2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The reference solution is provided by a calculation carried out using Code_Aster using the CALC_MATR_AJOU operator.
This is done by using a mesh on which the structural elements (inner and outer shells), the volume fluid elements (annular space), and the fluid-structure interface elements are defined. This method is fully described in [bib1].
On the interface elements, the zero normal speed limit condition is imposed, reflecting the condition of non-penetration of the fluid into the shells.
On the inlet and outlet sections of the annular space, the zero potential limit condition is imposed, translating the condition of zero disturbed pressure at the ends.
A first modal calculation of the structure in air is carried out. The operator CALC_MATR_AJOU then makes it possible to calculate the mass matrix added by the fluid, projected on the modal basis of the air structure. We can then recombine this added mass matrix with the generalized mass matrix of the air structure, and then solve a new modal problem that leads to the characteristics of the water system at rest. The results obtained for the natural frequencies of the system in water at rest constitute the reference solution.
Mesh characteristics: |
17 nodes on a vertical generator |
60 knots on a crown |
|
960 QUAD4 stitches on each shell |
|
=> 1920 meshes for the fluid-structure interface |
|
2880 HEXA8 meshes for the fluid domain |
|
180 QUAD4 meshes for the entrance section of the annular space |
|
180 QUAD4 meshes for the exit section of the annular space |
2.2. Benchmark results#
Mode number |
1 |
2 |
3 |
3 |
3 |
4 |
5 |
5 |
6 |
8 |
9 |
10 |
10 |
11 |
11 |
12 |
Freq. (\(\mathit{Hz}\)) |
5.65 |
5.65 |
5.65 |
6.48 |
6.48 |
6.48 |
9.34 |
20.82 |
20.82 |
28.22 |
28.22 |
28.22 |
28.22 |
28.22 |
31.48 |
2.3. Uncertainty about the solution#
In the case studied, the half-thicknesses of the shell represent 10% of the dimension of the annular gap. In the reference calculation performed by operator CALC_MATR_AJOU, half-thicknesses are neglected for the definition of the fluid domain. A systematic error of the order of 5% on the frequencies is therefore expected, as a result of this approximation.
In addition, the differences in modeling and resolution between the numerical method implemented in operator CALC_MATR_AJOU and the analytical model developed in operator CALC_FLUI_STRU induce an additional difference.
2.4. Bibliographical references#
ROUSSEAU: « Specifications and principle of carrying out the stiffness and damping calculation added in Code_Aster « , HP-51/96/005/B.
PEROTIN, « Model MOCCA_COQUE concept note « , HT-32/95/021/A.