Analysis of the non-parallel flow-based Green's function in the acoustic analogy for complex axisymmetric jets

Abstract

This paper considers how a complex axisymmetric jet modifies the structure of the propa- gator tensor in Goldstein’s generalized acoustic analogy. The jet flow we consider is in general a dual stream flow that operates either as a single jet or a complex co-axial jet flow. The latter of which is of interest to turbofan engine manufacturers. The form of the acoustic analogy that we use here is based on our recent work on jet noise modeling (Afsar et al. 2019, PhilTrans. A., vol. 377) that highlighted the importance of non-parallel flow effects in the correct calcu- lation of the propagator. The propagator calculation takes advantage of the fact that mean flow non-parallelism enters the lowest order asymptotic expansion of the former at sufficiently low frequencies of the same order as the jet spread rate. Whilst this might seem restrictive, our previously reported calculations at high subsonic and mildly supersonic jets indicate that the subsequent jet noise predictions remain accurate up to the peak frequency (typically at a Strouhal number based on jet velocity and diameter of ≈ 0.5 − 0.6) for the small angle acoustic radiation. One of critical assumptions of this approach is that the mean flow speed of sound squared is given by either the Crocco relation (in unheated jets) or the Crocco-Busemann relation for heated flows. Our analysis for the dual stream complex axisymmetric jet however shows that the latter assumption (in the form of Crocco-Busemann formula) is no longer an accurate representation of the speed of sound variation. We therefore present a more general form of the asymptotic analysis than that used in Afsar et al. (2019a & b). For the complex jet mean flow field, the mean flow speed of sound is otherwise arbitrary but must remain a single-valued function of the streamwise mean flow. The predictions based on this approach are shown to remain accurate up to the peak frequency. We discuss how to extend the range of validity by utilizing a suitable composite asymptotic solution for the Green’s function problem.