Extension of the concept of Ffowcs-Williams and Hawkings type wave extrapolation to non-trivial flow effects and exterior surfaces
Authors: Delfs Jan Werner , Mößner Michael , Proskurov Stanislav , Ewert Roland 
Published: September 2022

In appreciation of Ffowcs-Williams and Hawkings’ seminal contribution on describing the sound radiation from moving objects, this article discusses a concept of taking into account local non-trivial flow effects on the sound propagation. The approach is motivated by the fact that the numerical simulation of the sound propagation from complete full scale aircraft by means of volume-discretizing (CAA = Computational AeroAcoustics) methods is prohibitively expensive. In fact, a homogeneous use of such CAA approach would waste computational resources since for low speed conditions the sound propagation around the aircraft is subject to very mild flow effects almost everywhere and may be treated by more inexpensive methods. The part of the domain, where the sound propagation is subject to strong flow effects and thus requiring the use of CAA is quite restricted. These circumstances may be exploited given a consistent coupling of methods. The proposed concept is based on the strong (alternatively weak) coupling of a volume discretizing solver for the Acoustic Perturbation Equations (APE) and a modified Ffowcs-Williams and Hawkings (FW-H) type acoustic integral. The approach is established in the frequency domain and requires two basic ingredients, namely a) a volume discretizing solver for the APE, or for Möhring-Howe’s aeroacoustic analogy, to take into account strong non trivial flow effects like refraction at shear flows wherever necessary, and b) an aeroacoustic integral equation for the propagation part in areas where non-potential mean flow effects are negligible. The coupling of this aeroacoustic integral and the APE solver may be realized in a strong (i.e. two-ways) form in which both components feed back information into one another, or in a weak form (i.e. one-way), in which the sound field output data from the APE solver serves as given input for the integral equation. If an aircraft geometry has minor influence on the sound radiation to arbitrary observer positions, the aeroacoustic integrals may simply be evaluated explicitly. If on the other hand, the presence of the geometry has an important influence on the sound radiation, then the acoustic integral equation is implicit and requires some sort of numerical solution, in this case a Fast Multipole Boundary Element solver. While conceptually the weak coupling follows the spirit of the FW-H approach to describe sound propagation from aeroacoustic sources the underlying aeroacoustic integral is not based on Lighthill’s analogy, but the aeroacoustic analogy of Möhring-Howe. This is a consequence of the fact that in the two way-coupling the acoustic particle velocity in a moving medium needs to be determined, which is non-trivial based on an acoustic integral. As an important feature of the strong coupling the acoustic integral also provides practically perfect non-reflection boundary conditions even when the desireably small CAA domain does not extend into the far field. The validity of the presented computation approach is demonstrated in two example use cases.