# Jet Noise in Airframe Integration and Shielding
## 3 Options for Computation of Jet Noise
### Extension of the Near Field to the Far Field  
Since the FAA noise restrictions are based on sound measurements away from the aircraft, usually, the experimental data of the radiated noise is taken far from the aerodynamically generated noise sources. Hence, the far-field sound needs to be calculated. The noise generation process is usually nonlinear, and it is computationally expensive. On the other hand, the sound propagation is usually a linear process because the amplitude of the pressure waves is small. If the noise source is identified by some means, then several techniques can be used for calculating the associated radiated sound. _[PERSON]'s Analogy:_ In the pioneering work of [PERSON] [50], the governing Navier-Stokes equations are manipulated such that the left-hand side represents the wave equation while the right-hand side represents the sound source. The radiated sound field is then obtained as a volume integral of the time dependent Lighthill stress tensor (LST), \(\rho u_{i}u_{j}\). However, since the time fluctuation of Lighthill stresses cannot be obtained with RANS, only empirical models were used in utilizing [PERSON]'s analogy to estimate the far-field noise that led to questionable results. However, in the study by [PERSON] and [PERSON] [40], the Lighthill stress tensor was calculated by integrating the time-dependent NSE across the radius and using the nonlinear stability theory to obtain the shape of the radial profiles. Solution of the resulting ODE enabled calculating, for the first time, the noise sources in a round jet based on the unsteady NSE. This made it possible to conduct [PERSON]'s volume integration while using the time-dependent sources and with proper accounting for the retarded time. The results have shown that this wave-like large-scale structure is responsible for producing the peak noise at the measured spectra, and in explaining the forwarded quadrupole-type directivity pattern. The seminal work of [PERSON] was extended by [PERSON] and [PERSON] [51] to account for the presence of solid boundaries. The Ffowcs-Williams and Hawkins (FWH) method enabled tackling situations where solid boundaries exist, such as the prediction of rotor noise or jet-airframe interactions. _Kirchhoff Method:_ [PERSON] and [PERSON] [52] proposed the use of the Kirchhoff formulation to predict jet noise. In this formulation, an enclosed surface is established around the source regime. It is assumed that the pressure distribution and its derivatives on this surface are known and that the process outside of it is linear. A surface integral solution of the governing wave equation is then obtained for sound radiation in terms of both the pressure signal as well as its normal derivative on the surface. The method is simple compared to [PERSON]'s analogy, but both the pressure and its normal derivative are needed on the surface, which are obtained through a numerical solution of the governing equations inside [PERSON]'s surface. The computational results are usually not very accurate close to the boundaries because of the boundary treatment approximations, and this is particularly true for the normal derivatives. Figure 10: Effect of implementing various boundary treatments on the calculated acoustic field associate with an acoustic monopole in a uniform flow (Ref. [49]). Therefore, [PERSON] results suffer from the drawback of dependency on the location of Kirchhoff's surface. This drawback is resolved in [PERSON] et al. [53], wherein a new surface-integration formulation (SIF) is obtained, which requires only the pressure signal on the Kirchhoff's surface. Figure 11 shows the result of [PERSON] vs. [PERSON] et al. [53] in predicting the noise radiation by a point source. The figure shows that [PERSON] results are dependent on the accuracy of calculating the pressure derivative, which makes it strongly grid dependent. The [PERSON] et al. [53] method using only the pressure (without its derivative) does not suffer from this problem. For more details on integral methods, see the review by [PERSON] [54].