Discontinuous Galerkin Implementation of the Extended Helmholtz Resonator Model in Time Domain

Abstract

Summary For being the major source of noise pollution in aero-engines, fan forward and rearward noise is of primary importance in the aeronautics industry. The computation of aircraft engine noise is usually treated in frequency domain. Relevant quantities like acoustic power spectrum or sound directivity can be easily post processed out of such simulation. Considering that a constant number of grid points per wavelength is required, the size of the linear systems is growing like f 3 , if f is the frequency. Because the systems to solve are stifi, especially in the presence of a mean ∞ow, frequency domain solvers cannot handle very large problems. Typically, if D is the diameter of the engine nacelle and ‚ is the wavelength of the signal, frequency domain solvers are limited to a ratio kr = D=‚ of around 30. The limited e‐ciency and scalability of direct solvers do not make them good candidates for addressing problems where the Helmholtz number kr is as high as 50, especially on distributed memory computers. A possible way to increase the range of frequencies of the calculations is to switch to time domain. It is still possible to compute both power spectrum and sound directivity but not as directly as it was done in the frequency domain. So, a time domain approach has to be proven to be at least as e‐cient as the frequency domain one. In the framework of the Messiaen project(European collaborative project under the Sixth Framework Programme), an e‐cient high order discontinuous Galerkin method (DGM) has been developed that solves the linearized Euler equations. The DG method 1{4 is a popular scheme for the resolution of hyperbolic conservation laws. The properties of the quadrature free implementation of the method 5 allows to obtain an e‐ciency that is close to the peak e‐ciency of the processor. Moreover, good scalability properties are obtained in parallel. This method is now implemented in an industrial framewok (Actran DGM). It has been demonstrated that such a method is a good alternative to the frequency domain at high Helmholtz number. 6 The aim of our work is to participate to a more general research that aim is to flnd ways to reduce the noise of aircraft engines. One technology that enables noise reduction the use of acoustic liners. Liners notably reduce the noise power spectrum and allow to change the sound directivity. Liners are usually modeled in the frequency domain as a frequency dependant wall impedance. This model has to be translated in the time domain where all the frequencies are present, even when we try to solve for a flxed frequency input data. Therefore, an impedance model that matches the design impedance and that is valid for a range of frequency is needed. In 7 S.W. Rienstra has developed a modifled Helmholtz resonator model in time-domain. Starting from the quadrature free RK-DGM framework, we provide a description of the data structures and algorithms that are required to implement this time domain impedance model. We will discuss the conversion from the direct expression of the relation between acoustic pressure and normal velocity to boundary conditions implemented in the discontinuous Galerkin code for aeroacoustic problems. First, we will detail the non-∞ow case. Then, we will extend the model when a mean ∞ow is present. The