Penetrable Surfaces and the Extended Analogy

In all applications where the quadrupole term in eq. (3) is significant and must be calculated, which preferentially occurs at high speeds, the computations can be made cumbersome because the sources are distributed within a volume the boundaries of which are not precisely defined. In con­trast, the surface source terms are much simpler to compute and clearly delimited. If CFD must be used in a limited domain surrounding the sur­faces, and if the computations are able to reproduce the acoustic field in the vicinity of its sources, a more convenient way of solving the acoustic problem can be proposed by taking the information not on the physical sur­faces but on a penetrable control surface that can be user-defined at some distance away. This generalized form of Ffowcs Williams & Hawkings’ anal­ogy is widely used in modern Computational Aero-Acoustics (CAA). The continuity and momentum equations are now written as

dt + dx-(pVi) = {po Vsn + p(Vn – Vsn)} r(f) >

д d

dt (pVi) + (pVi V – ) = &vi (Vn – Vsn) – "ij «Л s(f)>

where n = Vf stands for the normal to the control surface and where Vsn = Vs • Vf and Vn = V – Vf are normal velocities. The new expression of Ffowcs Williams & Hawkings’ equation reads

2 d2P 52 ( VV i 2 г

c°dX2 = зХ~Щ (p Vi V – aij – CoP^ d

{[pVi(Vn – Vsn) – aij Uj 5(f)}

d

+ ^ {[P0 Vsn + p(Vn – Vsn)] S(f)} and all notations refer to the control surface, again of equation f = 0. Lighthill’s tensor only needs being evaluated outside the surface. Therefore the latter can be chosen in such a way that the quadrupole contribution becomes negligible. In counterpart since the control surface is penetrable, the surface source terms are more complicated than in the standard form of Ffowcs Williams & Hawkings’ equation.

As an example of the methodology, the sound radiated by a rotor operat­ing in free field can be computed from a fixed control surface embedding the rotor, using eq. (5), which seems far simpler than the integration over the
moving blades. But the CFD code applied to get the complete field inside the control surface must reproduce accurately the sound waves generated by the blades or in the vicinity of the blades and their propagation up to the surface, in order to avoid numerical errors. This may be challenging. Finally, the benefit in the solving of the acoustic equation of the analogy is at the price of a bigger computational effort in the simulation of the flow inside the control surface. No way to escape the intrinsic difficulties.

An important corollary of Ffowcs Williams & Hawkings’ formulation with penetrable surfaces is that it can take the non linear processes into ac­count more easily. If sound is generated close to surfaces at a very high level, it propagates initially with significant non-linear aspects. Since the analogy written on the physical surfaces is exact when no approximation is made and since the wave equation is linear, the non-linear mechanisms must be all grouped in the equivalent quadrupole sources; if the latter are discarded from the analysis, the non linearity is ignored. In contrast, non-linear effects are treated implicitly by the CFD code used to solve the internal problem when resorting to a penetrable control surface.