Lewy S., Gounet H.

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 would be very interesting to provide some information on the noise sources without requiring tedious spinning-mode experimental analyses. It has been shown in a previous article that such an approach is feasible. This paper suggests a constrained iterative least squares fitting for inverse prediction of the radiated spinning modes. 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. A numerical procedure using Matlab has been implemented. Two methods are found to be suitable for solving under-determined or over-determined systems of equations, i.e., either Matlab pseudo-inverse function or backslash operator. A constraint comes from the fact that squared modal amplitudes must be positive which involves an iterative process. Modes with negative squared amplitudes are removed from the following iteration. Several numerical simulations are discussed, and the condition numbers of the system matrices are analyzed. It is clearly shown that the solution depends on the selected method, and is not unique. This means that the initial set of modes needs to be carefully chosen according to the known physical properties of the acoustic sources. It is assessed that the iterative process is very fast and efficient, and well fits the simulated measured directivities. Issues for future improvements are raised in the conclusions.