An imaging method of acoustic spinning modes propagating within a circular duct simply with surface pressure information is introduced in this paper. The proposed method is developed in a theoretical way and is demonstrated by a numerical simulation case. Nowadays, the measurements within a duct have to be conducted using in-duct microphone array, which is unable to provide information of complete acoustic solutions across the test section. The proposed method can estimate immeasurable information by forming a so-called observer. The fundamental idea behind the testing method was originally developed in control theory for ordinary differential equations. Spinning mode propagation, however, is formulated in partial differential equations. A finite difference technique is used to reduce the associated partial differential equations to a classical form in control. The observer method can thereafter be applied straightforwardly. The algorithm is recursive and, thus, could be operated in real-time. A numerical simulation for a straight circular duct is conducted. The acoustic solutions on the test section can be reconstructed with good agreement to analytical solutions. The results suggest the potential and applications of the proposed method.
Received 02 January 2012Revised 31 December 2012Accepted 08 April 2013Published online 05 June 2013
This work is supported by the National Science Foundation Grant of China (Grant Nos. 11172007 and 11150110134), and Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
Article outline: I. INTRODUCTION II. BACKGROUND III. THEORETICAL DEVELOPMENT A. Mathematical model B. State space model IV. NUMERICAL TESTS AND DISCUSSION V. SUMMARY
2.F. O. Castres and P. F. Joseph, “Experimental investigation of an inversion technique for the determination of broadband duct mode amplitudes by the use of near-field sensor arrays,” J. Acoust. Soc. Am.122, 848–859 (2007).
11.D. R. Morgan and T. M. Smith, “Coherence effects on the detection performance of quadratic array processors, with applications to large-array matched-field beamforming,” J. Acoust. Soc. Am.87, 737–747 (1990).
13.P. Sijtsma, “Feasibility of in-duct beamforming,” AIAA Pap. 2007–3696 (2007).
14.C. R. Lowis, P. F. Joseph, and A. J. Kempton, “Estimation of the far-field directivity of broadband aeroengine fan noise using an in-duct axial microphone array,” J. Sound Vib.329, 3940–3957 (2010).