4. Model A results#
4.1. Tested values#
The comparisons focus on the resting water frequencies of the first 12 modes of the system.
Mode Numbers |
CALC_FLUI_STRU ( \(H=\mathrm{9cm}\) ) |
CALC_MATR_AJOU |
( \(H=\mathrm{10cm}\) ) |
gap |
1 and 2 |
5.30 Hz |
5.65 Hz |
6.2% |
|
3 and 4 |
6.12 Hz |
6.48 Hz |
5.5% |
|
5 and 6 |
8.69 Hz |
9.34 Hz |
6.9% |
|
7 and 8 |
21.96 Hz |
20.82 Hz |
— 5.5% |
|
9 and 10 |
29.34 Hz |
28.22 Hz |
—3.9% |
|
11 and 12 |
33.75 Hz |
31.48 Hz |
—7.2% |
For information, the values of the frequencies of these modes in air are given:
Mode Numbers |
Shell in Move |
Shell Order |
Beam Order |
Frequency |
|
1 and 2 |
external |
3 |
3 |
1 |
25.15 Hz |
3 and 4 |
internal |
3 |
3 |
1 |
26.12 Hz |
5 and 6 |
external |
4 |
1 |
31.91 Hz |
|
7 and 8 |
internal |
2 |
1 |
36.85 Hz |
|
9 and 10 |
internal |
4 |
1 |
37.42 Hz |
|
11 and 12 |
external |
2 |
1 |
39.49 Hz |
4.2. notes#
The results are in line with what could be expected. In fact, we observe:
a systematic error of the order of 5%, due to the fact that shell thicknesses were not taken into account for the definition of the fluid domain in the reference calculation;
a residual difference due to differences in modeling and resolution between the two operators CALC_MATR_AJOU and CALC_FLUI_STRU.
Modes 1 to 6 are modes for which the structure is strongly coupled with the fluid. In practice, these modes correspond to movements of the internal and external shells globally in phase opposition. For these modes, the terms of added mass are theoretically proportional to \(\frac{{\rho }_{f}\pi {R}^{3}}{H}\), where \(R\) refers to the mean radius and \(H\) the thickness of the annular space.
For these first six modes, the results provided by CALC_FLUI_STRU lead to natural frequencies that are lower than those calculated by CALC_MATR_AJOU. Indeed, \(H\) being lower, the terms of added mass are greater.
Modes 7 to 12 are modes for which the structure is weakly coupled with the fluid. In practice, these modes correspond to movements of the internal and external shells that are almost in phase. Thus, the terms of added mass are theoretically proportional to the mass of water entrained, i.e. to \({\rho }_{f}\pi RH\).
In this case, it is normal for the results provided by CALC_FLUI_STRU to lead to higher natural frequencies than those calculated by CALC_MATR_AJOU. Indeed, \(H\) being lower, the terms of added mass are smaller.