# Jet Noise in Airframe Integration and Shielding
## 4 Numerical Simulation of the Shielding Effect
### Numerical Approach  
_Computational Grid:_ The computational grid used in the current simulations contains hexahedrally dominant cells. The entire computational domain extends to \(80D\) downstream of the nozzle exit and \(10D\) upstream of the nozzle exit; it also extends radially up to \(25D\) from both the major and minor axis planes. The grid spacing on nozzle walls is chosen such that it ensures \(y^{+}\) to have a value of 30 on the wall, and to make sure the near-wall calculations of the boundary layer in the RANS region are accurate. This value for \(y^{+}\) is calculated considering the isentropic flow assumption along the nozzle and using the nozzle exhaust velocity, \(U_{j}\). The grid spacing near the nozzle walls is illustrated in Figure 22a-c. The fine grid spacing on the nozzle walls is gradually increased such that the volume inside the nozzle has the maximum element sizing of \(D/50\). Such grid spacing is kept consistent for both the baseline and with-plate (shielded) cases. This grid spacing is maintained and extended up to \(x/D=5\) in the jet axis direction to capture turbulent mixing near the nozzle exit and the shock cell structures in the plume. The grid spacing is then gradually increased up to \(D/40\) in the jet axis direction up to \(x/D=18\). Then, another refinement box is placed that is extended to \(x/D=30\), which gradually  
Figure 21: (a) Nozzle geometry cross-sectional (minor axis plane) and the location of far-field points. (b) Dimension of the flat plate. increases the cell size up to a maximum value of \(D/30\). Such refinement boxes are shown as blue boxes in Figure 23a-c. The grid spacing expands gradually in both the major and minor directions up to \(y/D=6\), and \(z/D=10.5\), respectively, and in the axial direction up to \(x/D=40\). This conservative coarsening is chosen to have a refined box to maintain acoustic waves. This refined nearfield acoustic region has the maximum grid spacing of \(D/10\) and is to be used for FWH acoustic predictions, and is illustrated with the red box in Figure 23a-c. Such grid spacing on the FWH surface would ensure the capturing of acoustic waves up to Strouhal number \(St=fD/U_{j}=0.35\), where \(f\) is the frequency. This maximum frequency represents up to 75% of the frequency range of the spectra reported in experimental studies and contains the main information in the spectral analysis of the acoustic signal, such as the peak frequency of \(St=0.1\). The maximum resolvable frequency is calculated based on the assumption that a minimum of 15 points (cells) per wavelength are required to capture the acoustic waves up to \(St=0.35\) with the current numerical scheme. Such a requirement has been tested for the prediction of waves using second-order finite volume schemes when applied to hexahedral cells [73]. _Numerical Scheme_: The _rhoCentralFoam_ solver in OpenFOAM is adopted for this study. OpenFOAM is an open source CFD software package consisting of a set of flexible C++ modules to resolve complex fluid flows. _rhoCentralFoam_ is an unsteady, compressible solver that uses semi-discrete, non-staggered, Godunov-type central (Ref. [74]) and upwind-central (Ref. [75]) schemes proposed by [PERSON] and [PERSON] [75]. These schemes avoid the explicit need for a Riemann solver, resulting in a numerical approach that is both simple and efficient. The directed convective fluxes mentioned above are interpolated using the vanAlbada scheme (Ref. [76]) to provide a second-order spatial discretization that, as a TVD scheme, is appropriate for capturing flow discontinuities, such as shocks, and the limiter automatically provides high-order stable solution. The finite volume method is applied for expressing the differential equations. In the application of the finite volume to polyhedral cells with an arbitrary number of faces, each face is assigned to an owner cell and a neighboring cell. This is explained in detail by [PERSON] et al. [77].