Modeling A
==============


Characteristics of modeling
-----------------------------------

The pipe is modelled using Timoshenko straight beam elements: POU_D_T. It is broken down into 270 elements spread over four sections. The sections :math:`\mathrm{AB}` and :math:`\mathrm{CD}` each include 61 elements, the section :math:`\mathrm{BC}`, 60 elements, and the section :math:`\mathrm{DE}`, 88. At points :math:`A` and :math:`D` the degrees of freedom are blocked in the directions :math:`x`, :math:`y` and :math:`z` in translation, and in the :math:`z` direction in rotation. In accordance with the results of the first series of tests, the fluid damping at rest is taken to be equal to 1.3% and the added mass coefficient is equal to 2.0711.


Characteristics of the mesh
----------------------------

The total number of nodes used for this mesh is 271.

The stitches, of type SEG2, are 270 in number.

The mesh file is written in ASTER format.


Calculation steps
----------------

The flow speed profile and the parameters taking into account fluid-structure coupling are defined using operators DEFI_FONC_FLUI and DEFI_FLUI_STRU. The modal parameters of the structure are calculated taking into account fluid-elastic forces using the operator CALC_FLUI_STRU. The definition of random excitation is carried out using the operator DEFI_SPEC_TURB. Excitation is projected on a modal basis using the PROJ_SPEC_BASE operator. Modal response interspectra are calculated using the DYNA_SPEC_MODAL operator. The autospectra of movement at the nodes are deduced by using the operator REST_SPEC_PHYS.

We can only calculate the spectral response for one flow speed at a time, so these last three steps (PROJ_SPEC_BASE, DYNA_SPEC_MODAL, REST_SPEC_PHYS) are carried out in a loop, in the command file, where we go through the list of flow velocities.


Tested values
---------------

The [:ref:`Figure 5.1-a <Figure 5.1-a>`] shows a comparison of the displacement RMS at point :math:`E`, expressed as a percentage of the outside diameter of the tube, between:


* on the one hand, the first series of tests ANL which focused on 3 of the tubes instrumentated in the tube bundle in the same way as a steam generator bundle,


*and on the other hand simulation using*Code_Aster*.

These results are presented as a function of the intertube speed of the incident flow, expressed in :math:`m/s`. The experimental results were obtained for discrete values of flow velocities. The calculation having been carried out for 201 speed values equally distributed over the interval :math:`[0.5‑2.5m/s]`, the calculated displacement appears as a continuous curve in the explored range of speeds. Experimentally as in the calculation, we note the existence of a strong increase in displacement RMS. This increase is associated with a vibratory instability of the dynamic system, associated with the cancellation of its damping. The flow rate at which this increase occurs is the instability rate of the system. The calculation allows this speed to be estimated at :math:`2m/s` while the experimental finding predicts a value greater than :math:`\mathrm{1,8}m/s` (instability tests are often stopped before instability is reached to avoid damage to structures). In this configuration, the calculation-measurement difference on this result, which is essential for the maintenance of steam generator tubes, is therefore of the order of 10%.


.. image:: images/Object_38.svg
    :width: 591
    :height: 407


.. _RefImage_Object_38.svg:

**Figure 5.1-a: Displacement RMS** **** at point E, as a function of the interflow pipe speed, expressed as a percentage of the outside diameter of the tube. **


notes
---------

The modal deformations under flow are assumed to remain unchanged compared to those calculated in fluid at rest.


Non-regression tests
-----------------------

To ensure the non-regression of the code, the operators TEST_FONCTION and TEST_TABLE are used in the command file. These two operators make it possible to test respectively, on the one hand, the values of natural frequencies and reduced damping, and on the other hand, the displacement RMS. The tolerance is set at 1E—03%.