# Simulation of Fluid Dynamics Monitoring Using Ultrasonic Measurements
## 3 Fluid Dynamics Modeling in PLAJEST Experiment and Frozen-Flow Hypothesis  
A large benchmark exercise was performed under the auspices of an international collaboration on thermal hydraulics for sodium-cooled fast-reactor development with participation from the Japan Atomic Energy Agency (JAEA), the U.S. Department of Energy (DOE) and the French Commissariat a l'Energie Atomique et aux Energies Alternatives (CEA). It was based on experiments performed to study the effects of thermal striping, where three differentially heated jets mix inside a cavity [25]. These experiments were carried out either in water or in liquid sodium, and velocity and temperature data were provided. The object of the benchmark was to numerically predict the results of these experiments and make comparisons with available measured data. For the French part, the numerical simulations were led using the TrioCFD simulation code (known as Trio_U by 2015). The large eddy simulation (LES) model was applied for the analyses. A computational domain reproducing the test sections was created. The discretization of the equations was based on unstructured staggered meshes and the resolution on a finite-volume approach. The numerical results were in good agreement with the experiments. Good results were obtained for the time-averaged fields and the power spectral densities of temperature fluctuations [27; 54; 55]. A schematic description of the three jets is drawn in the upper part of Figure 2. Three jets are mixed, with a cold jet between two hot jets. The three jets have a mean flow speed of 0.51 m/s. Two CFD maps, separated by 1 ms, are represented in this figure. The insonified area is represented, superimposed on the left map by a rectangular box. The box is centered at \(Z=0.09\) m, which corresponds to the most turbulent area. Letters E and R, respectively, represent the position of the virtual ultrasonic emitter and that of the virtual receiver. This area is 230 mm long, and the total time for the wave to propagate along this distance is less than 100 us, a tenth of the time between the two CFD maps. The emitter and the receiver correspond to a 1 MHz transducer with a diameter of 0.0254 m (i.e., 1 inch). A complete study of wave propagation is realized, and pressure time-series data are recorded on several planes in this volume, but not only in the plane of the virtual receiver (see Figure 3 thereafter). Figure 1: An example of 2D first-order finite element (**left**) and a fourth-order spectral element (**right**). When considering how the temperature field modifies the wave propagation (Step 10 in Figure 4), it implies taking into account the temperature-dependent properties of liquid sodium reported in [56]. [PERSON] et al. demonstrated that the density of liquid sodium varies linearly with the temperature. Within the temperature range between the normal melting point and the normal boiling point, the density is calculated with :  
\[\rho\ =1014-0.235\cdot T \tag{5}\]  
at normal atmospheric pressure, where \(\rho\) is the density of sodium, and \(T\) is the sodium temperature in Kelvin degrees. On the other hand, the sound velocity in liquid sodium decreases monotonically with temperature, because of the decrease in the number of interatomic interactions. In the normal melting-boiling point (371-1155 \({}^{\circ}\)K) range, the sound velocity in pure liquid sodium may be described based on this linear relation:  
\[c_{p}\ =2723-0.531\cdot T \tag{6}\]  
where \(c_{p}\) is the celerity of ultrasonic waves in meters per second. Figure 3: Definition of recorded planes in the insonified area. Figure 2: Two maps from the PLAJEST CFD simulation and representation of the insonified volume. In Figure 5, the difference between the two temperature maps is represented. The maximum temperature difference is lower than 2 \({}^{\circ}\)K in 1 ms.