# Jet Noise in Airframe Integration and Shielding
## 4 Numerical Simulation of the Shielding Effect
### Numerical Approach
The discretization of a general dependent tensor field, \(\Psi\), of any rank is described by values of \(\Psi_{P}\) at cell centers to values of \(\Psi_{f}\) at cell faces. In compressible fluid flows, properties are not only transported by the flow but also by the propagation of waves. This requires the construction of flux interpolations to consider that transport can occur in any direction [78]. The convective terms of the conservation equations in the forms of \(\
abla. [\rho\mathbf{u}]\), \(\
abla. [\mathbf{u}(\rho\mathbf{u})]\), \(\
abla. [\mathbf{u}(\rho E)]\), and \(\
abla. [\mathbf{u}p]\) integrated over a control volume and linearized. Figure 22: Computational grid near the nozzle exit. (**a**) Baseline case minor plane, (**b**) Shielded case minor plane, (**c**) Both baseline and shielded cases major plane. Figure 23: Computational grid and the refinement regions. (**a**) Baseline case minor plane, (**b**) Shielded case minor plane, (**c**) Both baseline and shielded cases major plane. The volumetric flux across a face is split into two components of \(f+\) and \(f-\), which are evaluated based on the cell values on either side of the face. The '+' and '-' sides refer to the owner and neighbor cells of a face, and a positive flux is in the direction of the face area normal vector, which points out of the owner cell ('+' side) and into the neighbor cell ('-' side). The contributions of the two flux components to the flux evaluation are controlled by the weighted coefficient, \(\alpha\), where it is calculated using the absolute speed of the fastest traveling waves in the respective directions. For example, \(\alpha=0.5\) corresponds to an entirely central scheme. The directed convective fluxes mentioned above are interpolated using the vanAlbada scheme. The gradient terms are calculated as:  
\[\int\limits_{V}\
abla\mathbf{\Psi}dV=\sum\limits_{f}S_{f}\mathbf{\Psi}_{f}= \sum\limits_{f}[\alpha S_{f+}\mathbf{\Psi}_{f+}+(1-\alpha)S_{f-}\mathbf{\Psi} _{f-}] \tag{1}\]  
The \(f+\) and \(f-\) interpolation uses the same limiter described for convective terms. Also, the discretization of Laplacian with the diffusion coefficient, \(\Gamma\), is described as:  
\[\int\limits_{V}\
abla.(\Gamma\
abla\mathbf{\Psi})dV=\sum\limits_{f}\Gamma_{f}S _{f}.(\
abla\mathbf{\Psi})_{f} \tag{2}\]  
where \(\Gamma_{f}\) is interpolated linearly from cell center values. In addition, second-order implicit temporal discretization [79] is used to ensure the overall second order of accuracy of the numerical simulations. Boundary ConditionsAt the nozzle inlet, a total pressure condition of 3.67 MPa is specified and the jet was expected to be ideally expanded with an NPR value of 3.67. The temperature at the inlet of the nozzle is prescribed to 900 K to ensure the \(TR=3.0\), where the ambient pressure is \(P_{a}=101325\) Pa and has a temperature value of \(T_{a}=300\) K. An advective far-field condition was imposed on the rest of the domain boundaries, which corresponds to \"wave transmissive\" boundary conditions in OpenFOAM. This non-reflecting condition is based on the same idea of the non-reflecting boundary condition as mentioned by [PERSON] and [PERSON] [80] without full inter-field coupling. The nozzle walls are prescribed as adiabatic no-slip conditions. Turbulence modellingIn this study, the \(k-\omega\)_SST_\(DES\) turbulence model is adopted, where the URANS models are employed in the boundary layer, while the LES treatment is applied everywhere else. Therefore, the computational cost is much more efficient compared to the full LES that requires extensive near wall treatment. For the current simulations, a statistically steady solution is achieved with the \(k-\omega\)_SST_\(RANS\) model first, then the DES simulations are carried out using the RANS results as an initial solution.