3. Modeling A#
3.1. Characteristics of modeling#
The tube is represented by 100 elements of straight Timoshenko beams (MECA_POU_D_T), supported by as many 2-node segment cells (SEG2). Two MECA_DIS_TR elements are added to the end nodes of the tube, allowing metal rods to be modelled by discrete rotational stiffness.
The beam elements are given the circular cross-section characteristics:
outer radius |
\({R}_{\mathrm{ext}}=\mathrm{6,5}{.10}^{–3}m\) |
|
thickness |
\(E=\mathrm{2,1}{.10}^{–3}m\) |
(cf paragraph [§1.1]) |
These elements are also assigned a ELAS behavior material:
Young’s module |
\(E=\mathrm{2,80}\mathrm{.}{10}^{9}\mathrm{Pa}\) |
|
Poisson’s ratio |
\(\nu =\mathrm{0,3}\) |
|
density |
\(\rho =1500\mathrm{kg}/{m}^{3}\) |
(cf paragraph [§1.2]) |
The discrete elements are assigned the same rotational stiffness around the two axes orthogonal to the neutral fiber of the tube:
\({K}_{r}=\mathrm{6,29}\mathrm{Nm}/\mathrm{rad}\) |
This rotation stiffness has been adjusted in order to regain the natural frequency of the first dual air mode.
The translational degrees of freedom \(\mathrm{DX}\) and \(\mathrm{DZ}\) of the end nodes \(\mathit{N1}\) and \(\mathrm{N101}\) are blocked in order to prevent a rigid body movement of the tube (axial translation movement). We also block \(\mathrm{DY}\) from node \(\mathit{N1}\). In addition, at each node, the degree of freedom of rotation \(\mathrm{DRY}\) is blocked, in order to prohibit any twisting movement.
The tube is immersed in a cylindrical chamber with an internal radius \(\mathrm{2,5}\mathrm{cm}\) (see paragraph [§1.1]). The density and kinematic viscosity profiles of the surrounding water are assumed to be constant along the tube:
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]) |
No axial force is applied to the tube, which is therefore not prestressed.
The changes in the frequency and the reduced damping of the first double mode of bending are calculated for a range of average flow velocities from \(0\) to \(8m/s\), in steps of \(1m/s\).
An initial reduced damping of the tube of 4.8% is taken into account.
3.2. Characteristics of the mesh#
The total number of nodes used for meshing is 101.
There are 100 meshes (type SEG2).
The mesh file is in ASTER format.
3.3. Calculation steps#
The validation of fluid-structure coupling operators, for configurations of the « bundle of tubes under axial flow » type, is carried out in two main steps.
The first consists in defining the parameters for taking into account fluid-structure coupling with the operator DEFI_FLUI_STRU followed by the keyword FAISCEAU_AXIAL.
The second is the calculation of modal frequency and reduced damping changes as a function of the average flow speed, with the operator CALC_FLUI_STRU and by implementing the MEFISTEAU model.
3.4. Tested values#
The tests focus on the frequency and the reduced damping of the first dual mode of tube bending, at the average flow speed of \(0m/s\) and \(4m/s\). Two types of tests are carried out:
a comparison test with the experimental measurements,
a test to guarantee the non-regression of the code.
3.4.1. Frequency of the first dual mode of bending#
Comparison test with the experiment, at the flow speed of \(0m/s\):
The tolerance for relative deviation from the experimental value is 0.1%.
Mode number |
Experimental value |
Calculated value |
Relative variance |
1 |
7 Hz |
7.000871 Hz |
1.2E -02% |
2 |
7 Hz |
7.000871 Hz |
1.2E -02% |
3.4.2. Reduced damping of the first double flexure mode#
Comparison test with the experiment, at a flow speed of 4 m/s:
The tolerance for relative deviation from the reference is equal to 1%.
Mode number |
Experimental value |
Calculated value |
Relative variance |
1 |
17% |
17% |
0.2% |
2 |
17% |
17% |
0.2% |
3.5. notes#
The reference values are those obtained by*Code_Aster* when restoring the test case, which therefore makes it possible to verify the non-regression of the code during its evolution.