Analytical Comparison of the Acoustic Analogy and Kirchhoff Formulation for Moving Surfaces

Abstract

The Lighthill acoustic analogy, as embodied in the Ffowcs Williams{Hawkings (FW{H) equation, is compared with the Kirchho formulation for moving surfaces. A comparison of the two governing equations reveals that the main Kirchho advantage (namely nonlinear ow eects are included in the surface integration) is also available to the FW{H method if the integration surface used in the FW{H equation is not assumed impenetrable. The FW{H equation is analytically superior for aeroacoustics because it is based upon the conservation laws of uid mechanics rather than the wave equation. This means that the FW{H equation is valid even if the integration surface is in the nonlinear region. This is demonstrated numerically in the paper. The Kirchho approach can lead to substantial errors if the integration surface is not positioned in the linear region. These errors may be hard to identify. Finally, new metrics based on the Sobolev norm are introduced which may be used to compare input data for both quadrupole noise calculations and Kirchho noise predictions.

Surface Integral Analogy Approaches to Computing Noise Generated by a 3D High-Lift Wing Configuration

“…This is an advantage since it brings less limitation on the location of the surface. 6,14,15 The surface can be placed in the area where nonlinear effects exist. That is, the distance can be closer to the walls or the intensive flow area in the FWH method than the Kirchhoff method.…”

“…6,14,15 The differences have been evaluated numerically to predict the noise from, for example, helicopter rotors 5,6,15 and jets.…”

“…7 In practice, this means that the nonlinear quantities such as ρu i u j on the integral surface must be considerably weaker than the linear terms p ′ , dp/dn and dp/dt. 6 The surface is required to be large enough to enclose most of the strong nonlinear structures of the flow. However, it is challenging to keep the accuracy of the numerical solution on the surface free from numerical contaminations, for example, the dissipation and the dispersion that are caused by the numerical scheme.…”

“…5,6 The term of the volume integral can be neglected if the permeable surface is considered to enclose most of the sources in the flow field. This method has been validated numerically, for instance, through predicting the noise from helicopter rotors.…”

“…This method has been validated numerically, for instance, through predicting the noise from helicopter rotors. 5,6 The Kirchhoff method is an alternative acoustic analogy of the surface integral. [7][8][9][10] The integral surface in this method should be located in the linear region of the flow where the wave propagation condition is fulfilled.…”