# Wavenumber-Frequency Analysis of Internal Aerodynamic Noise in Constriction-Expansion Pipe
## 4 Numerical Results
### Axisymmetric Steady Simulation  
The valve flow was simplified as a quasi-1D flow in a pipe consisting of uniform inlet and outlet ducts with the same diameter (0.292 m), and an orifice with a minimum diameter of 0.144 m, as shown in Figure 2. These dimensions were based on the data for a real valve in a pipe that is used in an existing ocean plant.  
The inlet and outlet flow conditions were prescribed using the following design data. The inlet total pressure was 9.0 bar, inlet static pressure was 7.0 bar, inlet temperature was 523 K, outlet pressure was 4.4 bar, and outlet temperature was 433 K. An axisymmetric boundary condition was applied along the center line of the pipe, and the no-slip boundary condition was applied on the inner pipe wall. The number of grid points in a typical calculation was 200,000.  
The results of the axisymmetric steady simulation were compared with those that were obtained from the quasi-1D theory. Figure 3 shows the convergence time of the simulation in terms of the inlet and outlet mass flows. The mass flow rates are seen to converge quickly after 5000 iterations to the same value (25.7 kg/s) at the inlet and outlet. Because the flow is an inflow at the inlet, the mass flow rate has a positive value, and vice versa at the outlet. All of the subsequent analyses were performed using the converged numerical results.  
Figure 4 shows the iso-contours of the static pressure, temperature, and velocity magnitude in the vicinity of the orifice. Their variations on the center line (\(y=0\)) of the pipe are also shown in Figure 5, where the velocity results are described in terms of the Mach number. Note that the orifice is located at \(x=1\) m. Upstream from the orifice, the total pressure is approximately 9.0 bar, and it continuously decreases as the flow approaches the orifice. Downstream of the orifice, the static pressure increases sharply up to 4.4 bar and remains almost constant far downstream. It is found that the shock wave is located at approximately \(x=1.1\) m. Other variables show similar behaviors. To verify the numerical results, these distributions along the center line were compared with those of the theoretical solutions. To improve the understanding of the numerical solutions, three conditions for the numerical results were compared with the theoretical ones: steady with friction, steady without friction, and unsteady with friction. Because the analytical solution was based on the 1D inviscid compressible Euler equations, the numerical results without viscous friction showed the closest agreement with the theoretical ones. The steady and unsteady RANS results showed similar behaviors, except for the oscillation of the steady results in the vicinity of the shock waves.  
However, there was excellent agreement between the numerical and theoretical results, except for the region just after the shock wave, where the complexity of the interaction between the shock wave and boundary layer invalidated the 1D approximation based on the theoretical approach. These results ensured the validity of the current numerical method.  
Figure 3: Time histories of mass flow rates.  
Figure 2: Grids in physical domain for pipe flow for steady and unsteady computational fluid dynamics (CFD) simulations (**a**) pipe inlet; (**b**) pipe throat; (**c**) pipe outlet.