3. Modeling A#

3.1. Characteristics of modeling#

Each of the tubes in the bundle is represented by 50 elements of a right Timoshenko beam (MECA_POU_D_T), supported by as many SEG2 meshes (segments with 2 knots).

A MECA_DIS_TR element is added at each end node; these elements make it possible to model metal rods by discrete rotational stiffness.

The circular cross-section characteristics are assigned to the elements of the tubes:

outer radius	\({R}_{\mathrm{ext}}=\mathrm{4,75}{.10}^{–3}m\)
thickness	\(e={5.10}^{-4}m\)	(cf paragraph [§1.1])

These elements are also assigned a ELAS behavior material:

Young’s module	\(E=\mathrm{6,89}{.10}^{10}\mathrm{Pa}\)
Poisson’s ratio	\(\nu =\mathrm{0,3 }\)
density	\(\rho =20450\mathrm{kg}/{m}^{3}\)	(cf paragraph [§1.2])

The discrete elements are assigned the same rotational stiffness around the two axes orthogonal to the direction axis of the beam:

\({K}_{r}=\mathrm{6,29}\mathrm{N.m}/\mathrm{rad}\)

This rotation stiffness has been adjusted in order to correctly find the frequency value of the first double mode of air bending of the beam.

Each piano string is represented by a MECA_POU_D_T element. These elements are assigned the characteristic of a solid circular section \(R={10}^{-3}m\) (see paragraph [§1.1]) and a material with a behavior of ELAS:

Young’s module	\(E=\mathrm{2,1}{.10}^{11}\mathrm{Pa}\)
Poisson’s ratio	\(\nu =\mathrm{0,3}\)
density	\(\rho =7800\mathrm{kg}/{m}^{3}\)	(cf paragraph [§1.2])

The degrees of freedom of translation in \(y\) and \(z\) (\(\mathrm{DY}\) and \(\mathrm{DZ}\)) of the nodes at the ends of each tube are blocked. In order to prohibit rigid body movement (axial translation movement), the degrees of freedom \(\mathrm{DX}\) of the nodes at the lower ends of each tube are also blocked. Finally, at each node, the degree of freedom of rotation \(\mathrm{DRX}\) is blocked to prevent any torsional movement.

An axial compression force of \(\mathrm{26,7 }N\) is applied to each of the nodes at the upper ends of the tubes. The intensity of the force was therefore readjusted in order to correctly find the value of the frequency of the first double mode of air bending of the beam. This readjustment can be explained by the summary modeling of the metal rods providing support and compression.

The elementary force vectors are deduced from the nodal forces, then an assembled vector is deduced, which is constructed according to the numbering of the degrees of freedom of the complete structure. The static deformation due to the compression of the tubes is then obtained by multiplying the assembled vector by the inverse of the structural stiffness matrix. Using this static deformation, a field of constraints to the elements is then calculated, from which a geometric rigidity matrix is deduced. This is then added to the structural stiffness matrix in order to obtain the stiffness matrix after compressing the tubes, which is finally used to calculate the modes in air.

The beam is immersed in a rectangular chamber with dimensions \(\mathrm{7,8 }\mathrm{cm}\times \mathrm{4,2 }\mathrm{cm}\) (cf. paragraph [§2.1]). The density and kinematic viscosity profiles of the surrounding water are constant along the tubes:

density	\({\rho }_{\mathrm{eau}}=1000\mathrm{kg}/{m}^{3}\)
kinematic viscosity	\({\nu }_{\mathrm{eau}}=\mathrm{1,1}{.10}^{-6}{m}^{2}/s\)	(cf paragraph [§1.2])

The changes in the frequency and the reduced damping of the first two modes of flexure of the beam are calculated for average flow velocities varying from \(0\) to \(\mathrm{10 }m/s\) in steps of \(\mathrm{1 }m/s\). An initial reduced amortization of 12.3% is taken into account.

3.2. Characteristics of the mesh#

The total number of nodes used for this mesh is 459.

The meshes are 470 in number and of type SEG2.

The mesh file is in ASTER format.

3.3. Calculation steps#

The functionalities that we want to validate are those of fluid-structure coupling operators, for configurations of the « bundle of tubes under axial flow » type.

First, the parameters for taking fluid-elastic coupling into account are defined, with the operator DEFI_FLUI_STRU (keyword factor FAISCEAU_AXIAL).

Next, the evolution of the frequencies and reduced modal damping is calculated as a function of the average flow speed, with the operator CALC_FLUI_STRU and by implementing the model MEFISTEAU.

Modeling A makes it possible to test these functionalities with the complete representation of the beam. In addition to fluid-structure coupling operators, other modules for resolution and mechanical calculation are used.

In our case, the displacement field at the nodes is calculated by inverting the structural stiffness matrix and multiplying the inverse matrix obtained by a force vector assembled with operators FACTORISER and RESOUDRE.

Then, the geometric rigidity matrix is calculated using a field of constraints to the elements with the operator CALC_MATR_ELEM, option RIGI_GEOM.

3.4. Tested values#

The tests focus on the reduced frequencies and damping of the first two modes of flexure of the beam, at the average flow speed of \(\mathrm{4 }m/s\).

The experimental measurements concern only the characteristics of the first mode of vibrating flexure along the longest side of the chamber. This mode is the first one determined by calculation. Two types of tests are carried out:

a comparison test with the experimental measurements on the first mode,
a test on the first two modes in order to guarantee the non-regression of the code.

3.4.1. Frequencies of the first two modes of beam bending#

Comparison test with the experience on the first mode:

The tolerance for relative deviation from the experimental value is equal to 10%.

Mode number	Experimental value	Calculated value	Relative deviation
1	4.47 \(\mathrm{Hz}\)	4.735 \(\mathrm{Hz}\)	5.90%

3.4.2. Reduced damping of the first two modes of beam bending#

Comparison test with the experience on the first mode:

The tolerance for relative deviation from the experimental value is 20%.

Mode number	Experimental value	Calculated value	Relative deviation
1	19%	22.6474%	+19.20%

3.5. notes#

The reference values are those obtained by Code_Aster when restoring the test case, which will therefore make it possible to verify the subsequent non-regression of the code during its evolution.