# Simulation of Fluid Dynamics Monitoring Using Ultrasonic Measurements
## 5 Temporal Temperature Fluctuation Measurement by Ultrasound  
The analysis of the temperature fluctuation was performed over a total duration of approximately 10 s, from 200.000 s to 210.197 s. Because the computation time allocated on the supercomputers that we used was limited, it was not possible to run the temperature field interpolation and wave propagation calculation processes for all of these CFD time steps. Instead, we had to extract several time steps from the temperature field with a wider time interval from the CFD results. In order to select the minimum time interval to be used for our acoustic simulation, we exploited the power spectral density (PSD) curve calculated in Figure 7[27]. This curve allows knowing the frequency of the temperature fluctuation at \(x=-0.015\) m (between the left and the central jets), \(y=0.09\) m (middle point on the \(y\) axis) and \(z=0.1\) m. This is a normalized PSD calculated by dividing the original PSD with the maximum PSD value. The peak of this PSD curve is found around 3 Hz and is in accordance with experimental results, as mentioned by [PERSON] in [27]. This peak is lower than 5 Hz. This peak is followed by a slope (Kolmogorov slope). In this work, all our simulations were performed on two of the largest supercomputers in Europe: CURIE (CEA TGCC) and OCCIGEN (CINES), both part of Grand Equipement National de Calcul Intensif (GENCI). Despite the use of these  
Figure 6: Examples of some interpolations of temperature fields from a tetrahedral mesh to a hexahedral mesh using the MEDCoupling pre-/postprocessing library. Figure 7: Normalized PSD curves of temperature history in the CFD calculation (blue line) at \(x=-0.015\) m, \(y=0.09\) m and \(z=0.1\) m (from STMF laboratory). that it would be difficult to perform calculations with a fine temporal step. In order to be able to follow the fluid dynamics, we wanted to confirm fluid fluctuations at a 3 Hz frequency as seen in PSD curves. We fixed the frequency limit to 5 Hz, then the Shannon sampling criterion imposed us to use a minimum sampling frequency of 10 Hz). Temperature fields were thus extracted from the complete volume with a 0.1 s interval. The computation domain was divided into 256 parts, and parallel calculations were carried out. The duration of the interpolation of a temperature field from a TrioCFD result to SPECFEM3D was approximately 20 h using a single CPU at one chosen altitude and one time step (the insonified area represented in Figure 2). The average duration for an acoustic simulation was about 26 min, excluding the mesh generation and the interpolation processes of the temperature fields, which were performed once and for all. Complete results, signals at each point in the volume, cannot be fully stored on a physical disc because of the high number of calculations. Thus the data resulting from SPECFEM3D calculations were recorded for several planes, as described in Figure 3.  
To analyze temporal temperature fluctuations within our heterogeneous medium and then to make the correlation with ultrasonic data, a PSD curve was calculated at three different points. These three points have the same \(y\) and \(z\) position (\(y\) = 0.09 m and \(z\) = 0.1 m) and a different \(x\) position (\(x\) = \(-\)0.035 m, 0.0 m and 0.035 m). The position \(y\) = 0.09 m corresponds to the middle of the jets along the \(y\) axis, and \(z\) = 0.1 m corresponds to an altitude with high turbulence. \(x\) = 0.0 m is in the middle of the central jet. \(x\) = \(-\)0.035 m and 0.035 m are two positions between the cold jet and the hot jet. These values are imposed by the initial choice of recorded planes (Figure 3). In these positions, temperature fluctuations are high but probably less than for positions \(x\) = \(-\)0.015 m and 0.015 m chosen by [PERSON] et al. Figure 8A,B shows the temperature PSD curve of the original CFD results (every millisecond) at three selected points and the same kind of curves but with only extracted time steps (every 0.1 s), respectively.