Code_Aster ®
Version
7.4
Titrate:
Modeling of the turbulent excitations
Date:
05/04/05
Author (S):
A. ADOBES, L. VIVAN Key
:
R4.07.02-B Page
: 1/26
Organization (S): EDF-R & D/MFTT, CS
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
Document: R4.07.02
Modeling of the turbulent excitations
Summary:
One describes the modeling of the turbulent excitations available in Code_Aster and the way in which these
last are taken into account in a calculation of dynamics. The turbulent excitations are characterized
by a spectral concentration of efforts, specified using operator DEFI_SPEC_TURB [U4.44.31]. Their catch in
count in a calculation of dynamics is done by projection of the spectrum on the basis of modal structure of which
one wants to calculate the answer. The operations of projection are carried out using the operator
PROJ_SPEC_BASE [U4. 63.14].
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the turbulent excitations
Date:
05/04/05
Author (S):
A. ADOBES, L. VIVAN Key
:
R4.07.02-B Page
: 2/26
Count
matters
1 Principle of calculation ................................................................................................................................... 3
1.1 Determination of a modal base of the system under flow and projection of the excitation ....... 3
1.2 Calculation of the response to the turbulent excitation: frequential resolution .......................................... 4
1.2.1 Introduction ............................................................................................................................. 4
1.2.2 Calculation of the interspectres of modal excitations ..................................................................... 4
1.2.3 Calculation of the interspectres of modal answer ......................................................................... 4
1.2.4 Recombination on physical basis ........................................................................................ 5
1.2.5 Statistical elements .............................................................................................................. 5
1.3 Calculation of the response to the turbulent excitation: temporal resolution ............................................. 6
1.3.1 Factorization of the density interspectrale ............................................................................... 6
1.3.2 Generation of the random modal excitations ..................................................................... 6
1.3.3 Modification of a modal base and projection ........................................................................ 6
1.3.4 Definition of the obstacles ......................................................................................................... 6
1.3.5 Dynamic resolution ............................................................................................................ 6
1.3.6 Projection of Ritz ................................................................................................................... 6
2 Models of turbulent excitation applicable to the telegraphic structures ....................................................... 7
2.1 General principles .......................................................................................................................... 7
2.1.1 Assumptions ............................................................................................................................. 7
2.1.2 Calculation of the interspectres of modal excitations ..................................................................... 7
2.2 Spectra of the type “length of correlation” ..................................................................................... 9
2.2.1 Key words ................................................................................................................................ 9
2.2.2 Definition of the model ............................................................................................................... 9
2.2.2.1 Density interspectrale ................................................................................................ 9
2.2.2.2 Modeling of the spectrum of turbulence by an expression with variables séparées..10
2.3 Model of turbulent excitation distributed .......................................................................................... 17
2.3.1 Key words .............................................................................................................................. 17
2.3.2 Decomposition on a family of functions of form ......................................................... 17
2.3.3 Setting in equations ................................................................................................................ 18
2.3.3.1 Application of a turbulent excitation distributed ....................................................... 18
2.3.3.2 Turbulent excitation identified on model GRAPPE1 ..................................... 18
2.3.3.3 Projection of the excitation on modal basis .............................................................. 19
2.4 Model of localized turbulent excitation ........................................................................................ 21
2.4.1 Key words .............................................................................................................................. 21
2.4.2 Bases .......................................................................................................................... 21
2.4.3 Setting in equations ................................................................................................................ 22
2.4.3.1 Application of a localized turbulent excitation ..................................................... 22
2.4.3.2 Turbulent excitation identified on model GRAPPE2 ..................................... 24
3 Bibliography ........................................................................................................................................ 25
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the turbulent excitations
Date:
05/04/05
Author (S):
A. ADOBES, L. VIVAN Key
:
R4.07.02-B Page
: 3/26
1
Principle of calculation
1.1 Determination of a modal base of the system under flow and
projection of the excitation
The calculation of the dynamic response of a system to a turbulent excitation induced by a flow
fluid is carried out by respecting the following stages:
1) initially, one calculates the modal base of the system except flow using
operator MODE_ITER_SIMULT [U4.52.03],
2) one defines then the characteristics of the studied configuration, for taking into account of
phenomenon of coupling fluid-structure, using operator DEFI_FLUI_STRU [U4.25.01].
This operator allows for example to inform the profiles speed associated with the zones
of fluid excitation, for configurations of the type “beam of tubes under flow
transverse ". It produces a concept of the type [type_flui_stru] intended to be used by
operators implemented downstream in the command file,
the 3) modal characteristics of the system under flow are then calculated using
operator CALC_FLUI_STRU [U4.66.02]. One has at output a modal base for
each rate of flow,
4) the definition of the turbulent excitation is done then by a call to the operator
DEFI_SPEC_TURB [U4.44.31]. Modelings available are as follows:
·
spectra of the type “length of correlation”, specific of the configurations of the type
“beam of tubes under transverse flow”, for the application to the vibrations of
tubes of Steam Generator. The key words corresponding factors are SPEC_LONG_COR_1,
SPEC_LONG_COR_2, SPEC_LONG_COR_3 and SPEC_LONG_COR_4. These spectra are
preset; however, the user can adjust the parameters of them. This part is
developed with the paragraph [§2.2],
·
model of turbulent excitation distributed. The key word factor corresponding is
SPEC_FONC_FORME. The spectrum of excitation is defined by its decomposition on one
family of functions of form while providing, on the one hand a matrix interspectrale, and
in addition a list of functions of form associated with this matrix. Concepts
[interspectre] and [function] associated must be generated upstream. In the case
component “control rod”, the user can also use a spectrum of
turbulence preset, identified on model GRAPPE1. This part is developed with
paragraph [§2.3],
·
model of localized turbulent excitation. The key word factor corresponding is
SPEC_EXCI_POINT. It is used in the case of a spectrum of excitation associated with one or
several specific forces and moments. The definition of the excitation is done then in
providing:
-
a matrix interspectrale of excitations (the concept [interspectre] associated
must be generated upstream),
-
the list of the nodes of application of these excitations,
-
the nature of the excitation applied of each one of these nodes (force or moment),
-
directions of application of the excitations thus defined.
This part is developed with the paragraph [§2.4].
5) The projection of the spectrum of turbulent excitation previously definite, on the basis of modal
structure under flow, is then carried out using operator PROJ_SPEC_BASE
[U4.63.14].
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
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Code_Aster ®
Version
7.4
Titrate:
Modeling of the turbulent excitations
Date:
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Author (S):
A. ADOBES, L. VIVAN Key
:
R4.07.02-B Page
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1.2 Calculation of the response to the turbulent excitation
: resolution
frequential
1.2.1 Introduction
The calculation of the frequential response of the structure or the system coupled fluid-structure is done in
three stages:
1) calculation of the interspectres of modal excitations,
2) calculation of the interspectres of modal answer,
3) recombination on the physical basis.
Initially, one introduces for each mode the transfer function of the mechanical system
(structure alone or system coupled fluid-structure). Each of the three stages above is then
detailed.
1.2.2 Calculation of the interspectres of modal excitations
Interspectres of modal excitations S
(F U
QiQj
,) are determined by projection of the spectrum
of turbulent excitation on the basis of modal system mechanical (structure alone or coupled system
fluid-structure). This stage of projection is detailed in paragraph [§2] for the various models
applicable to telegraphic structures.
1.2.3 Calculation of the interspectres of modal answer
Interspectres of modal displacements S
(F U
qiqj
,) result then from the interspectres
modal excitations S
(F U
QiQj
,) using the following relation:
S
*
Q Q
(F, U) = Hi (F, U) SQiQj (F, U) H J (F, U) éq
1.2.3-1
I J
where H * I (F, U) indicates the combined complex of the transfer function H (F U
I
,) of the system
mechanics considered. Being given a frequency F and a rate of flow U, the function of
transfer H (F U
I
,) of the mechanical system for mode I is defined by:
1
H (F, U
I
) =
éq
1.2.3-2
F 2
F
M
2
- + 2 J
I I
+
1
F
I
F
I
I
where Mi indicates the modal mass of mode I, and F
I
I indicate respectively, at the speed U,
pulsation and the Eigen frequency of mode I, I indicates, at the speed U, the reduced damping of
mode I, and J indicates the complex number such as J 2 = - 1.
The calculation of the interspectres of modal displacements starting from the interspectres of modal excitations
and of the transfer functions is carried out using operator DYNA_SPEC_MODAL [U4.53.23].
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
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Code_Aster ®
Version
7.4
Titrate:
Modeling of the turbulent excitations
Date:
05/04/05
Author (S):
A. ADOBES, L. VIVAN Key
:
R4.07.02-B Page
: 5/26
One deduces in particular from [éq 1.2.3-2] the relation binding the autospectres modal displacements to
autospectres of modal excitations:
S
2
qiqi (F, U) = Hi (F, U) SQiQi (F, U)
éq
1.2.3-3
2
where H (F U
I
,) the square of the module of H (F U indicates
I
,)
1.2.4 Recombination on physical basis
Being given a rate of flow U, the interspectre of physical displacement Known U (X, X, F
1
2
)
1 2
at the points of X-coordinates X and X
1
2, at the frequency F, is obtained by modal recombination. This
operation is written:
NR NR
S
(X, X, F) = (X) (X) S
J
2
Q Q (F, U
U U
I
)
éq
1.2.4-1
1 2
1
2
1
I J
I 1
= J 1
=
Where NR indicates the number of modes of the base; I (xk) is the component at the point of discretization
xk of the deformation of the ième mode following the direction of space considered.
The recombination on physical basis is carried out using operator REST_SPEC_PHYS
[U4.63.22]. The direction of space considered is specified at the time of the call to this operator.
1.2.5 Elements
statistics
The modal variance 2 ()
I U, associated at the speed U, is expressed as follows:
2i U
() = 2 S
Q Q (F, U) df
éq
1.2.5-1
I J
0
At the rate of flow U, value RMS
()
RMS X of response in an item X of the structure is
data by:
NR
(X)
2
= (X) 2
(U)
RMS
I
I
éq
1.2.5-2
I 1
=
Where NR indicates the number of modes of the base and ()
I X is the component as in point X of
deformation of the ième mode following the direction of space considered.
This operation is carried out by operator POST_DYNA_ALEA [U4.84.04].
Handbook of Référence
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Code_Aster ®
Version
7.4
Titrate:
Modeling of the turbulent excitations
Date:
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Author (S):
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:
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1.3
Calculation of the response to the turbulent excitation: temporal resolution
The temporal resolution proceeds according to the sequence of the following operations:
1.3.1 Factorization of the density interspectrale
Operator FACT_INTE_SPEC [U4.36.04] carries out the factorization of the density interspectrale
modal excitations S
(F U
QiQj
,), before application of the method of Monte Carlo.
1.3.2 Generation of the random modal excitations
Operator GENE_FONC_ALEA [U4.36.05] generates random modal excitations Q (T)
I
in
carrying out hard copies by the method of Monte Carlo. Operator RECU_FONCTION [U4.32.03]
allows to recover each evolution Q (T)
I
.
1.3.3 Modification of a modal base and projection
Operator MODI_BASE_MODALE [U4.66.21] modifies the modal base of the structure in substituent with
initial characteristics those obtained for a rate of flow considered.
Operator PROJ_MATR_BASE [U4.63.12] allows the projection of the matrices of mass and stiffness
assembled on the new modal basis previously definite.
1.3.4 Definition of the obstacles
The definition of the geometry of the obstacles is carried out, if necessary, using the operator
DEFI_OBSTACLE [U4.44.21].
1.3.5 Resolution
dynamics
Transitory dynamic calculation for mode I (1 I NR) is carried out using a diagram
of numerical integration with operator DYNA_TRAN_MODAL [U4.53.21].
M Q & (T) + C Q & (T) + K Q (T) = Q (T
II I
II I
II I
I
) éq
1.3.5-1
Where M, C and K
II
II
II indicate respectively the generalized mass, damping and stiffness
associated the ième mode; Q (T) and Q (T)
I
I
displacement and the excitation indicate respectively
generalized associated the ième mode.
1.3.6 Projection of Ritz
The restitution on physical basis is carried out using a projection of Ritz:
NR
U (X, T) = U (X) Q (T)
I
I
éq
1.3.6-1
i=1
U (X, T) indicates the assembled vector of physical displacements; U (X)
I
is the assembled vector
defining the ième modal form and Q (T)
I
generalized displacement following the ième mode.
This last operation is carried out using operator REST_BASE_PHYS [U4.63.21].
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
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Code_Aster ®
Version
7.4
Titrate:
Modeling of the turbulent excitations
Date:
05/04/05
Author (S):
A. ADOBES, L. VIVAN Key
:
R4.07.02-B Page
: 7/26
2
Models of turbulent excitation applicable to the structures
telegraphic
2.1 Principles
Generals
2.1.1 Assumptions
One supposes that the linear excitation induced on the telegraphic structure by turbulence of the flow
can be modelled in the form of a stationary process random ergodic Gaussian of
null average. This turbulent excitation thus is entirely characterized by its density
interspectrale S (X, X
F
1
2,), where X and X
1
2 are two unspecified points of the beam and indicates
the pulsation. The turbulent excitation applied to the structure is thus characterized by its density
interspectrale S F.
Moreover, one supposes that the turbulent forces are independent of the movement of the structure.
The turbulent excitation is identified in experiments on a model of reference. It is
then applicable to any real component in geometrical similarity with the model of reference.
2.1.2 Calculation of the interspectres of modal excitations
One indicates by F (X S
T
,) linear density of turbulent excitation exerted on the beam; X is
the current X-coordinate of a point of the beam and S the complex pulsation (variable of Laplace). They are done
additional assumptions following H1 and H2:
H1. The excited length is lower than the overall length L of the beam.
H2. The expression of F (X S
T
,) does not depend on the origin of the excited zone Xe; that is translated
by F (X, S) = F (X - X, S
T
T
E
).
In this case, one can express the linear density ft in the following form:
1
D D
F (X, S) =
U 2
D C,
,
, S
T
F
, Re
éq
2.1.2-1
2
D
L
R
H
E
X - X
sD
UD
with: =
E
S =
=
L
R
Re
U
E
Where the density of the fluid indicates, U is the mean velocity of flow of the fluid, D and
Dh are respectively the diameter of the structure and the hydraulic diameter, C F represents it
adimensional coefficient of turbulent force, X is the current X-coordinate of a point of the beam, Xe
indicate the X-coordinate of the origin of the excited zone, represents It the excited length, is the variable
of space reduced, S is the complex pulsation (variable of Laplace), Sr is the complex pulsation
reduced, is the kinematic viscosity of the fluid, finally “Re” indicates the Reynolds number.
Handbook of Référence
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Titrate:
Modeling of the turbulent excitations
Date:
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Author (S):
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:
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By geometrical assumption of similarity of the real component with the model of reference, one
obtains:
1
F (X, S) =
U 2
D C (, S
T
F
R, Re)
éq
2.1.2-2
2
Thus, the modal turbulent excitation Q (S)
I
can be written in the field of Laplace (assumption H2):
X + L
E
E
1
Q (X) =
F
(X, S) (X) dx = L F
(L, S) (L
+ X) D
I
T
I
E
T
E
I
E
E
éq
2.1.2-3
Xe
0
where ()
I X is the component of the ième modal deformation according to the direction of space in which
acts the turbulent excitation.
By means of the expression [éq 2.1.2-2], one deduces:
1
1
Q (S) = U 2 DLL C
(, S, Re) (L + X) D
I
E
F
R
I
E
E
éq
2.1.2-4
2
0
The densities interspectrales of modal turbulent excitations are expressed then in the form:
1
2
1 1
2
D
S
(F, U) = U DLL
T (, F
1
2
R, Re) (L
X) (L
X) D D
Q Q
E
2
+
+
U
I
1 E
E
J
2nd
E
I J
1
2
0 0
éq 2.1.2-4
with
1 I, J NR, where NR is the number of modes selected to determine the answer of
structure;
T: interspectre of cf enters and
1
2;
fD
Fr =
: reduced frequency.
U
Note:
In what follows, one preserves the assumptions H1 and H2 and one notes I (F
ij
R, Re) the integral:
1 1
Iij (F, Re
R
) =
T (1,2, F, Re
R
) (
I
1L + X) (
E
E
J
2 L + X)
D
E
E
1
D 2 éq
2.1.2-5
0 0
Handbook of Référence
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Titrate:
Modeling of the turbulent excitations
Date:
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Author (S):
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:
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Using this notation, the interspectres of modal excitations are written:
1
2
2
D
S
(F, U) = U DLL
I (F
Q Q
E
ij
R, Re)
éq
2.1.2-6
I J
2
U
The expression of the autospectres of modal excitations is similar:
1
2
2
D
S
(F, U) = U DLL
I (F
Q Q
E
II
R, Re)
éq
2.1.2-7
I I
2
U
2.2
Spectra of the type “length of correlation”
2.2.1 Key words
The key words factors SPEC_LONG_COR_i (I varying from 1 to 4) of operator DEFI_SPEC_TURB
[U4.44.31] give access spectra of the type “length of correlation”. These spectra,
specific of the configurations of the type “beam of tubes under transverse flow”, are
preset but the user can adjust the parameters of them.
2.2.2 Definition of the model
2.2.2.1 Density
interspectrale
In the case of spectra of the type “length of correlation”, the density interspectrale characterizing
the turbulent excitation is supposed to be able to be put in a form at separable variables such
that:
S (X, X,) = S () (X, X
I
1
2
0
0
1
2)
éq
2.2.2.1-1
In this expression, S ()
0 represent the autospectre turbulence and 0 (1
X, x2) indicates one
function of space correlation defined by:
- X - X
2
1
0 (1
X, x2) = exp éq
2.2.2.1-2
C
where X and X
1
2 indicate the X-coordinates of two points of observation and C represents the length of
correlation.
Four analytical expressions are available in operator DEFI_SPEC_TURB [U4.44.31]. These
expressions correspond each one to a particular representation of S ()
0.
The user defines a spectrum of turbulence by choosing one of these analytical forms, of which it
can adjust the parameters.
Handbook of Référence
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Titrate:
Modeling of the turbulent excitations
Date:
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Author (S):
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2.2.2.2 Modeling of the spectrum of turbulence by an expression with separate variables
·
General case
The function tt introduced into the relation is modelled by a form with variables
separated:
NS
(, F, Re) = (,
1
2
1
2)
T
R
N
N (F R, Re)
éq
2.2.2.2-1
N 1
=
Where NR S indicates the degree of the base of the functions of form N and N is a function
independent of the variable of space. These two functions are stored in the base of
data and can be selected by the user.
The autospectres of modal excitations are given by [éq 2.1.2-7] while introducing:
NS
I (F, Re) = L2 (F
II
R
nor
N
R, Re) éq
2.2.2.2-2
n=1
1 1
with: L2 = (1
, 2) (L + X
1
) (L + X
2
) D D
nor
N
I
E
E
I
E
E
1 2
éq
2.2.2.2-3
0 0
The principle of calculation is as follows: one first of all calculates the values of L2ni while realizing
the calculation of the double integrals; one calculates then N (Fr, Re) for all the values of
N; one obtains finally the expression of S
(F, U
Q Q
) using the equation [éq 2.1.2-4].
I I
·
Particular case: model used for the tubes of steam generator
The particular case of the study of the tubes of Steam Generator corresponds to a particular case of the general case
presented previously by posing NS = 1. The interspectre of turbulent excitation between two
points of reduced X-coordinates and
1
2 is then given by:
-
(, F
1
2
1
2
, Re) = exp -
L
T
R
E (F R, Re) éq
2.2.2.2-4
C
where C represents the length of correlation of the turbulent forces and is the length
excited. In general, one takes C of about 3 to 4 times the diameter external of the tube.
Handbook of Référence
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Titrate:
Modeling of the turbulent excitations
Date:
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Author (S):
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:
R4.07.02-B Page
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Spectra of autocorrelation of modal excitations, in the case of profiles speed and of
density constant, are given by:
2
D
SQiQi (F U)
1
,
2
=
U
E
DLL
III (F, Re
R
) éq 2.2.2.2-5
2
U
with:
1 1
-
I (F, Re) = (F, Re) .exp- 2
1 L
E.
I
(L + X
1 E
E) I
(L + X
2nd
E) .d D
II
R
R
1
2
C
0 0
éq 2.2.2.2-6
In the general case of profiles of density and unspecified rate of flow,
one a:
2
SQiQj (F U)
1
D
,
= D S (Fr)
2 U
X +L X +L
E
E E
E
x2 - 1
X
exp -
E (1
X) E (X)
2
2 U.E. (X) 2
1 U.E. (x2) I (1
X) I (x2) 1
dx dx2
X
X
C
E
E
éq 2.2.2.2-7
Where D is the diameter of the structure, is the length of the excited zone, Xe is
the X-coordinate of the origin of the excited zone, U is the mean velocity of the flow, S (Fr)
is a spectral concentration of separate excitation the mean velocity of the flow
U, X and X
1
2 are the curvilinear X-coordinates of two points of observation on the tube, (X
E
)
is the profile of density of the fluid along the tube, U (X)
E
is the profile speed
transverse of the flow along the tube and C indicates the length of correlation.
Adimensional profiles of density and transverse speed of the flow
external are in the following way defined:
()
E X indicating the evolution of the density of the external fluid along the zone
immersed Limm of the tube, one indicates by the density of the external fluid realized
on the immersed part of the tube:
X
+ L
imm
imm
1
=
(X) dx éq 2.2.2.2-8
L
E
imm
ximm
One indicates by R (X) the adimensional profile of density such as () =
E X
R (X).
Handbook of Référence
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Modeling of the turbulent excitations
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U (X)
E
indicating the evolution rate of flow of the external fluid over the length
excited of the tube, one indicates by U the rate of flow of the fluid realized on
excited length of the tube:
X + L
E
E
1
U =
U (X) dx
éq 2.2.2.2-9
L
E
E
Xe
One indicates by (
U X) adimensional profile transverse speed of the external flow,
such as U
X = U U X
E ()
().
By introducing the average sizes and the adimensional profiles into the expression
[éq 2.2.2.2-7], one obtains:
2
X + L X + L
1
2
D
E
E E
E
X - X
S
2
1
QiQj (F, U) =
U D.
S (Fr)
exp
2
-
U
C
X
X
éq
2.2.2.2-10
E
E
2
2
E
(x1) E
(x2) Ue (x1) Ue (x2) I (x1) J (x2) dx dx
1
2
X - X
After having noted =
E, it comes:
1 1
1
-
2
3
3 2
x2
1
X
S
(F, U) = U D L S (F) × exp -
R
QiQj
E
R
(1 + X) R (2 + X)
4
E
L
E
E
L
E
0 0
C
2
U (L + X
E
E) 2
1
U (2L + X
E
E) I (1L + X
E
E) I (2 L + X
E
E)] d1d 2
éq 2.2.2.2-11
Where S (Fr) represents the spectrum of turbulence, definite according to a reduced frequency Fr
(a Strouhal number). For a tube in interaction with a transverse flow, F R
is written:
fD
Fr =
U
where F is the dimensioned frequency, D is the diameter of the tube and U is speed
average of the flow.
The double integral of the expression [éq 2.2.2.2-11] is evaluated by the operator
PROJ_SPEC_BASE [U4.63.14].
Handbook of Référence
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7.4
Titrate:
Modeling of the turbulent excitations
Date:
05/04/05
Author (S):
A. ADOBES, L. VIVAN Key
:
R4.07.02-B Page
: 13/26
·
Case of multiple zones of excitation
If there are several zones of excitations, the notations are introduced
additional following:
The zone of excitation K being located by its X-coordinate xk beginning and its Lk length, one
note U (X)
K
profile speed transverse of the fluid flow on the level of this zone.
average transverse speed on the zone of excitation K is then given by:
X + L
K
K
1
U =
U (X) dx
K
L
K
K
xk
One deduces the adimensional profile from it transverse speed, standardized on the zone K:
U (X)
U (X
K
K
) =
The U.K.
K indicating the total number of zones of excitation, average transverse speed on
the whole of the zones of excitation is defined by:
K
1
U =
U
K
K
K =1
If Vgap is the speed intertube at the entry of the Steam Generator (the range speeds retailers is defined
in CALC_FLUI_STRU [U4.66.02] using key word VITE_FLUI), one proceeds to one
the second standardization; transverse speed in an item X located in the zone of excitation K
is given by:
U (X)
U
V (X) = V
K
= V
K U (X
K
gap
)
U
gap the U.K.
Thanks to this standardization, the arithmetic mean transverse speed on all them
zones of excitation is equal at the speed intertube; one has indeed:
+
1 K
1 X L
K
K
V (X) dx V
K
L
K
gap
=
K 1
=
K
X
K
The calculation of the interspectres of modal excitations, realized by operator PROJ_SPEC_BASE
[U4.63.14], is done by adding the contributions with each zone of excitation according to
the relation:
2 K
1
D
S
,
=
I J (F V
)
D
Lk
S
K
ij
(F
Q Q
gap
R)
2
×
×
V
K =1
K
Handbook of Référence
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Titrate:
Modeling of the turbulent excitations
Date:
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Author (S):
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:
R4.07.02-B Page
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with:
U
fD
V = V
K
K
K
gap ×
and F
=
U
R
Vk
X + L X + L
K
K K
K
- X - X
2
1
and Lk =
exp
(X) (X V
) 2 (X V
) 2 (X) (X) (X) dx dx
ij
E 1 E 2 K 1 K
2
I
1
I
2
1
2
C
X
X
K
K
that is to say:
X + L X + L
K
K K
K
- X - X
Lk = V 4 ×
exp
2
1 (X) (X) u2 (X) u2 (X) (X) (X) dx dx
ij
K
E 1 E 2 K 1 K 2 I 1 I
2
1
2
C
X
X
K
K
One poses:
X + L X + L
K
K K
K
- X - X
L K =
exp
2
1 (X) (X) u2 (X) u2 (X) (X) (X) dx dx
ij
E 1 E 2 K 1 K 2 I 1 I
2
1
2
C
X
X
K
K
The expression of the interspectres of modal excitations becomes then:
2 K
1
D
S
,
=
4
I J (F V
)
D
V
L K
S
K
K
ij
(F
Q Q
gap
R)
2
×
×
×
V
K =1
K
from where:
K
1
S
,
=
3 × 3 × ×
I J (F V
) D
(V lk S (F K
Q Q
gap
K
ij
R)
4
K =1
·
Analytical expressions of the spectra available for the user
Various analytical expressions of the spectra available in the operator
DEFI_SPEC_TURB [U4.44.31] are as follows:
·
SPEC_LONG_COR_1
Each Ui speed defined by the user by discretizing the range speeds
[U - U
kN
min
max] explored is initially standardized in the Ui form while applying
the equation:
The U.K.
U kN = U
I
I
U
where the U.K. and U respectively indicate the speed realized on the zone of excitation “K”, and
mean velocity on the unit of the zones of excitation.
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Titrate:
Modeling of the turbulent excitations
Date:
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Author (S):
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:
R4.07.02-B Page
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A Reynolds number “local” Rik
E, associated the zone “K” and the Ui speed is then
calculated starting from the local characteristics of the flow:
kN
U
D
Reik
I
=
The spectrum of turbulent excitation associated the zone “K” and the Ui speed is given under
the shape of a vector S ik, having as many components as of points used for
to discretize the frequential interval [F
- F
min
max], support of the excitation. J-ième
component S ikj of this vector is provided by the expression:
Sik
0
J =
éq
2.2.2.2-12
2
F ik 2
F ik 2
rj
rj
1 -
2
+ 4
F
F
rc
rc
F ik
rj is provided by:
F D
F ik
J
rj =
U kN
I
where:
F J is the value of frequency associated with the j-ième component in the discretization with
the frequential interval [F
- F
min
max], F rc is a cut-off frequency being worth 0.2; O,
depend on the Reynolds number according to equations' provided in the table
below:
Rik
E
O
] -; 1.5.104]
2.83504 10-4 3
0.7
] 1.5.104; 3.5.104]
Idem Idem
20 42
.
- 14 10-4 Rik - 9 8
. 110-8
2
Rik
1
+ 19
. 7 10 12
3
-
ik
R
13
. 10-4
E
E
E
-
ik
-
ik
- 35 95
. 10 17
4
R
+ 34 69
.
10 22
5
R
E
E
] 3.5.104; 5.104] Idem
4 0.3
] 5.104; 5.5.104] 50.18975
104 Idem
Idem
] 5.5.104; +]
Idem
4 0.6
Handbook of Référence
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Titrate:
Modeling of the turbulent excitations
Date:
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Author (S):
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:
R4.07.02-B Page
: 16/26
·
SPEC_LONG_COR_2
The spectrum of turbulent excitation is written:
S (F
0
R) =
éq
2.2.2.2-13
F
R
1 +
frc
The default values of the parameters are as follows:
- 3
0 = 15
. 10
= 2 7
.
F = 01
.
rc
·
SPEC_LONG_COR_3
The spectrum of turbulent excitation is written:
S (F
0
R) =
éq
2.2.2.2-14
F
R
with:
=
0
0 (F
)
rc
=
(F)
rc
The default values of the parameters are as follows: F rc = 2
If F F
R
rc, one a:
3
-
0 = 510
= 05
.
if not
- 5
0 = 4 10
= 35
.
·
SPEC_LONG_COR_4
The spectrum of turbulent excitation is written:
S (F
0
R) =
éq 2.2.2.2-15
F
R v
with:
1
0 =
10
- 2
6 8
. 10
Handbook of Référence
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Modeling of the turbulent excitations
Date:
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Author (S):
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:
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The other parameters are defined by:
=
To 0 5. -
B 15. -
C 2 5. -
D 35.
v
v
v
v
= 2
= 4
v the rate of vacuum indicates; v
=
is the volume throughput defined by
v
mU; m is the flow
mass and U indicates the mean velocity of the flow. Values of the coefficients of
polynomial in v are as follows:
To = 24 042
.
B = - 50 421
.
C = 63 483
.
D = 33 284
.
2.3
Model of turbulent excitation distributed
2.3.1 Key words
The key word factor SPEC_FONC_FORME of operator DEFI_SPEC_TURB [U4.44.31] makes it possible to define
a spectrum of excitation by its decomposition on a family of functions of form. The user with
possibility of defining the spectrum by providing a matrix interspectrale and a list of functions of
form associated. The concepts [interspectre] and [function] must then be generated in
upstream. In the case of the component “control rod”, the user can also use one
preset spectrum of turbulence, identified on model GRAPPE1.
2.3.2 Decomposition on a family of functions of form
The model of turbulent excitation distributed supposes that the instantaneous linear density of the forces
turbulent F (X T
T
,) can be broken up on a family of functions of form ()
K X of
dimension K in the following way:
K
F (X, T) =
T
K (X) K T () éq
2.3.2-1
k=1
Coefficients ()
K T at every moment define the decomposition of the turbulent excitation on
family of functions of form.
The density interspectrale of turbulent excitation between two points of the telegraphic structure of X-coordinates x1
and x2 is written then:
K K
S (X, X,) = (X) (X) S
F
K
L
kl ()
1
2
1
2
éq
2.3.2-2
K 1
= K 1
=
This formulation makes it possible to take into account an excitation whose space distribution is
unspecified.
Handbook of Référence
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Titrate:
Modeling of the turbulent excitations
Date:
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Author (S):
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:
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2.3.3 Setting in equations
2.3.3.1 Application of a turbulent excitation distributed
The length of application L is characterized in an intrinsic way by the field of definition of
functions of form associated with the excitation. The zone of application is determined by the data of
name of the node around of which it is centered.
xn indicating the X-coordinate locating this node, the turbulent excitation is imposed on
field [X - L 2, X + L
N
N
2].
The turbulent excitation being able, in addition, to be developed in a way correlated in both
directions Y and Z orthogonal with the axis of the telegraphic structure, the functions of form are a priori
vectors with two components.
One thus informs, by convention, these functions about the interval [0,2L], the field [0, L] being
associated the direction Y and the field [L, 2L] being associated direction Z.
2.3.3.2 Turbulent excitation identified on model GRAPPE1
The functions of form K are the first 12 modal deformations of inflection of the structure
identified in experiments, distributed according to two orthogonal directions' with the principal axis of
beam. The general analytical expression of these deformations is as follows:
R
(X)
Yk
K (X) =
éq
2.3.3.2-1
Zk (X)
with:
N
N
N
N
Yk
Yk
Yk
Yk
Yk (X) =
Yk
With cos
X
Yk
B
sin
X
Yk
C
CH
X
Yk
D
HS
X
L + L + L +
L éq 2.3.3.2-2
N
N
N
N
Zk
Zk
Zk
Zk
Zk (X) =
Zk
With cos
X
BZk sin
X
CZk CH
X
Zk
D
HS
X
L +
L +
L +
L éq 2.3.3.2-3
where nYk and nZk indicate numbers of waves, L is the length of application of the excitation and them
coefficients AYk, BYk, CYk, DYk, AZk, BZk, CZk, DZk are constant real coefficients
characteristics of the function of form considered.
The first 6 functions of form are associated the direction Y and AZk, BZk, CZk, DZk are
thus null, for 1 K 6.
The 6 last functions of form are associated direction Z and AYk, BYk, CYk, DYk are
thus null, for 7 K 12.
This family of functions of form is thus characterized by 5x12 = 60 real coefficients.
Handbook of Référence
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Modeling of the turbulent excitations
Date:
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:
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The turbulent excitation identified on model GRAPPE1 is homogeneous in the two directions
orthogonal with the axis of the telegraphic structure, turbulence being décorrélée between these two directions.
The matrix interspectrale [Skl] identified on model GRAPPE1 is thus a matrix of
dimension 12x12, consisted of two identical diagonal blocks of dimension 6:
[
S
0
S
O
K =
L]
[()] []
[]
0
[S ()
O
]
By square property of symmetry, this matrix is entirely defined by the data of the part
triangular higher (or lower) of [S ()
O], is 21 interspectres. For each one among them, them
characteristic parameters are the level of plate, the cut-off frequency and the slope of the spectrum
beyond this frequency.
The matrix interspectrale of turbulent excitation identified on model GRAPPE1 is thus
characterized by 63 real coefficients (3x21).
Note:
Excitations GRAPPE1 are available to two flows of reference. The whole of
data characterizing these excitations thus represents 246 real coefficients ([60+63] x2).
2.3.3.3 Projection of the excitation on modal basis
One notes:
DY (X)
I
I (X) =
the modal deformed i-éme of the structure.
DZi (X)
Are ik the co-ordinates of the modal i-éme deformed of the structure on the basis of the functions of
form K (X):
K
(X) =
I
ik
K (X)
éq
2.3.3.3-1
K =1
The interspectres of modal excitations SQ Q () applied to the structure are written then:
I J
K K
S
() =
S
Q Q
ik
jl
(
)
éq
2.3.3.3-2
I J
K L
K =1k =1
For each mode I of the structure, the coefficients ik are given by integrating the equation
[éq 2.3.3.3-1] prémultipliée by the functions J, on the applicability of the excitation. One
obtains as follows:
X + L
0
2
K
X + L
0
2
(X + L 2) (X) dx = (X + L 2)
J
I
ik
J
K (X + L 2) dx
X - L 2
K =1
X - L
0
0
2
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X + L
0
2
K
L
(X + L 2) (X) dx = (X)
J
I
ik J
K (X) dx
(
I, J)
éq
2.3.3.3-3
X - L 2
K =1
0
0
For each I, the equation [éq 2.3.3.3-3] is written in matric form:
[has] () = (B
jk
ik
ij)
éq
2.3.3.3-4
with:
L
has
=
(X) (X) dx
jk
J
K
0
that is to say:
L
has
= ((X) (X) + (X) (X))dx
jk
Yj
Yk
Zj
Zk
0
and
X + L
0
2
B =
(X + L 2) (X) dx
ij
J
I
X - L
0
2
that is to say:
X + L
0
2
B =
(DY (X) (X + L 2) + DZ (X) (X + L 2))dx
ij
I
Yj
I
Zj
X - L
0
2
The resolution of each linear system of equations leads to the ik.
The calculation of the scalar products is carried out in operator PROJ_SPEC_BASE [U4. 63.14].
Note:
1)
functions
K (X) represent, in practice, the modal deformations raised on
model. The system (), with dominating diagonal, is thus well conditioned. In
private individual, when the telegraphic structure model has a homogeneous linear density, them
functions K (X) are orthogonal and the matrix [jk has] is diagonal.
2) Tests comparing the applicability of the excitation with the field of definition of
the structure are carried out.
Handbook of Référence
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Modeling of the turbulent excitations
Date:
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:
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2.4
Model of localized turbulent excitation
2.4.1 Key words
The key word factor SPEC_EXCI_POINT of operator DEFI_SPEC_TURB [U4.44.31] is used in
cases of a spectrum of excitation associated with one or more forces and moments specific. The user
can define the spectrum while providing:
·
a matrix interspectrale of excitations (the concept [interspectre] associated must be
generated upstream),
·
the list of the nodes of application of these excitations,
·
the nature of the excitation applied of each one of these nodes (force or moment),
·
directions of application of the excitations thus defined.
It can also use a preset spectrum of turbulence, identified on model GRAPPE2.
2.4.2 Bases
The model of localized turbulent excitation is a particular case of the model of turbulent excitation
distributed. Thus, one supposes just as in paragraph [§2.3.2] that instantaneous linear density
turbulent forces F X T
T (,) can be broken up on a family of functions of form
K (X) in the following way:
K
F (X, T) = (X) (T
T
K
K
) éq
2.4.2-1
K =1
The coefficients K (T) at every moment define the decomposition of the turbulent excitation on
family of functions of form.
The density interspectrale of turbulent excitation between two points of the telegraphic structure of X-coordinates x1
and x2 is written then:
K K
S (X, X,) = (X) (X) S
F
K
L
(
)
1
2
1
2
éq
2.4.2-2
K L
K 1
= L 1
=
The characteristic of the model of localized turbulent excitation is due to the specificity of the functions of
form K (X):
(X) =
K
(X - xk) allows to represent a specific force applied to the point of X-coordinate
xk
(X) =
K
(X - xk) allows to represent one specific moment applied to the point of X-coordinate
xk
(X - xk) and (X - xk) the distribution of Dirac function and the derivative indicate respectively of
distribution of Dirac function at the point of X-coordinate xk.
Handbook of Référence
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Titrate:
Modeling of the turbulent excitations
Date:
05/04/05
Author (S):
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:
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Taking into account the specificity of the functions of form, the projection of a localized turbulent excitation
on modal basis is much simpler than in the general case (excitation distributed), since one
can analytically calculate the expression of the projected excitation.
2.4.3 Setting in equations
2.4.3.1 Application of a localized turbulent excitation
One considers a turbulent excitation applied to a structure telegraphic and made up of forces and of
specific moments. This excitation is entirely characterized by the following data:
·
list nodes of application of the forces and moments specific,
·
nature of the excitation applied in each node (force or moment),
·
direction of the excitation applied in each node.
K
M
Thus F (X, T) = F (S) (X - xk) nk - mm (S) (X - xm) N
T
K
m
éq 2.4.3.1-1
K =1
m=1
is the expression of a located turbulent excitation, characterized by K forces and M moments
specific, applied respectively to the nodes of X-coordinates xk and xm in the directions nk and
rnm.
0
One a: nk =
cos (K) and definite Nm in a similar way.
sin (
K)
represent the azimuth giving the direction of application of the force (or the moment) in the plan P
orthogonal with neutral fiber with the node of application, such as defined in the figure [Figure 2.4.3.1-a]
below:
Z
P
F
Nap
node of application
X
neutral fiber
Appear 2.4.3.1-a: Définition of the direction of application
Handbook of Référence
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Modeling of the turbulent excitations
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:
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The generalized excitation associated the ième mode of the structure, Q S
I (), being defined by:
L
Q (S) = (X) F (X, T) dx
I
I
T
éq
2.4.3.1-2
0
where L represents the length of the beam and I (X) the deformation of mode I, one obtains, taking into account
expression [éq 2.4.3.1-1]:
K
M
Q (S) = F (S) (X) N - M (S) (X
I
K
I
K
K
m
I
K) Nm éq
2.4.3.1-3
K =1
m=1
The calculation of the interspectres of modal excitations leads then to:
K
K
S
(S) = S
Q Q
(
) .n
. (
) .n
I J
F F (S) (
1
1
2
2
K
K
I xk
K)
J xk
K
K =1 K =
1
2
1
2 1
K
M
+ S
F M
(
) .n
. (
) .n
(S) (
1
1
2
2
K
m
I xk
K)
J xm
m
K =1 m =
1
2
1
2 1
éq
2.4.3.1-4
M
K
+ S
'
MR. F
(
) .n
. (
) .n
(S) (
X
I
m1
m1)
K
m
K
J xk 2
2
m =1 K =
1
2
1
2 1
M
M
+ S
MR. M
X
S ('(
) .n
I
m1
m1).
(
) .n
()
m
2
m
m
m
J X
2
m 1
= m 1
1
2
1
2 =
Note:
When the user defines the spectrum of turbulent excitation, it must inform the matrix
interspectrale of the specific excitations whose terms intervene above. This
stamp has as a dimension K+M (a number of forces and specific moments applied).
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the turbulent excitations
Date:
05/04/05
Author (S):
A. ADOBES, L. VIVAN Key
:
R4.07.02-B Page
: 24/26
2.4.3.2 Turbulent excitation identified on model GRAPPE2
The turbulent excitation identified on model GRAPPE2 is represented by a force and one
moment resulting, applied in the same node following the two orthogonal directions to the axis of
structure. The linear density of this excitation has as an expression:
0
1
F
2
2
T (X, S) =
U
H
D [LP tF (rs) (X - x0) - LP MT (rs) (X - x0)]
0
éq 2.4.3.2-1
2
1
Where is the density of the fluid, U is the mean velocity of the flow, Dh is it
hydraulic diameter, LP is the thickness of the plate of housing (corresponding to the length
S D
excited), x0 is the X-coordinate of the point of application of the excitation, Sr =
is the complex frequency
U
reduced, F (S
T
R) and M (S
T
R) are the adimensional coefficients representing the force and the moment
resulting.
The sizes, U, Dh and LP make it possible to dimension the excitation.
In substituent the expression [éq 2.4.3.2-3] in the relation [éq 2.4.3.1-4] defining the modal excitation
Q S
I (), one obtains:
0
0
1
2
2
Q (S)
U D L F (S)
(X
0)
0 + L M (S) '
=
(X
I
H
p
T
R
I
p
T
R
I
0)
0
éq 2.4.3.2-2
2
1
1
The specific force and moment identified on model GRAPPE2 being décorrélés, the calculation of
interspectres of modal excitations leads finally to:
0
0
1
2
2
D
S
=
U
D
L2
Q Q
H
p I
(x0)
1 × J (x0)
1 SF F (Sr)
I J
2
U
T T
1
1
éq
2.4.3.2-3
0
0
4
'
'
+ LP I (x0)
1 × J (x0)
1 S MR. M (Sr)
T
T
1
1
In this expression, D is the diameter external of the structure, S
(S
F F
R) and S
(S
MR. M
R)
T T
T
T
the adimensional autospectres of force and moment represent respectively identified on
model GRAPPE2. Operator PROJ_SPEC_BASE [U4.63.14] calculates the interspectres excitations
modal according to the relation [éq 2.4.3.2-3] above.
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the turbulent excitations
Date:
05/04/05
Author (S):
A. ADOBES, L. VIVAN Key
:
R4.07.02-B Page
: 25/26
Note:
1) Autospectres adimensional GRAPPE2 are usable to simulate it
behavior of any structure in similarity with the model; one then utilizes
geometrical parameters structural feature to dimension
the excitation. Model GRAPPE2 having been built in similarity with
configuration engine, the following reports/ratios are fixed and characteristic of this
geometry:
D
L
H
p
and
D
D
It is pointed out that Dh and D respectively indicate the hydraulic diameter and it
diameter external of the structure; LP is the thickness of the plate of housing,
corresponding to the excited length.
The data of, U and D are thus sufficient to dimension in manner
univocal the turbulent excitation starting from the autospectres adimensional.
2) Adimensional autospectres S
(S
F F
R) and S
(S
MR. M
R) one and the other being defined by
T T
T
T
three real coefficients (level of plate, reduced frequency of cut and slope beyond
of this frequency), only six constants make it possible to characterize the excitation
turbulent adimensional identified on model GRAPPE2.
Four configurations having been studied (ascending flow or going down, stem of
order centered or offset), the whole of the data characterizing the excitations
GRAPPE2 thus represents 24 real coefficients.
3 Bibliography
[1]
NR. GAY, T. FRIOU: Resorption of software FLUSTRU in ASTER. HT32/93/002/B
[2]
S. GRANGER, NR. GAY: Software FLUSTRU Version 3. Note principle. HT32/93/013/B
[3]
L. PEROTIN, Mr. LAINET: Integration of a general model of turbulent excitation in
Code_Aster: specifications. HT32/96/003/A
Handbook of Référence
R4.07 booklet: Coupling fluid-structure
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Modeling of the turbulent excitations
Date:
05/04/05
Author (S):
A. ADOBES, L. VIVAN Key
:
R4.07.02-B Page
: 26/26
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Handbook of Référence
R4.07 booklet: Coupling fluid-structure
HT-66/05/002/A
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