On the existence of an overlap region between the Green’s function for a locally parallel axi-symmetric jet and the leading order non-parallel flow solution

Sassanis, Vasilis and Afsar, Mohammed and Sescu, Adrian and Lele, Sanjiva (2017) On the existence of an overlap region between the Green’s function for a locally parallel axi-symmetric jet and the leading order non-parallel flow solution. In: The 70th Annual Meeting of the American Physical Society Division of Fluid Dynamics, 2017-11-19 - 2017-11-21, Colorado Convention Center.

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Abstract

We consider determination of the propagator within the generalized acoustic analogy for prediction of supersonic jet noise. The propagator is a tensor functional of the adjoint vector Green’s function that requires solution of the linearized Euler equations for a given mean flow. The exact form of these equations can be obtained for a spreading jet. However since high Reynolds number jets have small spread rates, ϵ < < O(1), this parameter can be exploited to formulate an asymptotic model that encompasses mean flow spatial evolution at leading order. Such a model was used by Afsar et al. (AIAA-2017-3030 for prediction of supersonic jet noise. We show the existence of an overlap between this solution (valid at low frequencies) and one based on a locally parallel (i.e. non-spreading) mean flow, valid at O(1) frequencies. It is clear that there must exist an overlap between these solutions, since the former non-parallel solution was determined at the distinguished limit where the scaled frequency Ω=ω/ϵ=O(1) was held fixed. Hence the inner equation shows that as Ω→∞, non-parallelism will be confined to a thin streamwise region of size O(Ω−1) and will, therefore, be subdominant at leading order when ΩY=Y¯=O(1).