## 4 Numerical and Experimental Results  
### Resonance Frequency Prediction of Flow-Induced Noise  
In order to validate the numerical method for resonance frequency prediction, the experimental system in Figure 3 is modelled, and the flow-induced resonance noise is numerically calculated. Figure 5 gives the numerical model for a cavity depth of 261 mm, with boundary conditions which are consistent with the experimental parameters. Numerical microphones (M1 and M2) are set at the leading and trailing edges of the cavity. In the numerical simulations, 905,769 structured grids over the whole area are used and the near-wall cells are refined. In addition, the time term is discretized in second-order implicit format. The momentum equation is discretized in bounded central difference format. The pressure-velocity coupling equation is solved using the Pressure Implicit with Splitting of Operator (PISO) algorithm with a calculated time step of \(\Delta t=5\times 10^{-5}\) s. Thus, its corresponding analysis frequency is 10 kHz.  
The pressure and velocity in the ventilation ducts are obtained with the commercial software FLUENT. The results of the flow field calculation for a 261 mm depth cavity are given in Figure 6. As can be seen from the flow velocity calculations, the areas where the flow velocity varies are mainly inside the cavity and at the boundary layer of the duct. The calculations show that the area of pressure variation is mainly at the mouth of the cavity. Further, a series of high and low pressures are distributed near the duct wall after the trailing edge of the cavity. This indicates that the downstream edge of the cavity is directly impacted by the unsteady flow, creating vortexes at the mouth of the cavity and causing fluid oscillations. Furthermore, the pressure pulses generated by the vortex-edge  
Figure 4: The experimental ANC system.  
Figure 5: The numerical model of the 261 mm depth cavity.  
interactions trigger the vortex shedding at the upstream edge and regulate the feedback loop. The feedback is provided by the acoustic particle velocity of the resonant standing wave in the duct systems with cavities. This particle velocity causes new perturbations in the unstable shear layer at the separation location. As the perturbations in the shear layer are amplified downstream, they interact with the acoustic field and generate acoustic energy, enhancing the resonance of the acoustic modes. The acoustic resonances of the ducts with side-wall cavities are caused by a feedback excitation mechanism referred in the literatures as the fluid-resonant mechanism [2]. The calculated noise power spectrums are shown in Figure 7, from which it can be observed that obvious resonance noise appear at 336 Hz for the leading edge microphone M1 and 337 Hz for the trailing edge microphone M2.  
The mesh-independent verification of grid is given in Table 1 together with a comparison between the numerical and experimental results, from which it can be seen that relatively accurate models could be achieved with more than 361,464 grids. It can also be observed that the numerical results for resonance noise frequencies are in good agreement with the experimental results. The amplitudes of the numerical results are closer to the experimental results when more small-scale vortices are identified as the number of grid encryption grids increases.  
Figure 6: Simulation results with 261 mm cavity: (**a**) the computed flow rate; (**b**) the computed static pressure.  
Figure 7: Calculation results of the noise power spectrum.