4. B modeling#

4.1. Characteristics of modeling#

Modeling B is identical to modeling A (see paragraph [§3.1]), but this time the pipe is prestressed under compression.

An axial compression force of \(\mathrm{23,7}N\) is applied to the end node \(\mathrm{N101}\). The intensity of the effort was therefore readjusted in relation to the experimental value provided for \(40N\), in order to correctly find the frequency value of the first dual mode in air (cf paragraphs [§1.2], [§1.3]). This readjustment can be applied by the summary modeling of the metal rods providing support and compression.

The elementary force vector is deduced from the nodal force, then an assembled vector is deduced, which is constructed according to the numbering of the degrees of freedom of the tube. The static deformation due to compression is then obtained by multiplying the assembled vector by the inverse of the structural stiffness matrix. Using this static deformation, a stress field at 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 compression tube stiffness matrix, which is finally used to calculate the air modes.

The changes in the frequency and the reduced damping of the first double mode of flexure are calculated for a range of average flow velocities from \(0\) to \(8m/s\), in steps of \(1m/s\). Account is taken of an initial reduced amortization of the tube of 4.3%.

4.2. Characteristics of the mesh#

The characteristics of the mesh in this second modeling are the same as those of modeling A, i.e.:

101 knots used and 100 SEG2 stitches.

The mesh file is in ASTER format.

4.3. Calculation steps#

As for modeling A, the functionalities to be validated are those of fluid-structure coupling operators for configurations of the « tube bundle under axial flow » type (see paragraph [§3.3]).

In addition, B modeling makes it possible to test other functionalities.

The first makes it possible to calculate a displacement field at the nodes by inverting the structural stiffness matrix and producing the inverse by an assembled force vector, with the operators FACTORISER and RESOUDRE.

The second allows the calculation of a geometric rigidity matrix using an element constraint field, with the operator CALC_MATR_ELEM, option RIGI_GEOM.

4.4. 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:

  1. a comparison test with the experimental measurements,

  2. a test to guarantee the non-regression of the code.

4.4.1. Frequency of the first dual mode of bending#

  1. Comparison test with the experiment, at the flow speed of \(0m/s\):

The tolerance for relative deviation from the reference is equal to 0.1%.

Mode number

Experimental value

Calculated value

Relative variance

1

5.1 Hz

5.10426 Hz

8.4E -02%

2

5.1 Hz

5.10426 Hz

8.4E -02%

4.4.2. Reduced damping of the first double flexure mode#

  1. Comparison test with the experiment, at the flow speed of \(4m/s\):

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

4.5. notes#

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