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


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

For the calculation of the added coefficients:

we show [:ref:`bib1 <bib1>`] that the added mass and damping coefficients depend, in each region where :math:`\rho` is constant, on the permanent potential of the fluid velocities :math:`\stackrel{ˉ}{\phi }` as well as on two fluctuating potentials :math:`{\phi }_{1}` and :math:`{\phi }_{2}`: these potentials are written in the case of the rotational movement of the external cylinder around the pivot C [:ref:`bib1 <bib1>`]:

For the region relating to

.. image:: images/Object_14.svg
    :width: 18
    :height: 21


.. _RefImage_Object_14.svg:

:

.. image:: images/Object_15.svg
    :width: 18
    :height: 21


.. _RefImage_Object_15.svg:

For the region relating to

.. image:: images/Object_16.svg
    :width: 18
    :height: 21


.. _RefImage_Object_16.svg:

:

.. image:: images/Object_17.svg
    :width: 18
    :height: 21


.. _RefImage_Object_17.svg:

However, the added modal coefficients projected onto this mode of rotation can be written as:


.. image:: images/Object_18.svg
    :width: 18
    :height: 21


.. _RefImage_Object_18.svg:

Suppose by separating the integral on two half-cylinders:


.. image:: images/Object_19.svg
    :width: 18
    :height: 21


.. _RefImage_Object_19.svg:


1. Digital applications:

An added damping calculation was made which corresponds, for the given speed, to a damped vibratory behavior of the structure:

speed

.. image:: images/Object_20.svg
    :width: 18
    :height: 21


.. _RefImage_Object_20.svg:

To :math:`4{\mathrm{m.s}}^{-1}`

The values of the mechanical system are:


.. image:: images/Object_21.svg
    :width: 18
    :height: 21


.. _RefImage_Object_21.svg:

The mass added by the flow is equal to:


.. image:: images/Object_22.svg
    :width: 18
    :height: 21


.. _RefImage_Object_22.svg:

(independent of the flow speed value)

The added damping applies with :math:`{V}_{0}=4{\mathrm{m.s}}^{-1}` (it is independent of the change in density):


.. image:: images/Object_24.svg
    :width: 18
    :height: 21


.. _RefImage_Object_24.svg:

Knowing that the amortization of the mechanical system is worth

.. image:: images/Object_25.svg
    :width: 18
    :height: 21


.. _RefImage_Object_25.svg:

, the total damping of the fluid/structure system is written as:


1. to

    .. image:: images/Object_26.svg
        :width: 18
        :height: 21


    .. _RefImage_Object_26.svg:

The flow does not amplify the vibrations. The internal structural damping is significant enough to dissipate the energy provided by the flow to the structure. **The system is still depreciated**.


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

Analytical result.


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


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