. A plate, representing the plate of housing, makes it possible to reproduce it
annular containment. The flow through the plate can be ascending or descendant. The stem
of control can be centered or offset (50% of the average play) on the level of the plate of
housing.
Four experimental configurations are thus possible, according to the direction of the flow and of
centering or not of the stem of control. The coefficients of forces fluid-rubber bands were identified
for each one of these configurations and are available in Code_Aster.
Model GRAPPE2 was dimensioned in geometrical, hydraulic similarity and of frequency
reduced compared to the configuration jet engine. The only data of the diameter of the stem of control
thus allows, in particular, to deduce the unit from the other geometrical magnitudes.
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3.2
Stages of calculation
·
The first stage of calculation consists in calculating the modal base of the water structure at rest,
locally induced effects of mass added to the level of the containment of the plate of
housing being neglected. This stage is carried out by the operator
MODE_ITER_SIMULT
.
With this intention, a homogeneous equivalent density is assigned to the whole of
structure, in order to take into account the apparent mass added by the fluid, except for
that induced by the effects of containment on the level of annular space. This mass
voluminal equivalent is defined by:
tube
F
eq
S
R
+
=
2
éq
3.2-1
where:
indicate an adimensional coefficient of containment depend on the configuration
studied;
= 1 is the value used for calculations of control rods. It
corresponds to a vibrating roller in an unlimited fluid field.
R
indicate the radius external of the tube,
S
indicate the surface of the cross-section of the tube,
tube
indicate the density of material constituting the vibrating tube.
·
The second stage is the taking into account of the coupling with the fluid flow. It is carried out with
the aid of the operator
CALC_FLUI_STRU
.
3.3
Representation of the excitation fluid-rubber band
That is to say
X
direction of neutral fiber of the tube. The excitation fluid-rubber band identified on the model
GRAPPE2 is represented by a resulting force and a moment, applied in the same point
of X-coordinate O
X
, corresponding to the central area of the passage of the stem of control through
plate housing. The excitation is thus defined, in the physical base, by the relation:
()
() (
)
() (
)
0
0
X
X
S
X
X
S
S
X
C
C
C
-
-
-
=
'
M
F
,
f^
éq
3.3-1
where
'
indicate the derivative compared to
X
distribution of Dirac function
.
The resulting force,
C
F
, acts thus under the effect of transverse displacements of the stem of control;
and resulting moment,
C
M
, acts under the effect of the rotation of the latter.
One notes
()
S
T
X
the vector of transverse displacements and
()
S
the vector of associated rotations,
defined by:
()
(
)
(
)




=
S
X
U
S
X
U
S
Z
y
T
,
,
X
0
0
0
éq 3.3-2
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()












=
, S)
(X
X
U
, S)
(X
X
U
S
Z
y
0
0
0
éq 3.3-3
The following relations are used to calculate the forces and the moments fluid-rubber bands
resulting starting from the added masses
2
1
Cm
Cm,
, of added depreciation
()
()
R
R
V
Cd
V
Cd
2
1
,
and
added stiffnesses
()
()
R
R
V
Ck
V
Ck
2
1
,
, adimensional coefficients identified on the model
GRAPPE2:
()
()
()
()
S
V
Ck
L
U
S
V
Cd
DUL
S
Cm
L
D
S
C
T
R
p
F
R
p
F
p
F
X
F




+
+
-
=
1
2
1
2
1
2
2
1
2
1
2
1
éq
3.3-4
()
()
() ()
S
V
Ck
L
U
S
V
Cd
DUL
S
Cm
L
D
S
C
R
p
F
R
p
F
p
F




+
+
-
=
2
3
2
2
3
2
2
3
2
2
1
2
1
2
1
M
éq
3.3-5
In order to simplify the writing of the equations, one notes thereafter:
()
() ()
()
() ()
S
S
H
S
C
S
S
H
S
C
T
=
=
2
1
and
M
X
F
Adimensional fallback speed
V
R
is defined here using the relation
sD
U
V
R
=
, where
S
indicate
variable of Laplace.
The expressions [éq 3.3-4] and [éq 3.3-5] utilize the thickness
p
L
plate of housing. This
thickness results from the value of the diameter of the stem of control,
D
, because of similarity
geometrical with the configuration jet engine. The effort fluid-rubber band
()
S
X
C
,
f^
is thus completely
characterized by the data of the following sizes:
F
Density of the fluid,
U
Rate of the average flow in annular space between stem of control and
plate housing,
D
Diameter of the stem of control,
1
Cm
Coefficient of added mass associated the translatory movement,
()
R
V
Cd
1
Added damping coefficient associated the translatory movement,
()
R
V
Ck
1
Coefficient of added stiffness associated the translatory movement,
2
Cm
Coefficient of added mass associated the rotational movement,
()
R
V
Cd
2
Added damping coefficient associated the rotational movement,
()
R
V
Ck
2
Coefficient of added stiffness associated the rotational movement.
Adimensional coefficients of added mass,
2
1
and
Cm
Cm
, allow the taking into account of
inertial effects induced by local containment of the stem of control on the level of the plate of
housing. These effects are estimated as follows.
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Titrate:
Coupling fluid-structure for the tubular structures and the hulls
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Author (S):
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Key
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That is to say
H
the thickness of the annular flow on the level of containment, deduced from
D
by similarity
geometrical compared to the configuration jet engine;
indicate the adimensional coefficient of
containment introduced by the relation [éq 3.2-1]. One obtains then [bib 4]:
(
)
3
2
4
2
4
2
1
2
4
4
8
2
1
3
2
2
2
2
3
2
2
2
3
1
2
p
F
L
O
F
p
F
p
F
p
F
F
p
F
L
H
D
D
dx
X
X
H
D
D
Cm
L
D
L
H
D
D
L
D
H
D
Cm
L
D
p




-
=
-




-
=




-
=






-
=
One deduces the values from them from
1
Cm
and
2
Cm
by:




-
=
H
D
Cm
2
2
1
éq
3.3-6




-
=
=
H
D
Cm
Cm
2
6
3
1
2
éq 3.3-7
Coefficients
2
1
1
Cd
Ck
Cd
,
,
and
2
Ck
are directly deduced from the measurement and are expressed in form
adimensional correlations.
3.4 Projection on modal basis and expression of the terms of the matrix
of transfer of effort fluid-rubber band
Decomposition of the movement on modal basis
One notes
()
X
J
modal deformation of the J
ème
mode of the structure. Decomposition of the vector of
displacements in the modal base is expressed in the form:
()
() ()
()
=
=






=
=
NR
J
NR
J
J
J
J
J
J
J
S
Q
X
DZ
X
DY
X
DX
S
Q
X
S
X
1
1
)
(
)
(
)
(
,
U
éq
3.4-1
Where
J
DX
,
J
DY
and
J
DZ
correspond to the three components of translation characterizing them
modal deformations calculated using Code_Aster.
Calculation of the generalized excitation associated mode I
The generalized excitation
()
S
I
Q
associated the mode
I
is defined by the relation:
()
() ()
dx
X
S
X
S
L
I
C
I
=
0
.
,
f^
Q
éq 3.4-2
where
L
indicate the length of the structure on which one wants to impose excitations GRAPPE2.
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Titrate:
Coupling fluid-structure for the tubular structures and the hulls
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Author (S):
T. KESTENS, Mr. LAINET
Key
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HI-86/02/008/A
Transfer functions
()
S
H
1
and
()
S
H
2
being defined starting from the relations [éq 3.3-4] and [éq 3.3-5],
one deduces some, taking into account the expressions [éq 3.3-1], [éq 3.3-4] and [éq 3.3-5]:
()
()
()
() () (
)
()
()
()
()
() () (
)
()
()
=
=




-






-




-




=
NR
J
I
I
O
J
O
J
O
J
L
NR
J
I
I
O
J
O
J
O
J
L
I
dx
X
DZ
X
DY
X
X
S
Q
X
DZ
X
DY
S
H
dx
X
DZ
X
DY
X
X
S
Q
X
DZ
X
DY
S
H
S
1 0
2
1 0
1
0
0
0
0
.
.
Q
'
'
'
éq
3.4-3
From where, after integration:
()
()
()
()
()
()
[
]
{
()
()
()
()
()
[
]
}
()
() ()
=
=
=
+
+
+
=
NR
J
J
ij
J
O
I
O
I
O
J
O
I
NR
J
O
J
O
I
O
J
O
I
I
S
Q
S
B
S
Q
X
DZ
X
DZ
X
DY
X
DY
S
H
X
DZ
X
DZ
X
DY
X
DY
S
H
S
1
2
1
1
'
'
'
'
.
.
.
.
Q
éq
3.4-4
Note:
()
()
O
I
O
I
X
DRZ
X
DY
=
'
and
()
()
O
I
O
I
X
DRY
X
DZ
-
=
'
3.5
Resolution of the modal problem under flow
The modal problem is solved by supposing, at first approximation, that the diagonal terms of
the matrix of transfer of the efforts fluid-rubber bands
()
[]
S
B
are dominating compared to the terms
extradiagonaux.
The matrix
()
[]
S
B
being thus reduced to its diagonal, the modal deformations are not disturbed
by the taking into account of the coupling fluid-rubber band; the only modified parameters are them
Eigen frequencies and modal reduced depreciation.
The modal problem under flow breaks up then into
NR
independent scalar problems,
solved by a method of the Broyden type:
(
)
()
(
)
()
(
)
0
2
=
+
+
+
+
+
S
S
S
S
II
aj
II
II
aj
II
II
aj
II
K
K
C
C
M
M
éq
3.5-1
where
II
aj
M
indicate the generalized mass added by the fluid,
()
S
II
aj
C
indicate the generalized damping added by the fluid,
()
S
II
aj
K
indicate the generalized stiffness added by the fluid.
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II
aj
M
,
()
S
II
aj
C
and
()
S
II
aj
K
are calculated using the relations:
()
()
{
}
()
()






+
+
+
+
=
O
I
O
I
p
O
I
O
p
F
II
aj
X
DZ
X
DY
Cm
L
X
DZ
X
DY
Cm
L
D
2
2
2
2
2
2
1
1
2
2
1
'
'
M
éq 3.5-2
()
()
()
()
{
}
()
()
()






+
+
+
-
=
O
I
O
I
R
p
O
I
O
R
p
F
II
aj
X
DZ
X
DY
V
Cd
L
X
DZ
X
DY
V
Cd
DUL
S
2
2
2
2
2
2
1
1
2
1
'
'
C
éq 3.5-3
()
()
()
()
{
}
()
()
()






+
+
+
-
=
O
I
O
I
R
p
O
I
O
R
p
F
II
aj
X
DZ
X
DY
V
Ck
L
X
DZ
X
DY
V
Ck
L
U
S
2
2
2
2
2
2
1
1
2
2
1
'
'
K
éq 3.5-4
II
aj
C
and
II
aj
K
depend implicitly on
S
via the fallback speed
sD
U
V
R
=
.
The three sizes necessary to dimension these terms are thus only
D
F
,
and
p
L
U,
being deduced from
D
thanks to the geometrical property of similarity.
Like that was indicated previously, the adimensional coefficients
()
()
()
R
R
R
V
Cd
V
Ck
V
Cd
2
1
1
,
,
and
()
R
V
Ck
2
are resulting from the identified empirical correlations
in experiments on model GRAPPE2.
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4 Excitation fluid-rubber band acting on two hulls
cylindrical coaxial under annular flow
(example: space tank/envelope of core)
The integration of this model of excitation fluid-rubber band in Code_Aster was approached in the note
specifications [bib2]. The note of principle of model MOCCA_COQUE [bib5] constitutes
theoretical documentation of reference.
4.1
Description of the studied configuration
The studied hardware configuration is made up of two coaxial, separate cylindrical hulls
by an annular space in which runs out a viscous incompressible monophasic fluid
[Figure 4.1-a]. The flow is done in the direction of the axis of revolution of the cylinders; to fix them
notations, one supposes in the continuation of the document that it is about the axis
X
.
One notes:
L
the common length of the two cylindrical hulls,
(
)
T
X
R
,
,
1
the interior radius of annular space,
(
)
T
X
R
,
,
2
the radius external of annular space,
(
)
T
X
R
,
,
the average radius
(
)
(
)
(
)


+
=
2
,
,
,
,
,
,
2
1
T
X
R
T
X
R
T
X
R
,
(
)
T
X
H
,
,
annular play
(
)
(
)
(
)
(
)
T
X
R
T
X
R
T
X
H
,
,
,
,
,
,
1
2
-
=
,
X
R
E
E
E
,
,
vectors of the base of cylindrical co-ordinates.
R
1
R
2
L
External hull
Internal hull
Appear 4.1-a: Schematic diagram coaxial hulls
4.2
Stages of calculation
·
The first stage of calculation consists in determining the modal base in air of the structure. This
operation is carried out by the operator
MODE_ITER_SIMULT
. This calculation is necessary because
decomposition of the matrix of transfer of the forces fluid-rubber bands
()
[]
S
B
is expressed in
this base.
·
The second stage relates to the taking into account of the forces fluid-rubber bands. It intervenes in
the operator
CALC_FLUI_STRU
. This stage breaks up into eight sub-tasks:
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4.2.1 Preprocessings
1°/Determination of the characteristic geometrical magnitudes, starting from the topology of
mesh: common length of the two hulls, average radius, average annular play.
2°/Characterization of the modal deformations in air: determination of the commands of hull, of
main plans, of the numbers of wave and the coefficients of associated deformations of beam
with each mode of the structure, both for the hull interns the external hull.
4.2.2 Resolution of the modal water problem at rest
3°/Calculation of the matrix of mass added by the fluid
[]
aj
M
in the modal base of
structure in air
4°/Calculation of the modal characteristics of the water structure at rest while solving:
[]
[]
(
)
[]
{
}
()
0
2
=
+
+
Q
K
M
M
I
aj
I
S
One obtains the new structural features out of water at rest
E
I
E
I
E
I
F
,
K
,
M
(generalized mass and stiffness, Eigen frequency of mode I) as well as the deformations
modal
E
I
, expressed in the base in air.
5°/Calculation of the water deformations at rest in the physical base, by basic change:
[] [] []
E
I
has
I
E
I
.
=
4.2.3 Resolution of the modal problem under flow
For each rate of flow:
6°/Calculation of
()
[]
S
B
in the modal base in air.
This calculation is carried out by solving the non stationary fluid problem according to the method
specified in the paragraph § 4.3.1.
7°/Calculation of the forces fluid-rubber bands induced by the effects of damping and stiffness
additions, in the modal water base at rest.
()
[]
[]
()
[]
[]
{
}
[]
E
I
aj
E
I
T
E
S
S
S
2
M
B
B
-
=
8°/Resolution of the modal problem by neglecting the extradiagonaux terms of the latter
stamp, by the method of Broyden (loops on the sub-tasks 6° and 7°).
()
0
2
=
-
+
+
S
S
S
E
II
E
I
E
I
E
I
B
K
C
M
Modal characteristics of the structure:
EC.
I
EC.
I
EC.
I
F
,
,
M
(generalized mass,
Eigen frequency and damping of mode I, under flow) are given.
modal deformations are supposed to be identical to those out of water at rest.
End of loop on the rates of flow
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Note:
·
The calculation of the terms of the matrix of transfer of the forces fluid-rubber bands requires
resolution of the non stationary fluid problem (sub-task 6°). This resolution is not
itself possible that if certain sizes beforehand were determined
geometrical characteristics of the configuration, as well as the coefficients of the forms
analytical of the modal deformations of the structures (preprocessings 1° and 2°).
·
If the user chooses to carry out the first stage (calculation of the modal base by
the operator
MODE_ITER_SIMULT
) by taking directly into account the effects of mass
added, those should not be taken any more into account by the operator
CALC_FLUI_STRU
.
For that, the key word
MASS_AJOU
control
DEFI_FLUI_STRU
must be
informed by
“NOT”
. The sub-tasks 3° with 7° become then:

/
Calculation of the effects of mass added by the fluid, in the modal base of
water structure, in order to be able to cut off these effects of the effort fluid-rubber band
total, since the terms of added mass are already taken into account.

/
Removed sub-task.

/
Removed sub-task.
For each rate of flow

/
Calculation of the matrix
()
[]
S
B
in the modal water base.

/
Calculation of the forces fluid-rubber bands induced by the effects of damping and of
stiffness added in the modal water base:
()
[]
()
[]
[]
2
S
S
S
aj
E
M
B
B
-
=
The sub-tasks 1°, 2° and 8° are not modified.
4.3
Resolution of the non stationary fluid problem
4.3.1 Assumptions
simplifying
Some assumptions on the nature of the flow make it possible to simplify the equations of Navier-
Stokes non stationary, at the base of the problem fluid-structure.
H1
It is supposed that the flow is the superposition of a stationary average flow, obtained
when the structures are fixed, and of a non stationary flow induced by the movement of
walls.
H2
It is supposed that the vibrations of structure are of low amplitude with respect to the thickness of
the average annular flow.
H3
One supposes that the disturbances speed induced by the vibratory movements are, in
average on a radius, primarily directed in the directions
and
X
: one supposes thus
that the vibratory movement induced a helicoid movement of fluid around the structures
rather than a radial movement compared to these last. These disturbances speed
define command 1.
H4
One supposes finally that the speed and pressure field is uniform, with command 1, in
radial direction.
These simplifying assumptions make it possible to solve the fluid problem analytically. The matrix
of transfer of the forces fluid-rubber bands
()
[]
S
B
is deduced from the non stationary flow resulting from this
resolution.
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4.3.2 Analyze in disturbances
With the help of the assumptions stated previously, the analysis in disturbances of the fluid problem
conduit to seek the non stationary flow in the form:
2
command
+
+
=
0
0
R
U
éq
4.3.2-1
(
)
2
command
+
+
=
T
X
U
U
,
,
~
0
éq
4.3.2-2
()
(
)
2
command
+
+
=
T
X
U
X
U
U
X
X
,
,
~
éq
4.3.2-3
() (
)
2
command
+
+
=
T
X
p
X
P
P
,
,
~
éq
4.3.2-4
with:
(
)
R
R
R
X T
1
1
1
=
+
~,
éq
4.3.2-5
(
)
R
R
R
X T
2
2
2
=
+
~
,
éq
4.3.2-6
The variables are defined
~
H
and
~
R
like:
~
~
~
H
R
R
=
-
2
1et
2
1
2
R
R
R
~
~
~
+
=
.
By limiting the development of the Navier-Stokes equations to the first command, one obtains two
systems of equations characterizing the stationary part and the disturbed part of the flow, it
second system being a linear system.
The resolution of the stationary fluid problem leads thus to:
()
U
X
U
=
constant and
2
1
U
C
H
X
P
F
-
=
éq
4.3.2-7
In the equation [éq 4.3.2-7],
indicate the density of the fluid and
C
F
the stationary part of
coefficient of friction to the wall. The incompressible fluid being supposed, its density is not
not broken up partly stationary and fluctuating part.
C
F
is deduced from the law of Nikuradzé
characterizing the flows in control:
(
)
(
)
,
E
R
m
E
E
fo
F
R
,
R
C
C
=
with
X
E
U
H
R
2
=
éq
4.3.2-8
where m indicates the value of an exponent,
indicate the kinematic viscosity of the fluid and
the roughness of
walls.
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It results from this:
(
)
(
)
()
()
2
command
2
+
+
=
-
=
=
U
U
C
m
R
C
R
C
C
R
R
C
C
X
F
E
F
E
F
F
R
m
E
E
fo
F
E
~
)
(
~
~
~
,
,
with
U
H
R
E
2
=
and
X
E
U
H
R
~
~
2
=
The linear differential connection of a nature 1 characterizing the non stationary part of the flow induced
by the movements of walls is written in the field of Laplace:








=
+


+
+
+
=
+


+
+


+
-


+
-
=
+
H
H
U
C
X
p
U
H
U
m
C
S
X
U
U
p
R
U
H
U
C
S
X
U
U
R
U
S
X
R
R
U
H
U
S
X
H
H
U
U
R
X
U
F
X
F
X
F
X
~
~
~
)
(
~
~
~
~
~
~
~
~
~
~
2
1
2
0
1
1
éq
4.3.2-9
Three boundary conditions of input-output make it possible to solve this system. The first of these
conditions is obtained by supposing that the flow is sufficiently regular upstream of space
annular, so that the tangential component the speed of input can be neglected:
0
=
U
in
0
=
X
éq
4.3.2-10
The two others are obtained by applying the conservation equation of the kinetic energy, under
its quasi-stationary form, between the infinite upstream and the input of annular space, then between the exit of
annular space and infinite downstream. One obtains then respectively, with the command disturbances:
(
)
(
)




=


-
-
+
=


+
+
+
R
R
L
X
rdr =
U
U
C
C
U
U
p
R
R
X
rdr =
U
U
C
C
U
U
p
D
D
X
D
D
X
S
S
E
E
2
1
2
1
in
in
0
2
1
1
0
0
2
1
1
2
2
~
~
~
~
~
~
éq
4.3.2-11
In these expressions,
E
D
C
and
S
D
C
the stationary parts of the loss ratios represent
of load singular of input and exit. They take into account the dissipation of induced energy,
when the walls are fixed, by possible abrupt evolutions of the geometry at the entry or
exit of annular space. In the majority of the cases, these coefficients can be estimated simply
using data of the literature (Idel' cik for example). When geometrical configuration
of input or exit is very particular, these coefficients can also be given with aid
of a two-dimensional code of mechanics of the fluids adapted to the study of the problems with fixed walls, of
type N3S.
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Titrate:
Coupling fluid-structure for the tubular structures and the hulls
Date:
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Author (S):
T. KESTENS, Mr. LAINET
Key
:
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HI-86/02/008/A
E
D
C~
and
S
D
C~
are the non stationary parts of the singular loss ratios of load. These
coefficients take into account the disturbances of the lines of separation induced by
movements of structure. They can be modelized thanks to a quasi-stationary approach of
even natural that that introduced for the estimate of the coefficient of friction of wall.
The system [éq 4.3.2-9] is solved analytically, using the limiting conditions [éq 4.3.2-10] and
[éq 4.3.2-11], by clarifying the functions
~
~
H
R
and
characterizing the second member.
Disturbances
(
)
S
X
R
,
,
~
1
and
(
)
S
X
R
,
,
~
2
defining the movement of the walls, the parts
disturbed annular play and average radius are then defined, in the field of Laplace, by:
(
) (
) (
)
S
X
R
S
X
R
S
X
H
,
,
~
,
,
~
,
,
~
1
2
-
=
éq
4.3.2-12
(
) (
) (
)
2
2
1
S
X
R
S
X
R
S
X
R
,
,
~
,
,
~
,
,
~
+
=
éq
4.3.2-13
4.3.3 Decomposition on modal basis
That is to say
NR
the number of oscillatory modes of the structure in the studied frequency band.
decomposition on the basis of modal movement of the walls is expressed in the following way:
(
)
(
)
[
]
() ()
S
X
R
K
S
X
R
I
I
NR
I
I
I
.
.
cos
,
,
~
*
1
1
1
1
1
=
-
=
éq
4.3.3-1
(
)
(
)
[
]
() ()
S
X
R
K
S
X
R
I
I
NR
I
I
I
.
.
cos
,
,
~
*
2
1
2
2
2
=
-
=
éq
4.3.3-2
where
I
K
1
and
I
K
2
the commands of hull of I represent
ème
mode for the respective movements
hulls internal and external,
I
1
and
I
2
allow to characterize the main plans of these modes,
()
X
R
I *
1
and
()
X
R
I *
2
are deduced from the deformations of beam of the structures internal and external
associated the mode considered,
and
()
S
I
represent generalized displacement.
Note:
Functions
()
X
R
I *
1
and
()
X
R
I *
2
are represented, within the framework of the analytical resolution,
in the form of linear combinations of sine, cosine, hyperbolic sine and cosine
hyperbolic:
()


+


+


+


=
X
L
HS
D
X
L
CH
C
X
L
B
X
L
With
X
R
I
I
I
I
I
I
I
I
I
1
1
1
1
1
1
1
1
1
sin
cos
*
éq
4.3.3-3
()


+


+


+


=
X
L
HS
D
X
L
CH
C
X
L
B
X
L
With
X
R
I
I
I
I
I
I
I
I
I
2
2
2
2
2
2
2
2
2
sin
cos
*
éq 4.3.3-4
with
I
1
and
I
2
numbers of wave of I
ème
mode for the movements of the hulls internal and
external respectively.
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Coupling fluid-structure for the tubular structures and the hulls
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Solutions of the fluid problem
~, ~
~
p U
U
X
and
are required in the form of decompositions
on modal basis deduced from those of
~
~
R
R
1
2
and
clarified by the relations [éq 4.3.3-1] and
[éq 4.3.3-2]. One obtains thus, in the field of Laplace:
(
)
()
(
)
[
]
()
(
)
[
]
()
S
K
K
S
X
p
K
K
S
X
p
S
X
p
I
NR
I
I
I
I
I
I
I
I
I
=




-
+
-
=
1
2
2
2
2
2
1
1
2
1
1
cos
,
cos
,
,
,
~
*
*
éq 4.3.3-5
(
)
()
(
)
[
]
()
(
)
[
]
(
)
()
S
K
S
X
U
K
S
X
U
S
X
U
I
NR
I
I
I
I
I
I
I
X
=
-
+
-
=
1
2
2
2
1
1
1
cos
,
cos
,
,
,
~
*
*
éq
4.3.3-6
(
)
()
(
)
[
]
()
(
)
[
]
()
S
K
K
S
X
v
K
K
S
X
v
S
X
U
I
NR
I
I
I
I
I
I
I
I
I
=




-
+
-
=
1
2
2
2
2
1
1
1
1
sin
,
sin
,
,
,
~
*
*
éq
4.3.3-7
4.3.4 Expression of the terms of the matrix of transfer of the forces fluid-rubber bands
The surface effort fluid-rubber band,
F
, is the resultant of the field of pressure and the stresses
viscous and turbulent exerted by the flow on the walls of the structure moving.
F
N
T
T
= -
+
+
P
X X
éq
4.3.4-1
The effort generalized fluid-rubber band associated I
ème
oscillatory mode of the structure,
()
S
I
Q
, is written as follows:
()
=
I
S
I
I
I
ds
S
X
.
F
Q
éq
4.3.4-2
Where
S
I
indicate the surface of the walls of the structure wet by the flow, and the vector
X
I
represent I
ème
vector deformed modal in this expression. The representation of the field of
speeds and of pressure and the representation in the form of a law of wall of the viscous stresses
and turbulent exerted on the structure moving allow to express the effort fluid-rubber band
generalized
()
S
I
Q
in the following way:
()
() ()
S
S
S
J
NR
J
ij
I
=
=
1
B
Q
éq
4.3.4-3
with
()
()
()
S
S
S
ij
ij
ij
2
1
B
B
B
+
=
()
S
ij
1
B
and
()
S
ij
2
B
the contributions of the hulls indicate respectively interior and external.
These contributions are defined by:
()
(
)
[
]
()
()
()
(
)
[
]
()
()
()


+
-
-


+
-
-
=
L
J
I
F
I
K
K
J
I
I
I
L
J
I
F
I
K
K
J
I
I
I
ij
dx
X
R
S
X
v
U
C
S
X
p
K
K
R
dx
X
R
S
X
v
U
C
S
X
p
K
K
R
S
J
I
J
I
0
1
2
2
1
2
2
2
2
1
0
1
1
1
1
1
1
2
1
1
1
2
1
2
1
1
2
1
1
*
*
*
*
*
*
.
,
,
cos
.
,
,
cos
B
éq 4.3.4-4
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Titrate:
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and
()
(
)
[
]
()
()
()
(
)
[
]
()
()
()


+
-
-


+
-
-
=
L
J
I
F
I
K
K
J
I
I
I
L
J
I
F
I
K
K
J
I
I
I
ij
dx
X
R
S
X
v
U
C
S
X
p
K
K
R
dx
X
R
S
X
v
U
C
S
X
p
K
K
R
S
J
I
J
I
0
2
2
2
2
2
2
2
2
2
0
2
1
1
2
1
1
2
1
2
2
2
1
2
1
2
2
2
1
*
*
*
*
*
*
.
,
,
cos
.
,
,
cos
B
éq 4.3.4-5
4.4
Resolution of the modal problem under flow
As one explained in the paragraph [§ 4.2], one solves beforehand the modal problem out of water with
rest, in order to take into account the inertial coupling between modes. One estimates the matrix thus of
mass added by the fluid, while calculating
()
[]
S
B
for a mean velocity of the flow null.
modal characteristics of the system under flow are then obtained by disturbing them
water characteristics at rest. One does not hold any more account but of damping and the stiffness
additions: the terms of mass added previously calculated are cut off from the matrix
()
[]
S
B
.
The coupling between modes is then neglected; consequently, the modal deformations remain
unchanged compared to those out of water at rest. Only parameters disturbed by the setting in
flow of the fluid are the frequency and reduced modal damping. These parameters are calculated
while solving
NR
nonlinear equations mode by mode, implementation of a method of the type
Broyden.
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5
Axial flow (example: fuel assemblies)
The integration of this model of excitation fluid-rubber band in Code_Aster was approached in the note
specifications [bib2]. The note of principle of model MEFISTEAU [bib6] constitutes documentation
theoretical of reference.
5.1
Description of the studied configuration
One considers a beam of
K
circular cylinders mobile in bending and subjected to a flow
incompressible of viscous fluid, limited by a cylindrical rigid chamber of circular section or
rectangular [Figure 5.1-a].
Circular chamber
Rectangular chamber
L
X
Z
Y
Appear 5.1-a: Beam under axial flow
The cylinders all parallel, are directed along the axis of the chamber. They have a common length,
noted
L
. To simplify the notations, it is supposed thereafter that
X
is the directing axis. The flow
stationary axial and is supposed to be uniform in each section. Density of the which can fluid
to be variable along the axis
X
(heat gradients), the rate of the stationary flow depends
also of the variable
X
.
5.2
Stages of calculation
·
The first stage relates to the determination of the modal base in air of the beam. This
operation is carried out by the operator
MODE_ITER_SIMULT
. This stage is essential because them
forces fluid-rubber bands are projected on this basis.
·
The second stage relates to the taking into account of the forces fluid-rubber bands with the operator
CALC_FLUI_STRU
. This stage breaks up into 7 sub-tasks:
5.2.1 Preprocessings
1°/By means of the topology of the mesh, deduction of the co-ordinates of the centers of the cylinders
beam then checking of the good provision of the cylinders ones compared to
others (it is checked in particular that there is not overlapping between two cylinders) and by
report/ratio with the rigid chamber.
2°/Determination the length of excitation of the fluid, commune to all the cylinders, like
of an associated discretization along the directing axis.
3°/Constitution of the tables giving the modal deformations in air of each cylinder of
beam, for each mode taken into account for the coupling fluid-structure. One
interpolate for that the deformations at the points of the discretization determined before.
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5.2.2 Resolution of the modal problem under flow
4°/Resolution of the disturbed fluid problem. Determination of the potential speeds
disturbed the inversion of linear systems of high natures requires calling the setting in
work of the method of Crout.
For each rate of flow
5°/Calculation of the matrices of mass, damping and stiffness added giving the matrix of
transfer of the forces fluid-rubber bands in the modal base in air:
()
[
]
[] [] []
has
has
has
ij
S
S
S
K
C
M
B
-
-
-
=
2
[]
has
M
full symmetrical;
[] []
has
has
K
C
and
a priori full and nonsymmetrical.
6°/Resolution of the modal problem under flow; one solves the complete problem with
vectors and with the clean ones
[] [] []
()
[
]
{
}
() ()
0
2
=
-
+
+
Q
.
B
K
C
M
S
S
S
ij
ij
ij
ij
One does not neglect the extradiagonaux terms of
()
[
]
S
ij
B
. After reformulation, the resolution is
carried out using algorithm QR: obtaining the masses, frequencies and depreciation
modal reduced under flow
EC.
I
EC.
I
EC.
I
F
,
,
M
, complex modal deformations
EC.
I
expressed
in the base in air; of these last, one retains only the real part after minimization of
imaginary part (calculation of a criterion on the imaginary part).
7°/Restitution of the deformations under flow in the physical base.
[]
[]
[]
EC.
I
I
EC.
I
=
[]
I
is the matrix whose columns are the modal deformations in air, expressed in base
physics.
End of loop on the rates of flow
Note:
·
The knowledge of the co-ordinates of the centers of the cylinders (preprocessing 1°) is
necessary to the resolution of the disturbed fluid problem (sub-task 4°). This resolution
conduit with the estimate of the terms of the matrix of transfer of the forces fluid-rubber bands
(sub-task 5°), which utilizes the disturbances of pressure and speed.
·
Determination a common length of excitation and the creation of a discretization
associated (preprocessing 2°) allow to define a field of integration on the structures
for the projection of the forces fluid-rubber bands on the modal basis. The interpolation of
modal deformations at the same points is thus necessary (preprocessing 3°).
·
The dynamic behavior of the beam under flow can also be studied with
the aid of a simplified representation of the beam (with equivalent tubes). Stages of
calculation for the taking into account of the coupling fluid-structure are then identical to those
described previously, only differences appearing in the preprocessings. This
second approach is described more precisely in the note [bib2]. In the stage 1° of
preprocessings, the co-ordinates of the centers of the cylinders of the beam are then specified
directly by the user, who also establishes the correspondence between the cylinders of
beam and beams of the simplified representation given by the mesh. In the stage 3°
preprocessings, this correspondence makes it possible to assign to the cylinders of the beam, with
points of discretization determined in the stage 2°, the modal deformations of the beams of
simplified representation.
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5.3
Resolution of the non stationary fluid problem
5.3.1 Assumption
simplifying
H1
The field non stationary fluid speeds is analytically given while supposing
that the disturbed flow
~
is potential in all the fluid field, and that the flow
stationary is uniform transversely, but function of the axial position
X
:
()
()
U
U U
X
= + =
+
~
~
U X
éq
5.3.1-1
Such a field speeds admits a slip on the walls of the cylinders which will allow
to calculate the viscous stress by a law of friction.
H2
The movement of the cylinders does not generate disturbances speed
()
~
~
U
=
that
radially and orthoradialement (assumption of the slim bodies):
~
~
~
U
y
Z
=
+
U
U
y
Z
H3
The field of pressure is broken up into parts stationary and disturbed according to
P
P
p
= +
~
The stationary field of pressure depends only on
X
and its gradient is worth:
()
()
X
.
G
+
-
-
=
U
U
D
C
X
dx
U
D
U
X
dx
P
D
H
fl
2
éq
5.3.1-2
where
H
D
indicate the hydraulic diameter of the beam,
fl
C
indicate the coefficient of local friction for stationary speed
U
. It depends on
Reynolds number, calculated using stationary speed
U
, of the hydraulic diameter of
beam and of the surface roughness. This coefficient is deduced from the law of Nikuradzé
characterizing the flows in control;
G
indicate the field of gravity. Its action on the stationary field of pressure depends on
the slope of the beam
()
G X
.
.
5.3.2 Determination of the potential disturbed speeds
One seeks an analytical solution for
(
)
~
,
R
X T
in the form of a superposition of
elementary singularities which are written:
()
()
()
()
{
}
C
X T R
N
D
X T R
N
nk
K N
K
nk
K N
K
N
NR
trunk
.
.cos
.
.sin
-
-
=
+
1
éq
5.3.2-1
in the center of each cylinder K and:
()
()
()
()
{
}
=
+
trunk
NR
N
O
N
O
N
O
N
O
N
N
R
T
X
B
N
R
T
X
With
1
sin
.
.
,
cos
.
.
,
éq
5.3.2-2
in the center of the rigid chamber when the aforementioned is circular where:
trunk
NR
indicate the command of truncation of the series of Laurent
(
)
NR
trunk
=
3
,
K
K
R
,
the polar co-ordinates in a plan perpendicular to the axis indicate
X
,
centered in the center of the cylinder K,
O
O
R
,
the polar co-ordinates in a plan perpendicular to the axis indicate
X
,
centered in the center of the circular rigid chamber.
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Coefficients
()
() ()
T
X
With
T
X
D
T
X
C
N
nk
nk
,
,
,
,
,
and
()
T
X
B
N
,
expressions [éq 5.3.2-1] and [éq 5.3.2-2]
are given by applying the boundary condition of nonpenetration:
·
on the contour of each mobile cylinder
K
, this condition is written:
(
)
() ()
() ()
K
K
K
K
K
K
K
K
T
X
Dt
Dz
T
X
Dt
Dy
R
R
R
sin
,
cos
,
~
,
+
=
=




where
()
T
X
y
K
,
and
()
T
X
Z
K
,
the components of the displacement of neutral fiber of the cylinder K indicate
with the X-coordinate
X
in the reference mark
()
y Z
,
,
R
K
K
and
the polar co-ordinates in the reference mark indicate
()
y Z
,
whose origin is taken with
center cylinder
K
,
R
K
indicate the radius of the cylinder
K
,
()
X
X
U
T
Dt
D
+
.
·
on the contour of a circular rigid chamber, she is written:
(
)
0
=
=




O
O
O
O
R
R
R
~
,
where
R
O indicates the radius of the chamber.
In the case of a rectangular rigid chamber, this condition is taken into account by a method
derived from the method of the “images” [bib6]; the fluid problem confined by the rectangular chamber
is made equivalent to the problem in infinite medium by creating images of the mobile cylinders of
beam compared to the sides of the chamber. This method results in introducing news
singularities of the form [éq 5.3.2-1], placed at the center of the cylinders “images”, in the expression of
~
. It does not add however an unknown factor to the problem since the coefficients for this news
singularities are derived from those of the mobile cylinders of the beam by the play of the images.
Finally, the potential disturbed speeds is written:
(
)
()
()
()
()
=




+
=
K
K
K
K
K
K
T
X
Dt
Dz
R
G
T
X
Dt
Dy
R
F
T
X
R
1
,
,
,
,
,
,
,
~
éq
5.3.2-3
Where
K
indicate the number of mobile cylinders of the beam. Functions
()
()
F R
G R
K
K
,
,
and
are
linear combinations of
()
()
,
sin
.
,
cos
.
N
R
N
R
N
N
-
-
()
N
R
N
cos
.
and
()
N
R
N
sin
.
of which them
coefficients are determined by the boundary conditions preceding. That requires the resolution of
linear systems of high natures and with full matrices. The inversions are carried out while putting in
work the method of Crout.
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5.3.3 Modeling of the fluid forces
One retains initially the forces due to the disturbances of pressure
~
p
, connected to the potential speeds
disturbed by:
~
~
p
D
Dt
= -
éq
5.3.3-1
The resultant of the field of pressure disturbed around each mobile cylinder is a linear force
F
~
p
acting according to
y
Z
and
. This force depends linearly on
2
2
Dt
y
D
K
and
2
2
Dt
Z
D
K
, thus generating
terms of mass, damping and stiffness added.
One then takes into account the forces related to the viscosity of the fluid.
In a quasi-static approach, one considers the action of the fluid field speed
(
)
U U
+
~
around
of a cylinder at the moment
T
: in the reference mark related to the cylinder, the flow, speed
U
with command 0,
present an angle of attack compared to the cylinder which is a function of the disturbances speed and of
movement of the cylinder itself. It results from it a force from drag and a force of lift. One
show that the components according to
y
Z
and
resulting linear force
F
v
are written, for
roll
L
:
()




-
-




-
-
=
L
L
L
L
L
L
L
y
p
y
fl
y
U
Dt
y
D
C
U
R
U
T
y
C
U
R
~
~
F
éq
5.3.3-2
()


-
-


-
-
=
L
L
L
L
L
L
L
Z
p
y
fl
Z
U
Dt
Dz
C
U
R
U
T
Z
C
U
R
~
~
F
éq
5.3.3-3
where
C
p
very indicate the slope with null angle of attack of the coefficient of lift around a cylinder
slightly tilted (
C
p
= 0,08).
~
U
y and
~
U
Z indicate the averages of the disturbances speed along the axes
y
and
Z
around
cylinders, which depend linearly on
Dt
Dy
K
and
Dt
Dz
K
(cf [éq 5.3.2-3]).
These forces generate terms of added damping and stiffness.
One finally takes into account the action of the stationary field of pressure on the mobile structures
deformations. It is shown that the resulting linear force pl
F
on the cylinder
L
has as components, with
command 1:
()


=
X
y
P
X
R
y
p
L
L
L
2
F
éq
5.3.3-4
()


=
X
Z
P
X
R
Z
p
L
L
L
2
F
éq
5.3.3-5
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Coupling fluid-structure for the tubular structures and the hulls
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These forces generate only terms of added stiffness and any coupling between cylinders.
The expressions [éq 5.3.3-1], [éq 5.3.3-2] and [éq 5.3.3-3] highlight the need for solving
the disturbed fluid problem before estimating the forces fluid-rubber bands.
5.3.4 Expression of the terms of the matrix of transfer of the forces fluid-rubber bands
Summary of the linear forces
For each cylinder
L
, the forces fluid-rubber bands are written according to
y
and
Z
:
F
F
F
F
L
L
L
L
=
+
+
~
p
p
éq
5.3.4-1
and are linear combinations of:
(
)
y
T
y
T
y
T X
y
X
Z
T
Z
T
Z
T X
Z
X
K
K
K
K
K
K
K
K
K
K
,
,
,
,
,
,
,
,
2
2
2
2
2
2
2
2
2
2
1




=
Decomposition of the movement on modal basis
The movement of the beam of cylinders is broken up according to
NR
modes of vibration in air. One notes
kj
(
1
K K
and
1
J NR
) deformations according to
y
Z
and
cylinder
K
corresponding to the J
ème
mode
beam. Components of the displacement of neutral fiber of the cylinder
K
with the X-coordinate
X
can then be written:
()
() ()
=
=
NR
J
kj
J
K
X
T
T
y
1
y
.
Q
éq
5.3.4-2
()
() ()
=
=
NR
J
kj
J
K
X
T
T
Z
1
Z
.
Q
éq 5.3.4-3
where
()
()
NR
J
J
,
Q
Q
1
=
=
is the vector of generalized displacements.
Projection of the forces on modal basis
·
One notes
()
T
I
F
the projection of the forces fluid-rubber bands according to I
ème
mode of the beam.
()
() ()
dx
X
T
X
T
K
K
L
K
I
K
I
=
=
1 0
.
,
F
F
éq
5.3.4-4
()
T
I
F
is a linear combination of
(
)
NR
J
J
J
J
,
Q
,
Q
,
Q
1
=
&&
&
·
One notes
()
T
F
the vector of the modal forces fluid-rubber bands:
()
()
()
NR
I
I
T
T
,
F
F
1
=
=
who is written:
()
[]
()
()
[]
()
()
[]
()
()
T
T
T
T
has
has
has
Q
K
Q
C
Q
M
F
-
-
-
=
&
&&
éq
5.3.4-5
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Where
[]
has
M
indicate the matrix of the terms of mass added by the fluid,
[]
has
C
indicate the matrix of the terms of damping added by the fluid,
[]
has
K
indicate the matrix of the terms of stiffness added by the fluid.
These matrices are square real of command
NR
and their terms are independent of the movement
structures. The matrix
[]
has
M
is symmetrical; matrices
[]
has
C
and
[]
has
K
are not it
necessarily.
·
The projection of the equations of the movement on modal basis provides:
[] []
(
) ()
()
[] []
(
) ()
()
[] []
(
) ()
() ()
0
=
+
+
+
+
+
T
T
T
has
II
has
II
has
II
Q
K
K
Q
C
C
Q
M
M
&
&&
éq 5.3.4-6
where
[] [] []
II
II
II
K
C
,
M
and
the matrices of masses, depreciation indicate and stiffnesses of
structure in air; these matrices are of command
NR
and diagonals.
In the field of Laplace, the relation [éq 5.3.4-6] becomes:
[] []
(
)
[] []
(
)
[] []
(
) ()
() ()
0
2
=
+
+
+
+
+
S
S
S
has
II
has
II
has
II
Q
K
K
C
C
M
M
éq
5.3.4-7
·
One introduces then the matrix of transfer of the forces fluid-rubber bands
()
[]
B S
defined by:
()
[] []
[] []
has
has
has
S
S
S
K
C
M
B
-
-
-
=
2
éq
5.3.4-8
And one finds the relation [éq 1.2-1] paragraph [§ 1.2]:
[]
[] []
()
[]
(
)
()
() ()
0
2
=
-
+
+
S
S
S
S
II
II
II
Q
B
K
C
M
5.4
Resolution of the modal problem under flow
The modal problem under flow is formulated by the relation [éq 5.3.4-7] of the preceding paragraph.
This problem is solved after rewriting in the form of a standard problem to the vectors and with
eigenvalues of the type
[]
() ()
X
X
With
=
.
The new formulation is as follows:
[] []
(
)
[] []
(
)
[] []
(
)
[] []
(
)




=








+
+
-
+
+
-
-
-
Q
Q
Q
Q
C
C
M
M
K
K
M
M
]
[
]
[
S
S
S
Id
has
II
has
II
has
II
has
II
0
1
1
éq 5.4-1
Note:
1) One doubles the dimension of the problem compared to that of the initial problem.
2) Properties of the matrices
[]
II
M
and
[]
has
M
allow the inversion.
The resolution of this problem is done by means of algorithm QR. Modules implemented by
the operator
CALC_FLUI_STRU
are the same ones as those used by
MODE_ITER_SIMULT
.
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The problem with the clean elements that one solves is a complex problem. One is thus obtained
numbers even combined complex eigenvalues two to two. One preserves only those of which
the imaginary part is positive or null.
The clean vectors complex, are defined except for constant a complex multiplicative. Like
one takes into account only real modes, it initially acts to determine, for each
clean vector, the constant which minimizes the imaginary part of the vector compared to its real part,
within the meaning of the euclidian norm. The clean vectors are then redefined compared to this standard.
Taking into account standardization used, it is then possible not to preserve in the concept
mode_meca
that the real part of the clean vectors. One restores however, in the file
MESSAGE
,
indicators on the relationship between imaginary part and real part of the clean vectors thus
normalized, so that the user can consider skew introduced by not taken into account of the part
imaginary of the normalized vectors.
5.5
Taking into account of the presence of the grids of the beam of tubes
Modeling described previously, of the forces induced by an axial flow on a beam of
cylinders, does not take into account the presence of the grids of the beam (for example, the grids of
mix and of hold of the fuel assemblies). A comparison between this model and
tests carried out on the model
CHAIR
(in the configuration of a beam of nine flexible tubes
comprising a grid) is presented in a note of synthesis [bib8]: it is noted that the coupling
fluid-rubber band between the grid and the axial flow is not negligible and that it generates one
increase in the reduced modal damping of the tubes. The object of this paragraph is description
additional effects due to the grids and of their taking into account in the model
MEFISTEAU
.
5.5.1 Description of the configuration of the grids
One restricts here the study with two types of grids:
·
the grids of hold which are located at the ends of the beam,
·
the grids of mixture which are distributed between the grids of hold.
The grids all are positioned perpendicular to the beam of cylinders and are presented
in the form of a prismatic network at square base on side
D
G and height
H
G (along axis X of
cylinders). The grids of the same type are characterized by identical dimensions.
5.5.2 Additional stages of calculation
·
The first additional stage relates to the specification of the type of configuration of the grids
by the operator
DEFI_FLUI_STRU
, then the checking of the good provision of the grids ones
compared to the others, and the ends of the beam.
·
The second stage relates to the resolution of the modal problem under flow. In the loop
on the rates of flow, the matrix of transfer of the forces fluid-rubber bands in the base
modal in air is supplemented by the calculation of a matrix of added damping and a matrix
of added stiffness, been dependant on the grids.
5.5.3 Modeling of the fluid forces exerted on the grids
Calculation of the jump of pressure
First of all, the presence of grids disturbs the stationary field of pressure
P X
()
; one considers
each grid like a singularity involving a jump of pressure, whose expression is put under
form:
)
(
)
(
)
(
)
(
G
G
G
G
G
G
G
X
K
X
U
X
X
P
2
2
1
=
éq 5.5.3-1
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where
G
K
indicate the loss ratio of load due to the grid,
G
U
indicate the stationary speed of the flow on the level of the grid,
G
indicate the density of the flow on the level of the grid,
G
X
indicate the axial position of the medium of the grid along the beam.
Density
)
(
G
G
X
is calculated by linear interpolation of the profile of density
)
(
X
flow in the absence of grid. Stationary speed
)
(
G
G
X
U
is calculated in
application of the conservation of the mass throughput, which results in the following equation:
F
O
O
Fg
G
G
G
G
With
U
With
X
U
X
=
)
(
)
(
where
O
and
O
U
respectively indicate the profile of density and stationary speed of
the flow in foot of beam,
F
With
indicate the fluid section of the beam in the absence of grid,
Fg
With
indicate the fluid section of the beam on the level of the grid:
G
F
Fg
With
With
With
-
=
with
G
With
solid section of the grid.
One deduces the expression from it:
O
O
G
G
F
G
G
G
U
X
With
With
X
U
)
(
)
(
1
1
1




-
=
The loss ratio of load
K
G is calculated starting from the expression of the hydrodynamic force
total which applies to the grid, and we obtain:








-
+
=
)
(
)
(
G
fl
m
G
F
G
G
dg
G
F
G
X
C
P
H
With
With
X
C
With
With
K
2
1
1
éq 5.5.3-2
The 1
Er
term (in
dg
G
C
With
) comes from the effort of drag;
)
(
G
dg
X
C
is the coefficient of drag of
roast. 2
ème
term (in
fl
m
C
P
) is a pitch corrector unit term of friction applied to
beam alone with the altitude of the grid (
P
m is the wet perimeter of the beam in the absence of grid).
By introducing the expression [éq 5.5.3-2] into the relation [éq 5.5.3-1], one thus obtains the expression of
jump of pressure
)
(
G
X
P
for each grid of altitude
G
X
. This jump of pressure is taken into account with
level of the calculation of the stationary field of pressure
)
(X
P
, in the following way:
)
(
)
(
)
(
G
I
I
X
P
X
P
X
P
-
=
+
1
]
,
[
I
I
G
X
X
X
1
+
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Calculation of the specific fluid forces exerted on each grid
According to the same quasi-static approach as that carried out in the paragraph [§5.3.3], one shows that
the action of the fluid field speed
(
)
U U
+
~
around a grid implies a force of drag and one
force lift, according to the angle of attack of the flow compared to the grid. The components y
and Z of the resulting specific force F G are thus written, for each basic cell
K
of one
roast:
()






-
+


-
-
=
K
y
K
pg
K
y
K
dg
G
G
G
y
G
U
Dt
Dy
C
U
T
y
C
K
With
U
~
~
F
2
1
()






-
+


-
-
=
K
Z
K
pg
K
Z
K
dg
G
G
G
Z
G
U
Dt
Dz
C
U
T
Z
C
K
With
U
~
~
F
2
1
where
pg
C
very indicate the slope with null angle of attack of the coefficient of lift around a grid
slightly tilted.
K
With
G
indicate the solid section of the basic cell
K
grid (which includes/understands some
K
).
These forces thus will generate additional terms of added damping and stiffness,
that one obtains after modal decomposition of the movement and projection of these forces on the basis
modal.
5.6
Catch in depreciation account in fluid at rest
Until now, the damping brought to a beam of tubes by the presence of a fluid at rest
was not taken into account in modeling. One thus proposes here a model of damping in
fluid at rest, of which appendix 1 of the note of synthesis of the tests
CHAIR
[bib8] constitutes
reference material.
5.6.1 Modeling of the fluid force at rest exerted on a beam of tubes
The method of calculation of damping in fluid at rest which is implemented here, is one
generalization of the method of CHEN [bib9].
It is a question of calculating the force resulting on each tube from the stresses due to shearing in
boundary layer. It is a nonlinear problem because the fluid damping coefficient depends on
frequency One thus introduces following simplifications:
·
the problem is written using the water frequencies at rest calculated without taking in
count fluid damping,
·
one neglects the coupling between modes.
The linear force
RF
ik
being exerted on the tube
K
subjected to a harmonic movement of the beam
according to the mode
I
at the frequency
F
I
is given by the following relation:
R
R R
F
U U R C
ik
ik
ik K
Dki
=
éq 5.6.1-1
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where
R
U
ik
indicate the speed of slip between the tube
K
and fluid at rest, on both sides of
the boundary layer, defined by:
)
(
Q T
U
U
I
K
im
K
I
&
R
R
=
éq 5.6.1-2
with
)
sin (
)
(
T
F
T
Q
I
I
2
=
and
R
U
im
K
depends on the averages
~
U
y
and
~
U
Z
disturbances speed around the cylinders,
calculated beforehand by the model.
C
Dki
indicate the coefficient of drag of a cylinder of radius
R
K
, subjected to a flow
harmonic of amplitude ad infinitum
R
R
U
U
F
ik
im
K
I
max
=
2
, and is defined by:
2
3
2
2
3
2
)
(
max
K
I
K
I
K
I
Dki
R
F
U
R
F
C
R
=
éq
5.6.1-3
where
indicate the kinematic viscosity of the fluid.
The relation obtained while replacing [éq 5.6.1-2] and [éq 5.6.1-3] in the equation [5.6.1-1] is linearized
by a development in Fourier series (of the term
)
(
Q
)
(
Q
T
T
I
I
&
&
) one retains only the first
term; it comes:
)
(
Q
)
(
T
F
U
R
F
I
I
K
im
K
K
I
&
R
R
2
2
Projection on modal basis
By projection on modal basis and by neglecting the coupling between modes, one obtains the force
generalized being exerted on the beam of tube following the mode
I
:
=
=




=
K
K
L
K
K
I
L
K
I
K
im
I
K
K
I
K
I
I
T
dz
Z
U
F
R
dz
Z
F
T
1 0
1
0
2
2
)
(
Q
)
(
.
)
(
)
(
.
)
(
F
&
R
R
R
R
)
(
F T
I
is thus proportional to
)
(
Q T
I
&
and the vector of modal force associated
(
)
NR
I
I
T
T
,
)
(
F
)
(
F
1
=
=
puts in the form:
[]
()
)
(
Q
C
)
(
F
T
T
has
&
-
=
where
[]
has
C
indicate the matrix of damping added by the fluid at rest.
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6 Bibliography
[1]
NR. GAY, T. FRIOU: Resorption of software FLUSTRU in ASTER. HT32/93/002/B
[2]
L. PEROTIN, Mr. LAINET: Integration of various models of excitations fluid-rubber bands
in Code_Aster
®
: specifications. HT-32/96/014/A
[3]
S. GRANGER, NR. GAY: Software FLUSTRU Version 3. Note principle. HT32/93/013/B
[4]
S. GRANGER: Theoretical complements for the interpretation of tests GRAPPE2 under
flow. HT32/92/025/A
[5]
L. PEROTIN: Note principle of model MOCCA_COQUE. HT32/95/021/A
[6]
F. BEAUD: Note principle of model MEFISTEAU. HT-32/96/005/A
[7]
S. GRANGER: “A Total Model For Flow-Induced Vibration Off Bundles Tube In Cross-country race-Flow”
ASME Journal off Presses Vessel Technology, 1991, vol. 113, pp. 446-458.
[8]
J-L. WISE, F. BEAUD, P. MANDOU: Synthesis of the tests CHAIR in axial flow and
interpretation with model MEFISTEAU. HT-32/99/003/A
[9]
R.D. BLEVINS: “Flow-Induced Vibrations”, Krieger Publishing Company, 1994, pp308-310.
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