On Acoustic Radiation by Rotating Body Sources in Frequency Domain

Abstract

For the complexity of the sound generated mechanism of rotating body source, this paper describes a frequency-domain numerical method for predicting sound radiation of rotating body sources based on the Kirchhoff integral equation with analytical Green's function of rotating monopole and dipole source in free space. The sound radiation model is established in free space and the relationship between characteristics of sound field and acoustic nature frequency of source, angular frequency and its harmonics can be revealed by the mathematical solution. Numerical simulation shows that sound field has a strong directivity, fundamental frequency transmitting in the rotary shaft direction and harmonics spreading along radial direction and frequency shift phenomena appearing clearly in higher rotating speed of source. The method has a theoretical significance for exploring the low-noise rotating machinery. Research about sound radiation of moving source originates from Lighthill. The first integral approach for acoustic propagation is the acoustic analogy. In the acoustic analogy, the governing Navier-Stokes equations are rearranged to be in wave-type form. There is some question as to which terms should be identified as part of the sound source and retained in the right-hand side of the equation and which terms should be in the left-hand side as part of the operator. The far-field sound pressure is then given in terms of a volume integral over the domain containing the sound source. However the major difficulty with the acoustic analogy is that the sound source is not compact in supersonic flows. Several modifications to Lighthill's original theory have been proposed to account for the moving sound source or other effects. The Ffowcs Williams and Hawkings (FW-H) equation was introduced to extend acoustic analogy in the case of solid surfaces. However, when acoustic sources are present in the flowfield a volume integration is needed. This volume integration of the quadrupole source term is difficult to compute and is usually neglected in most acoustic analogy codes. Errors could be encountered in calculating the sound field. Another alternative is the Kirchhoff method which assumes that the sound transmission is governed by the simple wave equation. Kirchhoff's method consists of the calculation of the nonlinear near- and mid-field, usually numerically, with the far-field solutions found from a linear Kirchhoff formulation evaluated on a control surface surrounding the nonlinear-field. The control surface is assumed to enclose all the nonlinear flow effects and noise sources. The sound pressure can be obtained in terms of a surface integral of the surface pressure and its normal and time derivatives. This approach has the potential to overcome some of the difficulties associated with the traditional acoustic analogy approach. The method is simple and accurate and accounts for the nonlinear quadrupole noise in the far-field. Full diffraction and focusing effects are included while eliminating the propagation of the reactive near-field. Kirchhoff's formula, published in 1882, is used in the theory of diffraction of light and in other electromaqnetic problems. It also has many applications to problems of wave in acoustics. One of the novel uses of this formula was proposed by Hawkinqs for predicting