Towards a Highly Accurate Implementation of the Kirchhoff Approach for Computational Aeroacoustics

Abstract

This paper describes the development of high-order Kirchhoff algorithms and confirms that high-order accuracy can be achieved with the Kirchhoff approach when high-order integration and interpolation are properly implemented. This paper also establishes guidelines for enhancing accuracy of a given order property when the Kirchhoff formula is applied to results obtained from a computational fluid dynamics (CFD) solution. Accuracy is shown to increase when the Kirchhoff surface size is minimized. Reduction of the Kirchhoff surface size also enhances efficiency of the calculation because the size of the relatively expensive CFD computation is reduced. The accuracy of the Kirchhoff approach is also enhanced by increasing the density of information along the Kirchhoff surface. This increase in information is necessary because evaluation of the Kirchhoff integrand at the retarded time demands higher spatial resolution than the integration of the time-dependent nonlinear equations of the CFD calculation. A procedure has been developed that addresses the Kirchhoff resolution requirements without sacrificing efficiency. The error reduction realized with this procedure matches that realized with CFD mesh refinement, with almost no increase in cost.

Surface Integral Methods in Computational Aeroacoustics—From the (CFD) Near-Field to the (Acoustic) Far-Field

“…Also, the codes sometimes require significant memory. Finally 62 , the variation of the source strength over a surface element in the retarded time can be very high at certain observer locations ( ) and near sonic velocities ( ) requiring a large number of points per wavelength.…”

A Potential-Theoretic Method for Far-Field Sound Radiation Calculations

“…Equations (13) and (14) impose continuity of the pressure and normal velocity, respectively, across the wake, while (15) ensures that u → u ∞ as x 1 → −∞.φ is a known function of the upstream disturbance. For a complete discussion of the development of this boundary value problem, see [19,20].…”

“…Current efforts involving the direct numerical simulation of far-field sound have been made by Tam [24], Tam and Webb [25], Hixon et al [10,11], Botteldooren [5], Fung et al [8], Mitchell et al [16], Djambazov et al [6], Freund [7], Meadows and Atkins [15], Lyrintzis and co-workers [13,14], and Stenger [21]. In addition, there is a sequence of numerical works by Atassi and his associates [2-4, 17, 18, 22, 23] dealing with frequencydomain problems.…”

The unsteady continuous adjoint method for minimizing flow-induced sound radiation

“…The generated noise can be transmitted to the near-and far-field, either by solving additional equations, such as the Linearized Euler Equations, or with acoustic analogies, by computing volume or surface integrals which are analytical solutions of the Lighthill equation [13]. In this work, the Kirchhoff Integral method [14], which offers simplicity in implementation and accuracy for non-rotating configurations [15], is utilized. In order to obtain the sound pressure at a receiver, the pressure and its time and space derivatives, weighted by directivity coefficients, are integrated over a control surface surrounding the noise sources, namely the body in free-stream.…”