2. Turbulent excitation models applicable to wire structures#
2.1. General principles#
2.1.1. Assumptions#
It is assumed that the linear excitation induced on the wire structure by the turbulence of the flow can be modelled in the form of a random stationary ergodic Gaussian process with zero mean. This turbulent excitation is therefore entirely characterized by its interspectral density \({S}_{f}({x}_{1},{x}_{2},\omega )\), where \({x}_{1}\) and \({x}_{2}\) are any two points on the beam and \(w\) refers to the pulsation. The turbulent excitation applied to the structure is therefore characterized by its interspectral density \({S}_{f}\).
In addition, it is assumed that the turbulent forces are independent of the movement of the structure. Turbulent excitation is identified experimentally on a reference model. It is then applicable to any real component in geometric similarity with the reference model.
2.1.2. Calculation of modal excitation interspectra#
\({f}_{t}(x,s)\) refers to the linear density of turbulent excitation exerted on the beam; \(x\) is the current abscissa of a point on the beam and \(s\) the complex pulsation (Laplace variable). The following additional assumptions \(\mathit{H1}\) and \(\mathit{H2}\) are made:
\(\mathit{H1}\). The excited length \({L}_{e}\) is less than the total length \(L\) of the beam.
\(\mathit{H2}\). The expression for \({f}_{t}(x,s)\) does not depend on where the excited zone \({x}_{e}\) comes from; it translates to \({f}_{t}(x,s)={f}_{t}(x-{x}_{e},s)\).
In this case, line density \({f}_{t}\) can be expressed in the following form:
\({f}_{t}(x,s)=\frac{1}{2}{\mathrm{\rho U}}^{2}D\cdot {C}_{f}(\alpha ,\frac{D}{{D}_{h}},\frac{D}{{L}_{e}},{s}_{r}\text{,Re})\) eq 2.1.2-1
with: \(\begin{array}{ccc}\alpha =\frac{x-{x}_{e}}{{L}_{e}}& {s}_{r}=\frac{\text{sD}}{U}& \text{Re}=\frac{\text{UD}}{u}\end{array}\)
Where \(\rho\) refers to the density of the fluid, \(U\) is the average fluid flow speed, \(D\) and \({D}_{h}\) are the diameter of the structure and the hydraulic diameter respectively, \({C}_{f}\) represents the dimensionless coefficient of turbulent force, \(x\) is the current abscissa of a point on the beam, \({x}_{e}\) refers to the abscissa of the origin of the excited zone, \({L}_{e}\) represents the excited length, \(\alpha\) is the reduced space variable, is the reduced space variable, \(s\) is the complex pulsation (Laplace variable), \({s}_{r}\) is the reduced complex pulsation, \(\nu\) is the kinematic viscosity of the fluid, and finally « \(\text{Re}\) » refers to the Reynolds number.
Assuming the geometric similarity of the real component with the reference model, we obtain:
\({f}_{t}(x,s)\mathrm{=}\frac{1}{2}{\mathit{\rho U}}^{2}D\mathrm{\cdot }{C}_{f}(\alpha ,{s}_{r}\text{,Re})\) eq 2.1.2-2
Thus, modal turbulent excitation \({Q}_{i}(s)\) can be written in the Laplace domain (hypothesis \(\mathit{H2}\)):
\({Q}_{i}(x)\mathrm{=}\underset{{x}_{e}}{\overset{{x}_{e}\text{+}{L}_{e}}{\mathrm{\int }}}{f}_{t}(x,s){\phi }_{i}(x)\text{dx}\mathrm{=}{L}_{e}\underset{0}{\overset{1}{\mathrm{\int }}}{f}_{t}(\alpha {L}_{e},s){\phi }_{i}(\alpha {L}_{e}+{x}_{e})d\alpha\) eq 2.1.2-3
where \({\phi }_{i}(x)\) is the component of the \(i\) th modal deformation in the spatial direction in which the turbulent excitation acts.
Using the expression [éq 2.1.2-2], we deduce:
\({Q}_{i}(s)\mathrm{=}\frac{1}{2}{\mathit{\rho U}}^{2}{\text{DL}}_{e}\underset{0}{\overset{1}{\mathrm{\int }}}{C}_{f}(\alpha ,{s}_{r}\text{,Re}){\phi }_{i}(\alpha {L}_{e}+{x}_{e})d\alpha\) eq 2.1.2-4
The interspectral densities of modal turbulent excitations are then expressed in the form:
\({S}_{{Q}_{i}{Q}_{j}}(f,U)\mathrm{=}{(\frac{1}{2}{\mathit{\rho U}}^{2}{\text{DL}}_{e})}^{2}\frac{D}{U}\underset{0}{\overset{1}{\mathrm{\int }}}\underset{0}{\overset{1}{\mathrm{\int }}}{\Phi }_{t}({\alpha }_{1},{\alpha }_{2},{f}_{r}\text{,Re}){\phi }_{i}({\alpha }_{1}{L}_{e}+{x}_{e}){\phi }_{j}({\alpha }_{2}{L}_{e}+{x}_{e})d{\alpha }_{1}d{\alpha }_{2}\) eq 2.1.2-4
with |
\(1\le i,j\le N\), where \(N\) is the number of modes used to determine the response of the structure; |
\({\Phi }_{t}\): interspectrum of \({C}_{f}\) between \({\alpha }_{1}\) and \({\alpha }_{2}\); |
|
\({f}_{r}=\frac{\text{fD}}{U}\): reduced frequency. |
Note:
In the following, we keep the hypotheses \(\mathit{H1}\) and \(\mathit{H2}\) and note \({I}_{\text{ij}}({f}_{r}\text{,Re})\) the integral:
\({I}_{\text{ij}}({f}_{r}\text{,Re})=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{\Phi }_{t}({\alpha }_{1},{\alpha }_{2},{f}_{r}\text{,Re}){\phi }_{i}({\alpha }_{1}{L}_{e}+{x}_{e}){\phi }_{j}({\alpha }_{2}{L}_{e}+{x}_{e}){\mathrm{d\alpha }}_{1}{\mathrm{d\alpha }}_{2}\) eq 2.1.2-5
Using this notation, the modal excitation interspectra are written as:
\({S}_{{Q}_{i}{Q}_{j}}(f,U)={(\frac{1}{2}{\mathrm{\rho U}}^{2}{\text{DL}}_{e})}^{2}\frac{D}{U}{I}_{\text{ij}}({f}_{r}\text{,Re})\) eq 2.1.2-6
The expression for modal excitation autospectra is analogous:
\({S}_{{Q}_{i}{Q}_{i}}(f,U)={(\frac{1}{2}{\mathrm{\rho U}}^{2}{\text{DL}}_{e})}^{2}\frac{D}{U}{I}_{\text{ii}}({f}_{r}\text{,Re})\) eq 2.1.2-7
2.2. « Correlation length » spectra#
2.2.1. Keywords#
The key factors SPEC_LONG_COR_i (\(i\) varying from 1 to 4) of the DEFI_SPEC_TURB [U4.44.31] operator provide access to « correlation length » spectra. These spectra, specific to configurations of the « cross-flow tube bundle » type, are predefined but the user can adjust the parameters.
2.2.2. Model definition#
2.2.2.1. Interspectral density#
In the case of « correlation length » spectra, the interspectral density characterizing the turbulent excitation is assumed to be able to be put into a form with separable variables such as:
\({S}_{i}({x}_{1},{x}_{2},\omega )={S}_{0}(\omega ){\varphi }_{0}({x}_{1},{x}_{2})\) eq 2.2.2.1-1
In this expression, \({S}_{0}(\omega )\) represents the turbulence autospectrum and \({\varphi }_{0}({x}_{1},{x}_{2})\) refers to a spatial correlation function defined by:
\({\varphi }_{0}({x}_{1},{x}_{2})=\text{exp}\cdot (\frac{-\mid {x}_{2}-{x}_{1}\mid }{{\lambda }_{c}})\) eq 2.2.2.1-2
where \({x}_{1}\) and \({x}_{2}\) designate the x-axes of two observation points and \({\lambda }_{c}\) represents the correlation length.
Four analytic expressions are available in operator DEFI_SPEC_TURB [U4.44.31]. These expressions each correspond to a particular representation of \({S}_{0}(\omega )\).
The user defines a turbulence spectrum by choosing one of these analytical forms, whose parameters he can adjust.
2.2.2.2. Modelling the turbulence spectrum using an expression with separate variables#
General case
The function \({\Phi }_{\text{tt}}\) introduced in the relationship is modeled by a form with separate variables:
\({\Phi }_{t}({\alpha }_{1},{\alpha }_{2},{f}_{r}\text{,Re})=\sum _{n=1}^{{N}_{s}}{\varphi }_{n}({\alpha }_{1},{\alpha }_{2}){\Phi }_{n}({f}_{r}\text{,Re})\) eq 2.2.2-1
Where \({N}_{s}\) refers to the degree of the base of the form functions \({\varphi }_{n}\) and \({\Phi }_{n}\) is a function that is independent of the space variable. These two functions are stored in the database and can be selected by the user.
The modal excitation autospectra are given by [éq 2.1.2-7] by introducing:
\({I}_{\text{ii}}({f}_{r}\text{,Re})=\sum _{n=1}^{{N}_{s}}{L}_{\text{ni}}^{2}\cdot {\Phi }_{n}({f}_{r}\text{,Re})\) eq 2.2.2.2-2
with: \({L}_{\text{ni}}^{2}=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{\varphi }_{n}({\alpha }_{1},{\alpha }_{2})\cdot {\phi }_{i}({\alpha }_{1}{L}_{e}+{x}_{e}){\phi }_{i}({\alpha }_{2}{L}_{e}+{x}_{e})\cdot {\mathrm{d\alpha }}_{1}{\mathrm{d\alpha }}_{2}\) eq 2.2.2.2-3
The principle of calculation is as follows: we first calculate the values of \({L}_{\text{ni}}^{2}\) by calculating double integrals; we then calculate \({\Phi }_{n}({f}_{r}\text{,Re})\) for all the values of n; we finally obtain the expression for \({S}_{{Q}_{i}{Q}_{i}}(f,U)\) using the equation [éq 2.1.2-4].
Special case: model used for steam generator tubes
The particular case of the study of GV tubes corresponds to a particular case of the general case presented previously by posing \(\text{Ns}=1\). The turbulent excitation interspectrum between two reduced abscissa points \({\alpha }_{1}\) and \({\alpha }_{2}\) is then given by:
\({\Phi }_{t}({\alpha }_{1},{\alpha }_{2},{f}_{r}\text{,Re})=\text{exp}(-\frac{\mid {\alpha }_{1}-{\alpha }_{2}\mid }{{\lambda }_{c}}\cdot {L}_{e})\cdot \Phi ({f}_{r}\text{,Re})\) eq 2.2.2.2-4
where \({\lambda }_{c}\) represents the correlation length of the turbulent forces and \({L}_{e}\) is the excited length. In general, \({\lambda }_{c}\) is taken in the order of 3 to 4 times the outside diameter of the tube.
The autocorrelation spectra of modal excitations, in the case of constant speed and density profiles, are given by:
\({S}_{\text{QiQi}}(f,U)={(\frac{1}{2}{\mathrm{\rho U}}^{2}{\text{DL}}_{e})}^{2}\cdot \frac{D}{U}{I}_{\text{ii}}({f}_{r}\text{,Re})\) eq 2.2.2-5
with:
\({I}_{\text{ii}}({f}_{r}\text{,Re})=\Phi ({f}_{r}\text{,Re})\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\text{.}\text{exp}(-\frac{\mid {\alpha }_{2}-{\alpha }_{1}\mid }{{\lambda }_{c}}{L}_{e})\text{.}{\phi }_{i}({\alpha }_{1}{L}_{e}+{x}_{e}){\phi }_{i}({\alpha }_{2}{L}_{e}+{x}_{e})\text{.}{\mathrm{d\alpha }}_{1}{\mathrm{d\alpha }}_{2}\)
eq 2.2.2-6
In the general case of any density and flow velocity profiles, we have:
\(\begin{array}{}{S}_{\text{QiQj}}(f,U)={(\frac{1}{2}D)}^{2}\cdot \frac{D}{U}S({f}_{r})\\ \underset{{x}_{e}}{\overset{{x}_{e}{\text{+L}}_{e}}{\int }}\underset{{x}_{e}}{\overset{{x}_{e}\text{+}{L}_{e}}{\int }}\text{exp}(-\frac{\mid {x}_{2}-{x}_{1}\mid }{{\lambda }_{c}})\cdot {\rho }_{e}({x}_{1}){\rho }_{e}({x}_{2})\cdot {U}_{e}^{2}({x}_{1}){U}_{e}^{2}({x}_{2}){\phi }_{i}({x}_{1}){\phi }_{i}({x}_{2}){\text{dx}}_{1}{\text{dx}}_{2}\end{array}\)
eq 2.2.2.2-7
Where \(D\) is the diameter of the structure, \({L}_{e}\) is the length of the excited zone, \({x}_{e}\) is the abscissa of the origin of the excited zone, \(U\) is the mean speed of the flow, \(S({f}_{r})\) is an excitation spectral density independent of the average speed of the flow \(U\), \({x}_{1}\) and \({x}_{2}\) are the Curvilinear abscissa of two observation points on the tube, \({\rho }_{e}(x)\) is the density profile of the fluid along the tube, \({U}_{e}(x)\) is the transverse velocity profile of the flow along the tube, and \({\lambda }_{c}\) refers to the correlation length.
The dimensionless profiles of density and transverse velocity of the external flow are defined as follows:
\({\rho }_{e}(x)\) designating the evolution of the density of the external fluid along the immersed zone \({L}_{\text{imm}}\) of the tube, \(\rho\) designates the density of the external fluid averaged over the submerged part of the tube:
\(\rho =\frac{1}{{L}_{\text{imm}}}\underset{{x}_{\text{imm}}}{\overset{{x}_{\text{imm}}\text{+}{L}_{\text{imm}}}{\int }}{\rho }_{e}(x)\text{dx}\) eq 2.2.2-8
\(r(x)\) refers to the dimensionless density profile such as \({\rho }_{e}(x)=\rho \cdot r(x)\).
\({U}_{e}(x)\) designating the evolution of the flow speed of the external fluid over the excited length \({L}_{e}\) of the tube, \(U\) designates the flow speed of the fluid averaged over the excited length of the tube:
\(U=\frac{1}{{L}_{e}}\underset{{x}_{e}}{\overset{{x}_{e}\text{+}{L}_{e}}{\int }}{U}_{e}(x)\text{dx}\) eq 2.2.2-9
\(u(x)\) refers to the dimensionless transverse velocity profile of the external flow, such as \({U}_{e}(x)=U\cdot u(x)\).
By introducing the mean quantities and the dimensionless profiles into the expression [éq2.2.2.2-7], we obtain:
\(\begin{array}{}{S}_{\text{QiQj}}(f,U)={(\frac{1}{2}{\mathrm{\rho U}}^{2}D)}^{2}\text{.}\frac{D}{U}S({f}_{r})\underset{{x}_{e}}{\overset{{x}_{e}\text{+}{L}_{e}}{\int }}\underset{{x}_{e}}{\overset{{x}_{e}\text{+}{L}_{e}}{\int }}\text{exp}(-\frac{\mid {x}_{2}-{x}_{1}\mid }{{\lambda }_{c}})\\ {\rho }_{e}({x}_{1}){\rho }_{e}({x}_{2}){U}_{e}^{2}({x}_{1}){U}_{e}^{2}({x}_{2}){\phi }_{i}({x}_{1}){\phi }_{j}({x}_{2}){\text{dx}}_{1}{\text{dx}}_{2}\end{array}\) eq 2.2.2.2-10
After noting \(\alpha =\frac{x-{x}_{e}}{{L}_{e}}\), it comes:
\(\begin{array}{}{S}_{\text{QiQj}}(f,U)=\frac{1}{4}{\rho }^{2}{U}^{3}{D}^{3}{L}_{e}^{2}S({f}_{r})\times \underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}[\text{exp}(-\frac{\mid {x}_{2}-{x}_{1}\mid }{{\lambda }_{c}})r({\alpha }_{1}{L}_{e}+{x}_{e})r({\alpha }_{2}{L}_{e}+{x}_{e})\\ {u}^{2}({\alpha }_{1}{L}_{e}+{x}_{e}){u}^{2}({\alpha }_{2}{L}_{e}+{x}_{e}){\phi }_{i}({\alpha }_{1}{L}_{e}+{x}_{e}){\phi }_{i}({\alpha }_{2}{L}_{e}+{x}_{e})]{\mathrm{d\alpha }}_{1}{\mathrm{d\alpha }}_{2}\end{array}\)
eq 2.2.2.2-11
Where \(S({f}_{r})\) represents the turbulence spectrum, defined as a function of a reduced frequency \({f}_{r}\) (Strouhal number). For a tube interacting with a transverse flow, \({f}_{r}\) is written:
\({f}_{r}=\frac{\text{fD}}{U}\)
where \(f\) is the dimensioned frequency, \(D\) is the diameter of the tube, and \(U\) is the mean flow speed.
The double integral of the expression [éq 2.2.2.2-11] is evaluated by the PROJ_SPEC_BASE [U4.63.14] operator.
Cases of multiple arousal zones
In the case where there are several excitation zones, the following additional notations are introduced:
Since the excitation zone \(k\) is identified by its starting abscissa \({x}_{k}\) and its length \({L}_{k}\), the transverse velocity profile of the fluid flow at the level of this zone is denoted \({U}_{k}(x)\). The mean transverse speed over the excitation zone \(k\) is then given by:
\({\stackrel{ˉ}{U}}_{k}=\frac{1}{{L}_{k}}\underset{{x}_{k}}{\overset{{x}_{k}\text{+}{L}_{k}}{\int }}{U}_{k}(x)\text{dx}\)
The dimensionless transverse speed profile, standardized on zone \(k\), is deduced from this:
\({u}_{k}(x)=\frac{{U}_{k}(x)}{{\stackrel{ˉ}{U}}_{k}}\)
\(K\) designating the total number of excitation zones, the mean transverse speed over all the excitation zones is defined by:
\(\stackrel{ˉ}{U}=\frac{1}{K}\sum _{k=1}^{K}{\stackrel{ˉ}{U}}_{k}\)
If \({V}_{\text{gap}}\) is the intertube speed at the input of the GV (the flow rate range is defined in CALC_FLUI_STRU [U4.66.02] using the keyword VITE_FLUI), a second normalization is carried out; the transverse speed at a point \(x\) located in the excitation zone \(k\) is given by:
\({V}_{k}(x)={V}_{\text{gap}}\frac{{U}_{k}(x)}{\stackrel{ˉ}{U}}={V}_{\text{gap}}\frac{{\stackrel{ˉ}{U}}_{k}}{\stackrel{ˉ}{U}}{u}_{k}(x)\)
Thanks to this normalization, the arithmetic mean of the transverse speed over all the excitation zones is equal to the inter-tube speed; in fact, we have:
\(\frac{1}{K}\sum _{k=1}^{K}(\frac{1}{{L}_{k}}\underset{{x}_{k}}{\overset{{x}_{k}\text{+}{L}_{k}}{\int }}{V}_{k}(x)\text{dx})={V}_{\text{gap}}\)
The calculation of the modal excitation interspectra, carried out by the operator PROJ_SPEC_BASE [U4.63.14], is done by adding up the contributions of each of the excitation zones according to the relationship:
\({S}_{{Q}_{i}{Q}_{j}}(f,{V}_{\text{gap}})={(\frac{1}{2}D)}^{2}\sum _{k=1}^{K}(\frac{D}{{\stackrel{ˉ}{V}}_{k}}\times {L}_{\text{ij}}^{k}\times S({f}_{r}^{k}))\)
with:
\({\stackrel{ˉ}{V}}_{k}={V}_{\text{gap}}\times \frac{{U}_{k}}{\stackrel{ˉ}{U}}\) and \({f}_{r}^{k}=\frac{\text{fD}}{{\stackrel{ˉ}{V}}_{k}}\)
and \({L}_{\text{ij}}^{k}=\underset{{x}_{k}}{\overset{{x}_{k}\text{+}{L}_{k}}{\int }}\underset{{x}_{k}}{\overset{{x}_{k}\text{+}{L}_{k}}{\int }}\text{exp}(\frac{-\mid {x}_{2}-{x}_{1}\mid }{{\lambda }_{c}}){\rho }_{e}({x}_{1}){\rho }_{e}({x}_{2}){V}_{k}^{2}({x}_{1}){V}_{k}^{2}({x}_{2}){\phi }_{i}({x}_{1}){\phi }_{i}({x}_{2}){\text{dx}}_{1}{\text{dx}}_{2}\)
That is: \({L}_{\text{ij}}^{k}={\stackrel{ˉ}{V}}_{k}^{4}\times \underset{{x}_{k}}{\overset{{x}_{k}\text{+}{L}_{k}}{\int }}\underset{{x}_{k}}{\overset{{x}_{k}\text{+}{L}_{k}}{\int }}\text{exp}(\frac{-\mid {x}_{2}-{x}_{1}\mid }{{\lambda }_{c}}){\rho }_{e}({x}_{1}){\rho }_{e}({x}_{2}){u}_{k}^{2}({x}_{1}){u}_{k}^{2}({x}_{2}){\phi }_{i}({x}_{1}){\phi }_{i}({x}_{2}){\text{dx}}_{1}{\text{dx}}_{2}\)
We ask:
\({l}_{\text{ij}}^{k}=\underset{{x}_{k}}{\overset{{x}_{k}\text{+}{L}_{k}}{\int }}\underset{{x}_{k}}{\overset{{x}_{k}\text{+}{L}_{k}}{\int }}\text{exp}(\frac{-\mid {x}_{2}-{x}_{1}\mid }{{\lambda }_{c}}){\rho }_{e}({x}_{1}){\rho }_{e}({x}_{2}){u}_{k}^{2}({x}_{1}){u}_{k}^{2}({x}_{2}){\phi }_{i}({x}_{1}){\phi }_{i}({x}_{2}){\text{dx}}_{1}{\text{dx}}_{2}\)
The expression for the modal excitation interspectra then becomes:
\({S}_{{Q}_{i}{Q}_{j}}(f,{V}_{\text{gap}})={(\frac{1}{2}D)}^{2}\sum _{k=1}^{K}(\frac{D}{{\stackrel{ˉ}{V}}_{k}}\times {\stackrel{ˉ}{V}}_{k}^{4}\times {l}_{\text{ij}}^{k}\times S({f}_{r}^{k}))\)
from where:
\({S}_{{Q}_{i}{Q}_{j}}(f,{V}_{\text{gap}})=\frac{1}{4}{D}^{3}\times \sum _{k=1}^{K}({\stackrel{ˉ}{V}}_{k}^{3}\times {l}_{\text{ij}}^{k}\times S({f}_{r}^{k}))\)
Analytical expressions of the spectra available to the user
The various analytical expressions of the spectra available in the operator DEFI_SPEC_TURB [U4.44.31] are as follows:
SPEC_LONG_COR_1
Each speed \({U}_{i}\) defined by the user by discretizing the speed range \(\left[{U}_{\text{min}}-{U}_{\text{max}}\right]\) explored is first normalized in the form \({U}_{i}^{\text{kn}}\) by applying the equation:
\({U}_{i}^{\text{kn}}={U}_{i}\frac{{\stackrel{ˉ}{U}}^{k}}{\stackrel{ˉ}{U}}\)
where \({\overline{U}}^{k}\) and \(\overline{U}\) respectively designate the speed averaged over the excitation zone « \(k\) « , and the average speed over all the excitation zones.
A « local » Reynolds number \({R}_{e}^{\text{ik}}\), associated with the zone « \(k\) » and with the speed \({U}_{i}\), is then calculated from the local characteristics of the flow:
\({\text{Re}}^{\text{ik}}\text{= -}\frac{{U}_{i}^{\text{kn}}\cdot D}{\nu }\)
The turbulent excitation spectrum associated with the zone « \(k\) » and with the speed \({U}_{i}\) is determined in the form of a vector \({S}^{\text{ik}}\), having as many components as points used to discretize the frequency interval \(\left[{f}_{\text{min}}-{f}_{\text{max}}\right]\), which supports the excitation. The \(j\) -th \({S}_{j}^{\text{ik}}\) component of this vector is provided by the expression:
\({S}_{j}^{\text{ik}}=\frac{{\phi }_{0}}{{(1-{(\frac{{f}_{\text{rj}}^{\text{ik}}}{{f}_{\text{rc}}})}^{})}^{2}+4{\epsilon }^{2}{(\frac{{f}_{\text{rj}}^{\text{ik}}}{{f}_{\text{rc}}})}^{}}\) eq 2.2.2.2-12
\({f}_{\text{rj}}^{\text{ik}}\) is provided by:
\({f}_{\text{rj}}^{\text{ik}}=\frac{{f}_{j}D}{{U}_{i}^{\text{kn}}}\)
where:
\({f}_{j}\) is the frequency value associated with the j-th component in the discretization of the frequency interval \(\left[{f}_{\text{min}}-{f}_{\text{max}}\right]\),, \({f}_{\text{rc}}\) is a cutoff frequency equal to 0.2; \({\phi }_{o}\), \(\beta\), \(\epsilon\) depend on the Reynolds number according to the equations provided in the table below:
\({R}_{e}^{\text{ik}}\) |
|
|
|
\(\left]\mathrm{-}\mathrm{\infty };1.5{10}^{4}\right]\) |
2.83504 10-4 |
3 |
0.7 |
\(\left]1.5{10}^{4};3.5{10}^{4}\right]\) |
|
Same |
Same |
\(\left]3.5{10}^{4};5{10}^{4}\right]\) |
Same |
4 |
0.3 |
\(\left]5{10}^{4};5.5{10}^{4}\right]\) |
|
Same |
Same |
\(\left]5.5{10}^{4};+\mathrm{\infty }\right]\) |
Same |
4 |
0.6 |
SPEC_LONG_COR_2
The turbulent excitation spectrum is written as:
\(S({f}_{r})=\frac{{f}_{0}}{1+{(\frac{{f}_{r}}{{f}_{\text{rc}}})}^{\beta }}\) eq 2.2.2.2-13
The values for the parameters by default are as follows:
\(\begin{array}{}{\phi }_{0}\text{=}1\text{.}5\cdot {\text{10}}^{\text{-}3}\\ \beta \text{=}2\text{.}7\\ {f}_{\text{rc}}\text{=}0\text{.}1\end{array}\)
SPEC_LONG_COR_3
The turbulent excitation spectrum is written as:
\(S({f}_{r})=\frac{{\phi }_{0}}{{f}_{r}^{\beta }}\) eq 2.2.2.2-14
with:
\(\begin{array}{c}{\phi }_{0}={\phi }_{0}({f}_{\text{rc}})\\ \beta =\beta ({f}_{\text{rc}})\end{array}\)
The default values for the parameters are as follows: \({f}_{\text{rc}}\) = 2
If \({f}_{r}\le {f}_{\text{rc}}\), we have:
\(\begin{array}{c}{\phi }_{0}\text{=}5\cdot {\text{10}}^{\text{-}3}\\ \beta \text{=}0\text{.}5\end{array}\)
If not
\(\begin{array}{c}{\phi }_{0}\text{=}4\cdot {\text{10}}^{\text{-}5}\\ \beta \text{=}3\text{.}5\end{array}\)
SPEC_LONG_COR_4
The turbulent excitation spectrum is written as:
\(S({f}_{r})\text{=}\frac{{\phi }_{0}}{{f}_{r}^{\beta }{\rho }_{v}^{g}}\) eq 2.2.2.2-15
with:
\({\phi }_{0}=\frac{1}{6\text{.}8\cdot {\text{10}}^{-2}}{\text{10}}^{\phi }\)
The other parameters are defined by:
\(\begin{array}{}\phi \text{=}{\mathrm{At}}_{v}^{0\text{.}5}\text{-}B{\tau }_{v}^{1\text{.}5}\text{-}C{\tau }_{v}^{2\text{.}5}\text{-}D{\tau }_{v}^{3\text{.}5}\\ \beta \text{=}2\\ \gamma \text{=}4\end{array}\)
\({\tau }_{v}\) refers to the vacuum rate; \({\rho }_{v}\) is the volume flow defined by \({\rho }_{v}\text{=}{\rho }_{m}U\); \({\rho }_{m}\) is the mass flow rate; and \(U\) refers to the average flow speed. The values of the polynomial coefficients in \({\tau }_{v}\) are as follows:
\(\begin{array}{}A\text{=}\text{24}\text{.}\text{042}\\ B\text{= -}\text{50}\text{.}\text{421}\\ C\text{=}\text{63}\text{.}\text{483}\\ D\text{=}\text{33}\text{.}\text{284}\end{array}\)
2.3. Distributed turbulent excitation model#
2.3.1. Keywords#
The keyword factor SPEC_FONC_FORME of the DEFI_SPEC_TURB [U4.44.31] operator makes it possible to define an excitation spectrum by its decomposition on a family of shape functions. The user can define the spectrum by providing an interspectral matrix and a list of associated shape functions. The concepts [interspectrum] and [function] must then have been generated beforehand. In the case of the « control cluster » component, the user can also use a predefined turbulence spectrum, identified on model GRAPPE1.
2.3.2. Decomposition on a family of form functions#
The distributed turbulent excitation model assumes that the instantaneous linear density of turbulent forces \({f}_{t}(x,t)\) can be decomposed on a family of form functions \({j}_{k}(x)\) of dimension \(K\) in the following way:
\({f}_{t}(x,t)=\sum _{k=1}^{K}{\varphi }_{k}(x){\alpha }_{k}(t)\) eq 2.3.2-1
The coefficients \({\alpha }_{k}(t)\) define at each moment the decomposition of the turbulent excitation on the family of shape functions.
The interspectral density of turbulent excitation between two points of the wire structure with abscissa \({x}_{1}\) and \({x}_{2}\) is then written:
\({S}_{f}({x}_{1},{x}_{2},\omega )=\sum _{k=1}^{K}\sum _{k=1}^{K}{\varphi }_{k}({x}_{1}){\varphi }_{l}({x}_{2}){\mathrm{S\alpha }}_{k}{\alpha }_{l}(\omega )\) eq 2.3.2-2
This formulation makes it possible to take into account an excitation whose spatial distribution is arbitrary.
2.3.3. Put into equations#
2.3.3.1. Application of distributed turbulent excitation#
The application length \(L\) is intrinsically characterized by the domain of definition of the form functions associated with the excitation. The area of application is determined by the name data of the node around which it is centered.
\({x}_{n}\) designating the abscissa locating this node, the turbulent excitation is imposed on the \(\left[{x}_{n}-L/2,{x}_{n}+L/2\right]\) domain.
Since turbulent excitation can, on the other hand, be developed in a correlated manner in the two directions \(Y\) and \(Z\) orthogonal to the axis of the wire structure, the shape functions are in principle vectors with two components.
By convention, two form functions are therefore entered in a table_function, the first is associated with the direction \(Y\) and the other with the direction \(Z\). Each of the two functions is set to the interval \(\left[\mathrm{0,}L\right]\).
2.3.3.2. Turbulent excitement identified on model GRAPPE1#
The shape functions \({\varphi }_{k}\) are the first 12 modal-flexural deformations of the structure identified experimentally, distributed along the two directions orthogonal to the main axis of the beam. The general analytical expression of these deformations is as follows:
\({\overrightarrow{\varphi }}_{k}(x)=(\begin{array}{c}{\varphi }_{\text{Yk}}(x)\\ {\varphi }_{\text{Zk}}(x)\end{array})\) eq 2.3.3.2-1
with:
\({\varphi }_{\text{Yk}}(x)={A}_{\text{Yk}}\cdot \text{cos}(\frac{{n}_{\text{Yk}}}{L}x)+{B}_{\text{Yk}}\cdot \text{sin}(\frac{{n}_{\text{Yk}}}{L}x)+{C}_{\text{Yk}}\cdot \text{ch}(\frac{{n}_{\text{Yk}}}{L}x)+{D}_{\text{Yk}}\cdot \text{sh}(\frac{{n}_{\text{Yk}}}{L}x)\) eq 2.3.3.2-2
\({\varphi }_{\text{Zk}}(x)={A}_{\text{Zk}}\cdot \text{cos}(\frac{{n}_{\text{Zk}}}{L}x)+{B}_{\text{Zk}}\cdot \text{sin}(\frac{{n}_{\text{Zk}}}{L}x)+{C}_{\text{Zk}}\cdot \text{ch}(\frac{{n}_{\text{Zk}}}{L}x)+{D}_{\text{Zk}}\cdot \text{sh}(\frac{{n}_{\text{Zk}}}{L}x)\) eq 2.3.3.2-3
where \({n}_{\text{Yk}}\) and \({n}_{\text{Zk}}\) denote wave numbers, \(L\) is the length of application of the excitation and the coefficients \({A}_{\text{Yk}}\), \({B}_{\text{Yk}}\), \({C}_{\text{Yk}}\), \({D}_{\text{Yk}}\), \({A}_{\text{Zk}}\), \({B}_{\text{Zk}}\), \({C}_{\text{Zk}}\),,,, \({D}_{\text{Zk}}\) are real coefficients constants characteristic of the form function in question.
The first 6 shape functions are associated with the direction \(Y\) and \({A}_{\text{Zk}}\), \({B}_{\text{Zk}}\), \({C}_{\text{Zk}}\), \({D}_{\text{Zk}}\) are therefore zero, for \(1\mathrm{\le }k\mathrm{\le }6\).
The last 6 form functions are associated with the direction \(Z\) and \({A}_{\text{Yk}}\), \({B}_{\text{Yk}}\), \({C}_{\text{Yk}}\), \({D}_{\text{Yk}}\) are therefore zero, for \(7\mathrm{\le }k\mathrm{\le }12\).
This family of shape functions is therefore characterized by \(5\mathrm{\times }12\mathrm{=}60\) real coefficients.
The turbulent excitation identified on model GRAPPE1 is homogeneous in the two directions orthogonal to the axis of the wire structure, the turbulence being decorrelated between these two directions.
The interspectral matrix \(\left[{\mathrm{S\alpha }}_{k}{\alpha }_{l}\right]\) identified on the model GRAPPE1 is therefore a matrix of dimension \(12\mathrm{\times }12\), consisting of two identical diagonal blocks of dimension 6:
\(\left[{\mathrm{S\alpha }}_{k}{\alpha }_{l}\right]=\left[\begin{array}{cc}\left[{S}_{o}(\omega )\right]& \left[0\right]\\ \left[0\right]& \left[{S}_{o}(\omega )\right]\end{array}\right]\)
By hermitic symmetry property, this matrix is entirely defined by the data for the upper triangular (or lower) part of \(\left[{S}_{o}(\omega )\right]\), i.e. 21 interspectra. For each of them, the characteristic parameters are the plateau level, the cutoff frequency and the slope of the spectrum beyond this frequency.
The interspectral turbulent excitation matrix identified on model GRAPPE1 is therefore characterized by 63 real coefficients (\(3\mathrm{\times }21\)).
Note:
GRAPPE1 excitations are available at two reference rates. The set of data characterizing these excitations therefore represents* **246 real coefficients ( \(\mathrm{[}60+63\mathrm{]}\mathrm{\times }2\) **) .*
2.3.3.3. Excitation projection on a modal basis#
We note:
\({\phi }_{i}(x)=(\begin{array}{c}{\text{DY}}_{i}(x)\\ {\text{DZ}}_{i}(x)\end{array})\) the \(i\) -th deformed modal structure.
Let \({\beta }_{\text{ik}}\) be the coordinates of the \(i\) -th modal deform of the structure based on the form functions \({\varphi }_{k}(x)\):
\({\phi }_{i}(x)=\sum _{k=1}^{K}{\beta }_{\text{ik}}\cdot {\varphi }_{k}(x)\) eq 2.3.3.3-1
The modal excitation interspectra \({S}_{{Q}_{i}{Q}_{j}}(\omega )\) applied to the structure are then written:
\({S}_{{Q}_{i}{Q}_{j}}(\omega )\text{=}\sum _{k=1}^{K}\sum _{k=1}^{K}{\beta }_{\text{ik}}\cdot {\beta }_{\text{jl}}\cdot {S}_{{\alpha }_{k}{\alpha }_{l}}(\omega )\) eq 2.3.3.3-2
For each mode \(i\) of the structure, the coefficients \({\beta }_{\text{ik}}\) are determined by integrating the equation [éq2.3.3.3-1] premultiplied by the functions \({\varphi }_{j}\), over the field of application of the excitation. We thus obtain:
\(\underset{{x}_{0}\text{-}L/2}{\overset{{x}_{0}\text{+}L/2}{\int }}{\varphi }_{j}(x\text{+}L/2)\cdot {\phi }_{i}(x)\cdot \text{dx}\text{=}\sum _{k=1}^{K}{\beta }_{\text{ik}}\cdot \underset{{x}_{0}\text{-}L/2}{\overset{{x}_{0}\text{+}L/2}{\int }}{\varphi }_{j}(x\text{+}L/2)\cdot {\varphi }_{k}(x\text{+}L/2)\cdot \text{dx}\)
\(\begin{array}{cc}\underset{{x}_{0}\text{-}L/2}{\overset{{x}_{0}\text{+}L/2}{\int }}{\varphi }_{j}(x\text{+}L/2)\cdot {\phi }_{i}(x)\cdot \text{dx}\text{=}\sum _{k=1}^{K}{\beta }_{\text{ik}}\cdot \underset{0}{\overset{L}{\int }}{\varphi }_{j}(x)\cdot {\varphi }_{k}(x)\cdot \text{dx}& \forall (i,j)\end{array}\) eq 2.3.3.3-3
For each \(i\), equation [éq 2.3.3.3-3] is written in matrix form:
\(\left[\begin{array}{c}{a}_{\text{jk}}\end{array}\right]\cdot ({\beta }_{\text{ik}})\text{=}({b}_{\text{ij}})\) eq 2.3.3.3-4
with:
\({a}_{\text{jk}}\text{=}\underset{0}{\overset{L}{\int }}{\varphi }_{j}(x)\cdot {\varphi }_{k}(x)\cdot \text{dx}\)
either:
\({a}_{\text{jk}}\text{=}\underset{0}{\overset{L}{\int }}({\varphi }_{\text{Yj}}(x)\cdot {\varphi }_{\text{Yk}}(\varphi )+{\varphi }_{\text{Zj}}(x)\cdot {\varphi }_{\text{Zk}}(x))\cdot \text{dx}\)
and
\({b}_{\text{ij}}=\underset{{x}_{0}\text{-}L/2}{\overset{{x}_{0}\text{+}L/2}{\int }}{\varphi }_{j}(x+L/2)\cdot {\phi }_{i}(x)\cdot \text{dx}\)
either:
\({b}_{\text{ij}}\text{=}\underset{{x}_{0}\text{-}L/2}{\overset{{x}_{0}\text{+}L/2}{\int }}({\text{DY}}_{i}(x)\cdot {\varphi }_{\text{Yj}}(x+L/2)+{\text{DZ}}_{i}(x)\cdot {\varphi }_{\text{Zj}}(x+L/2))\cdot \text{dx}\)
Solving each of the systems of linear equations leads to \({\beta }_{\text{ik}}\).
Scalar products are calculated in the operator PROJ_SPEC_BASE [U4. 63.14].
Notes:
The functions \({\varphi }_{k}(x)\) represent, in practice, the modal deformations found on the model. The system (), which has a preponderant diagonal, is therefore well packaged. In particular, when the wireframe structure has a homogeneous linear mass, the functions \({\varphi }_{k}(x)\) are orthogonal and the matrix \(\left[\begin{array}{c}{a}_{\text{jk}}\end{array}\right]\) is diagonal.
Tests comparing the field of application of the excitation to the domain of definition of the structure are carried out.
2.4. Localized turbulent excitation model#
2.4.1. Keywords#
The keyword factor SPEC_EXCI_POINT of the DEFI_SPEC_TURB [U4.44.31] operator is used in the case of an excitation spectrum associated with one or more specific forces and moments. The user can define the spectrum by providing:
an interspectral matrix of excitations (the associated [interspectrum] concept must be generated beforehand),
the list of the nodes for applying these excitations,
the nature of the excitation applied in each of these nodes (force or moment),
the directions of application of the excitations thus defined.
It can also use a predefined turbulence spectrum, identified on model GRAPPE2.
2.4.2. Foundations#
The localized turbulent excitation model is a particular case of the distributed turbulent excitation model. Thus, it is assumed as in paragraph [§2.3.2] that the instantaneous linear density of turbulent forces \({f}_{t}(x,t)\) can be decomposed on a family of form functions \({\varphi }_{k}(x)\) in the following way:
\({f}_{t}(x,t)=\sum _{k=1}^{K}{\varphi }_{k}(x){\alpha }_{k}(t)\) eq 2.4.2-1
The coefficients \({\alpha }_{k}(t)\) define at each moment the decomposition of the turbulent excitation on the family of shape functions.
The interspectral density of turbulent excitation between two points of the wire structure with abscissa \({x}_{1}\) and \({x}_{2}\) is then written:
\({S}_{f}({x}_{1},{x}_{2},\omega )\text{=}\sum _{k=1}^{K}\sum _{l=1}^{K}{\varphi }_{k}({x}_{1})\cdot {\varphi }_{l}({x}_{2})\cdot {S}_{{\alpha }_{k}{\alpha }_{l}}(\omega )\) eq 2.4.2-2
The particularity of the localized turbulent excitation model is due to the specificity of the shape functions \({\varphi }_{k}(x)\):
\({\varphi }_{k}(x)\text{=}\delta (x-{x}_{k})\) allows you to represent a point force applied to the abscissa point \({x}_{k}\)
\({\varphi }_{k}(x)\text{=}{\delta }^{\text{'}}(x-{x}_{k})\) allows you to represent a point moment applied to the \({x}_{k}\) abscissa point
\(\delta (x-{x}_{k})\) and \({\delta }^{\text{'}}(x-{x}_{k})\) respectively designate the Dirac distribution and the derivative of the Dirac distribution at the abscissa point \({x}_{k}\).
Given the specificity of the shape functions, the projection of a localized turbulent excitation on a modal basis is much simpler than in the general case (distributed excitation), since it is possible to calculate analytically the expression of the projected excitation.
2.4.3. Put into equations#
2.4.3.1. Application of localized turbulent excitation#
We consider a turbulent excitation applied to a wire structure and consisting of specific forces and moments. This excitement is fully characterized by the following data:
list of nodes for the application of forces and specific moments,
nature of the excitation applied in each node (force or moment),
direction of the excitation applied in each node.
\(\text{Ainsi}{f}_{t}(x,t)\text{=}\sum _{k=1}^{K}{F}_{k}(s)\cdot \delta (x-{x}_{k})\cdot {n}_{k}-\sum _{m=1}^{M}{M}_{m}(s)\cdot {\delta }^{\text{'}}(x-{x}_{m})\cdot {n}_{m}\) eq 2.4.3.1-1
is the expression of localized turbulent excitation, characterized by \(K\) forces and \(M\) point moments, applied respectively to the abscissa nodes \({x}_{k}\) and \({x}_{m}\) in the \({n}_{k}\) and \({\overrightarrow{n}}_{m}\) directions.
We have: \({n}_{k}=(\begin{array}{c}0\\ \text{cos}\cdot ({\theta }_{k})\\ \text{sin}\cdot ({\theta }_{k})\end{array})\) and \({n}_{m}\) defined in an analogous way.
\(\theta\) represents the azimuth giving the direction of application of the force (or moment) in the \(P\) plane orthogonal to the neutral fiber at the application node, as defined in figure [Figure 2.4.3.1-a] below:
Figure 2.4.3.1-a: Definition of the direction of application
The generalized arousal associated with the \(i\) th mode of the structure, \({Q}_{i}(s)\), being defined by:
\({Q}_{i}(s)=\underset{0}{\overset{L}{\int }}{\phi }_{i}(x)\cdot {f}_{t}(x,t)\cdot \text{dx}\) eq 2.4.3.1-2
where \(L\) represents the length of the beam and \({\phi }_{i}(x)\) represents the deformation of the \(i\) mode, we obtain, taking into account the expression [éq 2.4.3.1-1]:
\({Q}_{i}(s)=\sum _{k=1}^{K}{F}_{k}(s)\cdot {\phi }_{i}({x}_{k})\cdot {n}_{k}-\sum _{m=1}^{M}{M}_{m}(s)\cdot {\phi }_{i}({x}_{k})\cdot {n}_{m}\) eq 2.4.3.1-3
The calculation of the modal excitation interspectra then leads to:
\(\begin{array}{c}{S}_{{Q}_{i}{Q}_{j}}(s)\text{=}\sum _{{k}_{1}=1}^{K}\sum _{{k}_{2}=1}^{K}{S}_{{F}_{{k}_{1}}{F}_{{k}_{2}}(s)}({\phi }_{i}({x}_{{k}_{1}})\text{.}{n}_{{k}_{1}})\text{.}({\phi }_{j}({x}_{{k}_{2}})\text{.}{n}_{{k}_{2}})\\ \text{+}\sum _{{k}_{1}=1}^{K}\sum _{{m}_{2}=1}^{M}{S}_{{F}_{{k}_{1}}{M}_{{m}_{2}}(s)}({\phi }_{i}({x}_{{k}_{1}})\text{.}{n}_{{k}_{1}})\text{.}({\phi }_{{j}^{¢}}({x}_{{m}_{2}})\text{.}{n}_{{m}_{2}})\\ \text{+}\sum _{{m}_{1}=1}^{M}\sum _{{k}_{2}=1}^{K}{S}_{{M}_{{m}_{1}}{F}_{{k}_{2}}(s)}({\phi }_{i}^{\text{'}}({x}_{{m}_{1}})\text{.}{n}_{{m}_{1}})\text{.}({\phi }_{j}({x}_{{k}_{2}})\text{.}{n}_{{k}_{2}})\\ \text{+}\sum _{{m}_{1}=1}^{M}\sum _{{m}_{2}=1}^{M}{S}_{{M}_{{m}_{1}}{M}_{{m}_{2}}(s)}({\phi }_{i}^{\text{'}}({x}_{{m}_{1}})\text{.}{n}_{{m}_{1}})\text{.}({\phi }_{{j}^{¢}}({x}_{{m}_{2}})\text{.}{n}_{{m}_{2}})\end{array}\) eq 2.4.3.1-4
Note:
When the user defines the turbulent excitation spectrum, he must fill in the interspectral matrix of point excitations whose terms are used above. This matrix has the dimension \(K+M\) (number of forces and point moments applied) .
2.4.3.2. Turbulent excitement identified on model GRAPPE2#
The turbulent excitation identified on model GRAPPE2 is represented by a resultant force and moment, applied at the same node in the two directions orthogonal to the axis of the structure. The linear density of this excitation is expressed as:
\({f}_{t}(x,s)=\frac{1}{2}{\mathrm{\rho U}}^{2}{D}_{h}\left[{L}_{p}\cdot {F}_{t}({s}_{r})\cdot \delta (x-{x}_{0})-{L}_{p}^{2}\cdot {M}_{t}({s}_{r})\cdot {\delta }^{\text{'}}(x-{x}_{0})\right](\begin{array}{c}0\\ 0\\ 1\end{array})\) eq 2.4.3.2-1
Where \(\rho\) is the density of the fluid, \(U\) is the mean flow velocity, \({D}_{h}\) is the hydraulic diameter, is the hydraulic diameter, \({L}_{p}\) is the thickness of the housing plate (corresponding to the excited length), \({x}_{0}\) is the abscissa of the point of application of the excitation, \({s}_{r}\text{=}\frac{s\cdot D}{U}\) is the reduced complex frequency, is the reduced complex frequency, \({F}_{t}({s}_{r})\) and \({M}_{t}({s}_{r})\) are the dimensionless coefficients representing the resulting force and moment.
The quantities \(\rho\), \(U\), \({D}_{h}\) and \({L}_{p}\) make it possible to size the excitation.
By substituting the expression [éq 2.4.3.2-3] in the relationship [éq 2.4.3.1-4] defining modal excitation \({Q}_{i}(s)\), we obtain:
\({Q}_{i}(s)=\frac{1}{2}{\mathrm{\rho U}}^{2}{D}_{h}\left[{L}_{p}\cdot {F}_{t}({s}_{r})\cdot {\phi }_{i}({x}_{0})\cdot (\begin{array}{c}0\\ 0\\ 1\end{array})+{L}_{p}^{2}\cdot {M}_{t}({s}_{r})\cdot {\phi }_{i}^{\text{'}}({x}_{0})\cdot (\begin{array}{c}0\\ 0\\ 1\end{array})\right]\) eq 2.4.3.2-2
Since the specific force and moment identified on model GRAPPE2 are uncorrelated, the calculation of the modal excitation interspectra finally leads to:
\(\begin{array}{cc}{S}_{{Q}_{i}{Q}_{j}}={(\frac{1}{2}{\mathrm{\rho U}}^{2}{D}_{h})}^{2}\frac{D}{U}\cdot & [:ref:`{L}_{p}^{2}\cdot {\phi }_{i}({x}_{0})\cdot (\begin{array}{c}0\\ 1\\ 1\end{array})´{\phi }_{j}({x}_{0})\cdot (\begin{array}{c}0\\ 1\\ 1\end{array})\cdot {S}_{{F}_{t}{F}_{t}}({s}_{r})\\ & +{L}_{p}^{4}\cdot {\phi }_{i}^{\text{'}}({x}_{0})\cdot (\begin{array}{c}0\\ 1\\ 1\end{array})´{\phi }_{j}^{\text{'}}({x}_{0})\cdot (\begin{array}{c}0\\ 1\\ 1\end{array})\cdot {S}_{{M}_{t}{M}_{t}}({s}_{r}) <{L}_{p}^{2}\cdot {\phi }_{i}({x}_{0})\cdot (\begin{array}{c}0\\ 1\\ 1\end{array})´{\phi }_{j}({x}_{0})\cdot (\begin{array}{c}0\\ 1\\ 1\end{array})\cdot {S}_{{F}_{t}{F}_{t}}({s}_{r})\\ & +{L}_{p}^{4}\cdot {\phi }_{i}^{\text{'}}({x}_{0})\cdot (\begin{array}{c}0\\ 1\\ 1\end{array})´{\phi }_{j}^{\text{'}}({x}_{0})\cdot (\begin{array}{c}0\\ 1\\ 1\end{array})\cdot {S}_{{M}_{t}{M}_{t}}({s}_{r})>\)]end {array} `eq 2.4.3.2-3
In this expression, \(D\) is the outside diameter of the structure, \({S}_{{F}_{t}{F}_{t}}({s}_{r})\) and \({S}_{{M}_{t}{M}_{t}}({s}_{r})\) represent respectively the dimensionless force and moment autospectra identified on the model GRAPPE2. The operator PROJ_SPEC_BASE [U4.63.14] calculates the modal excitation interspectra according to the relationship [éq 2.4.3.2-3] above.
Notes:
1) |
The dimensionless autospectra GRAPPE2 can be used to simulate the behavior of any structure in similarity with the model; the geometric parameters characteristic of the structure are then used to size the excitation. Since model GRAPPE2 was built in similarity with the reactor configuration, the following relationships are fixed and characteristic of this geometry: \(\frac{{D}_{h}}{D}\text{et}\frac{{L}_{p}}{D}\) It is recalled that \({D}_{h}\) and \(D\) designate the hydraulic diameter and the external diameter of the structure respectively; \({L}_{p}\) * is the thickness of the housing plate, corresponding to the excited length. The data of \(\rho\) , \(U\) , and and \(D\) is therefore sufficient to univocally size the turbulent excitement from the dimensionless autospectra. |
2) |
Since the adimensional autospectrum \({S}_{{F}_{t}{F}_{t}}({s}_{r})\) and \({S}_{{M}_{t}{M}_{t}}({s}_{r})\) are both defined by three real coefficients (plateau level, reduced cutoff frequency and slope beyond this frequency), only six constants make it possible to characterize the dimensionless turbulent excitation identified on model GRAPPE2. Four configurations having been studied (ascending or descending flow, centered or eccentric control rod), the set of data characterizing excitations GRAPPE2 therefore represent 24 real coefficients. |