## 5 Conclusions  
Numerical simulations were computed with the EUROPLEXUS software to improve the understanding of fluid-structure interaction phenomena related to fuel cladding failure in a nuclear reactor and, thus, to design a failure localization method based on pressure signals analysis. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Point**} & **Point’s Position** & \multirow{2}{*}{
\begin{tabular}{c} **Time Delay Precursor** \\ **(10\({}^{-3}\) A.U.L.)** \\ \end{tabular} } & **Time Delay Precursor** \\  & **Primary Waves (\(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{  }}}}}}}}}}}}\))** & **\(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{    \mathbf{ }}}}}}}}}}}}}\))** \\ \end{tabular} } & **\(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{   }}}}}}}}}}}}}}\)** \\ \hline P1 & 607.3 & \(325\pm 3\) & 1062 & 3160 & \(152.2\pm 7.3\) \\  & & \(1052\pm 9\) & 3550 \(\pm 132\) & 121.0 \(\pm\) 19.3 \\ \hline P2 & 1874 & \(4000\pm 3\) & 1062 & 3160 & \(6.2\pm 7.3\) \\  & & \(1052\pm 9\) & 3550 \(\pm 132\) & 103.2 \(\pm\) 45.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Single-sensor localization with P1 or P2—results (actual source position: \(135.3\times 10^{-3}\) A.U.L.). Figure 14: Results of the different methods for source localization (source position in arbitrary unit of length, defined as the length of the device main section). The simulations allow the confirmation of several phenomenological assumptions. The observation of the spatial evolution of the pressure profile proves that the plane wave assumption is valid. It also shows that a geometrical singularity in the channel leads to reflections of the pressure waves and, consequently, transmission loss between the two sides of the singularity. Moreover, the time evolution of the pressure at different positions of the system exhibits two kinds of waves, referred as precursor and primary waves, according to the terminology used in water-hammer studies. The simulations also bring quantitative information about the velocities of precursor and primary waves, which can be estimated from the simulated signals. They were compared to analytically determined values and proved to be consistent with them. This quantitative information can be of interest for the analysis of experimental results, since the measurements that are necessary to estimate waves velocities may not be possible in a real reactor. From a practical perspective, the simulations show that precursor and primary waves appearing in pressure signals can be used to detect and locate the failure. Thus, detections and localizations can be achieved with a single sensor, using the time difference of arrival (TDOA) between the two kinds of waves, although using the TDOA between the primary waves of two different sensors' signals provides more accurate results. Moreover, precursor waves have a small amplitude compared to the primary waves, and although they are detectable in noiseless simulated signals, they might be hidden by noise in experimental signals. Despite the useful information provided by the simulations, some limits of the current numerical model have been identified:  
* Simulated phenomena are simplified: only the transient over-pressure due to the failure is simulated. Material failure or distortion and fuel pieces ejection are not simulated, even though they might happen in the real case. This simplification is related to the reproduction of the excitation phenomenon, but it is independent from the results related to propagation phenomena;
* Coolant fluid vaporization is not reproduced, although it affects the wave propagation.