Gwenael Gabard

Discontinuous Galerkin methods with plane waves for time-harmonic problems

A general framework for discontinuous Galerkin methods in the frequency domain with numerical flux is presented. The main feature of the method is the use of plane waves instead of polynomials to approximate the solution in each element. The method is formulated for a general system of linear hyperbolic equations and is applied to problems of aeroacoustic propagation by solving the two-dimensional linearized Euler equations. It is found that the method requires only a small number of elements per wavelength to obtain accurate solutions and that it is more efficient than high-order DRP schemes. In addition, the conditioning of the method is found to be high but not critical in practice. It is shown that the Ultra-Weak Variational Formulation is in fact a subset of the present discontinuous Galerkin method. A special extension of the method is devised in order to deal with singular solutions generated by point sources like monopoles or dipoles. Aeroacoustic problems with non-uniform flows are also considered and results are presented for the sound radiated from a two-dimensional jet.

Random particle methods applied to broadband fan interaction noise

Predicting broadband fan noise is key to reduce noise emissions from aircraft and wind turbines. Complete CFD simulations of broadband fan noise generation remain too expensive to be used routinely for engineering design. A more efficient approach consists in synthesizing a turbulent velocity field that captures the main features of the exact solution. This synthetic turbulence is then used in a noise source model. This paper concentrates on predicting broadband fan noise interaction (also called leading edge noise) and demonstrates that a random particle mesh method (RPM) is well suited for simulating this source mechanism. The linearized Euler equations are used to describe sound generation and propagation. In this work, the definition of the filter kernel is generalized to include non-Gaussian filters that can directly follow more realistic energy spectra such as the ones developed by Liepmann and von Karman. The velocity correlation and energy spectrum of the turbulence are found to be well captured by the RPM. The acoustic predictions are successfully validated against Amiets analytical solution for a flat plate in a turbulent stream. A standard Langevin equation is used to model temporal decorrelation, but the presence of numerical issues leads to the introduction and validation of a second-order Langevin model.

Synthetic turbulence applied to broadband interaction noise

The aim of this paper is to investigate the use of synthetic turbulence and its application to broadband interaction noise by introducing it as a source in the linearized Euler equations. The turbulence generator model is able to synthetize two-dimensional incompressible, isotropic velocity fields by filtering white noise. The filter is expressed in terms of either the correlation function or the energy spectrum. In contrast with most filter-based models, non-Gaussian filters such as those derived from the Liepmann and von Karman spectra are considered. Another difference with previous work is that a fully Lagrangian approach is used. Random vortices are launched upstream of the flat plate and convected with the mean flow. The turbulent velocity field is computed and its normal velocity imposed along the airfoil. Simulation results are presented for a two-dimensional flat plate interacting with isotropic homogeneous turbulence. Results are compared with the analytical solution proposed by Amiet.

Influence of mean flow gradients on fan exhaust noise predictions

Aft fan noise is becoming a more dominant source as engine bypass ratio is increased n this paper an assessment of the effect of the mean flow gradients on fan exhaust noise propagation is carried out using both analytical models for simplified problems and numerical methods for realistic configurations. Fan exhaust noise can be significantly refracted by the mean flow gradients in the jet mixing layer, especially at high operating conditions (i.e. during take off). The refraction effect is predicted using either Lilley’s equation or the linearized Euler equations. For parallel base flows, an issue with these linear models is the presence of Kelvin-Helmholtz instabilities whose unlimited exponential growth is unphysical and problematic for computational methods. This problem is less critical for developing mixing layer for instance where the growth of the vorticity thickness reduces the growth of the instability waves [1]. Various techniques have been used for suppressing the instability; these include adding non-linear terms to saturate the growth of the instability [2], using frequency domain analysis [3], or removing the mean flow gradient terms [4]. It is the last approach, termed Gradient Term Suppression (GTS), which is investigated in the present work.

A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow

The effect of flow over an acoustic liner is generally described by the Myers impedance condition. The use of this impedance condition in time-domain numerical simulations has been plagued by stability issues, and various ad hoc techniques based on artificial damping or filtering have been used to stabilise the solution. The theoretical issue leading to the ill-posedness of this impedance condition in the time domain is now well understood. For computational models, some trends have been identified, but no detailed investigation of the cause of the instabilities in numerical simulations has been undertaken to date. This paper presents a dispersion analysis of the complete numerical model, based on finite-difference approximations, for a two-dimensional model of a uniform flow above an impedance surface. It provides insight into the properties of the instability in the numerical model and clarifies the parameters that influence its presence. The dispersion analysis is also used to give useful information about the accuracy of the acoustic solution. Comparison between the dispersion analysis and numerical simulations shows that the instability associated with the Myers condition can be identified in the numerical model, but its properties differ significantly from that of the continuous model. The unbounded growth of the instability in the continuous model is not present in the numerical model due to the wavenumber aliasing inherent to numerical approximations. Instead, the numerical instability includes a wavenumber component behaving as an absolute instability. The trend previously reported that the instability is more likely to appear with fine grids is explained. While the instability in the numerical model is heavily dependent on the spatial resolution, it is well resolved in time and is not sensitive to the time step. In addition filtering techniques to stabilise the solutions are considered and it is found that, while they can reduce the instability in some cases, they do not represent a systematic or robust solution in general.

Broadband Fan Interaction Noise using Synthetic Inhomogeneous Non-stationary Turbulence

implementations of the method are discussed. Simulation results are presented for an isolated at plate interacting with inhomogeneous non-stationary turbulence. The sensitivity of noise levels to the upstream turbulence is investigated by considering trains of Gaussian wakes with dierent widths and separations. Far-eld noise levels are validated against Amiet’s analytical solution modied to include the eects of the wakes.

A high-order finite element method for the linearised euler equations

Sound propagation in complex non-uniform mean flows is an important feature of turbofan exhaust noise radiation. The Linearised Euler Equations are able to represent the strong shear layer refraction effects on the sound field, as well as multiple length scales. Frequency domain solvers are suitable for tonal noise and considered a way to avoid linear instabilities, which may occur with time domain solvers. However, the classical Finite Element Method suffers from dispersion error and high memory requirements. These shortcomings are particularly critical for high frequencies and for the Linearised Euler Equations, which involve up to five unknowns. In this paper, a high-order Finite Element Method is used to solve the Linearised Euler Equations in the frequency domain in order to overcome those issues. The model involves high-order polynomial shape functions, unstructured triangular meshes, numerical stabilisation and Perfectly Matched Layers. The acoustic radiation from a straight circular semi-infinite hard-wall duct with several mean flow configurations is computed. Comparisons with analytic solutions demonstrate the method accuracy. The acoustic and vorticity waves are well represented, as well as the refraction of the sound field across the jet shear layer. The high-order approach allows to use coarse meshes, while maintaining a sufficient accuracy. The benefits in terms of memory requirements are significant when compared to standard low-order Finite Element Method.

Mode-Matching Techniques for Sound Propagation in Lined Ducts with Flow

This paper discusses the use of mode-matching techniques to model sound propagation in lined ducts with flow. With no flow mode-matching methods are relatively well established and the continuity of acoustic pressure and axial velocity is generally applied. But with a base flow the behaviour of the solution at the transition between, say, a hard wall and a lined wall is not well understood. For this reason different assumptions can be made on the behaviour of the acoustic field and these lead to different models. This paper presents a comparison of different models and illustrates that significant differences can be observed. A modified mode-matching scheme based on conservation of mass and momentum is proposed. The conservation of sound power is investigated for different matching conditions. Finally, the link between the solutions obtained with mode-matching methods and finite element methods is discussed, and the present analysis can shed some light on the underlying assumptions used in finite element models.

Random-vortex-particle methods for broadband fan interaction noise

The aim of this paper is to predict broadband fan interaction noise by introducing synthetic turbulence as a source in the linearised Euler equations solved in the time domain. The random-vortex-particle method used in this work reproduces two-dimensional isotropic turbulent flows by filtering white noise. The filter is expressed in terms of either the correlation function or the energy spectrum. Non-Gaussian spectra can also be considered such as Liepmann and von Karman energy spectra. Simulation results are presented for a two-dimensional flat plate interacting with homogeneous isotropic turbulence. The linearised Euler equations are solved with a multi-block finite-difference code where vortex-particles are launched upstream of the airfoil and convected with the mean flow. The correlation in time is modelled in the random-vortex-particle method using first and second-order Langevin models. The turbulent velocity field is computed and implemented as a boundary condition on the airfoil. Results for the acoustic far field are compared against the analytical solution for airfoils interacting with homogeneous isotropic frozen turbulence.

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.