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The Ffowcs Williams - Hawkings (FWH) equation is often used in an inexact manner in numerical settings, because the amount of information available is limited. Generally the volume integral, or quadrupole term, is omitted even though the (permeable) FWH surface fails to enclose the turbulence region, which is unmanageably long for jet or bluff-body flows. This motivates a search for variants of the equation that are more forgiving of these practices, and thus more accurate at the same level of numerical effort. Two such variants are discussed, one proposed by Morfey in 1973 for other reasons, and the other used implicitly by Shur et al. since 2003. Both use functions of the pressure rather than the density in key terms, those which are retained in practice. The latter variant has similarities with proposals of Goldstein. They vastly reduce the need for cancellations between surface and volume terms, when entropy differences are present. There is no reason to use arbitrarily open surfaces. Sleeves can be made tight around the jet, without touching the vortical fluid, which is beneficial at high frequencies. There is little to choose between the two variants, as long as the quadrupoles are omitted. The benefits are illustrated in the case of a high-subsonic hot jet in co-flow, treated by Large-Eddy Simulation with extraction of the far-field sound by the classical FWH equation and by its variants, as well as with the Kirchhoff equation.
Spalart et al. (Sat,) studied this question.