1. Reference problem#
1.1. Geometry#
The meridional fluidielastic oscillator model schematizing fluid—structure interaction in the ring cell-core shell space is described below.
The meridian elastic fluid oscillator is a model of the tank-shell annular space of a reactor core; it consists of an oscillator (lateral piston-spring representing a moving wall) coupled with a compressible fluid contained in a channel with rigid and fixed walls.
The channel is crossed by a depressurization wave.
Figure [Figure 1.1-a] below shows the model described.
Figure 1.1-a: Overall diagram of the meridian fluid-elastic oscillator
The channel has a rectangular cross section with dimensions \(e\times H\); the rigid lateral piston moves along \(z\) perpendicular to a wall.
A depressurization wave arrives from the left; when heading to the right (with no possibility of return) this wave sucks in the piston which, by its resulting displacement, generates waves propagating towards the ends of the duct, which is supposed to be infinitely long so that there is no reflection.
A two-dimensional model of this system is designed, shown in Figure [Figure 1.1-b] below:
Figure 1.1-b: Theoretical two-dimensional mechanical representation
Its geometric characteristics are as follows:
piston wall length |
\(\mathrm{2L}=\mathrm{5,0}m\) |
fluid height |
\(e=\mathrm{0,5}m\) |
fluid width |
\(H=\mathrm{1,0}m\) |
fluid section |
\(S\mathrm{=}e\mathrm{.}H\) |
1.2. Material properties#
The physical characteristics of the fluid material (hot water) in the tube are as follows:
density \({\rho }_{f}\mathrm{=}0.75\mathrm{\cdot }{10}^{3}\mathit{kg}\mathrm{/}{m}^{3}\),
speed of sound \({c}_{f}\mathrm{=}{10}^{3}m\mathrm{/}s\).
The physical characteristics of the materials constituting the wall piston and the end pistons have only a formal role in the calculation of code_aster because the structure is considered to be perfectly rigid.
These physical characteristics of the material are as follows:
Young’s module \(E\mathrm{=}2.{10}^{12}\mathit{Pa}\),
Poisson’s ratio \(\nu \mathrm{=}0.3\),
density \({\rho }_{s}\mathrm{=}0\mathit{kg}\mathrm{/}{m}^{3}\).
1.3. Characteristics of springs, weights and shock absorbers#
The characteristics of the wall piston as an oscillator are as follows:
Stiffness |
\(K\mathrm{=}5.{10}^{10}N\mathrm{/}m\) |
Mass |
\(M\mathrm{=}200.{10}^{3}\mathit{kg}\) |
Depreciation |
\(A\mathrm{=}0\mathit{Ns}\mathrm{/}m\) |
The characteristics of end pistons as oscillators are as follows:
Stiffness |
\(k\mathrm{=}0N\mathrm{/}m\) |
Mass |
\(m\mathrm{=}0\mathit{kg}\) |
Depreciation |
\(a={\mathrm{\rho }}_{f}{c}_{f}S=37.5{10}^{4}\mathit{Ns}/m\) |
1.4. Boundary conditions and loads#
Piston with an infinitely rigid wall and can only be moved along the vertical axis.
Infinite fluid length therefore no end reflection of the waves: this boundary condition is simulated in the model by a piston at each end, of zero mass, moving only along the \(x\) axis and equipped with a shock absorber with adequate damping; these pistons are also infinitely rigid.
Total reflection of waves on infinitely rigid walls of the fluid tube: achieved simply by omitting to model the wall by structural elements.