Axial flow (example: fuel assemblies)
=====================================================

The integration of this fluid-elastic excitation model in*Code_Aster* was discussed in the specification note [:ref:`bib. 2 <bib. 2>`]. Model concept note MEFISTEAU [:ref:`bib. 6 <bib. 6>`] is the theoretical reference documentation.


Description of the studied configuration
------------------------

We consider a bundle of

.. image:: images/Object_321.svg
    :width: 17
    :height: 15


.. _RefImage_Object_321.svg:

circular cylinders that are mobile in bending and subject to an incompressible flow of viscous fluid, limited by a rigid cylindrical chamber of circular or rectangular cross section [:ref:`Figure 5.1-a <Figure 5.1-a>`].


.. image:: images/100044FC000069D500004F60E222264E598443E0.svg
    :width: 17
    :height: 15


.. _RefImage_100044FC000069D500004F60E222264E598443E0.svg:

**Figure 5.1-a: Beam under axial flow**

The cylinders are all parallel, directed along the axis of the chamber. They have a common length, noted

.. image:: images/Object_322.svg
    :width: 17
    :height: 15


.. _RefImage_Object_322.svg:

. To simplify the notations, it is subsequently assumed that

.. image:: images/Object_323.svg
    :width: 17
    :height: 15


.. _RefImage_Object_323.svg:

is the guiding axis. The steady flow is axial and assumed to be uniform in each section. The density of the fluid can be variable along the axis

.. image:: images/Object_324.svg
    :width: 17
    :height: 15


.. _RefImage_Object_324.svg:

(thermal gradients), the speed of the steady flow also depends on the variable

.. image:: images/Object_325.svg
    :width: 17
    :height: 15


.. _RefImage_Object_325.svg:

.


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


* The first step concerns the determination of the air modal base of the beam. This operation is carried out by the operator CALC_MODES. This step is essential because fluid-elastic forces are projected onto this base.


* The second step involves taking fluid-elastic forces into account by operator CALC_FLUI_STRU. This step is broken down into 7 sub-tasks:


Pre-treatments
~~~~~~~~~~~~~~~~


.. csv-table::

    "1°/", "By means of the topology of the mesh, deduction of the coordinates of the centers of the cylinders in the bundle and then verification of the correct arrangement of the cylinders in relation to each other (in particular, it is verified that there is no overlap between two cylinders) and in relation to the rigid chamber."
    "", ""
    "2°/", "Determination of the fluid excitation length, common to all cylinders, as well as of an associated discretization along the direction axis."
    "", ""
    "3°/", "Composition of the tables giving the modal deformations in air of each cylinder of the bundle, for each of the modes taken into account for the fluid-structure coupling. To do this, the deformed ones are interpolated at the points of the discretization determined previously."



Solving the modal problem under flow
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "4°/", "Solving the disturbed fluid problem. Determining the potential of disturbed velocities requires the reversal of high-order linear systems calling for the implementation of the Crout method."


For each flow speed


+---+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|5°/|Calculation of the added mass, damping and stiffness matrices giving the fluid-elastic force transfer matrix in the air modal base:                                                                                                                                                                                                                                                                                                                                                                                                |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   + .. image:: images/Object_326.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|   |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+   +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   | .. image:: images/Object_327.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+   +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|   |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   | full symmetric;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   + .. image:: images/Object_328.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|   |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+   +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   + seems to be full and not symmetric.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+---+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+---+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|6°/|Solving the modal problem under flow; we solve the complete problem with vectors and eigenics                                                                                                                                                                                                                                                                                                                                                                                                                                      |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   + .. image:: images/Object_329.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|   |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+   +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+---+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


We do not overlook the extra-diagonal terms of

.. image:: images/Object_330.svg
    :width: 17
    :height: 15


.. _RefImage_Object_330.svg:

. After reformulation, the resolution is carried out using the QR algorithm: obtaining the masses, frequencies and reduced modal damping under flow

.. image:: images/Object_331.svg
    :width: 17
    :height: 15


.. _RefImage_Object_331.svg:

, complex modal deformations

.. image:: images/Object_332.svg
    :width: 17
    :height: 15


.. _RefImage_Object_332.svg:

expressed in the air base; of these, only the real part is retained after minimizing the imaginary part (calculating a criterion on the imaginary part).


+---+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|7°/|Restitution of deformations under flow in the physical base.                                                                                                                                                                                                                                                                                                                        |
+   +                                                                                                                                                                                                                                                                                                                                                                                    +
|   |                                                                                                                                                                                                                                                                                                                                                                                    |
+   + .. image:: images/Object_333.svg                                                                                                                                                                                                                                                                                                                                                   +
|   |    :width: 17                                                                                                                                                                                                                                                                                                                                                                      |
+   +    :height: 15                                                                                                                                                                                                                                                                                                                                                                     +
|   |                                                                                                                                                                                                                                                                                                                                                                                    |
+   +                                                                                                                                                                                                                                                                                                                                                                                    +
|   |                                                                                                                                                                                                                                                                                                                                                                                    |
+   +                                                                                                                                                                                                                                                                                                                                                                                    +
|   |                                                                                                                                                                                                                                                                                                                                                                                    |
+   +                                                                                                                                                                                                                                                                                                                                                                                    +
|   | .. image:: images/Object_334.svg                                                                                                                                                                                                                                                                                                                                                   |
+   +    :width: 17                                                                                                                                                                                                                                                                                                                                                                      +
|   |    :height: 15                                                                                                                                                                                                                                                                                                                                                                     |
+   +                                                                                                                                                                                                                                                                                                                                                                                    +
|   |                                                                                                                                                                                                                                                                                                                                                                                    |
+   +                                                                                                                                                                                                                                                                                                                                                                                    +
|   |                                                                                                                                                                                                                                                                                                                                                                                    |
+   +                                                                                                                                                                                                                                                                                                                                                                                    +
|   | is the matrix whose columns are the modal deformations in air, expressed on a physical basis.                                                                                                                                                                                                                                                                                      |
+---+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


End of loop on flow velocities

**Notes:**


* *Knowledge of the coordinates of the centers of the cylinders (pre-treatment 1°) is necessary to solve the disturbed fluid problem (sub-task 4°). This resolution leads to the estimation of the terms of the fluid-elastic force transfer matrix (sub-task 5°), which involve pressure and speed disturbances.*


* *The determination of a common excitation length and the creation of an associated discretization (2° pre-treatment) make it possible to define a domain of integration on the structures for the projection of fluid-elastic forces on a modal basis. The interpolation of the modal deformations at the same points is therefore required (3° pre-treatment) .*


* The dynamic behavior of the beam under flow can also be studied using a simplified representation of the beam (with equivalent tubes). The calculation steps for taking into account the fluid-structure coupling are then identical to those described above, the only differences appearing in the pre-treatments. This second approach is described more precisely in the note [:ref:`bib. 2 <bib. 2>`]. In step 1° of the pre-treatments, the coordinates of the centers of the cylinders in the bundle are then specified directly by the user, who also establishes the correspondence between the cylinders of the beam and the beams in the simplified representation given by the mesh. In step 3° of the pre-treatments, this correspondence makes it possible to assign to the cylinders of the beam, to the discretization points determined in step 2°, the modal deformations of the beams in the simplified representation.


Solving the unsteady fluid problem
--------------------------------------------


Simplifying hypothesis
~~~~~~~~~~~~~~~~~~~~~~~~~~~~


+------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|**H1**|The field of unsteady fluid velocities is determined analytically by assuming that the disturbed flow                                                                                                                                                                                                                                                                                                                                                                                |
+      +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+      + .. image:: images/Object_335.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                    +
|      |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
+      +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                      +
|      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+      +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+      +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+      + is potential throughout the fluid domain, and that the steady flow is uniform transversely, but a function of the axial position                                                                                                                                                                                                                                                                                                                                                    +
|      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+      +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|      | .. image:: images/Object_336.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
+      +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                       +
|      |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
+      +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+      +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+      +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+      +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|      |:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
+------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+



.. image:: images/Object_337.svg
    :width: 17
    :height: 15


.. _RefImage_Object_337.svg:

 **eq. 5.3.1-1**

Such a speed field allows sliding on the walls of the cylinders which will make it possible to calculate the viscous stress by a law of friction.


+------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|**H2**|The movement of the cylinders only generates speed disturbances                                                                                                                                                                                                                                                                                          |
+      +                                                                                                                                                                                                                                                                                                                                                         +
|      |                                                                                                                                                                                                                                                                                                                                                         |
+      + .. image:: images/Object_338.svg                                                                                                                                                                                                                                                                                                                        +
|      |    :width: 17                                                                                                                                                                                                                                                                                                                                           |
+      +    :height: 15                                                                                                                                                                                                                                                                                                                                          +
|      |                                                                                                                                                                                                                                                                                                                                                         |
+      +                                                                                                                                                                                                                                                                                                                                                         +
|      |                                                                                                                                                                                                                                                                                                                                                         |
+      +                                                                                                                                                                                                                                                                                                                                                         +
|      |                                                                                                                                                                                                                                                                                                                                                         |
+      + only radially and orthoradially (hypothesis of slender bodies):                                                                                                                                                                                                                                                                                         +
|      |                                                                                                                                                                                                                                                                                                                                                         |
+      +                                                                                                                                                                                                                                                                                                                                                         +
|      | .. image:: images/Object_339.svg                                                                                                                                                                                                                                                                                                                        |
+      +    :width: 17                                                                                                                                                                                                                                                                                                                                           +
|      |    :height: 15                                                                                                                                                                                                                                                                                                                                          |
+      +                                                                                                                                                                                                                                                                                                                                                         +
|      |                                                                                                                                                                                                                                                                                                                                                         |
+      +                                                                                                                                                                                                                                                                                                                                                         +
|      |                                                                                                                                                                                                                                                                                                                                                         |
+------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+



+------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|**H3**|The pressure field is decomposed into stationary and disturbed parts according to                                                                                                                                                                                                                                                                                                                |
+      +                                                                                                                                                                                                                                                                                                                                                                                                 +
|      |                                                                                                                                                                                                                                                                                                                                                                                                 |
+      + .. image:: images/Object_340.svg                                                                                                                                                                                                                                                                                                                                                                +
|      |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                   |
+      +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                  +
|      |                                                                                                                                                                                                                                                                                                                                                                                                 |
+      +                                                                                                                                                                                                                                                                                                                                                                                                 +
|      |                                                                                                                                                                                                                                                                                                                                                                                                 |
+      +                                                                                                                                                                                                                                                                                                                                                                                                 +
|      |                                                                                                                                                                                                                                                                                                                                                                                                 |
+      + The stationary pressure field only depends on                                                                                                                                                                                                                                                                                                                                                   +
|      |                                                                                                                                                                                                                                                                                                                                                                                                 |
+      +                                                                                                                                                                                                                                                                                                                                                                                                 +
|      | .. image:: images/Object_341.svg                                                                                                                                                                                                                                                                                                                                                                |
+      +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                   +
|      |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                  |
+      +                                                                                                                                                                                                                                                                                                                                                                                                 +
|      |                                                                                                                                                                                                                                                                                                                                                                                                 |
+      +                                                                                                                                                                                                                                                                                                                                                                                                 +
|      |                                                                                                                                                                                                                                                                                                                                                                                                 |
+      +                                                                                                                                                                                                                                                                                                                                                                                                 +
|      | and its gradient is equal to:                                                                                                                                                                                                                                                                                                                                                                   |
+------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+



.. image:: images/Object_342.svg
    :width: 17
    :height: 15


.. _RefImage_Object_342.svg:

 **eq. 5.3.1-2**


+-----+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|where|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+     + .. image:: images/Object_343.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+     + refers to the hydraulic diameter of the bundle,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|     | .. image:: images/Object_344.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
+     +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|     | refers to the local friction coefficient for the stationary speed                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
+     + .. image:: images/Object_345.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+     +. It depends on the Reynolds number, calculated using the stationary speed                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    +
|     | .. image:: images/Object_346.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
+     +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|     |, the hydraulic diameter of the bundle, and the roughness of the walls. This coefficient is deduced from Nikuradze's law characterizing pipe flows;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+-----+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+     + .. image:: images/Object_347.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+     + refers to the gravity field. Its action on the stationary pressure field depends on the inclination of the                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|     | .. image:: images/Object_348.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
+     +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|     | beam.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
+-----+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+



Determining the potential of disturbed speeds
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We are looking for an analytical solution for

.. image:: images/Object_349.svg
    :width: 17
    :height: 15


.. _RefImage_Object_349.svg:

in the form of a superposition of elementary singularities that can be written as:


.. image:: images/Object_350.svg
    :width: 17
    :height: 15


.. _RefImage_Object_350.svg:

 **eq. 5.3.2-1**

at the center of each cylinder :math:`k` and:


.. image:: images/Object_351.svg
    :width: 17
    :height: 15


.. _RefImage_Object_351.svg:

 **eq. 5.3.2-2**

at the center of the rigid enclosure when the latter is circular where:


+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                                                   |refers to the truncation order of the Laurent series                                                                                                                                                                                             |
+ .. image:: images/Object_353.svg                                                  +                                                                                                                                                                                                                                                 +
|    :width: 17                                                                     |                                                                                                                                                                                                                                                 |
+    :height: 15                                                                    + .. image:: images/Object_354.svg                                                                                                                                                                                                                +
|                                                                                   |    :width: 335                                                                                                                                                                                                                                  |
+                                                                                   +    :height: 251                                                                                                                                                                                                                                 +
|                                                                                   |                                                                                                                                                                                                                                                 |
+                                                                                   +                                                                                                                                                                                                                                                 +
|                                                                                   |                                                                                                                                                                                                                                                 |
+                                                                                   +                                                                                                                                                                                                                                                 +
|                                                                                   |                                                                                                                                                                                                                                                 |
+                                                                                   +                                                                                                                                                                                                                                                 +
|                                                                                   |                                                                                                                                                                                                                                                 |
+                                                                                   +,                                                                                                                                                                                                                                                +
|                                                                                   |                                                                                                                                                                                                                                                 |
+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                                                   |designate the polar coordinates in a plane perpendicular to the axis                                                                                                                                                                             |
+ .. image:: images/Object_355.svg                                                  +                                                                                                                                                                                                                                                 +
|    :width: 335                                                                    |                                                                                                                                                                                                                                                 |
+                                                                                   + .. image:: images/Object_356.svg                                                                                                                                                                                                                +
|                                                                                   |    :width: 335                                                                                                                                                                                                                                  |
+                                                                                   +    :height: 251                                                                                                                                                                                                                                 +
|                                                                                   |                                                                                                                                                                                                                                                 |
+                                                                                   +                                                                                                                                                                                                                                                 +
|                                                                                   |                                                                                                                                                                                                                                                 |
+                                                                                   +                                                                                                                                                                                                                                                 +
|                                                                                   |                                                                                                                                                                                                                                                 |
+                                                                                   +, centered at the center of the cylinder k,                                                                                                                                                                                                      +
|                                                                                   |                                                                                                                                                                                                                                                 |
+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                                                   |designate the polar coordinates in a plane perpendicular to the axis                                                                                                                                                                             |
+ .. image:: images/Object_357.svg                                                  +                                                                                                                                                                                                                                                 +
|    :width: 335                                                                    |                                                                                                                                                                                                                                                 |
+                                                                                   + .. image:: images/Object_358.svg                                                                                                                                                                                                                +
|                                                                                   |    :width: 335                                                                                                                                                                                                                                  |
+                                                                                   +    :height: 251                                                                                                                                                                                                                                 +
|                                                                                   |                                                                                                                                                                                                                                                 |
+                                                                                   +                                                                                                                                                                                                                                                 +
|                                                                                   |                                                                                                                                                                                                                                                 |
+                                                                                   +                                                                                                                                                                                                                                                 +
|                                                                                   |                                                                                                                                                                                                                                                 |
+                                                                                   +, centered at the center of the circular rigid enclosure.                                                                                                                                                                                        +
|                                                                                   |                                                                                                                                                                                                                                                 |
+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


The coefficients

.. image:: images/Object_359.svg
    :width: 17
    :height: 15


.. _RefImage_Object_359.svg:

and

.. image:: images/Object_360.svg
    :width: 17
    :height: 15


.. _RefImage_Object_360.svg:

expressions [:ref:` **éq. 5.3.2-1**  < **éq. 5.3.2-1** >`] and [:ref:` **éq. 5.3.2‑2**  < **éq. 5.3.2‑2** >`] are determined by applying the condition to the non-penetration limits:


* on the outline of each mobile cylinder

    .. image:: images/Object_361.svg
        :width: 17
        :height: 15


    .. _RefImage_Object_361.svg:

, this condition is written as:


.. image:: images/Object_362.svg
    :width: 17
    :height: 15


.. _RefImage_Object_362.svg:


+-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|where|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+     + .. image:: images/Object_363.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                    +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+     + and                                                                                                                                                                                                                                                                                                                                                                                                                                                                               +
|     | .. image:: images/Object_364.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
+     +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|     | designate the components of the displacement of the neutral fiber from the cylinder k to the abscissa                                                                                                                                                                                                                                                                                                                                                                             |
+     + .. image:: images/Object_365.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                    +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+     + in the coordinate system                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|     | .. image:: images/Object_366.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
+     +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|     |,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+     + .. image:: images/Object_367.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                    +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+     + designate the polar coordinates in the                                                                                                                                                                                                                                                                                                                                                                                                                                            +
|     | .. image:: images/Object_368.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
+     +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|     | coordinate system whose origin is taken at the center of the cylinder                                                                                                                                                                                                                                                                                                                                                                                                             |
+     + .. image:: images/Object_369.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                    +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+     +,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+     + .. image:: images/Object_370.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                    +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+     + refers to the radius of the cylinder                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|     | .. image:: images/Object_371.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
+     +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|     |,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+     + .. image:: images/Object_372.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                    +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+     +.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+



* on the outline of a circular rigid enclosure, it is written:


.. image:: images/Object_373.svg
    :width: 17
    :height: 15


.. _RefImage_Object_373.svg:

where

.. image:: images/Object_374.svg
    :width: 17
    :height: 15


.. _RefImage_Object_374.svg:

refers to the radius of the enclosure.

In the case of a rectangular rigid enclosure, this condition is taken into account by a method derived from the "images" method [:ref:`bib. 6 <bib. 6>`]; the fluid problem confined by the rectangular enclosure is made equivalent to the problem in an infinite environment by creating images of the cylinders moving in the beam with respect to the sides of the enclosure. This method leads to the introduction of new singularities of the form [:ref:` **éq. 5.3.2-1**  < **éq. 5.3.2-1** >`], placed at the center of the "image" cylinders, in the expression of

.. image:: images/Object_375.svg
    :width: 17
    :height: 15


.. _RefImage_Object_375.svg:

. However, it does not add any unknown to the problem since the coefficients for these new singularities are derived from those of the moving cylinders of the beam through the interplay of images.

Finally, the potential of disturbed speeds is written as:


.. image:: images/Object_376.svg
    :width: 17
    :height: 15


.. _RefImage_Object_376.svg:

 **eq. 5.3.2-3**

Where

.. image:: images/Object_377.svg
    :width: 17
    :height: 15


.. _RefImage_Object_377.svg:

refers to the number of moving cylinders in the bundle. The functions

.. image:: images/Object_378.svg
    :width: 17
    :height: 15


.. _RefImage_Object_378.svg:

are linear combinations of

.. image:: images/Object_379.svg
    :width: 17
    :height: 15


.. _RefImage_Object_379.svg:


.. image:: images/Object_380.svg
    :width: 17
    :height: 15


.. _RefImage_Object_380.svg:

and

.. image:: images/Object_381.svg
    :width: 17
    :height: 15


.. _RefImage_Object_381.svg:

whose coefficients are determined by the preceding boundary conditions. This requires the resolution of linear systems of high orders and full matrices. The inversions are carried out by implementing the Crout method.


Fluid force modeling
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

First, the forces due to pressure disturbances are retained.

.. image:: images/Object_382.svg
    :width: 17
    :height: 15


.. _RefImage_Object_382.svg:

, related to the potential of speeds disturbed by:


.. image:: images/Object_383.svg
    :width: 17
    :height: 15


.. _RefImage_Object_383.svg:

 **eq. 5.3.3-1**

The resultant of the disturbed pressure field around each moving cylinder is a linear force

.. image:: images/Object_384.svg
    :width: 17
    :height: 15


.. _RefImage_Object_384.svg:

Acting next

.. image:: images/Object_385.svg
    :width: 17
    :height: 15


.. _RefImage_Object_385.svg:

. This force depends linearly on

.. image:: images/Object_386.svg
    :width: 17
    :height: 15


.. _RefImage_Object_386.svg:

and

.. image:: images/Object_387.svg
    :width: 17
    :height: 15


.. _RefImage_Object_387.svg:

, thus generating added terms of mass, damping and stiffness.

The forces linked to the viscosity of the fluid are then taken into account.

In a quasistatic approach, we consider the action of the fluid velocity field

.. image:: images/Object_388.svg
    :width: 17
    :height: 15


.. _RefImage_Object_388.svg:

around a cylinder at the moment

.. image:: images/Object_389.svg
    :width: 17
    :height: 15


.. _RefImage_Object_389.svg:

: in the coordinate system linked to the cylinder, the flow, of speed

.. image:: images/Object_390.svg
    :width: 17
    :height: 15


.. _RefImage_Object_390.svg:

in order 0, has an incidence in relation to the cylinder that is a function of speed disturbances and the movement of the cylinder itself. This results in drag force and lift force. It is shown that the following components

.. image:: images/Object_391.svg
    :width: 17
    :height: 15


.. _RefImage_Object_391.svg:

Of the resultant linear force

.. image:: images/Object_392.svg
    :width: 17
    :height: 15


.. _RefImage_Object_392.svg:

Spell, for the cylinder

.. image:: images/Object_393.svg
    :width: 17
    :height: 15


.. _RefImage_Object_393.svg:

:


.. image:: images/Object_394.svg
    :width: 17
    :height: 15


.. _RefImage_Object_394.svg:

 **eq. 5.3.3-2**


.. image:: images/Object_395.svg
    :width: 17
    :height: 15


.. _RefImage_Object_395.svg:

 **eq. 5.3.3-3**


+-----+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|where|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     + .. image:: images/Object_396.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     + refers to the zero incidence slope of the lift coefficient around a very slightly inclined cylinder (                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|     | .. image:: images/Object_397.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
+     +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     | = 0.08).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+-----+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     + .. image:: images/Object_398.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     + and                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       +
|     | .. image:: images/Object_399.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
+     +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     | designate the mean of the speed disturbances along the axes                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
+     + .. image:: images/Object_400.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     + and                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       +
|     | .. image:: images/Object_401.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
+     +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     | along the axes and around the cylinders, which depend linearly on the                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+     + .. image:: images/Object_402.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            +
|     |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     + and                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       +
|     | .. image:: images/Object_403.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
+     +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     | (cf. [:ref:` **éq. 5.3.2-3**  < **éq. 5.3.2-3** >`]).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+-----+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


These forces generate additional damping and stiffness terms.

Finally, the action of the stationary pressure field on deformed mobile structures is taken into account. It is shown that the resultant linear force

.. image:: images/Object_404.svg
    :width: 17
    :height: 15


.. _RefImage_Object_404.svg:

On the cylinder

.. image:: images/Object_405.svg
    :width: 17
    :height: 15


.. _RefImage_Object_405.svg:

has as components, in order 1:


.. image:: images/Object_406.svg
    :width: 17
    :height: 15


.. _RefImage_Object_406.svg:

 **eq. 5.3.3-4**


.. image:: images/Object_407.svg
    :width: 17
    :height: 15


.. _RefImage_Object_407.svg:

 **eq. 5.3.3-5**

These forces only generate terms of added stiffness and no coupling between cylinders.

The expressions [:ref:` **éq. 5.3.3-1**  < **éq. 5.3.3-1** >`], [:ref:` **éq. 5.3.3-2**  < **éq. 5.3.3-2** >`], and [:ref:` **éq. 5.3.3-3**  < **éq. 5.3.3-3** >`] highlight the need to solve the disturbed fluid problem before estimating fluid-elastic forces.


Expression of terms in the fluid-elastic force transfer matrix
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Summary of linear forces**

For each cylinder

.. image:: images/Object_408.svg
    :width: 17
    :height: 15


.. _RefImage_Object_408.svg:

, fluid-elastic forces are written as follows

.. image:: images/Object_409.svg
    :width: 17
    :height: 15


.. _RefImage_Object_409.svg:

and

.. image:: images/Object_410.svg
    :width: 17
    :height: 15


.. _RefImage_Object_410.svg:

:


.. image:: images/Object_411.svg
    :width: 17
    :height: 15


.. _RefImage_Object_411.svg:

 **eq. 5.3.4-1**

and are linear combinations of:


.. image:: images/Object_412.svg
    :width: 17
    :height: 15


.. _RefImage_Object_412.svg:

**Breakdown of the movement on a modal basis**

The movement of the cylinder bundle is broken down as follows:

.. image:: images/Object_413.svg
    :width: 17
    :height: 15


.. _RefImage_Object_413.svg:

air vibration modes. We note

.. image:: images/Object_414.svg
    :width: 17
    :height: 15


.. _RefImage_Object_414.svg:

(

.. image:: images/Object_415.svg
    :width: 17
    :height: 15


.. _RefImage_Object_415.svg:

and

.. image:: images/Object_416.svg
    :width: 17
    :height: 15


.. _RefImage_Object_416.svg:

) the following deformations

.. image:: images/Object_417.svg
    :width: 17
    :height: 15


.. _RefImage_Object_417.svg:

Of the cylinder

.. image:: images/Object_418.svg
    :width: 17
    :height: 15


.. _RefImage_Object_418.svg:

corresponding to the j th mode of the beam. The components of the displacement of the neutral fiber of the cylinder

.. image:: images/Object_419.svg
    :width: 17
    :height: 15


.. _RefImage_Object_419.svg:

At the abscissa

.. image:: images/Object_420.svg
    :width: 17
    :height: 15


.. _RefImage_Object_420.svg:

can then be written as:


.. image:: images/Object_421.svg
    :width: 17
    :height: 15


.. _RefImage_Object_421.svg:

 **eq. 5.3.4-2**


.. image:: images/Object_422.svg
    :width: 17
    :height: 15


.. _RefImage_Object_422.svg:

 **eq. 5.3.4-3**

where

.. image:: images/Object_423.svg
    :width: 17
    :height: 15


.. _RefImage_Object_423.svg:

is the vector of generalized movements.

**Force projection on a modal basis**


* Note

    .. image:: images/Object_424.svg
        :width: 17
        :height: 15


    .. _RefImage_Object_424.svg:

the projection of fluid-elastic forces according to the i th mode of the beam.


.. image:: images/Object_425.svg
    :width: 17
    :height: 15


.. _RefImage_Object_425.svg:

 **eq. 5.3.4-4**


.. image:: images/Object_426.svg
    :width: 17
    :height: 15


.. _RefImage_Object_426.svg:

is a linear combination of

.. image:: images/Object_427.svg
    :width: 17
    :height: 15


.. _RefImage_Object_427.svg:


* Note

    .. image:: images/Object_428.svg
        :width: 17
        :height: 15


    .. _RefImage_Object_428.svg:

the vector of modal fluid-elastic forces:

    .. image:: images/Object_429.svg
        :width: 17
        :height: 15


    .. _RefImage_Object_429.svg:

Which is written:


.. image:: images/Object_430.svg
    :width: 17
    :height: 15


.. _RefImage_Object_430.svg:

 **eq. 5.3.4-5**


+-----+-----------------------------------------------------------------------------------------------------------------------------------------------+
|Where|                                                                                                                                               |
+     + .. image:: images/Object_431.svg                                                                                                              +
|     |    :width: 17                                                                                                                                 |
+     +    :height: 15                                                                                                                                +
|     |                                                                                                                                               |
+     + refers to the matrix of terms of mass added by the fluid,                                                                                     +
|     |                                                                                                                                               |
+-----+-----------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                                                                                               |
+     + .. image:: images/Object_432.svg                                                                                                              +
|     |    :width: 17                                                                                                                                 |
+     +    :height: 15                                                                                                                                +
|     |                                                                                                                                               |
+     + refers to the matrix of damping terms added by the fluid,                                                                                     +
|     |                                                                                                                                               |
+-----+-----------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                                                                                               |
+     + .. image:: images/Object_433.svg                                                                                                              +
|     |    :width: 17                                                                                                                                 |
+     +    :height: 15                                                                                                                                +
|     |                                                                                                                                               |
+     + refers to the matrix of stiffness terms added by the fluid.                                                                                   +
|     |                                                                                                                                               |
+-----+-----------------------------------------------------------------------------------------------------------------------------------------------+


These matrices are real square in order

.. image:: images/Object_434.svg
    :width: 17
    :height: 15


.. _RefImage_Object_434.svg:

and their terms are independent of the movement of the structures. The matrix

.. image:: images/Object_435.svg
    :width: 17
    :height: 15


.. _RefImage_Object_435.svg:

is symmetric; the matrices

.. image:: images/Object_436.svg
    :width: 17
    :height: 15


.. _RefImage_Object_436.svg:

and

.. image:: images/Object_437.svg
    :width: 17
    :height: 15


.. _RefImage_Object_437.svg:

are not necessarily a step.


* The projection of the equations of motion on a modal basis provides:


.. image:: images/Object_438.svg
    :width: 17
    :height: 15


.. _RefImage_Object_438.svg:

 **eq. 5.3.4-6**


+-----+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|where|                                                                                                                                                                                                                                                                                              |
+     + .. image:: images/Object_439.svg                                                                                                                                                                                                                                                             +
|     |    :width: 17                                                                                                                                                                                                                                                                                |
+     +    :height: 15                                                                                                                                                                                                                                                                               +
|     |                                                                                                                                                                                                                                                                                              |
+     + refer to the matrices of masses, damping and stiffness of structures in air; these matrices are of order                                                                                                                                                                                     +
|     | .. image:: images/Object_440.svg                                                                                                                                                                                                                                                             |
+     +    :width: 17                                                                                                                                                                                                                                                                                +
|     |    :height: 15                                                                                                                                                                                                                                                                               |
+     +                                                                                                                                                                                                                                                                                              +
|     | and diagonal.                                                                                                                                                                                                                                                                                |
+-----+----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


In Laplace's domain, the [:ref:` **éq. 5.3.4-6**  < **éq. 5.3.4-6** >`] relationship becomes:


.. image:: images/Object_441.svg
    :width: 17
    :height: 15


.. _RefImage_Object_441.svg:

 **eq. 5.3.4-7**


* The fluid-elastic force transfer matrix is then introduced

    .. image:: images/Object_442.svg
        :width: 17
        :height: 15


    .. _RefImage_Object_442.svg:

defined by:


.. image:: images/Object_443.svg
    :width: 17
    :height: 15


.. _RefImage_Object_443.svg:

 **eq. 5.3.4-8**

And we find the relationship [:ref:` **éq. 1.2-1**  < **éq. 1.2-1** >`] from paragraph [:ref:`§ 1.2 <§ 1.2>`]:


.. image:: images/Object_444.svg
    :width: 17
    :height: 15


.. _RefImage_Object_444.svg:


Solving the modal problem under flow
--------------------------------------------

The modal problem under flow is formulated by the relationship [:ref:` **éq. 5.3.4-7**  < **éq. 5.3.4-7** >`] from the previous paragraph.

This problem is solved after rewriting as a standard problem with vectors and eigenvalues of the type

.. image:: images/Object_445.svg
    :width: 17
    :height: 15


.. _RefImage_Object_445.svg:

.

The new wording is as follows:


.. image:: images/Object_446.svg
    :width: 17
    :height: 15


.. _RefImage_Object_446.svg:

 **eq. 5.4-** 1

**Notes:**


1. *The dimension of the problem is doubled compared to that of the* *initial problem.*


2. *The properties of the matries*

    .. image:: images/Object_447.svg
        :width: 17
        :height: 15


    .. _RefImage_Object_447.svg:

*and*

    .. image:: images/Object_448.svg
        :width: 17
        :height: 15


    .. _RefImage_Object_448.svg:

*allow for reversal.*

The solution to this problem is done by means of the QR algorithm. The modules implemented by the CALC_FLUI_STRU operator are the same as those used by CALC_MODES.

The problem with its own elements that we solve is a complex problem. We therefore obtain an even number of complex eigenvalues conjugated two by two. We only keep those whose imaginary part is positive or zero.

Eigenvectors are complex, defined to within a complex multiplicative constant. As only real modes are taken into account, the first step is to determine, for each eigenvector, the constant that minimizes the imaginary part of the vector with respect to its real part, in the sense of the Euclidean norm. The eigenvectors are then redefined with respect to this norm. Given the normalization used, it is then possible to keep only the real part of the eigenvectors in the mode_meca concept. However, in file MESSAGE, indicators on the relationships between the imaginary part and the real part of the eigenvectors thus normalized are restored, so that the user can judge the bias introduced by not taking into account the imaginary part of the normalized vectors.


Taking into account the presence of the tube bundle grids
----------------------------------------------------------------

The modeling described above, of the forces induced by an axial flow on a bundle of cylinders, does not take into account the presence of the grids of the bundle (for example, the grids for mixing and maintaining fuel assemblies). A comparison of this model with tests carried out on model CHAISE (in the configuration of a bundle of nine flexible tubes including a grid) is presented in a summary note [:ref:`bib. 8 <bib. 8>`]: we note that the fluid-elastic coupling between the grid and the axial flow is not negligible step and that it generates an increase in the reduced modal damping of the tubes. The purpose of this paragraph is to describe the additional effects due to grids and how they are taken into account in model MEFISTEAU.


Grid configuration description
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The study is restricted here to two types of grids:


* the support grids which are located at the ends of the bundle,


* the mixing grids which are distributed between the holding grids.

The grids are all positioned perpendicular to the cylinder bundle and are in the form of a prismatic network with a square base on each side.

.. image:: images/Object_449.svg
    :width: 17
    :height: 15


.. _RefImage_Object_449.svg:

And in height

.. image:: images/Object_450.svg
    :width: 17
    :height: 15


.. _RefImage_Object_450.svg:

(along the :math:`x` axis of the cylinders). Grids of the same type are characterized by identical dimensions.


Additional calculation steps
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The first additional step concerns the specification of the type of configuration of the grids by the operator DEFI_FLUI_STRU, then the verification of the correct arrangement of the grids in relation to one another, and in relation to the ends of the beam.

The second step concerns the resolution of the modal problem under flow. In the flow velocity loop, the fluid-elastic force transfer matrix in the air modal base is completed by the calculation of an added damping matrix and an added stiffness matrix, linked to the grids.


Modeling the fluid forces exerted on the grids
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**Pressure jump calculation**

First of all, the presence of grids disrupts the stationary pressure field

.. image:: images/Object_451.svg
    :width: 17
    :height: 15


.. _RefImage_Object_451.svg:

; each grid is considered to be a singularity causing a pressure jump, whose expression is in the form:


.. image:: images/Object_452.svg
    :width: 17
    :height: 15


.. _RefImage_Object_452.svg:

 **eq. 5.5.3-1**


+-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|where|                                                                                                                                                                                                                                   |
+     + .. image:: images/Object_453.svg                                                                                                                                                                                                  +
|     |    :width: 17                                                                                                                                                                                                                     |
+     +    :height: 15                                                                                                                                                                                                                    +
|     |                                                                                                                                                                                                                                   |
+     +                                                                                                                                                                                                                                   +
|     | .. image:: images/Object_454.svg                                                                                                                                                                                                  |
+     +    :width: 17                                                                                                                                                                                                                     +
|     |    :height: 15                                                                                                                                                                                                                    |
+     +                                                                                                                                                                                                                                   +
|     | refers to the coefficient of pressure loss due to the grid,                                                                                                                                                                       |
+-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                                                                                                                                                                                   |
+     + .. image:: images/Object_455.svg                                                                                                                                                                                                  +
|     |    :width: 17                                                                                                                                                                                                                     |
+     +    :height: 15                                                                                                                                                                                                                    +
|     |                                                                                                                                                                                                                                   |
+     + refers to the stationary speed of the flow at the level of the grid,                                                                                                                                                              +
|     |                                                                                                                                                                                                                                   |
+-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                                                                                                                                                                                   |
+     + .. image:: images/Object_456.svg                                                                                                                                                                                                  +
|     |    :width: 17                                                                                                                                                                                                                     |
+     +    :height: 15                                                                                                                                                                                                                    +
|     |                                                                                                                                                                                                                                   |
+     + refers to the density of the flow at the level of the grid,                                                                                                                                                                       +
|     |                                                                                                                                                                                                                                   |
+-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                                                                                                                                                                                   |
+     + .. image:: images/Object_457.svg                                                                                                                                                                                                  +
|     |    :width: 17                                                                                                                                                                                                                     |
+     +    :height: 15                                                                                                                                                                                                                    +
|     |                                                                                                                                                                                                                                   |
+     + refers to the axial position of the middle of the grid along the beam.                                                                                                                                                            +
|     |                                                                                                                                                                                                                                   |
+-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


Density

.. image:: images/Object_458.svg
    :width: 17
    :height: 15


.. _RefImage_Object_458.svg:

is calculated by linear interpolation of the density profile

.. image:: images/Object_459.svg
    :width: 17
    :height: 15


.. _RefImage_Object_459.svg:

of the flow in the absence of a grate. Stationary speed

.. image:: images/Object_460.svg
    :width: 17
    :height: 15


.. _RefImage_Object_460.svg:

is calculated using the conservation of the mass flow rate, which results in the following equation:


.. image:: images/Object_461.svg
    :width: 17
    :height: 15


.. _RefImage_Object_461.svg:

**Calculation of the point fluid forces exerted on each grid**

Using the same quasistatic approach as that carried out in paragraph [:ref:`§5.3.3 <§5.3.3>`], we show that the action of the fluid velocity field

.. image:: images/Object_462.svg
    :width: 17
    :height: 15


.. _RefImage_Object_462.svg:

around a grid involves a drag force and a lift force, depending on the incidence of the flow in relation to the grid. The **y** and **z** components of the resulting point force **f**

.. image:: images/Object_463.svg
    :width: 17
    :height: 15


.. _RefImage_Object_463.svg:

are therefore written, for each elementary cell

.. image:: images/Object_464.svg
    :width: 17
    :height: 15


.. _RefImage_Object_464.svg:

Of a

+-----+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|where|                                                                                                                                                                                                                                                                                                                                                                  |
+     + .. image:: images/Object_465.svg                                                                                                                                                                                                                                                                                                                                 +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                    |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                   +
|     |                                                                                                                                                                                                                                                                                                                                                                  |
+     + and                                                                                                                                                                                                                                                                                                                                                              +
|     | .. image:: images/Object_466.svg                                                                                                                                                                                                                                                                                                                                 |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                    +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                   |
+     +                                                                                                                                                                                                                                                                                                                                                                  +
|     | respectively designate the density and stationary speed profile of the flow at the foot of the beam,                                                                                                                                                                                                                                                             |
+-----+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                                                                                                                                                                                                                                                                                                                  |
+     + .. image:: images/Object_467.svg                                                                                                                                                                                                                                                                                                                                 +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                    |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                   +
|     |                                                                                                                                                                                                                                                                                                                                                                  |
+     + designates the fluid section of the beam in the absence of a grid,                                                                                                                                                                                                                                                                                               +
|     |                                                                                                                                                                                                                                                                                                                                                                  |
+-----+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                                                                                                                                                                                                                                                                                                                  |
+     + .. image:: images/Object_468.svg                                                                                                                                                                                                                                                                                                                                 +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                    |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                   +
|     |                                                                                                                                                                                                                                                                                                                                                                  |
+     + designates the fluid section of the beam at the level of the grid:                                                                                                                                                                                                                                                                                               +
|     | .. image:: images/Object_469.svg                                                                                                                                                                                                                                                                                                                                 |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                                                    +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                                                   |
+     +                                                                                                                                                                                                                                                                                                                                                                  +
|     | with                                                                                                                                                                                                                                                                                                                                                             |
+     + .. image:: images/Object_470.svg                                                                                                                                                                                                                                                                                                                                 +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                                                    |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                                                   +
|     |                                                                                                                                                                                                                                                                                                                                                                  |
+     + the solid section of the grid.                                                                                                                                                                                                                                                                                                                                   +
|     |                                                                                                                                                                                                                                                                                                                                                                  |
+-----+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


From this we deduce the expression:


.. image:: images/Object_471.svg
    :width: 17
    :height: 15


.. _RefImage_Object_471.svg:

The pressure loss coefficient

.. image:: images/Object_472.svg
    :width: 17
    :height: 15


.. _RefImage_Object_472.svg:

is calculated using the expression for the total hydrodynamic force that applies to the grid, and we get:


.. image:: images/Object_473.svg
    :width: 17
    :height: 15


.. _RefImage_Object_473.svg:

 **eq. 5.5.3-2**

The 1st term (en

.. image:: images/Object_474.svg
    :width: 17
    :height: 15


.. _RefImage_Object_474.svg:

) comes from drag effort;

.. image:: images/Object_475.svg
    :width: 17
    :height: 15


.. _RefImage_Object_475.svg:

is the drag coefficient of the grid. The 2nd term (en

.. image:: images/Object_476.svg
    :width: 17
    :height: 15


.. _RefImage_Object_476.svg:

) is a term correcting the friction force applied to the beam alone at the altitude of the grid (

.. image:: images/Object_477.svg
    :width: 17
    :height: 15


.. _RefImage_Object_477.svg:

is the wet perimeter of the beam in the absence of a grid).

By introducing the expression [:ref:` **éq. 5.5.3-2**  < **éq. 5.5.3-2** >`] into the relationship [:ref:` **éq. 5.5.3-1**  < **éq. 5.5.3-1** >`], we therefore get the expression for the pressure jump

.. image:: images/Object_478.svg
    :width: 17
    :height: 15


.. _RefImage_Object_478.svg:

for each altitude grid

.. image:: images/Object_479.svg
    :width: 17
    :height: 15


.. _RefImage_Object_479.svg:

. This pressure jump is taken into account when calculating the stationary pressure field.

.. image:: images/Object_480.svg
    :width: 17
    :height: 15


.. _RefImage_Object_480.svg:

, in the following manner:


.. image:: images/Object_481.svg
    :width: 17
    :height: 15


.. _RefImage_Object_481.svg:


.. image:: images/Object_482.svg
    :width: 17
    :height: 15


.. _RefImage_Object_482.svg:

grill:


.. image:: images/Object_483.svg
    :width: 17
    :height: 15


.. _RefImage_Object_483.svg:


.. image:: images/Object_484.svg
    :width: 17
    :height: 15


.. _RefImage_Object_484.svg:


+-----+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|where|                                                                                                                                                                                                                                                                                                                                             |
+     + .. image:: images/Object_486.svg                                                                                                                                                                                                                                                                                                            +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                               |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                              +
|     |                                                                                                                                                                                                                                                                                                                                             |
+     + refers to the zero incidence slope of the lift coefficient around a very weakly inclined grid.                                                                                                                                                                                                                                              +
|     |                                                                                                                                                                                                                                                                                                                                             |
+-----+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                                                                                                                                                                                                                                                                                             |
+     + .. image:: images/Object_488.svg                                                                                                                                                                                                                                                                                                            +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                               |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                              +
|     |                                                                                                                                                                                                                                                                                                                                             |
+     + designates the solid section of the elementary cell                                                                                                                                                                                                                                                                                         +
|     | .. image:: images/Object_489.svg                                                                                                                                                                                                                                                                                                            |
+     +    :width: 17                                                                                                                                                                                                                                                                                                                               +
|     |    :height: 15                                                                                                                                                                                                                                                                                                                              |
+     +                                                                                                                                                                                                                                                                                                                                             +
|     | of the grid (which includes                                                                                                                                                                                                                                                                                                                 |
+     + .. image:: images/Object_490.svg                                                                                                                                                                                                                                                                                                            +
|     |    :width: 17                                                                                                                                                                                                                                                                                                                               |
+     +    :height: 15                                                                                                                                                                                                                                                                                                                              +
|     |                                                                                                                                                                                                                                                                                                                                             |
+     +).                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                                                                                                                                                                                                                                                                                             |
+-----+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


These forces will therefore generate additional terms of added damping and stiffness, which are obtained after modal decomposition of the movement and projection of these forces on the modal basis.


Taking into account the amortization of fluid at rest
-----------------------------------------------------

Until now, the damping brought to a bundle of tubes by the presence of a fluid at rest has not been taken into account in the modeling. We therefore propose here a model of damping in fluid at rest, for which annex 1 of the summary note of tests CHAISE [:ref:`bib8 <bib8>`] constitutes the reference documentation.


Modeling the fluid force at rest exerted on a bundle of tubes
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The method for calculating depreciation in fluid at rest, which is implemented here, is a generalization of the method of CHEN [:ref:`bib. 9 <bib. 9>`].

The aim is to calculate the resultant force on each pipe from the stresses due to shear in the boundary layer. This is a non-linear problem because the fluid damping coefficient depends on the frequency. The following simplifications are therefore introduced:


* the problem is written using the frequencies in water at rest calculated without taking into account fluid damping,


* the coupling between modes is neglected.

The linear force

.. image:: images/Object_491.svg
    :width: 17
    :height: 15


.. _RefImage_Object_491.svg:

Exercising on the tube

.. image:: images/Object_492.svg
    :width: 17
    :height: 15


.. _RefImage_Object_492.svg:

subject to a harmonic movement of the beam according to the mode

.. image:: images/Object_493.svg
    :width: 17
    :height: 15


.. _RefImage_Object_493.svg:

At the frequency

.. image:: images/Object_494.svg
    :width: 17
    :height: 15


.. _RefImage_Object_494.svg:

is given by the following relationship:


.. image:: images/Object_496.svg
    :width: 17
    :height: 15


.. _RefImage_Object_496.svg:

 **eq. 5.6.1-1**


+-----+-----------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|where|                                                                 |refers to the sliding speed between the tube                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
+     + .. image:: images/Object_497.svg                                +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |    :width: 335                                                  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 + .. image:: images/Object_498.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|     |                                                                 |    :width: 335                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
+     +                                                                 +    :height: 251                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 + and the fluid at rest, on either side of the boundary layer, defined by:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 | .. image:: images/Object_499.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
+     +                                                                 +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|     |                                                                 |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 | **eq. 5.6.1-2**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 | with                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 + .. image:: images/Object_500.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|     |                                                                 |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+     +                                                                 +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 + and                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 | .. image:: images/Object_501.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
+     +                                                                 +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|     |                                                                 |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 | depends on averages                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 + .. image:: images/Object_502.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|     |                                                                 |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+     +                                                                 +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 | .. image:: images/Object_503.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
+     +                                                                 +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|     |                                                                 |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 | speed disturbances around the cylinders, calculated beforehand by the model.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+-----+-----------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|     |                                                                 |Denotes the drag coefficient of a cylinder with radius                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
+     + .. image:: images/Object_504.svg                                +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |    :width: 335                                                  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 + .. image:: images/Object_506.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|     |                                                                 |    :width: 335                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
+     +                                                                 +    :height: 251                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +, subject to a harmonic flow with an amplitude of infinity                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 | .. image:: images/Object_507.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
+     +                                                                 +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|     |                                                                 |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |, and is defined by:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 + .. image:: images/Object_508.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|     |                                                                 |    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
+     +                                                                 +    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 + **eq. 5.6.1-3**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 + where                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 | .. image:: images/Object_509.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
+     +                                                                 +    :width: 17                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|     |                                                                 |    :height: 15                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+     +                                                                 +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|     |                                                                 | refers to the kinematic viscosity of the fluid.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
+-----+-----------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


The relationship obtained by replacing [:ref:` **éq. 5.6.1-2**  < **éq. 5.6.1-2** >`] and [:ref:` **éq. 5.6.1-3**  < **éq. 5.6.1-3** >`] in equation [:ref:` **5.6.1-1**  < **5.6.1-1** >`] is linearized by a Fourier series expansion (of the term

.. image:: images/Object_510.svg
    :width: 17
    :height: 15


.. _RefImage_Object_510.svg:

) from which we only remember the first term; it comes from:


.. image:: images/Object_511.svg
    :width: 17
    :height: 15


.. _RefImage_Object_511.svg:

**Projection on a modal basis**

By projection on a modal basis and by neglecting the coupling between modes, we obtain the generalized force exerted on the tube bundle according to the mode

.. image:: images/Object_512.svg
    :width: 17
    :height: 15


.. _RefImage_Object_512.svg:

:


.. image:: images/Object_513.svg
    :width: 17
    :height: 15


.. _RefImage_Object_513.svg:


.. image:: images/Object_514.svg
    :width: 17
    :height: 15


.. _RefImage_Object_514.svg:

is therefore proportional to

.. image:: images/Object_515.svg
    :width: 17
    :height: 15


.. _RefImage_Object_515.svg:

and the associated modal force vector

.. image:: images/Object_516.svg
    :width: 17
    :height: 15


.. _RefImage_Object_516.svg:

takes the form:


.. image:: images/Object_517.svg
    :width: 17
    :height: 15


.. _RefImage_Object_517.svg:

where

.. image:: images/Object_518.svg
    :width: 17
    :height: 15


.. _RefImage_Object_518.svg:

refers to the damping matrix added by the fluid at rest.