# Wavenumber-Frequency Analysis of Internal Aerodynamic Noise in Constriction-Expansion Pipe
## 4 Numerical Results
### Three-Dimensional Unsteady Simulation  
This section discusses an unsteady simulation that was performed using the LES to capture the compressible acoustic field, as well as the incompressible hydrodynamic flow field. For efficient computation, the 3D RANS steady simulation result was used as an initial condition for the LES simulation. The unsteady simulation was carried out with a time step of 0.05 ms and 8,220,000 grids. In the case of the unsteady LES analysis, the unsteady flow characteristics should be captured by the numerical method combined with its mesh. The important unsteady phenomena in the CAA simulation are two-fold: one is aerodynamic noise source mechanism and the other is acoustic wave propagation. Therefore, the quality of grid for CAA simulation is generally determined by the highest frequency of the concerned frequency range of the problem. In other words, as the grid-resolution increases, the resolved frequency range also increases. An important aerodynamic noise source mechanism in the current problem is closely related to the turbulence in the boundary layer, especially, downstream of the orifice. The important length scale related to the aeroacoustic source is \(y+\) so that the minimum grid length is determined, such that \(y+=10\). The length scale related to acoustic wave propagation is the wavelength of the highest frequency component, which is set to be 10,000 Hz in the current problem. The maximum grid length is matched to be \(\Delta x_{max}=\lambda_{min}/8\), where \(\lambda_{min}\) is the wavelength of the acoustic wave of 10,000 Hz. The geometry and boundary conditions were the same as those for the 3D RANS simulation, as shown in Figure 6.  
Figure 11 shows the variations in the mass flow rates at the inlet and outlet in the LES simulations. Because the 3D RANS result was used for the initial condition, a transient period is seen before the converged numerical solutions occur after 1500 iterations. Figure 12 shows the distributions of the static pressure, turbulent viscosity, and vorticity on the central cross-sectional plane and in an enlarged area downstream of the orifice. The strong turbulent fluctuation due to the complex interaction between the shock and the separated boundary layer is observed in the region just downstream of the shock wave.  
There are two types of excitation sources for the vibration of a pipe wall: the compressible pressure fluctuation that is associated with acoustic waves, and the incompressible pressure fluctuation, which is associated with a turbulent flow. As shown in Figure 12c, the incompressible pressure fluctuations associated with a strong vortex convected with the mean flow can induce pipe vibration. This phenomenon is called flow-induced vibration. In addition, the acoustic waves generated by the turbulent fluctuation and its interaction with the solid surface propagate at the velocity of sound plus the mean flow velocity. The compressible pressure fluctuation associated with the acoustic waves also causes pipe vibration, which is called AIV. However, because the compressible pressure fluctuation propagates at the speed of sound, it is difficult to directly observe it from the flow field. Therefore, the compressible pressure fluctuation and incompressible pressure fluctuation were separated using a wavenumber-frequency analysis.  
Figure 10: Vorticity contours near throat: (**a**) separation location due to shock-boundary layer interaction; (**b**) shock location at center line.