Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------


The advantage of the test here is to calculate and test the modal effort autospectrum obtained from a pressure spectrum characteristic of established turbulent flows.

For this case, the mode of travel is imposed. It is therefore not necessary to perform a modal calculation for the structure. 
The modal force autospectrum, calculated from the static deformation, is written as:

:math:`\mathit{DSP}(\omega )\mathrm{=}{S}_{p}(\omega ){J}_{{{A}_{\text{ij}}}_{}}^{2}(w)`

The spectrum chosen here is constant and then zero based on a cutoff frequency:


.. image:: images/10000200000006A9000005092E78E7DE07EF2274.png
    :width: 6.3591in
    :height: 4.8075in


.. _RefImage_10000200000006A9000005092E78E7DE07EF2274.png:

The pressure spectrum is:

:math:`{S}_{{p}_{}}(\omega )\mathrm{=}{K}^{2}({\mathit{\rho U}}^{2}{)}^{2}{d}^{3}` for :math:`\mathrm{0,1}<\frac{\omega d}{2\pi U}<10`

With the settings selected, :math:`{S}_{{p}_{}}(\omega )\mathrm{\approx }1.0`

The coherence function chosen in the case of this circular cylinder subjected to parallel flow comes from a AU_YANG model:

:math:`{r}^{(s)}(x\mathrm{-}x\text{',}\omega )\mathrm{=}{e}^{\mathrm{-}(x\mathrm{-}x\text{'})\mathrm{/}\lambda }\text{cos}(\omega (x\mathrm{-}x\text{'})\mathrm{/}{U}_{c})`

:math:`{r}^{(s)}(\theta \mathrm{-}\theta \text{',}\omega )\mathrm{=}{e}^{\mathrm{-}R(\theta \mathrm{-}\theta \text{'})\mathrm{/}\lambda \text{'}}`

The parameters :math:`\lambda` and :math:`\lambda \text{'}` are the correlation lengths along the axis and the orthoradial direction respectively.

:math:`{U}_{c}` is the axial convective speed of vortices: it is equal to the product of the axial speed coefficient by the fluid speed. 

The acceptance function is defined by:

:math:`{J}_{{{A}_{\text{ij}}}_{}}^{2}(\omega )\mathrm{=}\underset{A}{\mathrm{\int }}\underset{A}{\mathrm{\int }}r(x\mathrm{-}x\text{',}\omega ){f}_{{i}_{\alpha }}(x){f}_{{j}_{\alpha \text{'}}}(x\text{'}){n}_{\alpha }(x){n}_{\alpha \text{'}}(x\text{'})\text{dA}\text{dA}\text{'}`

         In our case it applies to:

:math:`{J}_{{{A}_{\text{nm}}}_{}}^{2}(\omega )\mathrm{=}\underset{A}{\mathrm{\int }}\underset{A}{\mathrm{\int }}{e}^{\mathrm{-}R\mathrm{\mid }\theta \mathrm{-}\theta \text{'}\mathrm{\mid }\mathrm{/}\lambda \text{'}}{e}^{\mathrm{-}R\mathrm{\mid }x\mathrm{-}x\text{'}\mathrm{\mid }\mathrm{/}\lambda }\text{cos}(\frac{\omega (x\mathrm{-}x\text{'})}{{U}_{c}})x\text{cos}\theta x\text{'}\text{cos}\theta \text{'}\mathit{dxRd}\theta \mathit{dx}\text{'}\mathit{Rd}\theta \text{'}`

:math:`{J}_{{{A}_{\text{nm}}}_{}}^{2}(\omega )\mathrm{=}\underset{0}{\overset{2\pi }{\mathrm{\int }}}\underset{0}{\overset{2\pi }{\mathrm{\int }}}{e}^{\mathrm{-}R\mathrm{\mid }\theta \mathrm{-}\theta \text{'}\mathrm{\mid }\mathrm{/}\lambda \text{'}}\text{cos}\theta \text{cos}\theta \text{'}{R}^{2}d\theta d\theta \text{'}\underset{0}{\overset{H}{\mathrm{\int }}}\underset{0}{\overset{H}{\mathrm{\int }}}{e}^{\mathrm{-}R\mathrm{\mid }x\mathrm{-}x\text{'}\mathrm{\mid }\mathrm{/}\lambda }\text{cos}(\frac{\omega (x\mathrm{-}x\text{'})}{{U}_{c}})xx\text{'}\text{dxdx}\text{'}`

     In the following table, values of this integral are given using Python:


.. csv-table::

    ":math:`\omega (\mathrm{rad}/s)` "," :math:`{I}_{T}(\omega )`"
    "0.06283", "252.701"
    "0.6283", "249.663"



Benchmark results
----------------------

Analytical result.


Bibliographical references
--------------------------


1. ROUSSEAU G., LUU H.T.: Mass, damping and stiffness added for a vibrating structure placed in a potential flow - Bibliography and implementation in the*Code_Aster* - HP-61/95/064.


2. BLEVINS R.D: Formulas for natural frequency and mode shape. Ed. Krieger 1984.


3. ROUSSEAU G. Specification of the acceptance calculation in the*Code_Aster*. Spectral response of structures to random turbulent excitation HP51 /97/027/A