Hadrien Beriot

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.

Prediction of sound transmission through an annular cascade Using an analytical cascade response function

The present study aims at developing and assessing an analytical model for the sound transmission through an annular blade row, by comparing with the Finite Element Method Solver of Virtual Lab Acoustics in a con guration without mean ow. The model reformulates a three-dimensional annular model dedicated to turbulence interaction noise to deal with the case of an incident acoustic mode of an annular duct. It is a strip theory approach based on a previously published analytical formulation for the unsteady blade loading in a rectilinear cascade. Three formulations are developed on the basis of two di erent de nitions of the incident gusts impinging on the rectilinear cascade. The latter are designed to match most of the properties of the incident mode in the annular case. The formulations are compared with the Finite Element Method Solver of Virtual Lab Acoustics and with a rectilinear cascade model in con gurations with no mean ow. The benchmarks consist in three annular ducts from very high to moderate hub-to-tip ratio containing a possibly staggered annular cascade. The radial mode order of the incident mode is varied. Both pressure eld and intensity coe cients are compared.

Computation of sound in a simplified HVAC duct based on aerodynamic pressure

Curle’s analogy provides a solution to Lighthill’s equation when solid boundaries are present, and characterizes these boundaries as dipole sources. It can be shown that Curle’s dipoles can be used to define appropriate boundary conditions of a boundary value problem of the Helmholtz equation. This is advantageous from a modeling perspective, because the resulting numerical problem does not require volume-distributed sources to characterize the sound production. In this work, this approach is applied to compute the sound generated by an internal flow and radiated to the exterior. The noise produced by a simplified HVAC duct is computed, and compared to experimental data available in the literature in order to assess the accuracy of several numerical techniques. Specific aspects related to the modeling of interior/exterior problems are investigated.

HIGH-ORDER CURVED MESH GENERATION BY USING A FINE LINEAR TARGET MESH

This paper examines different mesh boundary curving algorithms in the particular case where no exact geometric description is available. The starting point for the curving is typically a coarse linear mesh, whereas the target geometry is represented by a refined linear mesh. Both can be generated using a classical linear mesh generator. Two different boundary curving algorithms are examined. The first algorithm is based on Lagrange nodal high-order polynomial interpolation where the interpolation nodes are iteratively moved towards the target curve. Relocation steps are included in each iteration to approximately preserve the original node spacing. The second algorithm is based on hierarchic modal shape functions. In a reference frame, projection based interpolation is applied that minimizes the distance between the interpolating function and the local target function in the H-seminorm. The performance and accuracy of the two methods are evaluated and compared. Thereby, the area between the target and the approximating curve is used as error measure. In general, both methods exhibit similar levels of performance. A lower bound on the accuracy is observed that depends on the level of refinement of the target mesh. Differences lie in the applicability of the two methods. For the method based on modal interpolation, several initial requirements have to be fulfilled. The method based on nodal interpolation on the other hand is simpler but less robust. Overall, the modal interpolation approach is preferable, where applicable.

Performance of the DGM for the linearized Euler equations with non-uniform mean-flow

A dispersion analysis of the fully-discrete, nodal discontinuous Galerkin method (DGM) for the solution of the time-domain linearized Euler equations (LEE) is performed. Two dispersion analysis methods are developed, considering both uniform and non-uniform mean-flow effects. Convergence studies are performed for the dispersion, dissipation, and nodal solution errors of the acoustic, entropy, and vorticity modes. The accuracy and stability of the DGM are analyzed in the context of aeroacoustic applications, and guidelines are proposed for the choice of optimal discretizations. Computational costs are estimated for a model problem and related to the choice of the element size, polynomial order, and time step. Results indicate that temporal error can become a dominant source of error for high accuracy requirements and long distance wave propagation. The stability of the scheme is analyzed for a shear layer mean flow profile. Aliasing-type errors are found to contribute to the formation of numerical instabilities which are further strengthened by increases in the polynomial order.

A GWBEM method for high frequency acoustic scattering

This paper considers the problem of scattering of a time‐harmonic acoustic incident wave by a hard obstacle. The numerical solution to this problem is found using a Galerkin Wave Boundary Element Method (GWBEM) whereby the functional space is built as the product of conventional low order piecewise polynomials with a set of plane waves propagating in various directions. In this work we present strategies for finding the appropriate plane wave basis locally on each boundary element in order to deal efficiently with very irregularly meshed structures exhibiting both large smooth scattering surfaces as well as corners and small geometrical features. Numerical results clearly demonstrate that these improvements allow the handling of scatterers with complicated geometries while maintaining a low discretization level of 2.5 to 3 degrees of freedom per full wavelength.

High-order 2D mesh curving methods with a piecewise linear target and application to Helmholtz problems

Abstract High-order simulation techniques typically require high-quality curvilinear meshes. In most cases, mesh curving methods assume that the exact geometry is known. However, in some situations only a fine linear FEM mesh is available and the connection to the CAD geometry is lost. In other applications, the geometry may be represented as a set of scanned points. In this paper, two curving methods are described that take a piecewise fine linear mesh as input: a least squares approach and a continuous optimization in the H 1 -seminorm. Hierarchic, modal shape functions are used as basis for the geometric approximation. This approach allows to create very high-order curvilinear meshes efficiently ( q > 4 ) without having to optimize the location of non-vertex nodes. The methods are compared on two test geometries and then used to solve a Helmholtz problem at various input frequencies. Finally, the main steps for the extension to 3D are outlined.

A stabilised high-order finite element model for the linearised Euler equations

The propagation of sound in complex flows is a critical issue for many industries. When modeling turbomachinery noise radiating from engine exhausts, the jet shear layer induces a strong refraction of the sound waves. This can be described by the Linearised Euler Equations (LEE). Most of the difficulties associated with time-domain solutions of the LEE can be avoided by working in the frequency domain. Standard finite elements suffer from large dispersion errors and to improve the computational efficiency we resort here to highorder FEM. The FEM is also known to encounter stability issues for advection-diffusion problems that can be corrected by adding artificial diffusion terms in the formulation. In this paper, we aim at investigating dedicated high-order stabilisation schemes for the time-harmonic LEE. A dispersion analysis of the one-dimensional time-harmonic transport equation is provided. The optimal stabilisation parameter is derived so as to cancel the dispersion error, for each polynomial order of the shape functions. The performance of the resulting stabilised formulation is investigated on a two-dimensional test case with unstructured meshes. The steady parameter used in the literature for the LEE performs well in the high-resolution regime, as attested by the results of the sound propagation from a semi-infinite circular duct with non-uniform mean flow. The sound propagation and radiation are accurately described, as well as the interactions between the acoustic waves and the hydrodynamic field resulting in the vorticity shedding from the duct lip.

Implementation of a vortex sheet in a finite element model based on potential theory for exhaust noise predictions

Predicting sound propagation through the jet exhaust of an aero-engine presents the specific difficulty of representing the refraction effect of the mean flow shear. This is described by the linearised Euler equations, but this model remains rather expensive to solve numerically. The other model commonly used in industry, the linearised potential theory, is faster to solve but needs to be modified to represent a shear layer. This paper presents a way to describe a vortex sheet in a finite element model based on the linearised potential theory. The key issues to address are the continuity of pressure and displacement that have to be enforced across the vortex sheet, as well as the implementation of the Kutta condition at the nozzle lip. Validation results are presented by comparison with analytical results. It is shown that the discretization of the continuity conditions is crucial to obtain a robust and accurate numerical model.