3. Operands
3.1. Tags SPEC_LONG_COR_n
The definition of an excitation spectrum of the « correlation length » type can only be done by a single occurrence of one of the key factors SPEC_LONG_COR_n, corresponding to an area of the tube defined beforehand by the function specified in the operand PROF_VITE_FLUI in the operand of the command DEFI_FLUI_STRU [U4.25.01]. The speed profile associated with this zone, mentioned here under the operand PROF_VITE_FLUI, must be identical to that specified in DEFI_FLUI_STRU [U4.25.01]. The use of « correlation length » excitation spectra is limited to the « cross-flow tube bundle » configuration (keyword FAISCEAU_TRANS factor of the DEFI_FLUI_STRU [U4.25.01] operator).
To perform a calculation with several excitation zones, it is necessary to define as many spectra as there are zones. The contributions of the various spectra can then be added when the excitation is projected onto a modal basis by the PROJ_SPEC_BASE [U4.63.14] command. However, it is not possible in this command to combine spectra of the « correlation length » type with spectra of another type (SPEC_CORR_CONV_n, SPEC_FONC_FORME or SPEC_EXCI_POINT).
The four spectra of the « correlation length » type have values set by default. Defining new coefficients is difficult, in particular with regard to model 3 for which there are connection conditions between the lines determined by the coefficients.
The general analytical form of models 1 to 4 is as follows:
\(S({s}_{\mathrm{1,}}{s}_{\mathrm{2,}}{f}_{r})\mathrm{=}S({f}_{r})\mathrm{\cdot }\mathrm{exp}(\frac{\mathrm{-}∣{s}_{2}\mathrm{-}{s}_{1}∣}{{\lambda }_{c}})\)
with:
\(S({s}_{\mathrm{1,}}{s}_{\mathrm{2,}}{f}_{r})\) |
dimensionless interspectrum of turbulence between two curvilinear abscissa points \(({s}_{\mathrm{1,}}{s}_{2})\); |
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\(S({f}_{r})\) |
turbulence autospectrum; |
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\(\mathrm{exp}(\frac{-∣{s}_{2}-{s}_{1}∣}{{\lambda }_{c}})\) |
spatial correlation function, and \({\lambda }_{c}\) correlation length. |
The spectrum is defined as a function of a reduced frequency \({f}_{r}\) (Strouhal number). For a pipe under cross flow, the expression for \({f}_{r}\) is as follows:
\({f}_{r}=\frac{f\cdot \mathrm{de}}{{V}_{g}}\)
\(f\) is the dimensioned frequency, \(\mathrm{de}\) the external diameter of the tube \({V}_{g}\) and the average transverse speed of the fluid along the structure, which will be recovered in the operator PROJ_SPEC_BASE [U4.63.14] via the [melasflu] concept produced by the operator CALC_FLUI_STRU [U4.66.02].
3.1.1. Analytical expression of spectra of type SPEC_LONG_COR_1
Keyword factor corresponding to the first spectrum model with correlation length.
Correlation length.
The name of the speed profile corresponding to the zone where turbulent excitation is applied.
Kinematic viscosity of the fluid.
\(S({f}_{r})\mathrm{=}\frac{{\Phi }_{0}}{{\left[1\mathrm{-}{\left[\frac{{f}_{r}}{{f}_{\mathit{rc}}}\right]}^{\beta \mathrm{/}2}\right]}^{2}+4{e}^{2}{\left[\frac{{f}_{r}}{{f}_{\mathit{rc}}}\right]}^{\beta \mathrm{/}2}}\)
with: \({\Phi }_{0}\mathrm{=}{\Phi }_{0}({R}_{e})\) 5th degree polynomial.
\(\begin{array}{}\beta =\beta ({R}_{e})\\ \varepsilon =\varepsilon ({R}_{e})\\ {f}_{\mathrm{rc}}=\mathrm{0,2}\end{array}\)
If \(\mathrm{1,5}{.10}^{4}<{R}_{e}\le {5.10}^{4}\):
\(\begin{array}{c}{\Phi }_{0}\mathrm{=}\mathrm{1,3}{.10}^{\mathrm{-}4}\mathrm{[}\mathrm{20,42}\mathrm{-}{14.10}^{\mathrm{-}4}{R}_{e}\mathrm{-}\mathrm{9,81}{.10}^{\mathrm{-}8}{R}_{e}^{2}+\mathrm{11,97}{.10}^{\mathrm{-}12}{R}_{e}^{3}\\ \mathrm{-}\mathrm{35,95}{.10}^{\mathrm{-}17}{R}_{e}^{4}+\mathrm{34,69}{.10}^{\mathrm{-}22}{R}_{e}^{5}\mathrm{]}\end{array}\)
If \({R}_{e}>{5.10}^{4}\): \({\Phi }_{0}=\mathrm{38,6075}\)
If \({R}_{e}\le \mathrm{3,5}{.10}^{4}\) |
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\(\varepsilon =\mathrm{0,7}\) \(\beta =3\) |
Otherwise if \(\mathrm{3,5}{.10}^{4}<{R}_{e}\le \mathrm{5,5}{.10}^{4}\) |
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\(\varepsilon =\mathrm{0,3}\) \(\beta =4\) |
Otherwise |
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\(\varepsilon =\mathrm{0,6}\) \(\beta =4\) |
3.1.2. Analytical expression of spectra of type SPEC_LONG_COR_2
Keyword factor corresponding to the second spectrum model with correlation length.
Correlation length.
The name of the speed profile corresponding to the zone where turbulent excitation is applied.
Reduced cutoff frequency.
Spectrum coefficients.
- note
If the user enters one of these operands, he must fill in the other two operands, in order to have consistent values.
If the user does not fill in any of the three operands, the values by default are used.
\(S({f}_{r})\mathrm{=}\frac{{\Phi }_{0}}{1+{\left[\frac{{f}_{r}}{{f}_{\mathit{rc}}}\right]}^{\beta }}\)
The default parameter values are: \({\Phi }_{0}=\mathrm{1,5}{.10}^{-3}\), \(\beta =\mathrm{2,7}\), \({f}_{\mathrm{rc}}=\mathrm{0,1}\)
3.1.3. Analytical expression of spectra of type SPEC_LONG_COR_3
Keyword factor corresponding to the third spectrum model with correlation length.
Correlation length.
The name of the speed profile corresponding to the zone where turbulent excitation is applied.
Reduced cutoff frequency.
PHI0_1 = phi01
BETA_1 = beta1
PHI0_2 = phi02
BETA_2 = beta2
Spectrum coefficients.
- note
All five operands must be used simultaneously. If one is filled in, the others must also be filled in.
The default values are used when the user has not filled in any of the five operands.
\(S({f}_{r})=\frac{{\Phi }_{0}}{{f}_{r}^{\beta }}\) with \(\{\begin{array}{}{\Phi }_{0}={\Phi }_{0}({f}_{\mathrm{rc}})\\ \beta =\beta ({f}_{\mathrm{rc}})\end{array}\) where \({f}_{\mathrm{rc}}=\mathrm{0,2}\)
If \({f}_{r}\le {f}_{\mathrm{rc}}\) \({\Phi }_{0}={5.10}^{-3}\) \(\beta =\mathrm{0,5}\)
Otherwise \({\Phi }_{0}={4.10}^{-5}\) \(\beta =\mathrm{3,5}\)
3.1.4. Analytical expression of spectra of type SPEC_LONG_COR_4
Keyword factor corresponding to the fourth spectrum model with correlation length.
Correlation length.
The name of the speed profile corresponding to the zone where turbulent excitation is applied.
Vacuum rate (two-phase flow).
◊/BETA = beta
GAMMA = gamma
Spectrum coefficients.
- note
If the user enters one of these two operands, he must fill in the other.
If neither of the two operands is specified, the values by default are used.
\(S({f}_{r})=\frac{{\Phi }_{0}}{{({f}_{r})}^{\beta }({\rho }_{v}^{\gamma })}\) with \(\{\begin{array}{}{\Phi }_{0}=\frac{1}{\mathrm{6,8}{.10}^{-2}}\cdot {10}^{\Phi }\\ \Phi =A\mathrm{.}{\tau }_{v}^{\mathrm{0,5}}-B\mathrm{.}{\tau }_{v}^{\mathrm{1,5}}+\mathrm{C.}{\tau }_{v}^{\mathrm{2,5}}-\mathrm{D.}{\tau }_{v}^{\mathrm{3,5}}\end{array}\)
\({\tau }_{v}\) refers to the void rate;
\(A=\mathrm{24,042};B=-\mathrm{50,421};C=\mathrm{63,483};D=\mathrm{33,284}\)
The default values for exponents are \(\beta =2\) and \(\gamma =4\).
\({\rho }_{v}\) is the volume flow: \({\rho }_{v}\mathrm{=}{\rho }_{m}\mathrm{\times }V\mathrm{=}\mathrm{\sum }_{i\mathrm{=}{N}_{d}}^{{N}_{f}}{\rho }_{e}\frac{({x}_{i})}{{N}_{n}}\mathrm{\times }V\)
where \(V\) refers to the fluid speed for which the fluid-structure interaction study was conducted and \({N}_{n}\) the number of points taken into account on the excited length. The fluid speed will be recovered in operator PROJ_SPEC_BASE [U4.63.14] via the [melasflu] concept produced by operator CALC_FLUI_STRU [U4.66.02].
3.2. Tags SPEC_CORR_CONV_n
The keywords factors SPEC_CORR_CONV_1 and SPEC_CORR_CONV_2 make it possible to define boundary layer turbulence spectra and a function of any frequency respectively. SPEC_CORR_CONV_3 leaves the user in complete control of the definition of the inter-spectrum, using analytical functions gathered in a table.
Theoretical details:
In the case of a plane structure subjected to a parallel turbulent flow, whose spectral response to this excitation is desired, the correlation model of CORCOS introduces a correlation function between two points \(x\) and \(x\text{'}\) on the plane structure, of the type
\(r(\omega ,x,x\text{'})\mathrm{=}\mathrm{exp}(\frac{\mathrm{-}∣x\mathrm{-}x\text{'}∣}{{\lambda }_{1}})\mathrm{\times }\mathrm{exp}(\frac{\mathrm{-}∣y\mathrm{-}y\text{'}∣}{{\lambda }_{2}})\mathrm{\times }\mathrm{cos}(\frac{\omega (x\mathrm{-}x\text{'})}{{U}_{c}}+\frac{\omega (y\mathrm{-}y\text{'})}{{U}_{c}\text{'})\)
In the basic model of CORCOS (coded for SPEC_CORR_CONV -1), we have \(\mathrm{\{}\begin{array}{c}{\lambda }_{1}\mathrm{=}\frac{1}{{k}_{L}}\text{avec}{k}_{L}\mathrm{=}\mathrm{0,1}\mathrm{.}\frac{\omega }{{U}_{c}}\\ {\lambda }_{2}\mathrm{=}\frac{1}{{k}_{T}}\text{avec}{k}_{T}\mathrm{=}\mathrm{0,55}\mathrm{.}\frac{\omega }{{U}_{c}}\end{array}\)
It is possible to choose the CORCOS model and use correlation lengths defined by the user via the SPEC_CORR_CONV_2 operand.
* \(x\) is the axis parallel to the flow.
\(y\) is the axis perpendicular to the flow.
\({U}_{c}\) is the convective speed of vortices. It is accepted that it represents between 60 and 70% of the fluid speed. By default, it is taken equal to 65% of the fluid speed.
\(U{\text{'}}_{c}\) is the orthoradial convective speed of vortices. By default, this phenomenon is not taken into account because it is rarely observed experimentally. However, it is possible to define it if necessary.
In the case of a circular cylindrical structure subject to axial flow, the correlation model of AU_YANG introduces a correlation function between two points defined by:
\(r(\omega ,x,x\text{'})\mathrm{=}\mathrm{exp}(\frac{\mathrm{-}∣x\mathrm{-}x\text{'}∣}{{\lambda }_{1}})\mathrm{\times }\mathrm{\times }\mathrm{exp}(\mathrm{-}R\frac{∣\theta \mathrm{-}\theta \text{'}∣}{\lambda \text{'}})\mathrm{\times }\mathrm{cos}(\frac{\omega (x\mathrm{-}x\text{'})}{{U}_{c}}+\frac{\omega R(\theta \mathrm{-}\theta \text{'})}{U{\text{'}}_{c}})\)
\(\theta\) and \(\theta \text{'}\) correspond to the angular positions of the two points of the cylinder to be correlated,
\(x\) and \(x\text{'}\) designate the ratings of the points to be correlated,
\(R\) is the radius of the cylinder,
\({U}_{c}\) is the axial convective speed of vortices: it is equal to the product of the axial speed coefficient by the fluid speed,
\(U{\text{'}}_{c}\) is the orthoradial convective speed of vortices. By default, this phenomenon is not taken into account because it is rarely observed experimentally. However, it is possible to define it if necessary.
\(\lambda\) and \(\lambda \text{'}\) are the correlation lengths along the axis and the orthoradial direction respectively.
3.2.1. Definition of a boundary layer turbulence spectrum
Keyword factor corresponding to the first pressure spectrum model with correlation length and convection speed of vortices in the fluid.
First correlation length (along the axis parallel to the flow) for the AU_YANG method. A specific length is automatically chosen if the CORCOS method is selected.
Second correlation length for the AU_YANG method.
Velocity of the fluid along the structure under study.
Spectrum cutoff frequency. In the case of the CORCOS method, we use the value \({f}_{c}=10\frac{U}{d}\) (see notations below) by default.
Constant giving the amplitude of the pressure spectrum.
By default, \(k\) is \(\mathrm{5,8}{10}^{\mathrm{-}3}\) in SI units.
Hydraulic diameter used in the expression of the amplitude of the pressure spectrum.
Fluid density.
◊ COEF_VITE_FLUI_A = alpha
Coefficient of the convective speed of vortices in the axial direction (direction of flow) for the methods of CORCOS, of AU_YANG.
◊ COEF_VITE_FLUI_O = beta
Coefficient of the convective speed of vortices in the direction orthoradial to the cylinder.
◊ METHODE = 'CORCOS' or 'AU_YANG'
Correlation method determined by the type of structure whose vibrations generated by turbulence are to be studied.
- note
In the case of the CORCOS method, the correlation lengths of the base model are used for LONG_COR_1 and LONG_COR_2 (see [§3.2]).
The pressure spectrum used is of the \({S}_{p}(\omega )={K}^{2}{(\rho {U}^{2})}^{2}{d}^{3}\) type if \(f\le {f}_{c}\) and \(0\text{pour}f>{f}_{c}\).
\(K\) refers to the model constant, entered under the \(K\) operand. For the CORCOS model, \(K\) is determined experimentally and is equal to \(K=\mathrm{5,8}{.10}^{-3}{s}^{1/2}{m}^{-3/2}\);
\(\rho\) is the density of the fluid, entered under the operand RHO_FLUI;
U is the speed of the fluid, entered under the operand VITE_FLUI;
d is the hydraulic diameter, entered under the operand D_ FLUI.
3.2.2. Definition of a turbulence spectrum of any frequency function
Keyword: factor used to define a pressure spectrum and correlation lengths as a function of pulsation
A function-type concept defining the pressure spectrum as a function of pulsation, produced by one of the operators DEFI_FONCTION [U4.31.02], CALC_FONCTION [U4.32.04], or CALC_FONC_INTERP [U4.32.01].
A function-type concept defining the correlation length in the flow axis for any method.
A function-type concept defining the correlation length in the axis perpendicular to the flow for any method.
Velocity of the fluid along the structure under study.
Cutoff frequency beyond which the function defining the pressure spectrum is considered to be zero.
♦ METHODE = 'CORCOS' or 'AU_YANG'
Correlation method determined by the type of structure whose vibrations generated by turbulence are to be studied.
◊ COEF_VITE_FLUI_A = alpha
Coefficient of the convective speed of vortices in the axial direction (direction of flow) for the methods of CORCOS, of AU_YANG.
◊ COEF_VITE_FLUI_O = beta
Coefficient of the convective speed of vortices in the direction orthoradial to the cylinder. By default, this parameter takes the value of 999 and is therefore not considered for the calculation.
3.2.3. SPEC_CORR_CONV_3: any spectrum defined analytically
Keyword factor used to define a spectrum based on analytical functions.
A function table-type concept containing the analytical formulas that define the spectrum.
Example of use: we want to describe the pressure forces induced by axial flow along a fuel rod in the form of a « correlation length » spectrum and describing:
on the one hand the decrease in turbulent energy downstream of the grid,
on the other hand the phase difference due to the convection of the turbulence with the flow.
The correlation length, and the self-spectrum depend on the frequency. The proposed analytical formulation is as follows:
\({S}_{f}(\underline{{r}_{1}},\underline{{r}_{2}},\omega )\mathrm{=}\mathrm{\{}\begin{array}{c}{S}_{x}\mathrm{=}\text{exp}(\mathrm{-}\frac{\mathrm{\mid }\underline{{r}_{2}}\mathrm{-}\underline{{r}_{1}}\mathrm{\mid }}{{\lambda }_{\text{cx}}(\omega )})\text{.}\text{exp}(\mathit{j\omega }\frac{{z}_{2}\mathrm{-}{z}_{1}}{{U}_{c}}){S}_{f}(\underline{{r}_{1}},\underline{{r}_{1}},\omega )\\ {S}_{y}\mathrm{=}\text{exp}(\mathrm{-}\frac{\mathrm{\mid }\underline{{r}_{2}}\mathrm{-}\underline{{r}_{1}}\mathrm{\mid }}{{\lambda }_{\text{cy}}(\omega )})\text{.}\text{exp}(\mathit{j\omega }\frac{{z}_{2}\mathrm{-}{z}_{1}}{{U}_{c}}){S}_{f}(\underline{{r}_{1}},\underline{{r}_{1}},\omega )\end{array}\)
\(\underline{{r}_{1}}\) and \(\underline{{r}_{2}}\) are the vectors identifying the positions of the two points to be correlated, \(z\) is the direction parallel to the pencil axis. You can also add, if you wish, a correlation term between efforts according to \(x\) and \(y\).
The spectrum above is defined in the sdll148b test case with correlated efforts. We propose here modeling with uncorrelated efforts (no cross terms SXY and SYX). In this test case, the pencil is oriented in the Y direction.
SXX = FORMULE (NOM_PARA =( 'X1', 'Y1', 'Y1', 'Z1', 'X2', 'Y2', 'Z2', 'FREQ'),
VALE_C ='exp (- FREQ /freq0) *
exp (distance (X1, Y1, Z1, Z1, Z1, X2, Y2, Z2) /correl (FREQ)) *
complex (cos (2*pi* FREQ * (Y2-Y1) /Uc),
sin (2*pi* FREQ * (Y2-Y1) /Uc)) ',)
SYY =...
# INTER - SPECTRE AVEC EFFORTS X AND Y DECORRELES
INTESPEC = CREA_TABLE (
LISTE =(
_F (LISTE_K =( "SXX "," "," SYY "), PARA =" FONCTION_C "),
_F (LISTE_K =( "DX", "DY"), PARA =" NUME_ORDRE_I "),
_F (LISTE_K =( "DX", "DY"), PARA =" NUME_ORDRE_J "),
),
TYPE_TABLE =" TABLE_FONCTION ",
TITRE =" EXCITATION FLUIDE TURBULENTE ",
)
SPECTRE1 = DEFI_SPEC_TURB (SPEC_CORR_CONV_3 =_F (TABLE_FONCTION = INTESPEC))
The distance function was defined in Python and gives the distance between two points with coordinates \(({x}_{\mathrm{1,}}{y}_{\mathrm{1,}}{z}_{1})\) and \(({x}_{\mathrm{2,}}{y}_{\mathrm{2,}}{z}_{2})\), respectively. The correl function depends exponentially on frequency.
3.4. Keyword SPEC_EXCI_POINT
Keyword factor used to define an excitation spectrum associated with one or more specific forces and moments.
Interspectrum concept defining an interspectral matrix of point excitations. This concept can be produced by the operator LIRE_INTE_SPEC [U4.56.01] after reading the interspectral matrix on an external file.
List of text-type arguments defining the nature of the excitement in each of the application nodes. The legal arguments are “FORCE” or “MOMENT”.
List of angles defining the directions of the force and moment vectors at each of the application nodes (see diagram).
The force vector is directed in the \(P\) plane orthogonal to the neutral fiber. In this plane, azimuth \(\theta\) gives the direction of the vector. Angles should be given in degrees.
List of groups containing the nodes for applying the excitement.
- note
The dimension of the interspectral matrix is the number of forces and point moments applied. The diagonal terms of this matrix characterize the autospectra of these excitations.
The lists defining the application nodes and the nature and the direction of the imposed excitations must therefore be ordered in accordance with the structure of the interspectral excitations matrix.
/GRAPPE_2 = 'ASC_CEN' or 'ASC_EXC' or '' or 'DES_CEN' or 'DES_EXC'
Four possible choices corresponding to the different experimental configurations for which excitation GRAPPE2 has been identified:
flow ASCendant CENtrée control rod,
flow ASCendant EXCentrée control rod,
flow DEScendant CENtrée control rod,
flow DEScendant EXCentrée control rod.
Excitation GRAPPE2 is characterized by a specific force and moment applied at the same node, in a homogeneous manner in the two directions orthogonal to the axis of the wire structure.
Density of the fluid surrounding the structure.
Group containing the Excitation Application Node GRAPPE2.
Note:
When using a spectrum GRAPPE2prédéfini, the list of nodes expected under the operand GROUP_NOest reduced to a single element (a single application node) .
A cara_elem concept, produced by the operator AFFE_CARA_ELEM [U4.42.01], defines the geometric characteristics assigned to the elements of the structure.
Geometric characteristics are necessary to estimate the hydraulic diameter. In addition, the cara_elem type concept provides information about the orientations of the elements.
Model type concept, produced by the operator AFFE_MODELE [U4.41.01], defines the types of elements assigned to the cells of the structure.