Canonical sound fields in the frequency-domain theory of supersonic leading-edge noise

Abstract

This paper determines the three-dimensional structure of certain single-frequency canonical sound fields occurring in the theory of blade-vortex interaction when the flow velocity relative to the blade is supersonic. A relative velocity of this magnitude occurs at the outer part of the fan blades in an aeroengine, at which the incoming vorticity has either been ingested from the atmosphere or created in the aeroengine itself. The sound fields analysed are those produced by the leading edge of a flat-plate blade at zero angle of attack on being struck by a gust which is either (i) localised along the span, or (ii) non-localised but discontinuous. The canonical gusts of type (i) have either a deltafunction or Gaussian shape, and those of type (ii) are either anti-symmetric or described by a Heaviside function; these gusts give rise to the four basic canonical sound fields. The paper also analyses a fifth sound field, produced by a single-frequency top-hat gust. This sound field has a complex structure involving aspects of both (i) and (ii), but can nevertheless be analysed in terms of the canonical sound fields. The main results of the paper are exact and approximate analytical formulae giving the dependence of the acoustic field on gust-shape and flow parameters, and also a simple formula which is ideal for numerical work. The last of these is used to assess in detail the numerical accuracy of all the approximate formulae, which are of either Fresnel or Keller type. A key result is that Keller-type formulae, representing sound rays produced in accord with the geometrical theory of diffraction, have a very wide range of validity. A companion paper (Chapman & Powles 2019) in this special issue of Wave Motion determines the canonical sound fields in the corresponding time-domain theory.

Basic singular fields in the theory of impulsive supersonic leading-edge noise

“…Thus there are no surfaces of discontinuity or singularity in the frequency domain. For this reason, the canonical frequency-domain theory of supersonic leading edges is presented with its own development in the companion paper [17], in which Keller's geometrical theory of diffraction provides the unifying idea. In the present paper we have given complete details of Hadamard's and Friedlander's wavefront surfaces for a large number of impulsive sound fields produced by different types of gusts convected towards a supersonic edge, and we have provided explicit formulae for the pressure field in the different regions.…”

“…A basic reference for the geometry of wavefront surfaces, and for the pressure field nearby, is [18]. Taking the present paper and the companion paper [17] together, one cannot fail to notice the enduring significance of the research in wave theory carried out by J. B. Keller and F. G. Friedlander in the fifteen years or so up about 1960.…”

“…As in [17], we consider a flat-plate aerofoil of infinite span occupying the half-plane y = 0, x > 0 in a coordinate system (x, y, z) for which the downstream direction is x, the vertical direction is y, and the span direction is z. There is a mean flow at speed U in the positive x direction, as shown in Figure 1, so that any gust convected towards the aerofoil is chopped at the z-axis, i.e.…”

“…This theory is applied in the special case of two dimensions in [15], and in three dimensions in the PhD thesis [16], but most of the three-dimensional impulsive fields calculated in the thesis are unpublished. The aim of this paper is to present these fields, together with their derivations; a companion paper [17] gives a large number of corresponding non-impulsive fields, produced by single-frequency gusts.…”