1. Reference problem#
1.1. Geometry#
The tubes in the bundle are hollow cylinders whose characteristic dimensions are as follows:
length |
\(L=\mathrm{1,2}m\) |
outside diameter |
\({\phi }_{\text{ext}}=\mathrm{9,5}\mathrm{mm}\) |
inner diameter |
\({\phi }_{\text{int}}=\mathrm{8,5}\mathrm{mm}\) |
Piano strings holding the tubes halfway up are likened to full cylinders \(\mathrm{2 }\mathrm{mm}\) in diameter.
The beam is at a square step of \(\mathrm{12,6 }\mathrm{mm}\). It is composed of \(3\times 3\) tubes and is placed in the center of a rectangular enclosure of dimensions \(\mathrm{7,8 }\mathrm{cm}\times \mathrm{4,2 }\mathrm{cm}\).
The roughness of the walls of the tubes is \(\varepsilon ={10}^{–5}m\).
1.2. Material properties#
The physical characteristics of the aluminum constituting the tubes are as follows:
Young’s module |
\({E}_{\mathrm{alu}}=\mathrm{6,89}{.10}^{10}\mathrm{Pa}\) |
Poisson’s ratio |
\({\nu }_{\mathrm{alu}}=\mathrm{0,3}\) |
Since tubes contain lead pellets, an equivalent density must be defined in relation to their cross section: \({\rho }_{\mathrm{eq}}=20450\mathrm{kg}/{m}^{3}\)
The ropes holding the tubes halfway up are made of steel, whose physical characteristics are as follows:
Young’s module |
\({E}_{\mathrm{acier}}=\mathrm{2,1}{.10}^{11}\mathrm{Pa}\) |
Poisson’s ratio |
\({\nu }_{\mathrm{acier}}=\mathrm{0,3}\) |
density |
\({\rho }_{\mathrm{acier}}=7800\mathrm{kg}/{m}^{3}\) |
The surrounding water has the following properties:
density |
\({\rho }_{\mathrm{eau}}=1000\mathrm{kg}/{m}^{3}\) |
kinematic viscosity |
\({\nu }_{\mathrm{eau}}=\mathrm{1,1}{.10}^{-6}{m}^{2}/s\) |
1.3. Boundary conditions and loads#
The ends of each tube are connected to fixed supports by metal rods. The relative flexibility of these rods frees up the degrees of freedom (DDL) of rotation of the ends of each tube. It can therefore be estimated that the tubes are swivel and swivel, the metal rods introducing an additional rotational stiffness at each end.
In addition, these rods make it possible to apply an axial force to the tubes, which can thus be prestressed in tension or in compression. The configuration studied corresponds to the bundle of tubes prestressed in compression by applying an axial force of \(\mathrm{50 }N\) at each of the upper ends of the tubes.
1.4. Bibliographical references#
HOTTA, H. NIIBORI, M. TANAKA and K. FUJITA: « Parametric study on parallel flow‑induced damping of PWR fuel assembly », ASME Conference, Nashville, TN, TN, PVP Vol.191 (1990)