## Abstract

We present a numerical method to compute the acoustic scattering by an arbitrary geometry through resolvent analysis, allowing a classification of source distributions into optimal and suboptimal with respect to their acoustic radiation. A boundary element method (BEM) is applied to solve the Helmholtz equation for a discretisation of a source distribution, and the computed pressure for each source element is used to construct the resolvent operator, which is the product of a tailored Green’s function G_{T} and quadrature weights. To understand how the acoustic field can be modelled, the optimal harmonic forcing and the associated linear response of the flow are obtained using a singular value decomposition of the linear resolvent operator. This was performed for two different cases. The first case is a semi-infinite flat plate, a case that has an analytical tailored Green’s function available in the literature, so a numerical approximation of the resolvent operator is directly obtained by multiplying the integration weights by a discretisation of G_{T}. Results show the optimal forcing for sources close to the plate as a quadrupole distribution concentrated at the vicinity of the trailing edge. The associated responses have the expected cardioid directivity in the far field, with variations of this for higher-order resolvent modes. In the second case we analyse three different NACA airfoils, 0012, 4412 and 0018, where the tailored Green’s function was obtained numerically, and used to build the resolvent operator. Results for three different acoustic wavenumbers k_{0} show that the optimal forcing has dominant contributions from the leading and trailing edge regions, as well as a relevant region near the upper and lower surfaces, where typical boundary-layer fluctuations would act. The associated response for the largest k_{0} in the acoustic field is a scattering cardioid with secondary scattering due to the presence of a leading edge, as expected. Resolvent analysis can also be used to decompose model sources into forcing-mode components, which lead to far-field sound given by the response modes weighted by the gains; this is carried out here for a sample harmonic source. The present methods can be used as a novel analysis tool for aeroacoustic problems, isolating clearly dominant source mechanisms with high acoustic efficiency.

Original language | English |
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Title of host publication | 25th AIAA/CEAS Aeroacoustics Conference, 2019 |

Publisher | American Institute of Aeronautics and Astronautics |

Number of pages | 12 |

ISBN (Print) | 9781624105883 |

DOIs | |

Publication status | Published - 2019 |

Externally published | Yes |

Event | AIAA/CEAS Aeroacoustics Conference 2019 - Delft, Netherlands Duration: 20 May 2019 → 23 May 2019 Conference number: 25th https://arc.aiaa.org/doi/book/10.2514/MAERO19 |

### Publication series

Name | 25th AIAA/CEAS Aeroacoustics Conference, 2019 |
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### Conference

Conference | AIAA/CEAS Aeroacoustics Conference 2019 |
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Country/Territory | Netherlands |

City | Delft |

Period | 20/05/19 → 23/05/19 |

Internet address |