Numerical inverse method predicting acoustic spinning modes radiated by a ducted fan from free-field test data

Abstract

Spinning modes generated by a ducted turbofan at a given frequency determine the acoustic free-field directivity. An inverse method starting from measured directivity patterns is interesting in providing information on the noise sources without requiring tedious spinning-mode experimental analyses. According to a previous article, equations are based on analytical modal splitting inside a cylindrical duct and on a Rayleigh or a Kirchhoff integral on the duct exit cross section to get far-field directivity. Equations are equal in number to free-field measurement locations and the unknowns are the propagating mode amplitudes (there are generally more unknowns than equations). A MATLAB procedure has been implemented by using either the pseudoinverse function or the backslash operator. A constraint comes from the fact that squared modal amplitudes must be positive which involves an iterative least squares fitting. Numerical simulations are discussed along with several examples based on tests performed by Rolls-Royce in the framework of a European project. It is assessed that computation is very fast and it well fits the measured directivities, but the solution depends on the method and is not unique. This means that the initial set of modes should be chosen according to any known physical property of the acoustic sources.

The radiating part of circular sources

“…The first extension comes from the fact that B 2v+1 ≡ 0, for v < n, on the right hand side of Equation 20. This means that u q , q = 2v + 1, is uniquely defined by the lower order coefficients with q ≤ 2v .…”

“…A second extension comes from examination of Equation 20. The first few entries of the system of equations are:…”

“…The first results are a check on the calculation of the coefficients u q (r) comparing those computed using Equation 20 and those computed directly from exact closed-form expressions [10] for K(r, r 2 ) in the case when the radial source term is a monomial in radius s n = r γ 1 . Figure 3 compares the two sets of coefficients for γ = 0, 2, 4, with the plots terminated at a value of q where the difference between the two sets of results becomes noticeable, q ≈ 20.…”

Analysis of the radiated information in spinning sound fields

“…The source which generates such fields can be represented as a set of modes which vary with azimuth on a circular disk, whether or not the source includes a spinning element. Examples include rotating systems such as cooling fans, 1,2 helicopter rotors 3,4 duct terminations such as aircraft engine intakes [5][6][7][8][9][10] and jets 11 if a jet is modeled as a distribution of disk-shaped sources.…”

Jet noise: sound generation by disc and cylinder sources

“…This is especially relevant to the inverse problem, where acoustic field measurements are used to attempt to infer the nature of the source, such as for aircraft engines (Holste & Neise 1997;Gérard et al 2007;Castres & Joseph 2007a,b;Lewy 2008). It has been recognized that the inverse problem is poorly conditioned, requiring special numerical methods for its treatment.…”