The robustness of acoustic analogies

Acoustic analogies for the prediction of flow noise are exact rearrangements of the flow equations N(q^→) = 0 into a nominal sound source S(q^→) and sound propagation operator ℒ such that ℒq^→ = S(q^→). In practice, the sound source is typically modeled and the propagation operator inverted to make predictions. Since the rearrangement is exact, any sufficiently accurate model of the source and inversion of ℒ will yield the correct sound, so other factors must determine the merits of any particular fonnulation. Using data from a two-dimensional mixing layer direct numerical simulation (DNS), we evaluate the robustness of four analogy formulations to different errors intentionally introduced into the source. The motivation is that since S can not be perfectly modeled, analogies that are less sensitive to errors in S are preferable. Our assessment is made within the framework of Goldstein's generalized acoustic analogy, in which different choices of base flow used in constructing ℒ give different consistent sources S and thus different analogies. We evaluate a uniform base flow (Lighthill-like) formulation, a globally parallel base-flow (Lilley-like) formulation, a spreading mean-flow base-flow formulation, and a locally parallel approximation of this spreading mean-flow formulation, which is not a formally exact consequence of the flow equations. The Lighthill-like formulation is found to be considerably less robust to source errors than the others, which all perform about equally well.