## 4 Results and Discussion
### Two-Dimensional Axisymmetric Steady Simulation  
The compressible RANS simulation was performed for a two-dimensional axisymmetric critical nozzle and the accuracy of the simulation was verified through the comparison of the predicted mass flow rate with measured data. The critical nozzle targeted is the toroidal-throat Venturi nozzle as specified in ISO 9300. Its geometry and the numerical mesh with the specified boundary conditions are shown in Figure 5.  
The simulations were carried out for the two types of nozzle of which the throat diameter was 0.4 mm and 0.56 mm, respectively. The inlet and outlet pressure were specified on the inlet and outlet boundaries, respectively, the adiabatic and no-slip wall conditions were set on the wall, and the axisymmetric condition was applied along the centerline axis of the venturi nozzle. The grids were constructed by using a structured grid system, and y\({}^{+}\) was set to be around five, and the total number of grids was \(350\times 100\). The efficiency of the critical nozzle can be determined by observing the variation  
Figure 4: AIV measurement system at KRISS. Figure 5: The geometry of the toroidal-throat Venturi nozzle and the computational grids with boundary conditions applied for axisymmetric Reynold-Averaged Navier–Stokes (RANS) simulation. Figure 3: Experimental setup for acoustic induced vibration (AIV) measurement. of mass flow rate according to the change of pressure difference between the inlet and the outlet. In this study, the pressure of the outlet was varied to control the pressure difference between the inlet and outlet of the nozzles. The applied inlet and outlet pressure conditions are shown in Table 1[16]. Figure 6 shows the predicted iso-contours of the Mach number inside the nozzles of d = 0.4 m at the pressure ratio of 0.666 and d = 0.56 m at the pressure ratio of 0.778. It is seen that shock waves are generated around the throat in both cases, which implies that the choked flow appears. Figure 7 compares the predicted ones with the measured data in terms of the variation of the non-dimensional mass flow rate according to the pressure ratio between the inlet and outlet. The \(x\)-axis represents the ratio of the outlet static pressure to the total inlet pressure, and the \(y\)-axis represents the ratio of the mass flow rate(\(q_{m}\)) to the mass flow rate(\(q_{m,~{}ref}\)) when the outlet pressure is equal to the atmospheric pressure. In the case of the nozzle flow, as the pressure difference between the inlet and outlet becomes larger, the flow is more accelerated up to the nozzle throat and the mass flow rate increases. However, when the outlet pressure becomes equal to or less than the critical pressure at which the chocked flow starts to form, the Mach number in the throat reaches one and does not exceed one anymore, and thus the mass flow rate is kept constant. In Figure 6, because the outlet pressure in both nozzles is lower than the critical pressure, the Mach number is one in the nozzle throat. Then the flow is accelerated to supersonic in the diffusion region through the nozzle throat, and shock waves are generated. This phenomenon corresponds to a typical process for creating a supersonic flow in an internal flow. In Figure 7, it can be seen that the mass flow rate through the nozzle increases as the outlet pressure decreases to the critical pressure. As the pressure ratio is lower than the critical ratio, the choked flow occurs and the mass flow rate is kept constant. The numerical results are in excellent agreement with the experimental results [16].